The present disclosure generally relates to autonomous vehicles, and more particularly relates to systems and methods for path planning in an autonomous vehicle.
An autonomous vehicle is a vehicle that is capable of sensing its environment and navigating with little or no user input. It does so by using sensing devices such as radar, lidar, image sensors, and the like. Autonomous vehicles further use information from global positioning systems (GPS) technology, navigation systems, vehicle-to-vehicle communication, vehicle-to-infrastructure technology, and/or drive-by-wire systems to navigate the vehicle.
While recent years have seen significant advancements in autonomous vehicles, such vehicles might still be improved in a number of respects. For example, it is often difficult for an autonomous vehicle to quickly determine a suitable path (along with target accelerations and velocities) to maneuver through a region of interest while avoiding obstacles whose paths might intersect with the region of interest within some predetermined planning horizon. Such scenarios arise, for example, while taking an unprotected left turn, maneuvering around a double-parked car, merging into oncoming traffic, and the like.
Accordingly, it is desirable to provide systems and methods for path planning in autonomous vehicles. Furthermore, other desirable features and characteristics of the present invention will become apparent from the subsequent detailed description and the appended claims, taken in conjunction with the accompanying drawings and the foregoing technical field and background.
Systems and method are provided for controlling a first vehicle. In one embodiment, a method of path planning includes receiving sensor data relating to an environment associated with a vehicle, defining a region of interest and an intended path of the vehicle based on the sensor data, and determining a set of predicted paths of one or more objects likely to intersect the region of interest within a planning horizon. The method further includes defining, within a spatiotemporal path space associated with the region of interest and the planning horizon, a set of obstacle regions corresponding to the set of predicted paths, and defining a plurality of decision points for each of the obstacle regions. The method further includes defining a directed graph based on the plurality of decision points and a cost function applied to a set of path segments interconnecting the decision points, and performing, with a processor, a search of the directed graph to determine a selected path.
In one embodiment, defining the directed graph includes providing a directed edge between a first decision point to a second decision point if: the second decision point is subsequent in time to the first vertex; the second decision point corresponds to a greater distance than the first decision point; the directed edge would not pass through one of the obstacle regions; and the directed edge would not exceed a kinematic constraint associated with the vehicle.
In one embodiment, the cost function is based on at least one of occupant comfort, energy usage, and a distance between the vehicle and the objects.
In one embodiment, each obstacle region of the set of obstacle regions is a polygon and the decision points are located at vertices of the polygon.
In one embodiment, each obstacle region of the set of obstacle regions is a rectangle.
In one embodiment, the decision points associated with each obstacle region are located at opposite corners of the rectangle, and one of the corners corresponds to a point on the obstacle region corresponding to a minimum time along the intended path and a minimum distance along the intended path.
In one embodiment, the region of interest is associated with one of an unprotected left turn by the vehicle, entry of a traffic flow by the vehicle, or maneuvering around a double-parked vehicle by the vehicle.
A system for controlling a vehicle in accordance with one embodiment includes a region of interest determination module, an object path determination module, a path space determination module, and a graph definition and analysis module. The region of interest determination module configured to receive sensor data relating to an environment associated with a vehicle, and define a region of interest and an intended path of the vehicle based on the sensor data. The object path determination module configured to determine a set of predicted paths of one or more objects likely to intersect the region of interest within a planning horizon. The path space definition module configured to define, within a spatiotemporal path space associated with the region of interest and the planning horizon, a set of obstacle regions corresponding to the set of predicted paths, and define a plurality of decision points for each of the obstacle regions. The graph definition and analysis module configured to define a directed graph based on the plurality of decision points and a cost function applied to a set of path segments interconnecting the decision points, and perform, with a processor, a search of the directed graph to determine a selected path.
In one embodiment, the graph definition and analysis module defines the directed graph by providing a directed edge between a first decision point to a second decision point if: the second decision point is subsequent in time to the first vertex; the second decision point corresponds to a greater distance than the first decision point; the directed edge would not pass through one of the obstacle regions; and the directed edge would not exceed a kinematic constraint associated with the vehicle.
In one embodiment, the cost function is based on at least one of occupant comfort, energy usage, and a distance between the vehicle and the objects.
In one embodiment, each obstacle region of the set of obstacle regions is a polygon and the decision points are located at vertices of the polygon.
In one embodiment, each obstacle region of the set of obstacle regions is a rectangle.
