Patent Publication Number: US-9424752-B1

Title: Methods and systems for performing fleet planning based on coarse estimates of regions

Description:
BACKGROUND 
     Unless otherwise indicated herein, the materials described in this section are not prior art to the claims in this application and are not admitted to be prior art by inclusion in this section. 
     Computing devices such as personal computers, laptop computers, tablet computers, cellular phones, and countless types of Internet-capable devices are increasingly prevalent in numerous aspects of modern life. As such, the demand for data connectivity via the Internet, cellular data networks, and other such networks, is growing. However, there are many areas of the world where data connectivity is still unavailable, or if available, is unreliable and/or costly. Accordingly, additional network infrastructure is desirable. 
     SUMMARY 
     In one example, a method is provided that comprises receiving information indicative of a sequence of coverage requirements for a region over a period of time. The region is characterized by a plurality of landmarks and the period of time is divided into a plurality of time intervals, and an individual coverage requirement identifies a desired number of vehicles of a plurality of vehicles for each of the plurality of landmarks at a given time interval. The method also includes for one or more time intervals of the period of time, dividing, by a processor, the region over which vehicles of the plurality of vehicles may traverse into a plurality of sub-regions such that for each subsequent time interval a size of a given sub-region increases. The method also includes for at least one of the one or more time intervals of the period of time, determining by the processor vehicles of the plurality of vehicles that can reach a given landmark in a given sub-region by an end of the one or more time intervals, and based on the sequence of coverage requirements, generating, by the processor, a fleet plan for the time intervals based on the determined vehicles. 
     In another example, a computer readable storage medium having stored therein instructions, that when executed by a computing device, cause the computing device to perform functions is provided. The functions comprise receiving information indicative of a sequence of coverage requirements for a region over a period of time. The region is characterized by a plurality of landmarks and the period of time is divided into a plurality of time intervals, and an individual coverage requirement identifies a desired number of vehicles of a plurality of vehicles for each of the plurality of landmarks at a given time interval. The functions also comprise for one or more time intervals of the period of time, dividing the region over which vehicles of the plurality of vehicles may traverse into a plurality of sub-regions such that for each subsequent time interval a size of a given sub-region increases. The functions also comprise for at least one of the one or more time intervals of the period of time, determining vehicles of the plurality of vehicles that can reach a given landmark in a given sub-region by an end of the one or more time intervals, and based on the sequence of coverage requirements, generating a fleet plan for the time intervals based on the determined vehicles. 
     In still another example, a system is provided that comprises at least one processor, and data storage comprising program instructions executable by the at least one processor to cause the at least one processor to perform functions. The functions comprise receiving information indicative of a sequence of coverage requirements for a region over a period of time. The region is characterized by a plurality of landmarks and the period of time is divided into a plurality of time intervals, and an individual coverage requirement identifies a desired number of vehicles of a plurality of vehicles for each of the plurality of landmarks at a given time interval. The functions also comprise for one or more time intervals of the period of time, dividing the region over which vehicles of the plurality of vehicles may traverse into a plurality of sub-regions such that for each subsequent time interval a size of a given sub-region increases. The functions also comprise for at least one of the one or more time intervals of the period of time, determining vehicles of the plurality of vehicles that can reach a given landmark in a given sub-region by an end of the one or more time intervals, and based on the sequence of coverage requirements, generating a fleet plan for the time intervals based on the determined vehicles. 
     These as well as other aspects, advantages, and alternatives, will become apparent to those of ordinary skill in the art by reading the following detailed description, with reference where appropriate to the accompanying figures. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  is a block diagram illustrating an example balloon network. 
         FIG. 2  is a block diagram illustrating an example balloon-network control system. 
         FIG. 3  shows a high-altitude balloon according to an example embodiment. 
         FIG. 4  illustrates a conceptual region divided into a grid of cells and a conceptual path for a vehicle through the region. 
         FIGS. 5-6 and 7A-7B  illustrate conceptual regions divided into sub-regions and a time period divided into time increments. 
         FIG. 8  is an example block diagram of a method for performing fleet planning based on coarse estimates of regions, in accordance with at least some embodiments described herein. 
         FIG. 9  illustrates example trajectories for a balloon that is traveling from a plurality of starting landmarks to a landmark E. 
         FIG. 10  illustrates an example cost value map where location A is a starting location of a balloon, location B is a desired final location, and example cost values are assigned to cells surrounding location B. 
         FIG. 11  illustrates estimated destination cells for a balloon starting at cell X, and moving with one of the five wind vectors A 1 -A 5 . 
         FIGS. 12A-12B  illustrate examples of determining a set of starting landmarks from which a balloon could reach a landmark E. 
         FIG. 13  illustrates an example graph of possible routes between a plurality of landmarks in accordance with an example embodiment. 
         FIG. 14  illustrates an example directed graph in accordance with an example embodiment. 
         FIG. 15  illustrates an example piecewise linear cost function in accordance with an example embodiment. 
         FIG. 16  is an example block diagram of a method to determine trajectories for a fleet of balloons, in accordance with at least some embodiments described herein. 
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description describes various features and functions of the disclosed systems and methods with reference to the accompanying figures. In the figures, similar symbols identify similar components, unless context dictates otherwise. The illustrative system and method embodiments described herein are not meant to be limiting. It may be readily understood that certain aspects of the disclosed systems and methods can be arranged and combined in a wide variety of different configurations, all of which are contemplated herein. 
     Examples herein may be configured to provide control of vehicles within a fleet of vehicles. The vehicles may be any number of types of vehicles including, for example, autonomous or semi-autonomous vehicles, such as aerial vehicles, boats, underwater vehicles, satellites, aerostats, etc. for mapping/surveying, sensing, providing connectivity, etc. In other examples, methods described herein may be used to control or dispatch autonomous cars to maintain a well-distributed fleet of autonomous taxis in a city to support rapid dispatch of taxis. In still other examples, vehicles may include vehicles, mobile vehicles, objects, mobile elements, or mobile objects within a fleet of vehicles. 
     In still further examples, vehicles may be or include vehicles of a data network that includes a plurality of balloons, for example, configured as a mesh network formed by high-altitude balloons deployed in the stratosphere. Since winds in the stratosphere may affect locations of the balloons in a differential manner, each balloon in an example network may be configured to change its horizontal position by adjusting its vertical position (i.e., altitude). For example, by adjusting its altitude, a balloon may be able find winds that will carry the balloon horizontally (e.g., latitudinally and/or longitudinally) to a desired horizontal location. In further examples, balloons may include propellers for horizontal thrust rather than (or in addition to) wind. 
     Within examples, methods and systems are described with respect to vehicles of a fleet of vehicles being balloons in a fleet of balloons arranged as a data network. However, as mentioned above, methods and systems herein may be applicable to other types of vehicles or elements of a fleet of vehicles, and thus, examples described are not limiting. 
     In some examples, methods and systems for performing fleet planning based on coarse estimates of regions are described. Information about a sequence of coverage requirements for a region over a period of time may be received. To utilize coarse estimates of the region, the region can be divided into a plurality of sub-regions such that for each subsequent time interval a size of a given sub-region increases. In this manner, the region can be represented by a fewer number of sub-regions for periods of time further into the future. A detailed representation of the region divided into more sub-regions can be used for periods of time that occur sooner. For each time interval of the period of time, fleet planning of vehicles can be performed using the various representations of the regions for the various time intervals of the time period. A fleet plan can be generated by combining fleet plans per time intervals of the time period, for example. 
       FIG. 1  is a block diagram illustrating an example balloon network  100 . As shown, the balloon network  100  includes balloons  102 A to  102 F, which are configured to communicate with one another via free-space optical links  104 . Configured as such, balloons  102 A to  102 F may collectively function as a mesh network for packet-data communications. Further, at least some of balloons  102 A and  102 B may be configured for RF communications with ground-based stations  106  via respective RF links  108 , or for balloon-to-balloon RF communication. Yet further, some balloons, such as balloon  102 F, may be configured to communicate via optical link  110  with ground-based station  112 . 
     In an exemplary embodiment, balloons  102 A to  102 F are high-altitude balloons, which are deployed in the stratosphere. At moderate latitudes, the stratosphere includes altitudes between approximately 10 kilometers (km) and 50 km altitude above the surface. At the poles, the stratosphere starts at an altitude of approximately 8 km. In an exemplary embodiment, high-altitude balloons may be generally configured to operate in an altitude range within the stratosphere that has relatively low wind-speeds (e.g., at about 17.5 km to 20 km altitude, the global mean wind speed may reach about 30 to 35 mph for large portions of the year, while a max wind speed may be up to 200 mph; and at 26 km altitude, a global mean wind speed may reach about 60 to 65 mph and a max wind speed of about 300 mph). 
