Patent Publication Number: US-2022227367-A1

Title: Systems and methods for vehicle navigation

Description:
CROSS REFERENCES TO RELATED APPLICATIONS 
     This application claims the benefit of priority of U.S. Provisional Patent Application No. 62/858,082, filed on Jun. 6, 2019; U.S. Provisional Patent Application No. 62/864,203, filed on Jun. 20, 2019; and U.S. Provisional Patent Application No. 62/878,523, filed on Jul. 25, 2019. The foregoing applications are incorporated herein by reference in their entirety. 
    
    
     BACKGROUND 
     Technical Field 
     The present disclosure relates generally to autonomous vehicle navigation. 
     Additionally, this disclosure relates to systems and methods for navigating according to potential accident liability constraints. 
     Background Information 
     As technology continues to advance, the goal of a fully autonomous vehicle that is capable of navigating on roadways is on the horizon. Autonomous vehicles may need to take into account a variety of factors and make appropriate decisions based on those factors to safely and accurately reach an intended destination. For example, an autonomous vehicle may need to process and interpret visual information (e.g., information captured from a camera), information from radar or lidar, and may also use information obtained from other sources (e.g., from a GPS device, a speed sensor, an accelerometer, a suspension sensor, etc.). At the same time, in order to navigate to a destination, an autonomous vehicle may also need to identify its location within a particular roadway (e.g., a specific lane within a multi-lane road), navigate alongside other vehicles, avoid obstacles and pedestrians, observe traffic signals and signs, travel from one road to another road at appropriate intersections or interchanges, and respond to any other situation that occurs or develops during the vehicle&#39;s operation. Moreover, the navigational system may need to adhere to certain imposed constraints. In some cases, those constraints may relate to interactions between a host vehicle and one or more other objects, such as other vehicles, pedestrians, etc. In other cases, the constraints may relate to liability rules to be followed in implementing one or more navigational actions for a host vehicle. 
     In the field of autonomous driving, there are two important considerations for viable autonomous vehicle systems. The first is a standardization of safety assurance, including requirements that every self-driving car must satisfy to ensure safety, and how those requirements can be verified. The second is scalability, as engineering solutions that lead to unleashed costs will not scale to millions of cars and may prevent widespread or even not so widespread adoption of autonomous vehicles. Thus, there is a need for an interpretable, mathematical model for safety assurance and a design of a system that adheres to safety assurance requirements while being scalable to millions of cars. 
     SUMMARY 
     Embodiments consistent with the present disclosure provide systems and methods for autonomous vehicle navigation. The disclosed embodiments may use cameras to provide autonomous vehicle navigation features. For example, consistent with the disclosed embodiments, the disclosed systems may include one, two, or more cameras that monitor the environment of a vehicle. The disclosed systems may provide a navigational response based on, for example, an analysis of images captured by one or more of the cameras. The navigational response may also take into account other data including, for example, global positioning system (GPS) data, sensor data (e.g., from an accelerometer, a speed sensor, a suspension sensor, etc.), and/or other map data. 
     In an embodiment, a navigation system for a host vehicle may include at least one processor. The processor may be programmed to receive, from a camera, at least one captured image representative of features in an environment of the vehicle; identify at least a portion of an intersection represented in the at least one captured image; and identify a representation in the at least one captured image of a pedestrian in a vicinity of the intersection. The processor may be programmed to determine a navigational action for the vehicle relative to the intersection based, at least in part, on routing information for the vehicle; determine a predicted path for the vehicle relative to the intersection based on the determined navigational action; and determine a predicted path for the pedestrian based on analysis of the at least one captured image. The processor may further be programmed to determine whether the vehicle is projected to collide with the pedestrian based, at least in part, on the projected path for the vehicle and based on the predicted path of the pedestrian; and cause at least one system associated with the vehicle to implement a collision mitigation action in response to a determination that the vehicle is projected to collide with the pedestrian. 
     In an embodiment, a method for navigating a host vehicle may comprise receiving, from a camera, at least one captured image representative of features in an environment of the vehicle; identifying at least a portion of an intersection represented in the at least one captured image; and identifying a representation in the at least one captured image of a pedestrian in a vicinity of the intersection. The method may further comprise determining a navigational action for the vehicle relative to the intersection based, at least in part, on routing information for the vehicle; determining a predicted path for the vehicle relative to the intersection based on the determined navigational action; and determining a predicted path for the pedestrian based on analysis of the at least one captured image. The method may further comprise determining whether the vehicle is projected to collide with the pedestrian based, at least in part, on the projected path for the vehicle and based on the predicted path of the pedestrian; and causing at least one system associated with the vehicle to implement a collision mitigation action in response to a determination that the vehicle is projected to collide with the pedestrian. 
     Consistent with other disclosed embodiments, non-transitory computer-readable storage media may store program instructions, which are executable by at least one processing device and perform any of the steps and/or methods described herein. 
     The foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and constitute a part of this disclosure, illustrate various disclosed embodiments. In the drawings: 
         FIG. 1  is a diagrammatic representation of an exemplary system consistent with the disclosed embodiments. 
         FIG. 2A  is a diagrammatic side view representation of an exemplary vehicle including a system consistent with the disclosed embodiments. 
         FIG. 2B  is a diagrammatic top view representation of the vehicle and system shown in  FIG. 2A  consistent with the disclosed embodiments. 
         FIG. 2C  is a diagrammatic top view representation of another embodiment of a vehicle including a system consistent with the disclosed embodiments. 
         FIG. 2D  is a diagrammatic top view representation of yet another embodiment of a vehicle including a system consistent with the disclosed embodiments. 
         FIG. 2E  is a diagrammatic top view representation of yet another embodiment of a vehicle including a system consistent with the disclosed embodiments. 
         FIG. 2F  is a diagrammatic representation of exemplary vehicle control systems consistent with the disclosed embodiments. 
         FIG. 3A  is a diagrammatic representation of an interior of a vehicle including a rearview mirror and a user interface for a vehicle imaging system consistent with the disclosed embodiments. 
         FIG. 3B  is an illustration of an example of a camera mount that is configured to be positioned behind a rearview mirror and against a vehicle windshield consistent with the disclosed embodiments. 
         FIG. 3C  is an illustration of the camera mount shown in  FIG. 3B  from a different perspective consistent with the disclosed embodiments. 
         FIG. 3D  is an illustration of an example of a camera mount that is configured to be positioned behind a rearview mirror and against a vehicle windshield consistent with the disclosed embodiments. 
         FIG. 4  is an exemplary block diagram of a memory configured to store instructions for performing one or more operations consistent with the disclosed embodiments. 
         FIG. 5A  is a flowchart showing an exemplary process for causing one or more navigational responses based on monocular image analysis consistent with disclosed embodiments. 
         FIG. 5B  is a flowchart showing an exemplary process for detecting one or more vehicles and/or pedestrians in a set of images consistent with the disclosed embodiments. 
         FIG. 5C  is a flowchart showing an exemplary process for detecting road marks and/or lane geometry information in a set of images consistent with the disclosed embodiments. 
         FIG. 5D  is a flowchart showing an exemplary process for detecting traffic lights in a set of images consistent with the disclosed embodiments. 
         FIG. 5E  is a flowchart showing an exemplary process for causing one or more navigational responses based on a vehicle path consistent with the disclosed embodiments. 
         FIG. 5F  is a flowchart showing an exemplary process for determining whether a leading vehicle is changing lanes consistent with the disclosed embodiments. 
         FIG. 6  is a flowchart showing an exemplary process for causing one or more navigational responses based on stereo image analysis consistent with the disclosed embodiments. 
         FIG. 7  is a flowchart showing an exemplary process for causing one or more navigational responses based on an analysis of three sets of images consistent with the disclosed embodiments. 
         FIG. 8  is a block diagram representation of modules that may be implemented by one or more specifically programmed processing devices of a navigation system for an autonomous vehicle consistent with the disclosed embodiments. 
         FIG. 9  is a navigation options graph consistent with the disclosed embodiments. 
         FIG. 10  is a navigation options graph consistent with the disclosed embodiments. 
         FIGS. 11A, 11B, and 11C  provide a schematic representation of navigational options of a host vehicle in a merge zone consistent with the disclosed embodiments. 
         FIG. 11D  provide a diagrammatic depiction of a double merge scenario consistent with the disclosed embodiments. 
         FIG. 11E  provides an options graph potentially useful in a double merge scenario consistent with the disclosed embodiments. 
         FIG. 12  provides a diagram of a representative image captured of an environment of a host vehicle, along with potential navigational constraints consistent with the disclosed embodiments. 
         FIG. 13  provides an algorithmic flow chart for navigating a vehicle consistent with the disclosed embodiments. 
         FIG. 14  provides an algorithmic flow chart for navigating a vehicle consistent with the disclosed embodiments. 
         FIG. 15  provides an algorithmic flow chart for navigating a vehicle consistent with the disclosed embodiments. 
         FIG. 16  provides an algorithmic flow chart for navigating a vehicle consistent with the disclosed embodiments. 
         FIGS. 17A and 17B  provide a diagrammatic illustration of a host vehicle navigating into a roundabout consistent with the disclosed embodiments. 
         FIG. 18  provides an algorithmic flow chart for navigating a vehicle consistent with the disclosed embodiments. 
         FIG. 19  illustrates an example of a host vehicle driving on a multi-lane highway consistent with the disclosed embodiments. 
         FIGS. 20A and 20B  illustrate examples of a vehicle cutting in in front of another vehicle consistent with the disclosed embodiments. 
         FIG. 21  illustrates an example of a vehicle following another vehicle consistent with the disclosed embodiments. 
         FIG. 22  illustrates an example of a vehicle exiting a parking lot and merging into a possibly busy road consistent with the disclosed embodiments. 
         FIG. 23  illustrates a vehicle traveling on a road consistent with the disclosed embodiments. 
         FIGS. 24A, 24B, 24C, and 24D  illustrate four example scenarios consistent with the disclosed embodiments. 
         FIG. 25  illustrates an example scenario consistent with the disclosed embodiments. 
         FIG. 26  illustrates an example scenario consistent with the disclosed embodiments. 
         FIG. 27  illustrates an example scenario consistent with the disclosed embodiments. 
         FIGS. 28A and 28B  illustrate an example of a scenario in which a vehicle is following another vehicle consistent with the disclosed embodiments. 
         FIGS. 29A and 29B  illustrate example blame in cut-in scenarios consistent with the disclosed embodiments. 
         FIGS. 30A and 30B  illustrate example blame in cut-in scenarios consistent with the disclosed embodiments. 
         FIGS. 31A, 31B, 31C, and 31D  illustrate example blame in drifting scenarios consistent with the disclosed embodiments. 
         FIGS. 32A and 32B  illustrate example blame in two-way traffic scenarios consistent with the disclosed embodiments. 
         FIGS. 33A and 33B  illustrate example blame in route priority scenarios consistent with the disclosed embodiments. 
         FIGS. 34A and 34B  illustrate example blame in route priority scenarios consistent with the disclosed embodiments. 
         FIGS. 35A and 35B  illustrate example blame in route priority scenarios consistent with the disclosed embodiments. 
         FIGS. 36A and 36B  illustrate example blame in route priority scenarios consistent with the disclosed embodiments. 
         FIGS. 37A and 37B  illustrate example blame in route priority scenarios consistent with the disclosed embodiments. 
         FIGS. 38A and 38B  illustrate example blame in route priority scenarios consistent with the disclosed embodiments. 
         FIGS. 39A and 39B  illustrate example blame in route priority scenarios consistent with the disclosed embodiments. 
         FIGS. 40A and 40B  illustrate example blame in traffic light scenarios consistent with the disclosed embodiments. 
         FIGS. 41A and 41B  illustrate example blame in traffic light scenarios consistent with the disclosed embodiments. 
         FIGS. 42A and 42B  illustrate example blame in traffic light scenarios consistent with the disclosed embodiments. 
         FIGS. 43A, 43B, and 43C  illustrate example vulnerable road users (VRUs) scenarios consistent with the disclosed embodiments. 
         FIGS. 44A, 44B, and 44C  illustrate example vulnerable road users (VRUs) scenarios consistent with the disclosed embodiments. 
         FIGS. 45A, 45B, and 45C  illustrate example vulnerable road users (VRUs) scenarios consistent with the disclosed embodiments. 
         FIGS. 46A, 46B, 46C, and 46D  illustrate example vulnerable road users (VRUs) scenarios consistent with the disclosed embodiments. 
         FIG. 47  is an illustration of an example scenario in which collision prediction may be used, consistent with the disclosed embodiments 
         FIGS. 48, 49, and 50  illustrate example envelopes that may be determined for a host vehicle and/or pedestrian, consistent with the disclosed embodiments. 
         FIG. 51  is a diagram illustrating an example technique for determining a time of collision between circular envelopes, consistent with the disclosed embodiments. 
         FIG. 52  is an illustration of example timelines corresponding to collision predictions for circular envelopes, consistent with the disclosed embodiments 
         FIG. 53  is a diagram illustrating an example refined analysis to determine a time of collision, consistent with the disclosed embodiments. 
         FIG. 54  is a flowchart showing an example process for navigating a host vehicle, consistent with the disclosed embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description refers to the accompanying drawings. Wherever possible, the same reference numbers are used in the drawings and the following description to refer to the same or similar parts. While several illustrative embodiments are described herein, modifications, adaptations and other implementations are possible. For example, substitutions, additions or modifications may be made to the components illustrated in the drawings, and the illustrative methods described herein may be modified by substituting, reordering, removing, or adding steps to the disclosed methods. Accordingly, the following detailed description is not limited to the disclosed embodiments and examples. Instead, the proper scope is defined by the appended claims. 
     Autonomous Vehicle Overview 
     As used throughout this disclosure, the term “autonomous vehicle” refers to a vehicle capable of implementing at least one navigational change without driver input. A “navigational change” refers to a change in one or more of steering, braking, or acceleration/deceleration of the vehicle. To be autonomous, a vehicle need not be fully automatic (e.g., fully operational without a driver or without driver input). Rather, an autonomous vehicle includes those that can operate under driver control during certain time periods and without driver control during other time periods. Autonomous vehicles may also include vehicles that control only some aspects of vehicle navigation, such as steering (e.g., to maintain a vehicle course between vehicle lane constraints) or some steering operations under certain circumstances (but not under all circumstances), but may leave other aspects to the driver (e.g., braking or braking under certain circumstances). In some cases, autonomous vehicles may handle some or all aspects of braking, speed control, and/or steering of the vehicle. 
     As human drivers typically rely on visual cues and observations in order to control a vehicle, transportation infrastructures are built accordingly, with lane markings, traffic signs, and traffic lights designed to provide visual information to drivers. In view of these design characteristics of transportation infrastructures, an autonomous vehicle may include a camera and a processing unit that analyzes visual information captured from the environment of the vehicle. The visual information may include, for example, images representing components of the transportation infrastructure (e.g., lane markings, traffic signs, traffic lights, etc.) that are observable by drivers and other obstacles (e.g., other vehicles, pedestrians, debris, etc.). Additionally, an autonomous vehicle may also use stored information, such as information that provides a model of the vehicle&#39;s environment when navigating. For example, the vehicle may use GPS data, sensor data (e.g., from an accelerometer, a speed sensor, a suspension sensor, etc.), and/or other map data to provide information related to its environment while it is traveling, and the vehicle (as well as other vehicles) may use the information to localize itself on the model. Some vehicles can also be capable of communication among them, sharing information, altering the peer vehicle of hazards or changes in the vehicles&#39; surroundings, etc. 
     System Overview 
       FIG. 1  is a block diagram representation of a system  100  consistent with the exemplary disclosed embodiments. System  100  may include various components depending on the requirements of a particular implementation. In some embodiments, system  100  may include a processing unit  110 , an image acquisition unit  120 , a position sensor  130 , one or more memory units  140 ,  150 , a map database  160 , a user interface  170 , and a wireless transceiver  172 . Processing unit  110  may include one or more processing devices. In some embodiments, processing unit  110  may include an applications processor  180 , an image processor  190 , or any other suitable processing device. Similarly, image acquisition unit  120  may include any number of image acquisition devices and components depending on the requirements of a particular application. In some embodiments, image acquisition unit  120  may include one or more image capture devices (e.g., cameras, CCDs, or any other type of image sensor), such as image capture device  122 , image capture device  124 , and image capture device  126 . System  100  may also include a data interface  128  communicatively connecting processing unit  110  to image acquisition unit  120 . For example, data interface  128  may include any wired and/or wireless link or links for transmitting image data acquired by image acquisition unit  120  to processing unit  110 . 
     Wireless transceiver  172  may include one or more devices configured to exchange transmissions over an air interface to one or more networks (e.g., cellular, the Internet, etc.) by use of a radio frequency, infrared frequency, magnetic field, or an electric field. Wireless transceiver  172  may use any known standard to transmit and/or receive data (e.g., Wi-Fi, Bluetooth®, Bluetooth Smart, 802.15.4, ZigBee, etc.). Such transmissions can include communications from the host vehicle to one or more remotely located servers. Such transmissions may also include communications (one-way or two-way) between the host vehicle and one or more target vehicles in an environment of the host vehicle (e.g., to facilitate coordination of navigation of the host vehicle in view of or together with target vehicles in the environment of the host vehicle), or even a broadcast transmission to unspecified recipients in a vicinity of the transmitting vehicle. 
     Both applications processor  180  and image processor  190  may include various types of hardware-based processing devices. For example, either or both of applications processor  180  and image processor  190  may include a microprocessor, preprocessors (such as an image preprocessor), graphics processors, a central processing unit (CPU), support circuits, digital signal processors, integrated circuits, memory, or any other types of devices suitable for running applications and for image processing and analysis. In some embodiments, applications processor  180  and/or image processor  190  may include any type of single or multi-core processor, mobile device microcontroller, central processing unit, etc. Various processing devices may be used, including, for example, processors available from manufacturers such as Intel®, AMD®, etc. and may include various architectures (e.g., x86 processor, ARM®, etc.). 
     In some embodiments, applications processor  180  and/or image processor  190  may include any of the EyeQ series of processor chips available from Mobileye®. These processor designs each include multiple processing units with local memory and instruction sets. Such processors may include video inputs for receiving image data from multiple image sensors and may also include video out capabilities. In one example, the EyeQ 2 ® uses 90 nm-micron technology operating at 332 Mhz. The EyeQ2® architecture consists of two floating point, hyper-thread 32-bit RISC CPUs (MIPS32® 34K® cores), five Vision Computing Engines (VCE), three Vector Microcode Processors (VMP®), Denali 64-bit Mobile DDR Controller, 128-bit internal Sonics Interconnect, dual 16-bit Video input and 18-bit Video output controllers, 16 channels DMA and several peripherals. The MIPS34K CPU manages the five VCEs, three VMP™ and the DMA, the second MIPS34K CPU and the multi-channel DMA as well as the other peripherals. The five VCEs, three VMP® and the MIPS34K CPU can perform intensive vision computations required by multi-function bundle applications. In another example, the EyeQ3®, which is a third generation processor and is six times more powerful that the EyeQ2®, may be used in the disclosed embodiments. In other examples, the EyeQ4® and/or the the EyeQ5® may be used in the disclosed embodiments. Of course, any newer or future EyeQ processing devices may also be used together with the disclosed embodiments. 
     Any of the processing devices disclosed herein may be configured to perform certain functions. Configuring a processing device, such as any of the described EyeQ processors or other controller or microprocessor, to perform certain functions may include programming of computer executable instructions and making those instructions available to the processing device for execution during operation of the processing device. In some embodiments, configuring a processing device may include programming the processing device directly with architectural instructions. In other embodiments, configuring a processing device may include storing executable instructions on a memory that is accessible to the processing device during operation. For example, the processing device may access the memory to obtain and execute the stored instructions during operation. In either case, the processing device configured to perform the sensing, image analysis, and/or navigational functions disclosed herein represents a specialized hardware-based system in control of multiple hardware based components of a host vehicle. 
     While  FIG. 1  depicts two separate processing devices included in processing unit  110 , more or fewer processing devices may be used. For example, in some embodiments, a single processing device may be used to accomplish the tasks of applications processor  180  and image processor  190 . In other embodiments, these tasks may be performed by more than two processing devices. Further, in some embodiments, system  100  may include one or more of processing unit  110  without including other components, such as image acquisition unit  120 . 
     Processing unit  110  may comprise various types of devices. For example, processing unit  110  may include various devices, such as a controller, an image preprocessor, a central processing unit (CPU), support circuits, digital signal processors, integrated circuits, memory, or any other types of devices for image processing and analysis. The image preprocessor may include a video processor for capturing, digitizing and processing the imagery from the image sensors. The CPU may comprise any number of microcontrollers or microprocessors. The support circuits may be any number of circuits generally well known in the art, including cache, power supply, clock and input-output circuits. The memory may store software that, when executed by the processor, controls the operation of the system. The memory may include databases and image processing software. The memory may comprise any number of random access memories, read only memories, flash memories, disk drives, optical storage, tape storage, removable storage and other types of storage. In one instance, the memory may be separate from the processing unit  110 . In another instance, the memory may be integrated into the processing unit  110 . 
     Each memory  140 ,  150  may include software instructions that when executed by a processor (e.g., applications processor  180  and/or image processor  190 ), may control operation of various aspects of system  100 . These memory units may include various databases and image processing software, as well as a trained system, such as a neural network, or a deep neural network, for example. The memory units may include random access memory, read only memory, flash memory, disk drives, optical storage, tape storage, removable storage and/or any other types of storage. In some embodiments, memory units  140 ,  150  may be separate from the applications processor  180  and/or image processor  190 . In other embodiments, these memory units may be integrated into applications processor  180  and/or image processor  190 . 
     Position sensor  130  may include any type of device suitable for determining a location associated with at least one component of system  100 . In some embodiments, position sensor  130  may include a GPS receiver. Such receivers can determine a user position and velocity by processing signals broadcasted by global positioning system satellites. Position information from position sensor  130  may be made available to applications processor  180  and/or image processor  190 . 
     In some embodiments, system  100  may include components such as a speed sensor (e.g., a speedometer) for measuring a speed of vehicle  200 . System  100  may also include one or more accelerometers (either single axis or multiaxis) for measuring accelerations of vehicle  200  along one or more axes. 
     The memory units  140 ,  150  may include a database, or data organized in any other form, that indication a location of known landmarks. Sensory information (such as images, radar signal, depth information from lidar or stereo processing of two or more images) of the environment may be processed together with position information, such as a GPS coordinate, vehicle&#39;s ego motion, etc. to determine a current location of the vehicle relative to the known landmarks, and refine the vehicle location. Certain aspects of this technology are included in a localization technology known as REM&#39;, which is being marketed by the assignee of the present application. 
     User interface  170  may include any device suitable for providing information to or for receiving inputs from one or more users of system  100 . In some embodiments, user interface  170  may include user input devices, including, for example, a touchscreen, microphone, keyboard, pointer devices, track wheels, cameras, knobs, buttons, etc. With such input devices, a user may be able to provide information inputs or commands to system  100  by typing instructions or information, providing voice commands, selecting menu options on a screen using buttons, pointers, or eye-tracking capabilities, or through any other suitable techniques for communicating information to system  100 . 
     User interface  170  may be equipped with one or more processing devices configured to provide and receive information to or from a user and process that information for use by, for example, applications processor  180 . In some embodiments, such processing devices may execute instructions for recognizing and tracking eye movements, receiving and interpreting voice commands, recognizing and interpreting touches and/or gestures made on a touchscreen, responding to keyboard entries or menu selections, etc. In some embodiments, user interface  170  may include a display, speaker, tactile device, and/or any other devices for providing output information to a user. 
     Map database  160  may include any type of database for storing map data useful to system  100 . In some embodiments, map database  160  may include data relating to the position, in a reference coordinate system, of various items, including roads, water features, geographic features, businesses, points of interest, restaurants, gas stations, etc. Map database  160  may store not only the locations of such items, but also descriptors relating to those items, including, for example, names associated with any of the stored features. In some embodiments, map database  160  may be physically located with other components of system  100 . Alternatively or additionally, map database  160  or a portion thereof may be located remotely with respect to other components of system  100  (e.g., processing unit  110 ). In such embodiments, information from map database  160  may be downloaded over a wired or wireless data connection to a network (e.g., over a cellular network and/or the Internet, etc.). In some cases, map database  160  may store a sparse data model including polynomial representations of certain road features (e.g., lane markings) or target trajectories for the host vehicle. Map database  160  may also include stored representations of various recognized landmarks that may be used to determine or update a known position of the host vehicle with respect to a target trajectory. The landmark representations may include data fields such as landmark type, landmark location, among other potential identifiers. 
     Image capture devices  122 ,  124 , and  126  may each include any type of device suitable for capturing at least one image from an environment. Moreover, any number of image capture devices may be used to acquire images for input to the image processor. Some embodiments may include only a single image capture device, while other embodiments may include two, three, or even four or more image capture devices. Image capture devices  122 ,  124 , and  126  will be further described with reference to  FIGS. 2B-2E , below. 
     One or more cameras (e.g., image capture devices  122 ,  124 , and  126 ) may be part of a sensing block included on a vehicle. Various other sensors may be included in the sensing block, and any or all of the sensors may be relied upon to develop a sensed navigational state of the vehicle. In addition to cameras (forward, sideward, rearward, etc), other sensors such as RADAR, LIDAR, and acoustic sensors may be included in the sensing block. Additionally, the sensing block may include one or more components configured to communicate and transmit/receive information relating to the environment of the vehicle. For example, such components may include wireless transceivers (RF, etc.) that may receive from a source remotely located with respect to the host vehicle sensor based information or any other type of information relating to the environment of the host vehicle. Such information may include sensor output information, or related information, received from vehicle systems other than the host vehicle. In some embodiments, such information may include information received from a remote computing device, a centralized server, etc. Furthermore, the cameras may take on many different configurations: single camera units, multiple cameras, camera clusters, long FOV, short FOV, wide angle, fisheye, etc. 
     System  100 , or various components thereof, may be incorporated into various different platforms. In some embodiments, system  100  may be included on a vehicle  200 , as shown in  FIG. 2A . For example, vehicle  200  may be equipped with a processing unit  110  and any of the other components of system  100 , as described above relative to  FIG. 1 . While in some embodiments vehicle  200  may be equipped with only a single image capture device (e.g., camera), in other embodiments, such as those discussed in connection with  FIGS. 2B-2E , multiple image capture devices may be used. For example, either of image capture devices  122  and  124  of vehicle  200 , as shown in  FIG. 2A , may be part of an ADAS (Advanced Driver Assistance Systems) imaging set. 
     The image capture devices included on vehicle  200  as part of the image acquisition unit  120  may be positioned at any suitable location. In some embodiments, as shown in  FIGS. 2A-2E and 3A-3C , image capture device  122  may be located in the vicinity of the rearview mirror. This position may provide a line of sight similar to that of the driver of vehicle  200 , which may aid in determining what is and is not visible to the driver. Image capture device  122  may be positioned at any location near the rearview mirror, but placing image capture device  122  on the driver side of the mirror may further aid in obtaining images representative of the driver&#39;s field of view and/or line of sight. 
     Other locations for the image capture devices of image acquisition unit  120  may also be used. For example, image capture device  124  may be located on or in a bumper of vehicle  200 . Such a location may be especially suitable for image capture devices having a wide field of view. The line of sight of bumper-located image capture devices can be different from that of the driver and, therefore, the bumper image capture device and driver may not always see the same objects. The image capture devices (e.g., image capture devices  122 ,  124 , and  126 ) may also be located in other locations. For example, the image capture devices may be located on or in one or both of the side mirrors of vehicle  200 , on the roof of vehicle  200 , on the hood of vehicle  200 , on the trunk of vehicle  200 , on the sides of vehicle  200 , mounted on, positioned behind, or positioned in front of any of the windows of vehicle  200 , and mounted in or near light fixtures on the front and/or back of vehicle  200 , etc. 
     In addition to image capture devices, vehicle  200  may include various other components of system  100 . For example, processing unit  110  may be included on vehicle  200  either integrated with or separate from an engine control unit (ECU) of the vehicle. Vehicle  200  may also be equipped with a position sensor  130 , such as a GPS receiver and may also include a map database  160  and memory units  140  and  150 . 
     As discussed earlier, wireless transceiver  172  may and/or receive data over one or more networks (e.g., cellular networks, the Internet, etc.). For example, wireless transceiver  172  may upload data collected by system  100  to one or more servers, and download data from the one or more servers. Via wireless transceiver  172 , system  100  may receive, for example, periodic or on demand updates to data stored in map database  160 , memory  140 , and/or memory  150 . Similarly, wireless transceiver  172  may upload any data (e.g., images captured by image acquisition unit  120 , data received by position sensor  130  or other sensors, vehicle control systems, etc.) from system  100  and/or any data processed by processing unit  110  to the one or more servers. 
     System  100  may upload data to a server (e.g., to the cloud) based on a privacy level setting. For example, system  100  may implement privacy level settings to regulate or limit the types of data (including metadata) sent to the server that may uniquely identify a vehicle and or driver/owner of a vehicle. Such settings may be set by user via, for example, wireless transceiver  172 , be initialized by factory default settings, or by data received by wireless transceiver  172 . 
     In some embodiments, system  100  may upload data according to a “high” privacy level, and under setting a setting, system  100  may transmit data (e.g., location information related to a route, captured images, etc.) without any details about the specific vehicle and/or driver/owner. For example, when uploading data according to a “high” privacy setting, system  100  may not include a vehicle identification number (VIN) or a name of a driver or owner of the vehicle, and may instead transmit data, such as captured images and/or limited location information related to a route. 
     Other privacy levels are contemplated as well. For example, system  100  may transmit data to a server according to an “intermediate” privacy level and include additional information not included under a “high” privacy level, such as a make and/or model of a vehicle and/or a vehicle type (e.g., a passenger vehicle, sport utility vehicle, truck, etc.). In some embodiments, system  100  may upload data according to a “low” privacy level. Under a “low” privacy level setting, system  100  may upload data and include information sufficient to uniquely identify a specific vehicle, owner/driver, and/or a portion or entirely of a route traveled by the vehicle. Such “low” privacy level data may include one or more of, for example, a VIN, a driver/owner name, an origination point of a vehicle prior to departure, an intended destination of the vehicle, a make and/or model of the vehicle, a type of the vehicle, etc. 
       FIG. 2A  is a diagrammatic side view representation of an exemplary vehicle imaging system consistent with the disclosed embodiments.  FIG. 2B  is a diagrammatic top view illustration of the embodiment shown in  FIG. 2A . As illustrated in  FIG. 2B , the disclosed embodiments may include a vehicle  200  including in its body a system  100  with a first image capture device  122  positioned in the vicinity of the rearview mirror and/or near the driver of vehicle  200 , a second image capture device  124  positioned on or in a bumper region (e.g., one of bumper regions  210 ) of vehicle  200 , and a processing unit  110 . 
     As illustrated in  FIG. 2C , image capture devices  122  and  124  may both be positioned in the vicinity of the rearview mirror and/or near the driver of vehicle  200 . Additionally, while two image capture devices  122  and  124  are shown in  FIGS. 2B and 2C , it should be understood that other embodiments may include more than two image capture devices. For example, in the embodiments shown in  FIGS. 2D and 2E , first, second, and third image capture devices  122 ,  124 , and  126 , are included in the system  100  of vehicle  200 . 
     As illustrated in  FIG. 2D , image capture device  122  may be positioned in the vicinity of the rearview mirror and/or near the driver of vehicle  200 , and image capture devices  124  and  126  may be positioned on or in a bumper region (e.g., one of bumper regions  210 ) of vehicle  200 . And as shown in  FIG. 2E , image capture devices  122 ,  124 , and  126  may be positioned in the vicinity of the rearview mirror and/or near the driver seat of vehicle  200 . The disclosed embodiments are not limited to any particular number and configuration of the image capture devices, and the image capture devices may be positioned in any appropriate location within and/or on vehicle  200 . 
     It is to be understood that the disclosed embodiments are not limited to vehicles and could be applied in other contexts. It is also to be understood that disclosed embodiments are not limited to a particular type of vehicle  200  and may be applicable to all types of vehicles including automobiles, trucks, trailers, and other types of vehicles. 
     The first image capture device  122  may include any suitable type of image capture device. Image capture device  122  may include an optical axis. In one instance, the image capture device  122  may include an Aptina M9V024 WVGA sensor with a global shutter. In other embodiments, image capture device  122  may provide a resolution of 1280×960 pixels and may include a rolling shutter. Image capture device  122  may include various optical elements. In some embodiments one or more lenses may be included, for example, to provide a desired focal length and field of view for the image capture device. In some embodiments, image capture device  122  may be associated with a  6 mm lens or a 12 mm lens. In some embodiments, image capture device  122  may be configured to capture images having a desired field-of-view (FOV)  202 , as illustrated in  FIG. 2D . For example, image capture device  122  may be configured to have a regular FOV, such as within a range of 40 degrees to 56 degrees, including a 46 degree FOV, 50 degree FOV, 52 degree FOV, or greater. Alternatively, image capture device  122  may be configured to have a narrow FOV in the range of 23 to 40 degrees, such as a 28 degree FOV or 36 degree FOV. In addition, image capture device  122  may be configured to have a wide FOV in the range of 100 to 180 degrees. In some embodiments, image capture device  122  may include a wide angle bumper camera or one with up to a 180 degree FOV. In some embodiments, image capture device  122  may be a 7.2M pixel image capture device with an aspect ratio of about 2:1 (e.g., HxV=3800×1900 pixels) with about 100 degree horizontal FOV. Such an image capture device may be used in place of a three image capture device configuration. Due to significant lens distortion, the vertical FOV of such an image capture device may be significantly less than 50 degrees in implementations in which the image capture device uses a radially symmetric lens. For example, such a lens may not be radially symmetric which would allow for a vertical FOV greater than 50 degrees with 100 degree horizontal FOV. 
     The first image capture device  122  may acquire a plurality of first images relative to a scene associated with vehicle  200 . Each of the plurality of first images may be acquired as a series of image scan lines, which may be captured using a rolling shutter. Each scan line may include a plurality of pixels. 
     The first image capture device  122  may have a scan rate associated with acquisition of each of the first series of image scan lines. The scan rate may refer to a rate at which an image sensor can acquire image data associated with each pixel included in a particular scan line. 
     Image capture devices  122 ,  124 , and  126  may contain any suitable type and number of image sensors, including CCD sensors or CMOS sensors, for example. In one embodiment, a CMOS image sensor may be employed along with a rolling shutter, such that each pixel in a row is read one at a time, and scanning of the rows proceeds on a row-by-row basis until an entire image frame has been captured. In some embodiments, the rows may be captured sequentially from top to bottom relative to the frame. 
     In some embodiments, one or more of the image capture devices (e.g., image capture devices  122 ,  124 , and  126 ) disclosed herein may constitute a high resolution imager and may have a resolution greater than 5M pixel, 7M pixel, 10M pixel, or greater. 
     The use of a rolling shutter may result in pixels in different rows being exposed and captured at different times, which may cause skew and other image artifacts in the captured image frame. On the other hand, when the image capture device  122  is configured to operate with a global or synchronous shutter, all of the pixels may be exposed for the same amount of time and during a common exposure period. As a result, the image data in a frame collected from a system employing a global shutter represents a snapshot of the entire FOV (such as FOV  202 ) at a particular time. In contrast, in a rolling shutter application, each row in a frame is exposed and data is capture at different times. Thus, moving objects may appear distorted in an image capture device having a rolling shutter. This phenomenon will be described in greater detail below. 
     The second image capture device  124  and the third image capturing device  126  may be any type of image capture device. Like the first image capture device  122 , each of image capture devices  124  and  126  may include an optical axis. In one embodiment, each of image capture devices  124  and  126  may include an Aptina M9V024 WVGA sensor with a global shutter. Alternatively, each of image capture devices  124  and  126  may include a rolling shutter. Like image capture device  122 , image capture devices  124  and  126  may be configured to include various lenses and optical elements. In some embodiments, lenses associated with image capture devices  124  and  126  may provide FOVs (such as FOVs  204  and  206 ) that are the same as, or narrower than, a FOV (such as FOV  202 ) associated with image capture device  122 . For example, image capture devices  124  and  126  may have FOVs of 40 degrees, 30 degrees, 26 degrees, 23 degrees, 20 degrees, or less. 
     Image capture devices  124  and  126  may acquire a plurality of second and third images relative to a scene associated with vehicle  200 . Each of the plurality of second and third images may be acquired as a second and third series of image scan lines, which may be captured using a rolling shutter. Each scan line or row may have a plurality of pixels. Image capture devices  124  and  126  may have second and third scan rates associated with acquisition of each of image scan lines included in the second and third series. 
     Each image capture device  122 ,  124 , and  126  may be positioned at any suitable position and orientation relative to vehicle  200 . The relative positioning of the image capture devices  122 ,  124 , and  126  may be selected to aid in fusing together the information acquired from the image capture devices. For example, in some embodiments, a FOV (such as FOV  204 ) associated with image capture device  124  may overlap partially or fully with a FOV (such as FOV  202 ) associated with image capture device  122  and a FOV (such as FOV  206 ) associated with image capture device  126 . 
     Image capture devices  122 ,  124 , and  126  may be located on vehicle  200  at any suitable relative heights. In one instance, there may be a height difference between the image capture devices  122 ,  124 , and  126 , which may provide sufficient parallax information to enable stereo analysis. For example, as shown in  FIG. 2A , the two image capture devices  122  and  124  are at different heights. There may also be a lateral displacement difference between image capture devices  122 ,  124 , and  126 , giving additional parallax information for stereo analysis by processing unit  110 , for example. The difference in the lateral displacement may be denoted by d x , as shown in  FIGS. 2C and 2D . In some embodiments, fore or aft displacement (e.g., range displacement) may exist between image capture devices  122 ,  124 , and  126 . For example, image capture device  122  may be located  0 . 5  to  2  meters or more behind image capture device  124  and/or image capture device  126 . This type of displacement may enable one of the image capture devices to cover potential blind spots of the other image capture device(s). 
     Image capture devices  122  may have any suitable resolution capability (e.g., number of pixels associated with the image sensor), and the resolution of the image sensor(s) associated with the image capture device  122  may be higher, lower, or the same as the resolution of the image sensor(s) associated with image capture devices  124  and  126 . In some embodiments, the image sensor(s) associated with image capture device  122  and/or image capture devices  124  and  126  may have a resolution of 640×480, 1024×768, 1280×960, or any other suitable resolution. 
     The frame rate (e.g., the rate at which an image capture device acquires a set of pixel data of one image frame before moving on to capture pixel data associated with the next image frame) may be controllable. The frame rate associated with image capture device  122  may be higher, lower, or the same as the frame rate associated with image capture devices  124  and  126 . The frame rate associated with image capture devices  122 ,  124 , and  126  may depend on a variety of factors that may affect the timing of the frame rate. For example, one or more of image capture devices  122 ,  124 , and  126  may include a selectable pixel delay period imposed before or after acquisition of image data associated with one or more pixels of an image sensor in image capture device  122 ,  124 , and/or  126 . Generally, image data corresponding to each pixel may be acquired according to a clock rate for the device (e.g., one pixel per clock cycle). Additionally, in embodiments including a rolling shutter, one or more of image capture devices  122 ,  124 , and  126  may include a selectable horizontal blanking period imposed before or after acquisition of image data associated with a row of pixels of an image sensor in image capture device  122 ,  124 , and/or  126 . Further, one or more of image capture devices  122 ,  124 , and/or  126  may include a selectable vertical blanking period imposed before or after acquisition of image data associated with an image frame of image capture device  122 ,  124 , and  126 . 
     These timing controls may enable synchronization of frame rates associated with image capture devices  122 ,  124 , and  126 , even where the line scan rates of each are different. Additionally, as will be discussed in greater detail below, these selectable timing controls, among other factors (e.g., image sensor resolution, maximum line scan rates, etc.) may enable synchronization of image capture from an area where the FOV of image capture device  122  overlaps with one or more FOVs of image capture devices  124  and  126 , even where the field of view of image capture device  122  is different from the FOVs of image capture devices  124  and  126 . 
     Frame rate timing in image capture device  122 ,  124 , and  126  may depend on the resolution of the associated image sensors. For example, assuming similar line scan rates for both devices, if one device includes an image sensor having a resolution of 640×480 and another device includes an image sensor with a resolution of 1280×960, then more time will be required to acquire a frame of image data from the sensor having the higher resolution. 
     Another factor that may affect the timing of image data acquisition in image capture devices  122 ,  124 , and  126  is the maximum line scan rate. For example, acquisition of a row of image data from an image sensor included in image capture device  122 ,  124 , and  126  will require some minimum amount of time. Assuming no pixel delay periods are added, this minimum amount of time for acquisition of a row of image data will be related to the maximum line scan rate for a particular device. Devices that offer higher maximum line scan rates have the potential to provide higher frame rates than devices with lower maximum line scan rates. In some embodiments, one or more of image capture devices  124  and  126  may have a maximum line scan rate that is higher than a maximum line scan rate associated with image capture device  122 . In some embodiments, the maximum line scan rate of image capture device  124  and/or  126  may be 1.25, 1.5, 1.75, or 2 times or more than a maximum line scan rate of image capture device  122 . 
     In another embodiment, image capture devices  122 ,  124 , and  126  may have the same maximum line scan rate, but image capture device  122  may be operated at a scan rate less than or equal to its maximum scan rate. The system may be configured such that one or more of image capture devices  124  and  126  operate at a line scan rate that is equal to the line scan rate of image capture device  122 . In other instances, the system may be configured such that the line scan rate of image capture device  124  and/or image capture device  126  may be 1.25, 1.5, 1.75, or 2 times or more than the line scan rate of image capture device  122 . 
     In some embodiments, image capture devices  122 ,  124 , and  126  may be asymmetric. That is, they may include cameras having different fields of view (FOV) and focal lengths. The fields of view of image capture devices  122 ,  124 , and  126  may include any desired area relative to an environment of vehicle  200 , for example. In some embodiments, one or more of image capture devices  122 ,  124 , and  126  may be configured to acquire image data from an environment in front of vehicle  200 , behind vehicle  200 , to the sides of vehicle  200 , or combinations thereof. 
     Further, the focal length associated with each image capture device  122 ,  124 , and/or  126  may be selectable (e.g., by inclusion of appropriate lenses etc.) such that each device acquires images of objects at a desired distance range relative to vehicle  200 . For example, in some embodiments image capture devices  122 ,  124 , and  126  may acquire images of close-up objects within a few meters from the vehicle. Image capture devices  122 ,  124 , and  126  may also be configured to acquire images of objects at ranges more distant from the vehicle (e.g., 25 m, 50 m, 100 m, 150 m, or more). Further, the focal lengths of image capture devices  122 ,  124 , and  126  may be selected such that one image capture device (e.g., image capture device  122 ) can acquire images of objects relatively close to the vehicle (e.g., within 10 m or within 20 m) while the other image capture devices (e.g., image capture devices  124  and  126 ) can acquire images of more distant objects (e.g., greater than 20 m, 50 m, 100 m, 150 m, etc.) from vehicle  200 . 
     According to some embodiments, the FOV of one or more image capture devices  122 ,  124 , and  126  may have a wide angle. For example, it may be advantageous to have a FOV of 140 degrees, especially for image capture devices  122 ,  124 , and  126  that may be used to capture images of the area in the vicinity of vehicle  200 . For example, image capture device  122  may be used to capture images of the area to the right or left of vehicle  200  and, in such embodiments, it may be desirable for image capture device  122  to have a wide FOV (e.g., at least 140 degrees). 
     The field of view associated with each of image capture devices  122 ,  124 , and  126  may depend on the respective focal lengths. For example, as the focal length increases, the corresponding field of view decreases. 
     Image capture devices  122 ,  124 , and  126  may be configured to have any suitable fields of view. In one particular example, image capture device  122  may have a horizontal FOV of 46 degrees, image capture device  124  may have a horizontal FOV of 23 degrees, and image capture device  126  may have a horizontal FOV in between 23 and 46 degrees. In another instance, image capture device  122  may have a horizontal FOV of 52 degrees, image capture device  124  may have a horizontal FOV of 26 degrees, and image capture device  126  may have a horizontal FOV in between 26 and 52 degrees. In some embodiments, a ratio of the FOV of image capture device  122  to the FOVs of image capture device  124  and/or image capture device  126  may vary from 1.5 to 2.0. In other embodiments, this ratio may vary between 1.25 and 2.25. 
     System  100  may be configured so that a field of view of image capture device  122  overlaps, at least partially or fully, with a field of view of image capture device  124  and/or image capture device  126 . In some embodiments, system  100  may be configured such that the fields of view of image capture devices  124  and  126 , for example, fall within (e.g., are narrower than) and share a common center with the field of view of image capture device  122 . In other embodiments, the image capture devices  122 ,  124 , and  126  may capture adjacent FOVs or may have partial overlap in their FOVs. In some embodiments, the fields of view of image capture devices  122 ,  124 , and  126  may be aligned such that a center of the narrower FOV image capture devices  124  and/or  126  may be located in a lower half of the field of view of the wider FOV device  122 . 
       FIG. 2F  is a diagrammatic representation of exemplary vehicle control systems, consistent with the disclosed embodiments. As indicated in  FIG. 2F , vehicle  200  may include throttling system  220 , braking system  230 , and steering system  240 . System  100  may provide inputs (e.g., control signals) to one or more of throttling system  220 , braking system  230 , and steering system  240  over one or more data links (e.g., any wired and/or wireless link or links for transmitting data). For example, based on analysis of images acquired by image capture devices  122 ,  124 , and/or  126 , system  100  may provide control signals to one or more of throttling system  220 , braking system  230 , and steering system  240  to navigate vehicle  200  (e.g., by causing an acceleration, a turn, a lane shift, etc.). Further, system  100  may receive inputs from one or more of throttling system  220 , braking system  230 , and steering system  24  indicating operating conditions of vehicle  200  (e.g., speed, whether vehicle  200  is braking and/or turning, etc.). Further details are provided in connection with  FIGS. 4-7 , below. 
     As shown in  FIG. 3A , vehicle  200  may also include a user interface  170  for interacting with a driver or a passenger of vehicle  200 . For example, user interface  170  in a vehicle application may include a touch screen  320 , knobs  330 , buttons  340 , and a microphone  350 . A driver or passenger of vehicle  200  may also use handles (e.g., located on or near the steering column of vehicle  200  including, for example, turn signal handles), buttons (e.g., located on the steering wheel of vehicle  200 ), and the like, to interact with system  100 . In some embodiments, microphone  350  may be positioned adjacent to a rearview mirror  310 . Similarly, in some embodiments, image capture device  122  may be located near rearview mirror  310 . In some embodiments, user interface  170  may also include one or more speakers  360  (e.g., speakers of a vehicle audio system). For example, system  100  may provide various notifications (e.g., alerts) via speakers  360 . 
       FIGS. 3B-3D  are illustrations of an exemplary camera mount  370  configured to be positioned behind a rearview mirror (e.g., rearview mirror  310 ) and against a vehicle windshield, consistent with disclosed embodiments. As shown in  FIG. 3B , camera mount  370  may include image capture devices  122 ,  124 , and  126 . Image capture devices  124  and  126  may be positioned behind a glare shield  380 , which may be flush against the vehicle windshield and include a composition of film and/or anti-reflective materials. For example, glare shield  380  may be positioned such that it aligns against a vehicle windshield having a matching slope. In some embodiments, each of image capture devices  122 ,  124 , and  126  may be positioned behind glare shield  380 , as depicted, for example, in  FIG. 3D . The disclosed embodiments are not limited to any particular configuration of image capture devices  122 ,  124 , and  126 , camera mount  370 , and glare shield  380 .  FIG. 3C  is an illustration of camera mount  370  shown in  FIG. 3B  from a front perspective. 
     As will be appreciated by a person skilled in the art having the benefit of this disclosure, numerous variations and/or modifications may be made to the foregoing disclosed embodiments. For example, not all components are essential for the operation of system  100 . Further, any component may be located in any appropriate part of system  100  and the components may be rearranged into a variety of configurations while providing the functionality of the disclosed embodiments. Therefore, the foregoing configurations are examples and, regardless of the configurations discussed above, system  100  can provide a wide range of functionality to analyze the surroundings of vehicle  200  and navigate vehicle  200  in response to the analysis. 
     As discussed below in further detail and consistent with various disclosed embodiments, system  100  may provide a variety of features related to autonomous driving and/or driver assist technology. For example, system  100  may analyze image data, position data (e.g., GPS location information), map data, speed data, and/or data from sensors included in vehicle  200 . System  100  may collect the data for analysis from, for example, image acquisition unit  120 , position sensor  130 , and other sensors. Further, system  100  may analyze the collected data to determine whether or not vehicle  200  should take a certain action, and then automatically take the determined action without human intervention. For example, when vehicle  200  navigates without human intervention, system  100  may automatically control the braking, acceleration, and/or steering of vehicle  200  (e.g., by sending control signals to one or more of throttling system  220 , braking system  230 , and steering system  240 ). Further, system  100  may analyze the collected data and issue warnings and/or alerts to vehicle occupants based on the analysis of the collected data. Additional details regarding the various embodiments that are provided by system  100  are provided below. 
     Forward-Facing Multi-Imaging System 
     As discussed above, system  100  may provide drive assist functionality that uses a multi-camera system. The multi-camera system may use one or more cameras facing in the forward direction of a vehicle. In other embodiments, the multi-camera system may include one or more cameras facing to the side of a vehicle or to the rear of the vehicle. In one embodiment, for example, system  100  may use a two-camera imaging system, where a first camera and a second camera (e.g., image capture devices  122  and  124 ) may be positioned at the front and/or the sides of a vehicle (e.g., vehicle  200 ). Other camera configurations are consistent with the disclosed embodiments, and the configurations disclosed herein are examples. For example, system  100  may include a configuration of any number of cameras (e.g., one, two, three, four, five, six, seven, eight, etc.) Furthermore, system  100  may include “clusters” of cameras. For example, a cluster of cameras (including any appropriate number of cameras, e.g., one, four, eight, etc.) may be forward-facing relative to a vehicle, or may be facing any other direction (e.g., reward-facing, side-facing, at an angle, etc.) Accordingly, system  100  may include multiple clusters of cameras, with each cluster oriented in a particular direction to capture images from a particular region of a vehicle&#39;s environment. 
     The first camera may have a field of view that is greater than, less than, or partially overlapping with, the field of view of the second camera. In addition, the first camera may be connected to a first image processor to perform monocular image analysis of images provided by the first camera, and the second camera may be connected to a second image processor to perform monocular image analysis of images provided by the second camera. The outputs (e.g., processed information) of the first and second image processors may be combined. 
     In some embodiments, the second image processor may receive images from both the first camera and second camera to perform stereo analysis. In another embodiment, system  100  may use a three-camera imaging system where each of the cameras has a different field of view. Such a system may, therefore, make decisions based on information derived from objects located at varying distances both forward and to the sides of the vehicle. References to monocular image analysis may refer to instances where image analysis is performed based on images captured from a single point of view (e.g., from a single camera). Stereo image analysis may refer to instances where image analysis is performed based on two or more images captured with one or more variations of an image capture parameter. For example, captured images suitable for performing stereo image analysis may include images captured: from two or more different positions, from different fields of view, using different focal lengths, along with parallax information, etc. 
     For example, in one embodiment, system  100  may implement a three camera configuration using image capture devices  122 - 126 . In such a configuration, image capture device  122  may provide a narrow field of view (e.g., 34 degrees, or other values selected from a range of about 20 to 45 degrees, etc.), image capture device  124  may provide a wide field of view (e.g., 150 degrees or other values selected from a range of about 100 to about 180 degrees), and image capture device  126  may provide an intermediate field of view (e.g.,  46  degrees or other values selected from a range of about 35 to about 60 degrees). In some embodiments, image capture device  126  may act as a main or primary camera. Image capture devices  122 - 126  may be positioned behind rearview mirror  310  and positioned substantially side-by-side (e.g., 6 cm apart). Further, in some embodiments, as discussed above, one or more of image capture devices  122 - 126  may be mounted behind glare shield  380  that is flush with the windshield of vehicle  200 . Such shielding may act to minimize the impact of any reflections from inside the car on image capture devices  122 - 126 . 
     In another embodiment, as discussed above in connection with  FIGS. 3B and 3C , the wide field of view camera (e.g., image capture device  124  in the above example) may be mounted lower than the narrow and main field of view cameras (e.g., image devices  122  and  126  in the above example). This configuration may provide a free line of sight from the wide field of view camera. To reduce reflections, the cameras may be mounted close to the windshield of vehicle  200 , and may include polarizers on the cameras to damp reflected light. 
     A three camera system may provide certain performance characteristics. For example, some embodiments may include an ability to validate the detection of objects by one camera based on detection results from another camera. In the three camera configuration discussed above, processing unit  110  may include, for example, three processing devices (e.g., three EyeQ series of processor chips, as discussed above), with each processing device dedicated to processing images captured by one or more of image capture devices  122 - 126 . 
     In a three camera system, a first processing device may receive images from both the main camera and the narrow field of view camera, and perform vision processing of the narrow FOV camera to, for example, detect other vehicles, pedestrians, lane marks, traffic signs, traffic lights, and other road objects. Further, the first processing device may calculate a disparity of pixels between the images from the main camera and the narrow camera and create a 3D reconstruction of the environment of vehicle  200 . The first processing device may then combine the 3D reconstruction with 3D map data or with 3D information calculated based on information from another camera. 
     The second processing device may receive images from the main camera and perform vision processing to detect other vehicles, pedestrians, lane marks, traffic signs, traffic lights, and other road objects. Additionally, the second processing device may calculate a camera displacement and, based on the displacement, calculate a disparity of pixels between successive images and create a 3D reconstruction of the scene (e.g., a structure from motion). The second processing device may send the structure from motion based 3D reconstruction to the first processing device to be combined with the stereo 3D images. 
     The third processing device may receive images from the wide FOV camera and process the images to detect vehicles, pedestrians, lane marks, traffic signs, traffic lights, and other road objects. The third processing device may further execute additional processing instructions to analyze images to identify objects moving in the image, such as vehicles changing lanes, pedestrians, etc. 
     In some embodiments, having streams of image-based information captured and processed independently may provide an opportunity for providing redundancy in the system. Such redundancy may include, for example, using a first image capture device and the images processed from that device to validate and/or supplement information obtained by capturing and processing image information from at least a second image capture device. 
     In some embodiments, system  100  may use two image capture devices (e.g., image capture devices  122  and  124 ) in providing navigation assistance for vehicle  200  and use a third image capture device (e.g., image capture device  126 ) to provide redundancy and validate the analysis of data received from the other two image capture devices. For example, in such a configuration, image capture devices  122  and  124  may provide images for stereo analysis by system  100  for navigating vehicle  200 , while image capture device  126  may provide images for monocular analysis by system  100  to provide redundancy and validation of information obtained based on images captured from image capture device  122  and/or image capture device  124 . That is, image capture device  126  (and a corresponding processing device) may be considered to provide a redundant sub-system for providing a check on the analysis derived from image capture devices  122  and  124  (e.g., to provide an automatic emergency braking (AEB) system). Furthermore, in some embodiments, redundancy and validation of received data may be supplemented based on information received from one more sensors (e.g., radar, lidar, acoustic sensors, information received from one or more transceivers outside of a vehicle, etc.). 
     One of skill in the art will recognize that the above camera configurations, camera placements, number of cameras, camera locations, etc., are examples only. These components and others described relative to the overall system may be assembled and used in a variety of different configurations without departing from the scope of the disclosed embodiments. Further details regarding usage of a multi-camera system to provide driver assist and/or autonomous vehicle functionality follow below. 
       FIG. 4  is an exemplary functional block diagram of memory  140  and/or  150 , which may be stored/programmed with instructions for performing one or more operations consistent with the disclosed embodiments. Although the following refers to memory  140 , one of skill in the art will recognize that instructions may be stored in memory  140  and/or  150 . 
     As shown in  FIG. 4 , memory  140  may store a monocular image analysis module  402 , a stereo image analysis module  404 , a velocity and acceleration module  406 , and a navigational response module  408 . The disclosed embodiments are not limited to any particular configuration of memory  140 . Further, applications processor  180  and/or image processor  190  may execute the instructions stored in any of modules  402 - 408  included in memory  140 . One of skill in the art will understand that references in the following discussions to processing unit  110  may refer to applications processor  180  and image processor  190  individually or collectively. Accordingly, steps of any of the following processes may be performed by one or more processing devices. 
     In one embodiment, monocular image analysis module  402  may store instructions (such as computer vision software) which, when executed by processing unit  110 , performs monocular image analysis of a set of images acquired by one of image capture devices  122 ,  124 , and  126 . In some embodiments, processing unit  110  may combine information from a set of images with additional sensory information (e.g., information from radar) to perform the monocular image analysis. As described in connection with  FIGS. 5A-5D  below, monocular image analysis module  402  may include instructions for detecting a set of features within the set of images, such as lane markings, vehicles, pedestrians, road signs, highway exit ramps, traffic lights, hazardous objects, and any other feature associated with an environment of a vehicle. Based on the analysis, system  100  (e.g., via processing unit  110 ) may cause one or more navigational responses in vehicle  200 , such as a turn, a lane shift, a change in acceleration, and the like, as discussed below in connection with navigational response module  408 . 
     In one embodiment, monocular image analysis module  402  may store instructions (such as computer vision software) which, when executed by processing unit  110 , performs monocular image analysis of a set of images acquired by one of image capture devices  122 ,  124 , and  126 . In some embodiments, processing unit  110  may combine information from a set of images with additional sensory information (e.g., information from radar, lidar, etc.) to perform the monocular image analysis. As described in connection with  FIGS. 5A-5D  below, monocular image analysis module  402  may include instructions for detecting a set of features within the set of images, such as lane markings, vehicles, pedestrians, road signs, highway exit ramps, traffic lights, hazardous objects, and any other feature associated with an environment of a vehicle. Based on the analysis, system  100  (e.g., via processing unit  110 ) may cause one or more navigational responses in vehicle  200 , such as a turn, a lane shift, a change in acceleration, and the like, as discussed below in connection with determining a navigational response. 
     In one embodiment, stereo image analysis module  404  may store instructions (such as computer vision software) which, when executed by processing unit  110 , performs stereo image analysis of first and second sets of images acquired by a combination of image capture devices selected from any of image capture devices  122 ,  124 , and  126 . In some embodiments, processing unit  110  may combine information from the first and second sets of images with additional sensory information (e.g., information from radar) to perform the stereo image analysis. For example, stereo image analysis module  404  may include instructions for performing stereo image analysis based on a first set of images acquired by image capture device  124  and a second set of images acquired by image capture device  126 . As described in connection with  FIG. 6  below, stereo image analysis module  404  may include instructions for detecting a set of features within the first and second sets of images, such as lane markings, vehicles, pedestrians, road signs, highway exit ramps, traffic lights, hazardous objects, and the like. Based on the analysis, processing unit  110  may cause one or more navigational responses in vehicle  200 , such as a turn, a lane shift, a change in acceleration, and the like, as discussed below in connection with navigational response module  408 . Furthermore, in some embodiments, stereo image analysis module  404  may implement techniques associated with a trained system (such as a neural network or a deep neural network) or an untrained system. 
     In one embodiment, velocity and acceleration module  406  may store software configured to analyze data received from one or more computing and electromechanical devices in vehicle  200  that are configured to cause a change in velocity and/or acceleration of vehicle  200 . For example, processing unit  110  may execute instructions associated with velocity and acceleration module  406  to calculate a target speed for vehicle  200  based on data derived from execution of monocular image analysis module  402  and/or stereo image analysis module  404 . Such data may include, for example, a target position, velocity, and/or acceleration, the position and/or speed of vehicle  200  relative to a nearby vehicle, pedestrian, or road object, position information for vehicle  200  relative to lane markings of the road, and the like. In addition, processing unit  110  may calculate a target speed for vehicle  200  based on sensory input (e.g., information from radar) and input from other systems of vehicle  200 , such as throttling system  220 , braking system  230 , and/or steering system  240  of vehicle  200 . Based on the calculated target speed, processing unit  110  may transmit electronic signals to throttling system  220 , braking system  230 , and/or steering system  240  of vehicle  200  to trigger a change in velocity and/or acceleration by, for example, physically depressing the brake or easing up off the accelerator of vehicle  200 . 
     In one embodiment, navigational response module  408  may store software executable by processing unit  110  to determine a desired navigational response based on data derived from execution of monocular image analysis module  402  and/or stereo image analysis module  404 . Such data may include position and speed information associated with nearby vehicles, pedestrians, and road objects, target position information for vehicle  200 , and the like. Additionally, in some embodiments, the navigational response may be based (partially or fully) on map data, a predetermined position of vehicle  200 , and/or a relative velocity or a relative acceleration between vehicle  200  and one or more objects detected from execution of monocular image analysis module  402  and/or stereo image analysis module  404 . Navigational response module  408  may also determine a desired navigational response based on sensory input (e.g., information from radar) and inputs from other systems of vehicle  200 , such as throttling system  220 , braking system  230 , and steering system  240  of vehicle  200 . Based on the desired navigational response, processing unit  110  may transmit electronic signals to throttling system  220 , braking system  230 , and steering system  240  of vehicle  200  to trigger a desired navigational response by, for example, turning the steering wheel of vehicle  200  to achieve a rotation of a predetermined angle. In some embodiments, processing unit  110  may use the output of navigational response module  408  (e.g., the desired navigational response) as an input to execution of velocity and acceleration module  406  for calculating a change in speed of vehicle  200 . 
     Furthermore, any of the modules (e.g., modules  402 ,  404 , and  406 ) disclosed herein may implement techniques associated with a trained system (such as a neural network or a deep neural network) or an untrained system. 
       FIG. 5A  is a flowchart showing an exemplary process  500 A for causing one or more navigational responses based on monocular image analysis, consistent with disclosed embodiments. At step  510 , processing unit  110  may receive a plurality of images via data interface  128  between processing unit  110  and image acquisition unit  120 . For instance, a camera included in image acquisition unit  120  (such as image capture device  122  having field of view  202 ) may capture a plurality of images of an area forward of vehicle  200  (or to the sides or rear of a vehicle, for example) and transmit them over a data connection (e.g., digital, wired, USB, wireless, Bluetooth, etc.) to processing unit  110 . Processing unit  110  may execute monocular image analysis module  402  to analyze the plurality of images at step  520 , as described in further detail in connection with  FIGS. 5B-5D  below. By performing the analysis, processing unit  110  may detect a set of features within the set of images, such as lane markings, vehicles, pedestrians, road signs, highway exit ramps, traffic lights, and the like. 
     Processing unit  110  may also execute monocular image analysis module  402  to detect various road hazards at step  520 , such as, for example, parts of a truck tire, fallen road signs, loose cargo, small animals, and the like. Road hazards may vary in structure, shape, size, and color, which may make detection of such hazards more challenging. In some embodiments, processing unit  110  may execute monocular image analysis module  402  to perform multi-frame analysis on the plurality of images to detect road hazards. For example, processing unit  110  may estimate camera motion between consecutive image frames and calculate the disparities in pixels between the frames to construct a 3D-map of the road. Processing unit  110  may then use the 3D-map to detect the road surface, as well as hazards existing above the road surface. 
     At step  530 , processing unit  110  may execute navigational response module  408  to cause one or more navigational responses in vehicle  200  based on the analysis performed at step  520  and the techniques as described above in connection with  FIG. 4 . Navigational responses may include, for example, a turn, a lane shift, a change in acceleration, and the like. In some embodiments, processing unit  110  may use data derived from execution of velocity and acceleration module  406  to cause the one or more navigational responses. Additionally, multiple navigational responses may occur simultaneously, in sequence, or any combination thereof. For instance, processing unit  110  may cause vehicle  200  to shift one lane over and then accelerate by, for example, sequentially transmitting control signals to steering system  240  and throttling system  220  of vehicle  200 . Alternatively, processing unit  110  may cause vehicle  200  to brake while at the same time shifting lanes by, for example, simultaneously transmitting control signals to braking system  230  and steering system  240  of vehicle  200 . 
       FIG. 5B  is a flowchart showing an exemplary process  500 B for detecting one or more vehicles and/or pedestrians in a set of images, consistent with disclosed embodiments. 
     Processing unit  110  may execute monocular image analysis module  402  to implement process  500 B. At step  540 , processing unit  110  may determine a set of candidate objects representing possible vehicles and/or pedestrians. For example, processing unit  110  may scan one or more images, compare the images to one or more predetermined patterns, and identify within each image possible locations that may contain objects of interest (e.g., vehicles, pedestrians, or portions thereof). The predetermined patterns may be designed in such a way to achieve a high rate of “false hits” and a low rate of “misses.” For example, processing unit  110  may use a low threshold of similarity to predetermined patterns for identifying candidate objects as possible vehicles or pedestrians. Doing so may allow processing unit  110  to reduce the probability of missing (e.g., not identifying) a candidate object representing a vehicle or pedestrian. 
     At step  542 , processing unit  110  may filter the set of candidate objects to exclude certain candidates (e.g., irrelevant or less relevant objects) based on classification criteria. Such criteria may be derived from various properties associated with object types stored in a database (e.g., a database stored in memory  140 ). Properties may include object shape, dimensions, texture, position (e.g., relative to vehicle  200 ), and the like. Thus, processing unit  110  may use one or more sets of criteria to reject false candidates from the set of candidate objects. 
     At step  544 , processing unit  110  may analyze multiple frames of images to determine whether objects in the set of candidate objects represent vehicles and/or pedestrians. For example, processing unit  110  may track a detected candidate object across consecutive frames and accumulate frame-by-frame data associated with the detected object (e.g., size, position relative to vehicle  200 , etc.). Additionally, processing unit  110  may estimate parameters for the detected object and compare the object&#39;s frame-by-frame position data to a predicted position. 
     At step  546 , processing unit  110  may construct a set of measurements for the detected objects. Such measurements may include, for example, position, velocity, and acceleration values (relative to vehicle  200 ) associated with the detected objects. In some embodiments, processing unit  110  may construct the measurements based on estimation techniques using a series of time-based observations such as Kalman filters or linear quadratic estimation (LQE), and/or based on available modeling data for different object types (e.g., cars, trucks, pedestrians, bicycles, road signs, etc.). The Kalman filters may be based on a measurement of an object&#39;s scale, where the scale measurement is proportional to a time to collision (e.g., the amount of time for vehicle  200  to reach the object). Thus, by performing steps  540 - 546 , processing unit  110  may identify vehicles and pedestrians appearing within the set of captured images and derive information (e.g., position, speed, size) associated with the vehicles and pedestrians. Based on the identification and the derived information, processing unit  110  may cause one or more navigational responses in vehicle  200 , as described in connection with  FIG. 5A , above. 
     At step  548 , processing unit  110  may perform an optical flow analysis of one or more images to reduce the probabilities of detecting a “false hit” and missing a candidate object that represents a vehicle or pedestrian. The optical flow analysis may refer to, for example, analyzing motion patterns relative to vehicle  200  in the one or more images associated with other vehicles and pedestrians, and that are distinct from road surface motion. Processing unit  110  may calculate the motion of candidate objects by observing the different positions of the objects across multiple image frames, which are captured at different times. Processing unit  110  may use the position and time values as inputs into mathematical models for calculating the motion of the candidate objects. Thus, optical flow analysis may provide another method of detecting vehicles and pedestrians that are nearby vehicle  200 . Processing unit  110  may perform optical flow analysis in combination with steps  540 - 546  to provide redundancy for detecting vehicles and pedestrians and increase the reliability of system  100 . 
       FIG. 5C  is a flowchart showing an exemplary process  500 C for detecting road marks and/or lane geometry information in a set of images, consistent with disclosed embodiments. Processing unit  110  may execute monocular image analysis module  402  to implement process  500 C. At step  550 , processing unit  110  may detect a set of objects by scanning one or more images. To detect segments of lane markings, lane geometry information, and other pertinent road marks, processing unit  110  may filter the set of objects to exclude those determined to be irrelevant (e.g., minor potholes, small rocks, etc.). At step  552 , processing unit  110  may group together the segments detected in step  550  belonging to the same road mark or lane mark. Based on the grouping, processing unit  110  may develop a model to represent the detected segments, such as a mathematical model. 
     At step  554 , processing unit  110  may construct a set of measurements associated with the detected segments. In some embodiments, processing unit  110  may create a projection of the detected segments from the image plane onto the real-world plane. The projection may be characterized using a  3 rd-degree polynomial having coefficients corresponding to physical properties such as the position, slope, curvature, and curvature derivative of the detected road. In generating the projection, processing unit  110  may take into account changes in the road surface, as well as pitch and roll rates associated with vehicle  200 . In addition, processing unit  110  may model the road elevation by analyzing position and motion cues present on the road surface. Further, processing unit  110  may estimate the pitch and roll rates associated with vehicle  200  by tracking a set of feature points in the one or more images. 
     At step  556 , processing unit  110  may perform multi-frame analysis by, for example, tracking the detected segments across consecutive image frames and accumulating frame-by-frame data associated with detected segments. As processing unit  110  performs multi-frame analysis, the set of measurements constructed at step  554  may become more reliable and associated with an increasingly higher confidence level. Thus, by performing steps  550 - 556 , processing unit  110  may identify road marks appearing within the set of captured images and derive lane geometry information. Based on the identification and the derived information, processing unit  110  may cause one or more navigational responses in vehicle  200 , as described in connection with  FIG. 5A , above. 
     At step  558 , processing unit  110  may consider additional sources of information to further develop a safety model for vehicle  200  in the context of its surroundings. Processing unit  110  may use the safety model to define a context in which system  100  may execute autonomous control of vehicle  200  in a safe manner. To develop the safety model, in some embodiments, processing unit  110  may consider the position and motion of other vehicles, the detected road edges and barriers, and/or general road shape descriptions extracted from map data (such as data from map database  160 ). By considering additional sources of information, processing unit  110  may provide redundancy for detecting road marks and lane geometry and increase the reliability of system  100 . 
       FIG. 5D  is a flowchart showing an exemplary process  500 D for detecting traffic lights in a set of images, consistent with disclosed embodiments. Processing unit  110  may execute monocular image analysis module  402  to implement process  500 D. At step  560 , processing unit  110  may scan the set of images and identify objects appearing at locations in the images likely to contain traffic lights. For example, processing unit  110  may filter the identified objects to construct a set of candidate objects, excluding those objects unlikely to correspond to traffic lights. The filtering may be done based on various properties associated with traffic lights, such as shape, dimensions, texture, position (e.g., relative to vehicle  200 ), and the like. Such properties may be based on multiple examples of traffic lights and traffic control signals and stored in a database. In some embodiments, processing unit  110  may perform multi-frame analysis on the set of candidate objects reflecting possible traffic lights. For example, processing unit  110  may track the candidate objects across consecutive image frames, estimate the real-world position of the candidate objects, and filter out those objects that are moving (which are unlikely to be traffic lights). In some embodiments, processing unit  110  may perform color analysis on the candidate objects and identify the relative position of the detected colors appearing inside possible traffic lights. 
     At step  562 , processing unit  110  may analyze the geometry of a junction. The analysis may be based on any combination of: (i) the number of lanes detected on either side of vehicle  200 , (ii) markings (such as arrow marks) detected on the road, and (iii) descriptions of the junction extracted from map data (such as data from map database  160 ). Processing unit  110  may conduct the analysis using information derived from execution of monocular analysis module  402 . In addition, Processing unit  110  may determine a correspondence between the traffic lights detected at step  560  and the lanes appearing near vehicle  200 . 
     As vehicle  200  approaches the junction, at step  564 , processing unit  110  may update the confidence level associated with the analyzed junction geometry and the detected traffic lights. For instance, the number of traffic lights estimated to appear at the junction as compared with the number actually appearing at the junction may impact the confidence level. Thus, based on the confidence level, processing unit  110  may delegate control to the driver of vehicle  200  in order to improve safety conditions. By performing steps  560 - 564 , processing unit  110  may identify traffic lights appearing within the set of captured images and analyze junction geometry information. Based on the identification and the analysis, processing unit  110  may cause one or more navigational responses in vehicle  200 , as described in connection with  FIG. 5A , above. 
       FIG. 5E  is a flowchart showing an exemplary process  500 E for causing one or more navigational responses in vehicle  200  based on a vehicle path, consistent with the disclosed embodiments. At step  570 , processing unit  110  may construct an initial vehicle path associated with vehicle  200 . The vehicle path may be represented using a set of points expressed in coordinates (x, z), and the distance d i  between two points in the set of points may fall in the range of 1 to 5 meters. In one embodiment, processing unit  110  may construct the initial vehicle path using two polynomials, such as left and right road polynomials. Processing unit  110  may calculate the geometric midpoint between the two polynomials and offset each point included in the resultant vehicle path by a predetermined offset (e.g., a smart lane offset), if any (an offset of zero may correspond to travel in the middle of a lane). The offset may be in a direction perpendicular to a segment between any two points in the vehicle path. In another embodiment, processing unit  110  may use one polynomial and an estimated lane width to offset each point of the vehicle path by half the estimated lane width plus a predetermined offset (e.g., a smart lane offset). 
     At step  572 , processing unit  110  may update the vehicle path constructed at step  570 . Processing unit  110  may reconstruct the vehicle path constructed at step  570  using a higher resolution, such that the distance d k  between two points in the set of points representing the vehicle path is less than the distance d i  described above. For example, the distance d k  may fall in the range of 0.1 to 0.3 meters. Processing unit  110  may reconstruct the vehicle path using a parabolic spline algorithm, which may yield a cumulative distance vector S corresponding to the total length of the vehicle path (i.e., based on the set of points representing the vehicle path). 
     At step  574 , processing unit  110  may determine a look-ahead point (expressed in coordinates as (x i , z i )) based on the updated vehicle path constructed at step  572 . Processing unit  110  may extract the look-ahead point from the cumulative distance vector S, and the look-ahead point may be associated with a look-ahead distance and look-ahead time. The look-ahead distance, which may have a lower bound ranging from 10 to 20 meters, may be calculated as the product of the speed of vehicle  200  and the look-ahead time. For example, as the speed of vehicle  200  decreases, the look-ahead distance may also decrease (e.g., until it reaches the lower bound). The look-ahead time, which may range from 0.5 to 1.5 seconds, may be inversely proportional to the gain of one or more control loops associated with causing a navigational response in vehicle  200 , such as the heading error tracking control loop. For example, the gain of the heading error tracking control loop may depend on the bandwidth of a yaw rate loop, a steering actuator loop, car lateral dynamics, and the like. Thus, the higher the gain of the heading error tracking control loop, the lower the look-ahead time. 
     At step  576 , processing unit  110  may determine a heading error and yaw rate command based on the look-ahead point determined at step  574 . Processing unit  110  may determine the heading error by calculating the arctangent of the look-ahead point, e.g., arctan (x i /z i ). Processing unit  110  may determine the yaw rate command as the product of the heading error and a high-level control gain. The high-level control gain may be equal to: (2/look-ahead time), if the look-ahead distance is not at the lower bound. Otherwise, the high-level control gain may be equal to: (2*speed of vehicle 200/look-ahead distance). 
       FIG. 5F  is a flowchart showing an exemplary process  500 F for determining whether a leading vehicle is changing lanes, consistent with the disclosed embodiments. At step  580 , processing unit  110  may determine navigation information associated with a leading vehicle (e.g., a vehicle traveling ahead of vehicle  200 ). For example, processing unit  110  may determine the position, velocity (e.g., direction and speed), and/or acceleration of the leading vehicle, using the techniques described in connection with  FIGS. 5A and 5B , above. Processing unit  110  may also determine one or more road polynomials, a look-ahead point (associated with vehicle  200 ), and/or a snail trail (e.g., a set of points describing a path taken by the leading vehicle), using the techniques described in connection with  FIG. 5E , above. 
     At step  582 , processing unit  110  may analyze the navigation information determined at step  580 . In one embodiment, processing unit  110  may calculate the distance between a snail trail and a road polynomial (e.g., along the trail). If the variance of this distance along the trail exceeds a predetermined threshold (for example, 0.1 to 0.2 meters on a straight road, 0.3 to 0.4 meters on a moderately curvy road, and 0.5 to 0.6 meters on a road with sharp curves), processing unit  110  may determine that the leading vehicle is likely changing lanes. In the case where multiple vehicles are detected traveling ahead of vehicle  200 , processing unit  110  may compare the snail trails associated with each vehicle. Based on the comparison, processing unit  110  may determine that a vehicle whose snail trail does not match with the snail trails of the other vehicles is likely changing lanes. Processing unit  110  may additionally compare the curvature of the snail trail (associated with the leading vehicle) with the expected curvature of the road segment in which the leading vehicle is traveling. The expected curvature may be extracted from map data (e.g., data from map database  160 ), from road polynomials, from other vehicles&#39; snail trails, from prior knowledge about the road, and the like. If the difference in curvature of the snail trail and the expected curvature of the road segment exceeds a predetermined threshold, processing unit  110  may determine that the leading vehicle is likely changing lanes. 
     In another embodiment, processing unit  110  may compare the leading vehicle&#39;s instantaneous position with the look-ahead point (associated with vehicle  200 ) over a specific period of time (e.g., 0.5 to 1.5 seconds). If the distance between the leading vehicle&#39;s instantaneous position and the look-ahead point varies during the specific period of time, and the cumulative sum of variation exceeds a predetermined threshold (for example, 0.3 to 0.4 meters on a straight road, 0.7 to 0.8 meters on a moderately curvy road, and 1.3 to 1.7 meters on a road with sharp curves), processing unit  110  may determine that the leading vehicle is likely changing lanes. In another embodiment, processing unit  110  may analyze the geometry of the snail trail by comparing the lateral distance traveled along the trail with the expected curvature of the snail trail. The expected radius of curvature may be determined according to the calculation: (δ z   2 +δ x   2 )/2/(δ x ), where δ x  represents the lateral distance traveled and δ z  represents the longitudinal distance traveled. If the difference between the lateral distance traveled and the expected curvature exceeds a predetermined threshold (e.g., 500 to 700 meters), processing unit  110  may determine that the leading vehicle is likely changing lanes. In another embodiment, processing unit  110  may analyze the position of the leading vehicle. If the position of the leading vehicle obscures a road polynomial (e.g., the leading vehicle is overlaid on top of the road polynomial), then processing unit  110  may determine that the leading vehicle is likely changing lanes. In the case where the position of the leading vehicle is such that, another vehicle is detected ahead of the leading vehicle and the snail trails of the two vehicles are not parallel, processing unit  110  may determine that the (closer) leading vehicle is likely changing lanes. 
     At step  584 , processing unit  110  may determine whether or not leading vehicle  200  is changing lanes based on the analysis performed at step  582 . For example, processing unit  110  may make the determination based on a weighted average of the individual analyses performed at step  582 . Under such a scheme, for example, a decision by processing unit  110  that the leading vehicle is likely changing lanes based on a particular type of analysis may be assigned a value of “1” (and “0” to represent a determination that the leading vehicle is not likely changing lanes). Different analyses performed at step  582  may be assigned different weights, and the disclosed embodiments are not limited to any particular combination of analyses and weights. Furthermore, in some embodiments, the analysis may make use of trained system (e.g., a machine learning or deep learning system), which may, for example, estimate a future path ahead of a current location of a vehicle based on an image captured at the current location. 
       FIG. 6  is a flowchart showing an exemplary process  600  for causing one or more navigational responses based on stereo image analysis, consistent with disclosed embodiments. At step  610 , processing unit  110  may receive a first and second plurality of images via data interface  128 . For example, cameras included in image acquisition unit  120  (such as image capture devices  122  and  124  having fields of view  202  and  204 ) may capture a first and second plurality of images of an area forward of vehicle  200  and transmit them over a digital connection (e.g., USB, wireless, Bluetooth, etc.) to processing unit  110 . In some embodiments, processing unit  110  may receive the first and second plurality of images via two or more data interfaces. The disclosed embodiments are not limited to any particular data interface configurations or protocols. 
     At step  620 , processing unit  110  may execute stereo image analysis module  404  to perform stereo image analysis of the first and second plurality of images to create a 3D map of the road in front of the vehicle and detect features within the images, such as lane markings, vehicles, pedestrians, road signs, highway exit ramps, traffic lights, road hazards, and the like. Stereo image analysis may be performed in a manner similar to the steps described in connection with  FIGS. 5A-5D , above. For example, processing unit  110  may execute stereo image analysis module  404  to detect candidate objects (e.g., vehicles, pedestrians, road marks, traffic lights, road hazards, etc.) within the first and second plurality of images, filter out a subset of the candidate objects based on various criteria, and perform multi-frame analysis, construct measurements, and determine a confidence level for the remaining candidate objects. In performing the steps above, processing unit  110  may consider information from both the first and second plurality of images, rather than information from one set of images alone. For example, processing unit  110  may analyze the differences in pixel-level data (or other data subsets from among the two streams of captured images) for a candidate object appearing in both the first and second plurality of images. As another example, processing unit  110  may estimate a position and/or velocity of a candidate object (e.g., relative to vehicle  200 ) by observing that the object appears in one of the plurality of images but not the other or relative to other differences that may exist relative to objects appearing in the two image streams. For example, position, velocity, and/or acceleration relative to vehicle  200  may be determined based on trajectories, positions, movement characteristics, etc. of features associated with an object appearing in one or both of the image streams. 
     At step  630 , processing unit  110  may execute navigational response module  408  to cause one or more navigational responses in vehicle  200  based on the analysis performed at step  620  and the techniques as described above in connection with  FIG. 4 . Navigational responses may include, for example, a turn, a lane shift, a change in acceleration, a change in velocity, braking, and the like. In some embodiments, processing unit  110  may use data derived from execution of velocity and acceleration module  406  to cause the one or more navigational responses. Additionally, multiple navigational responses may occur simultaneously, in sequence, or any combination thereof. 
       FIG. 7  is a flowchart showing an exemplary process  700  for causing one or more navigational responses based on an analysis of three sets of images, consistent with disclosed embodiments. At step  710 , processing unit  110  may receive a first, second, and third plurality of images via data interface  128 . For instance, cameras included in image acquisition unit  120  (such as image capture devices  122 ,  124 , and  126  having fields of view  202 ,  204 , and  206 ) may capture a first, second, and third plurality of images of an area forward and/or to the side of vehicle  200  and transmit them over a digital connection (e.g., USB, wireless, Bluetooth, etc.) to processing unit  110 . In some embodiments, processing unit  110  may receive the first, second, and third plurality of images via three or more data interfaces. For example, each of image capture devices  122 ,  124 ,  126  may have an associated data interface for communicating data to processing unit  110 . The disclosed embodiments are not limited to any particular data interface configurations or protocols. 
     At step  720 , processing unit  110  may analyze the first, second, and third plurality of images to detect features within the images, such as lane markings, vehicles, pedestrians, road signs, highway exit ramps, traffic lights, road hazards, and the like. The analysis may be performed in a manner similar to the steps described in connection with  FIGS. 5A-5D and 6 , above. For instance, processing unit  110  may perform monocular image analysis (e.g., via execution of monocular image analysis module  402  and based on the steps described in connection with  FIGS. 5A-5D , above) on each of the first, second, and third plurality of images. Alternatively, processing unit  110  may perform stereo image analysis (e.g., via execution of stereo image analysis module  404  and based on the steps described in connection with  FIG. 6 , above) on the first and second plurality of images, the second and third plurality of images, and/or the first and third plurality of images. The processed information corresponding to the analysis of the first, second, and/or third plurality of images may be combined. In some embodiments, processing unit  110  may perform a combination of monocular and stereo image analyses. For example, processing unit  110  may perform monocular image analysis (e.g., via execution of monocular image analysis module  402 ) on the first plurality of images and stereo image analysis (e.g., via execution of stereo image analysis module  404 ) on the second and third plurality of images. The configuration of image capture devices  122 ,  124 , and  126 —including their respective locations and fields of view  202 ,  204 , and  206 —may influence the types of analyses conducted on the first, second, and third plurality of images. The disclosed embodiments are not limited to a particular configuration of image capture devices  122 ,  124 , and  126 , or the types of analyses conducted on the first, second, and third plurality of images. 
     In some embodiments, processing unit  110  may perform testing on system  100  based on the images acquired and analyzed at steps  710  and  720 . Such testing may provide an indicator of the overall performance of system  100  for certain configurations of image capture devices  122 ,  124 , and  126 . For example, processing unit  110  may determine the proportion of “false hits” (e.g., cases where system  100  incorrectly determined the presence of a vehicle or pedestrian) and “misses.” 
     At step  730 , processing unit  110  may cause one or more navigational responses in vehicle  200  based on information derived from two of the first, second, and third plurality of images. Selection of two of the first, second, and third plurality of images may depend on various factors, such as, for example, the number, types, and sizes of objects detected in each of the plurality of images. Processing unit  110  may also make the selection based on image quality and resolution, the effective field of view reflected in the images, the number of captured frames, the extent to which one or more objects of interest actually appear in the frames (e.g., the percentage of frames in which an object appears, the proportion of the object that appears in each such frame, etc.), and the like. 
     In some embodiments, processing unit  110  may select information derived from two of the first, second, and third plurality of images by determining the extent to which information derived from one image source is consistent with information derived from other image sources. For example, processing unit  110  may combine the processed information derived from each of image capture devices  122 ,  124 , and  126  (whether by monocular analysis, stereo analysis, or any combination of the two) and determine visual indicators (e.g., lane markings, a detected vehicle and its location and/or path, a detected traffic light, etc.) that are consistent across the images captured from each of image capture devices  122 ,  124 , and  126 . Processing unit  110  may also exclude information that is inconsistent across the captured images (e.g., a vehicle changing lanes, a lane model indicating a vehicle that is too close to vehicle  200 , etc.). Thus, processing unit  110  may select information derived from two of the first, second, and third plurality of images based on the determinations of consistent and inconsistent information. 
     Navigational responses may include, for example, a turn, a lane shift, a change in acceleration, and the like. Processing unit  110  may cause the one or more navigational responses based on the analysis performed at step  720  and the techniques as described above in connection with  FIG. 4 . Processing unit  110  may also use data derived from execution of velocity and acceleration module  406  to cause the one or more navigational responses. In some embodiments, processing unit  110  may cause the one or more navigational responses based on a relative position, relative velocity, and/or relative acceleration between vehicle  200  and an object detected within any of the first, second, and third plurality of images. Multiple navigational responses may occur simultaneously, in sequence, or any combination thereof. 
     Reinforcement Learning and Trained Navigational Systems 
     The sections that follow discuss autonomous driving along with systems and methods for accomplishing autonomous control of a vehicle, whether that control is fully autonomous (a self-driving vehicle) or partially autonomous (e.g., one or more driver assist systems or functions). As shown in  FIG. 8 , the autonomous driving task can be partitioned into three main modules, including a sensing module  801 , a driving policy module  803 , and a control module  805 . In some embodiments, modules  801 ,  803 , and  805  may be stored in memory unit  140  and/or memory unit  150  of system  100 , or modules  801 ,  803 , and  805  (or portions thereof) may be stored remotely from system  100  (e.g., stored in a server accessible to system  100  via, for example, wireless transceiver  172 ). Furthermore, any of the modules (e.g., modules  801 ,  803 , and  805 ) disclosed herein may implement techniques associated with a trained system (such as a neural network or a deep neural network) or an untrained system.- 
      Sensing module  801 , which may be implemented using processing unit  110 , may handle various tasks relating to sensing of a navigational state in an environment of a host vehicle. Such tasks may rely upon input from various sensors and sensing systems associated with the host vehicle. These inputs may include images or image streams from one or more onboard cameras, GPS position information, accelerometer outputs, user feedback, or user inputs to one or more user interface devices, radar, lidar, etc. Sensing, which may include data from cameras and/or any other available sensors, along with map information, may be collected, analyzed, and formulated into a “sensed state,” describing information extracted from a scene in the environment of the host vehicle. The sensed state may include sensed information relating to target vehicles, lane markings, pedestrians, traffic lights, road geometry, lane shape, obstacles, distances to other objects/vehicles, relative velocities, relative accelerations, among any other potential sensed information. Supervised machine learning may be implemented in order to produce a sensing state output based on sensed data provided to sensing module  801 . The output of the sensing module may represent a sensed navigational “state” of the host vehicle, which may be passed to driving policy module  803 . 
     While a sensed state may be developed based on image data received from one or more cameras or image sensors associated with a host vehicle, a sensed state for use in navigation may be developed using any suitable sensor or combination of sensors. In some embodiments, the sensed state may be developed without reliance upon captured image data. In fact, any of the navigational principles described herein may be applicable to sensed states developed based on captured image data as well as sensed states developed using other non-image based sensors. The sensed state may also be determined via sources external to the host vehicle. For example, a sensed state may be developed in full or in part based on information received from sources remote from the host vehicle (e.g., based on sensor information, processed state information, etc. shared from other vehicles, shared from a central server, or from any other source of information relevant to a navigational state of the host vehicle.) 
     Driving policy module  803 , which is discussed in more detail below and which may be implemented using processing unit  110 , may implement a desired driving policy in order to decide on one or more navigational actions for the host vehicle to take in response to the sensed navigational state. If there are no other agents (e.g., target vehicles or pedestrians) present in the environment of the host vehicle, the sensed state input to driving policy module  803  may be handled in a relatively straightforward manner. The task becomes more complex when the sensed state requires negotiation with one or more other agents. The technology used to generate the output of driving policy module  803  may include reinforcement learning (discussed in more detail below). The output of driving policy module  803  may include at least one navigational action for the host vehicle and may include a desired acceleration (which may translate to an updated speed for the host vehicle), a desired yaw rate for the host vehicle, a desired trajectory, among other potential desired navigational actions. 
     Based on the output from the driving policy module  803 , control module  805 , which may also be implemented using processing unit  110 , may develop control instructions for one or more actuators or controlled devices associated with the host vehicle. Such actuators and devices may include an accelerator, one or more steering controls, a brake, a signal transmitter, a display, or any other actuator or device that may be controlled as part of a navigation operation associated with a host vehicle. Aspects of control theory may be used to generate the output of control module  805 . Control module  805  may be responsible for developing and outputting instructions to controllable components of the host vehicle in order to implement the desired navigational goals or requirements of driving policy module  803 . 
     Returning to driving policy module  803 , in some embodiments, a trained system trained through reinforcement learning may be used to implement driving policy module  803 . In other embodiments, driving policy module  803  may be implemented without a machine learning approach, by using specified algorithms to “manually” address the various scenarios that may arise during autonomous navigation. Such an approach, however, while viable, may result in a driving policy that is too simplistic and may lack the flexibility of a trained system based on machine learning. A trained system, for example, may be better equipped to handle complex navigational states and may better determine whether a taxi is parking or is stopping to pick up or drop off a passenger; determine whether a pedestrian intends to cross the street ahead of the host vehicle; balance unexpected behavior of other drivers with defensiveness; negotiate in dense traffic involving target vehicles and/or pedestrians; decide when to suspend certain navigational rules or augment other rules; anticipate unsensed, but anticipated conditions (e.g., whether a pedestrian will emerge from behind a car or obstacle); etc. A trained system based on reinforcement learning may also be better equipped to address a state space that is continuous and high-dimensional along with an action space that is continuous. 
     
