Patent Publication Number: US-2023159054-A1

Title: Encoding homotopy constraints in spatio-temporal grids

Description:
BACKGROUND 
     A state of the art planning architecture for autonomous vehicles extracts multiple homotopies, computes candidate trajectories within the homotopies (hereinafter, also referred to as a “trajectory realization”), and selects the best candidate trajectory by cost scoring. A model-predictive control (MPC) trajectory realizer can be used to generate a sample-based trajectory realization by sampling the input homotopy constraints. Because the homotopy constraints are represented by a lane-centric, spatio-temporal tube with left and right bounds, multiple types of maneuvers may be disabled. For example, a vehicle traveling behind another vehicle in an adjacent lane which is initially behind the ego vehicle may not be expressed by a sample-based MPC trajectory realizer in part because decisions are defined across all lanes and the ego vehicle would thus always start outside of a valid homotopy. Also, assumptions are made on the vehicle state propagation through spatial sampling. This leads to a discrepancy between the actual homotopy constraints, representing the physical world and obstacles, and the constraints which the optimal control problem is solved for by the MPC. Large discrepancies especially occur in dynamic environments with fast moving traffic. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG.  1    is an example environment in which a vehicle including one or more components of an autonomous system can be implemented; 
         FIG.  2    is a diagram of one or more systems of a vehicle including an autonomous system; 
         FIG.  3    is a diagram of components of one or more devices and/or one or more systems of  FIGS.  1  and  2   ; 
         FIG.  4    is a diagram of certain components of an autonomous system; 
         FIG.  5    illustrates a sampling issue during biasing using a sample-based MPC trajectory realizer; 
         FIG.  6 A  illustrates a spatio-temporal grid for a first maneuver type; 
         FIG.  6 B  illustrates a spatio-temporal grid for a second maneuver type; 
         FIG.  7    illustrates using curve fitting to make the discrete grids shown in  FIGS.  6 A,  6 B  continuously-differentiable; 
         FIG.  8    illustrates unimodal corridor constraints representing a homotopy encoded as convex shapes with smooth edges in the spatio-temporal grid and probing points on the ego vehicle perimeter that contribute to a collision cost at each MPC prediction stage; 
         FIG.  9    is block diagram of a planning system that uses grid-based trajectory realization; and 
         FIG.  10    is a flow diagram of a process of encoding homotopy constraints into spatio-temporal grids. 
     
    
    