     More specifically, in a high-altitude-balloon network, balloons  102 A to  102 F may generally be configured to operate at altitudes between 18 km and 25 km (although other altitudes are possible). At such altitudes, for example, the layer of the stratosphere generally has relatively low wind speeds (e.g., winds between 5 and 20 mph) and relatively little turbulence. Further, while the winds between 18 km and 25 km may vary with latitude and by season, the variations can be modeled in a reasonably accurate manner. Additionally, altitudes above 18 km are typically above the maximum flight level designated for commercial air traffic. Therefore, interference with commercial flights is not a concern when balloons are deployed between 18 km and 25 km. 
     To transmit data to another balloon, a given balloon  102 A to  102 F may be configured to transmit an optical signal via an optical link  104 . In an exemplary embodiment, a given balloon  102 A to  102 F may use one or more high-power light-emitting diodes (LEDs) to transmit an optical signal. Alternatively, some or all of balloons  102 A to  102 F may include laser systems for free-space optical communications over optical links  104 . Other types of free-space optical communication are possible. Further, in order to receive an optical signal from another balloon via an optical link  104 , a given balloon  102 A to  102 F may include one or more optical receivers. 
     In a further aspect, balloons  102 A to  102 F may utilize one or more of various different RF air-interface protocols for communication with ground-based stations  106  via respective RF links  108 . For instance, some or all of balloons  102 A to  102 F may be configured to communicate with ground-based station  106  using protocols described in IEEE 802.11 (including any of the IEEE 802.11 revisions), various cellular protocols such as GSM, CDMA, UMTS, EV-DO, WiMAX, and/or LTE, and/or one or more propriety protocols developed for balloon-ground RF communication, among other possibilities. 
     In a further aspect in balloon network  100 , balloon  102 F is configured as a downlink balloon to provide a high-capacity air-ground link. Like other balloons in an exemplary network, a downlink balloon  102 F may be operable for optical communication with other balloons via optical links  104 . However, a downlink balloon  102 F may also be configured for free-space optical communication with a ground-based station  112  via an optical link  110 . Optical link  110  may therefore serve as a high-capacity link (as compared to an RF link  108 ) between the balloon network  100  and ground-based station  112 . 
     Note that in some implementations, a downlink balloon  102 F may additionally be operable for RF communication with ground-based station  106 . In other cases, a downlink balloon  102 F may only use an optical link for balloon-to-ground communications. Further, while the arrangement shown in  FIG. 1  includes just one downlink balloon  102 F, an exemplary balloon network can also include multiple downlink balloons. On the other hand, a balloon network can also be implemented without any downlink balloons. 
     In other implementations, a downlink balloon may be equipped with a specialized, high-bandwidth RF communication system for balloon-to-ground communications, instead of, or in addition to, a free-space optical communication system. The high-bandwidth RF communication system may take the form of an ultra-wideband system, which may provide an RF link with substantially the same capacity as one of the optical links  104 . 
     Ground-based stations, such as ground-based station  106  and/or  112 , may include components such as transceivers, transmitters, and/or receivers for communication via RF links and/or optical links with a balloon network. Further, a ground-based station may use various air-interface protocols in order to communicate with a balloon  102 A to  102 F over an RF link. As such, a ground-based station  106  may be configured as an access point via which various devices can connect to balloon network  100 . 
     In a further aspect, some or all balloons  102 A to  102 F could be configured to establish a communication link with space-based satellites in addition to, or as an alternative to, a ground-based communication link. In some embodiments, a balloon may communicate with a satellite via an optical link. 
     Further, some ground-based stations, such as ground-based station  106 , may be configured as gateways between balloon network  100  and one or more other networks. Such a ground-based station  106  may thus serve as an interface between the balloon network and the Internet, a cellular service provider&#39;s network, and/or other types of networks. 
     As noted, balloons  102 A to  102 F may collectively function as a mesh network. More specifically, since balloons  102 A to  102 F may communicate with one another using free-space optical links, the balloons may collectively function as a free-space optical mesh network. 
     In a mesh-network configuration, each balloon  102 A to  102 F may function as a node of the mesh network, which is operable to receive data directed to the balloon and to route data to other balloons. As such, data may be routed from a source balloon to a destination balloon by determining an appropriate sequence of optical links between the source balloon and the destination balloon. These optical links may be collectively referred to as a “lightpath” for the connection between the source and destination balloons. Further, each of the optical links may be referred to as a “hop” on the lightpath. 
     To operate as a mesh network, balloons  102 A to  102 F may employ various routing techniques and self-healing algorithms. In some embodiments, a balloon network  100  may employ adaptive or dynamic routing, where a lightpath between a source and destination balloon is determined and set-up when the connection is needed, and released at a later time. Further, when adaptive routing is used, the lightpath may be determined dynamically depending upon the current state, past state, and/or predicted state of the balloon network. 
     In addition, the network topology may change as the balloons  102 A to  102 F move relative to one another and/or relative to the ground. Accordingly, an exemplary balloon network  100  may apply a mesh protocol to update the state of the network as the topology of the network changes. For example, to address the mobility of the balloons  102 A to  102 F, balloon network  100  may employ and/or adapt various techniques that are employed in mobile ad hoc networks (MANETs). Other examples are possible as well. 
     In some implementations, a balloon network  100  may be configured as a transparent mesh network. For instance, the balloons may include components for physical switching that is entirely optical, without any electrical components involved in the physical routing of optical signals. Thus, in a transparent configuration with optical switching, signals travel through a multi-hop lightpath that is entirely optical. 
     In other implementations, the balloon network  100  may implement a free-space optical mesh network that is opaque. In an opaque configuration, some or all balloons  102 A to  102 F may implement optical-electrical-optical (OEO) switching. For example, some or all balloons may include optical cross-connects (OXCs) for OEO conversion of optical signals. Other opaque configurations are also possible. 
     In a further aspect, balloons in an exemplary balloon network  100  may implement wavelength division multiplexing (WDM), which may help to increase link capacity. When WDM is implemented with transparent switching, physical lightpaths through the balloon network may be subject to the “wavelength continuity constraint.” More specifically, because the switching in a transparent network is entirely optical, the same wavelength can be assigned for all optical links on a given lightpath. 
     An opaque configuration, on the other hand, may avoid the wavelength continuity constraint. In particular, balloons in an opaque balloon network may include the OEO switching systems operable for wavelength conversion. As a result, balloons can convert the wavelength of an optical signal at each hop along a lightpath. 
     Further, various routing algorithms may be employed in an opaque configuration. For example, to determine a primary lightpath and/or one or more diverse backup lightpaths for a given connection, exemplary balloons may apply or consider shortest-path routing techniques such as Dijkstra&#39;s algorithm and k-shortest path, and/or edge and node-diverse or disjoint routing such as Suurballe&#39;s algorithm, among others. Additionally or alternatively, techniques for maintaining a particular quality of service (QoS) may be employed when determining a lightpath. Other techniques are also possible. 
     In some examples, a balloon network  100  may have a non-uniform topology. For instance, exemplary embodiments may involve topologies where balloons are distributed more or less densely in certain areas, for various reasons. As an example, to help meet the higher bandwidth demands that are typical in urban areas, balloons may be clustered more densely over urban areas. For similar reasons, the distribution of balloons may be denser over land than over large bodies of water. Many other examples of non-uniform topologies are possible. 
       FIG. 2  is a block diagram illustrating an example balloon-network control system. In particular,  FIG. 2  shows a distributed control system, which includes a central control system  200  and a number of regional control-systems  202 A to  202 C. Such a control system may be configured to coordinate certain functionality for balloon network  204 , and as such, may be configured to control and/or coordinate certain functions for balloons  206 A to  2061 . 
     In the illustrated embodiment, central control system  200  may be configured to communicate with balloons  206 A to  2061  via a number of regional control systems  202 A to  202 C. These regional control systems  202 A to  202 C may be configured to receive communications and/or aggregate data from balloons in the respective geographic areas that the balloons cover, and to relay the communications and/or data to central control system  200 . Further, regional control systems  202 A to  202 C may be configured to route communications from central control system  200  to the balloons in their respective geographic areas. For instance, as shown in  FIG. 2 , regional control system  202 A may relay communications and/or data between balloons  206 A to  206 C and central control system  200 , regional control system  202 B may relay communications and/or data between balloons  206 D to  206 F and central control system  200 , and regional control system  202 C may relay communications and/or data between balloons  206 G to  2061  and central control system  200 . 