In one embodiment, the decision points associated with each obstacle region are located at opposite corners of the rectangle, and one of the corners corresponds to a point on the obstacle region corresponding to a minimum time along the intended path and a minimum distance along the intended path.
In one embodiment, the region of interest is associated with one of an unprotected left turn by the vehicle, entry of a traffic flow by the vehicle, or maneuvering around a double-parked vehicle by the vehicle.
An autonomous vehicle in accordance with one embodiment includes at least one sensor that provides sensor data, and a controller that, by a processor and based on the sensor data defines a region of interest and an intended path of the autonomous vehicle based on the sensor data, and determines a set of predicted paths of one or more objects likely to intersect the region of interest within a planning horizon. The processor further defines, within a spatiotemporal path space associated with the region of interest and the planning horizon, a set of obstacle regions corresponding to the set of predicted paths; defines a plurality of decision points for each of the obstacle regions; defines a directed graph based on the plurality of decision points and a cost function applied to a set of path segments interconnecting the decision points; and performs, with a processor, a search of the directed graph to determine a selected path.
In one embodiment, the controller defines the directed graph by providing a directed edge between a first decision point to a second decision point if: the second decision point is subsequent in time to the first vertex; the second decision point corresponds to a greater distance than the first decision point; the directed edge would not pass through one of the obstacle regions; and the directed edge would not exceed a kinematic constraint associated with the vehicle.
In one embodiment, the cost function is based on at least one of occupant comfort, energy usage, and a distance between the vehicle and the objects.
In one embodiment, each obstacle region of the set of obstacle regions is a rectangle and the decision points are located at vertices of the rectangle.
In one embodiment, the decision points associated with each obstacle region are located at opposite corners of the rectangle, and one of the corners corresponds to a point on the obstacle region corresponding to a minimum time along the intended path and a minimum distance along the intended path.
In one embodiment, the region of interest is associated with one of an unprotected left turn by the vehicle, entry of a traffic flow by the vehicle, or maneuvering around a double-parked vehicle by the vehicle.
The exemplary embodiments will hereinafter be described in conjunction with the following drawing figures, wherein like numerals denote like elements, and wherein:
The following detailed description is merely exemplary in nature and is not intended to limit the application and uses. Furthermore, there is no intention to be bound by any expressed or implied theory presented in the preceding technical field, background, brief summary, or the following detailed description. As used herein, the term “module” refers to any hardware, software, firmware, electronic control component, processing logic, and/or processor device, individually or in any combination, including without limitation: application specific integrated circuit (ASIC), a field-programmable gate-array (FPGA), an electronic circuit, a processor (shared, dedicated, or group) and memory that executes one or more software or firmware programs, a combinational logic circuit, and/or other suitable components that provide the described functionality.
Embodiments of the present disclosure may be described herein in terms of functional and/or logical block components and various processing steps. It should be appreciated that such block components may be realized by any number of hardware, software, and/or firmware components configured to perform the specified functions. For example, an embodiment of the present disclosure may employ various integrated circuit components, e.g., memory elements, digital signal processing elements, logic elements, look-up tables, or the like, which may carry out a variety of functions under the control of one or more microprocessors or other control devices. In addition, those skilled in the art will appreciate that embodiments of the present disclosure may be practiced in conjunction with any number of systems, and that the systems described herein is merely exemplary embodiments of the present disclosure.
For the sake of brevity, conventional techniques related to signal processing, data transmission, signaling, control, machine learning models, radar, lidar, image analysis, and other functional aspects of the systems (and the individual operating components of the systems) may not be described in detail herein. Furthermore, the connecting lines shown in the various figures contained herein are intended to represent example functional relationships and/or physical couplings between the various elements. It should be noted that many alternative or additional functional relationships or physical connections may be present in an embodiment of the present disclosure.
With reference to
As depicted in
In various embodiments, the vehicle 10 is an autonomous vehicle and the path planning system 100 is incorporated into the autonomous vehicle 10 (hereinafter referred to as the autonomous vehicle 10). The autonomous vehicle 10 is, for example, a vehicle that is automatically controlled to carry passengers from one location to another. The vehicle 10 is depicted in the illustrated embodiment as a passenger car, but it should be appreciated that any other vehicle, including motorcycles, trucks, sport utility vehicles (SUVs), recreational vehicles (RVs), marine vessels, aircraft, etc., can also be used.