     DETAILED DESCRIPTION 
     In the following description numerous specific details are set forth in order to provide a thorough understanding of the present disclosure for the purposes of explanation. It will be apparent, however, that the embodiments described by the present disclosure can be practiced without these specific details. In some instances, well-known structures and devices are illustrated in block diagram form in order to avoid unnecessarily obscuring aspects of the present disclosure. 
     Specific arrangements or orderings of schematic elements, such as those representing systems, devices, modules, instruction blocks, data elements, and/or the like are illustrated in the drawings for ease of description. However, it will be understood by those skilled in the art that the specific ordering or arrangement of the schematic elements in the drawings is not meant to imply that a particular order or sequence of processing, or separation of processes, is required unless explicitly described as such. Further, the inclusion of a schematic element in a drawing is not meant to imply that such element is required in all embodiments or that the features represented by such element may not be included in or combined with other elements in some embodiments unless explicitly described as such. 
     Further, where connecting elements such as solid or dashed lines or arrows are used in the drawings to illustrate a connection, relationship, or association between or among two or more other schematic elements, the absence of any such connecting elements is not meant to imply that no connection, relationship, or association can exist. In other words, some connections, relationships, or associations between elements are not illustrated in the drawings so as not to obscure the disclosure. In addition, for ease of illustration, a single connecting element can be used to represent multiple connections, relationships or associations between elements. For example, where a connecting element represents communication of signals, data, or instructions (e.g., “software instructions”), it should be understood by those skilled in the art that such element can represent one or multiple signal paths (e.g., a bus), as may be needed, to affect the communication. 
     Although the terms first, second, third, and/or the like are used to describe various elements, these elements should not be limited by these terms. The terms first, second, third, and/or the like are used only to distinguish one element from another. For example, a first contact could be termed a second contact and, similarly, a second contact could be termed a first contact without departing from the scope of the described embodiments. The first contact and the second contact are both contacts, but they are not the same contact. 
     The terminology used in the description of the various described embodiments herein is included for the purpose of describing particular embodiments only and is not intended to be limiting. As used in the description of the various described embodiments and the appended claims, the singular forms “a,” “an” and “the” are intended to include the plural forms as well and can be used interchangeably with “one or more” or “at least one,” unless the context clearly indicates otherwise. It will also be understood that the term “and/or” as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items. It will be further understood that the terms “includes,” “including,” “comprises,” and/or “comprising,” when used in this description specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
     As used herein, the terms “communication” and “communicate” refer to at least one of the reception, receipt, transmission, transfer, provision, and/or the like of information (or information represented by, for example, data, signals, messages, instructions, commands, and/or the like). For one unit (e.g., a device, a system, a component of a device or system, combinations thereof, and/or the like) to be in communication with another unit means that the one unit is able to directly or indirectly receive information from and/or send (e.g., transmit) information to the other unit. This may refer to a direct or indirect connection that is wired and/or wireless in nature. Additionally, two units may be in communication with each other even though the information transmitted may be modified, processed, relayed, and/or routed between the first and second unit. For example, a first unit may be in communication with a second unit even though the first unit passively receives information and does not actively transmit information to the second unit. 
     As another example, a first unit may be in communication with a second unit if at least one intermediary unit (e.g., a third unit located between the first unit and the second unit) processes information received from the first unit and transmits the processed information to the second unit. In some embodiments, a message may refer to a network packet (e.g., a data packet and/or the like) that includes data. 
     As used herein, the term “if” is, optionally, construed to mean “when”, “upon”, “in response to determining,” “in response to detecting,” and/or the like, depending on the context. Similarly, the phrase “if it is determined” or “if [a stated condition or event] is detected” is, optionally, construed to mean “upon determining,” “in response to determining,” “upon detecting [the stated condition or event],” “in response to detecting [the stated condition or event],” and/or the like, depending on the context. Also, as used herein, the terms “has”, “have”, “having”, or the like are intended to be open-ended terms. Further, the phrase “based on” is intended to mean “based at least partially on” unless explicitly stated otherwise. 
     Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings. In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the various described embodiments. However, it will be apparent to one of ordinary skill in the art that the various described embodiments can be practiced without these specific details. 
     In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments. 
     General Overview 
     In some aspects and/or embodiments, systems, methods, and computer program products described herein include and/or implement technology for encoding dynamic homotopy constraints in spatio-temporal grids. A “homotopy” is a unique space wherein any path starting with an initial AV state and ending at a terminal AV state can be continuously deformed The homotopy is encoded (e.g., rasterized) in spatio-temporal grids. 
     The dynamic homotopy constraints are input into a grid-based trajectory realizer, which generates one grid for each discrete timestep in the MPC prediction horizon to capture the exact geometries of the homotopy constraints originating from both lane geometry and other agents and obstacles, collectively defined as the grid-based dynamic homotopy constraints. Because the grid-based trajectory realizer is formulated as an optimization problem in Cartesian coordinates rather than curvilinear coordinates, and in conjunction with the grid-based dynamic homotopy constraints, the MPC can query the dynamic homotopy constraints more accurately. 
     In an embodiment, spline functions (e.g., B-splines) are used to make the discrete spatio-temporal grids continuously differentiable, where basis functions (e.g., cubic functions) are used to combine neighboring cell values (e.g., grid occupancy probabilities) in the spatio-temporal grids continuously, while providing the corresponding gradients necessary for MPC. Using splines, unimodal corridor constraints representing the homotopy are encoded in the spatio-temporal grids as convex shapes with smooth edges. 
     An MPC collision cost is introduced in the total MPC cost equation that includes a static term and a kinetic collision energy term based on a grid cell occupancy probability of a particular location. Probing points on the vehicle perimeter contribute to the MPC collision cost at each MPC prediction stage. The total MPC collision cost is part of the total MPC cost function in a trade-off with other objective functions, such as comfort and tracking cost functions. 
     In an embodiment, a method comprises: determining, with at least one processor, a plurality of dynamic homotopy constraints associated with a scenario based on one or more agents in an environment; embedding, with the at least one processor, dynamic homotopy constraints in a plurality of spatio-temporal grids a plurality of times for each timestep of a prediction horizon, wherein the embedding comprises: encoding, with the at least one processor, the plurality of dynamic homotopy constraints in the plurality of spatio-temporal grids that represent the environment; and generating, with the at least one processor, a plurality of trajectories based on the plurality of homotopy constraints embedded in the plurality of spatio-temporal grids; selecting, with the at least one processor, a particular trajectory from among the plurality of trajectories generated while updating the plurality of spatio-temporal grids; and controlling, with a control circuit of the vehicle, the vehicle based on the particular trajectory selected from among the plurality of trajectories. 
     In an embodiment, the plurality of trajectories are generated by a grid-based model-predictive control (MPC) trajectory realizer. 
     In an embodiment, the grid-based MPC trajectory realizer is formulated as an optimization problem in Cartesian coordinates. 
     In an embodiment, the optimization problem comprises: computing an MPC collision cost with static and kinetic collision energy terms based on the one or more agents in the environment encoded within a spatio-temporal grid; and contributing to the MPC collision cost probing points on a perimeter of the vehicle when generating the grid-based trajectory realization. 
     In an embodiment, the method further comprises: determining a total collision cost for a plurality of MPC prediction stages; evaluating an MPC cost function that includes the total collision cost and at least one other cost. 
     In an embodiment, the at least one other cost is associated with vehicle passenger comfort. 
     In an embodiment, the at least one other cost is associated with tracking error. 
     In an embodiment, at least one homotopy constraint is embedded in the spatio-temporal grid as a convex shape with smooth edges. 
     In an embodiment, the spatio-temporal grid includes a velocity and direction of travel of at least one agent. 
     In an embodiment, a prediction uncertainty for the at least one agent is encoded within the spatio-temporal grid and included in the optimization problem. 
     In an embodiment, the spatio-temporal grid is discretized and a spline function is applied to rows and columns of the spatio-temporal grid to make the spatio-temporal grid map continuously-differentiable. 
     In an embodiment, the spline function is a B-spline. 
     In an embodiment, cubic basis functions are used to combine neighboring cell values in the spatio-temporal grid continuously, while providing a corresponding gradient for the MPC and are twice differentiable. 
     In an embodiment, prediction uncertainty is included in the spatio-temporal grid by convolving the spatio-temporal grid with a Gaussian kernel. 
     By virtue of the implementation of systems, methods, and computer program products described herein, techniques for finding and accessing a vehicle provide at least the following advantages. 
     The issues discovered in sample-based MPC trajectory realizers are resolved using spatio-temporal grids, such that the enforced constraints within the MPC align with the actual homotopy constraints. With grid-based trajectory realizers, more driving behaviors can be supported than with sample-based trajectory realizers, including but not limited to: multiple lane changes, merging in tight traffic, accelerating to merge in front of vehicles coming from behind, and slowing down to merge behind vehicles coming from behind. The online generation of spatio-temporal grids makes it easy for designers to explore new cost structures leveraging the spatial continuity provided by the spatio-temporal grids without having to reformulate the optimization problem. 
     Referring now to  FIG.  1   , illustrated is example environment  100  in which vehicles that include autonomous systems, as well as vehicles that do not, are operated. As illustrated, environment  100  includes vehicles  102   a - 102   n , objects  104   a - 104   n , routes  106   a - 106   n , area  108 , vehicle-to-infrastructure (V2I) device  110 , network  112 , remote AV system  114 , fleet management system  116 , and V2I system  118 . Vehicles  102   a - 102   n , vehicle-to-infrastructure (V2I) device  110 , network  112 , AV system  114 , fleet management system  116 , and V2I system  118  interconnects (e.g., establish a connection to communicate and/or the like) via wired connections, wireless connections, or a combination of wired or wireless connections. In some embodiments, objects  104   a - 104   n  interconnect with at least one of vehicles  102   a - 102   n , vehicle-to-infrastructure (V2I) device  110 , network  112 , AV system  114 , fleet management system  116 , and V2I system  118  via wired connections, wireless connections, or a combination of wired or wireless connections. 