     In order to facilitate communications between the central control system  200  and balloons  206 A to  2061 , certain balloons may be configured as downlink balloons, which are operable to communicate with regional control systems  202 A to  202 C. Accordingly, each regional control system  202 A to  202 C may be configured to communicate with the downlink balloon or balloons in the respective geographic area it covers. For example, in the illustrated embodiment, balloons  206 A,  206 D, and  206 H are configured as downlink balloons. As such, regional control systems  202 A to  202 C may respectively communicate with balloons  206 A,  206 D, and  206 H via optical links  206 ,  208 , and  210 , respectively. 
     In a centralized control arrangement, such as that shown in  FIG. 2 , the central control system  200  (and possibly regional control systems  202 A to  202 C as well) may coordinate certain mesh-networking functions for balloon network  204 . For example, balloons  206 A to  2061  may send the central control system  200  certain state information, which the central control system  200  may utilize to determine the state of balloon network  204 . The state information from a given balloon may include location data, optical-link information (e.g., the identity of other balloons with which the balloon has established an optical link, the bandwidth of the link, wavelength usage and/or availability on a link, etc.), wind data collected by the balloon, and/or other types of information. Accordingly, the central control system  200  may aggregate state information from some or all of the balloons  206 A to  2061  in order to determine an overall state of the network. 
     The overall state of the network may then be used to coordinate and/or facilitate certain mesh-networking functions such as determining lightpaths for connections. For example, the central control system  200  may determine a current topology based on the aggregate state information from some or all of the balloons  206 A to  2061 . The topology may provide a picture of the current optical links that are available in the balloon network and/or the wavelength availability on the links. This topology may then be sent to some or all of the balloons so that a routing technique may be employed to select appropriate lightpaths (and possibly backup lightpaths) for communications through the balloon network  204 . 
     In a further aspect, the central control system  200  (and possibly regional control systems  202 A to  202 C as well) may also coordinate functions for balloon network  204 . For example, the central control system  200  may input state information that is received from balloons  206 A to  2061  to an energy function, which may effectively compare the current topology of the network to a desired topology, and provide a vector indicating a direction of movement (if any) for each balloon, such that the balloons can move towards the desired topology. Further, the central control system  200  may use altitudinal wind data to determine respective altitude adjustments that may be initiated to achieve the movement towards the desired topology. The central control system  200  may provide and/or support other station-keeping functions as well. 
     In some embodiments, control and coordination of a balloon network may be shared by a ground-based control system and a balloon network to varying degrees, depending upon the implementation. In fact, in some embodiments, there may be no ground-based control systems. In such an embodiment, all network control and coordination functions may be implemented by the balloon network itself. For example, certain balloons may be configured to provide the same or similar functions as central control system  200  and/or regional control systems  202 A to  202 C. Other examples are also possible. 
     Furthermore, control and/or coordination of a balloon network may be de-centralized. For example, each balloon may relay state information to, and receive state information from, some or all nearby balloons. Further, each balloon may relay state information that it receives from a nearby balloon to some or all nearby balloons. When all balloons do so, each balloon may be able to individually determine the state of the network. Alternatively, certain balloons may be designated to aggregate state information for a given portion of the network. These balloons may then coordinate with one another to determine the overall state of the network. 
     Further, in some aspects, control of a balloon network may be partially or entirely localized, such that it is not dependent on the overall state of the network. For example, individual balloons may implement functions that only consider nearby balloons. In particular, each balloon may implement an energy function that takes into account its own state and the states of nearby balloons. The energy function may be used to maintain and/or move to a desired position with respect to the nearby balloons, without necessarily considering the desired topology of the network as a whole. However, when each balloon implements such an energy function for station-keeping, the balloon network as a whole may maintain and/or move towards the desired topology. As an example, each balloon may receive distance information d 1  to d k  with respect to each of its k closest neighbors. Each balloon may treat the distance to each of the k balloons as a virtual spring with vector representing a force direction from the first nearest neighbor balloon toward itself and with force magnitude proportional to d i . The balloon may sum each of the k vectors to determine a summed vector of desired movement for itself. The balloon may attempt to achieve the desired movement by controlling its altitude. 
     In another embodiment, a similar process could be carried out for each of the k balloons and each balloon could transmit its planned movement vector to its local neighbors. Further rounds of refinement to each balloon&#39;s planned movement vector can be made based on the corresponding planned movement vectors of its neighbors. 
     Various types of balloon systems may be incorporated in an exemplary balloon network. As noted above, an exemplary embodiment may utilize high-altitude balloons, which typically operate in an altitude range between 18 km and 25 km.  FIG. 3  shows a high-altitude balloon  300 , according to an exemplary embodiment. As shown, the balloon  300  includes an envelope  302 , a skirt  304 , a payload  306 , and a cut-down system  308  that is attached between the balloon  302  and payload  306 . 
     The envelope  302  and skirt  304  may take various forms, for instance, the envelope  302  and/or skirt  304  may be made of metalized Mylar or BoPet. Alternatively or additionally, some or all of the envelope  302  and/or skirt  304  may be constructed from a highly-flexible latex material or a rubber material such as chloroprene. Further, the shape and size of the envelope  302  and skirt  304  may vary depending upon the particular implementation. Additionally, the envelope  302  may be filled with various different types of gases, such as helium and/or hydrogen. 
     The payload  306  of balloon  300  may include a processor  312  and on-board data storage, such as memory  314 . The memory  314  may take the form of or include a non-transitory computer-readable medium. The non-transitory computer-readable medium may have instructions stored thereon, which can be accessed and executed by the processor  312  in order to carry out the balloon functions described herein. 
     The payload  306  of balloon  300  may also include various other types of equipment and systems to provide a number of different functions. For example, payload  306  may include optical communication system  316 , which may transmit optical signals via an ultra-bright LED system  320 , and which may receive optical signals via an optical-communication receiver  322  (e.g., a photo-diode receiver system). Further, payload  306  may include an RF communication system  318 , which may transmit and/or receive RF communications via an antenna system  340 . The payload  306  may further include a positioning system  324  that can include a GPS or other location-based sensors. 
     The payload  306  may also include a power supply  326  to supply power to the various components of balloon  300 . The power supply  326  may include or take the form of a rechargeable battery. In addition, the balloon  300  may include a solar power generation system  327 . The solar power generation system  327  may include solar panels and could be used to generate power that charges and/or is distributed by the power supply  326 . 
     Further, payload  306  may include various types of other systems  328  and sensors  334 . For example, payload  306  may include one or more video and/or still cameras, a GPS system, various motion sensors (e.g., accelerometers, gyroscopes, and/or compasses), and/or various sensors for capturing environmental data. Further, some or all of the components within payload  306  may be implemented in a radiosonde, which may be operable to measure, e.g., pressure, altitude, geographical position (latitude and longitude), temperature, relative humidity, and/or wind speed and/or direction, among other information. 
     As noted, balloon  300  includes an ultra-bright LED system  320  for free-space optical communication with other balloons. As such, optical communication system  316  may be configured to transmit a free-space optical signal by modulating the ultra-bright LED system  320 . The optical communication system  316  may be implemented with mechanical systems and/or with hardware, firmware, and/or software. 
     In a further aspect, balloon  300  may be configured for altitude control. For instance, balloon  300  may include a variable buoyancy system, which is configured to change the altitude of the balloon  300  by adjusting the volume and/or density of the gas in the balloon  300 . A variable buoyancy system may take various forms, and may generally be any system that can change the volume and/or density of gas in envelope  302 . 
     In an exemplary embodiment, a variable buoyancy system may include a bladder  310  that is located inside of envelope  302 . The bladder  310  could be an elastic chamber configured to hold liquid and/or gas. Alternatively, the bladder  310  need not be inside the envelope  302 . For instance, the bladder  310  could be a rigid bladder that could be pressurized well beyond neutral pressure. The buoyancy of the balloon  300  may therefore be adjusted by changing the density and/or volume of the gas in bladder  310 . To change the density in bladder  310 , balloon  300  may be configured with systems and/or mechanisms for heating and/or cooling the gas in bladder  310 . Further, to change the volume, balloon  300  may include pumps or other features for adding gas to and/or removing gas from bladder  310 . Additionally or alternatively, to change the volume of bladder  310 , balloon  300  may include release valves or other features that are controllable to allow gas to escape from bladder  310 . Multiple bladders  310  could be implemented within the scope of this disclosure. For instance, multiple bladders could be used to improve balloon stability. 
     In an example embodiment, the envelope  302  could be filled with helium, hydrogen or other lighter-than-air material. The envelope  302  could thus have an associated upward buoyancy force. In such an embodiment, air in the bladder  310  could be considered a ballast tank that may have an associated downward ballast force. In another example embodiment, the amount of air in the bladder  310  could be changed by pumping air (e.g., with an air compressor) into and out of the bladder  310 . By adjusting the amount of air in the bladder  310 , the ballast force may be controlled. In some embodiments, the ballast force may be used, in part, to counteract the buoyancy force and/or to provide altitude stability. 