In an exemplary embodiment, the autonomous vehicle 10 corresponds to a level four or level five automation system under the Society of Automotive Engineers (SAE) “J3016” standard taxonomy of automated driving levels. Using this terminology, a level four system indicates “high automation,” referring to a driving mode in which the automated driving system performs all aspects of the dynamic driving task, even if a human driver does not respond appropriately to a request to intervene. A level five system, on the other hand, indicates “full automation,” referring to a driving mode in which the automated driving system performs all aspects of the dynamic driving task under all roadway and environmental conditions that can be managed by a human driver. It will be appreciated, however, the embodiments in accordance with the present subject matter are not limited to any particular taxonomy or rubric of automation categories. Furthermore, systems in accordance with the present embodiment may be used in conjunction with any vehicle in which the present subject matter may be implemented, regardless of its level of autonomy.
As shown, the autonomous vehicle 10 generally includes a propulsion system 20, a transmission system 22, a steering system 24, a brake system 26, a sensor system 28, an actuator system 30, at least one data storage device 32, at least one controller 34, and a communication system 36. The propulsion system 20 may, in various embodiments, include an internal combustion engine, an electric machine such as a traction motor, and/or a fuel cell propulsion system. The transmission system 22 is configured to transmit power from the propulsion system 20 to the vehicle wheels 16 and 18 according to selectable speed ratios. According to various embodiments, the transmission system 22 may include a step-ratio automatic transmission, a continuously-variable transmission, or other appropriate transmission.
The brake system 26 is configured to provide braking torque to the vehicle wheels 16 and 18. Brake system 26 may, in various embodiments, include friction brakes, brake by wire, a regenerative braking system such as an electric machine, and/or other appropriate braking systems.
The steering system 24 influences a position of the vehicle wheels 16 and/or 18. While depicted as including a steering wheel 25 for illustrative purposes, in some embodiments contemplated within the scope of the present disclosure, the steering system 24 may not include a steering wheel.
The sensor system 28 includes one or more sensing devices 40a-40n that sense observable conditions of the exterior environment and/or the interior environment of the autonomous vehicle 10 (such as the state of one or more occupants). Sensing devices 40a-40n might include, but are not limited to, radars (e.g., long-range, medium-range-short range), lidars, global positioning systems, optical cameras (e.g., forward facing, 360-degree, rear-facing, side-facing, stereo, etc.), thermal (e.g., infrared) cameras, ultrasonic sensors, odometry sensors (e.g., encoders) and/or other sensors that might be utilized in connection with systems and methods in accordance with the present subject matter.
The actuator system 30 includes one or more actuator devices 42a-42n that control one or more vehicle features such as, but not limited to, the propulsion system 20, the transmission system 22, the steering system 24, and the brake system 26. In various embodiments, autonomous vehicle 10 may also include interior and/or exterior vehicle features not illustrated in
The data storage device 32 stores data for use in automatically controlling the autonomous vehicle 10. In various embodiments, the data storage device 32 stores defined maps of the navigable environment. In various embodiments, the defined maps may be predefined by and obtained from a remote system (described in further detail with regard to
The controller 34 includes at least one processor 44 and a computer-readable storage device or media 46. The processor 44 may be any custom-made or commercially available processor, a central processing unit (CPU), a graphics processing unit (GPU), an application specific integrated circuit (ASIC) (e.g., a custom ASIC implementing a neural network), a field programmable gate array (FPGA), an auxiliary processor among several processors associated with the controller 34, a semiconductor-based microprocessor (in the form of a microchip or chip set), any combination thereof, or generally any device for executing instructions. The computer readable storage device or media 46 may include volatile and nonvolatile storage in read-only memory (ROM), random-access memory (RAM), and keep-alive memory (KAM), for example. KAM is a persistent or non-volatile memory that may be used to store various operating variables while the processor 44 is powered down. The computer-readable storage device or media 46 may be implemented using any of a number of known memory devices such as PROMs (programmable read-only memory), EPROMs (electrically PROM), EEPROMs (electrically erasable PROM), flash memory, or any other electric, magnetic, optical, or combination memory devices capable of storing data, some of which represent executable instructions, used by the controller 34 in controlling the autonomous vehicle 10. In various embodiments, controller 34 is configured to implement a path planning system as discussed in detail below.