       
         
           
             
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     There are of course many ways to solve the problem. One advantage of this approach is that it tackles the problem directly, and therefore often leads to good practical results. One potential disadvantage is that it often requires an “on-policy” training, namely, the learning of π is an iterative process, where at iteration j we have a non-perfect policy, π j , and to construct the next policy π j , we must interact with the environment while acting based on π j . 
     The system may also be trained through value based learning (learning Q or V functions). Suppose a good approximation can be learned to the optimal value function V*. An optimal policy may be constructed (e.g., by relying on the Bellman equation). Some versions of value based learning can be implemented offline (called “off-policy” training). Some disadvantages of the value-based approach may result from its strong dependence on Markovian assumptions and required approximation of a complicated function (it may be more difficult to approximate the value function than to approximate the policy directly). 
     Another technique may include model based learning and planning (learning the probability of state transitions and solving the optimization problem of finding the optimal V). Combinations of these techniques may also be used to train the learning system. In this approach, the dynamics of the process may be learned, namely, the function that takes (St, at) and yields a distribution over the next state s t+1 . Once this function is learned, the optimization problem may be solved to find the policy π whose value is optimal. This is called “planning”. One advantage of this approach may be that the learning part is supervised and can be applied offline by observing triplets (s t , a t , s t+1 ). One disadvantage of this approach, similar to the “imitation” approach, may be that small errors in the learning process can accumulate and to yield inadequately performing policies. 
       A⊂   2  
 