     In other embodiments, the envelope  302  could be substantially rigid and include an enclosed volume. Air could be evacuated from envelope  302  while the enclosed volume is substantially maintained. In other words, at least a partial vacuum could be created and maintained within the enclosed volume. Thus, the envelope  302  and the enclosed volume could become lighter than air and provide a buoyancy force. In yet other embodiments, air or another material could be controllably introduced into the partial vacuum of the enclosed volume in an effort to adjust the overall buoyancy force and/or to provide altitude control. 
     In another embodiment, a portion of the envelope  302  could be a first color (e.g., black) and/or a first material from the rest of envelope  302 , which may have a second color (e.g., white) and/or a second material. For instance, the first color and/or first material could be configured to absorb a relatively larger amount of solar energy than the second color and/or second material. Thus, rotating the balloon such that the first material is facing the sun may act to heat the envelope  302  as well as the gas inside the envelope  302 . In this way, the buoyancy force of the envelope  302  may increase. By rotating the balloon such that the second material is facing the sun, the temperature of gas inside the envelope  302  may decrease. Accordingly, the buoyancy force may decrease. In this manner, the buoyancy force of the balloon could be adjusted by changing the temperature/volume of gas inside the envelope  302  using solar energy. In such embodiments, it is possible that a bladder  310  may not be a necessary element of balloon  300 . Thus, various contemplated embodiments, altitude control of balloon  300  could be achieved, at least in part, by adjusting the rotation of the balloon with respect to the sun. 
     Further, a balloon  300  may include a navigation system (not shown). The navigation system may implement station-keeping functions to maintain position within and/or move to a position in accordance with a desired topology. In particular, the navigation system may use altitudinal wind data to determine altitudinal adjustments that result in the wind carrying the balloon in a desired direction and/or to a desired location. The altitude-control system may then make adjustments to the density of the balloon chamber in order to effectuate the determined altitudinal adjustments and cause the balloon to move laterally to the desired direction and/or to the desired location. 
     Alternatively, the altitudinal adjustments may be computed by a ground-based control system and communicated to the high-altitude balloon. As another alternative, the altitudinal adjustments may be computed by a ground-based or satellite-based control system and communicated to the high-altitude balloon. Furthermore, in some embodiments, specific balloons in a heterogeneous balloon network may be configured to compute altitudinal adjustments for other balloons and transmit the adjustment commands to those other balloons. 
     As shown, the balloon  300  also includes a cut-down system  308 . The cut-down system  308  may be activated to separate the payload  306  from the rest of balloon  300 . This functionality may be utilized anytime the payload needs to be accessed on the ground, such as when it is time to remove balloon  300  from a balloon network, when maintenance is due on systems within payload  306 , and/or when power supply  326  needs to be recharged or replaced. 
     In an exemplary embodiment, the cut-down system  308  may include a connector, such as a balloon cord, connecting the payload  306  to the envelope  302  and a means for severing the connector (e.g., a shearing mechanism or an explosive bolt). In an example embodiment, the balloon cord, which may be nylon, is wrapped with a nichrome wire. A current could be passed through the nichrome wire to heat it and melt the cord, cutting the payload  306  away from the envelope  302 . Other types of cut-down systems and/or variations on the illustrated cut-down system  308  are possible as well. 
     In an alternative arrangement, a balloon may not include a cut-down system. In such an arrangement, the navigation system may be operable to navigate the balloon to a landing location, in the event the balloon needs to be removed from the network and/or accessed on the ground. Further, it is possible that a balloon may be self-sustaining, such that it theoretically does not need to be accessed on the ground. In yet other embodiments, balloons may be serviced in-flight by specific service balloons or another type of service aerostat or service aircraft. 
     As described, movement and locations of balloons can vary since winds in the stratosphere may affect the locations of the balloons in a differential manner. A balloon in an example network may be configured to change its horizontal position by adjusting its vertical position (i.e., altitude). For example, by adjusting its altitude, a balloon may be able find winds that will carry the balloon horizontally (e.g., latitudinally and/or longitudinally) to a desired horizontal location. Wind speed varies with altitude, and since current wind speeds as well as weather forecasts are available, in principle, a balloon may be directed to a location by identifying an altitude at which a wind direction takes a balloon to along a desired trajectory. However, a balloon without other forms of propulsion may be constrained to follow the wind and there may not be a single altitude with winds taking the balloon along the desired trajectory. In addition, to control a fleet of balloons, movement of the balloons should occur from one location above the surface of the Earth to another in a predictable manner. 
     In some instances, when managing a large fleet of balloons or other autonomous or semi-autonomous vehicles, it may be difficult to manage where each vehicle will go. Instead, a fleet administrator may have goals that the fleet is directed to satisfy, such as six hours from now, provide at least ten vehicles in area A, five vehicles in area B, eighteen vehicles in area C, etc. Following, at twelve hours from now, provide five vehicles in area A, and ten vehicles in area B and twenty vehicles in area C, etc. Subsequently, eighteen hours from now, still further changes may be desired. 
     Translating goals or sequence of coverage requirements into trajectories or paths for each vehicle to follow may include considering a stating location of an vehicle, from a starting location a determination of which vehicles can reach a goal area by a deadline for a first set of goals (e.g., six hours from now), from a location at the first deadline a determination of which vehicles can reach goal areas for a deadline for a second set of goals, and so on. 
     Within examples considering a fleet of balloons, for each balloon in a fleet of balloons, path planning may be performed to determine of all locations the balloon may traverse, a location where the balloon should traverse to satisfy a goal. A next location or set of available tasks for a balloon is dependent on a task previously achieved (i.e., where a balloon can travel to next is dependent on where the balloon is coming from). 
     In some examples, fleet planning includes initially solving balloon trajectories for a general region, and using a solution as an input for solving balloon trajectories for sub-regions of the general region. In addition, fleet planning may be performed to initially solve balloon trajectories at a time distant in the future, and using a solution as an input for solving balloons trajectories backward in time. Alternatively, solutions may be determined forward in time. In some instances, the fleet planning may include staging balloons so as to maneuver a balloon into a location ahead of a deadline from which the balloon will be able to arrive at the deadline goal on time. Staging requirements may require a balloon to take a trajectory that does not contribute usefully to goals during, for example, the first and second deadlines so that the balloon can reach a desired target area for a subsequent deadline. Such long-range planning may be useful for goal areas spread non-uniformly throughout a coverage area of a fleet, and when there are large regions (relative to a maximum range of a vehicle between deadlines) requiring few or no goals, but which are traversed to satisfy the coverage areas. In one example, for a goal distribution encompassing the world that is proportional to population density, there are no people in the oceans, and few people on many parts of land, and many people in major cities, and thus, world fleet planning may be performed according to a non-uniform distribution. 
       FIG. 4  illustrates a conceptual region divided into a grid of cells and a conceptual path for a vehicle through the region. For example, the region is divided into a grid of cells (e.g.,  4  by  4  grid of cells). A balloon may initially be at a location labeled as A and a location labeled as B may be the desired destination. 
       FIGS. 5-6 and 7A-7B  illustrate conceptual regions divided into sub-regions and a time period divided into time increments. As one example,  FIG. 5  illustrates a  4  by  4  grid of cells labeled A through P, and illustrates a path of the balloon from cell C to cell H. The 4 by 4 grid of cells may be considered a region  500 , and the cells A through P may be considered sub-regions of the region  500 . The region  500  may be divided into any number of cells or sub-regions. At an initial time increment I 1  from time T=0 to T=X, where X is an increment (shown to be three in the example in  FIG. 5 ), balloon locations and trajectories across the cells C to cell H may be determined. It may not be possible for the balloon to traverse from cell C to cell H during the time increment I 1 . 
     Following, at a second increment, I 2 , which is shown to be twice the first increment, cells at a first increment may be combined to create cells AA through HH. As an example, cell AA may include cells A and B. In this example, the size of the cells or sub-regions of the region  500  may be doubled as the time increment is doubled. The cells AA through HH are a more coarse representation of cells A through P, such that, for example, any balloons present in cells A and B may be assumed to be within cell AA. Fleet planning may be performed at or for a time increment I 2 , such that locations and trajectories of balloons within cells AA through HH are estimated according to the coverage requirement of the balloon reaching a desired location in cell DD. 