The instructions may include one or more separate programs, each of which comprises an ordered listing of executable instructions for implementing logical functions. The instructions, when executed by the processor 44, receive and process signals from the sensor system 28, perform logic, calculations, methods and/or algorithms for automatically controlling the components of the autonomous vehicle 10, and generate control signals that are transmitted to the actuator system 30 to automatically control the components of the autonomous vehicle 10 based on the logic, calculations, methods, and/or algorithms. Although only one controller 34 is shown in
The communication system 36 is configured to wirelessly communicate information to and from other entities 48, such as but not limited to, other vehicles (“V2V” communication), infrastructure (“V2I” communication), networks (“V2N” communication), pedestrian (“V2P” communication), remote transportation systems, and/or user devices (described in more detail with regard to
With reference now to
The communication network 56 supports communication as needed between devices, systems, and components supported by the operating environment 50 (e.g., via tangible communication links and/or wireless communication links). For example, the communication network 56 may include a wireless carrier system 60 such as a cellular telephone system that includes a plurality of cell towers (not shown), one or more mobile switching centers (MSCs) (not shown), as well as any other networking components required to connect the wireless carrier system 60 with a land communications system. Each cell tower includes sending and receiving antennas and a base station, with the base stations from different cell towers being connected to the MSC either directly or via intermediary equipment such as a base station controller. The wireless carrier system 60 can implement any suitable communications technology, including for example, digital technologies such as CDMA (e.g., CDMA2000), LTE (e.g., 4G LTE or 5G LTE), GSM/GPRS, or other current or emerging wireless technologies. Other cell tower/base station/MSC arrangements are possible and could be used with the wireless carrier system 60. For example, the base station and cell tower could be co-located at the same site or they could be remotely located from one another, each base station could be responsible for a single cell tower or a single base station could service various cell towers, or various base stations could be coupled to a single MSC, to name but a few of the possible arrangements.
Apart from including the wireless carrier system 60, a second wireless carrier system in the form of a satellite communication system 64 can be included to provide uni-directional or bi-directional communication with the autonomous vehicles 10a-10n. This can be done using one or more communication satellites (not shown) and an uplink transmitting station (not shown). Uni-directional communication can include, for example, satellite radio services, wherein programming content (news, music, etc.) is received by the transmitting station, packaged for upload, and then sent to the satellite, which broadcasts the programming to subscribers. Bi-directional communication can include, for example, satellite telephony services using the satellite to relay telephone communications between the vehicle 10 and the station. The satellite telephony can be utilized either in addition to or in lieu of the wireless carrier system 60.
A land communication system 62 may further be included that is a conventional land-based telecommunications network connected to one or more landline telephones and connects the wireless carrier system 60 to the remote transportation system 52. For example, the land communication system 62 may include a public switched telephone network (PSTN) such as that used to provide hardwired telephony, packet-switched data communications, and the Internet infrastructure. One or more segments of the land communication system 62 can be implemented through the use of a standard wired network, a fiber or other optical network, a cable network, power lines, other wireless networks such as wireless local area networks (WLANs), or networks providing broadband wireless access (BWA), or any combination thereof. Furthermore, the remote transportation system 52 need not be connected via the land communication system 62, but can include wireless telephony equipment so that it can communicate directly with a wireless network, such as the wireless carrier system 60.
Although only one user device 54 is shown in
The remote transportation system 52 includes one or more backend server systems, not shown), which may be cloud-based, network-based, or resident at the particular campus or geographical location serviced by the remote transportation system 52. The remote transportation system 52 can be manned by a live advisor, an automated advisor, an artificial intelligence system, or a combination thereof. The remote transportation system 52 can communicate with the user devices 54 and the autonomous vehicles 10a-10n to schedule rides, dispatch autonomous vehicles 10a-10n, and the like. In various embodiments, the remote transportation system 52 stores store account information such as subscriber authentication information, vehicle identifiers, profile records, biometric data, behavioral patterns, and other pertinent subscriber information.
In accordance with a typical use case workflow, a registered user of the remote transportation system 52 can create a ride request via the user device 54. The ride request will typically indicate the passenger's desired pickup location (or current GPS location), the desired destination location (which may identify a predefined vehicle stop and/or a user-specified passenger destination), and a pickup time. The remote transportation system 52 receives the ride request, processes the request, and dispatches a selected one of the autonomous vehicles 10a-10n (when and if one is available) to pick up the passenger at the designated pickup location and at the appropriate time. The transportation system 52 can also generate and send a suitably configured confirmation message or notification to the user device 54, to let the passenger know that a vehicle is on the way.