     Vehicles  102   a - 102   n  (referred to individually as vehicle  102  and collectively as vehicles  102 ) include at least one device configured to transport goods and/or people. In some embodiments, vehicles  102  are configured to be in communication with V2I device  110 , remote AV system  114 , fleet management system  116 , and/or V2I system  118  via network  112 . In some embodiments, vehicles  102  include cars, buses, trucks, trains, and/or the like. In some embodiments, vehicles  102  are the same as, or similar to, vehicles  200 , described herein (see  FIG.  2   ). In some embodiments, a vehicle  200  of a set of vehicles  200  is associated with an autonomous fleet manager. In some embodiments, vehicles  102  travel along respective routes  106   a - 106   n  (referred to individually as route  106  and collectively as routes  106 ), vehicles  102   a - 102   n  (referred to individually as vehicle  102  and collectively as vehicles  102 ) include at least one device configured to transport goods and/or people. In some embodiments, vehicles  102  are configured to be in communication with V2I device  110 , remote AV system  114 , fleet management system  116 , and/or V2I system  118  via network  112 . In some embodiments, vehicles  102  include cars, buses, trucks, trains, and/or the like. In some embodiments, vehicles  102  are the same as, or similar to, vehicles  200 , described herein (see  FIG.  2   ). In some embodiments, a vehicle  200  of a set of vehicles  200  is associated with an autonomous fleet manager. In some embodiments, vehicles  102  travel along respective routes  106   a - 106   n  (referred to individually as route  106  and collectively as routes  106 ), as described herein. In some embodiments, one or more vehicles  102  include an autonomous system (e.g., an autonomous system that is the same as or similar to autonomous system  202 ).Vehicles  102   a - 102   n  (referred to individually as vehicle  102  and collectively as vehicles  102 ) include at least one device configured to transport goods and/or people. In some embodiments, vehicles  102  are configured to be in communication with V2I device  110 , remote AV system  114 , fleet management system  116 , and/or V2I system  118  via network  112 . In some embodiments, vehicles  102  include cars, buses, trucks, trains, and/or the like. In some embodiments, vehicles  102  are the same as, or similar to, vehicles  200 , described herein (see  FIG.  2   ). 
     In some embodiments, a vehicle  200  of a set of vehicles  200  is associated with an autonomous fleet manager. In some embodiments, vehicles  102  travel along respective routes  106   a - 106   n  (referred to individually as route  106  and collectively as routes  106 ), as described herein. In some embodiments, one or more vehicles  102  include an autonomous system (e.g., an autonomous system that is the same as or similar to autonomous system  202 ).6 and collectively as routes  106 ), as described herein. In some embodiments, one or more vehicles  102  include an autonomous system (e.g., an autonomous system that is the same as or similar to autonomous system  202 ). 
     Objects  104   a - 104   n  (referred to individually as object  104  and collectively as objects  104 ) include, for example, at least one vehicle, at least one pedestrian, at least one cyclist, at least one structure (e.g., a building, a sign, a fire hydrant, etc.), and/or the like. Each object  104  is stationary (e.g., located at a fixed location for a period of time) or mobile (e.g., having a velocity and associated with at least one trajectory). In some embodiments, objects  104  are associated with corresponding locations in area  108 . 
     Routes  106   a - 106   n  (referred to individually as route  106  and collectively as routes  106 ) are each associated with (e.g., prescribe) a sequence of actions (also known as a trajectory) connecting states along which an AV can navigate. Each route  106  starts at an initial state (e.g., a state that corresponds to a first spatiotemporal location, velocity, and/or the like) and a final goal state (e.g., a state that corresponds to a second spatiotemporal location that is different from the first spatiotemporal location) or goal region (e.g., a subspace of acceptable states (e.g., terminal states)). 
     In some embodiments, the first state includes a location at which an individual or individuals are to be picked-up by the AV and the second state or region includes a location or locations at which the individual or individuals picked-up by the AV are to be dropped-off. In some embodiments, routes  106  include a plurality of acceptable state sequences (e.g., a plurality of spatiotemporal location sequences), the plurality of state sequences associated with (e.g., defining) a plurality of trajectories. In an example, routes  106  include only high level actions or imprecise state locations, such as a series of connected roads dictating turning directions at roadway intersections. 
     Additionally, or alternatively, routes  106  may include more precise actions or states such as, for example, specific target lanes or precise locations within the lane areas and targeted speed at those positions. In an example, routes  106  include a plurality of precise state sequences along the at least one high level action sequence with a limited look ahead horizon to reach intermediate goals, where the combination of successive iterations of limited horizon state sequences cumulatively correspond to a plurality of trajectories that collectively form the high level route to terminate at the final goal state or region. 
     Area  108  includes a physical area (e.g., a geographic region) within which vehicles  102  can navigate. In an example, area  108  includes at least one state (e.g., a country, a province, an individual state of a plurality of states included in a country, etc.), at least one portion of a state, at least one city, at least one portion of a city, etc. In some embodiments, area  108  includes at least one named thoroughfare (referred to herein as a “road”) such as a highway, an interstate highway, a parkway, a city street, etc. Additionally, or alternatively, in some examples area  108  includes at least one unnamed road such as a driveway, a section of a parking lot, a section of a vacant and/or undeveloped lot, a dirt path, etc. In some embodiments, a road includes at least one lane (e.g., a portion of the road that can be traversed by vehicles  102 ). In an example, a road includes at least one lane associated with (e.g., identified based on) at least one lane marking. 
     Vehicle-to-Infrastructure (V2I) device  110  (sometimes referred to as a Vehicle-to-Infrastructure (V2X) device) includes at least one device configured to be in communication with vehicles  102  and/or V2I infrastructure system  118 . In some embodiments, V2I device  110  is configured to be in communication with vehicles  102 , remote AV system  114 , fleet management system  116 , and/or V2I system  118  via network  112 . In some embodiments, V2I device  110  includes a radio frequency identification (RFID) device, signage, cameras (e.g., two-dimensional (2D) and/or three-dimensional (3D) cameras), lane markers, streetlights, parking meters, etc. In some embodiments, V2I device  110  is configured to communicate directly with vehicles  102 . Additionally, or alternatively, in some embodiments V2I device  110  is configured to communicate with vehicles  102 , remote AV system  114 , and/or fleet management system  116  via V2I system  118 . In some embodiments, V2I device  110  is configured to communicate with V2I system  118  via network  112 . 
     Network  112  includes one or more wired and/or wireless networks. In an example, network  112  includes a cellular network (e.g., a long term evolution (LTE) network, a third generation (3G) network, a fourth generation (4G) network, a fifth generation (5G) network, a code division multiple access (CDMA) network, etc.), a public land mobile network (PLMN), a local area network (LAN), a wide area network (WAN), a metropolitan area network (MAN), a telephone network (e.g., the public switched telephone network (PSTN), a private network, an ad hoc network, an intranet, the Internet, a fiber optic-based network, a cloud computing network, etc., a combination of some or all of these networks, and/or the like. 
     Remote AV system  114  includes at least one device configured to be in communication with vehicles  102 , V2I device  110 , network  112 , remote AV system  114 , fleet management system  116 , and/or V2I system  118  via network  112 . In an example, remote AV system  114  includes a server, a group of servers, and/or other like devices. In some embodiments, remote AV system  114  is co-located with the fleet management system  116 . In some embodiments, remote AV system  114  is involved in the installation of some or all of the components of a vehicle, including an autonomous system, an autonomous vehicle compute, software implemented by an autonomous vehicle compute, and/or the like. In some embodiments, remote AV system  114  maintains (e.g., updates and/or replaces) such components and/or software during the lifetime of the vehicle. 
     Fleet management system  116  includes at least one device configured to be in communication with vehicles  102 , V2I device  110 , remote AV system  114 , and/or V2I infrastructure system  118 . In an example, fleet management system  116  includes a server, a group of servers, and/or other like devices. In some embodiments, fleet management system  116  is associated with a ridesharing company (e.g., an organization that controls operation of multiple vehicles (e.g., vehicles that include autonomous systems and/or vehicles that do not include autonomous systems) and/or the like). 
     In some embodiments, V2I system  118  includes at least one device configured to be in communication with vehicles  102 , V2I device  110 , remote AV system  114 , and/or fleet management system  116  via network  112 . In some examples, V2I system  118  is configured to be in communication with V2I device  110  via a connection different from network  112 . In some embodiments, V2I system  118  includes a server, a group of servers, and/or other like devices. In some embodiments, V2I system  118  is associated with a municipality or a private institution (e.g., a private institution that maintains V2I device  110  and/or the like). 
     The number and arrangement of elements illustrated in  FIG.  1    are provided as an example. There can be additional elements, fewer elements, different elements, and/or differently arranged elements, than those illustrated in  FIG.  1   . Additionally, or alternatively, at least one element of environment  100  can perform one or more functions described as being performed by at least one different element of  FIG.  1   . Additionally, or alternatively, at least one set of elements of environment  100  can perform one or more functions described as being performed by at least one different set of elements of environment  100 . 
     Referring now to  FIG.  2   , vehicle  200  includes autonomous system  202 , powertrain control system  204 , steering control system  206 , and brake system  208 . In some embodiments, vehicle  200  is the same as or similar to vehicle  102  (see  FIG.  1   ). In some embodiments, vehicle  102  have autonomous capability (e.g., implement at least one function, feature, device, and/or the like that enable vehicle  200  to be partially or fully operated without human intervention including, without limitation, fully autonomous vehicles (e.g., vehicles that forego reliance on human intervention), highly autonomous vehicles (e.g., vehicles that forego reliance on human intervention in certain situations), and/or the like). 
     For a detailed description of fully autonomous vehicles and highly autonomous vehicles, reference may be made to SAE International’s standard J3016: Taxonomy and Definitions for Terms Related to On-Road Motor Vehicle Automated Driving Systems, which is incorporated by reference in its entirety. In some embodiments, vehicle  200  is associated with an autonomous fleet manager and/or a ridesharing company. 
     Autonomous system  202  includes a sensor suite that includes one or more devices such as cameras  202   a , LiDAR sensors  202   b , radar sensors  202   c , and microphones  202   d . In some embodiments, autonomous system  202  can include more or fewer devices and/or different devices (e.g., ultrasonic sensors, inertial sensors, GPS receivers (discussed below), odometry sensors that generate data associated with an indication of a distance that vehicle  200  has traveled, and/or the like). In some embodiments, autonomous system  202  uses the one or more devices included in autonomous system  202  to generate data associated with environment  100 , described herein. The data generated by the one or more devices of autonomous system  202  can be used by one or more systems described herein to observe the environment (e.g., environment  100 ) in which vehicle  200  is located. In some embodiments, autonomous system  202  includes communication device  202   e , autonomous vehicle compute  202   f , and drive-by-wire (DBW) system  202   h . 