     Following, for increment I 3 , cells AA through HH may be combined into cells labeled AAA through DDD. Increment I 3  may be twice the second increment, and a size of cells AAA through DDD may each be twice the size of cells AA through HH. Fleet planning may be performed at time increment I 3  such that locations and trajectories of balloons within cells AAA through DDD are estimated according to the coverage requirement of the balloon reaching a desired location in cell BBB. 
     Cells may be combined in such a manner until only one cell remains, such that conceptually, at time increment I 4 , not shown, cells AAA through DDD could be combined into one cell to represent the overall region  500 . 
     Cells or sub-regions may also be combined in any way, such as by doubling in size as shown in the example in  FIG. 5 . In another example,  FIG. 6  illustrates a region  600  including an initial 4 by 4 grid of cells labeled A through P at time increment I 1 , and at time increment I 2  a size of the cells is quadrupled such that the region  600  includes four cells labeled AA through DD. Subsequently, at time increment I 3 , cells AA through DD are combined into one cell that represents the region  600 . For the example, shown in  FIG. 6 , the time increments are doubled from time increment I 1  to I 2  to I 3 . However, in other examples, increases in time increments may be any value and may vary from increment to increment. 
     In the examples in  FIGS. 5-6 , for each subsequent time interval a size of a given sub-region increases. In addition, for each subsequent time interval, a number of sub-regions decreases due to a size of an overall region remaining the same. 
       FIG. 7A  illustrates another example region  700  divided into sub-regions at various time increments. In  FIG. 7A , at time increment X, the region  700  is divided into sub-regions A through E. At time increment Y, the region  700  is divided into sub-regions A through E again, however, a size of the sub-regions is increased. At time increment Z, the region  700  is divided into sub-regions A through E again, however, the size of the sub-regions is increased again. In the example in  FIG. 7A , for each subsequent time interval a size of a given sub-region increases, and a size of the region  700  increases as well. 
       FIG. 7B  illustrates another example time period divided into increments. In the example shown in  FIG. 7B , at a first iteration, the planning period may be divided into two phases (Phase 1 and Phase 2). A solution can be solved for to determine a distribution of a fleet for these two phases. Following, during a second iteration, Phase 1 may be split into two phases (Phase A and Phase B). A solution is then solved for Phases A and B, and variables that were determined during the first iteration can be as inputs to determine solutions to Phases A and B. Then, during a third iteration, Phase A can be divided into A1 and A2, and the solution from the second iteration can be used as input to solve for the distribution for Phases A1 and A2. Using such a technique can determine distributions more quickly as compared to solving for all eight phases simultaneously. 
     For fleet planning, regions of an area (e.g., cells) may be characterized by landmarks or locations which a balloon can reach over a given time. As a size of sub-regions increases, a length of time needed for a balloon to traverse the sub-region increases as well, and with regard to fleet planning, a longer time period enables planning for balloons to traverse a larger distance. Thus, in some examples, a time span and a grid size can be increased at a same rate to solve for balloon locations and trajectories. In yet other examples, either the time span or the grid size can increase while maintaining the other one at fixed values. 
     A fleet plan may be determined at any time increment to determine a location where balloons need to be at such a time to satisfy a given goal, and then, fleet plans can be solved backward in time to determine locations at previous legs of a fleet plan. Thus, a recursive refinement can be performed to solve once with a full time span divided into two time increments, re-solve only a first half of the time span and with the first half divided into two smaller time increments, and repeat solving a first time span increment divided into two smaller increments until a desired resolution of the solution is obtained. By using a coarse estimation of an area further out in time, less computational power can be used to perform fleet planning, which can be useful in instances in which weather predictions are less reliable for further points in the future. 
       FIG. 8  is an example block diagram of a method for performing fleet planning based on coarse estimates of regions, in accordance with at least some embodiments described herein. Method  800  shown in  FIG. 8  presents an embodiment of a method that, for example, could be used with the balloon networks  100  and  204 , for example, and may be performed by a device, a server, a control station, a balloon, or a combination of any components of the balloon networks  100  and  204 . Method  800  may include one or more operations, functions, or actions as illustrated by one or more of blocks  802 - 808 . Although the blocks are illustrated in a sequential order, these blocks may in some instances be performed in parallel, and/or in a different order than those described herein. Also, the various blocks may be combined into fewer blocks, divided into additional blocks, and/or removed based upon the desired implementation. 
     In addition, for the method  800  and other processes and methods disclosed herein, the flowchart shows functionality and operation of one possible implementation of present embodiments. In this regard, each block may represent a module, a segment, or a portion of program code, which includes one or more instructions executable by a processor for implementing specific logical functions or steps in the process. The program code may be stored on any type of computer readable medium, for example, such as a storage device including a disk or hard drive. The computer readable medium may include a non-transitory computer readable medium, for example, such as computer-readable media that stores data for short periods of time like register memory, processor cache and Random Access Memory (RAM). The computer readable medium may also include non-transitory media, such as secondary or persistent long term storage, like read only memory (ROM), optical or magnetic disks, compact-disc read only memory (CD-ROM), for example. The computer readable media may also be any other volatile or non-volatile storage systems. The computer readable medium may be considered a computer readable storage medium, a tangible storage device, or other article of manufacture, for example. 
     In addition, for the method  800  and other processes and methods disclosed herein, each block in  FIG. 8  may represent circuitry that is wired to perform the specific logical functions in the process. 
     At block  802 , the method  800  includes receiving information indicative of a sequence of coverage requirements for a region over a period of time. The region may be characterized by a plurality of landmarks and the period of time may be divided into a plurality of time intervals. An individual coverage requirement identifies a desired number of vehicles of a plurality of vehicles for each of the plurality of landmarks at a given time interval. Note that the number of vehicles may be a minimum number of vehicles that is desired for each of the plurality of landmarks, and in some instances, it may be acceptable to have more than the minimum number of vehicles at each landmark. Additionally, in some instances, the minimum number of vehicles may be zero for one or more landmarks and/or one or more phases. 
     In one example, the vehicles include balloons configured to provide a data network, and a region may include any area, such as a city, a state, a country, or the Earth. Landmarks of a region may be locations within the region designated by a predetermined amount of distance between the locations. The sequence of coverage requirements may indicate, as one example, that six balloons are desired at a first location in two hours, seven balloons are desired at a second locations in four hours, and so on. 
     At block  804 , the method  800  includes, for one or more time intervals of the period of time, dividing the region over which vehicles of the plurality of vehicles may traverse into a plurality of sub-regions such that for each subsequent time interval a size of a given sub-region increases. A processor may divide the region into equal parts for a first time interval, and for subsequent time intervals, the processor may combine parts to increase the size of the sub-regions. In one example, a size of sub-regions may be based on a length of the time interval so that as the length of the time interval increases, the size of sub-regions increase. In another example, over time, sub-regions of the region may represent an increasing coarse representation of the region. In still further examples, for each subsequent time interval a number of sub-regions for which the region is divided into decreases. In yet further examples, a size of the sub-regions and a length between time intervals may increase at related rates. 
       FIGS. 5-6 and 7A-7B  illustrate example regions and divisions of regions into sub-regions over multiple time increments. 
     At block  806 , the method  800  includes, for at least one (or in other examples, for each) of the one or more time intervals of the period of time, determining vehicles of the plurality of vehicles that can reach a given landmark in a given sub-region by an end of the one or more time intervals. A processor may perform the determination by receiving information about respective starting locations of each of the vehicles, and determine vehicles that can traverse a distance from the respective starting locations to the given landmark by the end of the one or more time intervals. In an example where the vehicles include balloons within a data network operable to provide data communication via optical or radio-frequency (RF) links, information about an estimated wind direction and speed at given altitudes can be determined, and then a distance and direction that the balloons would travel based on the estimated wind direction and speed at the given altitudes can be estimated. A sub-region where the balloons would reside by traveling the estimated distance and direction can then be identified. 
     In one example, making the determination in block  806  may be considered determining a phase fleet plan (e.g., describing a fleet plan for a phase of the time period). To determine a phase fleet plan of a time interval, information of the phase fleet plan of a subsequent time interval may be considered. Thus, a phase fleet plan for a final time interval may be determined first, and fleet plans for previous time intervals can be determined in reverse order. As an example, a first phase fleet plan for the region being divided into a first number of sub-regions can be determined, and then based on the first phase fleet plan, a second phase fleet plan for the region being divided into a second number of sub-regions more than the first number of sub-regions can be determined. In this example, initially a phase fleet plan for a coarse estimation of the region can be determined and used to determine the phase fleet plan for a more detailed estimation of the region. 
     In some examples, for a time interval, estimates of end-of-interval positions for vehicles within sub-regions can be determined, and used as initial interval positions for the vehicles for a subsequent time interval. In other examples, estimates of initial interval positions for the vehicles within sub-regions can be determined and used as estimates as end-of-interval positions for a previous time interval. 