As can be appreciated, the subject matter disclosed herein provides certain enhanced features and functionality to what may be considered as a standard or baseline autonomous vehicle 10 and/or an autonomous vehicle based remote transportation system 52. To this end, an autonomous vehicle and autonomous vehicle based remote transportation system can be modified, enhanced, or otherwise supplemented to provide the additional features described in more detail below.
In accordance with various embodiments, controller 34 implements an autonomous driving system (ADS) 70 as shown in
In various embodiments, the instructions of the autonomous driving system 70 may be organized by function or system. For example, as shown in
In various embodiments, the computer vision system 74 synthesizes and processes sensor data and predicts the presence, location, classification, and/or path of objects and features of the environment of the vehicle 10. In various embodiments, the computer vision system 74 can incorporate information from multiple sensors (e.g., sensor system 28), including but not limited to cameras, lidars, radars, and/or any number of other types of sensors.
The positioning system 76 processes sensor data along with other data to determine a position (e.g., a local position relative to a map, an exact position relative to a lane of a road, a vehicle heading, etc.) of the vehicle 10 relative to the environment. As can be appreciated, a variety of techniques may be employed to accomplish this localization, including, for example, simultaneous localization and mapping (SLAM), particle filters, Kalman filters, Bayesian filters, and the like.
The guidance system 78 processes sensor data along with other data to determine a path for the vehicle 10 to follow. The vehicle control system 80 generates control signals for controlling the vehicle 10 according to the determined path.
In various embodiments, the controller 34 implements machine learning techniques to assist the functionality of the controller 34, such as feature detection/classification, obstruction mitigation, route traversal, mapping, sensor integration, ground-truth determination, and the like.
It will be understood that various embodiments of the path planning system 100 according to the present disclosure may include any number of sub-modules embedded within the controller 34 which may be combined and/or further partitioned to similarly implement systems and methods described herein. Furthermore, inputs to the path planning system 100 may be received from the sensor system 28, received from other control modules (not shown) associated with the autonomous vehicle 10, received from the communication system 36, and/or determined/modeled by other sub-modules (not shown) within the controller 34 of
In various embodiments, all or parts of the path planning system 100 may be included within the computer vision system 74, the positioning system 76, the guidance system 78, and/or the vehicle control system 80. As mentioned briefly above, the path planning system 100 of
Referring to
In accordance with various embodiments, solver module 420 itself includes a region of interest determination module 430, an object path determination module 440, a path space definition module 450, and a graph definition and analysis module 460.
Module 430 is generally configured to define or assist in defining a region of interest and an intended path of the vehicle based on the sensor data 401. Module 440 is generally configured to determine a set of predicted paths of one or more objects likely to intersect the region of interest within a planning horizon (e.g., a predetermined length of time). Module 450 is generally configured to define, within a spatiotemporal path space associated with the region of interest and the planning horizon, a set of obstacle regions corresponding to the set of predicted paths and a plurality of decision points for each of the obstacle regions. Module 460 is generally configured to construct a directed graph based on the plurality of decision points and a cost function applied to a set of path segments interconnecting the decision points, and then search the directed graph to determine a selected path 461 that substantially minimizes the cost function.
Output 421 might take a variety of forms, but will generally specify, as a function of time, a path in terms of positions, velocities, and accelerations of the type that might typically be produced by guidance system 78 of
One or more of the modules described above (e.g., modules 420, 430, 440, 450, and 460) may be implemented as one or more machine learning models that undergo supervised, unsupervised, semi-supervised, or reinforcement learning and perform classification (e.g., binary or multiclass classification), regression, clustering, dimensionality reduction, and/or such tasks. Examples of such models include, without limitation, artificial neural networks (ANN) (such as a recurrent neural networks (RNN) and convolutional neural network (CNN)), decision tree models (such as classification and regression trees (CART)), ensemble learning models (such as boosting, bootstrapped aggregation, gradient boosting machines, and random forests), Bayesian network models (e.g., naive Bayes), principal component analysis (PCA), support vector machines (SVM), clustering models (such as K-nearest-neighbor, K-means, expectation maximization, hierarchical clustering, etc.), linear discriminant analysis models. In some embodiments, training of any models incorporated into module 420 may take place within a system remote from vehicle 10 (e.g., system 52 in
Referring now to
In various embodiments, the method begins at 501, in which a “region of interest” and intended path of AV 10 are determined. In general, the phrase “region of interest” refers to any closed spatial region (e.g., roadway, intersection, etc.) through which AV 10 intends to traverse in the near term (e.g., within some predetermined time interval or “planning horizon”). This region may be determined, for example, by guidance system 78 of
Furthermore, it will be appreciated that the present systems and methods are not limited to unprotected left turn scenarios as depicted in
Referring again to
Computer vision system 74 of
Once the region of interest and possible obstacles are determined, a spatiotemporal path space is then defined by module 450 (at 503) based on the planning horizon and the region of interest. In accordance with one embodiment, the spatiotemporal path space is a planar Cartesian space (R2) in which one axis corresponds to the future travel distance (d) along the intended path of AV, and another axis corresponds to time (t).