     Cameras  202   a  include at least one device configured to be in communication with communication device  202   e , autonomous vehicle compute  202   f , and/or safety controller  202   g  via a bus (e.g., a bus that is the same as or similar to bus  302  of  FIG.  3   ). Cameras  202   a  include at least one camera (e.g., a digital camera using a light sensor such as a charge-coupled device (CCD), a thermal camera, an infrared (IR) camera, an event camera, and/or the like) to capture images including physical objects (e.g., cars, buses, curbs, people, and/or the like). In some embodiments, camera  202   a  generates camera data as output. In some examples, camera  202   a  generates camera data that includes image data associated with an image. In this example, the image data may specify at least one parameter (e.g., image characteristics such as exposure, brightness, etc., an image timestamp, and/or the like) corresponding to the image. In such an example, the image may be in a format (e.g., RAW, JPEG, PNG, and/or the like). 
     In some embodiments, camera  202   a  includes a plurality of independent cameras configured on (e.g., positioned on) a vehicle to capture images for the purpose of stereopsis (stereo vision). In some examples, camera  202   a  includes a plurality of cameras that generate image data and transmit the image data to autonomous vehicle compute  202   f  and/or a fleet management system (e.g., a fleet management system that is the same as or similar to fleet management system  116  of  FIG.  1   ). In such an example, autonomous vehicle compute  202   f  determines depth to one or more objects in a field of view of at least two cameras of the plurality of cameras based on the image data from the at least two cameras. In some embodiments, camera  202   a  is configured to capture images of objects within a distance from camera  202   a  (e.g., up to 100 meters, up to a kilometer, and/or the like). Accordingly, camera  202   a  include features such as sensors and lenses that are optimized for perceiving objects that are at one or more distances from cameras  202   a . 
     In an embodiment, camera  202   a  includes at least one camera configured to capture one or more images associated with one or more traffic lights, street signs and/or other physical objects that provide visual navigation information. In some embodiments, camera  202   a  generates traffic light data associated with one or more images. In some examples, camera  202   a  generates TLD data associated with one or more images that include a format (e.g., RAW, JPEG, PNG, and/or the like). In some embodiments, camera  202   a  that generates TLD data differs from other systems described herein incorporating cameras in that camera  202   a  can include one or more cameras with a wide field of view (e.g., a wide-angle lens, a fish-eye lens, a lens having a viewing angle of approximately 120 degrees or more, and/or the like) to generate images about as many physical objects as possible. 
     Laser Detection and Ranging (LiDAR) sensors  202   b  include at least one device configured to be in communication with communication device  202   e , autonomous vehicle compute  202   f , and/or safety controller  202   g  via a bus (e.g., a bus that is the same as or similar to bus  302  of  FIG.  3   ). LiDAR sensors  202   b  include a system configured to transmit light from a light emitter (e.g., a laser transmitter). Light emitted by LiDAR sensors  202   b  include light (e.g., infrared light and/or the like) that is outside of the visible spectrum. In some embodiments, during operation, light emitted by LiDAR sensors  202   b  encounters a physical object (e.g., a vehicle) and is reflected back to LiDAR sensors  202   b . In some embodiments, the light emitted by LiDAR sensors  202   b  does not penetrate the physical objects that the light encounters. LiDAR sensors  202   b  also include at least one light detector which detects the light that was emitted from the light emitter after the light encounters a physical object. In some embodiments, at least one data processing system associated with LiDAR sensors  202   b  generates an image (e.g., a point cloud, a combined point cloud, and/or the like) representing the objects included in a field of view of LiDAR sensors  202   b . In some examples, the at least one data processing system associated with LiDAR sensor  202   b  generates an image that represents the boundaries of a physical object, the surfaces (e.g., the topology of the surfaces) of the physical object, and/or the like. In such an example, the image is used to determine the boundaries of physical objects in the field of view of LiDAR sensors  202   b . 
     Radio Detection and Ranging (radar) sensors  202   c  include at least one device configured to be in communication with communication device  202   e , autonomous vehicle compute  202   f , and/or safety controller  202   g  via a bus (e.g., a bus that is the same as or similar to bus  302  of  FIG.  3   ). Radar sensors  202   c  include a system configured to transmit radio waves (either pulsed or continuously). The radio waves transmitted by radar sensors  202   c  include radio waves that are within a predetermined spectrum. In some embodiments, during operation, radio waves transmitted by radar sensors  202   c  encounter a physical object and are reflected back to radar sensors  202   c . In some embodiments, the radio waves transmitted by radar sensors  202   c  are not reflected by some objects. In some embodiments, at least one data processing system associated with radar sensors  202   c  generates signals representing the objects included in a field of view of radar sensors  202   c . For example, the at least one data processing system associated with radar sensor  202   c  generates an image that represents the boundaries of a physical object, the surfaces (e.g., the topology of the surfaces) of the physical object, and/or the like. In some examples, the image is used to determine the boundaries of physical objects in the field of view of radar sensors  202   c . 
     Microphones  202   d  includes at least one device configured to be in communication with communication device  202   e , autonomous vehicle compute  202   f , and/or safety controller  202   g  via a bus (e.g., a bus that is the same as or similar to bus  302  of  FIG.  3   ). Microphones  202   d  include one or more microphones (e.g., array microphones, external microphones, and/or the like) that capture audio signals and generate data associated with (e.g., representing) the audio signals. In some examples, microphones  202   d  include transducer devices and/or like devices. In some embodiments, one or more systems described herein can receive the data generated by microphones  202   d  and determine a position of an object relative to vehicle  200  (e.g., a distance and/or the like) based on the audio signals associated with the data. 
     Communication device  202   e  include at least one device configured to be in communication with cameras  202   a , LiDAR sensors  202   b , radar sensors  202   c , microphones  202   d , autonomous vehicle compute  202   f , safety controller  202   g , and/or DBW system  202   h . For example, communication device  202   e  may include a device that is the same as or similar to communication interface  314  of  FIG.  3   . In some embodiments, communication device  202   e  includes a vehicle-to-vehicle (V2V) communication device (e.g., a device that enables wireless communication of data between vehicles). 
     Autonomous vehicle compute  202   f  include at least one device configured to be in communication with cameras  202   a , LiDAR sensors  202   b , radar sensors  202   c , microphones  202   d , communication device  202   e , safety controller  202   g , and/or DBW system  202   h . In some examples, autonomous vehicle compute  202   f  includes a device such as a client device, a mobile device (e.g., a cellular telephone, a tablet, and/or the like) a server (e.g., a computing device including one or more central processing units, graphical processing units, and/or the like), and/or the like. In some embodiments, autonomous vehicle compute  202   f  is the same as or similar to autonomous vehicle compute  400 , described herein. Additionally, or alternatively, in some embodiments autonomous vehicle compute  202   f  is configured to be in communication with an autonomous vehicle system (e.g., an autonomous vehicle system that is the same as or similar to remote AV system  114  of  FIG.  1   ), a fleet management system (e.g., a fleet management system that is the same as or similar to fleet management system  116  of  FIG.  1   ), a V2I device (e.g., a V2I device that is the same as or similar to V2I device  110  of  FIG.  1   ), and/or a V2I system (e.g., a V2I system that is the same as or similar to V2I system  118  of  FIG.  1   ). 
     Safety controller  202   g  includes at least one device configured to be in communication with cameras  202   a , LiDAR sensors  202   b , radar sensors  202   c , microphones  202   d , communication device  202   e , autonomous vehicle computer  202   f , and/or DBW system  202   h . In some examples, safety controller  202   g  includes one or more controllers (electrical controllers, electromechanical controllers, and/or the like) that are configured to generate and/or transmit control signals to operate one or more devices of vehicle  200  (e.g., powertrain control system  204 , steering control system  206 , brake system  208 , and/or the like). In some embodiments, safety controller  202   g  is configured to generate control signals that take precedence over (e.g., overrides) control signals generated and/or transmitted by autonomous vehicle compute  202   f . 
     DBW system  202   h  includes at least one device configured to be in communication with communication device  202   e  and/or autonomous vehicle compute  202   f . In some examples, DBW system  202   h  includes one or more controllers (e.g., electrical controllers, electromechanical controllers, and/or the like) that are configured to generate and/or transmit control signals to operate one or more devices of vehicle  200  (e.g., powertrain control system  204 , steering control system  206 , brake system  208 , and/or the like). Additionally, or alternatively, the one or more controllers of DBW system  202   h  are configured to generate and/or transmit control signals to operate at least one different device (e.g., a turn signal, headlights, door locks, windshield wipers, and/or the like) of vehicle  200 . 
     Powertrain control system  204  includes at least one device configured to be in communication with DBW system  202   h . In some examples, powertrain control system  204  includes at least one controller, actuator, and/or the like. In some embodiments, powertrain control system  204  receives control signals from DBW system  202   h  and powertrain control system  204  causes vehicle  200  to start moving forward, stop moving forward, start moving backward, stop moving backward, accelerate in a direction, decelerate in a direction, perform a left turn, perform a right turn, and/or the like. In an example, powertrain control system  204  causes the energy (e.g., fuel, electricity, and/or the like) provided to a motor of the vehicle to increase, remain the same, or decrease, thereby causing at least one wheel of vehicle  200  to rotate or not rotate. 
     Steering control system  206  includes at least one device configured to rotate one or more wheels of vehicle  200 . In some examples, steering control system  206  includes at least one controller, actuator, and/or the like. In some embodiments, steering control system  206  causes the front two wheels and/or the rear two wheels of vehicle  200  to rotate to the left or right to cause vehicle  200  to turn to the left or right. 
     Brake system  208  includes at least one device configured to actuate one or more brakes to cause vehicle  200  to reduce speed and/or remain stationary. In some examples, brake system  208  includes at least one controller and/or actuator that is configured to cause one or more calipers associated with one or more wheels of vehicle  200  to close on a corresponding rotor of vehicle  200 . Additionally, or alternatively, in some examples brake system  208  includes an automatic emergency braking (AEB) system, a regenerative braking system, and/or the like. 
     In some embodiments, vehicle  200  includes at least one platform sensor (not explicitly illustrated) that measures or infers properties of a state or a condition of vehicle  200 . In some examples, vehicle  200  includes platform sensors such as a global positioning system (GPS) receiver, an inertial measurement unit (IMU), a wheel speed sensor, a wheel brake pressure sensor, a wheel torque sensor, an engine torque sensor, a steering angle sensor, and/or the like. 