     Further examples for performing functions of block  806  are described below with reference to  FIGS. 9-16 . 
     At block  808 , the method  800  includes, based on the sequence of coverage requirements, generating a fleet plan for the time intervals based on the determined vehicles. A processor may generate the fleet plan by determining which vehicles may be able to satisfy the coverage requirements by traversing sub-regions over requisite time intervals. The fleet plan may include information indicating which vehicles should be directed to or through specified sub-regions, and to do so, the fleet plan may also indicate certain altitudes at which to direct the balloons to enable the balloons to travel a desired direction. 
     The method  800  may further include receiving updates about the vehicles, such as an updated location of one or more vehicles or an update in wind conditions of the region, and responsively regenerating the fleet plan and for the time intervals corresponding to the region being divided into sub-regions having a size less than a threshold amount. In this example, when a change in wind conditions is received, and thus, a possible change in course of travel of balloons, a fleet plan may be regenerated. The fleet plan can be regenerated for all time intervals in the near future, which correspond to the region being divided into sub-regions of smaller sizes in contrast to a coarse estimation. The threshold amount may be any size and varies depending on the coverage requirement from a few square miles to hundreds or thousands of miles. 
     The method  800  may further include providing instructions to the vehicles to operate according to the fleet plan. The method  800  may be performed by a ground-based station and instructions may be provided wirelessly to balloons, for example. In other instances, the method  800  (or portions thereof) can be performed by a processor residing on or coupled to the vehicle, and the processor can provide instructions to the vehicle (or to a control mechanism of the vehicle) to operate according to the fleet plan. 
     In some examples, once a solution or fleet plan has been determined for a region divided in a coarse manner or for a time interval advanced into the future, a degree of long term planning has been resolved. However, a fleet plan resulting from the coarsened solution has less detail than solving for a fleet plan of a more granular or refined division of the region. As an example, referring to  FIG. 5 , by determining a fleet plan at time increment I 2 , balloon locations may be determined to a granularity level of being present within one of sub-regions AA through HH. However, a higher granularity level may be desired. 
     To regain detail, a coarse solution can be refined by independently re-planning the coarse problem at a desired resolution. To do so, in one example, each phase or time interval of the coarse problem can be considered. Conceptually, P may represent a coarse phase, and P 1  through P N  may represent phases from a first time increment through a final time increment that were merged to produce P when the problem was coarsened. All variables associated with P 1  through P N  can be obtained including locations of balloons at time intervals, wind conditions at certain locations, projected trajectories of the balloons, etc. Together, these variables, any constraints applied (e.g., a balloon can only travel a certain distance in a given time interval), and objective terms form a linear programming sub-problem. 
     As one example, sub-problems can be solved in sequence. The sub-problem corresponding to P may be solved first, and end-of-phase balloon positions for a final phase in P&#39;s sub-problem can be used as initial balloon locations for a first phase in the sub-problem corresponding to the coarse phase subsequent to P. 
     As another example, sub-problems may be solved in parallel by adding constraints to force a solution to match with that found in the coarse problem. If P is not the first phase in the coarse planning problem (that is, if P 1  is not the first phase in the original planning problem), then it is not known how many balloons are available at each location at the start of P 1 . Instead, these quantities can be treated as variables. Constraints can be added to ensure that the start-of-phase balloon count variables stay in accord with values in the optimized coarse plan. For each start-of-phase location L in P, let {L 1 } be the set of locations from P 1  that were gathered into L when the problem was coarsened. A constraint can be added to ensure that a sum of the start-of-phase vehicle count variables for the locations in {L 1 } equals the start-of-phase vehicle count in L. Similar constraints can be set up between the end-of-phase locations in P and the end-of-phase locations in P N . In this example, it is possible that the end-of-phase balloon counts for the last phase in one sub-problem might not exactly match the end-of-phase balloon counts for the first phase in the subsequent sub-problem. To resolve this, the sub-problem solutions can be used as an initialization for a linear program solver working on the full original problem. Alternately, the sub-problem solutions can be used as initializations for solving the sub-problems in sequence. 
     In some instances, it may be desirable to re-execute a planning process frequently as new information arrives about exact locations of balloon in the fleet, a health of a balloon, changes in environment to be navigated, etc. In such situations, it may be that only the first one or few coarse phases need to be refined into detailed plans. 
     Phase fleet planning, or generally fleet planning, may be performed using a number of example methods. Within examples, for a balloon trajectory from location A to location C, it is possible that there may be a wind at some altitude that could take a balloon from location A to location B, and wind at another altitude that could take the balloon from location B to location C. Thus, to traverse from location A to location C, an intermediate location B is needed. In other examples, to traverse from location A to location C may require passing from location A to location B 1  to location B 2  to location B 3  and so on until arriving at location C. A number of possibilities grow with a number of “way-points” along a path. To control a fleet of balloons, a simultaneous plan of trajectory for many balloons at many starting locations may be needed, and each balloon may take a different path. In addition, goals other than a location or trajectory may constrain how to control balloons including minimizing time or energy used, or maximizing time present over areas of interest, such as highly populated regions for example. 
     Within examples, a system may be configured to receive (or determine) a starting location of each balloon of a fleet of balloons and receive a sequence of coverage requirements for the region for a planning time period. As an example, the sequence of coverage requirements for the time period may specify a goal number of balloons for each of a plurality of landmarks within the region at various times T during the time period. The system may also be configured to divide the time period into phases based on distinct values of T in the sequence of coverage requirements: a first phase may be defined from a starting time of the time period to an earliest T; a second phase may be defined between the end of the first phase and a second earliest T, etc. Additionally, a set of landmarks may be associated with the start and end of each phase: the start of the first phase may use the initial location of each of the balloons as landmarks; the end of the first phase and the start of the second phase may share the set of landmarks associated with the earliest T, etc. 
     Based on the received information, the system may be further configured to determine trajectories that satisfy the sequence of coverage requirements for the region. To determine the trajectories, the system may initially establish possible routes for balloons within each phase. For each landmark at the start of a phase and for each landmark at the end of the phase, the system may determine how close to the end-of-phase landmark that a balloon starting from the start-of-phase landmark could get by traveling during the phase. 
       FIG. 9  illustrates example trajectories for a balloon that is traveling from a plurality of starting landmarks to a landmark E. Specifically,  FIG. 9  illustrates trajectories from each of landmark A, B, C, D, and F to an ending landmark E. As shown in  FIG. 9 , a region  900  has been divided into a plurality of subregions, and the landmarks A-L have been established at the center of each subregion. Note that in some examples coverage requirements may specify a goal number of balloons to be located at one or more locations within the region  900 , and the goal number of balloons may be added to the nearest landmark to determine the goal number of balloons for each landmark. For example, if an individual coverage requirement indicates that five balloons are desired at location Loc 1  and seven balloons are desired at location Loc 2 , the desired number of balloons for landmark A may be determined to be twelve. 
     Additionally, although the landmarks A-L have been distributed uniformly throughout the region  900 , the example is not meant to be limiting. In other instances, landmarks may be non-uniformly distributed within the region  900 . For example, if a region covers the entire Earth, one or more oceans or countries may not include any landmarks. 
     The example trajectories may be determined based on estimated or predicted wind conditions at one or more altitudes. In the example of  FIG. 9 , the wind conditions are assumed to generally include winds flowing from left-to-right with additional variation in the vertical direction from time T=0 to T=1. Therefore, it is assumed that a balloon could not flow from landmarks G-L to landmark E from time T=0 to T=1, and for convenience, only trajectories from landmarks A, B, C, D, and F are shown in  FIG. 9 . 
     In one example, a backward planner may be used to determine a measure of how close to a target location that a balloon could reach if the balloon was starting at another location within a region. For example, given a target time and target location as well as an initial time, the backward planner may be configured to determine, for a grid of starting locations within a region, a measure of how close to the target location a balloon could get, if the balloon was flying from initial time to target time. 
     The backward planner may consider a problem of minimizing a cost function associated with a balloon path. As an example, the cost function may be a measure of how close a balloon gets to a target location by following a path.  FIG. 10  illustrates an example cost value map where location A is a starting location of a balloon, location B is a desired final location, and example cost values are assigned to cells surrounding location B. For a time period from T=0 to T=N, the time period can be discretized in order to consider a situation of the balloon at times T=0, 1, 2, . . . , N. The region can also be discretized by dividing the region into a finite number of cells. For instance,  FIG. 10  illustrates a region that has been divided into a number of cells (e.g., a 10×7 grid of cells). 