     Automatic Cruise Control (ACC) policy, O ACC : S→A: this policy always outputs a yaw rate of 0 and only changes the speed so as to implement smooth and accident-free driving. 
     ACC+Right policy, O R : S→A: Same as O L , but the vehicle may be centered toward the middle of the right lane. 
     These policies may be referred to as “options”. Relying on these “options”, a policy can be learned that selects options, π O : S→O, where O is the set of available options. In one case, O={O ACC , O L , O R }. The option-selector policy, π O , defines an actual policy, π: S→A, by setting, for every s, π(s)=O π     O     (s) (s). 
     In practice, the policy function may be decomposed into an options graph  901 , as shown in  FIG. 9 . Another example options graph  1000  is shown in  FIG. 10 . The options graph can represent a hierarchical set of decisions organized as a Directed Acyclic Graph (DAG). There is a special node called the root node  903  of the graph. This node has no incoming nodes. The decision process traverses through the graph, starting from the root node, until it reaches a “leaf” node, which refers to a node that has no outgoing decision lines. As shown in  FIG. 9 , leaf nodes may include nodes  905 ,  907 , and  909 , for example. Upon encountering a leaf node, driving policy module  803  may output the acceleration and steering commands associated with a desired navigational action associated with the leaf node. 
     Internal nodes, such as nodes  911 ,  913 , and  915 , for example, may result in implementation of a policy that chooses a child among its available options. The set of available children of an internal node include all of the nodes associated with a particular internal node via decision lines. For example, internal node  913  designated as “Merge” in  FIG. 9  includes three children nodes  909 ,  915 , and  917  (“Stay,” “Overtake Right,” and “Overtake Left,” respectively) each joined to node  913  by a decision line. 
     Flexibility of the decision making system may be gained by enabling nodes to adjust their position in the hierarchy of the options graph. For example, any of the nodes may be allowed to declare themselves as “critical.” Each node may implement a function “is critical,” that outputs “True” if the node is in a critical section of its policy implementation. For example, a node that is responsible for a take-over, may declare itself as critical while in the middle of a maneuver. This may impose constraints on the set of available children of a node u, which may include all nodes v which are children of node u and for which there exists a path from v to a leaf node that goes through all nodes designated as critical. Such an approach may allow, on one hand, declaration of the desired path on the graph at each time step, while on the other hand, stability of a policy may be preserved, especially while critical portions of the policy are being implemented. 
     By defining an options graph, the problem of learning the driving policy π: S→A may be decomposed into a problem of defining a policy for each node of the graph, where the policy at internal nodes should choose from among available children nodes. For some of the nodes, the respective policy may be implemented manually (e.g., through if-then type algorithms specifying a set of actions in response to an observed state) while for others the policies may be implemented using a trained system built through reinforcement learning. The choice between manual or trained/learned approaches may depend on safety aspects associated with the task and on its relative simplicity. The option graphs may be constructed in a manner such that some of the nodes are straightforward to implement, while other nodes may rely on trained models. Such an approach can ensure safe operation of the system. 
     The following discussion provides further details regarding the role of the options graph of  FIG. 9  within driving policy module  803 . As discussed above, the input to the driving policy module is a “sensed state,” which summarizes the environment map, for example, as obtained from available sensors. The output of driving policy module  803  is a set of desires (optionally, together with a set of hard constraints) that define a trajectory as a solution of an optimization problem. 
     As described above, the options graph represents a hierarchical set of decisions organized as a DAG. There is a special node called the “root” of the graph. The root node is the only node that has no incoming edges (e.g., decision lines). The decision process traverses the graph, starting from the root node, until it reaches a “leaf” node, namely, a node that has no outgoing edges. Each internal node should implement a policy that picks a child among its available children. Every leaf node should implement a policy that, based on the entire path from the root to the leaf, defines a set of Desires (e.g., a set of navigational goals for the host vehicle). 
     The set of Desires, together with a set of hard constraints that are defined directly based on the sensed state, establish an optimization problem whose solution is the trajectory for the vehicle. The hard constraints may be employed to further increase the safety of the system, and the Desires can be used to provide driving comfort and human-like driving behavior of the system. The trajectory provided as a solution to the optimization problem, in turn, defines the commands that should be provided to the steering, braking, and/or engine actuators in order to accomplish the trajectory. 
     Returning to  FIG. 9 , options graph  901  represents an options graph for a two-lane highway, including with merging lanes (meaning that at some points, a third lane is merged into either the right or the left lane of the highway). The root node  903  first decides if the host vehicle is in a plain road scenario or approaching a merge scenario. This is an example of a decision that can be implemented based on the sensing state. Plain road node  911  includes three child nodes: stay node  909 , overtake left node  917 , and overtake right node  915 . Stay refers to a situation in which the host vehicle would like to keep driving in the same lane. The stay node is a leaf node (no outgoing edges/lines). Therefore, it the stay node defines a set of Desires. The first Desire it defines may include the desired lateral position—e.g., as close as possible to the center of the current lane of travel. There may also be a desire to navigate smoothly (e.g., within predefined or allowable acceleration maximums). The stay node may also define how the host vehicle is to react to other vehicles. For example, the stay node may review sensed target vehicles and assign each a semantic meaning, which can be translated into components of the trajectory. 
     Various semantic meanings may be assigned to target vehicles in an environment of the host vehicle. For example, in some embodiments the semantic meaning may include any of the following designations: 1) not relevant: indicating that the sensed vehicle in the scene is currently not relevant; 2) next lane: indicating that the sensed vehicle is in an adjacent lane and an appropriate offset should be maintained relative to this vehicle (the exact offset may be calculated in the optimization problem that constructs the trajectory given the Desires and hard constraints, and can potentially be vehicle dependent—the stay leaf of the options graph sets the target vehicle&#39;s semantic type, which defines the Desire relative to the target vehicle); 3) give way: the host vehicle will attempt to give way to the sensed target vehicle by, for example, reducing speed (especially where the host vehicle determines that the target vehicle is likely to cut into the lane of the host vehicle); 4) take way: the host vehicle will attempt to take the right of way by, for example, increasing speed; 5) follow: the host vehicle desires to maintain smooth driving following after this target vehicle; 6) takeover left/right: this means the host vehicle would like to initiate a lane change to the left or right lane. Overtake left node  917  and overtake right node  915  are internal nodes that do not yet define Desires. 
     The next node in options graph  901  is the select gap node  919 . This node may be responsible for selecting a gap between two target vehicles in a particular target lane that host vehicle desires to enter. By choosing a node of the form IDj, for some value of j, the host vehicle arrives at a leaf that designates a Desire for the trajectory optimization problem—e.g., the host vehicle wishes to make a maneuver so as to arrive at the selected gap. Such a maneuver may involve first accelerating/braking in the current lane and then heading to the target lane at an appropriate time to enter the selected gap. If the select gap node  919  cannot find an appropriate gap, it moves to the abort node  921 , which defines a desire to move back to the center of the current lane and cancel the takeover. 
     Returning to merge node  913 , when the host vehicle approaches a merge, it has several options that may depend on a particular situation. For example, as shown in  FIG. 11A , host vehicle  1105  is traveling along a two-lane road with no other target vehicles detected, either in the primary lanes of the two-lane road or in the merge lane  1111 . In this situation, driving policy module  803 , upon reaching merge node  913 , may select stay node  909 . That is, staying within its current lane may be desired where no target vehicles are sensed as merging onto the roadway. 
     In  FIG. 11B , the situation is slightly different. Here, host vehicle  1105  senses one or more target vehicles  1107  entering the main roadway  1112  from merge lane  1111 . In this situation, once driving policy module  803  encounters merge node  913 , it may choose to initiate an overtake left maneuver in order to avoid the merging situation. 
     In  FIG. 11C , host vehicle  1105  encounters one or more target vehicles  1107  entering main roadway  1112  from merge lane  1111 . Host vehicle  1105  also detects target vehicles  1109  traveling in a lane adjacent to the lane of the host vehicle. The host vehicle also detects one or more target vehicles  1110  traveling in the same lane as host vehicle  1105 . In this situation, driving policy module  803  may decide to adjust the speed of host vehicle  1105  to give way to target vehicle  1107  and to proceed ahead of target vehicle  1115 . This can be accomplished, for example, by progressing to select gap node  919 , which, in turn, will select a gap between ID 0  (vehicle  1107 ) and ID 1  (vehicle  1115 ) as the appropriate merging gap. In such a case, the appropriate gap of the merging situation defines the objective for a trajectory planner optimization problem. 
     As discussed above, nodes of the options graph may declare themselves as “critical,” which may ensure that the selected option passes through the critical nodes. Formally, each node may implement a function IsCritical. After performing a forward pass on the options graph, from the root to a leaf, and solving the optimization problem of the trajectory planner, a backward pass may be performed from the leaf back to the root. Along this backward pass, the IsCritical function of all nodes in the pass may be called, and a list of all critical nodes may be saved. In the forward path corresponding to the next time frame, driving policy module  803  may be required to choose a path from the root node to a leaf that goes through all critical nodes. 
       FIGS. 11A-11C  may be used to show a potential benefit of this approach. For example, in a situation where an overtake action is initiated, and driving policy module  803  arrives at the leaf corresponding to IDk, it would be undesirable to choose, for example, the stay node  909  when the host vehicle is in the middle of the takeover maneuver. To avoid such jumpiness, the IDj node can designate itself as critical. During the maneuver, the success of the trajectory planner can be monitored, and function IsCritical will return a “True” value if the overtake maneuver progresses as intended. This approach may ensure that in the next time frame, the takeover maneuver will be continued (rather than jumping to another, potentially inconsistent maneuver prior to completion of the initially selected maneuver). If, on the other hand, monitoring of the maneuver indicates that the selected maneuver is not progressing as intended, or if the maneuver has become unnecessary or impossible, the function IsCritical can return a “False” value. This can allow the select gap node to select a different gap in the next time frame, or to abort the overtake maneuver altogether. This approach may allow, on one hand, declaration of the desired path on the options graph at each time step, while on the other hand, may help to promote stability of the policy while in critical parts of the execution. 
     Hard constraints, which will be discussed in more detail below, may be differentiated from navigational desires. For example, hard constraints may ensure safe driving by applying an added layer of filtering of a planned navigational action. The implicated hard constraints, which may be programmed and defined manually, rather than through use of a trained system built upon reinforcement learning, can be determined from the sensed state. In some embodiments, however, the trained system may learn the applicable hard constraints to be applied and followed. Such an approach may promote driving policy module  803  arriving at a selected action that is already in compliance with the applicable hard constraints, which may reduce or eliminate selected actions that may require later modification to comply with applicable hard constraints. Nevertheless, as a redundant safety measure, hard constraints may be applied to the output of driving policy module  803  even where driving policy module  803  has been trained to account for predetermined hard constraints. 
     There are many examples of potential hard constraints. For example, a hard constraint may be defined in conjunction with a guardrail on an edge of a road. In no situation may the host vehicle be allowed to pass the guardrail. Such a rule induces a hard lateral constraint on the trajectory of the host vehicle. Another example of a hard constraint may include a road bump (e.g., a speed control bump), which may induce a hard constraint on the speed of driving before the bump and while traversing the bump. Hard constraints may be considered safety critical and, therefore, may be defined manually rather than relying solely on a trained system learning the constraints during training. 
     In contrast to hard constraints, the goal of desires may be to enable or achieve comfortable driving. As discussed above, an example of a desire may include a goal of positioning the host vehicle at a lateral position within a lane that corresponds to the center of the host vehicle lane. Another desire may include the ID of a gap to fit into. Note that there is not a requirement for the host vehicle to be exactly in the center of the lane, but instead a desire to be as close as possible to it may ensure that the host vehicle tends to migrate to the center of the lane even in the event of deviations from the center of the lane. Desires may not be safety critical. In some embodiments, desires may require negotiation with other drivers and pedestrians. One approach for constructing the desires may rely on the options graph, and the policy implemented in at least some nodes of the graph may be based on reinforcement learning. 
     For the nodes of options graph  901  or  1000  implemented as nodes trained based on learning, the training process may include decomposing the problem into a supervised learning phase and a reinforcement learning phase. In the supervised learning phase, a differentiable mapping from (s t , a t ) to ŝ t+1  can be learned such that ŝ t+1 ≈s t+1 . This may be similar to “model-based” reinforcement learning. However, in the forward loop of the network, ŝ t+1  may be replaced by the actual value of s t+1 , therefore eliminating the problem of error accumulation. The role of prediction of ŝ t+1  is to propagate messages from the future back to past actions. In this sense, the algorithm may be a combination of “model-based” reinforcement learning with “policy-based learning.” 
     An important element that may be provided in some scenarios is a differentiable path from future losses/rewards back to decisions on actions. With the option graph structure, the implementation of options that involve safety constraints are usually not differentiable. To overcome this issue, the choice of a child in a learned policy node may be stochastic. That is, a node may output a probability vector, p, that assigns probabilities used in choosing each of the children of the particular node. Suppose that a node has k children and let a (1) , . . . , a (k)  be the actions of the path from each child to a leaf. The resulting predicted action is therefore â=Σ i=1   k pia (i) , which may result in a differentiable path from the action top. In practice, an action a may be chosen to be a (i)  for i˜p, and the difference between a and a may be referred to as additive noise. 
     For the training of ŝ t+1  given s t , a t , supervised learning may be used together with real data. For training the policy of nodes simulators can be used. Later, fine tuning of a policy can be accomplished using real data. Two concepts may make the simulation more realistic. First, using imitation, an initial policy can be constructed using the “behavior cloning” paradigm, using large real world data sets. In some cases, the resulting agents may be suitable. In other cases, the resulting agents at least form very good initial policies for the other agents on the roads. Second, using self-play, our own policy may be used to augment the training. For example, given an initial implementation of the other agents (cars/pedestrians) that may be experienced, a policy may be trained based on a simulator. Some of the other agents may be replaced with the new policy, and the process may be repeated. As a result, the policy can continue to improve as it should respond to a larger variety of other agents that have differing levels of sophistication. 
     Further, in some embodiments, the system may implement a multi-agent approach. For example, the system may take into account data from various sources and/or images capturing from multiple angles. Further, some disclosed embodiments may provide economy of energy, as anticipation of an event which does not directly involve the host vehicle, but which may have an effect on the host vehicle can be considered, or even anticipation of an event that may lead to unpredictable circumstances involving other vehicles may be a consideration (e.g., radar may “see through” the leading vehicle and anticipation of an unavoidable, or even a high likelihood of an event that will affect the host vehicle). 
     Trained System with Imposed Navigational Constraints 
     In the context of autonomous driving, a significant concern is how to ensure that a learned policy of a trained navigational network will be safe. In some embodiments, the driving policy system may be trained using constraints, such that the actions selected by the trained system may already account for applicable safety constraints. Additionally, in some embodiments, an extra layer of safety may be provided by passing the selected actions of the trained system through one or more hard constraints implicated by a particular sensed scene in the environment of the host vehicle. Such an approach may ensure that that the actions taken by the host vehicle have been restricted to those confirmed as satisfying applicable safety constraints. 
     At its core, the navigational system may include a learning algorithm based on a policy function that maps an observed state to one or more desired actions. In some implementations, the learning algorithm is a deep learning algorithm. The desired actions may include at least one action expected to maximize an anticipated reward for a vehicle. While in some cases, the actual action taken by the vehicle may correspond to one of the desired actions, in other cases, the actual action taken may be determined based on the observed state, one or more desired actions, and non-learned, hard constraints (e.g., safety constraints) imposed on the learning navigational engine. These constraints may include no-drive zones surrounding various types of detected objects (e.g., target vehicles, pedestrians, stationary objects on the side of a road or in a roadway, moving objects on the side of a road or in a roadway, guard rails, etc.) In some cases, the size of the zone may vary based on a detected motion (e.g., speed and/or direction) of a detected object. Other constraints may include a maximum speed of travel when passing within an influence zone of a pedestrian, a maximum deceleration (to account for a target vehicle spacing behind the host vehicle), a mandatory stop at a sensed crosswalk or railroad crossing, etc. 
     Hard constraints used in conjunction with a system trained through machine learning may offer a degree of safety in autonomous driving that may surpass a degree of safety available based on the output of the trained system alone. For example, the machine learning system may be trained using a desired set of constraints as training guidelines and, therefore, the trained system may select an action in response to a sensed navigational state that accounts for and adheres to the limitations of applicable navigational constraints. Still, however, the trained system has some flexibility in selecting navigational actions and, therefore, there may exist at least some situations in which an action selected by the trained system may not strictly adhere to relevant navigational constraints. Therefore, in order to require that a selected action strictly adheres to relevant navigational constraints, the output of the trained system may be combined with, compared to, filtered with, adjusted, modified, etc. using a non-machine learning component outside the learning/trained framework that guarantees strict application of relevant navigational constraints. 
             s ˜P     ∈   R( s ) 
       R( s )=−r
 
       R( s )∈[−1, 1]R( s )  s −r r
 
     Observe that the effect of an accident on  [R( s )] is the additive term −pr where p is the probability mass of trajectories with an accident event. If this term is negligible, i.e., p«1/r, then the learning system may prefer a policy that performs an accident (or adopt in general a reckless driving policy) in order to fulfill the takeover maneuver successfully more often than a policy that would be more defensive at the expense of having some takeover maneuvers not complete successfully. In other words, if the probability of accidents is to be at most p, then r must be set such that r»1/p. It may be desireable to make p extremely small 
         [R( s )]r 
       R( s )pr 2 rr»1/p
 
     Lemma: Let π o  be a policy and let p and r be scalars such that with probability p, R( s )=−r is obtained, and with probability 1−p we have R( s )∈[−1, 1] is obtained. Then, 
       Var[ R (   s   )]≥ pr   2 −( pr +(1 −p )) 2 =( p−p   2 ) r   2 −2 p (1− p ) r −(1 −p ) 2   ≈p   2  
 
     where the last approximation holds for the case r≥1/p. 
     This discussion shows that an objection of the form  R( s )] may not ensure functional safety without causing a variance problem. The baseline subtraction method for variance reduction may not offer a sufficient remedy to the problem because the problem would shift from a high variance of R( s ) to an equally high variance of the baseline constants whose estimation would equally suffer numeric instabilities. Moreover, if the probability of an accident is p, then on average at least 1/p sequences should be sampled before obtaining an accident event. This implies a lower bound of 1/p samples of sequences for a learning algorithm that aims at minimizing  [R( s )]. The solution to this problem may be found in the architectural design described herein, rather than through numerical conditioning techniques. The approach here is based on the notion that hard constraints should be injected outside of the learning framework. In other words, the policy function may be decomposed into a learnable part and a nonlearnable part. Formally, the policy function may be structured as π 0 =π (T)   O π θ   (D) , where π 0   (D)  maps the (agnostic) state space into a set of Desires (e.g., desired navigational goals, etc.), while π (T)  maps the Desires into a trajectory (which may determine how the car should move in a short range). The function π θ   (D)  is responsible for the comfort of driving and for making strategic decisions such as which other cars should be over-taken or given way and what is the desired position of the host vehicle within its lane, etc. The mapping from sensed navigational state to Desires is a policy π θ   (D)  that may be learned from experience by maximizing an expected reward. The desires produced by π 0   (D)  may be translated into a cost function over driving trajectories. The function π (T) , not a learned function, may be implemented by finding a trajectory that minimizes the cost subject to hard constraints on 
     
       
      
       
       
      
     
         =[0, ν max   ]×L×{g, t, o}   n  
 
       (ν, l, c 1 , . . . , c n )∈ 
 
       (x 1 , y 1 ), . . . , (x k , y k ) (x i , y i ) 
       τ·i τ=0.1 sec
 
       ν∈[0, ν max ]
 
       Σ i=2   k (ν−[[( x   i   , y   i )−( x   i−1   , y   i−1 )]]/τ) 2 .
 
     Given desired lateral position, l∈L, the cost associated with desired lateral position is 
       Σ i=1   k  dist( x   i   , y   i   , l )
 
       (x′ 1 , y′ 1 ), . . . , (x′ k , y′ k )
 
       τi&gt;τj+0.5
 
       [τ(j−i)+0.5]+
 
       τj&gt;τi+0.5 [τ(i−j)+0.5]+
 
       π (T)  
 
     To summarize, the policy, can be decomposed into a mapping from the agnostic state to a set of Desires and a mapping from the Desires to an actual trajectory. The latter mapping is not based on learning and may be implemented by solving an optimization problem whose cost depends on the Desires and whose hard constraints may guarantee functional safety of the policy. 
     The following discussion describes mapping from the agnostic state to the set of Desires. As described above, to be compliant with functional safety, a system reliant upon reinforcement learning alone may suffer a high and unwieldy variance on the reward R( s ). This result may be avoided by decomposing the problem into a mapping from (agnostic) state space to a set of Desires using policy gradient iterations followed by a mapping to an actual trajectory which does not involve a system trained based on machine learning. 
     For various reasons, the decision making may be further decomposed into semantically meaningful components. For example, the size of   might be large and even continuous. In the double-merge scenario described above with respect to  FIG. 11D ,  =[0, ν max ]×[L×{g, t, o} n ). Additionally, the gradient estimator may involve the term Σ t=1   T ∇ θ π θ (a t |s t ). In such an expression, the variance may grow with the time horizon T. In some cases, the value of T may be roughly  250  which may be high enough to create significant variance. Supposing a sampling rate is in the range of 10 Hz and the merge area  1130  is 100 meters, preparation for the merge may begin approximately 300 meters before the merge area. If the host vehicle travels at 16 meters per second (about 60 km per hour), then the value of T for an episode may be roughly 250. 
     Returning to the concept of an options graph, an options graph that may be representative of the double merge scenario depicted in  FIG. 11D  is shown in  FIG. 11E . As previously discussed, an options graph may represent a hierarchical set of decisions organized as a Directed Acyclic Graph (DAG). There may be a special node in the graph called the “root” node  1140 , which may be the only node that has no incoming edges (e.g., decision lines). The decision process may traverse the graph, starting from the root node, until it reaches a “leaf” node, namely, a node that has no outgoing edges. Each internal node may implement a policy function that chooses a child from among its available children. There may be a predefined mapping from the set of traversals over the options graph to the set of desires,  . In other words, a traversal on the options graph may be automatically translated into a desire in  . Given a node, ν, in the graph, a parameter vector θ ν  may specify the policy of choosing a child of ν. If θ is the concatenation of all the θ ν  then π θ   (D)  may be defined by traversing from the root of the graph to a leaf, while at each node v using the policy defined by θ ν , to choose a child node. 
     In the double merge options graph  1139  of  FIG. 11E , root node  1140  may first decide if the host vehicle is within the merging area (e.g., area  1130  of  FIG. 11D ) or if the host vehicle instead is approaching the merging area and needs to prepare for a possible merge. In both cases, the host vehicle may need to decide whether to change lanes (e.g., to the left or to the right side) or whether to stay in the current lane. If the host vehicle has decided to change lanes, the host vehicle may need to decide whether conditions are suitable to go on and perform the lane change maneuver (e.g., at “go” node  1142 ). If it is not possible to change lanes, the host vehicle may attempt to “push” toward the desired lane (e.g., at node  1144  as part of a negotiation with vehicles in the desired lane) by aiming at being on the lane mark. Alternatively, the host vehicle may opt to “stay” in the same lane (e.g., at node  1146 ). Such a process may determine the lateral position for the host vehicle in a natural way. For example, 
     This may enable determination of the desired lateral position in a natural way. For example, if the host vehicle changes lanes from lane  2  to lane  3 , the “go” node may set the desired lateral position to  3 , the “stay” node may set the desired lateral position to 2, and the “push” node may set the desired lateral position to 2.5. Next, the host vehicle may decide whether to maintain the “same” speed (node  1148 ), “accelerate” (node  1150 ), or “decelerate” (node  1152 ). Next, the host vehicle may enter a “chain like” structure  1154  that goes over the other vehicles and sets their semantic meaning to a value in the set {g, t, o}. This process may set the desires relative to the other vehicles. The parameters of all nodes in this chain may be shared (similar to Recurrent Neural Networks). 
     A potential benefit of the options is the interpretability of the results. Another potential benefit is that the decomposable structure of the set   can be relied upon and, therefore, the policy at each node may be chosen from among a small number of possibilities. Additionally, the structure may allow for a reduction in the variance of the policy gradient estimator. 
     As discussed above, the length of an episode in the double merge scenario may be roughly T=250 steps. Such a value (or any other suitable value depending on a particular navigational scenario) may provide enough time to see the consequences of the host vehicle actions (e.g., if the host vehicle decided to change lanes as a preparation for the merge, the host vehicle will see the benefit only after a successful completion of the merge). On the other hand, due to the dynamic of driving, the host vehicle must make decisions at a fast enough frequency (e.g., 10 Hz in the case described above). 
     The options graph may enable a decrease in the effective value of T in at least two ways. First, given higher level decisions, a reward can be defined for lower level decisions while taking into account shorter episodes. For example, when the host vehicle has already chosen a “lane change” and the “go” node, a policy can be learned for assigning semantic meaning to vehicles by looking at episodes of 2-3 seconds (meaning that T becomes 20-30 instead of 250). Second, for high level decisions (such as whether to change lanes or to stay in the same lane), the host vehicle may not need to make decisions every 0.1 seconds. Instead, the host vehicle may be able to either make decisions at a lower frequency (e.g., every second), or implement an “option termination” function, and then the gradient may be calculated only after every termination of the option. In both cases, the effective value of T may be an order of magnitude smaller than its original value. All in all, the estimator at every node may depend on a value of T which is an order of magnitude smaller than the original 250 steps, which may immediately transfer to a smaller variance. 
     As discussed above, hard constraints may promote safer driving, and there may be several different types of constraints. For example, static hard constraints may be defined directly from the sensing state. These may include speed bumps, speed limits, road curvature, junctions, etc., within the environment of the host vehicle that may implicate one or more constraints on vehicle speed, heading, acceleration, breaking (deceleration), etc. Static hard constraints may also include semantic free space where the host vehicle is prohibited from going outside of the free space and from navigating too close to physical barriers, for example. Static hard constraints may also limit (e.g., prohibit) maneuvers that do not comply with various aspects of a kinematic motion of the vehicle, for example, a static hard constraint can be used to prohibit maneuvers that might lead to the host vehicle overturning, sliding, or otherwise losing control. 
     Hard constraints may also be associated with vehicles. For example, a constraint may be employed requiring that a vehicle maintain a longitudinal distance to other vehicles of at least one meter and a lateral distance from other vehicles of at least 0.5 meters. Constraints may also be applied such that the host vehicle will avoid maintaining a collision course with one or more other vehicles. For example, a time τ may be a measure of time based on a particular scene. The predicted trajectories of the host vehicle and one or more other vehicles may be considered from a current time to time τ. Where the two trajectories intersect, (t i   o , t i   l ) may represent the time of arrival and the leaving time of vehicle i to the intersection point. That is, each car will arrive at point when a first part of the car passes the intersection point, and a certain amount of time will be required before the last part of the car passes through the intersection point. This amount of time separates the arrival time from the leaving time. Assuming that t 1   a &lt;t 2   a  (i.e., that the arrival time of vehicle  1  is less than the arrival time of vehicle  2 ), then we will want to ensure that vehicle  1  has left the intersection point prior to vehicle  2  arriving. Otherwise, a collision would result. Thus, a hard constraint may be implemented such that t 1   l &gt;t 2   a . Moreover, to ensure that vehicle  1  and vehicle  2  do not miss one another by a minimal amount, an added margin of safety may be obtained by including a buffer time into the constraint (e.g., 0.5 seconds or another suitable value). A hard constraint relating to predicted intersection trajectories of two vehicles may be expressed as t 1   l &gt;t 2   a +0.5. 
     The amount of time τ over which the trajectories of the host vehicle and one or more other vehicles are tracked may vary. In junction scenarios, however, where speeds may be lower, τ may be longer, and τ may be defined such that a host vehicle will enter and leave the junction in less than τ seconds. 
     Applying hard constraints to vehicle trajectories, of course, requires that the trajectories of those vehicles be predicted. For the host vehicle, trajectory prediction may be relatively straightforward, as the host vehicle generally already understands and, indeed, is planning an intended trajectory at any given time. Relative to other vehicles, predicting their trajectories can be less straightforward. For other vehicles, the baseline calculation for determining predicted trajectories may rely on the current speed and heading of the other vehicles, as determined, for example, based on analysis of an image stream captured by one or more cameras and/or other sensors (radar, lidar, acoustic, etc.) aboard the host vehicle. 
     There can be some exceptions, however, that can simplify the problem or at least provide added confidence in a trajectory predicted for another vehicle. For example, with respect to structured roads in which there is an indication of lanes and where give-way rules may exist, the trajectories of other vehicles can be based, at least in part, upon the position of the other vehicles relative to the lanes and based upon applicable give-way rules. Thus, in some situations, when there are observed lane structures, it may be assumed that next-lane vehicles will respect lane boundaries. That is, the host vehicle may assume that a next-lane vehicle will stay in its lane unless there is observed evidence (e.g., a signal light, strong lateral movement, movement across a lane boundary) indicating that the next-lane vehicle will cut into the lane of the host vehicle. 
     Other situations may also provide clues regarding the expected trajectories of other vehicles. For example, at stop signs, traffic lights, roundabouts, etc., where the host vehicle may have the right of way, it may be assumed that other vehicles will respect that right of way. Thus, unless there is observed evidence of a rule break, other vehicles may be assumed to proceed along a trajectory that respects the rights of way possessed by the host vehicle. 
     Hard constraints may also be applied with respect to pedestrians in an environment of the host vehicle. For example, a buffer distance may be established with respect to pedestrians such that the host vehicle is prohibited from navigating any closer than the prescribed buffer distance relative to any observed pedestrian. The pedestrian buffer distance may be any suitable distance. In some embodiments, the buffer distance may be at least one meter relative to an observed pedestrian. 
     Similar to the situation with vehicles, hard constraints may also be applied with respect to relative motion between pedestrians and the host vehicle. For example, the trajectory of a pedestrian (based on a heading direction and speed) may be monitored relative to the projected trajectory of the host vehicle. Given a particular pedestrian trajectory, with every point p on the trajectory, t(p) may represent the time required for the pedestrian to reach point p. To maintain the required buffer distance of at least  1  meter from the pedestrian, either t(p) must be larger than the time the host vehicle will reach point p (with sufficient difference in time such that the host vehicle passes in front of the pedestrian by a distance of at least one meter) or that t(p) must be less than the time the host vehicle will reach point p (e.g., if the host vehicle brakes to give way to the pedestrian). Still, in the latter example, the hard constraint may require that the host vehicle arrive at point p at a sufficient time later than the pedestrian such that the host vehicle can pass behind the pedestrian and maintain the required buffer distance of at least one meter. Of course, there may be exceptions to the pedestrian hard constraint. For example, where the host vehicle has the right of way or where speeds are very slow, and there is no observed evidence that the pedestrian will decline to give way to the host vehicle or will otherwise navigate toward the host vehicle, the pedestrian hard constraint may be relaxed (e.g., to a smaller buffer of at least 0.75 meters or 0.50 meters). 
     In some examples, constraints may be relaxed where it is determined that not all can be met. For example, in situations where a road is too narrow to leave desired spacing (e.g., 0.5 meters) from both curbs or from a curb and a parked vehicle, one or more the constraints may be relaxed if there are mitigating circumstances. For example, if there are no pedestrians (or other objects) on the sidewalk one can proceed slowly at 0.1 meters from a curb. In some embodiments, constraints may be relaxed if doing so will improve the user experience. For example, in order to avoid a pothole, constraints may be relaxed to allow a vehicle to navigate closers to the edges of the lane, a curb, or a pedestrian more than might ordinarily be permitted. Furthermore, when determining which constrains to relax, in some embodiments, the one or more constraints chosen to relax are those deemed to have the least available negative impact to safety. For example, a constraint relating to how close the vehicle may travel to the curb or to a concrete barrier may be relaxed before relaxing one dealing with proximity to other vehicles. In some embodiments, pedestrian constraints may be the last to be relaxed, or may never be relaxed in some situations. 
       FIG. 12  shows an example of a scene that may be captured and analyzed during navigation of a host vehicle. For example, a host vehicle may include a navigation system (e.g., system  100 ), as described above, that may receive from a camera (e.g., at least one of image capture device  122 , image capture device  124 , and image capture device  126 ) associated with the host vehicle a plurality of images representative of an environment of the host vehicle. The scene shown in  FIG. 12  is an example of one of the images that may be captured at time t from an environment of a host vehicle traveling in lane  1210  along a predicted trajectory  1212 . The navigation system may include at least one processing device (e.g., including any of the EyeQ processors or other devices described above) that are specifically programmed to receive the plurality of images and analyze the images to determine an action in response to the scene. Specifically, the at least one processing device may implement sensing module  801 , driving policy module  803 , and control module  805 , as shown in  FIG. 8 . Sensing module  801  may be responsible for collecting and outputting the image information collected from the cameras and providing that information, in the form of an identified navigational state, to driving policy module  803 , which may constitute a trained navigational system that has been trained through machine learning techniques, such as supervised learning, reinforcement learning, etc. Based on the navigational state information provided to driving policy module  803  by sensing module  801 , driving policy module  803  (e.g., by implementing the options graph approach described above) may generate a desired navigational action for execution by the host vehicle in response to the identified navigational state. 
     In some embodiments, the at least one processing device may translate the desired navigation action directly into navigational commands using, for example, control module  805 . In other embodiments, however, hard constraints may be applied such that the desired navigational action provided by the driving policy module  803  is tested against various predetermined navigational constraints that may be implicated by the scene and the desired navigational action. For example, where driving policy module  803  outputs a desired navigational action that would cause the host vehicle to follow trajectory  1212 , this navigational action may be tested relative to one or more hard constraints associated with various aspects of the environment of the host vehicle. For example, a captured image  1201  may reveal a curb  1213 , a pedestrian  1215 , a target vehicle  1217 , and a stationary object (e.g., an overturned box) present in the scene. Each of these may be associated with one or more hard constraints. For example, curb  1213  may be associated with a static constraint that prohibits the host vehicle from navigating into the curb or past the curb and onto a sidewalk  1214 . Curb  1213  may also be associated with a road barrier envelope that defines a distance (e.g., a buffer zone) extending away from (e.g., by  0 . 1  meters,  0 . 25  meters,  0 . 5  meters,  1  meter, etc.) and along the curb, which defines a no-navigate zone for the host vehicle. Of course, static constraints may be associated with other types of roadside boundaries as well (e.g., guard rails, concrete pillars, traffic cones, pylons, or any other type of roadside barrier). 
     It should be noted that distances and ranging may be determined by any suitable method. For example, in some embodiments, distance information may be provided by onboard radar and/or lidar systems. Alternatively or additionally, distance information may be derived from analysis of one or more images captured from the environment of the host vehicle. For example, numbers of pixels of a recognized object represented in an image may be determined and compared to known field of view and focal length geometries of the image capture devices to determine scale and distances. Velocities and accelerations may be determined, for example, by observing changes in scale between objects from image to image over known time intervals. This analysis may indicate the direction of movement toward or away from the host vehicle along with how fast the object is pulling away from or coming toward the host vehicle. Crossing velocity may be determined through analysis of the change in an object&#39;s X coordinate position from one image to another over known time periods. 
     Pedestrian  1215  may be associated with a pedestrian envelope that defines a buffer zone  1216 . In some cases, an imposed hard constraint may prohibit the host vehicle from navigating within a distance of 1 meter from pedestrian  1215  (in any direction relative to the pedestrian). Pedestrian  1215  may also define the location of a pedestrian influence zone  1220 . Such an influence zone may be associated with a constraint that limits the speed of the host vehicle within the influence zone. The influence zone may extend 5 meters, 10 meters, 20 meters, etc., from pedestrian  1215 . Each graduation of the influence zone may be associated with a different speed limit. For example, within a zone of 1 meter to five meters from pedestrian  1215 , host vehicle may be limited to a first speed (e.g., 10 mph, 20 mph, etc.) that may be less than a speed limit in a pedestrian influence zone extending from 5 meters to 10 meters. Any graduation for the various stages of the influence zone may be used. In some embodiments, the first stage may be narrower than from 1 meter to five meters and may extend only from one meter to two meters. In other embodiments, the first stage of the influence zone may extend from 1 meter (the boundary of the no-navigate zone around a pedestrian) to a distance of at least 10 meters. A second stage, in turn, may extend from 10 meters to at least about 20 meters. The second stage may be associated with a maximum rate of travel for the host vehicle that is greater than the maximum rate of travel associated with the first stage of the pedestrian influence zone. 
     One or more stationary object constraints may also be implicated by the detected scene in the environment of the host vehicle. For example, in image  1201 , the at least one processing device may detect a stationary object, such as box  1219  present in the roadway. Detected stationary objects may include various objects, such as at least one of a tree, a pole, a road sign, or an object in a roadway. One or more predefined navigational constraints may be associated with the detected stationary object. For example, such constraints may include a stationary object envelope, wherein the stationary object envelope defines a buffer zone about the object within which navigation of the host vehicle may be prohibited. At least a portion of the buffer zone may extend a predetermined distance from an edge of the detected stationary object. For example, in the scene represented by image  1201 , a buffer zone of at least 0.1 meters, 0.25 meters, 0.5 meters or more may be associated with box  1219  such that the host vehicle will pass to the right or to the left of the box by at least some distance (e.g., the buffer zone distance) in order to avoid a collision with the detected stationary object. 
     The predefined hard constraints may also include one or more target vehicle constraints. For example, a target vehicle  1217  may be detected in image  1201 . To ensure that the host vehicle does not collide with target vehicle  1217 , one or more hard constraints may be employed. In some cases, a target vehicle envelope may be associated with a single buffer zone distance. For example, the buffer zone may be defined by a 1 meter distance surrounding the target vehicle in all directions. The buffer zone may define a region extending from the target vehicle by at least one meter into which the host vehicle is prohibited from navigating. 
     The envelope surrounding target vehicle  1217  need not be defined by a fixed buffer distance, however. In some cases the predefined hard constraints associate with target vehicles (or any other movable objects detected in the environment of the host vehicle) may depend on the orientation of the host vehicle relative to the detected target vehicle. For example, in some cases, a longitudinal buffer zone distance (e.g., one extending from the target vehicle toward the front or rear of the host vehicle—such as in the case that the host vehicle is driving toward the target vehicle) may be at least one meter. A lateral buffer zone distance (e.g., one extending from the target vehicle toward either side of the host vehicle—such as when the host vehicle is traveling in a same or opposite direction as the target vehicle such that a side of the host vehicle will pass adjacent to a side of the target vehicle) may be at least 0.5 meters. 
     As described above, other constraints may also be implicated by detection of a target vehicle or a pedestrian in the environment of the host vehicle. For example, the predicted trajectories of the host vehicle and target vehicle  1217  may be considered and where the two trajectories intersect (e.g., at intersection point  1230 ), a hard constraint may require t 1   l &gt;t 2   a  or t 1   l &gt;t 2   a +0.5 where the host vehicle is vehicle  1 , and target vehicle  1217  is vehicle  2 . Similarly, the trajectory of pedestrian  1215  (based on a heading direction and speed) may be monitored relative to the projected trajectory of the host vehicle. Given a particular pedestrian trajectory, with every point p on the trajectory, t(p) will represent the time required for the pedestrian to reach point p (i.e., point  1231  in  FIG. 12 ). To maintain the required buffer distance of at least 1 meter from the pedestrian, either t(p) must be larger than the time the host vehicle will reach point p (with sufficient difference in time such that the host vehicle passes in front of the pedestrian by a distance of at least one meter) or that t(p) must be less than the time the host vehicle will reach point p (e.g., if the host vehicle brakes to give way to the pedestrian). Still, in the latter example, the hard constraint will require that the host vehicle arrive at point p at a sufficient time later than the pedestrian such that the host vehicle can pass behind the pedestrian and maintain the required buffer distance of at least one meter. 
     Other hard constraints may also be employed. For example, a maximum deceleration rate of the host vehicle may be employed in at least some cases. Such a maximum deceleration rate may be determined based on a detected distance to a target vehicle following the host vehicle (e.g., using images collected from a rearward facing camera). The hard constraints may include a mandatory stop at a sensed crosswalk or a railroad crossing or other applicable constraints. 
     Where analysis of a scene in an environment of the host vehicle indicates that one or more predefined navigational constraints may be implicated, those constraints may be imposed relative to one or more planned navigational actions for the host vehicle. For example, where analysis of a scene results in driving policy module  803  returning a desired navigational action, that desired navigational action may be tested against one or more implicated constraints. 
     If the desired navigational action is determined to violate any aspect of the implicated constraints (e.g., if the desired navigational action would carry the host vehicle within a distance of 0.7 meters of pedestrian  1215  where a predefined hard constraint requires that the host vehicle remain at least 1.0 meters from pedestrian  1215 ), then at least one modification to the desired navigational action may be made based on the one or more predefined navigational constraints. 
     Adjusting the desired navigational action in this way may provide an actual navigational action for the host vehicle in compliance with the constraints implicated by a particular scene detected in the environment of the host vehicle. 
     After determination of the actual navigational action for the host vehicle, that navigational action may be implemented by causing at least one adjustment of a navigational actuator of the host vehicle in response to the determined actual navigational action for the host vehicle. Such navigational actuator may include at least one of a steering mechanism, a brake, or an accelerator of the host vehicle. 
     Prioritized Constraints 
     As described above, various hard constraints may be employed with a navigational system to ensure safe operation of a host vehicle. The constraints may include a minimum safe driving distance with respect to a pedestrian, a target vehicle, a road barrier, or a detected object, a maximum speed of travel when passing within an influence zone of a detected pedestrian, or a maximum deceleration rate for the host vehicle, among others. These constraints may be imposed with a trained system trained based on machine learning (supervised, reinforcement, or a combination), but they also may be useful with non-trained systems (e.g., those employing algorithms to directly address anticipated situations arising in scenes from a host vehicle environment). 
     In either case, there may be a hierarchy of constraints. In other words, some navigational constraints may have priority over other constraints. Thus, if a situation arose in which a navigational action was not available that would result in all implicated constraints being satisfied, the navigation system may determine the available navigational action that achieves the highest priority constraints first. For example, the system may cause the vehicle to avoid a pedestrian first even if navigation to avoid the pedestrian would result in a collision with another vehicle or an object detected in a road. In another example, the system may cause the vehicle to ride up on a curb to avoid a pedestrian. 
       FIG. 13  provides a flowchart illustrating an algorithm for implementing a hierarchy of implicated constraints determined based on analysis of a scene in an environment of a host vehicle. For example, at step  1301 , at least one processing device associated with the navigational system (e.g., an EyeQ processor, etc.) may receive, from a camera mounted on the host vehicle, a plurality of images representative of an environment of the host vehicle. Through analysis of an image or images representative of the scene of the host vehicle environment at step  1303 , a navigational state associated with the host vehicle may be identified. For example, a navigational state may indicate that the host vehicle is traveling along a two-lane road  1210 , as in  FIG. 12 , that a target vehicle  1217  is moving through an intersection ahead of the host vehicle, that a pedestrian  1215  is waiting to cross the road on which the host vehicle travels, that an object  1219  is present ahead in the host vehicle lane, among various other attributes of the scene. 
     At step  1305 , one or more navigational constraints implicated by the navigational state of the host vehicle may be determined. For example, the at least one processing device, after analyzing a scene in the environment of the host vehicle represented by one or more captured images may determine one or more navigational constraints implicated by objects, vehicles, pedestrians, etc., recognized through image analysis of the captured images. In some embodiments, the at least one processing device may determine at least a first predefined navigational constraint and a second predefined navigational constraint implicated by the navigational state, and the first predefined navigational constraint may differ from the second predefined navigational constraint. For example, the first navigational constraint may relate to one or more target vehicles detected in the environment of the host vehicle, and the second navigational constraint may relate to a pedestrian detected in the environment of the host vehicle. 
     At step  1307 , the at least one processing device may determine a priority associated with constraints identified in step  1305 . In the example described, the second predefined navigational constraint, relating to pedestrians, may have a priority higher than the first predefined navigational constraint, which relates to target vehicles. While priorities associated with navigational constraints may be determined or assigned based on various factors, in some embodiments, the priority of a navigational constraint may be related to its relative importance from a safety perspective. For example, while it may be important that all implemented navigational constraints be followed or satisfied in as many situations as possible, some constraints may be associated with greater safety risks than others and, therefore, may be assigned higher priorities. For example, a navigational constraint requiring that the host vehicle maintain at least a 1 meter spacing from a pedestrian may have a higher priority than a constraint requiring that the host vehicle maintain at least a 1 meter spacing from a target vehicle. This may be because a collision with a pedestrian may have more severe consequences than a collision with another vehicle. Similarly, maintaining a space between the host vehicle and a target vehicle may have a higher priority than a constraint requiring the host vehicle to avoid a box in the road, to drive less than a certain speed over a speed bump, or to expose the host vehicle occupants to no more than a maximum acceleration level. 
     While driving policy module  803  is designed to maximize safety by satisfying navigational constraints implicated by a particular scene or navigational state, in some situations it may be physically impossible to satisfy every implicated constraint. In such situations, the priority of each implicated constraint may be used to determine which of the implicated constraints should be satisfied first, as shown at step  1309 . Continuing with the example above, in a situation where it is not possible satisfy both the pedestrian gap constraint and the target vehicle gap constraint, but rather only one of the constraints can be satisfied, then the higher priority of the pedestrian gap constraint may result in that constraint being satisfied before attempting to maintain a gap to the target vehicle. Thus, in normal situations, the at least one processing device may determine, based on the identified navigational state of the host vehicle, a first navigational action for the host vehicle satisfying both the first predefined navigational constraint and the second predefined navigational constraint where both the first predefined navigational constraint and the second predefined navigational constraint can be satisfied, as shown at step  1311 . In other situations, however, where not all the implicated constraints can be satisfied, the at least one processing device may determine, based on the identified navigational state, a second navigational action for the host vehicle satisfying the second predefined navigational constraint (i.e., the higher priority constraint), but not satisfying the first predefined navigational constraint (having a priority lower than the second navigational constraint), where the first predefined navigational constraint and the second predefined navigational constraint cannot both be satisfied, as shown at step  1313 . 
     Next, at step  1315 , to implement the determined navigational actions for the host vehicle the at least one processing device can cause at least one adjustment of a navigational actuator of the host vehicle in response to the determined first navigational action or the determined second navigational action for the host vehicle. As in previous example, the navigational actuator may include at least one of a steering mechanism, a brake, or an accelerator. 
     Constraint Relaxation 
     As discussed above, navigational constraints may be imposed for safety purposes. The constraints may include a minimum safe driving distance with respect to a pedestrian, a target vehicle, a road barrier, or a detected object, a maximum speed of travel when passing within an influence zone of a detected pedestrian, or a maximum deceleration rate for the host vehicle, among others. These constraints may be imposed in a learning or non-learning navigational system. In certain situations, these constraints may be relaxed. For example, where the host vehicle slows or stops near a pedestrian, then progresses slowly to convey an intention to pass by the pedestrian, a response of the pedestrian can be detected from acquired images. If the response of the pedestrian is to stay still or to stop moving (and/or if eye contact with the pedestrian is sensed), it may be understood that the pedestrian recognizes an intent of the navigational system to pass by the pedestrian. In such situations, the system may relax one or more predefined constraints and implement a less stringent constraint (e.g., allow the vehicle to navigate within 0.5 meters of a pedestrian rather than within a more stringent 1 meter boundary). 
       FIG. 14  provides a flowchart for implementing control of the host vehicle based on relaxation of one or more navigational constraints. At step  1401 , the at least one processing device may receive, from a camera associated with the host vehicle, a plurality of images representative of an environment of the host vehicle. Analysis of the images at step  1403  may enable identification of a navigational state associated with the host vehicle. At step  1405 , the at least one processor may determine navigational constraints associated with the navigational state of the host vehicle. The navigational constraints may include a first predefined navigational constraint implicated by at least one aspect of the navigational state. At step  1407 , analysis of the plurality of images may reveal the presence of at least one navigational constraint relaxation factor. 
     A navigational constraint relaxation factor may include any suitable indicator that one or more navigational constraints may be suspended, altered, or otherwise relaxed in at least one aspect. In some embodiments, the at least one navigational constraint relaxation factor may include a determination (based on image analysis) that the eyes of a pedestrian are looking in a direction of the host vehicle. In such cases, it may more safely be assumed that the pedestrian is aware of the host vehicle. As a result, a confidence level may be higher that the pedestrian will not engage in unexpected actions that cause the pedestrian to move into a path of the host vehicle. Other constraint relaxation factors may also be used. For example, the at least one navigational constraint relaxation factor may include: a pedestrian determined to be not moving (e.g., one presumed to be less likely of entering a path of the host vehicle); or a pedestrian whose motion is determined to be slowing. The navigational constraint relaxation factor may also include more complicated actions, such as a pedestrian determined to be not moving after the host vehicle has come to a stop and then resumed movement. In such a situation, the pedestrian may be assumed to understand that the host vehicle has a right of way, and the pedestrian coming to a stop may suggest an intent of the pedestrian to give way to the host vehicle. Other situations that may cause one or more constraints to be relaxed include the type of curb stone (e.g., a low curb stone or one with a gradual slope might allow a relaxed distance constraint), lack of pedestrians or other objects on sidewalk, a vehicle with its engine not running may have a relaxed distance, or a a situation in which a pedestrian is facing away and/or is moving away from the area towards which the host vehicle is heading. 
     Where the presence of a navigational constraint relaxation factor is identified (e.g., at step  1407 ), a second navigational constraint may be determined or developed in response to detection of the constraint relaxation factor. This second navigational constraint may be different from the first navigational constraint and may include at least one characteristic relaxed with respect to the first navigational constraint. The second navigational constraint may include a newly generated constraint based on the first constraint, where the newly generated constraint includes at least one modification that relaxes the first constraint in at least one respect. Alternatively, the second constraint may constitute a predetermined constraint that is less stringent than the first navigational constraint in at least one respect. In some embodiments, such second constraints may be reserved for usage only for situations where a constraint relaxation factor is identified in an environment of the host vehicle. Whether the second constraint is newly generated or selected from a set of fully or partially available predetermined constraints, application of a second navigational constraint in place of a more stringent first navigational constraint (that may be applied in the absence of detection of relevant navigational constraint relaxation factors) may be referred to as constraint relaxation and may be accomplished in step  1409 . 
     Where at least one constraint relaxation factor is detected at step  1407 , and at least one constraint has been relaxed in step  1409 , a navigational action for the host vehicle may be determined at step  1411 . The navigational action for the host vehicle may be based on the identified navigational state and may satisfy the second navigational constraint. The navigational action may be implemented at step  1413  by causing at least one adjustment of a navigational actuator of the host vehicle in response to the determined navigational action. 
     As discussed above, the usage of navigational constraints and relaxed navigational constraints may be employed with navigational systems that are trained (e.g., through machine learning) or untrained (e.g., systems programmed to respond with predetermined actions in response to specific navigational states). Where trained navigational systems are used, the availability of relaxed navigational constraints for certain navigational situations may represent a mode switching from a trained system response to an untrained system response. For example, a trained navigational network may determine an original navigational action for the host vehicle, based on the first navigational constraint. The action taken by the vehicle, however, may be one that is different from the navigational action satisfying the first navigational constraint. Rather, the action taken may satisfy the more relaxed second navigational constraint and may be an action developed by a non-trained system (e.g., as a response to detection of a particular condition in the environment of the host vehicle, such as the presence of a navigational constraint relaxation factor). 
     There are many examples of navigational constraints that may be relaxed in response to detection in the environment of the host vehicle of a constraint relaxation factor. For example, where a predefined navigational constraint includes a buffer zone associated with a detected pedestrian, and at least a portion of the buffer zone extends a distance from the detected pedestrian, a relaxed navigational constraint (either newly developed, called up from memory from a predetermined set, or generated as a relaxed version of a preexisting constraint) may include a different or modified buffer zone. For example, the different or modified buffer zone may have a distance relative to the pedestrian that is less than the original or unmodified buffer zone relative to the detected pedestrian. As a result, in view of the relaxed constraint, the host vehicle may be permitted to navigate closer to a detected pedestrian, where an appropriate constraint relaxation factor is detected in the environment of the host vehicle. 
     A relaxed characteristic of a navigational constraint may include a reduced width in a buffer zone associated with at least one pedestrian, as noted above. The relaxed characteristic, however, may also include a reduced width in a buffer zone associated with a target vehicle, a detected object, a roadside barrier, or any other object detected in the environment of the host vehicle. 
     The at least one relaxed characteristic may also include other types of modifications in navigational constraint characteristics. For example, the relaxed characteristic may include an increase in speed associated with at least one predefined navigational constraint. 
     The relaxed characteristic may also include an increase in a maximum allowable deceleration/acceleration associated with at least one predefined navigational constraint. 
     While constraints may be relaxed in certain situations, as described above, in other situations, navigational constraints may be augmented. For example, in some situations, a navigational system may determine that conditions warrant augmentation of a normal set of navigational constraints. Such augmentation may include adding new constraints to a predefined set of constraints or adjusting one or more aspects of a predefined constraint. The addition or adjustment may result in more conservative navigation relative the predefined set of constraints applicable under normal driving conditions. Conditions that may warrant constraint augmentation may include sensor failure, adverse environmental conditions (rain, snow, fog, or other conditions associated with reduced visibility or reduced vehicle traction), etc. 
       FIG. 15  provides a flowchart for implementing control of the host vehicle based on augmentation of one or more navigational constraints. At step  1501 , the at least one processing device may receive, from a camera associated with the host vehicle, a plurality of images representative of an environment of the host vehicle. Analysis of the images at step  1503  may enable identification of a navigational state associated with the host vehicle. At step  1505 , the at least one processor may determine navigational constraints associated with the navigational state of the host vehicle. The navigational constraints may include a first predefined navigational constraint implicated by at least one aspect of the navigational state. At step  1507 , analysis of the plurality of images may reveal the presence of at least one navigational constraint augmentation factor. 
     An implicated navigational constraint may include any of the navigational constraints discussed above (e.g., with respect to  FIG. 12 ) or any other suitable navigational constraints. A navigational constraint augmentation factor may include any indicator that one or more navigational constraints may be supplemented/augmented in at least one aspect. Supplementation or augmentation of navigational constraints may be performed on a per set basis (e.g., by adding new navigational constraints to a predetermined set of constraints) or may be performed on a per constraint basis (e.g., modifying a particular constraint such that the modified constraint is more restrictive than the original, or adding a new constraint that corresponds to a predetermined constraint, wherein the new constraint is more restrictive than the corresponding constraint in at least one aspect). Additionally, or alternatively, supplementation or augmentation of navigational constraints may refer selection from among a set of predetermined constraints based on a hierarchy. For example, a set of augmented constraints may be available for selection based on whether a navigational augmentation factor is detected in the environment of or relative to the host vehicle. Under normal conditions where no augmentation factor is detected, then the implicated navigational constraints may be drawn from constraints applicable to normal conditions. On the other hand, where one or more constraint augmentation factors are detected, the implicated constraints may be drawn from augmented constraints either generated or predefined relative to the one or more augmentation factors. The augmented constraints may be more restrictive in at least one aspect than corresponding constraints applicable under normal conditions. 
     In some embodiments, the at least one navigational constraint augmentation factor may include a detection (e.g., based on image analysis) of the presence of ice, snow, or water on a surface of a road in the environment of the host vehicle. Such a determination may be based, for example, upon detection of: areas of reflectance higher than expected for dry roadways (e.g., indicative of ice or water on the roadway); white regions on the road indicating the presence of snow; shadows on the roadway consistent with the presence of longitudinal trenches (e.g., tire tracks in snow) on the roadway; water droplets or ice/snow particles on a windshield of the host vehicle; or any other suitable indicator of the presence of water or ice/snow on a surface of a road. 
     The at least one navigational constraint augmentation factor may also include detection of particulates on an outer surface of a windshield of the host vehicle. Such particulates may impair image quality of one or more image capture devices associated with the host vehicle. While described with respect to a windshield of the host vehicle, which is relevant for cameras mounted behind the windshield of the host vehicle, detection of particulates on other surfaces (e.g., a lens or lens cover of a camera, headlight lens, rear windshield, a tail light lens, or any other surface of the host vehicle visible to an image capture device (or detected by a sensor) associated with the host vehicle may also indicate the presence of a navigational constraint augmentation factor. 
     The navigational constraint augmentation factor may also be detected as an attribute of one or more image acquisition devices. For example, a detected decrease in image quality of one or more images captured by an image capture device (e.g., a camera) associated with the host vehicle may also constitute a navigational constraint augmentation factor. A decline in image quality may be associated with a hardware failure or partial hardware failure associated with the image capture device or an assembly associated with the image capture device. Such a decline in image quality may also be caused by environmental conditions. For example, the presence of smoke, fog, rain, snow, etc., in the air surrounding the host vehicle may also contribute to reduced image quality relative to the road, pedestrians, target vehicles, etc., that may be present in an environment of the host vehicle. 
     The navigational constraint augmentation factor may also relate to other aspects of the host vehicle. For example, in some situations, the navigational constraint augmentation factor may include a detected failure or partial failure of a system or sensor associate with the host vehicle. Such an augmentation factor may include, for example, detection of failure or partial failure of a speed sensor, GPS receiver, accelerometer, camera, radar, lidar, brakes, tires, or any other system associated with the host vehicle that may impact the ability of the host vehicle to navigate relative to navigational constraints associated with a navigational state of the host vehicle. 
     Where the presence of a navigational constraint augmentation factor is identified (e.g., at step  1507 ), a second navigational constraint may be determined or developed in response to detection of the constraint augmentation factor. This second navigational constraint may be different from the first navigational constraint and may include at least one characteristic augmented with respect to the first navigational constraint. The second navigational constraint may be more restrictive than the first navigational constraint, because detection of a constraint augmentation factor in the environment of the host vehicle or associated with the host vehicle may suggest that the host vehicle may have at least one navigational capability reduced with respect to normal operating conditions. Such reduced capabilities may include lowered road traction (e.g., ice, snow, or water on a roadway; reduced tire pressure; etc.); impaired vision (e.g., rain, snow, dust, smoke, fog etc. that reduces captured image quality); impaired detection capability (e.g., sensor failure or partial failure, reduced sensor performance, etc.), or any other reduction in capability of the host vehicle to navigate in response to a detected navigational state. 
     Where at least one constraint augmentation factor is detected at step  1507 , and at least one constraint has been augmented in step  1509 , a navigational action for the host vehicle may be determined at step  1511 . The navigational action for the host vehicle may be based on the identified navigational state and may satisfy the second navigational (i.e., augmented) constraint. The navigational action may be implemented at step  1513  by causing at least one adjustment of a navigational actuator of the host vehicle in response to the determined navigational action. 
     As discussed, the usage of navigational constraints and augmented navigational constraints may be employed with navigational systems that are trained (e.g., through machine learning) or untrained (e.g., systems programmed to respond with predetermined actions in response to specific navigational states). Where trained navigational systems are used, the availability of augmented navigational constraints for certain navigational situations may represent a mode switching from a trained system response to an untrained system response. For example, a trained navigational network may determine an original navigational action for the host vehicle, based on the first navigational constraint. The action taken by the vehicle, however, may be one that is different from the navigational action satisfying the first navigational constraint. Rather, the action taken may satisfy the augmented second navigational constraint and may be an action developed by a non-trained system (e.g., as a response to detection of a particular condition in the environment of the host vehicle, such as the presence of a navigational constraint augmented factor). 
     There are many examples of navigational constraints that may be generated, supplemented, or augmented in response to detection in the environment of the host vehicle of a constraint augmentation factor. For example, where a predefined navigational constraint includes a buffer zone associated with a detected pedestrian, object, vehicle, etc., and at least a portion of the buffer zone extends a distance from the detected pedestrian/object/vehicle, an augmented navigational constraint (either newly developed, called up from memory from a predetermined set, or generated as an augmented version of a preexisting constraint) may include a different or modified buffer zone. For example, the different or modified buffer zone may have a distance relative to the pedestrian/object/vehicle that is greater than the original or unmodified buffer zone relative to the detected pedestrian/object/vehicle. As a result, in view of the augmented constraint, the host vehicle may be forced to navigate further from the detected pedestrian/object/vehicle, where an appropriate constraint augmentation factor is detected in the environment of the host vehicle or relative to the host vehicle. 
     The at least one augmented characteristic may also include other types of modifications in navigational constraint characteristics. For example, the augmented characteristic may include a decrease in speed associated with at least one predefined navigational constraint. The augmented characteristic may also include a decrease in a maximum allowable deceleration/acceleration associated with at least one predefined navigational constraint. 
     Navigation Based on Long Range Planning 
     In some embodiments, the disclosed navigational system can respond not only to a detected navigational state in an environment of the host vehicle, but may also determine one or more navigational actions based on long range planning. For example, the system may consider the potential impact on future navigational states of one or more navigational actions available as options for navigating with respect to a detected navigational state. Considering the effects of available actions on future states may enable the navigational system to determine navigational actions based not just upon a currently detected navigational state, but also based upon long range planning. Navigation using long range planning techniques may be especially applicable where one or more reward functions are employed by the navigation system as a technique for selecting navigational actions from among available options. Potential rewards may be analyzed with respect to the available navigational actions that may be taken in response to a detected, current navigational state of the host vehicle. Further, however, the potential rewards may also be analyzed relative to actions that may be taken in response to future navigational states projected to result from the available actions to a current navigational state. As a result, the disclosed navigational system may, in some cases, select a navigational action in response to a detected navigational state even where the selected navigational action may not yield the highest reward from among the available actions that may be taken in response to the current navigational state. This may be especially true where the system determines that the selected action may result in a future navigational state giving rise to one or more potential navigational actions offering higher rewards than the selected action or, in some cases, any of the actions available relative to a current navigational state. The principle may be expressed more simply as taking a less favorable action now in order to produce higher reward options in the future. Thus, the disclosed navigational system capable of long range planning may choose a suboptimal short term action where long term prediction indicates that a short term loss in reward may result in long term reward gains. 
     In general, autonomous driving applications may involve a series of planning problems, where the navigational system may decide on immediate actions in order to optimize a longer term objective. For example, when a vehicle is confronted with a merge situation at a roundabout, the navigational system may decide on an immediate acceleration or braking command in order to initiate navigation into the roundabout. While the immediate action to the detected navigational state at the roundabout may involve an acceleration or braking command responsive to the detected state, the long term objective is a successful merge, and the long term effect of the selected command is the success/failure of the merge. The planning problem may be addressed by decomposing the problem into two phases. First, supervised learning may be applied for predicting the near future based on the present (assuming the predictor will be differentiable with respect to the representation of the present). Second, a full trajectory of the agent may be modeled using a recurrent neural network, where unexplained factors are modeled as (additive) input nodes. This may allow solutions to the long-term planning problem to be determined using supervised learning techniques and direct optimization over the recurrent neural network. Such an approach may also enable the learning of robust policies by incorporating adversarial elements to the environment. 
     Two of the most fundamental elements of autonomous driving systems are sensing and planning. Sensing deals with finding a compact representation of the present state of the environment, while planning deals with deciding on what actions to take so as to optimize future objectives. Supervised machine learning techniques are useful for solving sensing problems. Machine learning algorithmic frameworks may also be used for the planning part, especially reinforcement learning (RL) frameworks, such as those described above. 
     RL may be performed in a sequence of consecutive rounds. At round t, the planner (a.k.a. the agent or driving policy module  803 ) may observe a state, s t ∈S, which represents the agent as well as the environment. It then should decide on an action a t ∈A. After performing the action, the agent receives an immediate reward, r t ∈ , and is moved to a new state, s t+1 . As an example, the host vehicle may include an adaptive cruise control (ACC) system, in which the vehicle should autonomously implement acceleration/braking so as to keep an adequate distance to a preceding vehicle while maintaining smooth driving. The state can be modeled as a pair, s t =(x t , ν t )∈   2 , where x t  is the distance to the preceding vehicle and ν t  is the velocity of the host vehicle relative to the velocity of the preceding vehicle. The action a t ∈  will be the acceleration command (where the host vehicle slows down if a t &lt;0). The reward can be a function that depends on |a t | (reflecting the smoothness of driving) and on s t  (reflecting that the host vehicle maintains a safe distance from the preceding vehicle). The goal of the planner is to maximize the cumulative reward (maybe up to a time horizon or a discounted sum of future rewards). To do so, the planner may rely on a policy, π:S→A, which maps a state into an action. 
     Supervised Learning (SL) can be viewed as a special case of RL, in which s t  is sampled from some distribution over S, and the reward function may have the form r t =− (a t , y t ), where   is a loss function, and the learner observes the value of y t  which is the (possibly noisy) value of the optimal action to take when viewing the state s t . There may be several differences between a general RL model and a specific case of SL, and these differences can make the general RL problem more challenging. 
     In some SL situations, the actions (or predictions) taken by the learner may have no effect on the environment. In other words, s t+1  and a t  are independent. This can have two important implications. First, in SL, a sample (s 1 , y 1 ), . . . , (s m , y m ) can be collected in advance, and only then can the search begin for a policy (or predictor) that will have good accuracy relative to the sample. In contrast, in RL, the state s t+1  usually depends on the action taken (and also on the previous state), which in turn depends on the policy used to generate the action. This ties the data generation process to the policy learning process. Second, because actions do not affect the environment in SL, the contribution of the choice of a t  to the performance of π is local. Specifically, a t  only affects the value of the immediate reward. In contrast, in RL, actions that are taken at round t might have a long-term effect on the reward values in future rounds. 
     In SL, the knowledge of the “correct” answer, y t , together with the shape of the reward, r t =− (a t , y t ) may provide full knowledge of the reward for all possible choices of a t , which may enable calculation of the derivative of the reward with respect to a t . In contrast, in RL, a “one-shot” value of the reward may be all that can be observed for a specific choice of action taken. This may be referred to as a “bandit” feedback. This is one of the most significant reasons for the need of “exploration” as a part of long term navigational planning, because in RL-based systems, if only “bandit” feedback is available, the system may not always know if the action taken was the best action to take. 
     Many RL algorithms rely, at least in part, on the mathematically elegant model of a Markov Decision Process (MDP). The Markovian assumption is that the distribution of s t+1  is fully determined given s t  and a t . This yields a closed form expression for the cumulative reward of a given policy in terms of the stationary distribution over states of the MDP. The stationary distribution of a policy can be expressed as a solution to a linear programming problem. This yields two families of algorithms: 1) optimization with respect to the primal problem, which may be referred to as policy search, and 2) optimization with respect to a dual problem, whose variables are called the value function, V π . The value function determines the expected cumulative reward if the MDP begins from the initial state, s, and from there actions are chosen according to π. A related quantity is the state-action value function, Q π (s, a), which determines the cumulative reward assuming a start from state, s, an immediately chosen action a, and from there on actions chosen according to π. The Q function may give rise to a characterization of the optimal policy (using the Bellman&#39;s equation). In particular, the Q function may show that the optimal policy is a deterministic function from S to A (in fact, it may be characterized as a “greedy” policy with respect to the optimal Q function). 
     One potential advantage of the MDP model is that it allows coupling of the future into the present using the Q function. For example, given that a host vehicle is now in state, s, the value of Q π (s, a) may indicate the effect of performing action a on the future. Therefore, the Q function may provide a local measure of the quality of an action a, thus making the RL problem more similar to a SL scenario. 
     Many RL algorithms approximate the V function or the Q function in one way or another. Value iteration algorithms, e.g., the Q learning algorithm, may rely on the fact that the V and Q functions of the optimal policy may be fixed points of some operators derived from Bellman&#39;s equation. Actor-critic policy iteration algorithms aim to learn a policy in an iterative way, where at iteration t, the “critic” estimates Q πt  and based on this estimate, the “actor” improves the policy. 
     Despite the mathematical elegancy of MDPs and the convenience of switching to the Q function representation, this approach may have several limitations. For example, an approximate notion of a Markovian behaving state may be all that can be found in some cases. Furthermore, the transition of states may depend not only on the agent&#39;s action, but also on actions of other players in the environment. For example, in the ACC example mentioned above, while the dynamic of the autonomous vehicle may be Markovian, the next state may depend on the behavior of the driver of the other car, which is not necessarily Markovian. One possible solution to this problem is to use partially observed MDPs, in which it is assumed that there is a Markovian state, but an observation that is distributed according to the hidden state is what can be seen. 
     A more direct approach may consider game theoretical generalizations of MDPs (e.g., the Stochastic Games framework). Indeed, algorithms for MDPs may be generalized to multi-agents games (e.g., minimax-Q learning or Nash-Q learning). Other approaches may include explicit modeling of the other players and vanishing regret learning algorithms. Learning in a multi-agent setting may be more complex than in a single agent setting. 
     A second limitation of the Q function representation may arise by departing from a tabular setting. The tabular setting is when the number of states and actions is small, and therefore, Q can be expressed as a table with |S| rows and |A| columns. However, if the natural representation of S and A includes Euclidean spaces, and the state and action spaces are discretized, the number of states/actions may be exponential in the dimension. In such cases, it may not be practical to employ a tabular setting. Instead, the Q function may be approximated by some function from a parametric hypothesis class (e.g., neural networks of a certain architecture). For example, a deep-Q-network (DQN) learning algorithm may be used. In DQN, the state space can be continuous, but the action space may remain a small discrete set. There may be approaches for dealing with continuous action spaces, but they may rely on approximating the Q function. In any case, the Q function may be complicated and sensitive to noise, and, therefore, may be challenging to learn. 
     A different approach may be to address the RL problem using a recurrent neural network (RNN). In some cases, RNN may be combined with the notions of multi-agents games and robustness to adversarial environments from game theory. Further, this approach may be one that does not explicitly rely on any Markovian assumption. 
     The following describes in more detail an approach for navigation by planning based on prediction. In this approach, it may be assumed that the state space, S, is a subset of    d , and the action space, A, is a subset of    k . This may be a natural representation in many applications. As noted above, there may be two key differences between RL and SL: (1) because past actions affect future rewards, information from the future may need to be propagated back to the past; and (2) the “bandit” nature of rewards can blur the dependence between (state, action) and reward, which can complicate the learning process. 
     As a first step in the approach, an observation may be made that there are interesting problems in which the bandit nature of rewards is not an issue. For example, reward value (as will be discussed in more detail below) for the ACC application may be differentiable with respect to the current state and action. In fact, even if the reward is given in a “bandit” manner, the problem of learning a differentiable function, {circumflex over (r)}(s, a), such that {circumflex over (r)}(s t , a t )≈r t , may be a relatively straightforward SL problem (e.g., a one dimensional regression problem). Therefore, the first step of the approach may be to either define the reward as a function, {circumflex over (r)}(s, a), which is differentiable with respect to s and a, or to use a regression learning algorithm in order to learn a differentiable function, {circumflex over (r)}, that minimizes at least some regression loss over a sample with instance vector being (s t , a t )∈   d ×   k  and target scalar being r t . In some situations, in order to create a training set, elements of exploration may be used. 
     To address the connection between past and future, a similar idea may be used. For example, suppose a differentiable function {circumflex over (N)}(s, a) can be learned such that {circumflex over (N)}(s t , a t )≈s t+1 . Learning such a function may be characterized as an SL problem. {circumflex over (N)} may be viewed as a predictor for the near future. Next, a policy that maps from S to A may be described using a parametric function π θ : S→A. Expressing π θ  as a neural network, may enable expression of an episode of running the agent for T rounds using a recurrent neural network (RNN), where the next state is defined as s t+1 ={circumflex over (N)}(s t , a t )+ν t . Here, ν t ∈   d  may be defined by the environment and may express unpredictable aspects of the near future. The fact that s t+1  depends on s t  and a t  in a differentiable manner may enable a connection between future reward values and past actions. A parameter vector of the policy function, π θ , may be learned by back-propagation over the resulting RNN. Note that explicit probabilistic assumptions need not be imposed on ν t . In particular, there need not be a requirement for a Markovian relation. Instead, the recurrent network may be relied upon to propagate “enough” information between past and future. Intuitively, {circumflex over (N)}(s t , a t ) may describe the predictable part of the near future, while ν t  may express the unpredictable aspects, which may arise due to the behavior of other players in the environment. The learning system should learn a policy that will be robust to the behavior of other players. If ∥ν t ∥ is large, the connection between past actions and future reward values may be too noisy for learning a meaningful policy. Explicitly expressing the dynamic of the system in a transparent way may enable incorporation of prior knowledge more easily. For example, prior knowledge may simplify the problem of defining {circumflex over (N)}. 
     As discussed above, the learning system may benefit from robustness relative to an adversarial environment, such as the environment of a host vehicle, which may include multiple other drivers that may act in unexpected way. In a model that does not impose probabilistic assumptions on ν t , environments may be considered in which ν t  is chosen in an adversarial manner. In some cases, restrictions may be placed on μ t , otherwise the adversary can make the planning problem difficult or even impossible. One natural restriction may be to require that ∥μ t ∥ is bounded by a constant. 
     Robustness against adversarial environments may be useful in autonomous driving applications. Choosing μ t  in an adversarial way may even speed up the learning process, as it can focus the learning system toward a robust optimal policy. A simple game may be used to illustrate this concept. The state is s t Å , the action is a t ∈ , and the immediate loss function is 0.1|a t |+[|s t |−2]+, where [x]+=max{x, 0} is the ReLU (rectified linear unit) function. The next state is s t+1 =s t +a t +ν t , where ν t 531 [−0.5, 0.5] is chosen for the environment in an adversarial manner. Here, the optimal policy may be written as a two layer network with ReLU: a t =−[s t −1.5] + +[− t −1.5]+. Observe that when |s t |∈(1.5, 2], the optimal action may have a larger immediate loss than the action a=0. Therefore, the system may plan for the future and may not rely solely on the immediate loss. Observe that the derivative of the loss with respect to a t  is 0.1 sign(a t ), and the derivative with respect to s t  is 1[|s t |&gt;2]sign(s t ). In a situation in which s t ∈(1.5. 2], the adversarial choice of ν t  would be to set ν t =0.5 and, therefore, there may be a non-zero loss on round t+1, whenever a t &gt;1.5−s t . In such cases, the derivative of the loss may back-propagate directly to a t . Thus, the adversarial choice of ν t  may help the navigational system obtain a non-zero back-propagation message in cases for which the choice of at is sub-optimal. Such a relationship may aid the navigational system in selecting present actions based on an expectation that such a present action (even if that action would result in a suboptimal reward or even a loss) will provide opportunities in the future for more optimal actions that result in higher rewards. 
     Such an approach may be applied to virtually any navigational situation that may arise. The following describes the approach applied to one example: adaptive cruise control (ACC). In the ACC problem, the host vehicle may attempt to maintain an adequate distance to a target vehicle ahead (e.g., 1.5 seconds to the target car). Another goal may be to drive as smooth as possible while maintaining the desired gap. A model representing this situation may be defined as follows. The state space is  , and the action space is  . The first coordinate of the state is the speed of the target car, the second coordinate is the speed of the host vehicle, and the last coordinate is the distance between the host vehicle and target vehicle (e.g., location of the host vehicle minus the location of the target along the road curve). The action to be taken by the host vehicle is the acceleration, and may be denoted by a t . The quantity τ may denote the difference in time between consecutive rounds. While τ may be set to any suitable quantity, in one example, T may be 0.1 seconds. Position, s t , may be denoted as s t =(ν t   target , ν t   host , x t ), and the (unknown) acceleration of the target vehicle may be denoted as a t   target . 
     The full dynamics of the system can be described by: 
       ν t   target =[ν t−1   target   +τa   t−1   target ]+
 
       νhd t host =[ν t−1   host   +τa   t−1 ]+
 
         x   t   =[x   t−1 +τ(ν t−1   target −ν t−1   host )]+
 
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     The first vector is the predictable part, and the second vector is the unpredictable part. The reward on round t is defined as follows: 
       − r   t =0.1 |a   t   |+[|x   t   /x*   t −1|−0.3] +  where  x*   t =max{1, 1.5 ν t   host }
 
     The first term may result in a penalty for non-zero accelerations, thus encouraging smooth driving. The second term depends on the ratio between the distance to the target car, x t , and the desired distance, x* t , which is defined as the maximum between a distance of  1  meter and break distance of 1.5 seconds. In some cases, this ratio may be exactly 1, but as long as this ratio is within [0.7, 1.3], the policy may forego any penalties, which may allow the host vehicle some slack in navigation—a characteristic that may be important in achieving a smooth drive. 
     Implementing the approach outlined above, the navigation system of the host vehicle (e.g., through operation of driving policy module  803  within processing unit  110  of the navigation system) may select an action in response to an observed state. The selected action may be based on analysis not only of rewards associated with the responsive actions available relative to a sensed navigational state, but may also be based on consideration and analysis of future states, potential actions in response to the futures states, and rewards associated with the potential actions. 
       FIG. 16  illustrates an algorithmic approach to navigation based on detection and long range planning. For example, at step  1601 , the at least one processing device  110  of the navigation system for the host vehicle may receive a plurality of images. These images may capture scenes representative of an environment of the host vehicle and may be supplied by any of the image capture devices (e.g., cameras, sensors, etc.) described above. Analysis of one or more of these images at step  1603  may enable the at least one processing device  110  to identify a present navigational state associated with the host vehicle (as described above). 
     At steps  1605 ,  1607 , and  1609 , various potential navigational actions responsive to the sensed navigational state may be determined. These potential navigational actions (e.g., a first navigational action through an N th available navigational action) may be determined based on the sensed state and the long range goals of the navigational system (e.g., to complete a merge, follow a lead vehicle smoothly, pass a target vehicle, avoid an object in the roadway, slow for a detected stop sign, avoid a target vehicle cutting in, or any other navigational action that may advance the navigational goals of the system).    
     For each of the determined potential navigational actions, the system may determine an expected reward. The expected reward may be determined according to any of the techniques described above and may include analysis of a particular potential action relative to one or more reward functions. Expected rewards  1606 ,  1608 , and  1610  may be determined for each of the potential navigational actions (e.g., the first, second, and N th ) determined in steps  1605 ,  1607 , and  1609 , respectively. 
     In some cases, the navigational system of the host vehicle may select from among the available potential actions based on values associated with expected rewards  1606 ,  1608 , and  1610  (or any other type of indicator of an expected reward). For example, in some situations, the action that yields the highest expected reward may be selected. 
     In other cases, especially where the navigation system engages in long range planning to determine navigational actions for the host vehicle, the system may not choose the potential action that yields the highest expected reward. Rather, the system may look to the future to analyze whether there may be opportunities for realizing higher rewards later if lower reward actions are selected in response to a current navigational state. For example, for any or all of the potential actions determined at steps  1605 ,  1607 , and  1609 , a future state may be determined. Each future state, determined at steps  1613 ,  1615 , and  1617 , may represent a future navigational state expected to result based on the current navigational state as modified by a respective potential action (e.g., the potential actions determined at steps  1605 ,  1607 , and  1609 ). 
     For each of the future states predicted at steps  1613 ,  1615 , and  1617 , one or more future actions (as navigational options available in response to determined future state) may be determined and evaluated. At steps  1619 ,  1621 , and  1623 , for example, values or any other type of indicator of expected rewards associated with one or more of the future actions may be developed (e.g., based on one or more reward functions). The expected rewards associated with the one or more future actions may be evaluated by comparing values of reward functions associated with each future action or by comparing any other indicators associated with the expected rewards. 
     At step  1625 , the navigational system for the host vehicle may select a navigational action for the host vehicle based on a comparison of expected rewards, not just based on the potential actions identified relative to a current navigational state (e.g., at steps  1605 ,  1607 , and  1609 ), but also based on expected rewards determined as a result of potential future actions available in response to predicted future states (e.g., determined at steps  1613 ,  1615 , and  1617 ). The selection at step  1625  may be based on the options and rewards analysis performed at steps  1619 ,  1621 , and  1623 . 
     The selection of a navigational action at step  1625  may be based on a comparison of expected rewards associated with future action options only. In such a case, the navigational system may select an action to the current state based solely on a comparison of expected rewards resulting from actions to potential future navigational states. For example, the system may select the potential action identified at step  1605 ,  1607 , or  1609  that is associated with a highest future reward value as determined through analysis at steps  1619 ,  1621 , and  1623 . 
     The selection of a navigational action at step  1625  may also be based on comparison of current action options only (as noted above). In this situation, the navigational system may select the potential action identified at step  1605 ,  1607 , or  1609  that is associated with a highest expected reward,  1606 ,  1608 , or  1610 . Such a selection may be performed with little or no consideration of future navigational states or future expected rewards to navigational actions available in response to expected future navigational states. 
     On the other hand, in some cases, the selection of a navigational action at step  1625  may be based on a comparison of expected rewards associated with both future action options and with current action options. This, in fact, may be one of the principles of navigation based on long range planning. For example, expected rewards to future actions may be analyzed to determine if any may warrant a selection of a lower reward action in response to the current navigational state in order to achieve a potential higher reward in response to a subsequent navigational action expected to be available in response to future navigational states. As an example, a value or other indicator of an expected reward  1606  may indicate a highest expected reward from among rewards  1606 ,  1608 , and  1610 . On the other hand, expected reward  1608  may indicate a lowest expected reward from among rewards  1606 ,  1608 , and  1610 . Rather than simply selecting the potential action determined at step  1605  (i.e., the action giving rise to the highest expected reward  1606 ), analysis of future states, potential future actions, and future rewards may be used in making a navigational action selection at step  1625 . In one example, it may be determined that a reward identified at step  1621  (in response to at least one future action to a future state determined at step  1615  based on the second potential action determined at step  1607 ) may be higher than expected reward  1606 . Based on this comparison, the second potential action determined at step  1607  may be selected rather than the first potential action determined at step  1605  despite expected reward  1606  being higher than expected reward  1608 . In one example, the potential navigational action determined at step  1605  may include a merge in front of a detected target vehicle, while the potential navigational action determined at step  1607  may include a merge behind the target vehicle. While the expected reward  1606  of merging in front of the target vehicle may be higher than the expected reward  1608  associated with merging behind the target vehicle, it may be determined that merging behind the target vehicle may result in a future state for which there may be action options yielding even higher potential rewards than expected reward  1606 ,  1608 , or other rewards based on available actions in response to a current, sensed navigational state. 
     Selection from among potential actions at step  1625  may be based on any suitable comparison of expected rewards (or any other metric or indicator of benefits associated with one potential action over another). In some cases, as described above, a second potential action may be selected over a first potential action if the second potential action is projected to provide at least one future action associated with an expected reward higher than a reward associated with the first potential action. In other cases, more complex comparisons may be employed. For example, rewards associated with action options in response to projected future states may be compared to more than one expected reward associated with a determined potential action. 
     In some scenarios, actions and expected rewards based on projected future states may affect selection of a potential action to a current state if at least one of the future actions is expected to yield a reward higher than any of the rewards expected as a result of the potential actions to a current state (e.g., expected rewards  1606 ,  1608 ,  1610 , etc.). In some cases, the future action option that yields the highest expected reward (e.g., from among the expected rewards associated with potential actions to a sensed current state as well as from among expected rewards associated with potential future action options relative to potential future navigational states) may be used as a guide for selection of a potential action to a current navigational state. That is, after identifying a future action option yielding the highest expected reward (or a reward above a predetermined threshold, etc.), the potential action that would lead to the future state associated with the identified future action yielding the highest expected reward may be selected at step  1625 . 
     In other cases, selection of available actions may be made based on determined differences between expected rewards. For example, a second potential action determined at step  1607  may be selected if a difference between an expected reward associated with a future action determined at step  1621  and expected reward  1606  is greater than a difference between expected reward  1608  and expected reward  1606  (assuming +sign differences). In another example, a second potential action determined at step  1607  may be selected if a difference between an expected reward associated with a future action determined at step  1621  and an expected reward associated with a future action determined at step  1619  is greater than a difference between expected reward  1608  and expected reward  1606 . 
     Several examples have been described for selecting from among potential actions to a current navigational state. Any other suitable comparison technique or criteria, however, may be used for selecting an available action through long range planning based on action and reward analysis extending to projected future states. Additionally, while  FIG. 16  represents two layers in the long range planning analysis (e.g., a first layer considering the rewards resulting from potential actions to a current state, and a second layer considering the rewards resulting from future action options in response to projected future states), analysis based on more layers may be possible. For example, rather than basing the long range planning analysis upon one or two layers, three, four or more layers of analysis could be used in selecting from among available potential actions in response to a current navigational state. 
     After a selection is made from among potential actions in response to a sensed navigational state, at step  1627 , the at least one processor may cause at least one adjustment of a navigational actuator of the host vehicle in response to the selected potential navigational action. The navigational actuator may include any suitable device for controlling at least one aspect of the host vehicle. For example, the navigational actuator may include at least one of a steering mechanism, a brake, or an accelerator. 
     Navigation Based on Inferred Aggression of Others 
     Target vehicles may be monitored through analysis of an acquired image stream to determine indicators of driving aggression. Aggression is described herein as a qualitative or quantitative parameter, but other characteristics may be used: perceived level of attention (potential impairment of driver, distracted—cell phone, asleep, etc.). In some cases, a target vehicle may be deemed to have a defensive posture, and in some cases, the target vehicle may be determined to have a more aggressive posture. Navigational actions may be selected or developed based on indicators of aggression. For example, in some cases, the relative velocity, relative acceleration, increases in relative acceleration, following distance, etc., relative to a host vehicle may be tracked to determine if the target vehicle is aggressive or defensive. If the target vehicle is determined to have a level of aggression exceeding a threshold, for example, the host vehicle may be inclined to give way to the target vehicle. A level of aggression of the target vehicle may also be discerned based on a determined behavior of the target vehicle relative to one or more obstacles in a path of or in a vicinity of the target vehicle (e.g., a leading vehicle, obstacle in the road, traffic light, etc.). 
     As an introduction to this concept, an example experiment will be described with respect to a merger of the host vehicle into a roundabout, in which a navigational goal is to pass through and out of the roundabout. The situation may begin with the host vehicle approaches an entrance of the roundabout and may end with the host vehicle reaches an exit of the roundabout (e.g., the second exit). Success may be measured based on whether the host vehicle maintains a safe distance from all other vehicles at all times, whether the host vehicle finishes the route as quickly as possible, and whether the host vehicle adheres to a smooth acceleration policy. In this illustration, NTtarget vehicles may be placed at random on the roundabout. To model a blend of adversarial and typical behavior, with probability p, a target vehicle may be modeled by an “aggressive” driving policy, such that the aggressive target vehicle accelerates when the host vehicle attempts to merge in front of the target vehicle. With probability 1−p, the target vehicle may be modeled by a “defensive” driving policy, such that the target vehicle decelerates and lets the host vehicle merge in. In this experiment, p=0.5, and the navigation system of the host vehicle may be provided with no information about the type of the other drivers. The types of other drivers may be chosen at random at the beginning of the episode. 
     The navigational state may be represented as the velocity and location of the host vehicle (the agent), and the locations, velocities, and accelerations of the target vehicles. 
     Maintaining target acceleration observations may be important in order to differentiate between aggressive and defensive drivers based on the current state. All target vehicles may move on a one-dimensional curve that outlines the roundabout path. The host vehicle may move on its own one-dimensional curve, which intersects the target vehicles&#39; curve at the merging point, and this point is the origin of both curves. To model reasonable driving, the absolute value of all vehicles&#39; accelerations may be upper bounded by a constant. Velocities may also be passed through a ReLU because driving backward is not allowed. Note that by not allowing driving backwards, long-term planning may become a necessity, as the agent cannot regret on its past actions. 
     As described above, the next state, s t+1 , may be decomposed into a sum of a predictable part, {circumflex over (N)}(s t , d t ) and a non-predictable part, ν t . The expression, N(s t , a t ), may represent the dynamics of vehicle locations and velocities (which may be well-defined in a differentiable manner), while ν t  may represent the target vehicles&#39; acceleration. It may be verified that {circumflex over (N)}(s t , a t ) can be expressed as a combination of ReLU functions over an affine transformation, hence it is differentiable with respect to s t  and a t . The vector ν t  may be defined by a simulator in a non-differentiable manner, and may implement aggressive behavior for some targets and defensive behavior for other targets. Two frames from such a simulator are shown in  FIGS. 17A and 17B . In this example experiment, a host vehicle  1701  learned to slowdown as it approached the entrance of the roundabout. It also learned to give way to aggressive vehicles (e.g., vehicles  1703  and  1705 ), and to safely continue when merging in front of defensive vehicles (e.g., vehicles  1706 ,  1708 , and  1710 ). In the example represented by  FIGS. 17A and 17B , the navigation system of host vehicle  1701  is not provided with the type of target vehicles. Rather, whether a particular vehicle is determined to be aggressive or defensive is determined through inference based on observed position and acceleration, for example, of the target vehicles. In  FIG. 17A , based on position, velocity, and/or relative acceleration, host vehicle  1701  may determine that vehicle  1703  has an aggressive tendency and, therefore, host vehicle  1701  may stop and wait for target vehicle  1703  to pass rather than attempting to merge in front of target vehicle  1703 . In  FIG. 17B , however, target vehicle  1701  recognized that the target vehicle  1710  traveling behind vehicle  1703  exhibited defensive tendencies (again, based on observed position, velocity, and/or relative acceleration of vehicle  1710 ) and, therefore, completed a successful merge in front of target vehicle  1710  and behind target vehicle  1703 . 
       FIG. 18  provides a flowchart representing an example algorithm for navigating a host vehicle based on predicted aggression of other vehicles. In the example of  FIG. 18 , a level of aggression associated with at least one target vehicle may be inferred based on observed behavior of the target vehicle relative to an object in the environment of the target vehicle. For example, at step  1801 , at least one processing device (e.g., processing device  110 ) of the host vehicle navigation system may receive, from a camera associated with the host vehicle, a plurality of images representative of an environment of the host vehicle. At step  1803 , analysis of one or more of the received images may enable the at least one processor to identify a target vehicle (e.g., vehicle  1703 ) in the environment of the host vehicle  1701 . At step  1805 , analysis of one or more of the received images may enable the at least one processing device to identify in the environment of the host vehicle at least one obstacle to the target vehicle. The object may include debris in a roadway, a stoplight/traffic light, a pedestrian, another vehicle (e.g., a vehicle traveling ahead of the target vehicle, a parked vehicle, etc.), a box in the roadway, a road barrier, a curb, or any other type of object that may be encountered in an environment of the host vehicle. At step  1807 , analysis of one or more of the received images may enable the at least one processing device to determine at least one navigational characteristic of the target vehicle relative to the at least one identified obstacle to the target vehicle. 
     Various navigational characteristics may be used to infer a level of aggression of a detected target vehicle in order to develop an appropriate navigational response to the target vehicle. For example, such navigational characteristics may include a relative acceleration between the target vehicle and the at least one identified obstacle, a distance of the target vehicle from the obstacle (e.g., a following distance of the target vehicle behind another vehicle), and/or a relative velocity between the target vehicle and the obstacle, etc. 
     In some embodiments, the navigational characteristics of the target vehicles may be determined based on outputs from sensors associated with the host vehicle (e.g., radar, speed sensors, GPS, etc.). In some cases, however, the navigational characteristics of the target vehicles may be determined partially or fully based on analysis of images of an environment of the host vehicle. For example, image analysis techniques described above and in, for example, U.S. Pat. No. 9,168,868, which is incorporated herein by reference, may be used to recognize target vehicles within an environment of the host vehicle. And, monitoring a location of a target vehicle in the captured images over time and/or monitoring locations in the captured images of one or more features associated with the target vehicle (e.g., tail lights, head lights, bumper, wheels, etc.) may enable a determination of relative distances, velocities, and/or accelerations between the target vehicles and the host vehicle or between the target vehicles and one or more other objects in an environment of the host vehicle. 
     An aggression level of an identified target vehicle may be inferred from any suitable observed navigational characteristic of the target vehicle or any combination of observed navigational characteristics. For example, a determination of aggressiveness may be made based on any observed characteristic and one or more predetermined threshold levels or any other suitable qualitative or quantitative analysis. In some embodiments, a target vehicle may be deemed as aggressive if the target vehicle is observed to be following the host vehicle or another vehicle at a distance less than a predetermined aggressive distance threshold. On the other hand, a target vehicle observed to be following the host vehicle or another vehicle at a distance greater than a predetermined defensive distance threshold may be deemed defensive. The predetermined aggressive distance threshold need not be the same as the predetermined defensive distance threshold. Additionally, either or both of the predetermined aggressive distance threshold and the predetermined defensive distance threshold may include a range of values, rather than a bright line value. Further, neither of the predetermined aggressive distance threshold nor the predetermined defensive distance threshold must be fixed. Rather these values, or ranges of values, may shift over time, and different thresholds/ranges of threshold values may be applied based on observed characteristics of a target vehicle. For example, the thresholds applied may depend on one or more other characteristics of the target vehicle. Higher observed relative velocities and/or accelerations may warrant application of larger threshold values/ranges. Conversely, lower relative velocities and/or accelerations, including zero relative velocities and/or accelerations, may warrant application of smaller distance threshold values/ranges in making the aggressive/defensive inference. 
     The aggressive/defensive inference may also be based on relative velocity and/or relative acceleration thresholds. A target vehicle may be deemed aggressive if its observed relative velocity and/or its relative acceleration with respect to another vehicle exceeds a predetermined level or range. A target vehicle may be deemed defensive if its observed relative velocity and/or its relative acceleration with respect to another vehicle falls below a predetermined level or range. 
     While the aggressive/defensive determination may be made based on any observed navigational characteristic alone, the determination may also depend on any combination of observed characteristics. For example, as noted above, in some cases, a target vehicle may be deemed aggressive based solely on an observation that it is following another vehicle at a distance below a certain threshold or range. In other cases, however, the target vehicle may be deemed aggressive if it both follows another vehicle at less than a predetermined amount (which may be the same as or different than the threshold applied where the determination is based on distance alone) and has a relative velocity and/or a relative acceleration of greater than a predetermined amount or range. Similarly, a target vehicle may be deemed defensive based solely on an observation that it is following another vehicle at a distance greater than a certain threshold or range. In other cases, however, the target vehicle may be deemed defensive if it both follows another vehicle at greater than a predetermined amount (which may be the same as or different than the threshold applied where the determination is based on distance alone) and has a relative velocity and/or a relative acceleration of less than a predetermined amount or range. System  100  may make an aggressive/defensive if, for example, a vehicle exceeds 0.5G acceleration or deceleration (e.g., jerk 5 m/s3), a vehicle has a lateral acceleration of 0.5G in a lane change or on a curve, a vehicle causes another vehicle to do any of the above, a vehicle changes lanes and causes another vehicle to give way by more than 0.3G deceleration or jerk of 3 m/s3, and/or a vehicle changes two lanes without stopping. 
     It should be understood that references to a quantity exceeding a range may indicate that the quantity either exceeds all values associated with the range or falls within the range. Similarly, references to a quantity falling below a range may indicate that the quantity either falls below all values associated with the range or falls within the range. Additionally, while the examples described for making an aggressive/defensive inference are described with respect to distance, relative acceleration, and relative velocity, any other suitable quantities may be used. For example, a time to collision may calculation may be used or any indirect indicator of distance, acceleration, and/or velocity of the target vehicle. It should also be noted that while the examples above focus on target vehicles relative to other vehicles, the aggressive/defensive inference may be made by observing the navigational characteristics of a target vehicle relative to any other type of obstacle (e.g., a pedestrian, road barrier, traffic light, debris, etc.). 
     Returning to the example shown in  FIGS. 17A and 17B , as host vehicle  1701  approaches the roundabout, the navigation system, including its at least one processing device, may receive a stream of images from a camera associated with the host vehicle. Based on analysis of one or more of the received images, any of target vehicles  1703 ,  1705 ,  1706 ,  1708 , and  1710  may be identified. Further, the navigation system may analyze the navigational characteristics of one or more of the identified target vehicles. The navigation system may recognize that the gap between target vehicles  1703  and  1705  represents the first opportunity for a potential merge into the roundabout. The navigation system may analyze target vehicle  1703  to determine indicators of aggression associated with target vehicle  1703 . If target vehicle  1703  is deemed aggressive, then the host vehicle navigation system may choose to give way to vehicle  1703  rather than merging in front of vehicle  1703 . On the other hand, if target vehicle  1703  is deemed defensive, then the host vehicle navigation system may attempt to complete a merge action ahead of vehicle  1703 . 
     As host vehicle  1701  approaches the roundabout, the at least one processing device of the navigation system may analyze the captured images to determine navigational characteristics associated with target vehicle  1703 . For example, based on the images, it may be determined that vehicle  1703  is following vehicle  1705  at a distance that provides a sufficient gap for the host vehicle  1701  to safely enter. Indeed, it may be determined that vehicle  1703  is following vehicle  1705  by a distance that exceeds an aggressive distance threshold, and therefore, based on this information, the host vehicle navigation system may be inclined to identify target vehicle  1703  as defensive. In some situations, however, more than one navigational characteristic of a target vehicle may be analyzed in making the aggressive/defensive determination, as discussed above. Furthering the analysis, the host vehicle navigation system may determine that, while target vehicle  1703  is following at a non-aggressive distance behind target vehicle  1705 , vehicle  1703  has a relative velocity and/or a relative acceleration with respect to vehicle  1705  that exceeds one or more thresholds associated with aggressive behavior. Indeed, host vehicle  1701  may determine that target vehicle  1703  is accelerating relative to vehicle  1705  and closing the gap that exists between vehicles  1703  and  1705 . Based on further analysis of the relative velocity, acceleration, and distance (and even a rate that the gap between vehicles  1703  and  1705  is closing), host vehicle  1701  may determine that target vehicle  1703  is behaving aggressively. Thus, while there may be a sufficient gap into which host vehicle may safely navigate, host vehicle  1701  may expect that a merge in front of target vehicle  1703  would result in an aggressively navigating vehicle directly behind the host vehicle. Further, target vehicle  1703  may be expected, based on the observed behavior through image analysis or other sensor output, that target vehicle  1703  would continue accelerating toward host vehicle  1701  or continuing toward host vehicle  1701  at a non-zero relative velocity if host vehicle  1701  was to merge in front of vehicle  1703 . Such a situation may be undesirable from a safety perspective and may also result in discomfort to passengers of the host vehicle. For such reasons, host vehicle  1701  may choose to give way to vehicle  1703 , as shown in  FIG. 17B , and merge into the roundabout behind vehicle  1703  and in front of vehicle  1710 , deemed defensive based on analysis of one or more of its navigational characteristics. 
     Returning to  FIG. 18 , at step  1809 , the at least one processing device of the navigation system of the host vehicle may determine, based on the identified at least one navigational characteristic of the target vehicle relative to the identified obstacle, a navigational action for the host vehicle (e.g., merge in front of vehicle  1710  and behind vehicle  1703 ). To implement the navigational action (at step  1811 ), the at least one processing device may cause at least one adjustment of a navigational actuator of the host vehicle in response to the determined navigational action. For example, a brake may be applied in order to give way to vehicle  1703  in  FIG. 17A , and an accelerator may be applied along with steering of the wheels of the host vehicle in order to cause the host vehicle to enter the roundabout behind vehicle  1703 , as shown if  FIG. 17B . 
     As described in the examples above, navigation of the host vehicle may be based on the navigational characteristics of a target vehicle relative to another vehicle or object. Additionally, navigation of the host vehicle may be based on navigational characteristics of the target vehicle alone without a particular reference to another vehicle or object. For example, at step  1807  of  FIG. 18 , analysis of a plurality of images captured from an environment of a host vehicle may enable determination of at least one navigational characteristic of an identified target vehicle indicative of a level of aggression associated with the target vehicle. The navigational characteristic may include a velocity, acceleration, etc. that need not be referenced with respect to another object or target vehicle in order to make an aggressive/defensive determination. For example, observed accelerations and/or velocities associated with a target vehicle that exceed a predetermined threshold or fall within or exceed a range of values may indicate aggressive behavior. Conversely, observed accelerations and/or velocities associated with a target vehicle that fall below a predetermined threshold or fall within or exceed a range of values may indicate defensive behavior. 
     Of course, in some instances the observed navigational characteristic (e.g., a location, distance, acceleration, etc.) may be referenced relative to the host vehicle in order to make the aggressive/defensive determination. For example, an observed navigational characteristic of the target vehicle indicative of a level of aggression associated with the target vehicle may include an increase in relative acceleration between the target vehicle and the host vehicle, a following distance of the target vehicle behind the host vehicle, a relative velocity between the target vehicle and the host vehicle, etc. 
     Navigation Based on Accident Liability Constraint 
     As described in the sections above, planned navigational actions may be tested against predetermined constraints to ensure compliance with certain rules. In some embodiments, this concept may be extended to considerations of potential accident liability. As discussed below, a primary goal of autonomous navigation is safety. As absolute safety may be impossible (e.g., at least because a particular host vehicle under autonomous control cannot control the other vehicles in its surroundings--it can only control its own actions), the use of potential accident liability as a consideration in autonomous navigation and, indeed, as a constraint to planned actions may help ensure that a particular autonomous vehicle does not take any actions that are deemed unsafe—e.g., those for which potential accident liability may attach to the host vehicle. If the host vehicle takes only actions that are safe and that are determined not to result in an accident of the host vehicle&#39;s own fault or responsibility, then desired levels of accident avoidance (e.g., fewer than 10 −9  per hour of driving) may be achieved. 
     The challenges posed by most current approaches to autonomous driving include a lack of safety guarantees (or at least an inability to provide desired levels of safety), and also a lack of scalability. Consider the issue of guaranteeing multi-agent safe driving. As society will unlikely tolerate road accident fatalities caused by machines, an acceptable level of safety is paramount to the acceptance of autonomous vehicles. While a goal may be to provide zero accidents, this may be impossible since multiple agents are typically involved in an accident and one may envision situations where an accident occurs solely due to the blame of other agents. For example, as shown in  FIG. 19 , host vehicle  1901  drives on a multi-lane highway, and while host vehicle  1901  can control its own actions relative to the target vehicles  1903 ,  1905 ,  1907 , and  1909 , it cannot control the actions of the target vehicles surrounding it. As a result, host vehicle  1901  may be unable to avoid an accident with at least one of the target vehicles should vehicle  1905 , for example, suddenly cut in to the host vehicle&#39;s lane on a collision course with the host vehicle. To address this difficulty, a typical response of autonomous vehicle practitioners is to resort to a statistical data-driven approach where safety validation becomes tighter as data over more mileage is collected. 
     To appreciate the problematic nature of a data-driven approach to safety, however, consider first that the probability of a fatality caused by an accident per one hour of (human) driving is known to be 10 −6 . It is reasonable to assume that for society to accept machines to replace humans in the task of driving, the fatality rate should be reduced by three orders of magnitude, namely to a probability of 10 −9  per hour. This estimate is similar to the assumed fatality rate of air bags and from aviation standards. For example, 10 −9  is the probability that a wing will spontaneously detach from an aircraft in mid-air. Attempts to guarantee safety using a data-driven statistical approach that provides additional confidence with accumulating miles driven, however, is not practical. The amount of data required to guarantee a probability of 10 −9  fatality per hour of driving is proportional to its inverse (i.e., 10 9  hours of data) which is roughly on the order of thirty billion miles. Moreover, a multi-agent system interacts with its environment and likely cannot be validated offline (unless a realistic simulator emulating real human driving with all its richness and complexities such as reckless driving is available--but the problem of validating the simulator would be even more difficult than creating a safe autonomous vehicle agent). And any change to the software of planning and control will require a new data collection of the same magnitude, which is clearly unwieldy and impractical. Further, developing a system through data invariably suffers from lack of interpretability and explainability of the actions being taken—if an autonomous vehicle (AV) has an accident resulting in a fatality, we need to know the reason. Consequently, a model-based approach to safety is required, but the existing “functional safety” and ASIL requirements in the automotive industry are not designed to cope with multi-agent environments. 
     A second primary challenge in developing a safe driving model for autonomous vehicles is the need for scalability. The premise underlying AV goes beyond “building a better world” and instead is based on the premise that mobility without a driver can be sustained at a lower cost than with a driver. This premise is invariably coupled with the notion of scalability—in the sense of supporting mass production of AVs (in the millions) and more importantly of supporting a negligible incremental cost to enable driving in a new city. Therefore the cost of computing and sensing does matter, if AV is to be mass manufactured, the cost of validation and the ability to drive “everywhere” rather than in a select few cities is also a necessary requirement to sustain a business. 
     The issue with most current approaches lies in a “brute force” state of mind along three axes: (i) the required “computing density,” (ii) the way high-definition maps are defined and created, and (iii) the required specification from sensors. A brute-force approach goes against scalability and shifts the weight towards a future in which unlimited on-board computing is ubiquitous, where the cost of building and maintaining HD-maps becomes negligible and scalable, and exotic super advanced sensors would be developed, productized to automotive grade, and at a negligible cost. A future for which any of the above comes to fruition is indeed plausible but having all of the above hold is likely a low-probability event. Thus, there is a need to provide a formal model that pieces together safety and scalability into an AV program that society can accept and is scalable in the sense of supporting millions of cars driving anywhere in the developed countries. 
     The disclosed embodiments represent a solution that may provide the target levels of safety (or may even surpass safety targets) and may also be scalable to systems including millions of autonomous vehicles (or more). On the safety front, a model called “Responsibility Sensitive Safety” (RSS) is introduced that formalizes the notion of “accident blame,” is interpretable and explainable, and incorporates a sense of “responsibility” into the actions of a robotic agent. The definition of RSS is agnostic to the manner in which it is implemented—which is a key feature to facilitate a goal of creating a convincing global safety model. RSS is motivated by the observation (as in  FIG. 19 ) that agents play a non-symmetrical role in an accident where typically only one of the agents is responsible for the accident and therefore is to be responsible for it. The RSS model also includes a formal treatment of “cautious driving” under limited sensing conditions where not all agents are always visible (due to occlusions, for example). One primary goal of the RSS model is to guarantee that an agent will never make an accident of its “blame” or for which it is responsible. A model may be useful only if it comes with an efficient policy (e.g., a function that maps the “sensing state” to an action) that complies with RSS. For example, an action that appears innocent at the current moment might lead to a catastrophic event in the far future (“butterfly effect”). RSS may be useful for constructing a set of local constraints on the short-term future that may guarantee (or at least virtually guarantee) that no accidents will happen in the future as a result of the host vehicle&#39;s actions. 
     Another contribution evolves around the introduction of a “semantic” language that consists of units, measurements, and action space, and specification as to how they are incorporated into planning, sensing and actuation of the AV. To get a sense of semantics, in this context, consider how a human taking driving lessons is instructed to think about a “driving policy.” These instructions are not geometric—they do not take the form “drive 13.7 meters at the current speed and then accelerate at a rate of 0.8 m/s 2 ”. Instead, the instructions are of a semantic nature—“follow the car in front of you” or “overtake that car on your left.” The typical language of human driving policy is about longitudinal and lateral goals rather than through geometric units of acceleration vectors. A formal semantic language may be useful on multiple fronts connected to the computational complexity of planning that do not scale up exponentially with time and number of agents, to the manner in which safety and comfort interact, to the way the computation of sensing is defined and the specification of sensor modalities and how they interact in a fusion methodology. A fusion methodology (based on the semantic language) may ensure that the RSS model achieves the required 10 −9  probability of fatality, per one hour of driving, all while performing only offline validation over a dataset of the order of 10 5  hours of driving data. 
     For example, in a reinforcement learning setting, a Q function (e.g., a function evaluating the long term quality of performing an action a∈A when the agent is at state s∈S; given such a Q-function, a natural choice of an action may be to pick the one with highest quality, π(s)=argmax a  Q(s, a)) may be defined over a semantic space in which the number of trajectories to be inspected at any given time is bounded by 10 4  regardless of the time horizon used for planning. The signal to noise ratio in this space may be high, allowing for effective machine learning approaches to succeed in modeling the Q function. In the case of computation of sensing, semantics may allow for distinguishing between mistakes that affect safety versus those mistakes that affect the comfort of driving. We define a PAC model (Probably Approximate Correct (PAC)), borrowing Valiants PAC-learning terminology) for sensing which is tied to the Q-function and show how measurement mistakes are incorporated into planning in a manner that complies with RSS yet allows for optimization of the comfort of driving. The language of semantics may be important for the success of certain aspects of this model as other standard measures of error, such as error with respect to a global coordinate system, may not comply with the PAC sensing model. In addition, the semantic language may be an important enabler for defining HD-maps that can be constructed using low-bandwidth sensing data and thus be constructed through crowd-sourcing and support scalability. 
     To summarize, the disclosed embodiments may include a formal model that covers important ingredients of an AV: sense, plan, and act. The model may help ensure that from a planning perspective there will be no accident of the AV&#39;s own responsibility. And also through a PAC-sensing model, even with sensing errors, the described fusion methodology may require only offline data collection of a very reasonable magnitude to comply with the described safety model. Furthermore, the model may tie together safety and scalability through the language of semantics, thereby providing a complete methodology for a safe and scalable AV. Finally, it is worth noting that developing an accepted safety model that would be adopted by the industry and regulatory bodies may be a necessary condition for the success of AV. 
     The RSS model may generally follow a classic sense-plan-act robotic control methodology. The sensing system may be responsible for understanding a present state of the environment of a host vehicle. The planning part, which may be referred to as a “driving policy” and which may be implemented by a set of hard-coded instructions, through a trained system (e.g., a neural network), or a combination, may be responsible for determining what is the best next move in view of available options for accomplishing a driving goal (e.g., how to move from the left lane to a right lane in order to exit a highway). The acting portion is responsible for implementing the plan (e.g., the system of actuators and one or more controllers for steering, accelerating, and/or braking, etc. a vehicle in order to implement a selected navigational action). The described embodiments below focus primarily on the sensing and planning parts. 
     Accidents may stem from sensing errors or planning errors. Planning is a multi-agent endeavor, as there are other road users (humans and machines) that react to actions of an AV. The described RSS model is designed to address safety for the planning part, among others. This may be referred to as multi-agent safety. In a statistical approach, estimation of the probability of planning errors may be done “online.” Namely, after every update of the software, billions of miles must be driven with the new version to provide an acceptable level of estimation of the frequency of planning errors. This is clearly infeasible. As an alternative, the RSS model may provide a 100% guarantee (or virtually 100% guarantee) that the planning module will not make mistakes of the AV&#39;s blame (the notion of “blame” is formally defined). The RSS model may also provide an efficient means for its validation not reliant upon online testing. 
     Errors in a sensing system may be easier to validate, because sensing can be independent of the vehicle actions, and therefore we can validate the probability of a severe sensing error using “offline” data. But, even collecting offline data of more than  10   9 hours of driving is challenging. As part of the description of a disclosed sensing system, a fusion approach is described that can be validated using a significantly smaller amount of data.    
     The described RSS system may also be scalable to millions of cars. For example, the described semantic driving policy and applied safety constraints may be consistent with sensing and mapping requirements that can scale to millions of cars even in today&#39;s technology. 
     for a foundational building block of such a system is a thorough safety definition, that is, a minimal standard to which AV systems may need to abide. In the following technical lemma, a statistical approach to validation of an AV system is shown to be infeasible, even for validating a simple claim such as “the system makes N accidents per hour”. This implies that a model-based safety definition is the only feasible tool for validating an AV system. Lemma 1 Let X be a probability space, and A be an event for which Pr(A)=p 1 &lt;0.1. Assume we sample 
     
       
         
           
             m 
             = 
             
               1 
               
                 p 
                 1 
               
             
           
         
       
     
     i.i.d. samples from A, and let Z=Σ i=1   m  1 [x∈A] . Then 
         Pr ( Z= 0)≥ e   −2 .
 
     Proof We use the inequality 1−x≥e −2x  (proven for completeness in Appendix A.1), to get 
         Pr ( Z= 0)=(1 −p   1 ) m   ≥e   −2p     1     m   =e   −2 . 
     Corollary  1  Assume an AV system AV 1  makes an accident with small yet insufficient probability p 1 . Any deterministic validation procedure which is given 1/p 1  samples, will, with constant probability, not distinguish between AV 1  and a different AV system AV 0  which never makes accidents. 
     To gain perspective over the typical values for such probabilities, assume we desire an accident probability of 10 −9  per hour, and a certain AV system provides only 10 −8  probability. Even if the system obtains 10 8  hours of driving, there is constant probability that the validation process will not be able to indicate that the system is dangerous. 
     Finally, note that this difficulty is for invalidating a single, specific, dangerous AV system. A full solution cannot be a viewed as a single system, as new versions, bug fixes, and updates will be necessary. Each change, even of a single line of code, generates a new system from a validator&#39;s perspective. Thus, a solution which is validated statistically, must do so online, over new samples after every small fix or change, to account for the shift in the distribution of states observed and arrived-at by the new system. Repeatedly and systematically obtaining such a huge number of samples (and even then, with constant probability, failing to validate the system), is infeasible. 
     Further, any statistical claim must be formalized to be measured. Claiming a statistical property over the number of accidents a system makes is significantly weaker than claiming “it drives in a safe manner.” In order to say that, one must formally define what is safety. 
     Absolute Safety is Impossible 
     An action a taken by a car c may be deemed absolutely safe if no accident can follow the action at some future time. It can be seen that it is impossible to achieve absolute safety, by observing simple driving scenarios, for example, as depicted in  FIG. 19 . From the perspective of vehicle  1901 , no action can ensure that none of the surrounding cars will crash into it. Solving this problem by forbidding the autonomous car from being in such situations is also impossible. As every highway with more than two lanes will lead to it at some point, forbidding this scenario amounts to a requirement to remain in the garage. The implications might seem, at first glance, disappointing. Nothing is absolutely safe. However, such a requirement for absolute safety, as defined above, may be too harsh, as evident by the fact that human drivers do not adhere to a requirement for absolute safety. Instead, humans behave according to a safety notion that depends on responsibility. 
     Responsibility-Sensitive Safety 
     An important aspect missing from the absolute safety concept is the non-symmetry of most accidents—it is usually one of the drivers who is responsible for a crash, and is to be blamed. In the example of  FIG. 19 , the central car  1901  is not to be blamed if the left car  1909 , for example, suddenly drives into it. To formalize the fact that considering its lack of responsibility, a behavior of AV  1901  staying in its own lane can be considered safe. To do so, a formal concept of “accident blame” or accident responsibility, which can serve as the premise for a safe driving approach, is described. 
     As an example, consider the simple case of two cars c f , c r , driving at the same speed, one behind the other, along a straight road. Assume c f , the car at the front, suddenly brakes because of an obstacle appearing on the road, and manages to avoid it. Unfortunately, c r  did not keep enough of a distance from c f , is not able to respond in time, and crashes into c f &#39;s rear side. It is clear that the blame is on c r ; it is the responsibility of the rear car to keep safe distance from the front car, and to be ready for unexpected, yet reasonable, braking. 
     Next, consider a much broader family of scenarios: driving in a multi-lane road, where cars can freely change lanes, cut into other cars&#39; paths, drive at different speeds, and so on. To simplify the following discussion, assume a straight road on a planar surface, where the lateral, longitudinal axes are the x, y axes, respectively. This can be achieved, under mild conditions, by defining a homomorphism between the actual curved road and a straight road. Additionally, consider a discrete time space. Definitions may aid in distinction between two intuitively different sets of cases: simple ones, where no significant lateral maneuver is performed, and more complex ones, involving lateral movement. 
     Definition 1 (Car Corridor) The corridor of a car c is the range [c x,left ,c x,right ]×[±∞], where c x,left ,c x,right  are the positions of the leftmost, rightmost corners of c. 
     Definition 2 (Cut-in) A car c 1  (car  2003  in  FIGS. 20A and 20B ) cuts-in to car c 0 &#39;s (car  2001  in  FIGS. 20A and 20B ) corridor at time t if it did not intersect c 0 &#39;s corridor at time t−1, and does intersect it at time t. 
     A furtherdistinction may be made between front/back parts of the corridor. The term “the direction of a cut-in” may describe movement in the direction of the relevant corridor boundary. These definitions may define cases with lateral movement. For the simple case where there is no such occurrence, such as the simple case of a car following another, the safe longitudinal distance is defined: 
     Definition 3 (Safe longitudinal distance) A longitudinal distance  2101  ( FIG. 21 ) between a car c r  (car  2103 ) and another car c f  (car  2105 ) that is in c r &#39;s frontal corridor is safe w. r. t. a response time p if for any braking command a, |a|&lt;a max,brake , performed by c f , if c r  will apply its maximal brake from time p until a full stop then it won&#39;t collide with c f . 
     Lemma 2 below calculates d as a function of the velocities of c r , c f , the response time ρ, and the maximal acceleration a max,brake . Both ρ and a max,brake  are constants, which should be determined to some reasonable values by regulation. 
     Lemma 2 Let c r  be a vehicle which is behind c f  on the longitudinal axis. Let a max,brake , a max,accel  be the maximal braking and acceleration commands, and let ρ be c r &#39;s response time. Let ν r , ν f  be the longitudinal velocities of the cars, and let l f , l r  be their lengths. Define ν p,max =ν r +ρ·a max,accel , and define 
     
       
         
           
             
               T 
               r 
             
             = 
             
               
                 p 
                 + 
                 
                   
                     
                       v 
                       
                         p 
                         , 
                         max 
                       
                     
                     
                       a 
                       
                         max 
                         , 
                         brake 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     T 
                     f 
                   
                 
               
               = 
               
                 
                   
                     
                       
                         v 
                         f 
                       
                       
                         a 
                         
                           max 
                           , 
                           brake 
                         
                       
                     
                     . 
                     
                         
                     
                     ⁢ 
                     Let 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   L 
                 
                 = 
                 
                   
                     ( 
                     
                       
                         l 
                         r 
                       
                       + 
                       
                         l 
                         f 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     / 
                   
                   ⁢ 
                   2. 
                 
               
             
           
         
       
     
     Then, the minimal safe longitudinal distance for c r  is: 
     
       
         
           
             
               d 
               min 
             
             = 
             
               { 
               
                 
                   
                     
                       L 
                       ⁢ 
                       
                           
                       
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           T 
                           r 
                         
                       
                       ≤ 
                       
                         T 
                         f 
                       
                     
                   
                 
                 
                   
                     
                       L 
                       + 
                       
                         
                           T 
                           f 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   v 
                                   
                                     p 
                                     , 
                                     max 
                                   
                                 
                                 - 
                                 
                                   v 
                                   f 
                                 
                               
                               ) 
                             
                             + 
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 a 
                                 
                                   max 
                                   , 
                                   brake 
                                 
                               
                             
                           
                           ] 
                         
                       
                       - 
                       
                         
                           
                             ρ 
                             2 
                           
                           ⁢ 
                           
                             a 
                             
                               max 
                               , 
                               brake 
                             
                           
                         
                         2 
                       
                       + 
                       
                         
                           
                             ( 
                             
                               
                                 T 
                                 r 
                               
                               - 
                               
                                 T 
                                 f 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 v 
                                 
                                   p 
                                   , 
                                   max 
                                 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       T 
                                       f 
                                     
                                     - 
                                     ρ 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   a 
                                   
                                     max 
                                     , 
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                             ) 
                           
                         
                         2 
                       
                     
                   
                   
                     
                       otherwise 
                       ⁢ 
                       
                           
                       
                     
                   
                 
               
             
           
         
       
     
     Proof Let d t  be the distance at time t. To prevent an accident, we must have that d t &gt;L for every t. To construct d min  we need to find the tightest needed lower bound on d 0 . Clearly, d 0  must be at least L. As long as the two cars didn&#39;t stop after T≥ρ seconds, the velocity of the preceding car will be ν f −T a max,brake  while c r &#39;s velocity will be upper bounded by σ ρ,max −(T−ρ)a max,accel . So, the distance between the cars after T seconds will be lower bounded by: 
     
       
         
           
             
               
                 d 
                 T 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 := 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 d 
                 0 
               
             
             + 
             
               
                 T 
                 2 
               
               ⁢ 
               
                 ( 
                 
                   
                     2 
                     ⁢ 
                     
                       v 
                       f 
                     
                   
                   - 
                   
                     T 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       a 
                       
                         max 
                         , 
                         brake 
                       
                     
                   
                 
                 ) 
               
             
             - 
             
                 
               
                 
                   [ 
                   
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         v 
                         
                           p 
                           , 
                           max 
                         
                       
                     
                     + 
                     
                       
                         
                           T 
                           - 
                           ρ 
                         
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               v 
                               
                                 p 
                                 , 
                                 max 
                               
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 T 
                                 - 
                                 ρ 
                               
                               ) 
                             
                             ⁢ 
                             
                               a 
                               
                                 max 
                                 , 
                                 brake 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     d 
                     0 
                   
                   + 
                   
                     T 
                     ⁡ 
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               v 
                               f 
                             
                             - 
                             
                               v 
                               
                                 ρ 
                                 , 
                                 max 
                               
                             
                           
                           ) 
                         
                         - 
                         
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             a 
                             
                               max 
                               , 
                               brake 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       
                         ρ 
                         2 
                       
                       ⁢ 
                       
                         a 
                         
                           max 
                           , 
                           brake 
                         
                       
                     
                     2 
                   
                 
               
             
           
         
       
     
     Note that T r  is the time on which cr arrives to a full stop (a velocity of 0) and T f  is the time on which the other vehicle arrives to a full stop. Note that a max,brake (T r −T f )=ν p,max −ν f +ρa max,brake , so if T r ≤T f  it suffices to require that d 0 &gt;L. If T r &gt;T f  then 
     
       
         
           
             
               d 
               
                 T 
                 r 
               
             
             = 
             
               
                 d 
                 0 
               
               + 
               
                 
                   T 
                   f 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       ( 
                       
                         
                           v 
                           f 
                         
                         - 
                         
                           v 
                           
                             ρ 
                             , 
                             max 
                           
                         
                       
                       ) 
                     
                     - 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         a 
                         
                           max 
                           , 
                           brake 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   
                     ρ 
                     2 
                   
                   ⁢ 
                   
                     a 
                     
                       max 
                       , 
                       brake 
                     
                   
                 
                 2 
               
               - 
               
                 
                   
                     
                       ( 
                       
                         
                           T 
                           r 
                         
                         - 
                         
                           T 
                           f 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           v 
                           
                             ρ 
                             , 
                             max 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 T 
                                 f 
                               
                               - 
                               ρ 
                             
                             ) 
                           
                           ⁢ 
                           
                             a 
                             
                               max 
                               , 
                               brake 
                             
                           
                         
                       
                       ) 
                     
                   
                   2 
                 
                 . 
               
             
           
         
       
     
     Requiring d Tr &gt;L and rearranging terms concludes the proof. 
     Finally, a comparison operator is defined which allows comparisons with some notion of “margin”: when comparing lengths, velocities and so on, it is necessary to accept very similar quantities as “equal”. 
     Definition 4 (μ-comparison) The μ-comparison of two numbers a, b is a&gt; μ b if a&gt;b+μ, a&lt; μ b if a&lt;b−μ and a= μ b if |a−b|≤μ. 
     The comparisons (argmin, argmax, etc.) below are μ-comparisons for some suitable μs. Assume an accident occurred between cars c 1 , c 2 . To consider who is to blame for the accident, the relevant moment which needs to be examined is defined. This is some point in time which preceded the accident, and intuitively, was the “point of no return”; after it, nothing could be done to prevent the accident. 
     Definition 5 (Blame Time) The Blame Time of an accident is the earliest time preceding the accident in which:
         there was an intersection between one of the cars and the other&#39;s corridor, and   the longitudinal distance was not safe.       

     Clearly there is such a time, since at the moment of accident, both conditions hold. Blame Times may be split into two separate categories:
         Ones in which a cut-in also occurs, namely, they are the first moment of intersection of one car and the other&#39;s corridor, and it&#39;s in a non-safe distance.   Ones in which a cut-in does not occur, namely, there was intersection with the corridor already, in a safe longitudinal distance, and the distance had changed to unsafe at the Blame Time.       

     Definition 6 (μ-Losing by Lateral Velocity) Assume a cut-in occurs between cars c 1 , c 2 . We say that c 1 , μ-Loses by Lateral Velocity in case its lateral velocity w.r.t. the direction of the cut-in is higher by μ than that of c 2 . 
     It should be noted that the direction of the velocity is important: For example, velocities of −1, 1 (both cars crashing into each other) is a tie, however if the velocities are 1, 1+μ/2, the one with positive direction towards the other car is to be blamed. Intuitively, this definition will allow us to blame a car which drives laterally very fast into another. 
     Definition 7 ((μ 1 , μ 2 )—Winning by Lateral Position) Assume a cut-in occurs between cars c 1 , c 2 . We say that c 1  (μ 1 , μ 2 )—Wins by Lateral Position in case its lateral position w.r.t. the cut-in lane&#39;s center (the center closest to the cut-in relevant corridor) is smaller than μ 1  (in absolute value), and smaller by μ 2  than that of c 2 . 
     Intuitively, we will not blame a car if it&#39;s very close to the lane center (μ 1 ), and much closer than the other car (by μ 2 ). 
     Definition 8 (Blame) The Blame or responsibility for an accident between cars c 1 , c 2 , is a function of the state at the Blame Time, and is defined as follows:
         If the Blame Time is not a cut-in time, the blame is on the rear car.   If the Blame Time is also a cut-in time, the blame is on both cars, unless for one of the cars, w.l.o.g. c 1 , the two following conditions hold, for some predefined μs:
           It doesn&#39;t lose by Lateral Velocity,   It wins by Lateral Position.   
               

     In that case, c 1  is spared. In other words, if an unsafe cut-in occurs, both cars are to blame, unless one of the cars is not (significantly) laterally faster, and is (significantly) closer to the lane center. By this, the desired behavior is captured: if following a car, keep a safe distance, and if cutting into a corridor of a car which simply drives in its own lane, do it only at a safe distance. An automated controller-based system for following the safety guidelines described above should not lead to overly defensive driving, as discussed further below. 
     Dealing with Limited Sensing 
     After considering the highway example, a second example next addresses a problem of limited sensing. A very common human response, when blamed for an accident, falls into the “but I couldn&#39;t see him” category. It is, many times, true. Human sensing capabilities are limited, sometimes because of an unaware decision to focus on a different part of the road, sometimes because of carelessness, and sometimes because of physical limitations—it is impossible to see a pedestrian hidden behind a parked car. Of those human limitations, advanced automatic sensing systems may only be subject to the latter: 360° view of the road, along with the fact that computers are never careless, puts them above human sensing capabilities. Returning to the “but I couldn&#39;t see him” example, a fitting answer is “well, you should&#39;ve been more careful.” To formalize what is being careful with respect to limited sensing, consider the scenario, depicted in  FIG. 22 . Car  2201  (c 0 ) is trying to exit a parking lot, merging into a (possibly) busy road, but cannot see whether there are cars in the street because its view is obscured by building  2203 . Assume that this is an urban, narrow street, with a speed limit of 30 km/h. A human driver&#39;s behavior is to slowly merge onto the road, obtaining more and more field of view, until sensing limitations are eliminated. A significant moment in time should be defined—the first time the occluded object is exposed to us; after its exposure, one deal with it just like any other object that one can sense. 
     Definition 9 (Exposure Time) The Exposure Time of an object is the first time in which we see it. 
     Definition 10 (Blame due to Unreasonable Speed) Assume that at the exposure time or after it, car c 1  (car  2205 ) was driving at speed ν&gt;ν limit , and co wasn&#39;t doing so. Then, the blame is only on c 1 . We say that c 1  is blamed due to unreasonable speed. 
     This extension allows co to exit the parking lot safely. Using our previous responsibility-sensitive safety definitions, along with a dynamic ν limit  definition (which uses the road conditions and speed limit, plus reasonable margins), the only necessity is to check whether in the worst case, as illustrated in the figure, the cut-in is in a safe longitudinal distance, while assuming that c 1  will not exceed ν limit . Intuitively, this encourages c 0  to drive slower and further from the occluder, thus slowly increasing its field of view and later allowing for safe merging into the street. 
     Having extended the accident responsibility definition to this basic case of limited sensing, a family of extensions may address similar cases. Simple assumptions as to what can be occluded (a potentially fast car cannot be occluded between two closely parked cars, but a pedestrian can), and what is the worst case maneuver it can perform (a pedestrian&#39;s ν limit  is much smaller than that of a car), imply restrictions on driving—one must be prepared for the worst, and have the ability to respond if suddenly, the exposure time comes. A more elaborate example, in an urban scenario, can be taken from the scenario of a pedestrian which is possibly occluded by a parked car. Accident blame for accidents with a pedestrian may be defined: 
     Definition 11 (Accident-with-Pedestrian Blame) The Accident-with-Pedestrian Blame is always on the car, unless one of the following three holds:
         the car hits the pedestrian with the car&#39;s side, and the lateral velocity of the car is smaller than μ, w.r.t. the direction of the hit.   the pedestrian&#39;s velocity at the exposure time or later was larger than ν limit .   the car is at complete stop.       

     Informally, the car is not to blame only if a pedestrian runs into its side, while the car does not ride faster than μ into the pedestrian, or if the car is at stop, or if the pedestrian was running super-humanly fast, in some direction, not necessarily the hitting direction. 
     While the described system may not ensure absolute safety, it may lead to a scenario where very few (if any) accidents occur among autonomous vehicles. For example, where all cars (and other road users) are able to successfully verify that they will not be blamed for an accident as a result of an action taken, accidents may be eliminated. By definition, for every accident there is at least one responsible car. Thus, if no car takes an action for which it may be responsible for a resulting accident (according to the RSS model described above), there should never be any accidents, leading to the type of absolute safety or near absolute safety sought by unwieldy and impractical statistical methods. 
     Not all roads are of a simple structure. Some, like junctions and roundabouts, contain more complex situations, along with various right of way rules. Not all occluded objects are cars or pedestrians, with bicycles and motorcycles all legitimate road users to be considered. 
     The principles introduced in this section may be extended to these additional cases. 
     Efficiently Validated Conditions for Responsibility-Sensitive Safety 
     This section discusses implementation aspects of RSS. To begin, it should be noted that an action that is performed now may have a butterfly effect that will lead to a chain of events with an accident after 10 minutes of driving, for example. A “brute-force” approach of checking all possible future outcomes not only impractical, it is likely impossible. To overcome this challenge, the responsibility-sensitive safety definitions described above are now described together with computationally efficient methods to validate them. 
     Computationally Feasible Safety Verification. 
     The main mathematical tool for computationally feasible verification is “induction.” To prove a claim by induction, one begins with proving the claim for simple cases, and then, each induction step extends the proof to more and more involved cases. To illustrate how this induction tool can be helpful for safety verification, consider again the simple example of a car c r  following another, c f  ( FIG. 21 ). The following constraint may be applied on the policy of c r . At each time step t, the policy can pick any acceleration command such that even if c f  will apply a deceleration of −a max , the resulting distance between cr and c f  at the next time step will be at least the safe longitudinal distance (defined in Definition 3 and Lemma 2). If no such action exists, c r  must apply the deceleration −a max . The following lemma uses induction to prove that any policy that adheres to the above constraints will never make an accident with c f . 
     Lemma 3 Under the assumptions given in Definition 3, if the policy of c r  adheres to the constraints given above it will never make an accident with c f . 
     Proof The proof is by induction. For the induction base, start with an initial state in which the distance between the two cars is safe (according to Lemma 2). The induction step is as follows. Consider the distance between c r  and c f  at some time t. If there is an action that results in a safe distance (even with c f  making maximal deceleration), we are fine. If all actions cannot guarantee safe distance, let t′&lt;t be the maximal time in which we took an action which was not maximal deceleration. By the induction hypothesis, at time t′+1, we were in a safe distance, and from there on we performed maximal deceleration. Hence, by the definition of safe distance, there was no crash from time t′ till now, which concludes the proof 
     The above example demonstrates a more general idea: there&#39;s some emergency maneuver which can be performed by c r  in an extreme case, and lead it back to a “safe state.” It should be noted that the constraints on the policy we have described above depend on just one future time step, hence it can be verified in a computationally efficient manner. 
     To generalize those ideas of sufficient local properties for RSS, we firstly define a Default Emergency Policy (DEP), and use it as a building block for defining a local property of action-taking which we call “cautious”. It is then shown that taking only cautious commands is sufficient for RSS. 
     Definition 12 (Default Emergency Policy) The Default Emergency Policy (DEP) is to apply maximum braking power, and maximum heading change towards 0 heading w.r.t. the lane. The maximum braking power and heading change are derived from physical parameters of the car (and maybe also from weather and road conditions). Definition 13 (Safe state) A state s is safe if performing DEP starting from it will not lead to an accident of our blame. As in the simple case of a car following another, we define a command to be cautious if it leads to a safe state. Definition 14 (Cautious command) Suppose we are currently at state so. A command a is cautious if the next state, s 1 , will be safe with respect to a set A of possible commands that other vehicles might perform now. The above definition depends on the worst-case commands, in the set A, other vehicles might perform. We will construct the set A based on reasonable upper bounds on maximum braking/acceleration and lateral movements. 
     The following theorem proves, by induction again, that if we only issue cautious commands then there will be no accidents of our blame. Theorem 1 Assume that in time  0 , c is in a safe state, and for every time step, c only issues cautious commands, where if no cautious command exists at some time step, c applies DEP. Then, c will never make accidents of its blame. Proof By induction. The base of the induction follows from the definition of a safe state and step from the definition of a cautious command. 
     One benefit of this approach is that there may be no need to check infinite future, as we can quickly return to a safe state, and continue safely from there. Moreover, given the fact we will plan again at t+1, and hence be able to perform DEP then if necessary, we should only check the command we are giving at time t, and not a possible longer plan we might have in mind—we can change that plan at t+1. Now, incorporating a learning component in a system, when it is verified at run time by this transparent model, is made possible. Finally, this local verification implies full future RSS, which is our desired goal. An implementation obstacle is that the cautiousness definition involves all trajectories another agent can perform until t brake , which, even for moderate t brake , is a huge space. To tackle this, we next turn to develop an efficiently computable way to verify cautiousness, and hence RSS, in a scalable manner. 
     Efficient Cautiousness Verification 
     A first observation is that a state is not safe if and only if there exists a specific vehicle, c, which can perform commands from the set A which lead to an accident of our blame while we execute the DEP. Therefore, in a scene with a single target car, denoted {tilde over (c)}, and in the general case, the procedure may be executed sequentially, for each of the other vehicles in the scene. 
     When considering a single target car, an action a is not cautious if and only if there is a sequence of commands for {tilde over (c)}, denoted ã 1 , . . . , ãt brake , all in the set A, that results in an accident of c&#39;s blame. As already proven, if at time  0 , it holds that {tilde over (c)} is in the frontal corridor of c, there is a simple way to check the cautiousness of a—we just need to verify that even if {tilde over (c)} will apply maximal brake for one time step (and we&#39;ll perform a), the resulting longitudinal distance will remain safe. The lemma below gives a sufficient condition for cautiousness in the more involved cases, where lateral maneuvers are to be considered too. 
     Lemma 4 Assume that at time T=0, {tilde over (c)} is not in the frontal corridor of c. Then, if at every T∈(0, t brake ], there is no non-safe cut in of c&#39;s blame, then a is cautious. 
     Proof Suppose that a is not cautious, namely, there exists ã 1 , . . . , ãt brake  that leads to an accident of c&#39;s blame. Before the accident there must be a cut-in time, T. Assume first that T&gt;0. If this cut-in was at a safe longitudinal distance, then there cannot be an accident of our blame due to the fact that DEP is executed by deceleration of −a max  and based on the definition of safe longitudinal distance (and we assume here that the response time ρ is larger than the time resolution of steps). If the cut-in was non-safe, by the assumption of the lemma it was not of c&#39;s blame, hence the accident is also not of c&#39;s blame. 
     Finally, if T&lt;0, by the assumption of the lemma, at the moment of cutting, {tilde over (c)} was at the back corridor of c. By induction, c performed only safe cut-ins in the past, and hence either the cut-in was safe or it was c&#39;s blame. In both cases, the current accident is not of c&#39;s blame. 
     In light of Lemma 4, the problem of checking whether there can be a non-safe cut in of c&#39;s blame remains. An efficient algorithm is presented below for checking the possibility of a non-safe cut-in at time t. To validate the entire trajectory we discretize the time interval [0, t brake ] and apply the algorithm on all time intervals (with a slightly larger value of p in the definition of safe distance to ensure that the discretization doesn&#39;t hurt). Let {tilde over (c)} diag  be the length of a diagonal of a minimal rectangle bounding {tilde over (c)}. For each time t∈[0, t brake ], define c length (t) to be the longitudinal “span” of c in time t, and let 
     
       
         
           
             
               L 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       c 
                       ~ 
                     
                     diag 
                   
                   + 
                   
                     c 
                     
                       length 
                       ⁡ 
                       
                         [ 
                         t 
                         ] 
                       
                     
                   
                 
                 2 
               
               . 
             
           
         
       
     
     Define c width [t] in similar fashion and 
     
       
         
           
             
               W 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       c 
                       ~ 
                     
                     diag 
                   
                   + 
                   
                     c 
                     
                       width 
                       ⁡ 
                       
                         [ 
                         t 
                         ] 
                       
                     
                   
                 
                 2 
               
               . 
             
           
         
       
     
     
       
         
           
               
             
               
                   
               
               
                 Algorithim 1: Check the possibility of a non-safe cut-in at  
               
               
                 time t 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 input: 
               
               
                  {tilde over (y)}[0], {tilde over (υ)} y [0], {tilde over (x)}[0], {tilde over (υ)} x [0] are longitudinal/lateral position/velocity of     at time 0 
               
               
                  y[t], υ y [t], x[t] , υ x [t] are longitudinal/lateral position/velocity of      at time t 
               
               
                  a y,min , a y,max  are longitudinal acceleration boundaries 
               
               
                  a x,max  is a lateral acceleration absolute value boundary 
               
               
                  L = L(t), W = W(t) 
               
               
                 check longitudinal feasibility: 
               
               
                  let {tilde over (y)} min  = {tilde over (y)}[0] + {tilde over (υ)}[0]t + ½ a y,min t 2   
               
               
                  let {tilde over (y)} max  = {tilde over (y)}[0] + {tilde over (υ)}[0]t + ½ a y,max t 2   
               
               
                  if [{tilde over (y)} min , {tilde over (y)} max ] ∩ [y[t] − L, y[t] + L ] ≠ θ 
               
               
                   continue to check lateral feasibility 
               
               
                  else if {tilde over (y)} min  &gt; y[t] + L and ({tilde over (y)} min , {tilde over (υ)} y [0] +a y,min t) is not longitudinally safe w.r.t (y[t], υ y [t])  
               
               
                   continue to check lateral feasibility 
               
               
                  else if {tilde over (y)} max  &lt; y[t] − L and ({tilde over (y)} max , {tilde over (υ)} y [0] +a y,maz t) is not longitudinally safe w.r.t (y[t], υ y [t])  
               
               
                   continue to check lateral feasibility 
               
               
                  return “non-feasible” 
               
               
                 check lateral feasibility: 
               
               
                  w.l.o.g. assume x[t] = 0 and {tilde over (x)}[0] ≤ 0, and {tilde over (υ)} x [0] ≥ 0. 
               
               
                  if a positionx[t] is not ocnsidered (μ 1 , μ 2 )-Winning by Lateral Position w.r.t. a position x[t] − W 
               
               
                   w.l.o.g. assume υ x [t] = −μ, as in Definition 6. 
               
               
                  else 
               
               
                   w.l.o.g. assume υ x [t] = −2μ, as in Definition 6. 
               
               
                  let t top  = 0.5(t − {tilde over (υ)} x [0]/a x,max ) 
               
               
                  if t top  &lt; 0 
               
               
                   return “non-feasible” 
               
               
                  let x max  = {tilde over (x)}[0] + 2({tilde over (υ)}[0]t top  + 0.5a x,max t top   2 ) + {tilde over (υ)} x [0] 2 /(a x,max ) 
               
               
                  if x max  &lt; −W 
               
               
                   return “non-feasible” 
               
               
                  return “feasible” 
               
               
                   
               
               
                     indicates data missing or illegible when filed 
               
            
           
         
       
     
     The following theorem proves the correctness of the above algorithm. 
     Theorem 2 If Algorithm 1 returns “non-feasible” then there cannot be a non-safe cut-in of the ego vehicle&#39;s blame at time t. To prove the theorem, we rely on the following key lemmas, which prove the correctness of the two building blocks of Algorithm 1. Start with the longitudinal feasibility: 
     Lemma 5 Under the notation of Algorithm 1, if the check longitudinal feasibility procedure is concluded by returning “non-feasible”, then there cannot be a non-safe cut-in of the ego vehicle&#39;s blame at time t. 
     Proof Ignoring the lateral aspect of a cut-in maneuver, we examine the mere possibility that the longitudinal distance between c and {tilde over (c)} will be unsafe. It is clear that the positions ({tilde over (y)} min , {tilde over (y)} max ) are bounding the position which can be attained by {tilde over (c)} at time t. By the fact [{tilde over (y)} min , {tilde over (y)} max ]∩[y[t]−L, y[t]+L]=Ø, we obtain that any longitudinally non-safe distance which is attainable, is ≥L. Assume {tilde over (y)} min &gt;y[t]+L, and assume by contradiction that a non-safe longitudinal position and velocity, denoted {tilde over (y)} bad  [t], {tilde over (ν)} ybad [t], are attainable using acceleration commands bounded by a y,min , a y,max . By definition of {tilde over (y)} min , we have {tilde over (y)} bad [t]&gt;{tilde over (y)} min , and hence the distance between the cars is larger, namely {tilde over (y)} bad [t]−(y[t]+L)&gt;{tilde over (y)} min −(y[t]+L). Since ({tilde over (y)} min ,{tilde over (ν)} y [0]+a y,min   t ) is longitudinally safe w.r.t. (y[t], ν y [t]), by definition of longitudinal non-safety, it follows that the attained velocity {tilde over (ν)} ybad [t] must be smaller than {tilde over (ν)} y [0]+a y,min t. However, it is clear that in order to achieve lesser speed, {tilde over (c)} must use average acceleration which is lesser than a y,min  throughout the time window [0, t], thus contradicting the fact that the longitudinal non-safety was attained using commands bounded by a y,min , a y,max . By considering a symmetrical argument for the case {tilde over (y)} max &lt;y[t]−L, the proof is completed. 
     Next, the lateral feasibility. 
     Lemma 6 Under the notation of Algorithm 1, if the check lateral feasibility procedure is concluded by returning “non-feasible”, then there cannot be a non-safe cut-in of the ego vehicle&#39;s blame at time t. 
     Proof First, it is clear that there is no loss of generality by the assumptions x[t]=0, {tilde over (x)}[0]≤0 and the ones regarding ν x [t], by a simple change of coordinates and consideration of relative velocity. Moreover, by similar arguments it is simple to extend to the case when {tilde over (ν)} x [0]≤0. 
     Note that the positions of the cars involved in the cut-in, which in our case, are (x[t], x[t]−W), imply some (μ 1 , μ 2 )-Winning by Lateral Position property, affecting the blame. By our assumptions over ν x [t], we obtain that the maximal lateral velocity {tilde over (c)} may use at time t, under assumption that c will be blamed, is 0: either in order to μ—“tie” lateral velocity (in the case c does not (μ 1 , μ 2 )-Win by Lateral Position, this can be enough in order to put the blame on it), or to μ-Win lateral velocity (in the case c does (μ 1 , μ 2 )-Win by Lateral Position, this is necessary in order to put the blame on it). It is left to check whether exists a maneuver starting at {tilde over (ν)} x [0], ending at {tilde over (ν)} x [t]=0, using lateral accelerations bounded by a x,max , with final position {tilde over (x)}[t]≥x[t]−W. In words, a cut-in which ends at the desired lateral velocity, 0. 
     Recall the definition t top =0.5(t−{tilde over (ν)} x [0]/a x,max ) from the algorithm. Assume t top &lt;0. This implies that the time needed by {tilde over (c)} in order to reach lateral velocity 0 when using maximal lateral acceleration, namely, {tilde over (ν)} x [0]/a x,max , is lesser than t. This implies that there is no manoeuvre which it can perform in order to reach the desired velocity in time, and hence no problematic maneuver exists. Therefore, if the procedure returned “non-feasible” due to t top &lt;0, indeed, there is no feasibility of a non-safe cut-in of c&#39;s blame. 
     Consider the case t top &gt;0. Then, the procedure returned “non-feasible” due to x max &lt;−W. Consider a family of lateral velocity profiles for {tilde over (c)} in the time range [0, t], denoted U={u a : a&gt;{tilde over (ν)} x [0]/t} and parameterized by a. We define, for each a, in similar fashion to the one used in the algorithm, t top (a):=0.5(t−{tilde over (ν)} x [0]/a). Note that t top (a)&gt;0 for all a&gt;{tilde over (ν)} x [0]/t. We now define the velocity profile ua for all times t′0 [0, t] as follows: 
     
       
         
           
             
               
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     First, it can be seen that ua satisfies the constraints u d (0)={tilde over (ν)} x [0], u d (t)={tilde over (ν)} x [t]. Second, the distance travelled while using ua can be calculated, as this amounts to integration of a piecewise linear function. Define the arrived-at position as {tilde over (x)} u     a   , and note that x max  defined in the algorithm is precisely 
     
       
         
           
             
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     there exists a value of a for which x={tilde over (x)} u     a   . In particular, for x=x[t]−W, such a value exists, and we denote it by a cut . 
     Observe that since x max , defined in the algorithm, is &lt;x[t]−W, we have that a cut &gt;a x,max . This in not sufficient in order to show no valid maneuver can lead to position ≥x[t]−W; this is implied only for members of the family U. We now prove that even outside of U, all velocity profiles which attain a final position of x[t]−W must use an acceleration value of at least a cut  on the way, making them invalid, and hence completing the proof. 
     Assume some velocity profile u satisfies the boundary constraints u(0)={tilde over (ν)} x [0], u(t)={tilde over (ν)} x [t]. Moreover, assume it attains some final position {tilde over (x)}&gt;x[t]−W. We thus obtain: 
       ∫ 0   t   u (τ) dτ≥∫   0   t   u   a     cut   (τ) dτ.  
 
     Assume u≥u a     cut    for all τ. In particular, u(t top (a cut ))≥u a     cut   (t top (a cut )). From the mean value theorem, there exists ζ0 [0, t top (a cut )] s.t. 
     
       
         
           
             
               
                 
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     implying infeasibility of u, as it uses acceleration (that is, u′, the derivative of the velocity) which exceeds a x,max . 
     Now, assume u≥u a     cut    does not hold for all τ. Then, due to the fact ∫ 0   t u(τ)dτ≥∫ 0   t u a     cut   (τ)dτ, there must be a point where u&gt;u a     cut   . If such point τ large  exists in [0,t top (a cut )], then we can easily use the mean value theorem in the same manner as above, to obtain ζ0[0,τ large ] where too large an acceleration is used. If such point only exists in [t top (a cut ),t], similar argument will give us a point ζ0 [τ large ,t] in which an acceleration value lesser than −a x,max  was used, concluding the proof. Equipped with the above lemmas, Theorem 2&#39;s proof is immediate. 
     Safety Verification—Occlusions 
     In similar fashion to dealing with observed objects, we can define the extension for cautiousness w.r.t. occluded objects, with a similar theorem to Theorem 1, proving that cautiousness implies there are never accidents of our blame. 
     Definition 15 (Cautiousness w.r.t. Occluded Objects) A command given at time t is Cautious w.r.t. Occluded Objects if in the case that the exposure time of the object is t+1, and we command a Default Emergency Policy (DEP) at t+1, there will not be an accident of our blame. 
     Lemma 7 If we only give cautious, w. r. t. occluded objects and non occluded objects, commands, there will never be an accident of our blame. 
     Proof Assume that an accident of our blame occurred at time t, with the exposure time being t′≤t. By cautiousness assumption, the command given at t′−1 allowed us to command a DEP at t′ without being blamed for an accident. As there was an accident of our blame, we apparently did not command a DEP at time t′. But from t′ on, we were safe w.r.t. non occluded objects, hence the command we gave was safe, and there was no accident of our blame. 
     Here too, we provide efficient ways to checking cautiousness with respect to a worst-case assumption over occluded objects, allowing for feasible, scalable, RSS. 
     Driving Policy 
     A driving policy is a mapping from a sensing state (a description of the world around us) into a driving command (e.g., the command is lateral and longitudinal accelerations for the coming second, which determines where and at what speed should the car be in one second from now). The driving command is passed to a controller, which aims at actually moving the car to the desired position/speed. 
     In the previous sections a formal safety model and proposed constraints on the commands issued by the driving policy that guarantee safety were described. The constraints on safety are designed for extreme cases. Typically, we do not want to even need these constraints, and would like to construct a driving policy that leads to a comfortable ride. The focus of this section is on how to build an efficient driving policy, in particular, one that requires computational resources that can scale to millions of cars. For now, this discussion does not address the issue of how to obtain the sensing state and assume an utopic sensing state, that faithfully represents the world around us without any limitations. Later sections discuss the effect of inaccuracies in the sensing state on the driving policy. 
     The problem of defining a driving policy is cast in the language of Reinforcement Learning (RL), as discussed in the sections above. At each iteration of RL, an agent observes a state describing the world, denoted s t , and should pick an action, denoted a t , based on a policy function, π, that maps states into actions. As a result of its action and other factors out of its control (such as the actions of other agents), the state of the world is changed to s t +1. We denote a (state, action) sequence by  s =((s 1 ,a 1 ), (s 2 ,a 2 ), . . . , (s len( s ) ,a len( s ) )). Every policy induces a probability function over (state, action) sequences. This probability function may be affected by the actions taken by the agent, but also depends on the environment (and in particular, on how other agents behave). We denote by P π  the probability over (state, action) sequences induced by π. The quality of a policy is defined to be      s ˜P     π   [ρ( s )], where ρ( s ) is a reward function that measures how good the sequence  s  is. In most case, ρ( s ) takes the form ρ( s )=Σ t=1   len( s )  ρ(s t , a t ), where ρ(s, a) is an instantaneous reward function that measures the immediate quality of being at state s and performing action a. For simplicity, we stick to this simpler case. 
     To cast the driving policy problem in the above RL language, let st be some representation of the road, and the positions, velocities, and accelerations, of the ego vehicle as well as other road users. Let a t  be a lateral and longitudinal acceleration command. The next state, s t+1 , depends on a t  as well as on how the other agents will behave. The instantaneous reward, ρ(s t , a t ), may depend on the relative position/velocities/acceleration to other cars, the difference between our speed and the desired speed, whether we follow the desired route, whether our acceleration is comfortable etc. 
     One challenge in deciding what action should the policy take at time t stems from the fact that one needs to estimate the long term effect of this action on the reward. For example, in the context of driving policy, an action that is taken at time t may seem a good action for the present (that is, the reward value ρ(s t , a t ) is good), but might lead to an accident after 5 seconds (that is, the reward value in 5 seconds would be catastrophic). We therefore need to estimate the long term quality of performing an action a when the agent is at state s. This is often called the Q-function, namely, Q(s, a) should reflect the long term quality of performing action a at time s. Given such a Q-function, the natural choice of an action is to pick the one with highest quality, π(s)=argmax a  Q(s, a). 
     The immediate questions are how to define Q and how to evaluate Q efficiently. Let us first make the (completely non-realistic) simplifying assumption that s t +1 is some deterministic function of (s t , a t ), namely, s t +1=f(s t , a t ). One familiar with Markov Decision Processes (MDPs), will notice that this assumption is even stronger than the Markovian assumption of MDPs (i.e., that s t +1 is conditionally independent of the past given (s t , a t )). As noted in [5], even the Markovian assumption is not adequate for multi-agent scenarios, such as driving, and we will therefore later relax the assumption. 
     Under this simplifying assumption, given st, for every sequence of decisions for T steps, (a t , . . . , a t + T ), we can calculate exactly the future states (s t +1, . . . , s t + T +1) as well as the reward values for times t, . . . , T. Summarizing all these reward values into a single number, e.g. by taking their sum Σ τ=t   T ρ(s rτ a τ ), we can define Q(s, a) as follows: 
     
       
         
           
             
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     That is, Q(s, a) is the best future we can hope for, if we are currently at state s and immediately perform action a. 
     Let us discuss how Q may be calculated. The first idea is to discretize the set of possible actions, A, into a finite set Â, and simply traverse all action sequences in the discretized set. Then, the runtime is dominated by the number of discrete action sequences, |Â| T . If Â represents 10 lateral accelerations and 10 longitudinal accelerations, we obtain 100 T  possibilities, which becomes infeasible even for small values of T. While there are heuristics for speeding up the search (e.g. coarse-to-fine search), this brute-force approach requires tremendous computational power. 
     The parameter Tis often called the “time horizon of planning”, and it controls a natural tradeoff between computation time and quality of evaluation—the larger T is, the better our evaluation of the current action (since we explicitly examine its effect deeper into the future), but on the other hand, a larger T increases the computation time exponentially. To understand why we may need a large value of T, consider a scenario in which we are 200 meters before a highway exit and we should take it. When the time horizon is long enough, the cumulative reward will indicate if at some time T between t and t+T we have arrived to the exit lane. On the other hand, for a short time horizon, even if we perform the right immediate action we will not know if it will lead us eventually to the exit lane. 
     A different approach attempts to perform offline calculations in order to construct an approximation of Q, denoted {circumflex over (Q)}, and then during the online run of the policy, use {circumflex over (Q)} as an approximation to Q, without explicitly rolling out the future. One way to construct such an approximation is to discretize both the action domain and the state domain. Denote by Â, Ŝ these discretized sets. An offline calculation may evaluate the value of Q(s, a) for every (s, a) 0 Ŝ H Â. Then, for every a 0 Â we define {circumflex over (Q)}(s t , a) to be Q(s, a) for s=argmin s0Ŝ  ∥s−s t ∥. Furthermore, based on the pioneering work of Bellman [2, 3], we can calculate Q(s, a) for every (s, a) 0 Ŝ H Â, based on dynamic programming procedures (such as the Value Iteration algorithm), and under our assumptions, the total runtime is order of T |Â||Ŝ|. The main problem with this approach is that in any reasonable approximation, Ŝ is extremely large (due to the curse of dimensionality). Indeed, the sensing state should represent 6 parameters for every other relevant vehicle in the sense—the longitudinal and lateral position, velocity, and acceleration. Even if we discretize each dimension to only 10 values (a very crude discretization), since we have 6 dimensions, to describe a single car we need 10 6  states, and to describe k cars we need 10 6k  states. This leads to unrealistic memory requirements for storing the values of Q for every (s, a) in Ŝ H Â. 
     One approach for dealing with this curse of dimensionality is to restrict Q to come from a restricted class of functions (often called a hypothesis class), such as linear functions over manually determined features or deep neural networks. For example, consider a deep neural network that approximates Q in the context of playing Atari games. This leads to a resource-efficient solution, provided that the class of functions that approximate Q can be evaluated efficiently. However, there are several disadvantages of this approach. First, it is not known if the chosen class of functions contains a good approximation to the desired Q function. Second, even if such function exists, it is not known if existing algorithms will manage to learn it efficiently. So far, there are not many success stories for learning a Q function for complicated multi-agent problems, such as the ones we are facing in driving. There are several theoretical reasons why this task is difficult. As mentioned regarding the Markovian assumption, underlying existing methods are problematic. But, a more severe problem is a very small signal-to-noise ratio due to the time resolution of decision making, as explained below. 
     Consider a simple scenario in which a vehicle needs to change lane in order to take a highway exit in 200 meters and the road is currently empty. The best decision is to start making the lane change. Decisions may be made every 0.1 second, so at the current time t, the best value of Q(s t , a) should be for the action a corresponding to a small lateral acceleration to the right. Consider the action a′ that corresponds to zero lateral acceleration. Since there is a very little difference between starting the change lane now, or in  0 . 1  seconds, the values of Q(s t , a) and Q(s t , a′) are almost the same. In other words, there is very little advantage for picking a over a′. On the other hand, since we are using a function approximation for Q, and since there is noise in measuring the state s t , it is likely that our approximation to the Q value is noisy. This yields a very small signal-to-noise ratio, which leads to an extremely slow learning, especially for stochastic learning algorithms which are heavily used for the neural networks approximation class. However, as noted in, this problem is not a property of any particular function approximation class, but rather, it is inherent in the definition of the Q function. 
     In summary, available approaches can be roughly divided into two camps. The first one is the brute-force approach which includes searching over many sequences of actions or discretizing the sensing state domain and maintaining a huge table in memory. This approach can lead to a very accurate approximation of Q but requires unlimited resources, either in terms of computation time or in terms of memory. The second one is a resource efficient approach in which we either search for short sequences of actions or we apply a function approximation to Q. In both cases, we pay by having a less accurate approximation of Q that might lead to poor decisions. 
     The approach described herein includes constructing a Q function that is both resource-efficient and accurate is to depart from geometrical actions and to adapt a semantic action space, as described in the next subsection. 
     Semantic Approach 
     As a basis for the disclosed semantic approach, consider a teenager that just got his driving license. His father sits next to him and gives him “driving policy” instructions. These instructions are not geometric—they do not take the form “drive 13.7 meters at the current speed and then accelerate at a rate of 0.8 m/s 2 ”. Instead, the instructions are of semantic nature−“follow the car in front of you” or “quickly overtake that car on your left.” We formalize a semantic language for such instructions, and use them as a semantic action space. We then define the Q function over the semantic action space. We show that a semantic action can have a very long time horizon, which allows us to estimate Q(s, a) without planning for many future semantic actions. Yes, the total number of semantic actions is still small. This allows us to obtain an accurate estimation of the Q function while still being resource efficient. Furthermore, as we show later, we combine learning techniques for further improving the quality function, while not suffering from a small signal-to-noise ratio due to a significant difference between different semantic actions. 
     Now define a semantic action space. The main idea is to define lateral and longitudinal goals, as well as the aggressiveness level of achieving them. Lateral goals are desired positions in lane coordinate system (e.g., “my goal is to be in the center of lane number 2”). Longitudinal goals are of three types. The first is relative position and speed with respect to other vehicles (e.g., “my goal is to be behind car number 3, at its same speed, and at a distance of 2 seconds from it”). The second is a speed target (e.g., “drive at the allowed speed for this road times 110%”). The third is a speed constraint at a certain position (e.g., when approaching a junction, “speed of 0 at the stop line”, or when passing a sharp curve, “speed of at most 60 kmh at a certain position on the curve”). For the third option, we can instead apply a “speed profile” (few discrete points on the route and the desired speed at each of them). A reasonable number of lateral goals is bounded by 16=4×4 (4 positions in at most 4 relevant lanes). A reasonable number of longitudinal goals of the first type is bounded by 8×2×3=48 (8 relevant cars, whether to be in front or behind them, and 3 relevant distances). A reasonable number of absolute speed targets are  10 , and a reasonable upper bound on the number of speed constraints is 2. To implement a given lateral or longitudinal goal, we need to apply acceleration and then deceleration (or the other way around). The aggressiveness of achieving the goal is a maximal (in absolute value) acceleration/deceleration to achieve the goal. With the goal and aggressiveness defined, we have a closed form formula to implement the goal, using kinematic calculations. The only remaining part is to determine the combination between the lateral and longitudinal goals (e.g., “start with the lateral goal, and exactly at the middle of it, start to apply also the longitudinal goal”). A set of 5 mixing times and 3 aggressiveness levels seems more than enough. All in all, we have obtained a semantic action space whose size is ≈10 4 . 
     It is worth mentioning that the variable time required for fulfilling these semantic actions is not the same as the frequency of the decision making process. To be reactive to the dynamic world, we should make decisions at a high frequency—in our implementation, every 100 ms. In contrast, each such decision is based on constructing a trajectory that fulfills some semantic action, which will have a much longer time horizon (say, 10 seconds). We use the longer time horizon since it helps us to better evaluate the short term prefix of the trajectory. The next subsection discusses the evaluation of semantic actions, but before that, we argue that semantic actions induce a sufficient search space. 
     As discussed above, a semantic action space induces a subset of all possible geometrical curves, whose size is exponentially smaller (in 7) than enumerating all possible geometrical curves. The first immediate question is whether the set of short term prefixes of this smaller search space contains all geometric commands that we will ever want to use. This is indeed sufficient in the following sense. If the road is free of other agents, then there is no reason to make changes except setting a lateral goal and/or absolute acceleration commands and/or speed constraints on certain positions. If the road contains other agents, we may want to negotiate the right of way with the other agents. In this case, it suffices to set longitudinal goals relatively to the other agents. The exact implementation of these goals in the long run may vary, but the short term prefixes will not change by much. Hence, we obtain a very good cover of the relevant short term geometrical commands. 
     Constructing an evaluation function for semantic actions 
     We have defined a semantic set of actions, denoted by A s . Given that we are currently in state, s, we need a way to choose the best a s  0 A s . To tackle this problem, we follow a similar approach to the options mechanism of [6]. The basic idea is to think of a s  as a meta-action (or an option). For each choice of a meta-action, we construct a geometrical trajectory (s 1 , a 1 ), . . . , (s T , a T ) that represents an implementation of the meta-action, a s . To do so we of course need to know how other agents will react to our actions, but for now we are still relying on (the non-realistic) assumption that s t+1 =f (s t , a t ) for some known deterministic function f. We can now use 
     
       
         
           
             
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     as a gooa approximation of the quality of performing the semantic action a s  when we are at state s 1 . 
     This approach can yield a powerful driving policy. However, in some situations a more sophisticated quality function may be needed. For example, suppose that we are following a slow truck before an exit lane, where we need to take the exit lane. One semantic option is to keep driving slowly behind the truck. Another one is to overtake the truck, hoping that later we can get back to the exit lane and make the exit on time. The quality measure described previously does not consider what will happen after we will overtake the truck, and hence we will not choose the second semantic action even if there is enough time to make the overtake and return to the exit lane. Machine learning can help us to construct a better evaluation of semantic actions that will take into account more than the immediate semantic actions. As previously discussed, learning a Q function over immediate geometric actions is problematic due to the low signal-to-noise ratio (the lack of advantage). This is not problematic when considering semantic actions, both because there is a large difference between performing the different semantic actions and because the semantic time horizon (how many semantic actions we take into account) is very small (probably less than three in most cases). 
     Another potential advantage of applying machine learning is for the sake of generalization: we may set an adequate evaluation function for every road, by a manual inspection of the properties of the road, which may involve some trial and error. But, can we automatically generalize to any road? Here, a machine learning approach, as discussed above, can be trained on a large variety of road types so as to generalize to unseen roads as well. The semantic action space according to the disclosed embodiments may allow for potential benefits: semantic actions contain information on a long time horizon, hence we can obtain a very accurate evaluation of their quality while being resource efficient. 
     The dynamics of the Other Agents 
     So far, we have relied on the assumption that s t+1  is a deterministic function of s t  and a t . As emphasized previously, this assumption is not completely realistic as our actions affect the behavior of other road users. While we do take into account some reactions of other agents to our actions (for example, we assume that if we will perform a safe cut-in than the car behind us will adjust its speed so as not to hit us from behind), it is not realistic to assume that we model all of the dynamics of other agents. The solution to this problem is to re-apply the decision making at a high frequency, and by doing this, constantly adapt our policy to the parts of the environment that are beyond our modeling. In a sense, one can think of this as a Markovization of the world at every step. 
     Sensing 
     This section describes the sensing state, which is a description of the relevant information of the scene, and forms the input to the driving policy module. By and large, the sensing state contains static and dynamic objects. The static objects are lanes, physical road delimiters, constraints on speed, constraints on the right of way, and information on occluders (e.g., a fence that occludes relevant part of a merging road). Dynamic objects are vehicles (e.g., bounding box, speed, acceleration), pedestrians (bounding box, speed, acceleration), traffic lights, dynamic road delimiters (e.g.. cones at a construction area), temporary traffic signs and police activity, and other obstacles on the road (e.g., an animal, a mattress that fell from a truck, etc.). 
     In any reasonable sensor setting, we cannot expect to obtain the exact sensing state, s. Instead, we view raw sensor and mapping data, which we denote by x 0 X, and there is a sensing system that takes x and produces an approximate sensing state. 
     Definition 16 (Sensing system) Let S denotes the domain of sensing state and let X be the domain of raw sensor and mapping data. A sensing system is a function ŝ: X→S. 
     It is important to understand when we should accept ŝ(x) as a reasonable approximation to s. The ultimate way to answer this question is by examining the implications of this approximation on the performance of our driving policy in general, and on the safety in particular. Following our safety-comfort distinction, here again we distinguish between sensing mistakes that lead to non-safe behavior and sensing mistakes that affect the comfort aspects of the ride. Before addressing the details, the type of errors a sensing system might make include:
         False negative: the sensing system misses an object   False positive: the sensing system indicates a “ghost” object   Inaccurate measurements: the sensing system correctly detects an object but incorrectly estimates its position or speed   Inaccurate semantic: the sensing system correctly detects an object but misinterpret its semantic meaning, for example, the color of a traffic light       

     Comfort 
     Recall that for a semantic action a, we have used Q(s, a) to denote our evaluation of a given that the current sensing state is s. Our policy picks the action π(s)=argmax a  Q(s, a). If we inject ŝ(x) instead of s then the selected semantic action would be π(ŝ(x))=argmax a  Q(s(x), a). If π(ŝ(x))=π(s) then ŝ(x) should be accepted as a good approximation to s. But, it is also not bad at all to pick π(ŝ(x)) as long as the quality of π(ŝ(x)) w.r.t. the true state, s, is almost optimal, namely, Q(s, π(ŝ(x)))≥Q(s, π(s))−∈, for some parameter ∈. We say that ŝ is ∈-accurate w.r.t. Q in such case. Naturally, we cannot expect the sensing system to be ∈−accurate all the time. We therefore also allow the sensing system to fail with some small probability δ. In such a case we say that ŝ is Probably (w.p. of at least 1−δ), Approximately (up to ∈), Correct, or PAC for short (borrowing Valiant&#39;s PAC learning terminology). 
     We may use several (∈, δ) pairs for evaluating different aspects of the system. For example, we can choose three thresholds, ∈ 1 &lt;∈ 2 &lt;∈ 3  to represent mild, medium, and gross mistakes, and for each one of them set a different value of δ. This leads to the following definition. 
     Definition  17  (PAC sensing system) Let ((∈ 1 , δ 1 ), . . . , (∈ k , δ k )) be a set of (accuracy, confidence) pairs, let S be the sensing state domain, let X be the raw sensor and mapping data domain, and let D be a distribution over X×S. Let A be an action space, Q: S×A→|be a quality function, and π: S→A be such that π(s)∈argmax a  Q(s, a). A sensing system, ŝ: X&#39;S, is Probably-Approximately-Correct (PAC) with respect to the above parameters if for every i∈{1, . . . , k} we have that P (x,s)˜D [Q(s, π(ŝ(x)))≥Q(s, π(s))−∈ 1 ]≥1−δ i . 
     Here, the definition depends on a distribution D over X×S. It is important to emphasize that we construct this distribution by recording data of many human drivers but not by following the particular policy of our autonomous vehicle. While the latter seems more adequate, it necessitates online validation, which makes the development of the sensing system impractical. Since the effect of any reasonable policy on D is minor, by applying simple data augmentation techniques we can construct an adequate distribution and then perform offline validation after every major update of the sensing system. The definition provides a sufficient, but not necessary, condition for comfort ride using ŝ. It is not necessary because it ignores the important fact that short term wrong decisions have little effect on the comfort of the ride. For example, suppose that there is a vehicle  100  meters ahead, and it is slower than the host vehicle. The best decision would be to start accelerating slightly now. If the sensing system misses this vehicle, but will detect it in the next time (after  100  mili-seconds), then the difference between the two rides will not be noticeable. To simplify the presentation, we have neglected this issue and required a stronger condition. The adaptation to a multi-frame PAC definition is conceptually straightforward, but is more technical. 
     We next derive design principles that follow from the above PAC definition. Recall that we have described several types of sensing mistakes. For mistakes of types false negative, false positive, and inaccurate semantic, either the mistakes will be on non-relevant objects (e.g., a traffic light for left turn when we are proceeding straight), or they will be captured by the δ part of the definition. We therefore focus on the “inaccurate measurements” type of errors, which happens frequently. 
     Somewhat surprisingly, we will show that the popular approach of measuring the accuracy of a sensing system via ego-accuracy (that is, by measuring the accuracy of position of every object with respect to the host vehicle) is not sufficient for ensuring PAC sensing system. We will then propose a different approach that ensures PAC sensing system, and will show how to obtain it efficiently. We start with some additional definitions. 
     For every object o in the scene, let p(o), {circumflex over (p)}(o) be the positions of o in the coordinate system of the host vehicle according to s, ŝ(x), respectively. Note that the distance between o and the host vehicle is ∥p∥. The additive error of {circumflex over (p)} is ∥p(o)−{circumflex over (p)}(o)∥. The relative error of {circumflex over (p)}(o), w.r.t. the distance between o and the host vehicle, is the additive error divided by ∥p(o)∥, namely 
     
       
         
           
             
               
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     It is not realistic to require that the additive error is small for far away objects. Indeed, consider o to be a vehicle at a distance of 150 meters from the host vehicle, and let ∈ be of moderate size, say ∈=0.1. For additive accuracy, it means that we should know the position of the vehicle up to 10 cm of accuracy. This is not realistic for reasonably priced sensors. On the other hand, for relative accuracy we need to estimate the position up to 10%, which amounts to 15 m of accuracy. This is feasible to achieve (as described below). 
     A sensing system, ŝ, positions a set of objects, O, in an ∈-ego-accurate way, if for every o∈O, the (relative) error between p(o) and {circumflex over (p)}(o) is at most ∈. The following example demonstrates that an ∈-ego-accurate sensing state does not guarantee PAC sensing system with respect to every reasonable Q. Indeed, consider a scenario in which the host vehicle drives at a speed of 30 m/s, and there is a stopped vehicle  150  meters in front of it. If this vehicle is in the ego lane, and there is no option to change lanes in time, we must start decelerating now at a rate of at least 3 m/s 2  (otherwise, we will either not stop in time or we will need to decelerate strongly later). On the other hand, if the vehicle is on the side of the road, we don&#39;t need to apply a strong deceleration. Suppose that p(o) is one of these cases while {circumflex over (p)}(o) is the other case, and there is a 5 meters difference between these two positions. Then, the relative error of {circumflex over (p)}(o) is 
     
       
         
           
             
               
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     That is, the sensing system may be ∈-ego-accurate for a rather small value of ∈ (less than 3.5% error), and yet, for any reasonable Q function, the values of Q are completely different since we are confusing between a situation in which we need to brake strongly and a situation in which we do not need to brake strongly. 
     The above example shows that ∈-ego-accuracy does not guarantee that our sensing system is PAC. Whether there is another property that is sufficient for PAC sensing system depends on Q. We will describe a family of Q functions for which there is a simple property of the positioning that guarantees PAC sensing system. The problem of ∈-ego-accuracy is that it might lead to semantic mistakes—in the aforementioned example, even though ŝ was ∈-ego-accurate with ∈&lt;3.5%, it mis-assigned the vehicle to the correct lane. To solve this problem, we rely on semantic units for lateral position. 
     Definition 18 (semantic units) A lane center is a simple natural curve, namely, it is a differentiable, injective, mapping  : [a, b]→| 3 , where for every a≤t 1 ≤t 2 ≤b we have that the length Length(t 1 , t 2 ):=∫ r=t     1     t     2   | ′(τ)|dτ equals to t 2 −t 1 . The width of the lane is a function w: [a, b]→| + . The projection of a point x∈| 3  onto the curve is the point on the curve closest to x, namely, the point  (t x ) for t x =argmin t∈[a, b] ∥ (t)−x∥. The semantic longitudinal position of x w.r.t. the lane is t x  and the semantic lateral position of x w.r.t. the lane is  (t x )/w(t x ). Semantic speed and acceleration are defined as first and second derivatives of the above. 
     Similarly to geometrical units, for semantic longitudinal distance we use relative error: if ŝ induces a semantic longitudinal distance of {circumflex over (p)}(o) for some object, while the true distance is p(o), then the relative error is 
     
       
         
           
             
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     (where the maximum in the denominator deals with cases in which the object has almost the same longitudinal distance (e.g., a car next to us on another lane). Since semantic lateral distances are small we can use additive error for them. This leads to the following definition: 
     Definition 19 (error in semantic units) Let   be a lane and suppose that the semantic longitudinal distance of the host vehicle w.r.t. the lane is 0. Let x∈   3  be a point and let p lat (x), p lon (x) be the semantic lateral and longitudinal distances to the point w.r.t. the lane. Let {circumflex over (p)} lat (x), {circumflex over (p)} lon (x) be approximated measurements. The distance between {circumflex over (p)} and p w.r.t. x is defined as 
     
       
         
           
             
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     The distance of the lateral and longitudinal velocities is defined analogously. 
     Equipped with the above definition, we are ready to define the property of Q and the corresponding sufficient condition for PAC sensing system. 
     Definition 20 (Semantically-Lipschitz Q) A Q function is L-semantically-Lipschitz if for every a, s, ŝ, |Q(s, a)−Q(ŝ(x), a)|≤L max o  d({circumflex over (p)}, p; o), where {circumflex over (p)}, p are the measurements induced by s, ŝ on an object o. 
     As an immediate corollary we obtain: 
     Lemma 8 If Q is L-semantically-Lipschitz and a sensing system ŝ produces semantic measurements such that with probability of at least 1−δ we have d({circumflex over (p)}, p; o)≤0/L, then ŝ is a PAC sensing system with parameters 0, δ. 
     Safety 
     This section discusses the potential for sensing errors that can lead to unwanted behaviors. As mentioned before, the policy is provably safe, in the sense that it won&#39;t lead to accidents of the host AV&#39;s blame. Such accidents might still occur due to hardware failure (e.g., a breakdown of all the sensors or exploding tire on the highway), software failure (a significant bug in some of the modules), or a sensing mistake. Our ultimate goal is that the probability of such events will be extremely small—a probability of 10 −9  for such an accident per hour. To appreciate this number, the average number of hours driver in the U.S. spends on the road is (as of 2016) less than 300 hours. So, in expectation, one would need to live 3.3 million years to be in an accident resulting from one of these types of events. 
     We first define what is a safety-related sensing error. Recall that at every step, our policy picks the value of a that maximizes Q(s, a), namely, π(s)=argmax, Q(s, a). We ensure safety by letting Q(s, a)=−∞ for every action a that is not cautious (see Definition 14). Therefore, the first type of safety-critic sensing mistake is if our sensing system leads to picking a non-safe action. Formally, letting π(ŝ(x))=argmax a  Q(ŝ(x), a) be the decision according to ŝ, we say that ŝ leads to a safety-critic miss if Q(s, π(ŝ(x)))=−∞. The second type of safety-critic sensing mistake is if all the actions are non-safe according to ŝ(x), and we must apply the standard emergency policy (e.g., braking hard), while according to s there is a safe action, namely, max, Q(s, a)&gt;−∞. This is dangerous when our speed is high and there is a car behind us. We call such mistake a safety-critic ghost. 
     Usually, a safety-critic miss is caused by a false negative while a safety-critic ghost is caused by a false positive. Such mistakes can also be caused from significantly incorrect measurements, but in most cases, our comfort objective ensures we are far away from the boundary of the safety definitions, and therefore reasonable measurements errors are unlikely to lead to a safety-critic mistake. How can we ensure that the probability of safety-critic mistakes will be very small, say, smaller than 10 −9  per hour? As followed from Lemma 1, without making further assumptions we need to check our system on more than 10 9  hours of driving. This is unrealistic (or at least extremely challenging)—it amounts to recording the driving of 3.3 million cars over a year. Furthermore, building a system that achieves such a high accuracy is a great challenge. A solution for both the system design and validation challenges is to rely on several sub-systems, each of which is engineered independently and depends on a different technology, and the systems are fused together in a way that ensures boosting of their individual accuracy. 
     Suppose we build three sub-systems, denoted, s 1 , s 2 , s 3  (the extension to more than 3 is straightforward). Each sub-system receives a and should output safe/non-safe. Actions for which the majority of the sub-systems (2 in our case) accept as safe are considered safe. If there is no action that is considered safe by at least 2 sub-systems, then the default emergency policy is applied. The performance of this fusion scheme is analyzed as follows based on the following definition: 
     Definition 21 (One side c-approximate independent) Two Bernoulli random variables r 1 , r 2  are called one side c-approximate independent if 
         [r 1 Λr 2 ]≤c [r 1 ] [r 2 ].
 
     For i 0 {1,2,3}, denote by e i   m , e i   g  the Bernoulli random variables that indicate if sub-system i performs a safety-critic miss/ghost respectively. Similarly, e m , e g  indicate a safety-critic miss/ghost of the fusion system. We rely on the assumption that for any pair i≠j the random variables e i   m , e j   m  are one sided c-approximate independent, and the same holds for e i   g , e j   g . Before explaining why this assumption is reasonable, let us first analyze its implication. We can bound the probability of e m  by: 
     
       
         
           
             
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     Therefore, if all sub-systems have  [e i   m ]≤p then  [e m ]≤3cp 2 . The exact same derivation holds for the safety-critic ghost mistakes. By applying a union bound we therefore conclude: 
     Corollary 2 Assume that for any pair i≠j the random variables e i   m , e j   m  are one sided c-approximate independent, and the same holds for e i   g , e j   g . Assume also that for every i,  [e i   m ]≤p and  [e i   g ]≤p. Then, 
         [e m Λe g ]≤6cp 2  
 
     This corollary allows us to use significantly smaller data sets in order to validate the sensing system. For example, if we would like to achieve a safety-critic mistake probability of 10 −9 , instead of taking order of 10 9  examples, it suffices to take order of 10 5  examples and test each system separately. 
     There may be pairs of sensors that yield non-correlated errors. For example, radar works well in bad weather conditions but might fail due to non-relevant metallic objects, as opposed to camera that is affected by bad weather but is not likely to be affected by metallic objects. Seemingly, camera and lidar have common sources of error—e.g., both are affected by foggy weather, heavy rain, or snow. However, the type of errors for camera and lidar would be different—a camera might miss objects due to bad weather and lidar might detect a ghost due to reflections from particles in the air. Since we have distinguished between the two types of errors, the approximate independency is still likely to hold. 
     Our definition of safety-important ghost requires that all actions are non-safe by at least two sensors. Even in difficult conditions (e.g., heavy fog), this is unlikely to happen. 
     The reason is that in such situations, systems that are affected by the difficult conditions (e.g., the lidar), will dictate a very defensive driving, as they can declare high velocity and lateral maneuver to be non-safe actions. As a result, the host AV will drive slowly, and then even if an emergency stop is required, it is not dangerous due to the low speed of driving. Therefore, our definition yields an adaptation of the driving style to the conditions of the road. 
     Building a Scalable Sensing System 
     The requirements from a sensing system, both in terms of comfort and safety, have been described. Next, an approach for building a sensing system that meets these requirements while being scalable is described. There are three main components of the sensing system. The first is long range, 360 degrees coverage, of the scene based on cameras. The three main advantages of cameras are: (1) high resolution, (2) texture, (3) price. The low price enables a scalable system. The texture enables to understand the semantics of the scene, including lane marks, traffic light, intentions of pedestrians, and more. The high resolution enables a long range of detection. Furthermore, detecting lane marks and objects in the same domain enables excellent semantic lateral accuracy. The two main disadvantages of cameras are: (1) the information is 2D and estimating longitudinal distance is difficult, (2) sensitivity to lighting conditions (low sun, bad weather). We overcome these difficulties using the next two components of our system. 
     The second component of our system is a semantic high-definition mapping technology, called Road Experience Management (REM) (which involves navigation based on target trajectories predetermined and stored for road segments along with an ability to determine precise locations along the target trajectories based on the location (e.g., in images) of recognized landmarks identified in the environment of the host vehicle). A common geometrical approach to map creation is to record a cloud of 3D points (obtained by a lidar) in the map creation process, and then, localization on the map is obtained by matching the existing lidar points to the ones in the map. There are several disadvantages of this approach. First, it requires a large memory per kilometer of mapping data, as we need to save many points. This necessitates an expensive communication infrastructure. Second, not all cars may be equipped with lidar sensors, and therefore, the map is updated very infrequently. This is problematic as changes in the road can occur (construction zones, hazards), and the “time-to-reflect-reality” of lidar-based mapping solutions is large. In contrast, REM follows a semantic-based approach. The idea is to leverage the large number of vehicles that are equipped with cameras and with software that detects semantically meaningful objects in the scene (lane marks, curbs, poles, traffic lights, etc.). Nowadays, many new cars are equipped with ADAS systems which can be leveraged for crowd-sourced map creation. Since the processing is done on the vehicle side, only a small amount of semantic data should be communicated to the cloud. This allows a very frequent update of the map in a scalable way. In addition, the autonomous vehicles can receive the small sized mapping data over existing communication platforms (the cellular network). Finally, highly accurate localization on the map can be obtained based on cameras, without the need for expensive lidars. 
     REM may be used for several purposes. First, it gives us a foresight on the static structure of the road (we can plan for a highway exit way in advance). Second, it gives us another source of accurate information of all of the static information, which together with the camera detections yields a robust view of the static part of the world. Third, it solves the problem of lifting the 2D information from the image plane into the 3D world as follows. The map describes all of the lanes as curves in the 3D world. Localization of the ego vehicle on the map enables to trivially lift every object on the road from the image plane to its 3D position. This yields a positioning system that adheres to the accuracy in semantic units. A third component of the system may be complementary radar and lidar systems. These systems may serve two purposes. First, they can offer extremely high levels of accuracy for augmenting safety. Second, they can give direct measurements on speed and distances, which further improves the comfort of the ride. 
     The following sections include technical lemmas and several practical considerations of the RSS system. 
     Lemma 9 For all x 0 [0,0.1], it holds that 1−x&gt;e −2x . Proof Let f (x)=1−x−e −2x . Our goal is to show it is ≥0 for ≥0 [0,0.1]. Note that f (0)=0, and it is therefore sufficient to have that f (x)≥0 in the aforementioned range. Explicitly, f(x)=−1+2e −2x . Clearly, f(0)=1, and it is monotonically decreasing, hence it is sufficient to verify that f(0.1)&gt;0, which is easy to do numerically, f(0.1)≈0.637. 
     Efficient Cautiousness Verification—Occluded Objects 
     As with non-occluded objects, we can check whether giving the current command, and after it, the DEP, is RSS. For this, we unroll our future until t brake , when assuming the exposure time is 1 and we then command DEP, which suffices for cautiousness by definition. For all t′ 0 [0,t brake ], we check whether a blameful accident can occur—when assuming worst case over the occluded object. We use some of our worst case maneuvers and safe distance rules. 
     We take an occluder based approach to find interest points—namely, for each occluding object, we calculate the worst case. This is a crucial efficiency-driven approach—a pedestrian, for example, can be hidden in many positions behind a car, and can perform many maneuvers, but there&#39;s a single worst case position and maneuver it can perform. 
     Next, consider the more elaborate case, that of the occluded pedestrian. 
     Consider an occluded area behind a parked car. The closest points in an occluded area and the front/side of our car c may be found, for example, by their geometrical properties (triangles, rectangles). Formally, we can consider the occluded area as a union of a small number of convex regions of simple shape, and treat each of them separately. Furthermore, it can be seen that a pedestrian can run into the front of the car (under the ν limit  constraint) from the occluded area IFF he can do it using the shortest path possible. Using the fact that the maximal distance which can be travelled by the pedestrian is ν limit ·t′, we obtain a simple check for a frontal hit possibility. As for a side hit, we note that in case the path is shorter than ν limit ·t′, we are responsible IFF our lateral velocity is greater than μ, in the direction of the hit. Disclosed is an algorithm for cautiousness verification with respect to occluded pedestrians, which is described here in free pseudo code. The crucial part, that of checking existence of possibility of blameful accident with a pedestrian occluded by a vehicle, is done in the simple manner described above. 
     
       
         
           
               
             
               
                   
               
               
                 Algorithm 2: Check cautiousness w.r.t. an occluded pedestrian 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 for t′ ∈ [0, t brake ] 
               
               
                  Roll self future until t′ 
               
               
                  if Exists possibility of blameful accident with a pedestrian occluded by a 
               
               
                  vehicle 
               
               
                   return “non-cautious” 
               
               
                 return “cautious” 
               
               
                   
               
            
           
         
       
     
     On the Problem of Validating a Simulator 
     As previously discussed, multi-agent safety may be difficult to validate statistically as it should be done in an “online” manner. One may argue that by building a simulator of the driving environment, we can validate the driving policy in the “lab.” However, validating that the simulator faithfully represents reality is as hard as validating the policy itself. To see why this is true, suppose that the simulator has been validated in the sense that applying a driving policy π in the simulator leads to a probability of an accident of {circumflex over (p)}, and the probability of an accident of π in the real world is p, with |p−{circumflex over (p)}|&lt;0. (We need that 0 will be smaller than 10 −9 .) Next replace the driving policy to be π′. Suppose that with probability of 10 −8 , π′ performs a weird action that confuses human drivers and leads to an accident. It is possible (and even rather likely) that this weird action is not modeled in the simulator, without contradicting its superb capabilities in estimating the performance of the original policy π. This proves that even if a simulator has been shown to reflect reality for a driving policy π, it is not guaranteed to reflect reality for another driving policy. 
     The Lane-Based Coordinate System 
     One simplifying assumption that can be made in the RSS definition is that the road is comprised by adjacent, straight lanes, of constant width. The distinction between lateral and longitudinal axes, along with an ordering of longitudinal position, may play a significant role in RSS. Moreover, the definition of those directions is clearly based on the lane shape. A transformation from (global) positions on the plane, to a lane-based coordinate system reduces the problem to the original, “straight lane of constant width,” case. 
     Assume that the lane&#39;s center is a smooth directed curve r on the plane, where all of its pieces, denoted r (1) , . . . , r (k) , are either linear, or an arc. Note that smoothness of the curve implies that no pair of consecutive pieces can be linear. Formally, the curve maps a “longitudinal” parameter, Y 0 [Y min , Y max ]⊂|, into the plane, namely, the curve is a function of the form r: [Y min , Y max ]→| 2 . We define a continuous lane-width function w: [Y min , Y max ]→| + , mapping the longitudinal position Y into a positive lane width value. For each Y, from smoothness of r, we can define the normal unit-vector to the curve at position Y, denoted r ⊥ (Y). We naturally define the subset of points on the plane which reside in the lane as follows: 
         R={r ( Y )+α w ( Y ) r   195  ( Y )| Y 0 [Y   min   , Y   max ], α0 [±1/2]}
 
     Informally, our goal is to construct a transformation ϕ of R into such that for two cars which are on the lane, their “logical ordering” will be preserved: if c r  is “behind” c f  on the curve, then ϕ(c r ) y &lt;ϕ(c f ) y . If c l  is “to the left of” c r  on the curve, then ϕ(c i ) x &lt;ϕ(c r ) x . Where, as in RSS, we will associate they-axis with the “longitudinal” axis, and the x-axis with the “lateral”. 
     To define ϕ, we rely on the assumption that for all i, if r (i)  is an arc of radius ρ, then the width of the lane throughout r (i)  is ≤ρ/2. Note that this assumption holds for any practical road. The assumption trivially implies that for all (x′,y′) 0 R, there exists a unique pair Y′ 0 [Y min , Y max ], α′ 0 [±1/2], s.t. (x′,y′)=r(Y′)+α′w(Y′)r ⊥ (Y′). We can now define ϕ: R→| 2  to be ϕ(x′, y′)=(Y′, α′), where (Y′, α′) are the unique values that satisfy (x′, y′)=r(Y′)+α′w(Y′)r ⊥ (Y′). 
     This definition captures the notion of a “lateral maneuver” in lane&#39;s coordinate system. Consider, for example, a widening lane, with a car driving exactly on one of the lane&#39;s boundaries (see  FIG. 23 ). The widening of lane  2301  means that the car  2303  is moving away from the center of the lane, and therefore has lateral velocity with respect to the lane. However, this doesn&#39;t mean it performs a lateral maneuver. The definition of ϕ(x′, y′) x =α′, namely, the lateral distance to the lane&#39;s center in w(Y′)-units, implies that the lane boundaries have a fixed lateral position of ±1/2. Hence, a car that sticks to one of the lane&#39;s boundaries is not considered to perform any lateral movement. Finally, it can be seen that ϕ is a homomorphism. The term lane-based coordinate system is used when discussing ϕ(R)=[Y min , Y max ]×[±1/2]. We have thus obtained a reduction from a general lane geometry to a straight, longitudinal/lateral, coordinate system. 
     Extending RSS to General Road Structure 
     In this section a complete definition of RSS that holds for all road structures is described. This section deals with the definition of RSS and not on how to efficiently ensure that a policy adheres to RSS. The concept of route priority is next introduced to capture any situation in which more than a single lane geometry exists, for example junctions. 
     The second generalization deals with two-way roads, in which there can be two cars driving at opposite directions. For this case, the already established RSS definition is still valid, with the minor generalization of “safe distance” to oncoming traffic. Controlled junctions (that use traffic lights to dictate the flow of traffic), may be fully handled by the concepts of route priority and two-way roads. Unstructured roads (for example parking areas), where there is no clear route definition may also be handled with RSS. RSS is still valid for this case, where the only needed modification is a way to define virtual routes and to assign each car to (possibly several) routes. 
     Route Priority 
     The concept of route priority is now introduced to deal with scenarios in which there are multiple different road geometries in one scene that overlap in a certain area. Examples, as shown in  FIGS. 24A-D , include roundabouts, junctions, and merge into highways. A way to transform general lane geometry into a lane-based one, with coherent meaning for longitudinal and lateral axes, has been described. Now scenarios in which multiple routes of different road geometry exists are addressed. It follows that when two vehicles approach the overlap area, both perform a cut-in to the frontal corridor of the other one. This phenomenon cannot happen when two routes have the same geometry (as is the case of two adjacent highway lanes). Roughly speaking, the principle of route priority states that if routes r 1 , r 2  overlap, and r 1  has priority over r 2 , then a vehicle coming from r 1  that enters into the frontal corridor of a vehicle that comes from r 2  is not considered to perform a cut-in. 
     To explain the concept formally, recall that the blame of an accident depends on geometrical properties which are derived from the lane&#39;s coordinate system, and on worst case assumptions which rely on it too. Let r 1 , . . . , r k  be the routes defining the road&#39;s structure. As a simple example, consider the merge scenario, as depicted in  FIG. 24A . Assume two cars,  2401  (c 1 ) and  2402  (c 2 ) are driving on routes r 1 , r 2  respectively, and ri is the prioritized route. For example, suppose that r 1  is a highway lane and r 2  is a merging lane. Having defined the route-based coordinate systems for each route, a first observation is that we can consider any maneuver in any route&#39;s coordinate system. For example, if we use r 2 &#39;s coordinate system, driving straight on r 1  seems like a merge into r 2 &#39;s left side. One approach to definition of RSS could be that each of the cars can perform a maneuver IFF ∀i 0 {1, 2}, if it is safe with respect to r i . However, this implies that c 1 , driving on the prioritized route, should be very conservative w.r.t. r 2 , the merging route, as c 2  can drive exactly on the route, and hence can win by lateral position. This is unnatural, as cars on the highway have the right-of-way in this case. To overcome this problem, we define certain areas in which route priority is defined, and only some of the routes are considered as relevant for safety. 
     Definition 22 (Accident Responsibility with Route Priority) Suppose r 1 , r 2  are two routes with different geometry that overlap. We use r 1 &gt; [b,e] r 2  to symbolize that r 1  has priority over r 2  in the longitudinal interval [b, e] ( FIG. 25 ) of r 1  coordinate system. Suppose there is an accident between cars c 1 , c 2 , driving on routes r 1 , r 2 . For i0 {1, 2}, let b i  ⊂{1, 2} indicate the cars to blame for the accident if we consider the coordinate system of r i . The blame for the accident is as follows:
         If r 1 &gt; [b, e] r 2  and on the blame time w.r.t. r 1 , one of the cars was in the interval [b,e] of the r 1 -system&#39;s longitudinal axis, then the blame is according to b 1 .   Otherwise, the blame is according to b 1  ∪b2.       

     To illustrate the definition, consider again the merge into highway example. The lines denoted “b” and “e” in  FIG. 25  indicate the values of b, e for which r 1 &gt; [b,e] r 2 . Thus, we allow cars to drive naturally on the highway, while implying merging cars must be safe with respect to those cars. In particular, observe that in the case a car  2401  c 1  drives at the center of the prioritized lane, with no lateral velocity, it will not be blamed for an accident with a car  2402  c 2  driving on a non-prioritized lane, unless  2402  c 2  has cut-in into  2401  c 1 &#39;s corridor at a safe distance. Note, that the end result is very similar to the regular RSS—this is exactly the same as a case where a car, on a straight road, tries to perform a lane change. Note that there may be cases where the route used by another agent is unknown. For example, in  FIG. 26 , car  2601  may not be able to determine whether a car  2602  will take path “a” or path “b”. In such cases, RSS may be obtained by iteratively checking all possibilities. 
     Two-Way Traffic 
     To deal with two-way traffic, the modification to the blame definition comes through sharpening the parts which rely on rear/front relationships, as those are of a slightly different meaning in such cases. Consider two cars c 1 , c 2  driving on some straight two lane road, in opposite longitudinal directions, namely, ν 1,long ·νν 2,long &lt;0. The direction of driving with respect to a lane may be negative in reasonable urban scenarios, such as a car deviating to the opposite lane to overtake a parked truck, or a car reversing into a parking spot. It is therefore required to extend the definition of the safe longitudinal distance which we have introduced for cases where negative longitudinal velocity was assumed to be un-realistic. Recall that a distance between c r , c f  was safe if a maximal brake by c f  would allow enough of response time for c r  to brake before crashing into c f . In our case, we again consider the “worst-case” by the opposite car, in a slightly different manner: of course we do not assume that the “worst case” is that it speeds up towards us, but that it indeed will brake to avoid a crash—but only using some reasonable braking power. In order to capture the difference in responsibility between the cars, when one of them clearly drives at the opposite direction, we start by defining a “correct” driving direction. 
     In the RSS definition for parallel lanes, the relevant lane has been defined as the one whose center is closest to the cut-in position. We can now reduce ourselves to consideration of this lane (or, in the case of symmetry, deal with the two lanes separately, as in Definition 22). In the definition below, the term “heading” denotes the arc tangent (in radians) of the lateral velocity divided by the longitudinal velocity. 
     Definition 23 ((μ 1 , μ 2 , μ 3 )—Winning by Correct Driving Direction) Assume c 1 , c 2  are driving in opposite directions, namely ν 1,long ·ν 2,long &lt;0. Let x i ,h i  be their lateral positions and headings w.r.t. the lane. We say that c 1  (μ 1 , μ 2 , μ 3 )-Wins by Correct Driving Direction if all of the following conditions hold:
         |h 1 |&lt;μ 1 ,   |h 2 −π|&lt;μ 2 ,   |x 1 |&lt;μ 3 .       

     The indicator of this event is denoted by W CDD (i). 
     In words, c 1  wins if it drives close to the lane canter, in the correct direction, while c 2  takes the opposite direction. At most one car can win, and it can be that none of the cars does so. Intuitively, assume there is a crash in the discussed situation. It is reasonable to put more responsibility over a car c 1 , that loses by Correct Driving Direction. This is done by re-defining amax,brake for the case that a car wins by correct driving direction. 
     Definition 24 (Reasonable Braking Power) Let αhd max,brake,wcdd&gt;0 be a constant, smaller than a max,brake . Assume c 1 , c 2  are driving in opposite directions. The Reasonable Braking Power of each car c i , denoted RBP i  is a max,brake,wcdd  if c i  (μ 1 , μ 2 , μ 3 )-Wins by Correct Driving Direction and a max,brake  otherwise. 
     The exact values of a max,brake,wcdd , a max,brake , for the cases of winning/not-winning by correct driving direction, are constants which should be defined, and can depend on the type of road and the lanes driven by each car. For example, in a narrow urban street, it may be the case that winning by a correct driving direction does not imply a much lesser brake value: in dense traffic, we do expect a car to brake at similar forces, either when someone clearly deviated into its lane or not. However, consider an example of a rural two way road, where high speeds are allowed. When deviating to the opposite lane, cars which drive at the correct direction cannot be expected to apply a very strong braking power to avoid hitting the host vehicle—the host vehicle will have more responsibility than them. Different constants can be defined for the case when two cars are at the same lane, with one of them reversing into a parking spot. 
     The safety distance between cars which are driving in opposite directions, and immediately derive its exact value, is next defined. 
     Definition 25 (Safe Longitudinal Distance—Two-Way Traffic) A longitudinal distance between a car c 1  and another car c 2  which are driving in opposite directions and are both in the frontal corridors of each other, is safe w.r.t. a response time ρ if for any acceleration command a, |a|&lt;a max,accel , performed by c 1 , c 2  until time ρ, if c 1  and c 2  will apply their Reasonable Braking Power from time ρ until a full stop then they won&#39;t collide. 
     Lemma 10 Let c 1 , c 2  as in Definition 25. Let RBP i , a max,accel  be the reasonable braking (for each i) and acceleration commands, and let ρ be the cars&#39; response time. Let ν 1 , ν 2  be the longitudinal velocities of the cars, and let l 1 , l 2  be their lengths. Define ν iρ,max =|ν i |+ρ·a max,accel . Let L=(l r +l f )/2. Then, the minimal safe longitudinal distance is: 
     
       
         
           
             
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     It can be seen that the term in the sum is the maximal distance travelled by each car until it reaches full stop, when performing the maneuver from Definition 25. Therefore, in order for the full stop to be at a distance greater than L, the initial distance must be larger than this sum and an additional term of L. 
     The same blame time definition of RSS, with the non-safe longitudinal distance as defined in Definition 25, is used to define the blame/accident responsibility for a two-way traffic scenario. 
     Definition 26 (Blame in Two-Way Traffic) The Blame in Two-Way Traffic of an accident between cars c 1 , c 2  driving in opposite directions, is a function of the state at the Blame Time, and is defined as follows:
         If the Blame Time is also a cut-in time, the blame is defined as in the regular RSS definition.   Otherwise, for every i, the blame is on c i  if at some t that happens after the blame time, c i  was not braking at a power of at least RBP i .       

     For example, assume a safe cut-in occurred before the blame time. For example, c 1  has deviated to the opposite lane, performed a cut-in into c 2 &#39;s corridor, at a safe distance. Note that c 2  wins by correct driving direction, and hence this distance can be very large—we do not expect c 2  to perform strong braking power, but only the Reasonable Braking Power. Then, both cars have responsibility not to crash into each other. However, if the cut-in was not in a safe-distance, we use the regular definition, noting that c 2  will not be blamed if it drove in the center of its lane, without lateral movement. The blame will be solely on c 1 . This allows a car to drive naturally at the center of its lane, without worrying about traffic which may unsafely deviate into its corridor. On the other hand, safe deviation to the opposite lane, a common maneuver required in dense urban traffic, is allowed. Considering the example of a car which initiates a reverse parking maneuver, it should start reversing while making sure the distance to cars behind it is safe. 
     Traffic Lights 
     In scenarios that include intersections with traffic lights, one might think that the simple rule for traffic lights scenarios is “if one car&#39;s route has the green light and the other car&#39;s route has a red light, then the blame is on the one whose route has the red light”. However, this is not the correct rule, especially in all cases. Consider for example the scenario depicted in  FIG. 27 . Even if the car  2701  is on a route having a green light, we do not expect it to ignore car  2703  that is already in the intersection. The correct rule is that the route that has a green light has priority over routes that have a red light. Therefore, we obtain clear reduction from traffic lights to the route priority concept we have described previously. 
     Unstructured Road 
     Turning to roads where no clear route geometry can be defined, consider first a scenario where there is no lane structure at all (e.g. a parking lot). A way to ensure that there will be no accidents can be to require that every car will drive in a straight line, while if a change of heading occurs, it must be done when there are no close cars in my surrounding. The rationale behind this is that a car can predict what other cars will do, and behave accordingly. If other cars deviate from this prediction (by changing heading), it is done with a long enough distance and therefore there may be enough time to correct the prediction. When there is lane structure, it may enable smarter predictions on what other cars will do. If there is no lane structure at all, a car will continue according to its current heading. Technically speaking, this is equivalent to assigning every car to a virtual straight route according to its heading. Next, consider the scenario in a large unstructured roundabout (e.g., around the Arc de Triomphe in Paris). Here, a sensible prediction is to assume that a car will continue according to the geometry of the roundabout, while keeping its offset. Technically, this is equivalent to assigning every car to a virtual arc route according to its current offset from the center of the roundabout. 
     The above described driving policy system (e.g., the RL system) may be implemented together with one or more of the described accident liability rules to provide a navigational system that takes into account potential accident liability when deciding on a particular navigational instruction to implement. Such rules may be applied during the planning phase; e.g., within a set of programmed instructions or within a trained model such that a proposed navigational action is developed by the system already in compliance with the rules. For example, a driving policy module may account for or be trained with, for example, one or more navigational rules upon which RSS is based. Additionally or alternatively, the RSS safety constraint may be applied as a filter layer through which all proposed navigational actions proposed by the planning phase are tested against the relevant accident liability rules to ensure that the proposed navigational actions are in compliance. If a particular action is in compliance with the RSS safety constraint, it may be implemented. Otherwise, if the proposed navigational action is not in compliance with the RSS safety constraint (e.g., if the proposed action could result in accident liability to the host vehicle based on one or more of the above-described rules), then the action is not taken. 
     In practice, a particular implementation may include a navigation system for a host vehicle. The host vehicle may be equipped with an image capture device (e.g., one or more cameras such as any of those described above) that, during operation, captures images representative of an environment of the host vehicle. Using the image information, a driving policy may take in a plurality of inputs and output a planned navigational action for accomplishing a navigational goal of the host vehicle. The driving policy may include a set of programmed instructions, a trained network, etc., that may receive various inputs (e.g., images from one or more cameras showing the surroundings of the host vehicle, including target vehicles, roads, objects, pedestrians, etc.; output from LIDAR or RADAR systems; outputs from speed sensors, suspension sensors, etc.; information representing one or more goals of the host vehicle—e.g., a navigational plan for delivering a passenger to a particular location, etc.). Based on the input, the processor may identify a target vehicle in the environment of the host vehicle, e.g., by analyzing camera images, LIDAR output, RADAR output, etc. In some embodiments, the processor may identify a target vehicle in the environment of the host vehicle by analyzing one or more inputs, such as one or more camera images, LIDAR output, and/or RADAR output. 
     Further, in some embodiments, the processor may identify a target vehicle in the environment of the host vehicle based on an agreement of a majority or combination of sensor inputs (e.g., by analyzing one or more camera images, LIDAR output, and/or RADAR output, and receiving a detection result identifying the target vehicle based on a majority agreement or combination of the inputs). 
     Based on the information available to the driving policy module, an output may be provided in the form of one or more planned navigational actions for accomplishing a navigational goal of the host vehicle. In some embodiments, the RSS safety constraint may be applied as a filter of the planned navigational actions. That is, the planned navigational action, once developed, can be tested against at least one accident liability rule (e.g., any of the accident liability rules discussed above) for determining potential accident liability for the host vehicle relative to the identified target vehicle. And, as noted, if the test of the planned navigational action against the at least one accident liability rule indicates that potential accident liability may exist for the host vehicle if the planned navigational action is taken, then the processor may cause the host vehicle not to implement the planned navigational action. On the other hand, if the test of the planned navigational action against the at least one accident liability rule indicates that no accident liability would result for the host vehicle if the planned navigational action is taken, then the processor may cause the host vehicle to implement the planned navigational action. 
     In some embodiments, the system may test a plurality of potential navigational actions against the at least one accident liability rule. Based on the results of the test, the system may filter the potential navigational actions to a subset of the plurality of potential navigational actions. For example, in some embodiments, the subset may include only the potential navigational actions for which the test against the at least one accident liability rule indicates that no accident liability would result for the host vehicle if the potential navigational actions were taken. The system may then score and/or prioritize the potential navigational actions without accident liability and select one of the navigational actions to implement based on, for example, an optimized score, or a highest priority. The score and/or priority may be based, for example, one or more factors, such as the potential navigational action viewed as being the most safe, most efficient, the most comfortable to passengers, etc. 
     In some instances, the determination of whether to implement a particular planned navigational action may also depend on whether a default emergency procedure would be available in a next state following the planned action. If a DEP is available, the RSS filter may approve the planned action. On the other hand, if a DEP would not be available, the next state may be deemed an unsafe one, and the planned navigational action may be rejected. In some embodiments, the planned navigational action may include at least one default emergency procedure. 
     One benefit of the described system is that to ensure safe actions by the vehicle, only the host vehicle&#39;s actions relative to a particular target vehicle need be considered. Thus, where more than one target vehicle is present, the planned action for the host vehicle may be tested for an accident liability rule sequentially with respect to the target vehicles in an influence zone in the vicinity of the host vehicle (e.g., within 25 meters, 50 meters, 100 meters, 200 meters, etc.). In practice, the at least one processor may be further programmed to: identify, based on analysis of the at least one image representative of an environment of the host vehicle (or based on LIDAR or RADAR information, etc.), a plurality of other target vehicles in the environment of the host vehicle and repeat the test of the planned navigational action against at least one accident liability rule for determining potential accident liability for the host vehicle relative to each of the plurality of other target vehicles. If the repeated tests of the planned navigational action against the at least one accident liability rule indicate that potential accident liability may exist for the host vehicle if the planned navigational action is taken, then the processor may cause the host vehicle not to implement the planned navigational action. If the repeated tests of the planned navigational action against the at least one accident liability rule indicate that no accident liability would result for the host vehicle if the planned navigational action is taken, then the processor may cause the host vehicle to implement the planned navigational action. 
     As noted, any of the rules described above can be used as the basis for the RSS safety test. In some embodiments, the at least one accident liability rule includes a following rule defining a distance behind the identified target vehicle within which the host vehicle may not proceed without a potential for accident liability. In other cases, the at least one accident liability rule includes a leading rule defining a distance forward of the identified target vehicle within which the host vehicle may not proceed without a potential for accident liability. 
     While the system described above can apply the RSS safety test to a single planned navigational action to test compliance with the rule that the host vehicle should not take any action for which it would be liable for a resulting accident, the test may be applied to more than one planned navigational action. For example, in some embodiments, the at least one processor, based on application of at least one driving policy may determine two or more planned navigational actions for accomplishing a navigational goal of the host vehicle. In these situations, the processor may test each of the two or more planned navigational actions against at least one accident liability rule for determining potential accident liability. And, for each of the two or more planned navigational actions, if the test indicates that potential accident liability may exist for the host vehicle if a particular one of the two or more planned navigational actions is taken, the processor may cause the host vehicle not to implement the particular one of the planned navigational actions. On the other hand, for each of the two or more planned navigational actions, if the test indicates that no accident liability would result for the host vehicle if a particular one of the two or more planned navigational actions is taken, then the processor may identify the particular one of the two or more planned navigational actions as a viable candidate for implementation. Next, the processor may select a navigational action to be taken from among the viable candidates for implementation based on at least one cost function and cause the host vehicle to implement the selected navigational action. 
     Because an implementation of RSS relates to a determination of relative potential liability for accidents between the host vehicle and one or more target vehicles, along with testing planned navigational actions for safety compliance, the system may track accident liability potential for encountered vehicles. For example, not only may the system be able to avoid taking an action for which a resulting accident would result in liability to the host vehicle, but the host vehicle systems may also be able to track one or more target vehicles and identify and track which accident liability rules have been broken by those target vehicles. In some embodiments, an accident liability tracking system for a host vehicle may include at least one processing device programmed to receive, from an image capture device, at least one image representative of an environment of the host vehicle and analyze the at least one image to identify a target vehicle in the environment of the host vehicle. Based on analysis of the at least one image, the processor may include programming to determine one or more characteristics of a navigational state of the identified target vehicle. The navigational state may include various operational characteristics of the target vehicle, such as vehicle speed, proximity to a center of a lane, lateral velocity, direction of travel, distance from the host vehicle, heading, or any other parameter that may be used to determine potential accident liability based on any of the rules described above. The processor may compare the determined one or more characteristics of the navigational state of the identified target vehicle to at least one accident liability rule (e.g., any of the rules described above, such as winning by lateral velocity, directional priority, winning by proximity to lane center, following or leading distance and cut-in, etc.). Based on comparison of the state to one or more rules, the processor may store at least one value indicative of potential accident liability on the part of the identified target vehicle. And in the case of an accident, the processor may provide an output of the stored at least one value (e.g., via any suitable data interface, either wired or wireless). Such an output may be provided, for example, after an accident between the host vehicle and at least one target vehicle, and the output may be used for or may otherwise provide an indication of liability for the accident. 
     The at least one value indicative of potential accident liability may be stored at any suitable time and under any suitable conditions. In some embodiments, the at least one processing device may assign and store a collision liability value for the identified target vehicle if it is determined that the host vehicle cannot avoid a collision with the identified target vehicle. 
     The accident liability tracking capability is not limited to a single target vehicle, but rather can be used to track potential accident liability for a plurality of encountered target vehicles. For example, the at least one processing device may be programmed to detect a plurality of target vehicles in the environment of the host vehicle, determine navigational state characteristics for each of the plurality of target vehicles, and determine and store values indicative of potential accident liability on the part of respective ones of the plurality of target vehicles based on comparisons of the respective navigational state characteristics for each of the target vehicles to the at least one accident liability rule. As noted, the accident liability rules used as the basis for liability tracking may include any of the rules described above or any other suitable rule. For example, the at least one accident liability rule may include a lateral velocity rule, a lateral position rule, a driving direction priority rule, a traffic light-based rule, a traffic sign-based rule, a route priority rule, etc. The accident liability tracking function may also be coupled with safe navigation based on RSS considerations (e.g., whether any action of the host vehicle would result in potential liability for a resulting accident). 
     In addition to navigating based on accident liability considerations according to 
     RSS, navigation can also be considered in terms of vehicle navigational states and a determination of whether a particular, future navigational state is deemed safe (e.g., whether a DEP exists such that accidents may be avoided or any resulting accident will not be deemed the fault of the host vehicle, as described in detail above). The host vehicle can be controlled to navigate from safe state to safe state. For example, in any particular state, the driving policy may be used to generate one or more planned navigational actions, and those actions may be tested by determining if the predicted future states corresponding to each planned action would offer a DEP. If so, the planned navigational action or actions providing the DEP may be deemed safe and may qualify for implementation. 
     In some embodiments, a navigation system for a host vehicle may include at least one processing device programmed to: receive, from an image capture device, at least one image representative of an environment of the host vehicle; determine, based on at least one driving policy, a planned navigational action for accomplishing a navigational goal of the host vehicle; analyze the at least one image to identify a target vehicle in the environment of the host vehicle; test the planned navigational action against at least one accident liability rule for determining potential accident liability for the host vehicle relative to the identified target vehicle; if the test of the planned navigational action against the at least one accident liability rule indicates that potential accident liability may exists for the host vehicle if the planned navigational action is taken, then cause the host vehicle not to implement the planned navigational action; and if the test of the planned navigational action against the at least one accident liability rule indicates that no accident liability would result for the host vehicle if the planned navigational action is taken, then cause the host vehicle to implement the planned navigational action. 
     In some embodiments, a navigation system for a host vehicle, may include at least one processing device programmed to: receive, from an image capture device, at least one image representative of an environment of the host vehicle; determine, based on at least one driving policy, a plurality of potential navigational actions for the host vehicle; analyze the at least one image to identify a target vehicle in the environment of the host vehicle; test the plurality of potential navigational actions against at least one accident liability rule for determining potential accident liability for the host vehicle relative to the identified target vehicle; select one of the potential navigational actions for which the test indicates that no accident liability would result for the host vehicle if the selected potential navigational action is taken; and cause the host vehicle to implement the selected potential navigational action. In some instances, the selected potential navigational action may be selected from a subset of the plurality of potential navigational actions for which the test indicates that no accident liability would result for the host vehicle if any of the subset of the plurality of potential navigational action were taken. Further, in some instances, the selected potential navigational action may be selected according to a scoring parameter. 
     In some embodiments, a system for navigating a host vehicle may include at least one processing device programmed to receive, from an image capture device, at least one image representative of an environment of the host vehicle and determine, based on at least one driving policy, a planned navigational action for accomplishing a navigational goal of the host vehicle. The processor may also analyze the at least one image to identify a target vehicle in the environment of the host vehicle; determine a next-state distance between the host vehicle and the target vehicle that would result if the planned navigational action was taken; determine a current maximum braking capability of the host vehicle and a current speed of the host vehicle; determine a current speed of the target vehicle and assume a maximum braking capability of the target vehicle based on at least one recognized characteristic of the target vehicle; and implement the planned navigational action if, given the maximum braking capability of the host vehicle and current speed of the host vehicle, the host vehicle can be stopped within a stopping distance that is less than the determined next-state distance summed together with a target vehicle travel distance determined based on the current speed of the target vehicle and the assumed maximum braking capability of the target vehicle. The stopping distance may further include a distance over which the host vehicle travels during a reaction time without braking. 
     The recognized characteristic of the target vehicle upon which the maximum braking capability of the target vehicle is determined may include any suitable characteristic. In some embodiments, the characteristic may include a vehicle type (e.g., motorcycle, car, bus, truck, each of which may be associated with different braking profiles), vehicle size, a predicted or known vehicle weight, a vehicle model (e.g., that may be used to look up a known braking capability), etc. 
     In some cases, the safe state determination may be made relative to more than one target vehicle. For example, in some cases a safe state determination (based on distance and braking capabilities) may be based on two or more identified target vehicles leading a host vehicle. Such a determination may be useful especially where information regarding what is ahead of the foremost target vehicle is not available. In such cases, it may be assumed for purposes of determining a safe state, safe distance, and/or available DEP that the foremost detectable vehicle will experience an imminent collision with an immovable or nearly immovable obstacle, such that the target vehicle following may reach a stop more quickly than its own braking profile allows (e.g., the second target vehicle may collide with a first, foremost vehicle and therefore reach a stop more quickly than expected max braking conditions). In such cases, it may be important to base the safe state, safe following distance, DEP determination upon the location of the foremost identified target vehicle relative to the host vehicle. 
     In some embodiments, such a safe state to safe state navigation system may include at least one processing device programmed to receive, from an image capture device, at least one image representative of an environment of the host vehicle. Here, as with other embodiments, the image information captured by an image capture device (e.g., a camera) may be supplemented with information obtained from one or more other sensors, such as a LIDAR or RADAR system. In some embodiments, the image information used to navigate may even originate from a LIDAR or RADAR system, rather than from an optical camera. The at least one processor may determine, based on at least one driving policy, a planned navigational action for accomplishing a navigational goal of the host vehicle. The processor may analyze the at least one image (e.g., obtained from any of a camera, a RADAR, a LIDAR, or any other device from which an image of the environment of the host vehicle may be obtained, whether optically based, distance map-based, etc.) to identify a first target vehicle ahead of the host vehicle and a second target vehicle ahead of the first target vehicle. The processor may then determine a next-state distance between the host vehicle and the second target vehicle that would result if the planned navigational action was taken. Next, the processor may determine a current maximum braking capability of the host vehicle and a current speed of the host vehicle. The processor may implement the planned navigational action if, given the maximum braking capability of the host vehicle and the current speed of the host vehicle, the host vehicle can be stopped within a stopping distance that is less than the determined next-state distance between the host vehicle and the second target vehicle. 
     That is, if the host vehicle processor determines that there is enough distance to stop in a next-state distance between the leading visible target vehicle and the host vehicle, without collision or without collision for which responsibility would attach to the host vehicle and assuming the leading visible target vehicle will suddenly at any moment come to a complete stop, then the processor of the host vehicle may take the planned navigational action. On the other hand, if there would be insufficient room to stop the host vehicle without collision, then the planned navigational action may not be taken. 
     Additionally, while the next-state distance may be used as a benchmark in some embodiments, in other cases, a different distance value may be used to determine whether to take the planned navigational action. In some cases, as in the one described above, the actual distance in which the host vehicle may need to be stopped to avoid a collision may be less than the predicted next-state distance. For example, where the leading, visible target vehicle is followed by one or more other vehicles (e.g., the first target vehicle in the example above), the actual predicted required stopping distance would be the predicted next-state distance less the length of the target vehicle(s) following the leading visible target vehicle. If the leading visible target vehicle comes to an immediate stop, it may be assumed that the following target vehicles would collide with the leading visible target vehicle and, therefore, they too would need to be avoided by the host vehicle to avoid a collision. Thus, the host vehicle processor can evaluate the next-state distance less the summed lengths of any intervening target vehicles between the host vehicle and the leading, visible/detected target vehicle to determine whether there would be sufficient space to bring the host vehicle to a halt under max braking conditions without a collision. 
     In other embodiments, the benchmark distance for evaluating a collision between the host vehicle and one or more leading target vehicles may be greater than the predicted next-state distance. For example, in some cases, the leading visible/detected target vehicle may come to a quick, but not immediate stop, such that the leading visible/detected target vehicle travels a short distance after the assumed collision. For example, if that vehicle hits a parked car, the colliding vehicle may still travel some distance before coming to a complete stop. 
     The distance traveled after the assumed collision may be less than an assumed or determined minimum stopping distance for the relevant target vehicle. Thus, in some cases, the processor of the host vehicle may lengthen the next-state distance in its evaluation of whether to take the planned navigational action. For example, in this determination, the next-state distance may be increased by 5%, 10%, 20%, etc. or may be supplemented with a predetermined fixed distance (10 m, 20 m, 50 m, etc.) to account for a reasonable distance that the leading/visible target vehicle may travel after an assumed imminent collision. 
     In addition to lengthening the next-state distance in the evaluation by an assumed distance value, the next-state distance may be modified by both accounting for a distance traveled after collision by the leading visible/detected target vehicle and the lengths of any target vehicles following the leading visible/detected target vehicle (which may be assumed to pile up with the leading visible/detected vehicle after its sudden stop). 
     In addition to basing the determination on whether to take the planned navigational action on the next-state distance between the host vehicle and the leading visible/detected target vehicle (as modified by considering the post-collision movement of the leading visible/detected target vehicle and/or the lengths of vehicles following the leading visible/detected target vehicle), the host vehicle may continue to account for the braking capability of one or more leading vehicles in its determination. For example, the host vehicle processor may continue to determine a next-state distance between the host vehicle and the first target vehicle (e.g., a target vehicle following the leading visible/detected target vehicle) that would result if the planned navigational action was taken; determine a current speed of the first target vehicle and assume a maximum braking capability of the first target vehicle based on at least one recognized characteristic of the first target vehicle; and not implement the planned navigational action if, given the maximum braking capability of the host vehicle and the current speed of the host vehicle, the host vehicle cannot be stopped within a stopping distance that is less than the determined next-state distance between the host vehicle and the first target vehicle summed together with a first target vehicle travel distance determined based on the current speed of the first target vehicle and the assumed maximum braking capability of the first target vehicle. Here, as in the examples described above, the recognized characteristic of the first target vehicle may include a vehicle type, a vehicle size, a vehicle model, etc. 
     In some cases (e.g., through actions of other vehicles), the host vehicle may determine that a collision is imminent and unavoidable. In such cases, the processor of the host vehicle may be configured to select a navigational action (if available) for which the resulting collision would result in no liability to the host vehicle. Additionally or alternatively, the processor of the host vehicle may be configured to select a navigational action that would offer less potential damage to the host vehicle or less potential damage to a target object than the current trajectory or relative to one or more other navigational options. Further, in some cases, the host vehicle processor may select a navigational action based on considerations of the type of object or objects for which a collision is expected. For example, when faced with a collision with a parked car for a first navigational action or with an immovable object for a second navigational action, the action offering the lower potential damage to the host vehicle (e.g., the action resulting in a collision with the parked car) may be selected. When faced with a collision with a car moving in a similar direction as the host vehicle for a first navigational action or with a parked car for a second navigational action, the action offering the lower potential damage to the host vehicle (e.g., the action resulting in a collision with the moving car) may be selected. 
     When faced with a collision with a pedestrian as a result of a first navigational action or with any other object for a second navigational action, the action offering any alternative to colliding with a pedestrian may be selected. 
     In practice, the system for navigating a host vehicle may include at least one processing device programmed to receive, from an image capture device, at least one image representative of an environment of the host vehicle (e.g., a visible image, LIDAR image, RADAR image, etc.); receive from at least one sensor an indicator of a current navigational state of the host vehicle; and determine, based on analysis of the at least one image and based on the indicator of the current navigational state of the host vehicle, that a collision between the host vehicle and one or more objects is unavoidable. The processor may evaluate available alternatives. For example, the processor may determine, based on at least one driving policy, a first planned navigational action for the host vehicle involving an expected collision with a first object and a second planned navigational action for the host vehicle involving an expected collision with a second object. The first and second planned navigational actions may be tested against at least one accident liability rule for determining potential accident liability. If the test of the first planned navigational action against the at least one accident liability rule indicates that potential accident liability may exist for the host vehicle if the first planned navigational action is taken, then the processor may cause the host vehicle not to implement the first planned navigational action. If the test of the second planned navigational action against the at least one accident liability rule indicates that no accident liability would result for the host vehicle if the second planned navigational action is taken, then the processor may cause the host vehicle to implement the second planned navigational action. The objects may include other vehicles or non-vehicle objects (e.g., road debris, trees, poles, signs, pedestrians, etc.). 
     The following figures and discussion provide examples of various scenarios that may occur when navigating and implementing the disclosed systems and methods. In these examples, a host vehicle may avoid taking an action that would result in blame attributable to the host vehicle for a resulting accident if the action is taken. 
       FIGS. 28A and 28B  illustrate example following scenarios and rules. As shown in  FIG. 28A , the region surrounding vehicle  2804  (e.g., a target vehicle) represents a minimum safe distance corridor for vehicle  2802  (e.g., a host vehicle), which is traveling in the lane a distance behind vehicle  2804 . According to one rule consistent with disclosed embodiments, to avoid an accident in which blame is attributable to vehicle  2802 , vehicle  2802  must maintain a minimum safe distance by remaining in the region surrounding vehicle  2802 . In contrast, as shown in  FIG. 28B , if vehicle  2804  brakes, then vehicle  2802  will be at fault if there is an accident. 
       FIGS. 29A and 29B  illustrate example blame in cut-in scenarios. In these scenarios, safe corridors around vehicle  2902  determine the fault in cut-in maneuvers. As shown in  FIG. 29A , vehicle  2902  is cutting in front of vehicle  2902 , violating the safe distance (depicted by the region surrounding vehicle  2902 ) and therefore is at fault. As show in  FIG. 29B , vehicle  2902  is cutting in front of vehicle  2902 , but maintains a safe distance in front of vehicle  2904 . 
       FIGS. 30A and 30B  illustrate example blame in cut-in scenarios. In these scenarios, safe corridors around vehicle  3004  determine whether vehicle  3002  is at fault. In  FIG. 30A , vehicle  3002  is traveling behind vehicle  3006  and changes into the lane in which target vehicle  3004  is traveling. In this scenario, vehicle  3002  violates a safe distance and therefore is at fault if there is an accident. In  FIG. 30B , vehicle  3002  cuts in behind vehicle  3004  and maintains a safe distance. 
       FIGS. 31A-31D  illustrate example blame in drifting scenarios. In  FIG. 31A , the scenario starts with a slight lateral maneuver by vehicle  3104 , cutting-in to the wide corridor of vehicle  3102 . In  FIG. 31B , vehicle  3104  continues cutting into the normal corridor of the vehicle  3102 , violating a safe distance region. Vehicle  3104  is to blame if there is an accident. In Fig.  31 C, vehicle  3104  maintains its initial position, while vehicle  3102  moves laterally “forcing” a violation of ae normal safe distance corridor. Vehicle  3102  is to blame if there is an accident. In  FIG. 31B , vehicle  3102  and  3104  move laterally towards each other. The blame is shared by both vehicles if there is an accident. 
       FIGS. 32A and 32B  illustrate example blame in two-way traffic scenarios. In  FIG. 32A , vehicle  3202  overtakes vehicle  3206 , and vehicle  3202  has performed a cut-in maneuver maintaining a safe distance from vehicle  3204 . If there is an accident, vehicle  3204  is to blame for not braking with reasonable force. In  FIG. 32B , vehicle  3202  cuts-in without keeping safe longitudinal distance from vehicle  3204 . In case of an accident, vehicle  3202  is to blame. 
       FIGS. 33A and 33B  illustrate example blame in two-way traffic scenarios. In  FIG. 33A , vehicle  3302  drifts into the path of oncoming vehicle  3204 , maintaining a safe distance. In case of an accident, vehicle  3204  is to blame for not braking with reasonable force. In  FIG. 33B , vehicle  3202  drifts into the path of the oncoming vehicle  3204 , violating a safe longitudinal distance. In case of an accident, vehicle  3204  is to blame. 
       FIGS. 34A and 34B  illustrate example blame in route priority scenarios. In  FIG. 34A , vehicle  3202  runs a stop sign. Blame is attributed to vehicle  3202  for not respecting the priority assigned to vehicle  3204  by the traffic light. In  FIG. 34B , although vehicle  3202  did not have priority, it was already in the intersection when vehicle  3204 ′s light turned green. If vehicle  3204  hits  3202 , vehicle  3204  would be to blame. 
       FIGS. 35A and 35B  illustrate example blame in route priority scenarios. In  FIG. 35A , vehicle  3502  backing-up into the path of an oncoming vehicle  3504 . Vehicle  3502  performs a cut-in maneuver maintaining a safe distance. In case of an accident, vehicle  3504  is to blame for not braking with reasonable force. In  FIG. 35B , vehicle  3502  car cuts-in without keeping a safe longitudinal distance. In case of an accident, vehicle  3502  is to blame. 
       FIGS. 36A and 36B  illustrate example blame in route priority scenarios. In  FIG. 36A , vehicle  3602  and vehicle  3604  are driving in the same direction, while vehicle  3602  turns left across the path of vehicle  3604 . Vehicle  3602  performs cut-in maneuver maintaining safe distance. In case of an accident, vehicle  3604  is to blame for not braking with reasonable force. In  FIG. 36B , vehicle  3602  cuts-in without keeping safe longitudinal distance. In case of an accident, vehicle  3602  is to blame. 
       FIGS. 37A and 37B  illustrate example blame in route priority scenarios. In  FIG. 37A , vehicle  3702  wants to turn left, but must give way to the oncoming vehicle  3704 . Vehicle  3702  turns left, violating safe distance with respect to vehicle  3704 . Blame is on vehicle  3702 . In  FIG. 37B , vehicle  3702  turns left, maintaining a safe distance with respect to vehicle  3704 . In case of an accident, vehicle  3704  is to blame for not braking with reasonable force. 
       FIGS. 38A and 38B  illustrate example blame in route priority scenarios. In  FIG. 38A , vehicle  3802  and vehicle  3804  are driving straight, and vehicle  3802  has a stop sign. Vehicle  3802  enters the intersection, violating a safe distance with respect to vehicle  3804 . Blame is on vehicle  3802 . In  FIG. 38B , vehicle  3802  enters the intersection while maintaining a safe distance with respect to vehicle  3804 . In case of an accident, vehicle  3804 is to blame for not braking with reasonable force. 
       FIGS. 39A and 39B  illustrate example blame in route priority scenarios. In  FIG. 39A , vehicle  3902  wants to turn left, but must give way to vehicle  3904  coming from its right. Vehicle  3902  enters the intersection, violating the right-of-way and a safe distance with respect to vehicle  3904 . Blame is on vehicle  3902 . In  FIG. 39B , vehicle  3902  enters the intersection while maintaining the right-of-way and a safe distance with respect to vehicle  3904 . In case of an accident, vehicle  3904  is to blame for not braking with reasonable force. 
       FIGS. 40A and 40B  illustrate example blame in traffic light scenarios. In  FIG. 40A , vehicle  4002  is running a red light. Blame is attributed to vehicle  4002  for not respecting the priority assigned to vehicle  4004  by the traffic light. In  FIG. 40B , although vehicle  4002  did not have priority, it was already in the intersection when the light for vehicle  4004  turned green. If vehicle  4004  hits vehicle  4002 , vehicle  4004  would be to blame. 
       FIGS. 41A and 41B  illustrate example blame in traffic light scenarios. Vehicle  4102  is turning left across the path of the oncoming vehicle  4104 . Vehicle  4104  has priority. In  FIG. 41 , vehicle  4102  turns left, violating a safe distance with respect to vehicle  4104 . Blame is attributed to vehicle  4102 . In  FIG. 41B , vehicle  4102  turns left, maintaining a safe distance with respect to vehicle  4104 . In case of an accident, vehicle  4104  is to blame for not braking with reasonable force. 
       FIGS. 42A and 42B  illustrate example blame in traffic light scenarios. In  FIG. 42A , vehicle  4202  is turning right, cutting into the path of vehicle  4204 , which is driving straight. Right-on-red is assumed to be a legal maneuver, but vehicle  4204  has right of way, as vehicle  4202  violates a safe distance with respect to vehicle  4204 . Blame is attributed to vehicle  4202 . In  FIG. 42B , vehicle  4202  turns right, maintaining a safe distance with respect to vehicle  4204 . In case of an accident, vehicle  4204  is to blame for not braking with reasonable force. 
       FIGS. 43A-43C  illustrate example vulnerable road users (VRUs) scenarios. Accidents with animals or VRUs where the vehicle performs a maneuver are treated as a variation of a cut-in, where the default blame is on the car, with some exceptions. In  FIG. 43A , vehicle  4302  cuts into the path of an animal (or VRU) while maintaining safe distance and ensuring an accident can be avoided. In  FIG. 43B , vehicle  4302  cuts into the path of an animal (or VRU) violating safe distance. Blame is attributed to vehicle  4302 . In  FIG. 43C , vehicle  4302  notices the animal and stops, giving the animal sufficient time to stop. If the animal hits the car, the animal is to blame. 
       FIGS. 44A-44C  illustrate example vulnerable road users (VRUs) scenarios. In  FIG. 44A , vehicle  4402  is turning left at a signalized intersection and encounters a pedestrian in the crosswalk. Vehicle  4402  has a red light and the VRU has a green. Vehicle  4402  is at fault. In  FIG. 44B , vehicle  4402  has a green light, and the VRU has a red light. If the VRU enters the crosswalk, the VRU is at fault. In  FIG. 44C , vehicle  4402  has a green light, and the VRU has a red light. If the VRU was already in the crosswalk, vehicle  4402  is it fault. 
       FIGS. 45A-45C  illustrate example vulnerable road users (VRUs) scenarios. In  FIG. 45A , vehicle  4402  is turning right and encounters a cyclist. The cyclist has a green light. Vehicle  4502  is at fault. In  FIG. 45B , the cyclist has a red light. If the cyclist enters the intersection, the cyclist is at fault. In  FIG. 45C , the cyclist has a red light, but was already in the intersection. Vehicle  4502  is at fault. 
       FIGS. 46A-46D  illustrate example vulnerable road users (VRUs) scenarios. Accidents with VRUs where a vehicle does not perform performance a maneuver are blamed on the car by default, with some exceptions. In  FIG. 46A , vehicle  4602  must always make sure to maintain safe distance and ensuring an accident can be avoided with a VRU. In  FIG. 46B , if vehicle  4602  does not maintain safe distance, vehicle  4602  is to blame. In  FIG. 46C , if vehicle  4602  does not maintain sufficiently low speed as to avoid colliding with a VRU that is potentially occluded by vehicle  5604 , or drives above the legal limit, vehicle  4602  is to blame. In  FIG. 46D , in another scenario with a potential occlusion of a VRU by vehicle  4604 , if vehicle  4602  maintains sufficiently low speed, but the VRUs speed is above a reasonable threshold, the VRU is to blame. 
     As disclosed herein, RSS defines a framework for multi-agent scenarios. Accidents with static objects, road departures, loss of control or vehicle failure are blamed on the host. RSS defines cautious maneuvers, which will not allow accidents with other objects, unless other objects maneuvers dangerously into the path of the host (in which case they are to blame). In the case of a sure-collision where blame is on the target, the host will apply its brakes. The system may consider evasive steering only if the maneuver is “cautious” (perceived not to cause another accident). 
     Non-collision incidents include accidents initiated by vehicle fires, potholes, falling objects, etc. In these cases, the blame may default to the host, except for scenarios which the host can avoid, such as potholes and falling objects, which may be classified as a “static object” scenario, assuming they become visible at safe distance or that a cautious evasive maneuver exists. In multi-agent scenarios where the host vehicle was stationary, the host is not to blame. In this case the target essentially performed a non-safe cut-in. For example, if a cyclist rides into a stationary car, the host is not to blame. 
     RSS also includes guidelines for assigning blame where the road is not structured clearly, such as parking lots or wide roundabouts without lane marks. In these unstructured road scenarios, blame is assigned by examining deviations of each vehicle from its path to determine if they allowed sufficient distance to allow other objects in the area to adjust. 
     Navigation Scenarios and Collision Prediction 
     As described above, an autonomous or partially autonomous vehicle may use a variety of techniques for navigating through an environment of the vehicle. To ensure the safety of drivers or passengers of the host vehicle, as well as drivers or passengers of other vehicles, or pedestrians, the vehicle&#39;s navigation system may be configured to handle a wide variety of driving scenarios, from those commonly experienced during vehicle drives to less common scenarios or even unexpected scenarios. Certain scenarios, including situations where the host vehicle is turning, may present unique challenges for accurately predicting collisions with objects or pedestrians. The navigation system may need to make accurate assessments of objects within the environment of the vehicle and quickly and efficiently make navigation action decisions. Accordingly, advanced collision prediction techniques may be provided to address these scenarios. 
     Certain scenarios may serve as benchmarks for evaluating the response and performance of an autonomous or partially autonomous driving system. In some cases, such benchmark scenarios may be developed and distributed by regulatory boards, government entities, or the like. One such program is the European New Car Assessment Program (Euro NCAP), which provides a five-star safety rating system to help consumers, their families, and businesses compare vehicles and to help them identify the safest choice for their needs. The safety rating is determined from a series of vehicle tests, designed and carried out by Euro NCAP. These tests may represent important real-life accident scenarios that could result in injured or killed car occupants or other road users. While a vehicle navigation system should be capable of responding and navigating safely relative to a wide range of encountered driving scenarios, the Euro NCAP tests may provide a set of discrete driving scenarios relative to which the performance of an AV may be evaluated. One such scenario is the CPTA (car to pedestrian turning adult) situation in which a host vehicle is approaching an intersection from one direction and a pedestrian is crossing the intersection in an opposite direction. The host vehicle may turn either left or right at the intersection and may cross the path of the pedestrian. This scenario may be similar to the scenario illustrated in  FIG. 47  and described in greater detail below. 
     Certain challenges may be associated with accident prediction in such a scenario. For example, collision prediction may depend on a predictive behavior of the host vehicle (e.g., whether the host vehicle will continue to increase yaw during the turn, how the yaw will increase, whether the host vehicle speed will change, etc.). Using the disclosed techniques, the vehicle navigation system may be configured to predict the path of the host vehicle and the pedestrian through the intersection. In some embodiments, the system may vary the predicted path of the host vehicle depending on the speed. For example, a higher speed may result in a larger turning radius relative to lower speeds. The vehicle navigation system may be configured to predict collisions between the host vehicle and the pedestrian even with changing yaw of the host vehicle due to the turning maneuver. In some embodiments, the collision may be predicted using a simplified circle-based technique, described in detail below. This technique may allow for the vehicle navigations system to efficiently analyze the scenario by identifying particular times during the scenario where a collision is most likely to occur and performing a detailed analysis on those particular times. While the CPTA scenario is used herein by way of example, it is to be understood that the present disclosure is not limited to any particular scenario and may apply to a variety of situations a vehicle navigation system may encounter. 
       FIG. 47  is an illustration of an example scenario in which collision prediction may be used, consistent with the disclosed embodiments.  FIG. 47  depicts a road that includes a junction or intersection  4700 . Host vehicle  4710  may be approaching intersection  4700  along a road segment  4702 , as shown. Host vehicle  4710  may be planning to turn at intersection  4700  onto road segment  4704  along a path  4730 . A pedestrian  4720  may be crossing intersection  4700  such that the path  4740  of pedestrian  4720  crosses path  4730  of vehicle  4710 . In some embodiments, path  4740  of pedestrian  4720  may be in a direction that is opposite to the direction from which vehicle  4710  is approaching intersection  4700 , as shown. 
     In some embodiments, the scenario shown in  FIG. 47  may conform to the specifications of the CPTA scenario as prescribed by Euro NCAP, however the present disclosure is not limited to any particular scenario. Accordingly, it is to be understood that the scenario illustrated in  FIG. 47  is used by way of example and the disclosed techniques may apply to other scenarios. For example, pedestrian  4720  may be crossing road segment  4706  or  4708 , and vehicle  4710  may be turning onto road segment  4706  (e.g., along path  4734 ) or continuing straight (e.g., along path  4732 ). Further, pedestrian  4720  may be crossing roadway  4704  at an angle or in an opposite direction than shown. In some embodiments, pedestrian  4720  may be another moving object, such as an animal, a cyclist, another vehicle, another form of vulnerable road user (VRU), etc. 
     Host vehicle  4710  may be configured to capture one or more images, as described in various embodiments above. For example, host vehicle  4710  may correspond to vehicle  200  and may capture images using image acquisition unit  120 . Accordingly, any of the embodiments or descriptions above with respect to vehicle  200  or other host vehicles may similarly apply to host vehicle  4170 . In some embodiments, host vehicle  4710  may analyze captured images to identify a representation of intersection  4700  and a representation of pedestrian  4720  within the vicinity of intersection  4700  in the captured images. Host vehicle  4710  may be further be configured to predict a path for pedestrian  4720 . In some embodiments, the path may be based on images captured by host vehicle  4710 . For example, based on the images, the vehicle navigation system may identify a position of pedestrian  4720 , and the vehicle navigation system may assume a straight path of travel based on the current position of pedestrian  4720 . In some embodiments, path  4740  of pedestrian  4720  may be assumed to extend perpendicular to road section  4704  such that pedestrian  4720  is walking straight across the road, as show in  FIG. 47 . In some embodiments, a facing direction of the pedestrian may also be determined and path  4740  may extend along a straight line in the facing direction of pedestrian  4720 . The position and facing direction of pedestrian  4720  may be determined based on any suitable method, including one or more image analysis algorithms or a trained system that was trained to receive an image and provide as an output a facing direction of the pedestrian. In some embodiments, other sensors, such as LIDAR, RADAR, etc. may be used in addition to, or as an alternative to, image-based detection. 
     Consistent with the embodiments of the present disclosure, the predicted path  4740  for pedestrian  4720  may be determined using other methods. For example, the vehicle navigation system may track the movement of pedestrian  4720  over time. This may include tracking a movement of pedestrian  4720  over a sequence of images captured by host vehicle  4710 . In some embodiments, path  4740  may be predicted based on a determined position of pedestrian  4720  within the images. This may include identifying a point of contact between the pedestrian and a road or ground surface for each captured image. A road model may be developed that maps an image coordinate (e.g., represented as (x,y)) to a distance Z assuming the point in the image is on a surface of the road. This model can be a simple planar model where Z is a function of the camera height, focal length and the vanishing line of the road plane. In some embodiments, a more complex model which models a curved surface may be used. Accordingly, the mapping may give the point (X,Y,Z) where the vector (x,y,f) intersects the road plane model. 
     In some embodiments, the vehicle navigation system may be configured to use a consistent set of coordinates across each of the images used to determine a position of pedestrian  4720 . For example, the system may use a coordinate system based on an initial frame (frame  0 ). For a later frame (frame i) the system may use the ego motion of host vehicle  4710  (which may be tracked using various sensors as described above) to convert the vector (x i ; y i ; f) to a vector in the coordinates of frame  0 . Based on the determined positions of pedestrian  4720  in the sequence of images, a path of the pedestrian may be identified. For example, the path may be represented as a spline along the surface of the road. Determining a predicted path  4740  of pedestrian  4720  may include extrapolating the detected path of pedestrian  4740  across intersection  4700  either as a straight path or a curved path, depending on the spline. 
     Accordingly, path  4740  may assume that pedestrian  4720  maintains the same general trajectory detected in the images. In some embodiments, a trained system may be used. For example, a neural network or other machine learning algorithms may be trained over a large set of images and corresponding (future) pedestrian trajectories in various scenarios and scenes (e.g., used a ground truth). The trained model may provide an estimated future trajectory for a pedestrian based on at least one image of the pedestrian. 
     As another example, the path  4740  of pedestrian  4720  may be predicted by tracking features associated with pedestrian  4740  in the images over time. For example, the scale or expansion of the distance between any two points on the pedestrian across a sequence of images may be used to determine the change in distance from host vehicle  4710  to pedestrian  4720 . Assuming w 0  is the image distance between two points (e.g., measured in pixels or other units of measurement relative to the image) at a time t 0  when pedestrian  4720  is an estimated distance of Z 0  from host vehicle  4710 , and w i  is the image distance between the two points at time t i , the distance to pedestrian  4720  at time t i  may be represented as 
     
       
         
           
             
               
                 Z 
                 i 
               
               = 
               
                 
                   
                     w 
                     0 
                   
                   ⁢ 
                   
                     Z 
                     0 
                   
                 
                 
                   w 
                   i 
                 
               
             
             . 
           
         
       
     
     Various points associated with pedestrian  4720  may be used to determine distances w 0  and w i . In some embodiments the image distance from the head to the feet of pedestrian  4720  may be used. This relatively large distance may provide more accurate measurements than a shorter distance (e.g., two points on the torso of pedestrian  4720 ). However, because the feet are connected to pedestrian  4720 &#39;s body by an articulated member (i.e., the legs), this may add noise to the measured scale of pedestrian  4720 . Accordingly, other points, such as the distance from the waist to the head may be used. Because this distance is shorter, errors due to tracking might be relatively larger but errors due to lack of rigidity between the points may be decreased. Various other suitable points may be used. 
     In some embodiments, the same process may be repeated using multiple sets of points and an average may be used to track changes in distance from host vehicle  4710  to pedestrian  4720 . For example, a sequence of multiple images may be analyzed to identify points to be used for scale detection. In some embodiments, features may be identified by partitioning a rectangle bounding a representation of pedestrian  4720  into a 3×13 grid of sub-rectangles. In each sub-rectangle a “best” point may be selected, for example, using a Harris point detector. Alternatively, or additionally, feature points may be detected using a scale-invariant feature transform (SIFT) or a trained neural network. These points may then be matched with points on pedestrian  4720  in a subsequent image. Such matching can performed, for example, using a sum square distance (SSD), a sum absolute distance (SAD), a normalized correlation, or other suitable means. This can be performed on particular region of the subsequent image where the feature is expected (e.g., a 7×7 patch of pixels centered around the point, or other sizes). A random feature point from the top 8 rows of sub-rectangles and one from the bottom 5 rows may be used to compute a scale using the equation above. This may be repeated using several other sets of points and a confidence level may be determined based on the degree of correlation between the determined scale determinations. 
     In the example scenario shown in  FIG. 47 , a distance Z from host vehicle  4710  to pedestrian  4720  in camera coordinates may not be well defined and the two points on the image may move due to rotation of host vehicle  4710  along path  4730 . To correct this, the ego motion of host vehicle  4710  may be used to rotate the coordinate frame of a given image (e.g., image I i ) to be aligned with the coordinate frame of an initial image I 0 . Accordingly, the distance Z i  determined based on image I i  may be represented in terms of the coordinate frame of initial image I 0 . Based on the distances Z calculated in a sequence of images, the path of pedestrian  4720  may be tracked over time. Accordingly, the predicted path  4740  of pedestrian  4720  may be determined by extrapolating the detected path of pedestrian  4740  either as a straight path or a curved path, as described above. In some embodiments, the determined positions of and distances to pedestrian  4720  may also be used to calculate a projected speed of pedestrian  4720 . 
     Host vehicle  4710  may similarly determine a path  4730  for host vehicle  4710 . In some embodiments, path  4730  depend on a navigational action for the vehicle relative to intersection  4700 . For example, the navigational action may define whether host vehicle  4710  will turn left onto road segment  4704 , turn right onto road segment  4706 , or continue straight along road segment  4708 . The navigational action may be based, at least in part, on routing information for host vehicle  4710 . For example, the vehicle navigational system may determine that road segment  4704  provides the best route for reaching a desired destination of host vehicle  4710  and thus the navigational system may determine a navigational action corresponding to a left turn. 
     In some embodiments, the vehicle navigation system may predict path  4730  based on information obtained by accessing a road navigation map. For example, the map may be stored in map database  160 , as described above. In some embodiments, the map may include a sparse data model including polynomial representations of target trajectories for the host vehicle relative to intersection  4700 . The vehicle navigation system may select a route consistent with the determined navigational action as predicted path  4730 . 
     In some embodiments, the predicted path  4730  may differ in some respects to the target trajectories stored in the map. For example, path  4736  may represent the target trajectory associated with a left turn at intersection  4700 , which may differ from predicted path  4730 . In some embodiments, the difference may depend on a speed of vehicle  4710 . For example, target trajectory  4730  may be associated with a speed that is lower than a current speed of host vehicle  4710 . Accordingly, host vehicle  4710  may determine a predicted path with a smaller turn radius than target trajectory  4736 , such as path  4730 . Similarly, host vehicle  4710  may determine a predicted path with a larger turn radius, such as path  4738  based on a higher speed. Various other factors may influence the predicted path in relation to the target trajectory. For example, predicted path  4730  may differ from target trajectory  4736  based on a driving policy. In some embodiments, path  4730  may differ from target trajectory  4736  based on a user input. For example, a remote operator (e.g., if vehicle  4710  is a taxi) a driver, a system administrator, etc. may input information to define path  4730 . Other factors, such as a surface of the road, obstacles within intersection  4700 , a maintenance condition of vehicle  4710  or similar factors may also affect predicted path  4730 . 
     The predicted path  4730  may include a plurality of segments. For example, as shown in  FIG. 47 , path  4730  may include a first straight trajectory along road segment  4702 , an intermediate curved trajectory through intersection  4700 , and a second straight trajectory along road segment  4704 . In some embodiments, additional segments may be included in path  4730 , including a turn initiation segment, a turn completion segment, one or more intermediary trajectories (either straight or curved), etc. The intermediate curved trajectory may have a constant radius of curvature as shown in  FIG. 47  (which may be consistent with the CPTA scenario discussed above). In other embodiments, the curved trajectory may have a variable radius of curvature. For example, host vehicle  4710  may decrease the radius of curvature of path  4730  as host vehicle  4710  decelerates through the acceleration (or, similarly, increase the radius of curvature through an acceleration). Further by way of example, the behavior of the vehicle through any segment of a turn can influence the predicted behavior of the vehicle through one or more future turn segments. For example, the turn behavior prediction may be estimated for each turn segment, and may be influenced by factors such as the geometry of the road or junction, characteristics of the vehicle, conditions of the vehicle  4710  or various vehicle systems, ambient conditions (e.g., weather conditions, etc.), a condition of the road surface, the lighting conditions, user preferences, historical data (e.g., indicating risk or safety of the junction, an indication of average pedestrian traffic in the junction or at that particular cross in general or at a similar time), supplementary information(e.g., from social media networks, an indication of a potential for irregular heavy pedestrian traffic in the area or at this particular crossing, etc.). In some embodiments, the radius of curvature may have a constant rate of change along path  4730 . 
     Based on the predicted paths of host vehicle  4710  and pedestrian  4720 , the vehicle navigation system may determine whether host vehicle  4710  is projected to collide with pedestrian  4720 . In this context, a collision may be defined as any contact between host vehicle  4710  and pedestrian  4720  (or envelopes associated with vehicle  4710  and pedestrian  4720 ). A collision may be predicted according to various methods described in greater detail below. In the event that host vehicle  4710  is projected to collide with pedestrian  4720 , the vehicle navigation system may implement a collision mitigation action in response to the projected collision. In some embodiments, this may include a navigational action by host vehicle  4710 . For example, the collision mitigation action may include steering host vehicle  4710  away from path  4730  (e.g., in front of pedestrian  4720 , behind pedestrian  4720 , etc.). In some embodiments, the collision mitigation action may include applying the brakes of host vehicle  4710 , for example, to cause host vehicle  4710  come to a stop, or to slow down such that host vehicle  4710  is no longer projected to collide with pedestrian  4720 . In some embodiments, the collision mitigation action may include generating an alert or notification. For example, the vehicle navigation system may issue an audible or visual alert (e.g., through user interface  170 ) indicating a projected collision. This may prompt a driver of host vehicle  4710  to intervene and perform a driver-initiated navigational response (e.g., steering, applying the brake, etc.). 
     A variety of techniques and methods may be implemented to determine whether host vehicle  4710  is projected to collide with pedestrian  4720 . In some embodiments, a projected collision may be detected based on an envelope associated with host vehicle  4710  and/or pedestrian  4720 .  FIGS. 48, 49, and 50  illustrate example envelopes that may be determined for host vehicle  4710  and/or pedestrian  4720 , consistent with the disclosed embodiments. Although the envelopes shown in the examples are generally circular in the shape, one skilled in the art will appreciate that an envelope may have other shapes. For example, an envelope may be circular, oval, rectangular, square, triangular, or may have an irregular shape. In some embodiments, a partial envelop may be used (e.g., an envelope defined on a particular side of an object or person) and the envelope may partially surround an object or person. For example, in some embodiments, and envelope can be used only on the side where that is estimated to be closest to the pedestrian at an estimated closet point of approach or contact. For example, in case the vehicle is turning left, the envelope and the pedestrian is coming from an opposite direction and to the left of the vehicle, such that the closest side of the vehicle at an estimated closet point of approach or contact with the pedestrian is the left front of the vehicle the envelope may be added around that portion of the vehicle only. In some embodiments, an envelope that is partial may include two or more non-contiguous portions. 
     As shown in  FIG. 48 , an envelope  4810  may be determined for host vehicle  4710 . As used herein, an envelope may include any 2-dimensional or 3-dimensional shape representing an outer boundary of an object for purposes of defining the object in space. As shown in  FIG. 48 , envelope  4810  may include a bounding box representative of host vehicle  4710 . The bounding box may include six faces representing outermost edges of host vehicle  4710  along multiple axes. Envelope  4810  may include a predetermined frontal area  4812  representing a frontmost portion of host vehicle  4710  (e.g., the frontmost point on a bumper of host vehicle  4710 , etc.). Similarly, an exposed area  4820  of pedestrian  4720  may also be determined. The determination of whether host vehicle  4710  is projected to collide with pedestrian  4720  may be determined based on whether frontal area  4812  is predicted to collide with exposed area  4820 . Although not shown in  FIG. 48 , an envelope, such as a bounding box, may similarly be determined for pedestrian  4720 . Further, while a rectangular prism is used to represent envelope  4810  in  FIG. 48 , this is provided by way of example. Various other shapes and/or configuration so envelope  4810  may be used. 
     In some embodiments, envelope  4810  may include a predetermined buffer distance relative to the edges of host vehicle  4710 .  FIG. 49  is a block diagram illustrating an envelope  4914  including a predetermined buffer distance, consistent with the disclosed embodiments. Envelope  4912  may represent an envelope associated with edges of host vehicle  4710 , similar to envelope  4810 , as described above. Envelope  4914  may include a predetermined buffer distance (D 1 ) from envelope  4912 , as shown in  FIG. 49 . A collision between host vehicle  4710  may be determined based on whether envelope  4914  will intersect with any part of pedestrian  4720 . Pedestrian  4720  may have an associated envelope  4920  and the determination of whether host vehicle  4710  and pedestrian  4720  are projected to collide may be based on whether envelope  4920  collides with envelope  4914 . In some embodiments, envelope  4914  may be a predefined envelope associated with host vehicle  4710 . For example, a driving policy associated with host vehicle  4710  may define a distance D 1  to be used for collision detection purposes. Buffer distance D 1  may be any suitable distance (e.g., 1 meter, 2 meters, 3 meters, etc.). In some embodiments, buffer distance D 1  may be context-dependent. For example, buffer distance D 1  may vary depending on a current speed of host vehicle  4710 . In such embodiments, a higher speed may represent a higher risk of injury to pedestrian  4720  and/or passengers of host vehicle  4710 , and thus a greater buffer distance may be used. The greater buffer distance may further account for increased reaction times required to perform a collision mitigation action at a higher speed. In some embodiments, the buffer distance D 1  may depend, at least partially, on the type of object projected to collide with host vehicle  4710 . For example, pedestrians may represent a greater safety risk and thus may be associated with a greater buffer distance D 1  than a stationary object, an animal, or other target objects. 
     In some embodiments, the envelopes may be configured to account for multiple poses of host vehicle  4710  and/or pedestrian  4720 . As used herein, a pose may refer to a particular stance or orientation of an object. With respect to vehicle  4710 , the pose may refer to possible orientations of host vehicle  4710  at a given location and/or time. In the example shown in  FIG. 49 , envelope  4912  may account for a single pose of host vehicle  4710 . If vehicle  4710  were rotated into different orientations, and thus different poses, part of vehicle  4710  would extend beyond the bounds of envelope  4710 . This may also occur as a tire of host vehicle  4710  rotates or turns outwards (e.g., as the vehicle is turning, etc.). Similarly, if pedestrian  4720  were rotated to a different orientation or had a different stance (e.g., having his or her arms or legs extended further outwards, etc.) a portion of pedestrian  4720  may extend beyond envelope  4920 . 
     To improve the reliability of a collision prediction, an envelope may be configured to account for multiple poses of a pedestrian or vehicle. For example, envelope  4912  may be drawn to encompass multiple orientations of host vehicle  4710 . This may be accomplished by rotating vehicle  4710  by a predetermined angle in each direction (e.g., 5 degrees, 10 degrees, 20 degrees, 45 degrees, 90 degrees, etc.) and expanding envelope  4710  to include multiple poses or all possible poses within these ranges. This may be especially important in scenarios such as the scenario shown in  FIG. 47  where vehicle  4710  is turning and thus the exact pose of vehicle  4710  may be unknown. A similar process may be performed for pedestrian  4720 . In addition to different orientations of pedestrian  4720  envelope  4920  may further account for different stances (e.g., different poses along a stride of pedestrian  4720 , etc.). In some embodiments, the position of pedestrian  4720  or vehicle  4710  along their respective predicted paths may be used to identify the poses to account for in the envelope. For example, host vehicle  4710  may be assumed to have a base orientation defined by a line tangent to path  4730  at a given position and the range of poses may be based on the base orientation. In some embodiments a predetermined pedestrian envelope may be used, which may account for typical poses of a pedestrian. Based on these expanded envelopes, the vehicle navigation system may account for multiple poses of pedestrian  4720  and/or host vehicle  4710 . 
     In some embodiments, the determination of whether host vehicle  4710  is projected to collide with pedestrian  4720  may be based on one or more circle-shaped envelopes.  FIG. 50  illustrates example circle-shaped envelopes associated with host vehicle  4710  and pedestrian  4720 . For example, envelope  5010  may be a circle drawn such that it circumscribes vehicle  4710 . Similarly, envelope  5020  may be a circle configured to circumscribe pedestrian  4720 . Accordingly, no matter which direction host vehicle  4710  is facing (e.g., as the yaw of host vehicle  4710  changes along path  4730 ), envelope  5010  may maintain a uniform same size and shape. Therefore, circle-shaped envelopes  5010  and  5020  may account for multiple poses (e.g., within a certain range or across a certain arc) or all possible poses of pedestrian  4720  and/or host vehicle  4710 . While example envelopes are illustrated in  FIGS. 48-50 , it is to be understood that the present disclosure is not limited to any particular shape or configuration of envelopes. Further, in some embodiments, multiple envelopes may be determined. For example, both rectangular envelope  4810  and circular envelope  5010  may be drawn relative to host vehicle  4710 . The different envelopes may be used in different analyses. 
     Based on one or more of the determined envelopes, the navigation system of host vehicle  4710  may determine whether host vehicle  4710  and pedestrian  4720  are projected to collide. This may be determined, for example, by predicting a position and orientation of host vehicle  4710  and pedestrian  4720  at various points in time. For example, a position and orientation of host vehicle  4710  and pedestrian  4720  may be predicted and analyzed at fixed intervals throughout the movement of host vehicle  4710  along path  4730 . If the envelopes associated with host vehicle  4710  and pedestrian  4720  are determined to intersect at any of these intervals, a collision may be detected. 
     In some instances, a problem may arise in how these intervals are defined. A low resolution analysis (e.g., using relatively long time intervals) may result in inaccurate detection of a collision. For example, the first interval where an interference between host vehicle  4710  and pedestrian  4720  is detected may not represent the actual time of impact. Further, the model may miss detecting a collision altogether if the collision occurs between two points in time that are analyzed. In other words, a fast-moving object may appear to “teleport” through another object if the interference between the objects occurs between two time intervals. Conversely, a higher resolution analysis (e.g., using relatively short intervals) may require prohibitively long processing times and may be overly demanding in terms of computational load. For example, the navigation system may need to calculate potential collisions for dozens of scenarios at each interval. Accordingly, such calculations at very short intervals may not be feasible. 
     To overcome these and other problems, a multiphase analysis may be implemented. In a first phase, an analytical calculation may be used to find a close approximation of a potential collision. Then, based on the approximation, a refined analysis may be performed to narrow the results and identify a potential collision. As a result, the vehicle navigation system may identify projected collisions with a high degree of accuracy with a relatively low computational load or processing time. 
     In some embodiments, the initial analysis may be performed using a circle-based envelope technique. As described above, circular envelopes  5010  and  5020  may be drawn such that they circumscribe host vehicle  4710  and pedestrian  4720 , respectively. Through an initial analysis, the system may determine whether envelopes  5010  and  5020  collide. 
     Accordingly, the rotation of host vehicle  4710  and/or pedestrian  4720  may be neglected in the initial analysis. If a collision is projected to occur between host vehicle  4710  and pedestrian  4720 , there will necessarily be a collision between envelopes  5010  and  5020 . Similarly, if no collision is projected to occur, there will be no collision between envelopes  5010  and  5020 . Further, if a collision between host vehicle  4710  and pedestrian  4720  is projected to occur, it will necessarily occur after a time t c′  at which envelopes  5010  and  5020  collide. Accordingly, a refined analysis can begin at time tc. 
       FIG. 51  is a block diagram illustrating an example technique for determining a time of collision between circular envelopes, consistent with the disclosed embodiments. As shown in  FIG. 51 , host vehicle  4710  and pedestrian  4720  may be predicted to travel along paths  4730  and  4740 , respectively. At an initial time, to, host vehicle  4710  may be represented by a circular envelope  5110  and pedestrian  4720  may be represented by a circular envelope  5120 . A projected corridor for the circular envelopes along paths  4730  and  4740  may also be determined. For example, as shown in  FIG. 51 , vehicle  4710  may be associated with a corridor  5130  and pedestrian  4720  may be associated with a corridor  5140 . The vehicle navigation system may determine a range of times when circular envelope  5110  intersects with corridor  5140 , and a range of times when circular envelope  5120  intersects with corridor  5130 . 
     In order to determine collision times between the circular envelopes, a projected speed at which each of these objects travels may also be determined. The projected speed of host vehicle  4710  may be determined based on a current speed of host vehicle  4710  (e.g., using onboard speed sensors, etc.). In some embodiments, the navigation system may assume host vehicle  4710  maintains the current speed. In other embodiments, the navigation system may assume a different speed. For example, the projected speed may be lower than the current speed of host vehicle  4710 , assuming host vehicle  4710  slows down through the turn. In some embodiments, the projected speed of host vehicle  4710  may vary along path  4730 . For example, the system may assume a constant deceleration by host vehicle  4710  along path  4730 . Various other speed profiles may also be assumed. 
     The projected speed of pedestrian  4720  may be determined in various ways. In some embodiments, the projected speed of pedestrian  4720  may be determined based on analysis of images captured by host vehicle  4710 . For example, the change in position of pedestrian  4720  may be analyzed between two or more images and, based on the time between the captured images, a speed of the pedestrian may be detected. The projected speed of pedestrian  4720  may correspond to this detected speed, or may be adjusted (e.g., assuming the pedestrian speeds up or slows down). In some embodiments, the projected speed may be determined based on a stored value. For example, the projected speed for pedestrian  4720  may correspond to a predetermined average pedestrian speed used for the analysis. 
     Based on the projected speed of host vehicle  4710 , the navigational system may determine a time t v1  at which circular envelope  5110  first intersects corridor  5140 , as represented by circle  5112 . Similarly, the navigational system may determine a time t v2  at which circular envelope  5110  no longer intersects with corridor  5140 , as represented by circle  5114 . Similar times may be determined for pedestrian  4720  relative to corridor  5130 . More specifically, based on the projected speed of pedestrian  4720 , the navigational system may determine a time t p1  at which circular envelope  5120  first intersects corridor  5130 , as represented by circle  5122 , and a time t p2  at which circular envelope  5120  no longer intersects with corridor  5130 , as represented by circle  5124 . 
     Based on these times, the vehicle navigation system may narrow the determination of whether or not a collision is projected to occur.  FIG. 52  is an illustration of example timelines  5210  and  5220  corresponding to collision predictions for circular envelopes, consistent with the disclosed embodiments. As described above, a range between t v1  and t v2  may represent times where at least part of circular envelope  5110  intersects with corridor  5140 . 
     Accordingly, if there is a collision between envelopes  5110  and  5120 , it will occur within this time range. Similarly, any collision between envelopes  5110  and  5120  must also occur within a time range between t p1  and t p2  where at least part of circular envelope  5120  intersects with corridor  5130 . In some instances, as shown in timeline  5210 , the two ranges may not overlap. In other words, time t v2  at which circular envelope  5110  no longer intersects with corridor  5140  may occur prior to time t p1  at which circular envelope  5120  first intersects corridor  5130 . Therefore, there may be no risk of collision between circular envelopes  5110  and  5120 , and thus no risk of collision between host vehicle  4710  and pedestrian  4720 . Accordingly, the navigational system may conclude the analysis at this stage. 
     In other instances, the two ranges may overlap. For example, as shown in timeline  5220 , time to may occur prior to time t v2 . This may not necessarily mean that a collision between circular envelopes  5110  and  5120  will occur. However, if there is a collision between the circular envelopes (at time t c′ ), it will necessarily occur between t p1  and t v2  (or between t v1  and t p2 , depending on which circular envelope enters the opposing corridor first). 
     Based on this shorter range of potential collisions, time t c′  may be determined using various methods. As one example, a Newton-Raphson method may be performed beginning at the first time the two time ranges intersect to find a root of an equation defining the paths of the circular envelopes. The point at which the Newton-Raphson method converges may represent time t c′ . Various other methods for determining time t c′  may also be used. 
     Based on the time t c′  at which the circular envelopes  5110  and  5120  are projected to collide, the vehicle navigation system may perform further analysis to determine whether host vehicle  4710  and pedestrian  4720  are also projected to collide. If there is a time t c  representing a collision between host vehicle  4710  and pedestrian  4720 , it will necessarily occur after time t c′  and before time t v2  (or the last time ranges t v1 −t v2  and t p1 −t p2  overlap), as represented by time interval  5230 . Accordingly, a more detailed analysis, taking into account rotation of host vehicle  4710  and pedestrian  4720 , may focus on points in time included in time interval  5230 . 
     Although interval  5230  may be a significantly shorter interval to analyze than the full path  4730 , similar problems may occur based on the resolution of points in time that are analyzed. For example, there may still be a risk of a “teleportation” effect in which host vehicle  4710  is modeled to pass through pedestrian  4720  due to a potential collision occurring between two points in time that are analyzed. Accordingly, the vehicle navigation system may use a step approach within interval  5230  to converge on time t c . 
       FIG. 53  is a block diagram illustrating an example refined analysis to determine a time of collision, consistent with the disclosed embodiments. The vehicle navigation system may be configured to determine a time t c  at which bounding rectangle  5310  (representing host vehicle  4710 ) is projected to collide with bounding rectangle  5320  (representing pedestrian  4720 ). In some embodiments, bounding rectangle  5310  may correspond to envelope  4810 , as shown in  FIG. 48 . In other embodiments, rectangle  5310  may include a buffer distance (such as buffer distance D 1 ) and thus may correspond to envelope  4914 , as shown in  FIG. 49 . In some embodiments, for simplicity of calculation, rectangle  5310  may be reduced to a single line  5312  representing the edge of vehicle  4710  facing pedestrian  4720 . Similarly, rectangle  5320  may be reduced to line  5322 . Line  5312  may correspond to predetermined frontal area  4812  and line  5322  may correspond to exposed area  4820 . 
     At a given time tn, the vehicle navigation system may determine whether rectangles  5310  and  5320  intersect. If not, time t n  does not equal t c  and thus the vehicle navigation system may calculate the next time interval t n+1  for performing the next iteration. Initially, t n  may correspond to t c′ , identified as described above. The next time interval may be determined in a variety of ways. In some embodiments, the analysis may include identifying a derived joint motion between host vehicle  4710  and pedestrian  4720 . The derived joint motion may represent a motion at time t n  of host vehicle  4710  and pedestrian  4720  relative to each other. For example, the derived joint motion may be represented as a motion vector based on the combined motion vectors of host vehicle  4710  and pedestrian  4720  at time t n . In some embodiments, the derived joint motion may be expressed as a motion vector of one of the two objects relative to the other. For example, pedestrian  4720  may be assumed to be fixed and host vehicle  4710  may be represented as having a motion vector towards pedestrian  4720  with a speed, ν, based on the derived joint motion. 
     The vehicle navigation system may then determine a distance from rectangle  5310  to rectangle  5320 . This may include drawing lines from edges of each shape in the direction of the derived joint motion. In particular, lines  5330  and  5334  may be drawn from the endpoints of line  5312 , and lines  5332  and  5336  may be drawn from the endpoints of line  5322  in the direction of the derived joint motion. The minimum distance d (in this example, represented as line  5332 ) may then be determined. Assuming a constant speed and neglecting rotation, 
     
       
         
           
             d 
             v 
           
         
       
     
     will always be shorter than the time between t n  and t c . Accordingly, t n+1  may be defined as 
     
       
         
           
             
               t 
               
                 n 
                 + 
                 1 
               
             
             = 
             
               
                 t 
                 n 
               
               + 
               
                 d 
                 v 
               
             
           
         
       
     
     to avoid risk of missing a projected collision. The vehicle navigation system may repeat the previous steps at the next time interval to converge on time t c . 
     Based on the above methods, collisions between host vehicle  4710  and pedestrian  4720  may be predicted at a high degree of accuracy with a relatively low demand on computational resources. If a projected collision is detected, the vehicle navigation system may implement a control action, such as a steering or braking maneuver, as described above. In some embodiments, the determined action may depend on the time to collision (i.e., the time until t c ). For example, the system may apply a steering maneuver if the time to collision is too short to stop host vehicle  4710  based on the current speed. In some embodiments, the system may determine a degree to which the collision mitigation action must be performed (e.g., how hard to apply the brakes) based on the time to collision. The system may further report the time to collision, for example, as part of a notification or alert displayed to a driver or passenger. 
       FIG. 54  is a flowchart showing an example process  5400  for navigating a host vehicle, consistent with the disclosed embodiments. Process  5400  may be performed by at least one processing device, such as processing unit  110 , described above. It is to be understood that throughout the present disclosure, the term “processor” is used as a shorthand for “at least one processor.” In other words, a processor may include one or more structures that perform logic operations whether such structures are collocated, connected, or disbursed. In some embodiments, a non-transitory computer readable medium may contain instructions that when executed by a processor cause the processor to perform process  5400 . Further, process  5400  is not necessarily limited to the steps shown in  FIG. 54 , and any steps or processes of the various embodiments described throughout the present disclosure may also be included in process  5400 , including those described above with respect to  FIGS. 47-53 . 
     In step  5410 , process  5400  may include receiving, from a camera of a vehicle, at least one image captured from an environment of the vehicle. For example, image acquisition unit  120  may capture one or more images representing an environment of host vehicle  4710 . 
     In step  5420 , process  5400  may include identifying at least a portion of an intersection represented in the at least one captured image. For example, host vehicle  4710  may identify some or all of intersection  4700  represented in a captured image. Intersection  4700  may be detected using any suitable method. For example, various visual detection algorithms (such as object detectors, motion detectors, edge detectors, etc.), may be used to detect intersection  4700 . 
     In step  5430 , process  5400  may include identifying a representation in the at least one captured image of a pedestrian in a vicinity of the intersection. For example, host vehicle  4710  may identify pedestrian  4720  represented in the captured image in the vicinity of intersection  4700 . Various image visual detection algorithms (such as object detectors, person detectors, face detectors, motion detectors, edge detectors, etc.), may be used to detect intersection pedestrian  4720 . 
     In step  5440 , process  5400  may include determining a navigational action for the vehicle relative to the intersection. For example, the navigational action for the vehicle may include one of a left turn, a right turn, or a continue straight action relative to the intersection. In some embodiments, the navigational action may be determined, at least in part, based on routing information for the vehicle. For example, step  5440  may include accessing a road navigational map to determine a navigational action that is consistent with an intended or planned route of host vehicle  4710 . 
     In step  5450 , process  5400  may include determining a predicted path for the vehicle relative to the intersection based on the determined navigational action. For example, the predicted path may correspond to path  4730 , as shown in  FIG. 47 . The predicted path of the vehicle may be determined in various ways. In some embodiments, the predicted path of host vehicle  4710  may be divided into two or more segments. For example, determining the predicted path of the vehicle may include determining a predetermined straight line first trajectory extending from a current position of the vehicle, a predetermined straight line second trajectory extending from a projected second position associated with the intersection, and an intermediate, curved trajectory extending between the first and second trajectories. The curved trajectory may have a constant radius of curvature, which may be consistent with the Euro NCAP CPTA scenario as described above. In other embodiments, the curved trajectory may have a variable radius of curvature, such as a constantly changing radius of curvature, or a radius of curvature that changes based on other factors. 
     In some embodiments, determining the predicted path of the vehicle may include accessing a map storing a plurality of predetermined target trajectories for the intersection. For example, the vehicle navigation system may access map database  160 , which may include a plurality of trajectories associated with intersection  4700 . The plurality of predetermined target trajectories may be associated with viable paths for the vehicle along two or more different road segments extending from the intersection. For example, target trajectories  4732 ,  4734 , and  4736  may correspond to viable paths for host vehicle  4710  along road segments  4708 ,  4706 , and  4704 , respectively. In some embodiments, determining the predicted path of the vehicle may further include selecting one of the plurality of target trajectories based on the determined navigational action for the vehicle. For example, target trajectory  4736  may be selected as the projected path for vehicle  4710  based on a determined navigational action of a left turn. In some embodiments, the predicted path may include at least one difference from the selected target trajectory. The difference may be determined based on a speed of the vehicle, as described above. In some embodiments, a speed of the vehicle above a predetermined threshold speed may result in a predicted path having a turning radius greater than a turning radius associated with the selected target trajectory. For example, a speed over 20 mph (or any other suitable speed) may result in path  4738  being selected. Similarly, a speed of the vehicle below a predetermined threshold speed may result in a predicted path having a turn radius less than a turning radius associated with the selected target trajectory (e.g. path  4730 ). 
     In step  5460 , process  5400  may include determining a predicted path for the pedestrian based on analysis of the at least one captured image. For example, based on images captured from image acquisition unit  210 , processor  110  may be configured to determine a path of pedestrian  4720 . In some embodiments, the predicted path of the pedestrian may include a predetermined straight path of travel relative to a current detected position of the pedestrian. The straight path may extend perpendicular to at least one drivable path associated with the intersection, as shown in  FIG. 47 . In some embodiments, the determination of the predicted path of the pedestrian may further be based on a facing direction for the pedestrian. The facing direction may be detected based on analysis of the at least one captured image, for example, using various object detection algorithms or other forms of detection algorithms. 
     In some embodiments, the predicted path of the pedestrian may be based on a detected path of the pedestrian in the at least one captured image. For example, the at least one captured image may include a plurality of captured images and determining the predicted path of the pedestrian may include determining a position of the pedestrian in the plurality of images. The position of the pedestrian may be determined relative to a lower portion of the pedestrian, such as based on the feet of the pedestrian in contact with a surface of a road, as described above. The position of the pedestrian in the plurality of images may be determined in reference to a common coordinate system. For example, the coordinate system may be defined based on an initial image, and the position of the pedestrian determined in subsequent images may be converted to the coordinate system of the initial frame. 
     In some embodiments, the predicted path of the pedestrian may be determined based on a measured distance from the vehicle to the pedestrian, as described above. For example, the at least one captured image may include a plurality of captured images and determining the predicted path of the pedestrian may include determining a distance from the vehicle to the pedestrian in the plurality of images. In some embodiments, the distance to the pedestrian may be determined based on a dimension of the pedestrian in the plurality of images. As described in further detail above, the dimension may comprise a distance between at least two tracked points on the representation of the pedestrian in the plurality of images. The distance to the pedestrian tracked over a sequence of images may be used to predict the path of the pedestrian. 
     In step  5470 , process  5400  may include determining whether the vehicle is projected to collide with the pedestrian based, at least in part, on the projected path for the vehicle and based on the predicted path of the pedestrian. Step  5470  may be performed in various ways. In some embodiments, the determination of whether the vehicle is projected to collide with the pedestrian is based on a predetermined frontal area of the vehicle. For example, the vehicle navigation system may determine frontal area  4812  for purposes of collision detection. Similarly, the determination of whether the vehicle is projected to collide with the pedestrian may further be based on an exposed area of the pedestrian, such as exposed area  4820 , as shown in  FIG. 48  and as described above. 
     In some embodiments, the determination of whether the vehicle is projected to collide with the pedestrian may be based on an envelope associated with the vehicle. For example, the envelope associated with the vehicle may include a bounding box associated with one or more edges of the vehicle, as shown by envelope  4810  in  FIG. 48 . In some embodiments, the envelope associated with the vehicle may include a predetermined buffer distance relative to the edges of the vehicle. The determination of whether the vehicle is projected to collide with the pedestrian may include a determination of whether the envelope associated with the vehicle will intersect with any part of the pedestrian. For example, the envelope  4914  may include a buffer of a distance D 1  (e.g., 0.5 meters, 1 meter, 2 meters, 3 meters, etc.) as shown in  FIG. 49 . In some embodiments, the predetermined buffer distance may vary with respect to a speed of the vehicle or other factors. 
     Consistent with the embodiments of the present disclosure, determining whether the vehicle is projected to collide with the pedestrian may include determining whether the predicted path for the vehicle crosses the predicted path of the pedestrian. For example, step  5470  may include determining whether path  4730  intersects with path  4740 . In some embodiments, this may also be based on timing. For example, determining whether the vehicle is projected to collide with the pedestrian may further include determining whether the vehicle and the pedestrian are projected to arrive at the crossing of the predicted paths within a predetermined time window. 
     To determine the predicted timings the determination of whether the vehicle is projected to collide with the vehicle may further be based on a projected speed of the pedestrian and a projected speed of the vehicle. In some embodiments, the projected speed of the pedestrian may be equal to a detected speed of the pedestrian. For example, the at least one captured image may include a plurality of captured images, and the detected speed of the pedestrian may be determined based on analyzing movement of the pedestrian represented by the plurality of captured images. Alternatively, or additionally, the projected speed of the pedestrian may be associated with a predetermined average pedestrian speed. For example, the average speed of a pedestrian may be stored in memory (e.g., memory  140  or memory  150 ) and may be used as the projected speed of the pedestrian. In some embodiments, the projected speed of the vehicle is equal to a current speed of the vehicle. The projected speed may also be an adjusted speed of the vehicle. For example, the projected speed of the vehicle may be slower or faster than a current speed of the vehicle. 
     In some embodiments, the determination of whether the vehicle is projected to collide with the pedestrian may be based on a plurality of poses of the vehicle at an estimated time of collision, as described in further detail above. Similarly, the determination of whether the vehicle is projected to collide with the pedestrian is based on a plurality of poses of the pedestrian at the estimated time of collision. In some embodiments, the plurality of poses may include multiple poses or all possible poses of the vehicle and/or the pedestrian (e.g., based on a circular envelope, etc.). In some embodiments, the estimated time of collision may correspond to a time t c′  or t c  in which an envelope associated with host vehicle  4710  is determined to collide or intersect with an envelope of pedestrian  4720 . In some embodiments, the estimated time of collision may comprise a time at which the vehicle is closest to the pedestrian as the vehicle and pedestrian travel along their respective predicted paths. Accordingly, if envelopes accounting for multiple poses or all possible poses of host vehicle  4710  and pedestrian  4720  do not intersect at their closest positions, no collision will occur. In some embodiments, the plurality of poses may be determined based on at least one of the predicted path of the vehicle or the predicted path of the pedestrian, as described above. 
     In some embodiments, the determination of whether the vehicle is projected to collide with the pedestrian may be based on a circle-shaped envelope associated with the vehicle and a circle-shaped envelope associated with the pedestrian. The determination of whether a collision is projected may be based on a determination of whether the circle-shaped envelope associated with the vehicle is projected to intersect with the circle-shaped envelope associated with the pedestrian as a result of movement of the circle-shaped envelopes associated with the pedestrian and vehicle along their respective predicted paths, as described above with respect to  FIGS. 51 and 52 . For example, step  5470  may include determining a range of times when the circle-shaped envelope of the vehicle is predicted to intersect with a corridor of the circle-shaped envelope of the pedestrian, and vice versa. A predicted time of collision of the circle-shaped envelopes may be determined based on the ranges of times. In some embodiments, the determination of whether the vehicle is projected to collide with the pedestrian may further include a refined analysis in an area of potential intersection between the circle-shaped envelopes. The refined analysis may rely upon a refined envelope associated with at least one of the pedestrian or the vehicle. For example, the refined analysis may be based on rectangles  5310  and  5320 , as described above with respect to  FIG. 53 . 
     In step  5480 , process  5400  may include causing at least one system associated with the vehicle to implement a collision mitigation action in response to a determination that the vehicle is projected to collide with the pedestrian. In some embodiments, the collision mitigation action may include a notification or alert to a user of the vehicle, such as a driver, passenger, owner, etc. For example, the collision mitigation action includes issuing an audible or visual warning to the user of the vehicle. In some embodiments, the collision mitigation action may include applying brakes of the vehicle. Alternatively, or additionally, the collision mitigation action steering the vehicle away from the predicted path for the vehicle. For example, the collision mitigation action may include steering the vehicle toward a path determined to pass behind the pedestrian. 
     The foregoing description has been presented for purposes of illustration. It is not exhaustive and is not limited to the precise forms or embodiments disclosed. Modifications and adaptations will be apparent to those skilled in the art from consideration of the specification and practice of the disclosed embodiments. Additionally, although aspects of the disclosed embodiments are described as being stored in memory, one skilled in the art will appreciate that these aspects can also be stored on other types of computer readable media, such as secondary storage devices, for example, hard disks or CD ROM, or other forms of RAM or ROM, USB media, DVD, Blu-ray, 4K Ultra HD Blu-ray, or other optical drive media. 
     Computer programs based on the written description and disclosed methods are within the skill of an experienced developer. The various programs or program modules can be created using any of the techniques known to one skilled in the art or can be designed in connection with existing software. For example, program sections or program modules can be designed in or by means of .Net Framework, .Net Compact Framework (and related languages, such as Visual Basic, C, etc.), Java, C++, Objective-C, HTML, HTML/AJAX combinations, XML, or HTML with included Java applets. 
     Moreover, while illustrative embodiments have been described herein, the scope of any and all embodiments having equivalent elements, modifications, omissions, combinations (e.g., of aspects across various embodiments), adaptations and/or alterations as would be appreciated by those skilled in the art based on the present disclosure. The limitations in the claims are to be interpreted broadly based on the language employed in the claims and not limited to examples described in the present specification or during the prosecution of the application. The examples are to be construed as non-exclusive. Furthermore, the steps of the disclosed methods may be modified in any manner, including by reordering steps and/or inserting or deleting steps. It is intended, therefore, that the specification and examples be considered as illustrative only, with a true scope and spirit being indicated by the following claims and their full scope of equivalents.