With continued reference to
Thus, the goal of AV 10 will generally be to reach lane end 710 within the planning horizon (topmost horizontal line in
It will be appreciated that AV 10 may be subject to a set of kinematic constraints, which will generally vary depending upon the nature of AV 10. Such kinematic constraints (which may be embodied as settings configurable by an operator) might include, for example, maximum acceleration, minimum acceleration, maximum speed, minimum speed, and maximum jerk (i.e., rate of change of acceleration).
In this regard, it will be appreciated that the slope of a curve at any point within visualization 801 corresponds to the instantaneous velocity of an object (e.g., AV 10), and the rate of change of slope corresponds to the instantaneous acceleration of that object. Thus,
Referring again to
While regions 910 and 920 are illustrated as rectangles, the range of embodiments is not so limited. The dashed lines within regions 910 and 920 represent the actual paths likely to be taken by vehicles 601 and 602, respectively. Thus, any convenient polygon or curvilinear shape that encompasses these likely paths may be employed. Rectangles, however, are advantageous in that they can easily be modeled and represented, and can be used to generate decision points as described in further detail below.
Once the obstacle regions (e.g., regions 910 and 920) have been defined, system 100 then defines (at 505) decision points (within the spatiotemporal path space) for one or more of the obstacle regions. As used herein, the term “decision point” means a point on the perimeter of (or within some predetermined distance of) an obstacle region as defined previously at 504. In various embodiments—for example, in which the obstacle regions are polygons—the decision points are defined at one or more vertices. In various embodiments, the decision points are defined at (or near) a point on the obstacle region that is a minimum with respect to time (e.g., the leftmost point in a spatiotemporal space as described above), a maximum with respect to time, a minimum with respect to distance (i.e., the topmost point in a spatiotemporal space as described above), and/or a maximum with respect to distance. That is, the left and right boundaries substantially correspond to the end of the points where vehicles 601 and 602 would likely interfere with AV 10.
Referring to
It will be appreciated that the decision points as shown in visualization 803 of
Accordingly, at 506, module 460 defines a graph (e.g., a directed acyclic graph) wherein the vertices of the graph correspond to the decision points (or a subset of the decision points) defined at 505, and the edges of the graph correspond to a particular path segment between the decision points. System 100 further defines a cost value associated with each of the edges, which quantifies the relative desirability of AV following that path segment based on some predetermined cost function.
Referring to
Referring to the graph of
In according to various embodiments, a cost function value (or simply “cost”) is assigned to each of the edges of the graph, and a final path is selected to reduce the sum of these costs. For example, referring to
In some embodiments, the cost function is configured to penalize not making it through the intersection. In other embodiments, the cost function penalizes sitting still in an intersection. In some embodiments, the graph search terminates when it has found any valid path, or when it has found the best path, or after it has exhausted a fixed budget of search time.
In order to more fully describe the manner in which graphs are constructed based on decision points,
In the interest of clarity, the individual path segments have not been separately numbered in
In order to construct graph 1300, an edge is drawn between a first vertex and a second vertex if an only if (a) the second vertex is subsequent in time to the first vertex, (b) the second vertex has a greater distance d than the first vertex, (c) the resulting edge would not pass through an obstacle region, and (d) the resulting edge would not exceed a kinematic constraint (such as maximum speed). Thus, for example, decision point 962 is connected to both decision points 982 and 991, but is not connected to decision point 972 (which would require reaching an unreachable speed) or decision point 1203 (which would require passing through obstacle region 990).
Note that three “endpoints” are illustrated in
Referring again to
For example, referring again to the scenario illustrated in
While at least one exemplary embodiment has been presented in the foregoing detailed description, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or exemplary embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the disclosure in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing the exemplary embodiment or exemplary embodiments. It should be understood that various changes can be made in the function and arrangement of elements without departing from the scope of the disclosure as set forth in the appended claims and the legal equivalents thereof.