     At time T=N (e.g., a final time of a planning phase), there is no planning to do since the balloon will be at a final location at this time, and a cost value can be assigned to each cell based on a final location of the balloon. For time T=N, cost values may be systematically assigned such that a cell in which the desired location resides has a low cost and cells further away may be assigned a higher cost value (e.g., the farther away from the desired location, the higher the cost value). Thus, as shown, a cell in which the desired location B resides can be assigned a cost value of zero, and cost values for surrounding cells can be assigned higher cost values based on a distance away from the desired location cell. Cost values for all cells can be populated in this manner by increasing the cost value for a given cell based on the distance from the desired location. 
     Following, cost value maps of all cells may be generated for each time step T=0, 1, 2, . . . , N−1. To construct the cost value maps, a map may be generated for time T=N−1, and maps for previous times may be generated working backwards, such as next generating a map for time T=N−2, followed by T=N−3, and so on. 
     After generating the map at time T=N, to generate a map for a next previous time step, possible altitudes and corresponding possible locations that can be reached over the time interval from T=N−1 to T=N for a balloon present in every cell at time T=N−1 can be estimated. For each cell and for one or more altitudes, an estimation of where the balloon will arrive by starting at that cell and flying at that altitude during the time interval can be determined. For example, weather predictions at time T=N−1 may indicate that, for a given cell, different wind conditions exist for different altitudes.  FIG. 11  illustrates an area divided into cells (e.g., a 10 by 7 grid of cells), and for a given cell labeled X, example wind vectors A 1 -A 5  that are determined based on wind conditions at five different altitude levels. 
     In some examples, wind vectors associated with different altitudes may be the same for each cell. In other examples, wind vectors associated with different altitudes may vary based on the location of the cell. For each cell, an estimation of where a balloon would travel to (e.g., destination cell) at the end of the time period if the balloon started at the cell and moved with a given wind vector is determined. To do so, a direction and magnitude (e.g., speed) of the estimated wind vector can be used to estimate a trajectory and distance that the balloon travels, and to determine estimated destination cells (e.g., A 1(EST) -A 5(EST) ).  FIG. 11  illustrates estimated destination cells for a balloon starting at cell X, and moving with one of the five wind vectors A 1 -A 5 . Example simulations can be performed to simulate movement of the balloons due to the estimated wind directions and speed, for example. 
     In some examples, further estimations may be made by a balloon traveling at a first altitude for a portion of a time interval and then changing to a second altitude for a remainder of the time interval. It could be possible that by traveling at any given number of combinations of altitudes, the winds may be able to carry the balloon to a desired location over the time interval. 
     As shown in the example in  FIG. 11 , wind vector A 3  results in the balloon reaching a closest proximity to the desired final location B at time T=N. Cost values for the balloon traveling along the wind vectors A 1 -A 5  may be determined based on the stored cost values at T=N for the destination cells A 1(EST) -A 5(EST) . For example, the wind vector A 3  may be assigned a value of zero, and cost values for a remainder of the wind vectors may be assigned higher cost values for causing the balloon to travel to a location further away from the desired destination (e.g., such as a A 4  having cost of one, A 5  having cost of four, A 2  having cost of two, and A 1  having cost of three). 
     An altitude that results in a smallest cost value can be determined, and the altitude and cost value can be stored for the cell X for time T=N−1. Thus, in the example described, for cell X at time T=N−1, a balloon in cell X that has a desired destination location B at time T=N can be instructed to travel at altitude A 3 . The same determinations may be performed for each cell of the area. 
     The process above may be repeated to determine cost value maps for time interval T=N−2 based on prediction wind conditions for T=N−2 to T=N−1 and the stored cost values for T=N−1. Further, the process may be repeated for time intervals T=N−3, T=N−4, and so on until T=0. In some examples, each cell of a given map has estimates of directions that are based on estimates of directions as determined for a future time interval. This provides a feedback loop to plan at a first time interval for the balloon to take a given path so that the balloon will be at a location needed during the next time interval to take appropriate action, and so on. 
     Referring back to the example illustrated by  FIG. 9 , the backward planner may be able to determine, for each starting landmark A-D and F-L, a minimum distance from landmark E that a balloon could get, if the balloon was flying from T=0 to T=1. For example, a cost value for each of landmarks A-D and F-L may be proportional to a distance such that a given cost value can be converted to a given three-dimensional distance. The backward planner may also be able to determine, for each starting landmark A-D and F-L, which altitude a balloon should fly at T=0 in order to get as close as possible to the landmark E by T=1. 
     The backward planner is provided as one example mechanism for establishing possible routes between landmarks during a phase. However, other mechanisms that can determine, for a particular staring location/time and an ending location/time, how close to the ending location a balloon can get in the time period allotted, based on predicted wind velocities at one or more altitudes may also be utilized. 
     In some examples, filtering heuristics may also be applied to further refine the possible routes between landmarks during a phase.  FIGS. 12A-12B  illustrate examples of determining a set of starting landmarks from which a balloon could reach a landmark E. 
     In one example, thresholding may be used to impose a notion of how close to a target landmark that a balloon needs to be able to get in order to be considered to be capable of reaching the landmark. For example, a set of starting landmarks may be selected based on a comparison between the minimum distances associated with paths from landmarks A-D and F and a threshold distance. Based on the trajectories shown in  FIG. 12A , balloons traveling from landmarks A, B, and F could get within a threshold distance  1202  of the landmark E. Therefore, the landmarks A, B, and F may be retained as a set of starting landmarks from which a balloon could reach the landmark E, while the landmarks D and C may be removed. 
     In another example, a set of starting landmarks may be selected based on a confidence of reaching a target landmark from a starting landmark and/or a cost value associated with flying to the target landmark from a starting landmark. As shown in  FIG. 12B , each path from starting landmarks A-D and F includes an associated probability, as indicated by a percentage on a scale of 1 to 100. The probability may be determined based on the variability of one or more wind conditions (e.g., wind direction and/or speed). In one example, based on the associated probabilities, landmarks A and B and C may be selected as a set of possible starting landmarks because their associated probabilities are greater than a confidence threshold (e.g., 85%). 
     Additionally, as shown in  FIG. 12B , each path from starting landmarks A-D and F includes an associated cost, as indicated by a dollar amount. In one example, the cost may be associated with a cost in power that is necessary to adjust the altitude of a balloon to reach one or more altitudes during the path. Accordingly, a set of starting landmarks may be selected based on a comparison between the cost values associated with each starting landmark and a cost threshold. For instance, starting landmark B may be selected as the only possible starting landmark because the path associated with landmark B has the only associated cost that is below 3$. 
     In still another example, a set of starting landmarks may be determined from among the possible starting landmarks by retaining the top N number of starting landmarks (e.g., sorted to prefer minimum distances from within the target landmark that a balloon could get). It is contemplated that one or any combination of the described filtering heuristics may be used to determine a set of starting landmarks from which a target landmark may be reached. 
       FIG. 13  illustrates an example graph  1300  of possible routes between a plurality of landmarks in accordance with an example embodiment. Continuing with the example from  FIG. 4 , landmarks A-F are provided. Each edge between two landmarks represents a trajectory a balloon could follow to get from a landmark at the beginning of a phase to a landmark at the end of the phase. For instance, a balloon at landmark A at time T=0 may be able to get to landmark D or E by time T=1. Additionally, a balloon at landmark A at time T=1 may be able to get to landmark D or E by time T=2. Note that although the edges for phase 1 are the same as the edges for phase 2 in  FIG. 13 , in other examples, edges may vary among phases. 
     Once the graph is constructed, an optimization method may be used to choose how many balloons flow across each connection in the graph (i.e., from each starting landmark to each ending landmark) during each phase. The optimization method may be configured to obey one or more consistency constraints. For example, a number of vehicles leaving a landmark in phase P may be set to be the same as the number of balloons that arrived at that landmark at the end of phase P−1. As another example, a number of vehicles at any landmark or traveling along any edge cannot be negative. 
     If the sequence of coverage requirements is treated as strict requirements, the problem of determining trajectories for the fleet may be considered a constraint satisfaction problem, where optimization means solving the constraint graph subject to the constraint that each goal is satisfied. If it is allowable to sometimes leave some goals unsatisfied, the problem may be considered an optimization problem: find the configuration that, for example, minimizes the number of unsatisfied goals, or some other measure of the goals&#39; dissatisfaction. 
     Such a problem can be solved using a variety of methods, including as a linear program, as a minimum-cost-flow problem, via gradient descent, via stochastic optimization, or using other existing or future developed methods. 