     Referring now to  FIG.  3   , illustrated is a schematic diagram of a device  300 . As illustrated, device  300  includes computer processor  304 , memory  306 , storage component  308 , input interface  310 , output interface  312 , communication interface  314 , and bus  302 . In some embodiments, device  300  corresponds to at least one device of vehicles  102  (e.g., at least one device of a system of vehicles  102 ), at least one device of, and/or one or more devices of network  112  (e.g., one or more devices of a system of network  112 ). In some embodiments, one or more devices of vehicles  102  (e.g., one or more devices of a system of vehicles  102 ), and/or one or more devices of network  112  (e.g., one or more devices of a system of network  112 ) include at least one device  300  and/or at least one component of device  300 . As shown in  FIG.  3   , device  300  includes bus  302 , computer processor  304 , memory  306 , storage component  308 , input interface  310 , output interface  312 , and communication interface  314 . 
     Bus  302  includes a component that permits communication among the components of device  300 . In some embodiments, computer processor  304  is implemented in hardware, software, or a combination of hardware and software. In some examples, computer processor  304  includes a computer processor (e.g., a central processing unit (CPU), a graphics processing unit (GPU), an accelerated processing unit (APU), and/or the like), a microphone, a digital signal processor (DSP), and/or any processing component (e.g., a field-programmable gate array (FPGA), an application specific integrated circuit (ASIC), and/or the like) that can be programmed to perform at least one function. Memory  306  includes random access memory (RAM), read-only memory (ROM), and/or another type of dynamic and/or static storage device (e.g., flash memory, magnetic memory, optical memory, and/or the like) that stores data and/or instructions for use by computer processor  304 . 
     Storage component  308  stores data and/or software related to the operation and use of device  300 . In some examples, storage component  308  includes a hard disk (e.g., a magnetic disk, an optical disk, a magneto-optic disk, a solid state disk, and/or the like), a compact disc (CD), a digital versatile disc (DVD), a floppy disk, a cartridge, a magnetic tape, a CD-ROM, RAM, PROM, EPROM, FLASH-EPROM, NV-RAM, and/or another type of computer readable medium, along with a corresponding drive. 
     Input interface  310  includes a component that permits device  300  to receive information, such as via user input (e.g., a touchscreen display, a keyboard, a keypad, a mouse, a button, a switch, a microphone, a camera, and/or the like). Additionally, or alternatively, in some embodiments input interface  310  includes a sensor that senses information (e.g., a global positioning system (GPS) receiver, an accelerometer, a gyroscope, an actuator, and/or the like). Output interface  312  includes a component that provides output information from device  300  (e.g., a display, a speaker, one or more light-emitting diodes (LEDs), and/or the like). 
     In some embodiments, communication interface  314  includes a transceiver-like component (e.g., a transceiver, a separate receiver and transmitter, and/or the like) that permits device  300  to communicate with other devices via a wired connection, a wireless connection, or a combination of wired and wireless connections. In some examples, communication interface  314  permits device  300  to receive information from another device and/or provide information to another device. In some examples, communication interface  314  includes an Ethernet interface, an optical interface, a coaxial interface, an infrared interface, a radio frequency (RF) interface, a universal serial bus (USB) interface, a Wi-Fi® interface, a cellular network interface, and/or the like. 
     In some embodiments, device  300  performs one or more processes described herein. Device  300  performs these processes based on computer processor  304  executing software instructions stored by a computer-readable medium, such as memory  305  and/or storage component  308 . A computer-readable medium (e.g., a non-transitory computer readable medium) is defined herein as a non-transitory memory device. A non-transitory memory device includes memory space located inside a single physical storage device or memory space spread across multiple physical storage devices. 
     In some embodiments, software instructions are read into memory  306  and/or storage component  308  from another computer-readable medium or from another device via communication interface  314 . When executed, software instructions stored in memory  306  and/or storage component  308  cause computer processor  304  to perform one or more processes described herein. Additionally, or alternatively, hardwired circuitry is used in place of or in combination with software instructions to perform one or more processes described herein. Thus, embodiments described herein are not limited to any specific combination of hardware circuitry and software unless explicitly stated otherwise. 
     Memory  306  and/or storage component  308  includes data storage or at least one data structure (e.g., a database and/or the like). Device  300  is capable of receiving information from, storing information in, communicating information to, or searching information stored in the data storage or the at least one data structure in memory  306  or storage component  308 . In some examples, the information includes network data, input data, output data, or any combination thereof. 
     In some embodiments, device  300  is configured to execute software instructions that are either stored in memory  306  and/or in the memory of another device (e.g., another device that is the same as or similar to device  300 ). As used herein, the term “module” refers to at least one instruction stored in memory  306  and/or in the memory of another device that, when executed by computer processor  304  and/or by a computer processor of another device (e.g., another device that is the same as or similar to device  300 ) cause device  300  (e.g., at least one component of device  300 ) to perform one or more processes described herein. In some embodiments, a module is implemented in software, firmware, hardware, and/or the like. 
     The number and arrangement of components illustrated in  FIG.  3    are provided as an example. In some embodiments, device  300  can include additional components, fewer components, different components, or differently arranged components than those illustrated in  FIG.  3   . Additionally, or alternatively, a set of components (e.g., one or more components) of device  300  can perform one or more functions described as being performed by another component or another set of components of device  300 . 
     Referring now to  FIG.  4   , illustrates is an example block diagram of an AV compute  400  (sometimes referred to as an “AV stack”). As illustrated, AV compute  400  includes perception system  402  (sometimes referred to as a perception module), planning system  404  (sometimes referred to as a planning module), localization system  406  (sometimes referred to as a localization module), control system  408  (sometimes referred to as a control module), and database  410 . In some embodiments, perception system  402 , planning system  404 , localization system  406 , control system  408 , and database  410  are included and/or implemented in an autonomous navigation system of a vehicle (e.g., autonomous vehicle compute  202   f  of vehicle  200 ). Additionally, or alternatively, in some embodiments perception system  402 , planning system  404 , localization system  406 , control system  408 , and database  410  are included in one or more standalone systems (e.g., one or more systems that are the same as or similar to autonomous vehicle compute  400  and/or the like). In some examples, perception system  402 , planning system  404 , localization system  406 , control system  408 , and database  410  are included in one or more standalone systems that are located in a vehicle and/or at least one remote system as described herein. In some embodiments, any and/or all of the systems included in autonomous vehicle compute  400  are implemented in software (e.g., in software instructions stored in memory), computer hardware (e.g., by microprocessors, microcontrollers, application-specific integrated circuits [ASICs], Field Programmable Gate Arrays (FPGAs), and/or the like), or combinations of computer software and computer hardware. 
     It will also be understood that, in some embodiments, autonomous vehicle compute  400  is configured to be in communication with a remote system (e.g., an autonomous vehicle system that is the same as or similar to remote AV system  114 , a fleet management system  116  that is the same as or similar to fleet management system  116 , a V2I system that is the same as or similar to V2I system  118 , and/or the like). 
     In some embodiments, perception system  402  receives data associated with at least one physical object (e.g., data that is used by perception system  402  to detect the at least one physical object) in an environment and classifies the at least one physical object. In some examples, perception system  402  receives image data captured by at least one camera (e.g., cameras  202   a ), the image associated with (e.g., representing) one or more physical objects within a field of view of the at least one camera. In such an example, perception system  402  classifies at least one physical object based on one or more groupings of physical objects (e.g., bicycles, vehicles, traffic signs, pedestrians, and/or the like). In some embodiments, perception system  402  transmits data associated with the classification of the physical objects to planning system  404  based on perception system  402  classifying the physical objects. 
     In some embodiments, planning system  404  receives data associated with a destination and generates data associated with at least one route (e.g., routes  106 ) along which a vehicle (e.g., vehicles  102 ) can travel along toward a destination. In some embodiments, planning system  404  periodically or continuously receives data from perception system  402  (e.g., data associated with the classification of physical objects, described above) and planning system  404  updates the at least one trajectory or generates at least one different trajectory based on the data generated by perception system  402 . In some embodiments, planning system  404  receives data associated with an updated position of a vehicle (e.g., vehicles  102 ) from localization system  406  and planning system  404  updates the at least one trajectory or generates at least one different trajectory based on the data generated by localization system  406 . 
     In some embodiments, localization system  406  receives data associated with (e.g., representing) a location of a vehicle (e.g., vehicles  102 ) in an area. In some examples, localization system  406  receives LiDAR data associated with at least one point cloud generated by at least one LiDAR sensor (e.g., LiDAR sensors  202   b ). In certain examples, localization system  406  receives data associated with at least one point cloud from multiple LiDAR sensors and localization system  406  generates a combined point cloud based on each of the point clouds. In these examples, localization system  406  compares the at least one point cloud or the combined point cloud to two-dimensional (2D) and/or a three-dimensional (3D) map of the area stored in database  410 . Localization system  406  then determines the position of the vehicle in the area based on localization system  406  comparing the at least one point cloud or the combined point cloud to the map. In some embodiments, the map includes a combined point cloud of the area generated prior to navigation of the vehicle. In some embodiments, maps include, without limitation, high-precision maps of the roadway geometric properties, maps describing road network connectivity properties, maps describing roadway physical properties (such as traffic speed, traffic volume, the number of vehicular and cyclist traffic lanes, lane width, lane traffic directions, or lane marker types and locations, or combinations thereof), and maps describing the spatial locations of road features such as crosswalks, traffic signs or other travel signals of various types. In some embodiments, the map is generated in real-time based on the data received by the perception system. 
     In another example, localization system  406  receives Global Navigation Satellite System (GNSS) data generated by a global positioning system (GPS) receiver. In some examples, localization system  406  receives GNSS data associated with the location of the vehicle in the area and localization system  406  determines a latitude and longitude of the vehicle in the area. In such an example, localization system  406  determines the position of the vehicle in the area based on the latitude and longitude of the vehicle. In some embodiments, localization system  406  generates data associated with the position of the vehicle. In some examples, localization system  406  generates data associated with the position of the vehicle based on localization system  406  determining the position of the vehicle. In such an example, the data associated with the position of the vehicle includes data associated with one or more semantic properties corresponding to the position of the vehicle. 
     In some embodiments, control system  408  receives data associated with at least one trajectory from planning system  404  and control system  408  controls operation of the vehicle. In some examples, control system  408  receives data associated with at least one trajectory from planning system  404  and control system  408  controls operation of the vehicle by generating and transmitting control signals to cause a powertrain control system (e.g., DBW system  202   h , powertrain control system  204 , and/or the like), a steering control system (e.g., steering control system  206 ), and/or a brake system (e.g., brake system  208 ) to operate. In an example, where a trajectory includes a left turn, control system  408  transmits a control signal to cause steering control system  206  to adjust a steering angle of vehicle  200 , thereby causing vehicle  200  to turn left. Additionally, or alternatively, control system  408  generates and transmits control signals to cause other devices (e.g., headlights, turn signal, door locks, windshield wipers, and/or the like) of vehicle  200  to change states. 
     In some embodiments, perception system  402 , planning system  404 , localization system  406 , and/or control system  408  implement at least one machine learning model (e.g., at least one multilayer perceptron (MLP), at least one convolutional neural network (CNN), at least one recurrent neural network (RNN), at least one autoencoder, at least one transformer, and/or the like). In some examples, perception system  402 , planning system  404 , localization system  406 , and/or control system  408  implement at least one machine learning model alone or in combination with one or more of the above-noted systems. In some examples, perception system  402 , planning system  404 , localization system  406 , and/or control system  408  implement at least one machine learning model as part of a pipeline (e.g., a pipeline for identifying one or more objects located in an environment and/or the like). 
     Database  410  stores data that is transmitted to, received from, and/or updated by perception system  402 , planning system  404 , localization system  406  and/or control system  408 . In some examples, database  410  includes a storage component (e.g., a storage component that is the same as or similar to storage component  308  of  FIG.  3   ) that stores data and/or software related to the operation and uses at least one system of AV compute  400 . In some embodiments, database  410  stores data associated with 2D and/or 3D maps of at least one area. In some examples, database  410  stores data associated with 2D and/or 3D maps of a portion of a city, multiple portions of multiple cities, multiple cities, a county, a state, a State (e.g., a country), and/or the like). In such an example, a vehicle (e.g., a vehicle that is the same as or similar to vehicles  102  and/or vehicle  200 ) can drive along one or more drivable regions (e.g., single-lane roads, multi-lane roads, highways, back roads, off road trails, and/or the like) and cause at least one LiDAR sensor (e.g., a LiDAR sensor that is the same as or similar to LiDAR sensors  202   b ) to generate data associated with an image representing the objects included in a field of view of the at least one LiDAR sensor. 
     In some embodiments, database  410  can be implemented across a plurality of devices. In some examples, database  410  is included in a vehicle (e.g., a vehicle that is the same as or similar to vehicles  102  and/or vehicle  200 ), an autonomous vehicle system (e.g., an autonomous vehicle system that is the same as or similar to remote AV system  114 , a fleet management system (e.g., a fleet management system that is the same as or similar to fleet management system  116  of  FIG.  1   , a V2I system (e.g., a V2I system that is the same as or similar to V2I system  118  of  FIG.  1   ) and/or the like. 
     Spatio-Temporal Grids 
       FIG.  5    illustrates issues during biasing using a sample-based MPC trajectory realizer, according to an embodiment. Three sets of lanes  500   a ,  500   b  and  500   c  are shown, where a first set of lanes  500   a  shows soft dynamic homotopy constraints  505   a  at a given MPC time step, a second set of lanes  500   b  shows the intended sampled soft MPC constraints  505   b  at the given time step and a third set of lanes  500   c  shows the incorrect actual imposed soft MPC constraints  505   c  at the given time step. 
     In an embodiment, each dynamic homotopy constraint is defined by station-time and station-spatial-time constraints. Soft constraints are constraints that should be followed by the AV but can be violated to, for example, complete a trip to a destination or to avoid a collision. Some examples of “soft” constraints include but are not limited to: passenger comfort constraints and a minimum threshold of lateral clearance from a pedestrian who is crossing the street (“jaywalking”) to ensure that the pedestrian and the AV passenger are comfortable with the AV’s maneuvering. In an embodiment, soft constraints can be embodied in one or more hierarchical or non-hierarchical rulebooks. Soft constraints can include spatial constraints that change over time (e.g., lanes that open up as traffic proceeds) or station constraints that change over time (e.g., a traffic light turning from red to green). A spatial constraint can be, for example, a drivable area. 
     By contrast, hard constraints are logical constraints that must not be violated because, if violated, the AV would collide with another object, such as a pedestrian who may be “jaywalking” across the road. Note that hard constraints do not imply “do not collide” since they also capture intended maneuver decisions such as: “keep ego vehicle in current lane”, in which case the hard constraint would limit the vehicle to the current lane. 
     For example, a hard constraint can defined by time instances, such that an example wording could be: “at t+1 seconds the vehicle has to be within these bounds, defined around the centerline of the lane, both laterally and longitudinally (i.e., station constraints).” 
     A sample-based trajectory realization  504  is generated by a sample-based MPC trajectory realizer. Trajectory realization  504  allows vehicle  501  to change from first lane  502   a  to second lane  502   b  to travel around agent  503  (e.g., a parked vehicle) which is in front of vehicle  501  in first lane  502   a . As shown in set of lanes  500   a , biasing  508  of trajectory realization  504  occurs too early due to sampling issues related to the sample-based trajectory realizer, resulting in incorrect constraints being imposed on sample-based trajectory realization  504 . To avoid incorrect constraints being imposed on a trajectory realization, the homotopy constraints are encoded into continuously-differentiable, spatio-temporal grids that are incorporated into an MPC, as described in reference to  FIGS.  6 - 9   . 
       FIG.  6 A  illustrates a spatio-temporal grid for a first homotopy. The first homotopy is vehicle  600  overtaking object  603  by performing a lane change from the right lane into a pocket in the left lane between agents  601 ,  602 . Spatio-temporal grids are shown for discrete time steps t=0 second (s), t=1 s and t=2 s. There is a spatio-temporal grid for each discrete time step in the MPC prediction horizon. The white areas in the grids indicate areas occupied by agents  601 ,  602  and object  603 . The remaining areas in the grid collectively define a unimodal corridor representing the first dynamic homotopy constraints, and thus the area vehicle  600  can occupy during the maneuver at the particular time step. For example, at t=0 s, there is a large pocket between agents  601 ,  602  for vehicle  600  to merge into. At t=1 s, agent  602  is about even with object  603  and the pocket is smaller. At t=3 s, agent  602  overtakes object  603  and vehicle  600  must now merge in front of agent  601  before agent  601  overtakes object  603 . 
       FIG.  6 B  illustrates a spatio-temporal grid for a second homotopy. The second homotopy is vehicle  600  overtaking object  603  by performing a lane change from the right lane behind agents  601 ,  602 . Spatio-temporal grids are shown for the second homotopy for discrete time steps t=0 s, t=1 s and t=2 s. There is a spatio-temporal grid for each discrete time step in the MPC prediction horizon. The white areas in the grids indicate areas occupied by agents  601 ,  602  and object  603 . The remaining areas (dark) in the grid collectively define a unimodal corridor representing dynamic constraints of the second homotopy, and thus the area in that vehicle  600  can occupy during the maneuver for the particular time step. For example, at t=0 s agents  601 ,  602  are constraining vehicle  600  to the right lane. At t=1s, agents  601 ,  602  continue in the left lane with agent  602  being on the left side of obstacle  603 , leaving room for vehicle  600  to slow down in the right lane. 
     As can be observed from  FIGS.  6 A and  6 B , the dynamic homotopy constraints for these lane change maneuvers have been encoded into unimodal corridor constraints representing the homotopy at each time step in the MPC prediction horizon. However, because the unimodal corridor constraints are encoded in a discrete grid, it is desirable to make the discrete grid continuously-differentiable before it is incorporated into a grid MPC trajectory optimizer, as described in reference to  FIG.  9   . In an embodiment, this can be done using spline functions, as described in reference to  FIG.  7   . 
       FIG.  7    Illustrates using spline functions for the continuously-differentiable querying of the discrete spatio-temporal grids shown in  FIGS.  6 A,  6 B . In an embodiment, a non-rational B-spline (i.e. a simple non-weighted B-spline) is used to combine neighboring cell values continuously while providing the corresponding gradients and differentiability necessary for the MPC trajectory realizer, where the knots for the B-splines in the grid are uniformly distributed so that the control points and knots are organized in a 2D space. The result is that the unimodal corridor constraints representing the dynamic homotopy constraints are encoded as convex shapes with smooth edges in the grid, as shown in  FIG.  8   . In an embodiment, the dynamic homotopy constraints are encoded as non-convex shapes where regions with agents and other obstacles are blacked out. 
     Referring again to  FIG.  7   , each grid cell value is a grid cell occupancy value P (u grid , v grid ), where u grid  represents a column of patch  700  in the grid and v grid  represents a row of patch  700 , thereby indicating a position of the cell in patch  700 . In an embodiment, querying the grid cell occupancy value ρ (u grid , v grid ) includes summing the weighted contributions from all the values within patch  700  along the u-axis using the weights from cubic basis functions  702 A, resulting in the column vector  701 , and then summing the elements of  701  using the weights from cubic basis functions  702 B. In an embodiment, grid occupancy values can be binary, (e.g., a “1” or “0”) indicating whether the cell is included in the unimodal corridor or outside the unimodal corridor, respectively. In an embodiment, the grid occupancy values can be scalar values (e.g., floating point precision values) which are used to indicate cost or collision probabilities. 
       FIG.  8    illustrates patch  800  of a spatio-temporal grid with unimodal corridor  801  that has been made continuously-differentiable using spline functions, as described in reference to  FIG.  7   . As shown in  FIG.  8   , unimodal corridor  801  the edges along the value axis are smoothed, i.e. there are no value discontinuities. If for example patch  800  would only have binary values (0 or 1), when querying an occupancy value near a transition boundary the resulting queried value will gradually increase from 0 to 1 due to the smoothing effect of the cubic basis functions  702 A,  702 B. Thus, the smooth edges shown in  FIG.  8    are not required, however in an embodiment the smooth edges help to increase the non-zero gradient span useful for the MPC. 
     In an embodiment the smooth edges help to increase the non-zero continuously-differentiable gradient span, thus making the grid more suitable for incorporation into the grid MPC trajectory optimizer, which relies on both the first and second order derivatives (e.g., gradients and Hessian) to perform the optimization. 
     In an embodiment, probing points  802   a - 802   d  contribute to a collision cost at each MPC prediction stage. For example, an MPC collision cost is introduced with a static and kinetic collision energy term based on the grid occupancy value of a particular location (x, y), as shown in Equation [1]: 
     
       
         
           
             ζ 
             
               
                 v 
                 , 
                 x 
                 , 
                 y 
               
             
             = 
             
               
                 
                   c 
                   
                     s 
                     t 
                     a 
                     t 
                     i 
                     c 
                   
                 
                 + 
                 
                   c 
                   
                     k 
                     i 
                     n 
                     e 
                     t 
                     i 
                     c 
                   
                 
                 
                   v 
                   2 
                 
               
             
             ρ 
             
               
                 x 
                 , 
                 y 
               
             
             , 
           
         
       
     
      where v is the velocity of the vehicle, (x, y) is the position of the vehicle in the grid, c static  and c kinetic  are the static and kinetic energy cost parameters, respectively, and p(x,y) is the queried grid occupancy value at position (x, y) in the grid. 
     Each probing point  802   a - 802   d  contributes to the MPC collision cost C k , at each MPC prediction stage k, as shown in Equation [2]: 
     
       
         
           
             
               c 
               k 
             
             
               
                 x 
                 , 
                 y 
                 , 
                 v 
               
             
               
             = 
             
               ∑ 
               
                 
                     
                   
                     p 
                     = 
                     0 
                   
                   
                     n 
                     − 
                     1 
                   
                 
                 ζ 
               
             
             
               
                 v 
                 , 
                 x 
                 , 
                 + 
                 
                   x 
                   p 
                 
                 , 
                 y 
                 + 
                 
                   y 
                   p 
                 
               
             
             , 
           
         
       
     
      where (x p , y p ) represents the offset of the probing point relative to the vehicle position in the grid and n is the number of probing points (e.g., n=4). 
     In an embodiment, the total collision cost c k (x,y, v) over N prediction stages is part of the total MPC cost function in a trade-off with other objective functions (e.g., comfort, safety, tracking) according to Equation [3]: 
     
       
         
           
             C 
             = 
             
               ∑ 
               
                 
                     
                   
                     k 
                     = 
                     0 
                   
                   N 
                 
                 … 
                 + 
                 
                   c 
                   k 
                 
                 
                   
                     x 
                     
                       k 
                     
                     , 
                     y 
                     
                       k 
                     
                     , 
                     v 
                     
                       k 
                     
                   
                 
                 . 
               
             
           
         
       
     
       FIG.  9    is a block diagram of planning and control system  900 , in accordance with one or more embodiments. Planning and control system  900  includes route planner  901 , logical constraint generator  902 , homotopy extractor  903 , grid-based trajectory realizer  910 , trajectory evaluator  906 , tracking controller  907  and AV  908 . Grid-based trajectory realizer  910  further includes, for each homotopy of N extracted homotopies, spatio-temporal grid generator  904  and grid MPC trajectory generator  905  (e.g., goal location). 
     In an embodiment, route planner  901 : 1) receives an initial and terminal state (e.g., goal location) for AV  908 ; 2) plans a desired sequence of geometric blocks of road data (“roadblocks”) forming lanes with a lane router; 3) divides the route into road segments based on a lane change, such that a segment does not contain a lane change; 4) selects road segments in which the AV  908  is located based on the AV’s physical state (obtained from dynamic world model  909 ) which is projected on the roadblocks; 5) extracts anchor paths for selected road segments (which can be marked as anchor “desired” in case a lane change is desired); and 6) trims anchor paths based on maximal/minimal length. In case there is no lane change required, the adjacent anchor path is extracted and labeled only as “optional,” meaning the AV  908  can use the lane if needed for collision avoidance. 
     In an embodiment, logical constraint generator  902  includes generating at least one of “hard” constraints or “soft” constraints previously defined above. In some embodiments, different constraints are sampled differently. For example, homotopy extractor  903  can operate at 10 Hz and the trajectory optimization can be performed twice as fast at 20 Hz. 
     In an embodiment, homotopy extractor  903  receives a route plan from route planner  901  which contains an “anchor path.” The “anchor path” is the best estimate of the lane that AV  908  is located in, and an optional path (a potentially desired path) which can be used by AV  908  when performing a lane change. In an embodiment, the route plan also contains speed squared constraints and spatial constraints encoding the lane geometry (e.g., lane width) which are computed along the anchor path. In an embodiment, homotopy extractor  903  generates a graphical representation of the operating environment of AV  908 , the physical state of AV  908  based on sensor data (e.g., speed, position) and possible outcomes. In an embodiment, the graphical representation is a decision tree (e.g. a directed acyclic graph) that includes a number of nodes where each node represents a sample of the decision space for a particular driving scenario for AV  908 , such as, for example, a plurality of maneuvers related to other vehicles and objects and environmental constraints (e.g., drivable area, lane markings). 
     Given an initial state of AV  908 , a terminal state of AV  908  on the anchor path, a map representation and predictions of other agents in the scene, the homotopy extractor  903  finds all approximately feasible maneuvers the AV can perform. Note that in this context the resulting maneuvers might not be dynamically feasible but the homotopy extractor  903  guarantees that the resulting constraint set describing the maneuver is not an empty set (considering also the AV footprint). An AV maneuver is described by the homotopy, which is a unique space wherein any path starting with an initial AV state and ending at a terminal AV state can be continuously deformed. To find these maneuvers, homotopy extractor  903  iterates over all possible decisions the AV can take with respect to other agents, e.g. pass on the left/right side, pass before or after or just stay behind, and outputs constraint sets describing the maneuvers for each homotopy. In short, an output of homotopy extractor  903  is a set of all possible maneuvers which can be performed when considering the road and all surrounding agents, where each maneuver is described as a spatio-temporal space that the AV is allowed to occupy. Although this can be a computationally expensive search, a set of simple checks is used to eliminate all infeasible combinations. 
     To be able to describe constraints representing where the other agents are located, and what a collision of AV  908  with these agents mean, every agent is converted into a station-time obstacle or station-spatial-time obstacle. The station-time constraint is a constraint parameterized over time and the station-spatial-time constraint is a constraint parameterized over both station and time, wherein in both cases “station” refers to the traveled distance along the lane centerline (i.e., anchor path). 
     In an embodiment, grid-based trajectory realizer  910  takes as input the station-time constraints and the station-spatial-time constraints for each extracted homotopy (collectively, the “dynamic homotopy constraints”). For each extracted homotopy, spatial-temporal grid generator  904  encodes the dynamic homotopy constraints for the homotopy in a temporal sequence of continuously-differentiable grids, as previously described above in reference to  FIGS.  6 A,  6 B  above. The continuously-differentiable spatio-temporal grids are input into grid MPC trajectory optimizer  905 , which outputs an optimized trajectory based on the station-time and station-spatial-time constraints. 
     In an embodiment, trajectory evaluator  905  uses one or more rulebooks, one or more machine learning models  909  and/or one or more safety maneuver models to score the optimized trajectories, and then uses the scores to select the trajectory that is the most compliant with the rules in the one or more rulebooks (hereinafter, the “selected trajectory”). 
     Tracking controller  907  (e.g., DBW system  202   h  in  FIG.  2   ) is configured to receive the selected trajectory and generate and/or transmit control signals to operate one or more devices of vehicle  908  (e.g., powertrain control system  204 , steering control system  206 , brake system  208 , and/or the like) in accordance with the selected trajectory. 
       FIG.  10    is a flow diagram of process  1000  of grid-based trajectory realization by encoding dynamic homotopy constraints into spatio-temporal grids. Process  1000  can be implemented by, for example, AV compute  202   f  shown in  FIG.  2   , and in particular planning system  404  and control system  408  shown in  FIG.  4   . 
     Process  1000  includes the steps of determining, with at least one processor, a plurality of dynamic homotopy constraints associated with a scenario involving a vehicle in an environment, embedding, with the at least one processor, the dynamic homotopy constraints in a plurality of spatio-temporal grids, where each spatio-temporal grid includes individual grids for each timestep of a prediction horizon; generating, with the at least one processor, a plurality of trajectories based on the plurality of dynamic homotopy constraints embedded in the plurality of spatio-temporal grids ( 1003 ), selecting, with the at least one processor, a particular trajectory from among the plurality of trajectories ( 1004 ); and controlling, with a control circuit of the vehicle, the vehicle in the environment based on the particular trajectory selected from among the plurality of trajectories ( 1005 ). The details of each of the foregoing steps are described above in reference to  FIGS.  5 - 9   . 
     In the foregoing description, aspects and embodiments of the present disclosure have been described with reference to numerous specific details that can vary from implementation to implementation. Accordingly, the description and drawings are to be regarded in an illustrative rather than a restrictive sense. The sole and exclusive indicator of the scope of the invention, and what is intended by the applicants to be the scope of the invention, is the literal and equivalent scope of the set of claims that issue from this application, in the specific form in which such claims issue, including any subsequent correction. Any definitions expressly set forth herein for terms contained in such claims shall govern the meaning of such terms as used in the claims. In addition, when we use the term “further comprising,” in the foregoing description or following claims, what follows this phrase can be an additional step or entity, or a sub-step/sub-entity of a previously-recited step or entity.