     For instance, the problem may be solved as a linear program. In one case, the goal for the linear program is to take a time-varying description of how many balloons are desired at each landmark (“the goal distribution”), the initial location of each balloon, and a summary of which goal landmarks are reachable from which other landmarks at a given time, along with a measure of how accurately a balloon can be expected to reach a goal landmark, and determine a fleet-wide policy that maximizes a likelihood of satisfying the goal distribution. Note that the goal distribution is not concerned with determining where any particular balloon goes, but rather that the correct numbers of balloons are in the correct areas at each phase. 
     To solve the problem as a linear program, constants, variables, and constraints may be defined. 
     The constants may include:
         startingAtOrigin[origin]=the initial distribution; the number of balloons at each location at the beginning of a planning period   requiredAtGoal[phase, goal]=the goal distribution; how many balloons desired at each goal at the end of each phase   score[phase, origin, goal]=accuracy estimate for the path from origin to goal during phase.       

     The variables may include:
         A [phase, origin, goal]=how many vehicles move from origin to goal during phase; defined only if it is possible to reach the goal from the origin given the wind at that phase.   atGoal[phase, goal]=the number of balloons at goal at the end of phase   deficit[phase, goal]=the number of balloons by which the goal distribution was under-satisfied at goal at the of phase, that is: max(0, requiredAtGoal−atGoal).       

     The constraints may include:
         Non-negative flow
           ∀origin, ∀goal, ∀phase: 0≦A[phase, origin, goal]   
           Conservation on the origin side
           ∀origin: Σ goal  A[0, origin, goal]=startingAtOrigin[origin]   ∀origin, ∀phase&gt;0: Σ goal  A[phase,origin,goal]=atGoal[phase−1,origin]   
           Conservation on the goal side
           ∀goal, ∀phase: atGoal[phase,goal]=Σ origin A[phase, origin, goal]   
           Non-negative deficits
           ∀goal, ∀phase: 0 deficit[phase, goal]   ∀goal, ∀phase: requiredAtGoal[phase, goal] atGoal[phase, goal]++deficit[phase, goal]   
               

     Based on the constants, variables, and constraints defined above, a linear program solver may be used to minimize the example objective function shown below:
 
α* c   error (Σ phase Σ goal deficit[phase,goal])+(1−α)(Σ phase Σ origin Σ goal score[phase,origin,goal] A [phase,origin,goal])
 
     The objective function penalizes fleet plans that put less than the minimum number of balloons at some landmark; for each balloon fewer than desired, the penalty goes up. The optimal plan, therefore, is the plan that minimizes the objective function. An additional term can also be included that minimizes a measure of how close to a target landmark that each balloon can get. For instance, the additional term seeks to minimize the sum of the score constant for each path traversed. The additional term causes the linear program solver to prefer accurate trajectories and penalize trajectories that do not reliably deliver the balloon directly to a goal landmark. 
     In the example objective function, c error , is a weight parameter and α may be another parameter that is used to adjust the relative importance of minimizing the number of missed goals versus preferring accurate trajectories. For instance, increasing α*c error  may cause the linear program solver to prefer fleet trajectories that satisfy a goal distribution better, at the cost of being willing to use less accurate flight paths. Also, increasing (1−α) may cause the linear program solver to prefer fleet plans that use accurate/reliable flight paths, at the cost of being willing to leave more of a goal distribution unsatisfied. 
     In further examples, the score constant may be a measure of a confidence associated with whether a balloon traveling from origin could reach goal during phase. For instance, the measure of confidence may be a probability that is determined based on the variability of one or more wind conditions (e.g., wind direction and/or speed). In addition, the configuration of the linear program could further be modified such that the number of balloons atGoal is an expected number of balloons that is determined based on the probabilities of each balloon reaching the goal landmark. For example, if a balloon has a 75% chance of reaching a goal landmark, the balloons contribution to atGoal may be 0.75 balloons instead of 1 balloon. 
     In another variant, the score constant may be replaced or augmented with a measure of cost associated with a path from origin to goal during phase. For instance, the cost may be a cost-in-dollars, cost-in-power, or other cost measure. The linear program solver may then be configured to minimize total cost in the determined trajectories. 
     The optimization problem may also be solved as a minimum-cost-flow problem. For instance, the linear program described above may be reduced to a minimum-cost-flow problem. A minimum-cost-flow problem seeks to find the least costly way of sending a certain amount of flow through a flow network. The flow network is often represented as a directed graph with one or more source nodes and one or more sink nodes. Every edge in the network then specifies a maximum capacity of flow for the edge and a positive cost per unit of flow. 
       FIG. 14  illustrates an example directed graph  1400  in accordance with an example embodiment. The directed graph  1400  may be formed in a manner similar to the graph  1300  of  FIG. 13 . For convenience, only two phases, phase P and phase P+1, are shown. During “Phase P flow” and “Phase P+1 flow”, each edge represents a path between two landmarks. These edges may be considered to all have unlimited capacity. The cost for these edges is used to penalize trajectories that do not reliably deliver a balloon directly to a goal landmark. Similar to the linear program described above, each edge has a cost of: (1−α)*score[phase,origin,goal]. 
     Each initial balloon location is defined as a source node. Therefore, the number of balloons entering the directed graph  1400  corresponds to the number of balloons starting at that location. The directed graph  1400  also includes a sink node, where the number of balloons exiting the directed graph  1400  equals the total number of balloons in the fleet. This allows the balloons to end up in any configuration at the end of the final phase, then all get accounted for as exiting exactly once. 
     The minimum-cost-flow problem class can be configured to consider the physical constraints defined in the linear program as givens or initial inputs. In particular, non-negative flow and conservation of balloons may be required features for minimum-cost-flow problems. 
     To include consideration of deficits in the minimum-cost-flow problem, the directed graph  1400  includes a space to apply the deficit costs. Deficits are defined at the nodes in between phase flows by splitting each landmark node  1402  into two nodes. Two edges are then included between the two nodes. Specifically,  FIG. 14  includes a “Phase P deficit checks” column and “Phase P+1 deficit checks” column, where each landmark node  1402  has been split into two nodes that are connected by two edges. 
     The deficits can be defined according to a piecewise linear cost function.  FIG. 15  illustrates an example piecewise linear cost function  1500  in accordance with an example embodiment. The piecewise linear cost function  1500  allows the minimum-cost-flow problem to account for unlimited capacity at the landmark node  1402 . Because the cost of the top edge has a lower cost than the bottom edge, balloons will prefer to go across the top edge, up to the capacity limit of required[phase, goal] (which may be equal to requiredAtGoal as defined above). If more than a required number of balloons travel across the landmark node  1402 , the additional balloons in excess of the desired number of balloons for the landmark node  1402  will incur a cost of α*c error  while traveling across the infinite capacity bottom edge. Therefore, a solution to the minimum-cost-flow problem will seek to avoid having more than the desired number of balloons travel to each landmark node  1402 . 
       FIG. 16  is an example block diagram of a method  1600  to determine trajectories for a fleet of balloons, in accordance with at least some embodiments described herein. At block  1602 , the method  1600  includes receiving an initial location of each balloon of a fleet of balloons. At block  1604 , the method  1600  includes receiving a sequence of coverage requirements for a region and an associated period of time. The region may be divided into a plurality of landmarks and the period of time may be divided into a plurality of phases. An individual coverage requirement may then specify a desired number of balloons for each of the plurality of landmarks at an end of a given phase. 
     At block  1606 , the method  1600  includes, determining, for each phase and each respective landmark, a set of starting landmarks from which a balloon could reach the respective landmark by following one or more estimated winds at one or more altitudes during the phase. For instance, a backward planner may be used to determine, for each landmark, a minimum distance from each other landmark that a balloon could get, if the balloon was following one or more predicted winds during the phase. At block  1608 , the method  1600  includes, based on the initial location of each balloon, the sequence of coverage requirements, and the determined set of starting landmarks for each phase and each respective landmark, determining which respective landmark for each balloon to travel to during each phase. For instance, a linear objective function may be minimized to determine which respective landmark that each balloon should travel to. As another example, the determination regarding which respective landmark that each balloon should travel to during each phase may be made by solving a minimum-cost-flow problem. 
     It should be understood that arrangements described herein are for purposes of example only. As such, those skilled in the art will appreciate that other arrangements and other elements (e.g. machines, interfaces, functions, orders, and groupings of functions, etc.) can be used instead, and some elements may be omitted altogether according to the desired results. Further, many of the elements that are described are functional entities that may be implemented as discrete or distributed components or in conjunction with other components, in any suitable combination and location. 
     While various aspects and embodiments have been disclosed herein, other aspects and embodiments will be apparent to those skilled in the art. The various aspects and embodiments disclosed herein are for purposes of illustration and are not intended to be limiting, with the true scope being indicated by the following claims, along with the full scope of equivalents to which such claims are entitled. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting.