Shared control of semi-autonomous vehicles including collision avoidance in multi-agent scenarios

[0001] This application claims the benefit of U.S. Provisional Application No. 61/878,308, filed Sep. 16, 2013, which is incorporated herein by reference in its entirety.

[0002] 1. Field of the Description

[0003] The present description relates, in general, to techniques for remotely controlling movements and operations of vehicles such as ground-based vehicles (e.g., a two-to-four or more wheeled vehicle) and aerial vehicles (e.g., a multi-copter, a drone, or the like), and, more particularly, to methods and systems adapted for providing shared control (i.e., both remote and local operator control) over a semi-autonomous vehicle.

[0004] 2. Relevant Background

[0005] There are many settings where it is desirable to provide shared control over the operations of a robot such as a terrestrial vehicle or an aerial vehicle. For example, an amusement park may provide a ride in which a passenger vehicle is driven along a race track or through an obstacle course, and it may be desirable to allow the passenger to “drive” the vehicle by providing user input indicating a direction they wish to travel and a velocity or speed they want their vehicle to move in the selected direction.

[0006] However, the vehicle control is preferably shared to provide collision avoidance with static obstacles and also with other vehicles (which may be driven by other ride participants). In other cases, a human operator may remotely control a flying drone such as a multi-copter in a particular air space, but the control is shared with a central controller to avoid collisions or achieve other goals.

[0007] Human-robot shared control has been applied in a variety of settings, but there remains a need for improved shared-control methods that enhance human control such as by providing the human operator more local control or operational freedom while still safely avoiding collisions. For example, shared control has been applied in the field of tele-robot operation for tasks including space exploration and surgery. In the case of mobile robots, methods have been developed for human interaction with a forklift, and shared control has been used with a formation of aerial vehicles via a haptic device. Work has also progressed for human-robot shared control with semi-autonomous vehicles such as wheelchairs or similar vehicles and with self-driving cars and other urban transporters.

[0008] The present description provides a shared control method (and relative system for implementing this control method) for a vehicle (e.g., any wheeled or terrestrial vehicle, any aerial vehicle, and the like). A human driver, who may be in the vehicle or remote to the vehicle, commands (i.e., provides user input to provide) a preferred velocity. The method involves transforming this user input into a collision-free local motion that respects actuator constraints (e.g., vehicle kinematics) and allows for smooth and safe control. Collision-free local motions are achieved with an extension of velocity obstacles that takes into account dynamic constraints and also a grid-based map representation of the space through which the vehicle is traveling under share controlled.

[0009] To limit the freedom of control of the driver or human operator, a global guidance trajectory (or “ghost vehicle”) can be used in the shared control method. The global guidance trajectory may specify the areas (or volumes) where the vehicle is allowed to drive or move in each time instance. Further, to improve operations in many environments, schedules are imposed along with or as part of the global guidance trajectory to ensure the vehicles, even under human control, achieve broadly stated goals of reaching particular places (move from a first or start position or location to a second or finish position or location) under predefined time schedules.

[0010] The low computational complexity of the method makes it well suited for multi-agent settings and high update rates, and a centralized function and a distributed function (or centralized and distributed algorithms) are described herein that allow for real-time, concurrent control of many vehicles (e.g., several to 10 to 40 or more vehicles). Extensive experiments have been performed, and the results of these experiments (using a motorized, robotic wheelchair as the vehicle) at relatively high speeds in tight dynamic scenarios are presented in the following description.

[0011] More particularly, a method is described for achieving shared control over movement of a vehicle within a space (2D or 3D space depending on whether the vehicle is a wheeled/tracked vehicle or boat or is an aerial vehicle). The shared control method includes receiving user input related to a velocity (e.g., speed and direction) for the vehicle within the space. The method then includes processing the user input with a local motion planner selectively adjusting the velocity and the direction based on a set of predefined constraints (and the presence of neighboring or other vehicles within the space) to generate a trajectory (which may also be considered a velocity as it defines speed and direction) for the vehicle in the space. The method further includes communicating motion control signals including the trajectory to the vehicle and then operating drive mechanisms in the vehicle based on the motion control signals to move the vehicle from a first position to a second position within the space during an upcoming time period.

[0012] In some cases, the trajectory defines a preferred speed for the vehicle and a direction for movement in the space. The set of predefined constraints may include a grid map (e.g., image) of the space that defines locations of obstacles in the space (physical objects such as walls and also, in some cases, virtual objects that may be projected into the space or be displayed on a monitor/display device in the vehicle). There are at least two ways to treat virtual objects: (1) if the virtual object is fixed, it can be placed in the grid map and treated like a fixed physical object; and (2) if the virtual object is created for a transient period only or if it is moving, it can be treated like another vehicle. The processing of the user input includes searching the grid map (e.g., with the vehicle's current position in the space) to identify the obstacles and to define a path from the first position to the second position in the space that avoids the obstacles.

[0013] In some implementations of the method, the set of predefined constraints includes a guidance trajectory defining a passageway through the space that can be traveled by the vehicle without collision with the obstacles, and the trajectory is defined so as to control movement of the vehicle to retain the vehicle within the passageway. The passageway may include a set of interconnected waypoints in the space and an active area in which the vehicle may move is associated with each of the waypoints. At least some of the active areas differ in size or shape defining differing areas in which the vehicle may move within the space. In some embodiments, the set of predefined constraints includes kinematic data for the vehicle, and the processing based on neighboring vehicles in the space includes processing data for collision avoidance for other vehicles moving in the space, velocity predictions for ones of the other vehicles neighboring the vehicle in the space, and/or predictions of movements of the other vehicles during an upcoming time period.

[0042] In brief, methods and systems are described herein that are designed to provide shared control of semi-autonomous vehicles (terrestrial or ground based vehicles and/or aerial vehicles) based, in many cases, on velocity space optimization. The method for shared control of a semi-autonomous vehicle is performed based on or using: (1) real-time input from a driver or operator of the vehicle; (2) a guidance trajectory that specifies the areas (which may be 3D spaces or volumes for aerial vehicles) where the vehicle is allowed to drive or move in each time instance; (3) avoidance constraints with respect to other agents (which may be other semi-autonomous vehicles) and/or with respect to a grid map defining the location, size, and shape of static obstacles (real or virtual such as in the form of projected images in the vehicle's drive path/space, images displayed in a vehicle's head's up display, and the like); and (4) motion continuity constraints. The local planner may provide local planning of vehicle movements using efficient centralized and/or distributed algorithms based on convex optimization and non-convex search to achieve real-time local planning where an exact grid-based representation of the map can be used in conjunction with a velocity obstacles framework. This may be desirable to allow for real-time control of a larger number of vehicles (e.g., 10 to 40 or more vehicles in a space).

[0043] FIG. 1 illustrates a shared control system 100 for use in controlling movement (velocity, direction, and the like) of semi-autonomous vehicles 110 within a physical drive space 105. For example, the physical drive space 105 may be areas of an amusement park ride where it is acceptable for an operator of one of the vehicles 110 to drive or the space 105 may be a volume of air space through which the vehicles 110 may fly when controlled (e.g., remotely in the case of a drone or multi-copter). The driver/operator (not shown) may be seated in or supported on the vehicle 110 or may be at a remote location (e.g., user input 114 may be a remote device communicating velocity inputs via wireless communications).

[0044] The vehicle 110 may include a processor 112 running a local controller 116 that controls operations of a drive mechanism(s)/actuator 118 to cause the vehicle 110 to move in a particular direction and velocity within the space 105. This control may be based on motion control signals 138 from a central control system 130 and also based on input from user input devices 114. For example, a driver/operator 114 may move a steering wheel and accelerator pad to cause the local controller 116 to operate the drive mechanism/actuator 118 within an active area defined/allowed by the motion control signal 138 from central control system to avoid collision and to follow a schedule as the vehicle 110 moves along a guidance trajectory 144. The user input device 114 may also take the form of a joystick that can be manipulated by the driver/operator of vehicle 110.

[0045] The local controller 116 and/or other components on the vehicle 110 may operate to transmit (wirelessly in most cases) user input data 119 in response to operations of the user input device(s) 114 by the driver/operator, and the user input data 119 typically will include a user-preferred velocity (e.g., direction and speed for the vehicle 110 in drive space 105). The communicated data 119 may also include sensor or tracking system data that is useful for determining the present location of other/neighbor vehicles 110 in the space and, in some cases, their present velocity, both of which can be used by the control system 120 in controlling the vehicle 110 via motion control signals 138 to avoid collisions among the vehicles 110 in the space 105.

[0046] The system 100 further includes a central control system 120 that may take the form of a computer or computer-based system specially configured to control the vehicles 110 via motion control signals 138. Particularly, the control system 120 includes one or more processors 122 managing operations of input/output (I/O) devices such as a monitor, a keyboard, a mouse, a touchpad, a touchscreen, and the like for receiving input from an operator and for displaying data (such as displaying present locations of vehicles 110 in space 105 and the like). The processor 122 also runs or executes code in a computer readable medium to provide a controller module 130, which provides all or a large portion of the shared motion control described herein. The controller module 130 may include a local motion planner 132 to generate the motion control signals 138, and, to this end, the local motion planner 132 may include a trajectory controller 134 along with an optimization engine 136 whose functions and design are described in detail below.

[0047] The processor 122 manages one or data storages devices or memory 140 to store (at least temporarily) data used by the controller module 130 to provide the shared control of the vehicles 110. As shown, for example, the memory 140 may be used to store received user input and sensor data 142 from the vehicle 110. Further, the memory 140 may store a guidance trajectory 144 defined by the controller module 130 for controlling the vehicle 110 including defining active areas 145 within the drive space 105 in which the vehicle 110 may be driven by an operator/driver while following a ride path or other planned trajectory. The memory 140 also stores a number of optimization constraints 150 that are used by the optimization engine 136. The constraints 150 may include a grid map 152 that maps physical obstacles 154 and, in some embodiments, virtual objects 156 (e.g., displayed 2D or 3D images of objects projected into space 105 or displayed to within or nearby the vehicle 110 to appear to be in the space 105) to locations or positions within the drive space 105. The controller module 130 may search the grid map 152 based on a present location of the vehicle 110 to generate a motion control signal 138 that avoids collision with obstacles.

[0048] Additionally, the memory 140 may be used to store vehicle properties and/or kinematics 158 for the vehicle 110 that can be used to determine the motion control signal 138 (e.g., an acceptable local trajectory for a next control cycle or at a present time). Other constraints 150 may include data on other vehicles/agents 160 that may be used to avoid collisions and may be determined from sensor data 119 from the vehicle 110 or from an external tracking system (not shown) operating to determine present locations of each of the vehicles 110 in the drive space 105. The optimization constraints 150 may include velocity predictions 164 for other or neighbor vehicles 110 that may be nearby to the vehicle 110 (“ego vehicle”) in the drive space 105. This information can be used by the optimization engine 136 to control the vehicle 110 via signals 138 so as to avoid collisions with other flying or airborne vehicles 110 in the 3D space 105.

[0049] With the system 100 understood and in mind, it may be useful to provide an overview of the shared control method that may be provided by operation of the system 100 and similar systems. Then, the description can proceed with explanation of a useful vehicle model and control, an exemplary guidance trajectory, a shared control technique, local motion planning, algorithms for implementing the methods taught herein, and experimental results.

[0050] Consider a group of n vehicles of which m are controlled, either in a distributed or centralized way. The position, velocity, and acceleration of a robot, i, are denoted by p_i∈², v_i={dot over (p)}_i, and a_i={umlaut over (p)}_i respectively. A disk, D(p,r), of radius r_i centered at p_i defines its enveloping shape. The enlarged obstacle map is given by a static grid map O_r of the positions p where a disk D(p,r) is in collision with a static obstacle.

[0051] The driver/operator of the vehicle (or robot) commands the vehicle through the vehicle's user input devices by specifying a desired velocity. A guidance trajectory along with or together with active areas (i.e., areas (or spaces) where the vehicle is allowed to drive (or fly)) can be specified to: (1) guide the motion of the vehicle; (2) limit its movements; (3) guarantee that the vehicle has a certain performance even in the case where the driver does not specify a velocity; and (4) control the throughput of vehicles throughout a drive space or ride area. In brief, the desired velocity and the guidance trajectory (two inputs to the controller module) are combined and a local trajectory is computed (which is used to generate motion control signals, e.g., after optimization, for the vehicle's drive mechanism/actuator).

[0052] For each vehicle, the set of local trajectories is given by a controller (of sufficient continuity, C², in this description) towards a straight-line reference trajectory at velocity u∈² (see FIG. 2). This provides a low dimensional parameterization (given by u) of the local motions and allows for an efficient optimization in reference velocity space to achieve collision-free motions that respect the dynamic constraints of the vehicle. In each time step of the local planner, the motion is obtained by computing an optimal reference velocity, u*, such that its associated trajectory is collision free, satisfies the motion and guidance constraints, and minimizes the distance to a preferred velocity, ū. The preferred velocity takes into account the driver input, a guidance trajectory, and the neighboring obstacles.

[0053] Throughout this description, x denotes scalars, x vectors, x=∥x∥ its Euclidean norm, X matrices, X sets, and X⊕y their Minkowski sum. The super index *^k indicates the value at time t^k and subindex *_i indicates agent i and x_ij=x_i−x_j relative vectors. Time relative to the current time step is employed when appropriate and denoted by {tilde over (t)}=t−t^k. In practice, estimated states are used, but, to simplify the notation, no distinction is made herein between estimated and real states.

[0054] Turning now to a vehicle model and control, given a reference trajectory at constant velocity p_ref(t)=p^k+u{tilde over (t)} for {tilde over (t)}∈[0,∞), with p^k being the position of i at the current time-step k and u being the reference velocity, one can consider a trajectory tracking controller ƒ(z^k,u,t)=p(t) that is continuous in the initial state, z^k=[p^k,{dot over (p)}^k,{umlaut over (p)}^k, . . . ] of the agent and that is converging to the reference trajectory. The set of feasible motions is given by:
(z^k)={u∈², such that the trajectory ƒ(z^k,u,t) respects all dynamic restraints}  Eq. (1)
This set can be pre-computed by forward simulation and depends on {dot over (p)}^k and {umlaut over (p)}^k. A trajectory controller appropriate for the vehicle kinematics can then be formulated.

[0055] With regard to such a trajectory controller, and recalling the methods herein described are independent of the chosen controller, one such example is as follows. If the system is modeled as the second order integrator (continuity in acceleration) and the appropriate state feedback control low towards the reference trajectory is formulated, the trajectory ƒ(z^k,u,t) is given (for t<0) by:
p(t)=p^k+u t+(({dot over (p)}^k−u+(w₁−w₀)F){tilde over (t)}−F)e^{−w0 t}+Fe^{−w1 t}  Eq. (2)
with F=({umlaut over (p)}^k+2w₀({dot over (p)}^k−v_∞))/(w₀−w₁)² and w₀ and w₁ being design parameters related to the decay of the initial velocity and acceleration. Saturation limits are added, e.g., maximum linear and angular speed (|v₁|≦v_1,max, |v₂|≦v_2,max) and maximum linear and angular accelerations (|{dot over (v)}₁|≦v_1,max, |{dot over (v)}₂|≦{dot over (v)}_2,max).

[0056] Turning now to kinematics of the vehicle, the inventors employed wheelchairs for the semi-autonomous vehicles that can have shared control. With regard to wheelchair (vehicle, in this example) kinematics, the unicycle model is used, with p the point in-between both tractor wheels, δ the orientation of the vehicle, and v₁, v₂ the linear and angular speed of the platform. The graph 200 of FIG. 2, which shows the wheelchair/vehicle 210 with its two rear driving wheels 212 and to front castor wheels 214 and which shows a reference trajectory given by u and an executed trajectory, and the kinematics can be stated as:
[ {dot over (p)},{dot over (δ)}]′=[cos δ, sin δ,0]′v₁+[0,0,1]′v₂  Eq. (3)
A (fully controllable or holonomic) point p= p+[cos δ, sin δ]^TΔ to the front of the vehicle axles can then be considered, where Δ>0 defines the maneuverability of the vehicle. The following relations hold:
{dot over (p)}=ξ,v₁=ξ·[cos δ, sin δ]^T,v₂=ξ·[−sin δ, cos δ]^T/Δ  Eq. (4)
The local trajectory is applied at this point, and a circle of radius r is considered as the enveloping shape of the vehicle.

[0057] With regard to use of a guidance trajectory, it may be desirable for some applications of shared control to have higher control over the motion of the vehicles. In these cases, a roadmap or set of guidance trajectories can be created. A method 300 of creating a roadmap or set of guidance trajectories is shown in FIG. 3. The method 300 starts at step 305 such as with selecting a drive area or space in which a vehicle is allowed to move under driver control/input such as with regard to direction and speed. The method 300 continues at 310 by providing a static map of the selected drive space that includes locations, shapes, and sizes of obstacles. The roadmap may be computed automatically or be created via waypoints connected by straight lines as shown in FIG. 3. This representation does not require one to account for the dynamics of the vehicle nor to perfectly fit the map since both of these constraints or parameters are considered by the local motion planner.

[0058] At step 320, a set of waypoints is specified, and, for each waypoint, an arrival time, T_l, (or, alternatively, a speed over the segment) and an active area (where the vehicle is allowed to freely move) of a predefined radius, ρ_l, around the waypoint is provided. At 340, the method 300 continues with connecting consecutive waypoints. Then, at 350, the method 300 involves generating or defining the guidance trajectory along with the active area around it (e.g., by interpolation or the like). The method 300 then ends at 390. In practice, when two or more active areas (e.g., from different trajectories) intersect, the vehicle can be assigned to an active area depending on its current position and specified attributes of the guidance trajectories, such as if more than one vehicle can follow that trajectory or if the vehicle must stay in the current one.

[0059] In the method 300, one can select the trajectory, s, that minimizes the weighted distance to the current point on the guidance trajectory, q^s, relative to the current radius of the active area, ρ^s:
arg_s^min(1−σ^s)∥p−q^s∥/ρ^s  Eq. (5)
where σ^s∈(0,1) is an attribute indicating the predilection to stay in one trajectory. This framework provides an intuitive way to guide a team of vehicles throughout the drive space while giving the vehicles and their drivers/operators limited freedom in their movement via the active areas. An active area with a large radius, ρ, implies greater freedom of movement for the vehicle while an active area with a small radius, ρ, imposes less freedom and accurate tracking of the vehicle along and within the guidance trajectory.

[0060] FIG. 4 illustrates a grid map 400 of a drive space 410 that may be generated with the method 300 of FIG. 3. The drive space 410 includes several rooms and corridors with these available spaces/areas for driving shown at 426 (with lighter or no shading) and occupied spaces or physical/virtual objects shown at 420 (with darker shading). In this example, nine waypoints, w_i, are created as shown with exemplary waypoints 430 with arrival times, T_i, and a number of different radii, ρ_i, for their associated active areas 434. The waypoints 430 are connected by straight lines as shown at 438, and, in between waypoints, the ideal trajectory and active area is given by interpolation. In the grid map 400 with the guidance trajectories and active areas, waypoints 3a and 3b represent a bifurcation and waypoint 7 has a large active area where the vehicles are allowed to freely drive.

[0061] Now, with regard to shared control, in each time step, for each vehicle i, a preferred velocity ū_i is computed, based on the input from the driver, the guidance trajectory, and the neighboring obstacles as follows:
ū_i=μ₁(μ₂u_i^joy+u_i^traj)+(1−μ₁)u_i^A*+u_i^rep  Eq. (6)
where μ₂=1 if the vehicle is inside the active area and μ₂=0 otherwise, and μ₁=1 if q is in line of sight of the vehicle and μ₁=0 otherwise (see FIG. 5). In practice, a small hysteresis is introduced to μ₁ and μ₂ to avoid oscillations. In normal operations, μ₁=μ₂=1. Each velocity term is computed as follows. The input velocity u_i^joy is proportional to the joystick position (in relative frame of the vehicle). The guidance velocity u_i^traj (optional term) is given by a proportional controller to the guidance trajectory as:
u_i^traj=K_p(q_i−p_i)+{dot over (q)}_i  Eq. (7)
where q_i is the ideal position on the guidance trajectory at the current time. The repulsive velocity u_i^rep is computed with a potential field approach that pushes the vehicle away from both static and dynamic obstacles. A linear function weighting the distance to other vehicles is used as follows:

[0062] $\begin{array}{cc}{u}_i^{\mathrm{rep}}=\mathrm{Vr}\frac{\stackrel{\_}{D}-d\left(p_i𝒪\right)}{\stackrel{\_}{D}}{n}_{i,𝒪}+\sum _{j❘p_{\mathrm{ij}}<D}\mathrm{Vr}\frac{D-p_{\mathrm{ij}}}{D-r_i-r_j}\frac{p_{\mathrm{ij}}}{p_{\mathrm{ij}}}& \mathrm{Eq}.\left(8\right)\end{array}$
where V_r is the maximum repulsive velocity, D and D are the maximum distance from which a repulsive force is added, d(p_i,O) is the distance to the closest static obstacle within the O_rj map, and n_i,O, is the normal vector from it. In order to avoid corner issues, not only the closest point is considered but also an average over the closest points. The norm of u_i^rep is then limited to ∥u_i^rep∥≦V_r.

[0063] In cases where the guidance point, q_i, is not in direct line of view from the current position, p_i, a preferred velocity solely based on the previous terms could be prone to deadlocks. To avoid this, a term, u_i^A*, computed from a global path from p_i to q_i taking into account the map O_r but ignoring the kinematic constraints and other vehicles may be added. This term is given by the first control input of a standard A* search in the grid map.

[0064] FIG. 5 illustrates a graph 500 representing implementation of such shared control. Components of the preferred reference velocity are shown for a vehicle 510 within the active area 520 of the current point, q, on the ideal trajectory within the drive space. An obstacle 530 is shown in the graph with the shared control functioning to avoid collision with the obstacle.

[0065] With regard to local motion planning, in each control loop, given the preferred velocity, ū_i, of the ego vehicle and the current position and velocity of neighboring vehicles, a local trajectory is given by an optimal reference velocity, u*_i, is computed by solving an optimization problem in reference velocity space. The optimization problem may be formulated as a combination of a convex optimization with quadratic cost and linear and quadratic constraints and a non-convex optimization.

[0066] Two alternative formulations are presented: (1) a centralized optimization where the optimal reference velocities of all vehicles, u*_1:m=[u*₁, . . . , u*_m], are jointly computed and (2) a distributed optimization where each vehicle, i, independently solves an optimization where its optimal reference velocity, u*_i, is computed. The position p_j and velocity v_j of neighboring agents is known. The optimization cost is given in two parts. The first one is a regularizing term penalizing changes in velocity, and the second one minimizing the deviation to the preferred reference velocity. For the centralized cast the cost may be provided as:
C(u_1:m):=K_o∥u_1:m−v_1:m∥²+∥u_1:m−ū_1:m∥²  Eq. (9)
where K_o is a design constant. For the distributed case, the cost may be provided as:
C(u_i):=K_o∥u_i−v_i∥²+∥u_i−ū_i∥²  Eq. (10)
The kinematic constraints of the vehicle are included where the vehicle's radius is enlarged by a value ∈>0 and the local trajectories are limited to those with a tracking error below ∈ with respect to the reference trajectory (parameterized by u).

[0067] With regard to a first constraint (“Constraint 1”) concerning avoidance of other agents, for every pair of neighboring agents i≦m and j≦n, the constraint is given by the reference velocities such that ∥p_i−p_j+(u_i−u_j)t∥≧r_i+j=r_i+∈_i+r_j+∈_j, for all t∈[0,τ]. This constraint can be rewritten as u_i−u_j ∉ VO_ij^τ=∪_t=0^τ((D(p_j,r_j+∈_j) ⊕ D(pi,ri+∈i/t) representing a truncated cone. For example, FIG. 6 illustrates with graph 600 the implementation of the constraint of avoidance of other vehicles or agents. This constraint is shown as area 630 for avoidance between two vehicles 610 and 620, and vehicle avoidance constraint is approximated by three linear constraints as shown in the graph 600. This constraint can then be computed if the distance between the two agents is below a threshold (p_ij<K_d). The non-convex constraint ²\VO_ij^τ is approximated by three linear constraints of the form n_ij^l·u_ij−b_ij^l as:

[0068] $\begin{array}{cc}\left[\begin{array}{c}\cos \left({\gamma }^{+}\right)\\ \sin \left({\gamma }^{+}\right)\end{array}\right]u_{\mathrm{ij}}\le 0,-\frac{p_{\mathrm{ij}}}{\mathrm{pij}}·u_{\mathrm{ij}}\le \frac{p_{\mathrm{ij}}-r_{i+j}}{\tau },\left[\begin{array}{c}\cos \left({\gamma }^{-}\right)\\ \sin \left({\gamma }^{-}\right)\end{array}\right]u_{\mathrm{ij}}\le 0& \mathrm{Eq}.\left(11\right)\end{array}$
where γ⁺=α+β, γ⁻=α−β, α=a tan 2(−p_ij) and β=a cos (r_i+j/p_ij). The first and last constraints represent avoidance to the right and to the left, and the middle one a head-on maneuver (collision-free up to t=τ). The linearization of ²\VO_ij^τ is obtained by selecting the linear constraint with maximum constraint satisfaction for the current relative velocity (min_l(n_ij^lv_ij−b_ij^l)). Other sensible choices include linearization with respect to the preferred velocity ū_ij or fixed avoidance side (right/left). For j≦m, this linear constraint is added to the centralized optimization. For j>m (dynamic obstacles) or when distributed, it can be rewritten as follows.

[0069] In the distributed case, all agents are considered as independent decision-making agents solving their independent optimizations. To globally maintain the constraint satisfaction and avoid collisions, an assumption on agent j's reference velocity should be determined. Variable sharing of avoidance effort might be considered with Δv_i=λΔv_ij and the assumption that Δv_j=−(1−λ)Δv_ij. For collaborative agents that equally share the avoidance effort, λ=0.5. If it is considered that agent j ignores agent i and continues with its current velocity, then λ=1 and full avoidance is performed by agent i. For λ∈[0,1], the constraint is given by:

[0070] $\begin{array}{cc}{n}_{\mathrm{ij}}^l·u_{\mathrm{ij}}=\frac{n_{\mathrm{ij}}^l}{\lambda }·\left(u_i-\left(1-\lambda \right)v_i-\lambda v_i\right)\le b_{\mathrm{ij}}^l& \mathrm{Eq}.\left(12\right)\end{array}$

[0071] With regard to a second constraint (“Constraint 2”) concerning active areas, this constraint guarantees that the agent locally remains within the active area. For agent i≦m, the constraint is given by ∥p_i−q_i+(u_i+{dot over (q)}_i)τ∥²≦p_i² with q_i being the point on the guidance trajectory for the current time, {dot over (q)}_i being speed, and ρ_i the radius of its associated active area (by interpolation between the previous and next waypoint). With the constant velocity assumption for the guidance trajectory, if the vehicle is initially within the active area, this constraint guarantees that the vehicle remains inside for time t≦τ. If the vehicle is initially outside, the constraint guarantees that the vehicle enters within time τ. The constraint can be written as:

[0072] $\begin{array}{cc}{u}_i^T{u}_i+2{\left(\frac{p_i-q_i}{\tau }-{\stackrel{.}{q}}_i\right)}^T{u}_i+{\left(\frac{p_i-q_i}{\tau }-{\stackrel{.}{q}}_i\right)}^2\le \frac{{\rho }_i^2}{{\tau }^2}& \mathrm{Eq}.\left(13\right)\end{array}$
This is a quadratic constant (circle in reference velocity space) that is ignored if no active areas are given.

[0073] With regard to a third constraint (“Constraint 3”) concerning avoidance of static obstacles, for agent i≦m, the constraint is given by the reference velocities such that the new positions are not in collision with the enlarged obstacle map, p_i+u_it ∉ O_ri+∈i, for all t∈[0,τ]. This constraint is kept in its grid-based form (obstacles in FIG. 3) to allow the vehicle full navigation capability including being able to move close to the static obstacles and to go inside and/or through small openings. Given a known map, O₀, the enlarged map, O_ri+∈i, can be precomputed for several values of ∈_i.

[0074] With regard to a fourth constraint (“Constraint 4”) concerning dynamic restrictions, each agent should remain within ∈_i of its reference trajectory. For a maximum tracking error, ∈_i, and a current state, z_i=[p_i,{dot over (p)}_i,{umlaut over (p)}_i], the set of reference velocities, u_i, that can be achieved with position error below ∈_i is given by R_i:=R(z_i,∈_i) such that:
R_i:={u_i∈(z_i)|∥(p_i+tu_i)−ƒ(z_i,u_i,t)∥≦∈_i,∀t>0}  Eq. (14)

[0075] If the trajectory controller defined in part by Eq. (2) is used, the set of reachable reference velocities, R_i, depends on {dot over (p)}_i, {umlaut over (p)}_i, and ∈_i. A mapping, γ, from initial state, {dot over (p)}, {umlaut over (p)}, and the reference velocity, u, to a maximum tracking error is precomputed and stored in a look up table as:

[0076] $\begin{array}{cc}\gamma \left(u,\stackrel{.}{p},\ddot{p}\right)=\begin{array}{c}\max \\ t>0\end{array}\left(p+\mathrm{tu}\right)-f\left(z,u,t\right)& \mathrm{Eq}.\left(15\right)\end{array}$

[0077] An example of this constraint is shown in FIG. 7. FIG. 7 illustrates with graph or plot 700 the implementation of the dynamic restrictions constrain with a trajectory controller. As shown, a vehicle may have a current velocity as shown at 710, and the areas 720 and 730 represent reachable reference velocities, R_i, for the representative current velocity 710. The plot/graph 700 is in reference velocity space aligned with the vehicle axis (e.g., tangential/normal), with forward velocities being displayed. A bounding box is given by four linear constraints of the form H_l={u_i|A_lu_i≦b_l} is computed such that R(z_i,∈_i) ⊂ ∪_i=1⁴H_l. These linear constraints are included in both the centralized and decentralized optimizations to reduce the search space.

[0078] With regards to a useful optimization algorithm and guarantees, the optimization includes two types of constraints: (1) convex (linear and quadratic) and non-convex (grid-based). To efficiently find a solution, the optimization is divided into two parts. First, a convex subproblem is solved resulting in u_i^c, and, second, a search within the grid-based constraints is performed that is restricted to the convex area defined by the linear and quadratic constraints. The set of convex constraints (Constraints 1 and 2 and bounding box of Constraint 4) is denoted by C. The set of non-convex constraints (Constraints 3 and 4 for agent I, with respect to a grid of the same resolution) is denoted by {tilde over (C)}_i. For both the centralized (N=m) and the distributed (N=1, without loss of generality) optimizations, the algorithm may be stated as:

[0079]

Input distributed: z1, ū1 and r1 + ε1; pj, {dot over (p)}j and rj ∀j neighbor of 1. Consider
εj = ε1.
Input centralized: zi, ūi and ri + εi∀i ≦ m; pj, {dot over (p)}j and rj ∀j > m neighbor of
i ≦ m.
Result: u1:N* = [u1*, ..., uN*]
Compute constraints (See Eq. (5))
u1:Nc ← solution 2N-dimensional convex optimization with quadratic cost
(Eqs. (9)-(10)) and convex constraints C.
//Wave expansion from u1:Nc within convex area C.
Initialize shorted list L (increasing cost) with u1:Nc
while L = Ø do
u1:N ← first point in L; L := L\u1:N; feasible := ‘true’;
for i = 1 to N do
if feasibleDynamics(ui)=’false’ or feasibleMap(ui)=’false’
then
L ← expandNeighbors(L, u1:N, i, 1);
feasible := ‘false’; break;
end if
end for
if feasible = ‘true’ then return u1:N* = u1:N end if
end while; return 0.

[0080] Function feasibleDynamics(u_i) checks in a precomputed grid if the tracking error is below ∈_i given the initial state of the vehicle:
if u_i∈R_i (See Eq. 14) then return ‘true’; else ‘false’  Eq. (21)

[0081] Function feasibleMap(ui) checks if ui leads to a trajectory in collision with static obstacles given by the grid map O. This is efficiently checked in the precomputed dilated map, O_ri+∈i, (see Constraint 3):
if segment(p_i,p_i+u_iτ)∩O_ri+∈i=Ø then return ‘true’  Eq. (22)

[0082] The function expandNeighbors adds the neighboring grid points if they are within the convex region defined by the convex constraints in C, and they were not previously explored. This algorithm is recursive for N>1, although only one level of recursion is used to reduce computational time at the expense of optimality:

[0083]

Function L ← expandNeighbors(L, u1:Nin, k, rec);
for each 8-connected grid neighbor uk of ukin do
Consider u1:N = [u1in, ..., uk−1in, uk, uk+1in, ..., uNin].
if {u1:N not previously add to L} and
{uk not checked as unfeasible w.r.t. constr. {tilde over (C)}i} then
if u1:N satisfies convex constraints C
add u1:N to L
else
if {rec=1} and {∃ constr. type 1 unfeasible} then
Select constr. which maximizes n · u1:N − b > 0.
j := “index of second agent in the selected constraint”
if j ≦ N then L ← expandNeighbors(L, u1:N, j, 0)
end if; end if; end if; end for

[0084] If the optimization is unfeasible, constraints 2 are removed and the optimization is recomputed. If still unfeasible, agents i≦N decelerate on their last feasible local trajectory following the time remap:
γ(t)=−t_ƒ²/(2τ)+(1+t_ƒ/τ)t−t²/(2τ)  Eq. (16)
where t_ƒ is the time where the last feasible local motion was obtained. This reparameterization of the trajectories guarantees that the vehicles stop within the time horizon of the local planner, unless the optimization becomes feasible in a later time step.

[0085] With regard to computational complexity, with a limit in the number of linear constraints for collision avoidance, the complexity of the algorithm can be kept linear with the number of agents, although the centralized is of higher cost. If distributed, for each agent, the optimization is made up of two variables, one quadratic constraint, four linear constraints (bounding box of Constraint 5), a maximum of m linear constraints of Type 1 (in practice limited to a constant K_c) and a wave expansion within the 2D grid. If centralized, the optimization is made up of 2n variables, n quadratic constraints, less than

[0086] $4n+\frac{n\left(n-1\right)}{2}+n\left(m-n\right)$
linear constraints (limited to 4n+nk_c) and a joint wave expansion in n 2D grid maps.

[0087] With regard to safety guarantees, if feasible, the local trajectories are collision free up to time τ, with the assumption that other vehicles follow the same algorithm or maintain constant velocity. If unfeasible, no collision-free solution exists that respects all the constraints. If the time horizon is longer than the required time to stop, safety is preserved if all vehicles drive their last feasible trajectory with a time reparameterization to reach stop before a collision arises:
{dot over (γ)}(t_ƒ+τ)=(1+t_ƒ/τ)−(t_ƒ+τ)/τ=0  Eq. (17)

[0088] Note, though, that the optimization can be unfeasible due to several causes. First, there may not be enough time to find the solution within the allocated time. Second, there may be differences between the model vehicle and the real world vehicle. Third, there may be noise in localization and estimation of the vehicle's state. Fourth, optimization may be unfeasible due to the limited local planning horizon together with over simplification of motion capabilities by reducing them to the set of local motions. Fifth, if distributed, unfeasibility may be caused given the use of pair-wise partitions of velocity space with either the assumption of equal effort in the avoidance or constant speed. Hence, not all world constraints are taken into account for the neighboring agents/vehicles. As a result, a vehicle may have conflicting partitions with respect to different neighbors, static obstacles, and/or kinematics that render optimization unfeasible for this vehicle.

[0089] With regards to deadlock-free guarantees, for a single agent/vehicle, deadlock-free navigation is achieved in part due to the u_A* term in the optimization that drives the vehicle toward its goal position or guidance trajectory. In the multi-agent (or vehicle) scenario, deadlocks are still possible although in degenerated situations. Giving priority to those agents/vehicles further in their guidance trajectory can help resolve this situation, though. The input from the driver of the vehicle can also act as a deadlock-breaking input.

[0090] Turning to experimental results, the algorithms have been extensively tested in simulation with larger groups of robots/agents. Due to space limitations in the test setup, the analysis relied upon real wheelchairs for the vehicles with a driver (see FIG. 2). Each vehicle had an enclosing radius of 1 m, limits v_1,max=3 m/s, v_2,max=2 rad/s, {dot over (v)}_1,max=2 m/s², {dot over (v)}_2,max=20 rad/s², {dot over (v)}_1,min=−1.1 m/s², and a maximum sideways acceleration of 1.5 m/s². Vehicles were tracked by an overhead tracking system and controlled from a central computer at 30 Hz. Velocity commands from the driver were transmitted to the central computer at the same rate. Computations were performed using an i5 3 GHZ PC.

[0091] As part of the experiment with shared control, semi-autonomous driving was tested without a guidance trajectory and where each vehicle was driven by a human that inputted a desired velocity, u^joy. Over 50 hours of experiments showed that the algorithms described above are safe and collisions with walls and other obstacles as well as with other vehicles were avoided. FIGS. 8A-8F illustrate graphs 810, 820, 830, 840, 850, and 860 that show, respectively, examples of robot (or vehicle) and obstacle (or wall) collision avoidance, robot-to-robot (or vehicle-to-vehicle collision avoidance, distance velocity, relative velocity (negative when towards each other and with time-steps where the optimization was unfeasible are marked in gray in the background and where a zero distance indicates that two objects touch), and vehicle position (with the input velocity from the driver being displayed with dashed arrows and the safe vehicle command with solid arrows).

[0092] FIGS. 8A-8F show two representative examples of the distributed collision avoidance, where the driver-commanded velocity, u^joy, is shown with arrows in the graphs 850 and 860. In graph 850, safe and smooth velocity between 1 and 2.5 m/s was achieved in very close proximity to a wall when the driver was commanding the vehicle towards the wall. The vehicle slightly slows down as it gets closer to the wall (obstacle). In the graph 860, a frontal collision with another vehicle is avoided where a relative velocity of about 5 m/s in very close proximity (e.g., below 6 m) with a neighboring vehicle was handled. In this example, the optimization became unfeasible during 0.15 seconds, which was mostly due to lack of reactiveness in the low level controller and unmodeled dynamics that slowed the vehicle. This renders the algorithm feasible in subsequent time steps.

[0093] FIGS. 9A-9D show four histograms 910, 920, 930, and 940 over the shared control experiments Linear and angular velocities are shown with histograms 910 and 920 while histograms 930 and 940 show input velocity from the driver relative to the vehicle. From these histograms, it can be seen that even if the velocity input by the driver seems to follow a bang-bang control of zero (e.g., maximum velocity as seen in histogram 930 of FIG. 9C), the algorithm adapts the velocity of the vehicle in the range of −1 to 3 m/s to remain safe (see, for example, FIG. 9A and histogram 910). The driver stops at some times as can be seen from the experiment results. The angular velocity is smooth and within the limits (see, for example, FIG. 9B and histogram 920), while the orientation of the desired velocity (see FIG. 9B in vehicle reference frame) is mostly centered in the forward direction and presents clear peaks at 0, +90, and 180 degrees.

[0094] FIGS. 10A and 10B shows histograms 1010 and 1020 (distributed and centralized, respectively) in log scale of the computational time of the collision avoidance algorithm. If no solution is found in 35 ms in this experiment, the algorithm was configured to return unfeasible for that time step. FIGS. 10A and 10B show, in log scale to emphasize the infrequent worst cases, statistics of the computational time for the collision avoidance optimization. For the distributed algorithm as shown in histogram 1010 (per vehicle), a solution is usually found in less than about 1.5 ms. Higher times, e.g., below 6 ms in all experiments, depend on the non-convex search within the convex region. Similar computational times per vehicle were observed for experiments with larger numbers of vehicles in simulation. For the centralized algorithm as shown in histogram 1020 (two vehicles), higher computational times were observed for the worst case, and the inventors believe this may have been due to inefficient handling of high-dimensional sorted lists and other implementation inefficiencies.

[0095] With regard to constrained driving experiments using a guidance trajectory along with a grid map of obstacles, FIGS. 11A-11C show graphs 1110, 1120, and 1130 providing vehicle distance and linear velocities over time (in FIGS. 11A and 11B, respectively) and vehicle position (in FIG. 11C) with obstacles shown at 1132, areas covered by vehicle at 1134, and lines 1136 providing a time indicator. FIGS. 12A and 12B show histograms 1210 and 1220 (with and without a driver, respectively) of distance from vehicle to guidance trajectory relative to active area radius. FIGS. 12A and 12B show statistics over several experiments of vehicle distance to the guidance trajectory relative to the active area radius. For a vehicle without a driver (as seen in histogram 1210), the 10 percent offset is due to the proportional tracking controller employed. For a vehicle with a driver (as seen in histogram 1220), it can be observed that the driver is able to freely move within the active area, but the optimization algorithm successfully constrains the motion of the vehicle so as to avoid collisions with obstacles or other vehicles.

[0096] As will be appreciated based on the above description, a method is taught for shared control of a vehicle. In this shared control method, the driver commands the vehicle by specifying a preferred velocity. The specified velocity is then transformed into a local motion that respects the actuator limits of the vehicle and is collision free with respect to other moving vehicles and static obstacles (physical or virtual) given by (or defined by) a grid map of the drive space. This allows for smooth and safe control of the vehicle. In order to limit the freedom of the driver, a global guidance trajectory can be included in some embodiments of the shared control method, and the global guidance trajectory specifies the areas (or spaces) where the vehicle is allowed to drive in each time instance. Good performance has been observed in extensive experimental tests at speeds of up to 3 m/s and in close proximity to other vehicles and obstacles (e.g., walls). Further, the low computational time of the optimization algorithm allows for real-time control of numerous vehicles in the drive space.

[0097] The above description teaches the inventors' shared control for vehicles that are wheeled or driving on the ground in a 2D drive space. The inventors further explored application and/or extension of their shared control method (and associated system) to collision avoidance for aerial vehicles especially in multi-agent (or multi-aerial vehicle) scenarios. The following paragraphs describe an investigation of local motion planning and collision avoidance for a set of decision-making agents (vehicles) navigating or flying in 3D space. The shared control method is applicable to agents that are heterogeneous or homogeneous in size, dynamics, and aggressiveness. The method may use and expand upon the concept of velocity obstacles (VO), which characterizes the set of trajectories that lead to a collision between interacting agents.

[0098] Motion continuity constraints are satisfied by using a trajectory tracking controller and by constraining the set of available local trajectories in an optimization. Collision-free motion is obtained by selecting a feasible trajectory from the VO's complement, where reciprocity can also be encoded. The following three algorithms for local motion planning are presented herein: (1) a centralized convex optimization in which a joint quadratic cost function is minimized subject to linear and quadratic constraints; (2) a distributed convex optimization derived from the centralized convex optimization; and (3) a centralized non-convex optimization with binary variables in which the global optimum can be found but at a higher computational cost. The following description also presents results from experiments with up to four vehicles/agents in the form of physical quadrotors (or multi-copters) flying in close proximity and with two quadrotors avoiding a human in the 3D space (or flying space or air space).

[0099] The last decade has seen significant interest in unmanned aerial vehicles (UAVs) including multi-rotor helicopters (or multi-copters) due to their mechanical simplicity and agility. Successful navigational control builds on the interconnected competences of localization, mapping, and motion planning and/or control. The real-time computation of global collision-free trajectories in a multi-agent setting had proven challenging. One approach used was to hierarchically decompose the problem into a global planner, which may omit motion constraints, and a local reactive component that locally modifies the trajectory to incorporate all relevant constraints. The local reactive component is one topic of the following paragraphs, and this description teaches a real-time reactive local motion planner that takes into account motion constraints, static obstacles, and other flying vehicles or agents. Further, the important case where other agents/vehicles are themselves decision-making vehicles is addressed with the shared control method (and system).

[0100] Among the inventors' contributions provided in their shared control method is a derivation and evaluation of three methods for local planning in 3D environments or air spaces (e.g., based on recent extensions of the VO concept). A first method is a distributed convex optimization in which a quadratic cost function is minimized subject to linear and quadratic constraints. Only knowledge of relative position, velocity, and size of neighboring agents/vehicles is utilized. A second method for local planning uses a centralized convex optimization in which a joint quadratic cost function that is minimized subject to quadratic and linear constraints. This variant scales well with the number of agents and presents real-time capability but may provide, in some cases, a sub-optimal but still useful solution. The third method uses a centralized non-convex optimization in which a joint quadratic cost function is minimized subject to linear, quadratic, and non-convex constraints. This optimization may be formulated as a mixed integer quadratic program, and this variant explores the entire solution space but can, in some situations, scale poorly with the number of agents/vehicles. Each of the optimization methods can: (1) characterize agents as decision making agents that collaborate to achieve collision avoidance; (2) incorporate motion continuity constraints; and (3) handle heterogeneous groups of agents. The following description also provides a complexity analysis and a discussion of the advantages and disadvantages of each optimization method.

[0101] The control problem can be thought of as a consideration of a group of n agents freely moving in 3D space. For each agent, i, p_i∈³ denotes its position, v_i={dot over (p)}_i its velocity, and a_i={umlaut over (p)}_i its acceleration. The goal is to compute for each such agent, at high frequency (e.g., 10 Hz or the like), a local motion that respects the kinematics and dynamic constraints of the ego agent or aerial vehicle. The local motion should also be collision free with respect to neighboring agents or aerial vehicles in the 3D space for a short time horizon (e.g., for 2 to 10 or more seconds), which may be thought of as providing local motion that provides collision avoidance.

[0102] Two scenarios are discussed herein: (1) a centralized one where a central unit computes local motions for all agents simultaneously and (2) a distributed one where each agent or vehicle computes independently its local motion. For the second case, no communication between the agents is required, but each agent is able to maintain an estimate of the position and velocity of its neighbors. For the distributed case, neighboring agents are labeled as dynamic obstacles or decision-making agents, which employ an algorithm that is identical to that in the ego agent or vehicle. The local motion planning problem is first formulated as an efficient convex optimization (either centralized or distributed), followed by a non-convex optimization (centralized) that trades off efficiency for richness of the solution space.

[0103] The shape of each agent or vehicle is modeled by an enveloping cylinder of radius, r_i, and height, 2h_i, centered at the agent or vehicle. The local motion planning framework is applied to quadrotor helicopters, which are useful in many applications due to their agility and mechanical simplicity. Implementation details, including an appropriate control framework, are described in the coming paragraphs followed by discussion of extensive experimental results using up to four quadrotors along with a human moving through the 3D space.

[0104] As an overview to local motion planning, it may be useful to first introduce the terminology used to describe motion planning and then follow this with a definition of the local trajectories and an overview of the optimization problem solved to obtain collision-free motions. With regard to terminology, FIGS. 13A and 13B illustrate schemas 1310, 1320 of local motion planning including examples of candidate local trajectories and examples of a local trajectory, respectively, for an agent 1330 (e.g., a quadrotor helicopter).

[0105] A “global trajectory” is a trajectory between two configurations embedded in a cost field with simplified dynamics and constraints.” A “preferred velocity” is an ideal velocity computed by a guidance system to track the global velocity, and it may be denoted as ū∈³. A “local trajectory” is the trajectory executed by the agent or vehicle for a short time horizon and is selected from a set of candidate local trajectories. A “candidate local trajectory” is one that respects the kinematics and dynamic constraints of the agent or vehicle and is computed from a candidate reference trajectory via a bijective mapping or the like. A “candidate reference trajectory” is a trajectory given by a constant velocity, u∈³, that indicates the target direction and velocity of the agent or aerial vehicle. An “optimal reference trajectory” is a trajectory given by the optimal reference velocity, u*∈³, selected from the set of candidate reference velocities and which minimizes the deviation to ū∈³. Further, this trajectory defines the local trajectory.

[0106] One focus of the present description is on local motion planning using a preferred velocity, ū, computed by a guidance or control system in order to track a global trajectory. An optimal reference trajectory is computed such that its associated local trajectory is collision free. This provides a parameterization of local motions given by u and allows for an efficient optimization in candidate reference velocity space, ³, to achieve collision-free motions.

[0107] Each candidate reference trajectory is that of an omni-directional agent moving at a constant velocity, u, and starting at the current position, p^k, of the agent as shown by:
P_ref(t)=p^k+u(t−t^k),t∈[t^k,∞)  Eq. (18)
A trajectory tracking controller, respecting the kinematics and dynamic constraints of the agent, q({tilde over (t)})=ƒ(z^k,u,{tilde over (t)}) is considered, continuous in the initial state z^k=[p^k,{dot over (p)}^k,{umlaut over (p)}^k, . . . ] of the agent and converging to the candidate reference trajectory. This defines the candidate local trajectories given by q({tilde over (t)}) and a bijective mapping (for a given z^k) from candidate reference velocities, u.

[0108] As an overview of the optimization problem, a collision-free local trajectory (described above and parameterized by u*) is obtained via an optimization in candidate reference velocity space where the deviation to the preferred velocity, ū, is minimized. In the following description, two alternative formulations are presented as a centralized optimization and a distributed optimization, formulated as a convex optimization with quadratic cost and linearized constraints for the following: (a) Motion Continuity: The position error between the candidate reference trajectory and the local trajectory should be below a limit, ∈, to guarantee that the local trajectory is collision free if the candidate reference trajectory is. Continuity in position, velocity, acceleration, and further derivatives (optional) is guaranteed by construction; and (2) Collision Avoidance: The candidate reference velocities leading to a collision are described by the velocity obstacle, a cone in relative candidate reference velocity space for agents of ∈-enlarged radius. Consider C_i the set of collision avoidance constraints for an agent, i, with respect to its neighbors and C=∪_i∈AC_i.

[0109] These non-convex constraints can be linearized leading to a convex optimization with quadratic cost and linear quadratic constraints. The optimization problem is formulated below, and the convex optimization is described for the case of heterogeneous groups of agents of unknown dynamics. Further, an algorithm is described to solve the original non-convex optimization along with an extension for homogeneous groups of agents (same dynamics and controller). In the centralized case, a single convex optimization is solved where the optimal reference velocity of all agents, u*_1:m=[u*₁, . . . , u*_m], are jointly computed. In the distributed case, each agent, i∈A, independently solves an optimization where its optimal reference velocity, u*_i, is computed. In this case, it can be considered that agents collaborate in the avoidance and apply the same algorithm, otherwise they are treated as dynamic obstacles. Only information about the neighbor's current position, velocity, and shape is utilized in the optimization.

[0110] Now, it may be appropriate to further describe a formulation for local motion planning, and the following discusses in detail the optimization cost and the optimization constraints used in local motion planning. With regard to preferred velocity, in each local planning iteration, a preferred velocity, ū, can be computed given the current position and velocity of the vehicle so as to follow a global plan to achieve either a goal or a trajectory. With regard to achieving a trajectory, the preferred velocity, ū, is given by a trajectory tracking controller for omnidirectional agents.

[0111] With regard to achieving a goal, the preferred velocity, ū, is given by a proportional controller towards the goal position, g_i, saturated at the agent's or vehicle's preferred speed, V>0, and decreasing when in the neighborhood as shown in the following:

[0112] $\begin{array}{cc}{\stackrel{\_}{u}}_i=\stackrel{\_}{V}\min \left(1,\frac{g^i-p^i}{K_{\stackrel{\_}{v}}}\right)\frac{g_i-p_i}{g_i-p_i}& \mathrm{Eq}.\left(19\right)\end{array}$
where K _v>0 is the distance to the goal from which the preferred velocity is reduced linearly. Alternatively, it can be given by an LQR controller.

[0113] With regard to repulsive velocity, formulating the local motion planning as an optimization where avoidance constraints are given by Velocity Obstacles produces collision-free motions but typically reduces inter-agent distance to zero. Although optimal, this property can lead to unsafe motions in real scenarios with sensor noise. To appropriately maintain a minimum inter-agent distance, the preferred velocity, ū, is corrected by adding a repulsive velocity, , given by a vector field such as those shown in one of the two examples of repulsive velocities shown in the graph 1400 of FIG. 14 and that pushes agents apart from each other when at a certain distance. Example equations for repulsive velocity that may be utilized in the shared control method are provided below.

[0114] The optimization cost can be given by two parts, with the first one being a regularizing term penalizing changes in reference velocity and the second one being chosen for minimizing the deviation to the preferred velocity, ū_i, corrected by the repulsive velocity, _i. For example, it has been shown that pedestrians prefer to maintain a constant velocity in order to minimize energy, and, hence, preference was given to turning instead of changing speed. Furthermore, penalizing changes in speed leads to a reduction of deadlock situations. This idea is formalized as an elliptical cost, higher in the direction parallel to ū_i+_i, and lower perpendicular to it. One can let λ>0 represent the relative weight and then define L=diag(λ,1,1) and D_i the rotation matrix of the transformation onto the desired reference frame.

[0115] In the distributed case, the quadratic cost can be given by:
C(u_i):=K_o∥u_i−v_i∥²+∥L^1/2D_i(u_i−(ū_i+_i))∥²  Eq. (20)
where, with the exception of the optimization variables, u_i, the values of all other variables are given for the current time instance and K_o is a design constant.

[0116] In the centralized case, the quadratic cost can be given as:
C(u_1:m):=K_oΣ_i=1^m w_i∥u_i−v_i∥²+Σ_i=1^m w_i∥L^1/2Dⁱ(u_i−(ū_i+_i)∥²  Eq. (21)
where w_i represents relative weights between different agents to take into account avoidance preference between agents, which can be interpreted as aggressive (high w_i) versus shy (low) w_i) behavior.

[0117] Turning now to motion continuity constraints, in order to guarantee collision-free motions, it can be guaranteed that each agent remains within ∈_i of its reference trajectory, with ∈_i<min_j∥p_j−p_i∥/2. So, a first constraint (“Constraint 1”) for motion continuity can be stated: for a maximum tracking error, ∈_i, and current state, z_i=[p_i,{dot over (p)}_i,{umlaut over (p)}_i], the set of candidate reference velocities, u_i, that can be achieved with position error lower than the maximum tracking error, ∈_i, is given by:
R_i=R(z_i,∈_i)={u_i|∥(p_i+{tilde over (t)}u_i)−ƒ(z_i,u_i,{tilde over (t)})∥≦∈_i,∀{tilde over (t)}>0}  Eq. (22)

[0118] For arbitrary ƒ(z_i,u_i,{tilde over (t)}), the set Ri can be precomputed. For quadrotors freely moving in 3D space, an LQR controller can be used as the function ƒ(z_i,u_i,{tilde over (t)}). In the present description, poison control and local trajectory are decoupled and ƒ(z_i,u_i,{tilde over (t)}) is given by Eq. (40). If the agents were omnidirectional, the set Ri would be given by a sphere centered at zero and of radii v_max (no continuity in velocity). For the case of study of this description, given a maximum tracking error, ∈_i, and an initial state, z_i, the limits can be obtained from the computed data with:

[0119] $\begin{array}{cc}\delta R_{i,{\varepsilon }_i}=\left\{u_i{❘}_{i>0}^{\max }\left(p_i+\tilde{t}{u}_i\right)-f\left(z_i,u_i,\tilde{t}\right)={\varepsilon }_i\right\}& \mathrm{Eq}.\left(23\right)\end{array}$
To reduce computational complexity, δR_i,∈i is then approximated by the largest ellipse (quadratic constraint of the form u_i^TQ_iu_i+1_i·u_i≦a_i, with Q_i∈^3×3 positive semidefinite, l_i∈) fully contained inside δR_i,∈i.

[0120] With regard to collision avoidance constraints, each agent or vehicle should avoid colliding with static obstacles, other controlled agents or vehicles, and dynamic obstacles. A maximum time horizon, τ, can be considered for which collision-free motion is guaranteed. A static obstacle is given by a region O ⊂³. The vehicles or agents are modeled by an enclosing vertical cylinder. The disk of radius r centered at c is denoted by D_c,r ⊂², and the cylinder of radius r and height 2h centered at c is denoted by C_c,r,h ⊂ ³.

[0121] The collision avoidance constraints are computed with respect to an agent or vehicle following the reference trajectory. Since the agents are not able to perfectly track the reference trajectory, their enveloping cylinder should be enlarged by the maximum tracking error, ∈_i. For simplicity, the following notation is used: r_i:=r_i+∈_i and h_i:=h_i+∈_i for the enlarged radius and the cylinder height with p_i being the current position of an agent, i. Then a second constraint (“Constraint 2”) regarding avoidance of static obstacles can be stated as: for every agent i∈A and neighboring obstacle O, the constraint is given by the reference velocities for which the intersection between O and the agent is empty for all future time instances, i.e., p_i+u_i{tilde over (t)} ∉ (O ⊕ C_0,ri,h_i),∀{tilde over (t)}∈[0,τ].

[0122] This constraint can be given by a cone in the 3D candidate reference velocity space generated by O ⊕ C_0,ri,h_i−p_i and truncated at (O ⊕ C_0,ri,hi−p_i)/τ. An example of this constraint for a rectangular object is shown in the fly space or graph 1500 of FIG. 15. Boundary wall constraints are directly added, given by n·u_i≦d(wall,p_i)/τ, with n being the normal vector to the wall and d(wall,p_i) being the distance to it.

[0123] A third constraint (“Constraint 3”) for inter-agent collision avoidance can be stated: for every pair of neighboring agents i,j∈A, the constraint is given by the relative reference velocities for which the agents' enveloping shape does not intersect within the time horizon, i.e., u_i−u_j ∉ VO_ij^τ=∪_{{tilde over (t)}=0}^τ((C_pj,rj,hj ⊕ C_−pi,ri,hi)/{tilde over (t)}),∀{tilde over (t)}∈[0,τ]. For cylindrical-shaped agents, the constraint can be separated in the horizontal plane ∥p_i^H−p_j^H+(u_i^H−u_j^H){tilde over (t)}∥≧r_i+r_j, and the vertical component |p_i³−p_j³+(u_i³−u_j³){tilde over (t)}|≧h_i+h_j for all the {tilde over (t)}∈[0,τ]. The constraint is a truncated cone as shown in the graphs 1610, 1620, and 1630 of FIGS. 16A-16C (particularly in graph 1610), with graph 1610 illustrating a velocity obstacle in relative velocity, graph 1620 illustrating projection onto the horizontal plane, and graph 1630 showing projection onto a vertical plane.

[0124] Turning now to approximation by linear constraints, since the volume of reference velocities leading to a collision is convex, the aforementioned collision avoidance constraints are non-convex. To obtain a convex optimization, they can be linearized. For increased control over the avoidance behavior and topology, each non-convex constraint can be approximated by five linear constraints representing to the right, to the left, over and under the obstacle or another agent, and a head-on maneuver, which remains collision-free up to t=τ. In particular, for the feasible space ³\VO_ij^τ consider five (l∈[1,5]) linear constraints (half-spaces) H_ij^l of the form n_ij^l(u_i−u_j)≦b_ij^l, with n_ij^l∈³ and b_ij^l∈, and verifying that if u_i−u_j satisfies any of the linear constraints, it is feasible with respect to the original constraint that:
∪_l=1⁵H_ij^l⊂³\VO_ij^τ  Eq. (23)
An example of this process is provided in FIGS. 16A-16C, and the equations of the linear constraints are provided below. To linearize the original constraint, only one of the linear constraints is typically selected and added to the convex optimization.

[0125] Now, with regard to selection of a linear constraint, each collision avoidance constraint can be linearized by selecting one of the five linear constraints. First, one may choose a fixed side for avoidance. If agents are moving towards each other, (v_ij·p_ij<0), one may wish to avoid on a predefined side such as on the left (l=1). If agents are not moving towards each other, the constraint perpendicular to the apex of the cone (l=5) can be selected to maximize maneuverability. Second, the linear constraint chosen may be maximum constraint satisfaction for the current relative velocity:

[0126] $\begin{array}{c}\arg \min \\ l\end{array}\left(n_{\mathrm{ij}}^l·\left(v_i-v_j\right)-b_{\mathrm{ij}}^l\right).$
This selection maximizes the feasible area of the optimization, when taking into account the motion continuity constraints discussed above.

[0127] A third choice may be maximum constraint satisfaction for the preferred velocity:

[0128] $\begin{array}{cc}\begin{array}{c}\mathrm{argmin}\\ l\end{array}\left(n_{\mathrm{ij}}^l·\left({\stackrel{\_}{u}}_i-{\stackrel{\_}{u}}_j\right)-b_{\mathrm{ij}}^l\right)& \left(1\right)\end{array}$
if centralized and

[0129] $\begin{array}{cc}\begin{array}{c}\mathrm{argmin}\\ l\end{array}\left(n_{\mathrm{ij}}^i·\left({\stackrel{\_}{u}}_i-v_j\right)-b_{\mathrm{ij}}^l\right)& \left(2\right)\end{array}$
if distributed. This selection may provide faster progress towards the goal position, but the optimization may become quickly infeasible if the agent greatly deviates from its preferred trajectory. The first choice or option provides the best coordination results as it incorporates a social rule. The second choice or option maximizes the feasible area of the optimization, and the third choice or option may provide faster convergence to the ideal trajectory. An experimental evaluation of these options is given below. Without loss of generality, consider n_ij·u_ij≦b_ij the selected linearization of ³\VO_ij^τ, which can be directly added in the centralized optimization of Algorithm 1 discussed below.

[0130] With regard to partition, in the distributed case, the reference velocity of an agent, j, is typically unknown, and only its current velocity, v_i, can be inferred. With the assumption that every agent follows the same optimization algorithm, a partition should be found such that if both agents select new velocities independently their relative reference velocity, u_i−u_j, is collision free. The idea of reciprocal avoidance can be followed. The change in velocity is denoted by Δv_i=u_i−v_i and the relative change of velocity by Δv_ij=Δv_i−Δv_j. In the distributed case, all agents are considered as independent decision makers solving their independent optimizations. To globally maintain the constraint satisfaction and avoid collisions, an assumption on an agent's, j's, velocity can be made.

[0131] Variable sharing of avoidance effort might be considered in some cases leading to Δv_i=λΔv_ij with the assumption that Δv_j=−(1−λ)Δv_ij. For collaborative agents that equally share the avoidance effort, λ=0.5. If it is considered that an agent, j, ignores another agent, i, and continues with its current velocity, then λ=1 and full avoidance is performed by the agent, i. With the assumption that both agents share the avoidance effort (change in velocity) for λ∈[0,1], the linear constraint can be written as:

[0132] $\begin{array}{cc}\begin{array}{c}{n}_{\mathrm{ij}}·u_{\mathrm{ij}}=n_{\mathrm{ij}}·\left(u_i-v_j-\Delta v_j\right)\\ =n_{\mathrm{ij}}·(u_i-v_j+\left(1-\lambda \right)\Delta v_{\mathrm{ij}}\\ =n_{\mathrm{ij}}·\left(u_i-v_j+\left(1-\lambda \right)\left(u_i-v_i\right)/\lambda \right)\\ =\frac{n_{\mathrm{ij}}}{\lambda }·\left(u_i\left(1-\lambda \right)v_i-\lambda v_j\right)\le b_{\mathrm{ij}}\end{array}& \mathrm{Eq}.\left(24\right)\end{array}$
which leads to:
n_ij·u_i≦b_i=λb_ij+n_ij·((1−λ)v_i+λv_j  Eq. (25)
which is a fair partition of the velocity space for both agents and provides avoidance guarantees.

[0133] At this point, it may be useful to provide exemplary algorithms for use in convex optimization. To achieve low computational time, the inherently non-convex optimization can be approximated by a convex optimization with quadratic cost and linear and quadratic constraints. The following two algorithms can be used in this regard with one being for use in the centralized implementation and one in the distributed implementation.

[0134] The first algorithm (“Algorithm 1”) for centralized convex optimization can be stated as: a single convex optimization is solved where the optimal reference velocity of all agents, u*_1:m=[u*₁, . . . , u*_m], are jointly computed as:

[0135] $\begin{array}{cc}{u}_{1:m}^{*}=\begin{array}{c}\mathrm{argmin}\\ u_{1:m}\end{array}C\left(u_{1:m}\right)s.t.u_i\le v_{\max },\forall i\in A\mathrm{quadratic}\mathrm{Constraint}1,\forall i\in A\mathrm{linearized}\mathrm{Constraint}2,\forall i\in A\mathrm{linearized}\mathrm{Constraint}3,\mathrm{of}\mathrm{the}\mathrm{form}{n}_{ij}·\left(u_i-u_j\right)\le b_{\mathrm{ij}},\forall i,j\in A\mathrm{neighbors}& \mathrm{Eq}.\left(26\right)\end{array}$
where the optimization by Eq. (21).

[0136] The second algorithm (“Algorithm 2”) for distributed convex optimization can be states as: each agent, i∈A, independently solves an optimization where its optimal reference velocity, u*_i, is computed as:

[0137] $\begin{array}{cc}{u}_i^{*}=\begin{array}{c}\arg \min \\ u_i\end{array}C\left(u_i\right)s.t.u_i\le v_{\max },\mathrm{quadratic}\mathrm{Constraint}1,\mathrm{agent}i\mathrm{linearized}\mathrm{Constraint}2,\mathrm{agent}i\mathrm{linearized}\mathrm{Constraint}3,\mathrm{of}\mathrm{the}\mathrm{from}{n}_{\mathrm{ij}}·u_i\le b_i,\forall j\in A\mathrm{neighbor}\mathrm{of}i& \mathrm{Eq}.\left(27\right)\end{array}$
where the optimization cost is given by Eq. (20). These 2m and 2-dimensional quadratic optimizations with linear and quadratic constraints can be solved efficiently using presently available solvers.

[0138] If the optimization is feasible, a solution is found that guarantees collision-free motion up to a predefined time horizon, τ. If the optimization is infeasible, no solution exists for the linearized problem. In this case, the involved agents drive or fly their last feasible trajectory (obtained at time t_ƒ) with a time re-parameterization given by:
γ(t)=(−t_ƒ²+(2τ+2t_ƒ)t−t²)/(2τ),t∈[t^ƒ,t^ƒ+τ],γ(t)=t_ƒ+τ/2,t<t_ƒ+τ  Eq. (28)
This time re-parameterization guarantees that the involved agents reach a stop on their previously computed paths within the time horizon of the collision avoidance algorithm. Since this computation is performed at a high frequency, each individual agent is able to adapt to changing situations.

[0139] With these two convex optimization algorithms in mind, it may now be useful to discuss several theoretical guarantees. A first theoretical guarantee or remark pertains to computational complexity and can be stated as follows: linear in the number of variables and constraints although the centralized is of higher complexity. If distributed, for each agent, the optimization utilizes three variables, one quadratic constrain, and a maximum of m linear constraints for collision avoidance (in practice, limited to a constant value, K_c). If centralized, the optimization uses 3n variables, n quadratic constraints, and less than

[0140] $\frac{n\left(n-1\right)}{2}+n\left(m-n\right)$
linear constraints (usually limited to nk_c).

[0141] A second theoretical guarantee or remark pertains to safety guarantees and can be stated as: safety is preserved in normal operation. If feasible, collision-free motion is guaranteed for the local trajectory up to a particular time, τ (optimal reference trajectory is collision-free for agents of radii enlarged by the maximum tracking error, ∈, and agents stay within ∈ of it) with the assumption that all interacting agents either continue with their previous velocity or compute a new one following the same algorithm and assumptions. If one lets p_i(t) denote the position of an agent, i, at time t≧t^k and recalls that {tilde over (t)}=t−t^k, then:
∥p_i(t)−p_j(t)∥=∥ƒ(z_i^k,u_i,{tilde over (t)})−ƒ(z_j^k,u_j,{tilde over (t)})∥≧∥(p_i^k+u_i{tilde over (t)})−(p_j^k+u_j{tilde over (t)})∥−∈_i−∈_j≧r_i+∈_i+r_j+∈_j−∈_i−∈_j=r_i+r_j  Eq. (29)
where the first inequality holds from the triangular inequality and Constraint 1 (u_i∈R_i and u_j∈R_j). The second inequality holds from Constraint 3 (u_i−u_j ∉ VO_ij^τ).

[0142] If infeasible, no collision-free solution exists that respects all the constraints. If the time horizon is longer than the required time to stop, passive safety is preserved if the last feasible trajectories of all agents are driven with a time re-parameterization to reach stop before a collision arises (see Eq. 28). This time re-parameterization implies a slowdown of the agent, which in turn may render the optimization feasible in a later time step. Since this computation is performed at a high frequency, each individual agent can adapt to changing situations.

[0143] A third theoretical guarantee or remark pertains to infeasibility and can be stated as: the optimization can be infeasible due to several causes. First, there may not be enough time to find the solution within the allocated time. Second, there may be differences between the model and the real vehicle. Third, there may be noise in localization and estimation of the agents' states. Fourth, the optimization can be infeasible due to the limited local planning horizon together with an over simplification of motion capabilities by reducing them to the set of local motions. Fifth, if distributed and given the use of pair-wise partitions of velocity, space with either the assumption of equal effort in the avoidance or constant speed, the optimization may be infeasible when not all world constraints are taken into account for the neighboring agents. Thus, an agent may have conflicting partitions with respect to different neighbors rendering its optimization infeasible.

[0144] A fourth theoretical guarantee or remark pertains to deadlock-free guarantees and can be stated as: as for other local methods, deadlocks may still appear. With this in mind, a global planner for guidance may be useful in complex scenarios.

[0145] Now, it may be useful to provide an exemplary algorithm for use in non-convex optimization. Earlier, a method for local motion planning was described for heterogeneous groups of aerial vehicles (or agents), which was formulated as a convex optimization problem. The following presents an extension to a non-convex optimization where the global optimum can be found but at an increased computational cost. As taught earlier in this description, each non-convex constraint can be approximated by five linear constraints of which only one is introduced in the optimization, which leads to a local optimum. The full solution space can be explored by the addition of binary variables and reformulating the problem as a mixed integer quadratic program (MIQP).

[0146] One binary variable, ξ_c^l={0,1}, is added for each linear constraint, l∈[1:5], approximation of a nonconvex constraint, c∈C. Only one out of the five linear constraints is active, thus Σ_l=1⁵ξ_c^l=4,∀c∈C. An algorithm (“Algorithm 3”) for non-convex optimization (or for centralized mixed-integer optimization) can now be formulated. A single MIQP optimization is solved where the optimal reference velocity of all agents, as well as the active constraints, ξ=∪_{l∈[1,5],c∈C}ξ_c^l, are jointly computed. A vector, ƒ _ξ^T, can be defined with (optional) weights for each binary variable, for instance, to give preference to avoidance on one side, and then Algorithm 3 can presented as:

[0147]

argmin
	C ⁡ ( u 1 ⁢ : ⁢ m ) + f ⁢ T ξ ⁢ ξ _
u1:m
s.t.	∥ui∥ ≦ umax, ∀i ε A
	quadratic Constraint 1, ∀i ε A
	linear Constraints 2, of the form
	nil · ui − Nξil ≦ bil, ∀i ε A, ∀l ε [1,5]
	linear Constraints 3, of the form
	nijl · (ui − uj) − Nξijl ≦ bijl,
	∀i, j ε A, neighbors, ∀l ε [1,5]
	Σi5 ξcl = 4, ∀c ε C     Eq.(30)

where N>>0 is a large constant of arbitrary value. The linear cost vector f _ξ is a zero vector if no preference between linear constraints for each VO exists. Different weights can be added for each linear constraint to include preference for a particular avoidance side.

[0148] This optimization can be solved via branch and bound using state of the art solvers. Although the number of variables and constraints can be bounded to be linear with the number of agents, the number of branches to be explored increases exponentially. In practice, a bound in the resolution time or number of explored branches (avoidance topologies) should be set. A distributed MIQP optimization can also be formulated, but collision avoidance guarantees would be lost since this can lead to a disagreement in the avoidance side and reciprocal dances appear.

[0149] With all the above teaching in mind, a control framework can be presented that is useful with aerial vehicles, and the inventors implemented and experimentally evaluated the control framework for providing shared control over quadrotor helicopters as the aerial vehicles. Particularly, FIG. 17 illustrates a control framework or controller 1700 that was implemented to provide control over a quadrotor vehicle 1710, and, to this end, the controller 1700 includes a number of submodules (e.g., programs executed by a processor of a central unit in a centralized implementation) 1720-1790 whose functionality and/or operations are described in the following paragraphs.

[0150] Motion planning may be divided into global planning and local planning. With regard to global planning, in complex environments, a global planner may be desirable for convergence, but in this discussion a fixed trajectory or goal position may be considered or used (as shown at 1720). With regard to local planning, the controller 1700 may compute (e.g., with a 10 Hz frequency) a collision-free local motion given a desired global trajectory or a goal position. To this end, a guidance system 1730 may output a preferred velocity, ū, given a desired global trajectory or a goal position and the current state of the vehicle. Additionally, a collision avoidance module 1740 receives the preferred velocity, ū, and computes the optimal reference velocity, u*, i.e., the closest collision-free candidate reference velocity that satisfies the motion continuity constraints.

[0151] With regard to control, a position controller 1760 controls the position of the quadrotor helicopter (or other aerial vehicle) to the local trajectory and runs at a desired frequency such as 50 Hz offboard in a real-time computer. The local trajectory interpreter 1750 receives the optimal reference velocity, u*, at a number of time instances, t^k. The module 1750 then outputs the control states, q, {dot over (q)}, {umlaut over (q)}, to satisfy motion continuity at a particular time, and asymptotic convergence to the optimal reference trajectory as may be given by p^k+u*{tilde over (t)}. The position controller 1760 receives setpoints in position, q, velocity, {dot over (q)}, and acceleration, {umlaut over (q)}, and outputs the desired force, f.

[0152] The vehicle 1710 may be configured to provide a processor, memory, and software to provide a low-level controller that runs onboard (e.g., a 200 Hz or the like) and provides accurate attitude control at high frequency. The low-level controller may abstract the nonlinear quadrotor model as a point-mass for the high-level control. It may be composed of the following three submodules: (1) an attitude controller 1780 that receives a desired force, f (collective thrust and desired orientation) and outputs desired angular rates, w_x, w_y, w_z; (2) a body angular rate controller that receives the desired body angular rates, and collective thrust, T, and provides an output in the form of a rotation speed setpoint for each of the motors [ω₁, . . . , ω₄]; and (3) a motor controller that receives the desired rotation speeds of the motors [ω₁, . . . , ω₄], with some implementations of vehicle 1710 using off-the-shelf motor controllers to interface the four motors of the quadrotor.

[0153] A quadrotor is a highly dynamic vehicle. To account for delays, the state used for local planning may not be the current state of the vehicle but, instead, a predicted one (e.g., as may be provided by forward simulation of the previous local trajectory) at the time when the new local trajectory is sent. A quadrotor is an under-actuated system in which translation and rotation are coupled. Its orientation (attitude) is defined by the rotation matrix, R_I^B=[e_x, e_y, e_z]∈SO(3), which describes the transformation from the inertial coordinates frame, I, to the body fixed coordinate frame, B. The following paragraphs provide an overview of the model and control.

[0154] With regard to a quadrotor model (vehicle model), each of the four motors of the quadrotor has an angular speed, w_i, and produces a force, F_i=c_tw_i², and a moment, M_i=c_yw_i², where c_t and c_y are model-specific constants. Since the motor dynamics are fast, it is assumed that the desired force, F_i, is reached instantaneously. The control input, o=[T, o₁], with o₁=[M_x, M_y, M_z], T being the total thrust, and M_x, M_y, and M_z being the moments around the body axis, is given by:

[0155] $\begin{array}{cc}o=\left[\begin{array}{c}T\\ o_1\end{array}\right]\underset{\underset{U}{︸}}{\left[\begin{array}{cccc}{c}_t& c_t& c_t& c_t\\ 0& c_{a}1& 0& -c_{a}1\\ -c_{a}1& 0& c_{a}1& 0\\ c_y& -c_y& c_y& -c_y\end{array}\right]}\underset{\underset{w_m}{︸}}{\left[\begin{array}{c}{w}_1^2\\ w_2^2\\ w_3^2\\ w_4^2\end{array}\right]}& \mathrm{Eq}.\left(31\right)\end{array}$
where l is the length from the center of gravity of the quadrotor to the rotor axis of the motor and c_a is a model constant.

[0156] The angular velocity, w, and its associated skew anti-symmetric matrix, w, are denoted by:

[0157] $\begin{array}{cc}w:=\left[\begin{array}{ccc}0& -w_z& w_y\\ w_z& 0& -w_x\\ -w_y& w_z& 0\end{array}\right],w:=\left[\begin{array}{c}{w}_x\\ w_y\\ w_z\end{array}\right]& \mathrm{Eq}.\left(32\right)\end{array}$
The angular acceleration around the center of gravity is given by:
{dot over (w)}=J⁻¹[−w×Jw+o₁]  Eq. (33)
with J being the moment of inertia matrix reference to the center of mass along the body axes of the quadrotor. If R_B^I(R_I^B)⁻¹ denotes the transformation from the body to the inertial frame, the rotational velocity of the quadrotor's fixed body frame with respect to the inertial frame can be given by {dot over (R)}_B^I=wR_B^I.

[0158] A point-mass model can be used for the position dynamics. The accelerations of the quadrotor's point-mass in the inertial frame, {umlaut over (p)}_I, are given by:

[0159] $\begin{array}{cc}{\ddot{p}}_I={R_B^I\left[\begin{array}{ccc}0& 0& -\frac{T}{m}\end{array}\right]}^T+{\left[\begin{array}{ccc}0& 0& g\end{array}\right]}^T& \mathrm{Eq}.\left(34\right)\end{array}$
where two forces are applied, gravity in the z direction in the inertial frame is given by F_G=mg, and T is the thrust in the negative z direction in the quadrotor body-frame.

[0160] With regard to attitude control, an approach may be followed that involves directly converting differences of rotation matrices into body angular rates. A desired force setpoint, F, and a desired yaw angle, ψ, given by a high-level controller are transformed to a desired rotation matrix R=[ē_x, ē_y, ē_z] with

[0161] $\begin{array}{cc}{\stackrel{\_}{e}}_z=\frac{\stackrel{\_}{F}}{\stackrel{\_}{F}},{\stackrel{\_}{e}}_y=\frac{{\stackrel{\_}{x}}_{\psi }\times {\stackrel{\_}{e}}_z}{{\stackrel{\_}{x}}_{\psi }\times {\stackrel{\_}{e}}_z},{\stackrel{\_}{e}}_x={\stackrel{\_}{e}}_y\times {\stackrel{\_}{e}}_z& \mathrm{Eq}.\left(35\right)\end{array}$
and x_ψ=[cos ψ, sin ψ, 0]^T is a vector pointing in the desired yaw direction. The desired body angular rates can then be controlled with a standard PD controller and given by:

[0162] $\begin{array}{cc}\stackrel{\_}{w}=f\left(\frac{1}{2}\left({\stackrel{\_}{R}}_I^B{\hat{R}}_B^I-{\hat{R}}_I^B{\stackrel{\_}{R}}_B^I\right)\right)& \mathrm{Eq}.\left(36\right)\end{array}$
where {circumflex over (R)}** indicates the estimated rotation matrices.

[0163] Now, with regard to position control, according to the double integrator dynamics of Eq. (34), the discrete LTI model can be written as:
X_k+1=A_χk+B_u y=C_χk  Eq. (37)
with state vector χ:=[p_x, v_x, p_y, v_y, p_z, v_z]^T, input vector ρ=[F_x, F_y, F_z]^T, and system matrices given as:

[0164] $A:=I_3\oplus \left[\begin{array}{cc}1& dt\\ 0& 1\end{array}\right],B:=I_3\oplus \left[\begin{array}{c}\mathrm{st}\\ dt\end{array}\right],C:=I_3\oplus \left[\begin{array}{cc}1& 0\end{array}\right]$
with I being the identity matrix,

[0165] $\mathrm{st}=\frac{1}{2}d{t}^2,$
and ⊕ being the Kronecker product. To have an integral action for the LQR controller, the system given in Eq. (37) is augmented to the discrete LTI system:

[0166] $\tilde{X}:=\left[\begin{array}{c}X\\ q\end{array}\right],\tilde{A}:=\left[\begin{array}{cc}A& 0\\ C& I\end{array}\right],\tilde{B}:=\left[\begin{array}{c}B\\ 0\end{array}\right],C=\left[\begin{array}{cc}C& 0\end{array}\right]$
The control input, ρ, is then given with the state feedback law ρ=−K_{{tilde over (X)}} with K being the solution to the Discrete Algebraic Riccati equation (DARE) with state and input cost matrices.

[0167] With regard to motion primitives, a point-mass M-order integrator model can be used or considered, decoupled in each component:
q^(M+1)({tilde over (t)})=v  Eq. (38)
Then, one can let v∈³ be piecewise continuous and (.)^(M+1) represent the derivative of order M+1 with respect to time. The resulting solution q({tilde over (t)}) is C^M differentiable at the initial point. For M>0 the state feedback control law is given by:
v({tilde over (t)})=−a_Mq^(M)− . . . −a₂{umlaut over (q)}+a₁(u−{dot over (q)})+a₀(p^k+u{tilde over (t)}−q)  Eq. (39)

[0168] It has been shown that the under-actuated model of a quadrotor can be written as a differentially flat system, with its trajectory parameterized by its position and yaw angle and their derivatives up to forth degree in position and second degree in yaw angle. In the following, the yaw angle is considered constant. Although the method presented herein extends to M=4, in order to simplify the formulations given the difficulty of accurately estimating the high order derivatives and their marginal effect, only continuity in acceleration (M=2) is imposed in the following discussion. For M=2 (continuity in initial acceleration), the motion primitive is:
q({tilde over (t)})=p^k+u{tilde over (t)}+Fe^{−w1{tilde over (t)}}+(({dot over (p)}^k−u+(w₁−w₀)F){tilde over (t)}−F)e^{−w0{tilde over (t)}}  Eq. (40)
with F=({umlaut over (p)}^k+2w₀({dot over (p)}^k−v∞))/(w₀−w₁)² and w₀, w₁ being design parameters related to the decay of the initial velocity and acceleration, which can potentially be different for each of the three axes.

[0169] At this point in the discussion, it may be useful to describe results of experiments performed by the inventors with the shared control method of aerial vehicles. Particularly, experiments were performed both in simulation and with physical quadrotors. Simulation utilized a model of the dynamics and enabled experiments for larger groups of vehicles. As an example, FIG. 18 illustrates with graph 1800 position exchange for ten quadrotors in a simulation. The top of graph 1800 shows a horizontal (x-y) view and the bottom of graph 1800 shows a vertical (x-z) view. Vehicle position is depicted for timestamps 0 s:4 s:6 s:8 s:12 s in an arena of size 8 m by 4 m by 8 m. Convergence to the goal configuration was achieved in about 15 seconds in this simulation. All other results provided herein are for physical quadrotor helicopters.

[0170] The experimental space utilized was a rectangular space that was approximately 5 m by 5.5 m by 2 m in height. Quadrotor position was measured by an external motion tracking system running at 250 Hz. Each vehicle was modeled with the following parameters: w₀=3, w₁=3.5, r_i=0.35 m, h_i=0.5 m, v_max=2 m/s, a_max=2 m/s², ∈=0.1 m, and τ=3 s. The control system 1700 was used to control the quadrotors. A low-level attitude estimator and controller were run on each quadrotor's IMU, and the quadrotor was abstracted as a point-mass for the position controller. The position controller was run on a real-time computer and communication between the real-time computer and the quadrotor was done with a 50 Hz UART bridge. The overlying collision avoidance was run in a normal operating system environment and sent a collision-free optimal reference trajectory to the position controller over a communication channel. The position controller ensured that the quadrotors stayed on their local trajectories as computed by the collision avoidance algorithm. This hard real-time control structure ensured the stability of the overall system and kept the quadrotors on given position references.

[0171] The benchmark performance of the system was evaluated for a single quadrotor running the full framework, including local planning in which continuity up to acceleration was imposed. FIG. 19 illustrates with graphs 1910, 1920, 1930, and 1940 a single quadrotor tracking a circular motion given by a sinusoid reference signal in all three position components. FIG. 19 shows with graph 1910 the projection of the position p of the quadrotor onto the horizontal plane and shows with graph 1920 the projection onto a vertical plane. The reference circular motion is infeasible due to the space limitation of the experiment room, but the local planning successfully avoided colliding with the walls causing the flattened edges in the y component. The graph 1930 shows each component of the position p with respect to time as a continuous line, while the point-mass local trajectory q is shown by a dashed line. Good tracking performance was observed, although with a small delay due to unmodeled system delays and higher order dynamics. A steady-state error is further observed due to the use of an LQR controller without integral part in the experiments. Steady-state error is greatest in the vertical component, as a thrust calibration is not performed before flight, and its value varies from quadrotor to quadrotor. To handle steady state error, continuity s imposed in the local trajectory instead of on the real state of the quadrotor. This approach adapts well to poorly calibrated systems, like the one used in the experiments.

[0172] FIG. 19 in graph 1940 shows the measured velocity v of the quadrotor as a continuous line and the velocity of its point-mass local trajectory as a dashed line with respect to time. Good tracking performance was observed, although the delay was more apparent, especially for abrupt changes in velocity, which may be partially due to the unmodeled high-order dynamics of the quadrotor. The smoothness of the velocity profile can be adjusted by the parameters w₀ and w₁ in Eq. (40), which can be tuned for overall good performance.

[0173] Position swaps were performed with sets of two and four quadrotors, with about fifty transitions to different goal configurations (both antipodal and random). The centralized algorithm (Algorithm 1) and the distributed algorithm (Algorithm 2) were both tested, with the three different linearization options for the collision avoidance constraints described above. The joint MIQP optimization (Algorithm 3) was also tested, but it proved too slow for real-time performance at 10 Hz in the experimental implementation. However, in simulation, this approach provided good results.

[0174] FIG. 20 shows a representative position exchange for two quadrotors. In graphs 2010 and 2020, FIG. 20 shows the projection of their paths on the horizontal and vertical planes. The slight change in height is due to the dynamics of the quadrotors. FIG. 20 in graph 2030 shows the velocity profile for both quadrotors. The preferred speed of the quadrotors is 2 m/s, and the position exchange takes a few seconds. Very similar performance was obtained in the distributed and centralized cases and with the different linearization options.

[0175] Two experiments were performed in which four quadrotors were transitioned to antipodal positions while also changing height. Two different linearization strategies were employed, with one being the imposition of avoidance and with one being a case when collision avoidance constraints are linearized with respect to the measured velocity. FIG. 21 with graphs 2110, 2120, 2130, 2140, 2150, and 2160 the results these two experiments. Graphs 2110, 2120, and 2130 illustrate results of use of the first linearization strategy while graphs 2140, 2150, and 2160 illustrate results of use of the second linearization strategy. Graphs 2110, 2120, 2140, and 2150 show the projection of the position of all quadrotors onto the horizontal plane and a vertical plane. Graphs 2130 and 2160 show the velocity profile for all quadrotors. The preferred speed was 2 m/s, and the dimensions of the experiment room (fly space) only allowed the quadrotors to fly at a height between 0 and 2 meters. The position exchanges were collision free due in part to the local planning algorithm even for this more complex case in which the preferred velocity always points towards the goal positions and, therefore, is not collision free.

[0176] For the case when avoidance to the right was imposed, the behavior was highly predictable and smooth, with the rotation of all vehicles to one side being imposed in the linearization. This shows the control method provided good results both in the centralized case and the distributed case. Although this characteristic is effective for cases of symmetry, in arbitrary configuration, it can lead to longer trajectories. For the case of linearization with respect to the current velocity, the behavior can be different for different runs and emerges from non-deterministic characteristics of the motion. It was observed that in this particular example that two quadrotors chose to pass over each other using the control method. Although this linearization option presents very good performance in not too crowded environments, its performance may degrade in the case of high symmetry and very crowded scenarios (for example, due to disagreements in avoidance side especially in the distributed case).

[0177] Now, it may be useful to discuss the results of experiments performed to test the method for trajectory tracking, e.g., in the case of a moving goal. Two representative scenarios are described, and vertical motion is not displayed in the results because, without loss of generality, motion was approximately in a horizontal plane. FIG. 22 illustrates with graphs 2210, 2220, and 2230 the results of an experiment in which two quadrotors were tracking intersecting trajectories while avoiding collision locally. One quadrotor followed a circular motion (see line 2211) and one followed a diagonal motion (see line 2213 such that there are two intersection points in position and time. The one following a circular motion finally transitioned to a rest position. In graph 2210, FIG. 22 shows that both controlled quadrotors often deviate from their ideal trajectories in order to avoid collision. In graphs 2220 and 2230, the velocity plot shows that no deadlock appeared and both quadrotors were moving at a velocity of about 1.5 m/s except towards the end of the experiment where the quadrotor following a circular motion approached a static goal position. The trajectory for the other quadrotor then becomes feasible, and it is well tracked as shown by the velocity values after about 100 seconds.

[0178] FIG. 23 with graphs 2310 and 2320 shows the results of control experiments in which two quadrotors were tracking non-colliding circular trajectories with 180 degrees phase shift while a human (see line 2311) moved in the arena (test flying space). The human was considered as a dynamic obstacle. The two quadrotors avoided collisions locally while tracking, as closely as possible, their ideal trajectories. Graph 2310 shows the trajectories, and graph 2320 shows the measured velocity for all agents. No collision or deadlock was observed in the experiment, although in some cases a quadrotor had to stop as the optimization became infeasible. In this experiment, the human moved at a speed similar to that of the quadrotors (<2 m/s) and was slightly predictable in movements, due to a constant velocity assumption of the experiment.

[0179] At this point in the description, it may be useful to provide a discussion of the overall behavior of the control techniques over all the performed experiments based on statistics collected and processed by the inventors. Deadlock was not observed in the experiments, but it is possible from a theoretical standpoint and can occur either due to a stop condition being the optimal solution with respect to the optimization objective or due to the optimization problem being infeasible. The optimization can be infeasible for two main reasons: (1) for experimental reasons such as the unmodeled high-order dynamics of the vehicle and other uncalibrated parameters such as delays and (2) for reasons inherent in the method (and the trade-off between richness of planning versus low computational time) such as the limited horizon of the local planning or the convexification of the optimization via the linearization of the collision avoidance constraint.

[0180] The local planning optimization, which was computed at 10 Hz, was infeasible in under nine percent of the of the cumulated time steps over all experiments (over 40,000 data points at 10 Hz over more than 50 experimental runs). On encountering this condition, the speed of the quadrotor is reduced at a higher than normal rate following Eq. (28), and the quadrotor can, if necessary, be brought to a halt before a collision due to the finite local planning horizon. In all cases, due to the reduction in speed, the optimization became feasible after a further one or more time steps, e.g., below 0.5 seconds.

[0181] FIG. 24 illustrates with graphs 2410, 2420, and 2430 optimal reference velocity, u*, resulting from the optimization with respect to time for one of the experiments with four quadrotors. Six instances in which the optimization is infeasible are observed between the times 65 seconds and 68 seconds. These are points where the optimal reference velocity becomes zero before the quadrotors have converged to their goal positions. The effect of these infeasible time steps is to slow down the quadrotors, which in turn has the effect of rendering the local planning feasible in subsequent time steps. This extreme situation was mostly not observed in experiments with only two quadrotors.

[0182] FIG. 25 illustrates a histogram 2500 showing statistics accumulated for the inter-agent distance over the experiments run by the inventors with quadrotor helicopters as the controlled aerial vehicles in a fly space (or 3D space). There is a step decrease in the number of data points with distance below one meter. This is due to the fact that the quadrotors used in the experiment were considered to have a radius of 0.35 meters enlarged by ∈=0.1 meters to account for the dynamics, which can decrease towards zero when in close proximity. Note, a few data points appear to be in collision (<0.7 meters), which can happen in cases where the optimization becomes infeasible especially when a human/dynamic obstacle is moving through the test fly or 3D space. In the fully controlled case, no collision was visually observed and all situations were successfully resolved. In the case with a human present, the quadrotor halted before a collision in a few cases, and this happened, for example, when the velocity estimation is inaccurate or the human moves too fast or with an abrupt change in direction.

[0183] With regard to evaluation of the algorithm, good performance was observed throughout the experiments especially for the centralized convex optimization. The distributed optimization performs well in simulation and in experiments with two quadrotors. If the collision avoidance constraints are linearized with respect to the current velocity, the distributed approach suffers from inaccuracies in the velocity estimation, which can lead to performance degradation due to disagreements in the linearization. The centralized, non-convex optimization performs well in simulation but may be too slow in some cases for real-time control/performance.

[0184] In support of the above discussion, it may be useful to provide equations or pseudocode used for determining repulsive velocity with the controller modules of the present shared control system. For the repulsive velocity field of FIG. 14, the repulsive velocity for agent i∈A is given by:

[0185]

for j = 1 : m, j ≠ i do
if pijH < ri + rj and |pji3| < (hi + hj + Dh) then
u ij 3 = V r ⁡ ( r i + r j - p ij H r i + r j ) ⁢ ( h i + h j + D h - | p j 3 - p i 3 | D h ) ⁢ p ji 3 | p ij 3 |
end if
if pijH < ri + rj + Dr and |pji3| < (hi + hj) then
u ° ij H = V r ⁡ ( h i + h j - | p ij 3 | h i + h j ) ⁢ ( r i + r j + D r - p ij H D r ) ⁢ p ji H p ji H
end if
end for
u ° i ⁢ ⁢ := ⁢ ⁢ Σ j = 1 m ⁢ ⁢ u ij ° ; u ° i = min ⁡ ( V r ∥ u ° i ∥ , 1 ) ⁢ u ° i

where V_r is the maximum repulsive force and D_r, and D_h are the preferred minimal inter-agent distances in the X-Y plane and the Z component, respectively.

[0186]

if pijH > rij then
∝ = atan2(pijH), ∝n = acos(rij/pijH)
if |pij3| > hij then
∝ 1 ⁢ = atan ⁡ ( p ij 3 + h ij p ij H - sign ⁡ ( p ij 3 ) ⁢ r ij - π 2 )
∝ 2 ⁢ = atan ⁡ ( p ij 3 - h ij p ij H + sign ⁡ ( p ij 3 ) ⁢ r ij + π 2 )
O = [pijH − rij, pij3 − sign(pij3)hij]
else
∝ 1 ⁢ = atan ⁡ ( p ij 3 + sign ⁡ ( p ij 3 ) ⁢ h ij p ij H - r ij - sign ⁡ ( p ij 3 ) ⁢ π 2 )
∝ 2 ⁢ = atan ⁡ ( p ij 3 - sign ⁡ ( p ij 3 ) ⁢ h ij p ij H - r ij + sign ⁡ ( p ij 3 ) ⁢ π 2 )
O = [pijH − rij, 0]
end if
∝O = atan2(O)
nij1 = [cos(∝ − ∝n), sin(∝ − ∝n), 0]
nij2 = [cos(∝ + ∝n), sin(∝ + ∝n), 0]
nij3 = [pijHcos(∝1)/pijH, sin(∝1)]
nij4 = [pijHcos(∝2)/pijH, sin(∝2)]
nij5 = [pijHcos(∝O)/pijH, sin(∝O)]
cij1 = cij2 = cij3 = cij4 = 0; cij5 = ∥O∥/τ
else if |pij3| > hij then
∝ = atan ⁢ ⁢ 2 ⁢ ( p ij H ) , β = atan ⁡ ( | p ij 3 | - h ij r ij )
∝ 1 ⁢ = π 2 - atan ⁡ ( | p ij 3 | - h ij r ij - p ij H )
∝ 2 ⁢ = π 2 - atan ⁡ ( | p ij 3 | - h ij r ij + p ij H )
nij1= [cos ∝ cos ∝1, sin ∝ cos ∝1, sin ∝1]
nij2= [cos ∝ cos ∝2, sin ∝ cos ∝2, sin ∝2]
n ij 3 = [ cos ⁡ ( ∝ + π 2 ) ⁢ sin ⁢ ⁢ β , sin ⁡ ( ∝ + π 2 ) ⁢ sin ⁢ ⁢ β , cos ⁢ ⁢ β ]
n ij 4 = [ - cos ⁡ ( ∝ + π 2 ) ⁢ sin ⁢ ⁢ β , - sin ⁡ ( ∝ + π 2 ) ⁢ sin ⁢ ⁢ β , cos ⁢ ⁢ β ]
nij5 = [0, 0, 1]
cij1 = cij2 = cij3 = cij4 = 0; cij5 = (|pij3| − hij)/τ
if pij3 < 0 then
nijs[3] = −nijs[3] ∀s
end if
else
Agents i and j are in collision
nijl = pij/pij
cijl = −(rij − pijH + hij − |pj3 − pi3|/τ
end if

where H_ij¹ and H_ij² represent avoidance to the right/left, H_ij³ and H_ij⁴ above/below and H_ij⁵ represents a head-on maneuver, which remains collision-free up to t=τ.

[0187] The reader may also benefit from being aware of the equations for linearization of VO utilized in one implementation of the control system by the inventors. One can denote h_ij=h_i+h_j and r_ij=r_i+r_j. The non-convex constraint ³\VO_ij^T is linearized to obtain a convex problem. For an approximation with five linear constraints, as in FIG. 16, the linear constraints are given by:

[0188]

if pijH > rij then
∝ = atan2(pijH), ∝n = acos(rij/pijH)
if |pij3| > hij then
∝ 1 ⁢ = atan ⁡ ( p ij 3 + h ij p ij H - sign ⁡ ( p ij 3 ) ⁢ r ij - π 2 )
∝ 2 ⁢ = atan ⁡ ( p ij 3 - h ij p ij H + sign ⁡ ( p ij 3 ) ⁢ r ij + π 2 )
O = [pijH − rij, pij3 − sign(pij3)hij]
else
∝ 1 ⁢ = atan ⁡ ( p ij 3 + sign ⁡ ( p ij 3 ) ⁢ h ij p ij H - r ij - sign ⁡ ( p ij 3 ) ⁢ π 2 )
∝ 2 ⁢ = atan ⁡ ( p ij 3 - sign ⁡ ( p ij 3 ) ⁢ h ij p ij H - r ij + sign ⁡ ( p ij 3 ) ⁢ π 2 )
O = [pijH − rij, 0]
end if
∝O = atan2(O)
nijl = [cos(∝ − ∝n), sin(∝ − ∝n), 0]
nij2 = [cos(∝ + ∝n), sin(∝ + ∝n), 0]
nij3 = [pijHcos(∝1)/pijH, sin(∝1)]
nij4 = [pijHcos(∝2)/pijH, sin(∝2)]
nij5 = [pijHcos(∝O)/pijH, sin(∝O)]
cijl = cij2 = cij3 = cij4 = 0; cij5 = ∥O∥/τ
else if |pij3| > hij then
∝ = atan ⁢ ⁢ 2 ⁢ ( p ij H ) , β = atan ⁡ ( | p ij 3 | - h ij r ij )
∝ 1 ⁢ = π 2 - atan ⁡ ( | p ij 3 | - h ij r ij - p ij H )
∝ 2 ⁢ = π 2 - atan ⁡ ( | p ij 3 | - h ij r ij + p ij H )
nij1= [cos ∝ cos ∝1, sin ∝ cos ∝1, sin ∝1]
nij2= [cos ∝ cos ∝2, sin ∝ cos ∝2, sin ∝2]
n ij 3 = [ cos ⁡ ( ∝ + π 2 ) ⁢ sin ⁢ ⁢ β , sin ⁡ ( ∝ + π 2 ) ⁢ sin ⁢ ⁢ β , cos ⁢ ⁢ β ]
n ij 4 = [ - cos ⁡ ( ∝ + π 2 ) ⁢ sin ⁢ ⁢ β , - sin ⁡ ( ∝ + π 2 ) ⁢ sin ⁢ ⁢ β , cos ⁢ ⁢ β ]
nij5 = [0, 0, 1]
cij1 = cij2 = cij3 = cij4 = 0; cij5 = (|pij3| − hij)/τ
if pij3 < 0 then
nijs[3] = −nijs[3] ∀s
end if
else
Agents i and j are in collision
nijl = pij/pij
cijl = −(rij − pijH + hij − |pj3 − pi3|/τ
end if

where H_ij¹ and H_ij² represent avoidance to the right/left, H_ij³ and H_ij⁴ represent avoidance above/below, and H_ij⁵ represents a head-on maneuver, which remains collision-free up to t=τ.

[0189] Control systems, such as system 100 of FIG. 1, and shared control methods, such as the method 300 of FIG. 3, can be used and/or expanded to provide cooperative collision avoidance among decision-making agents (or vehicles such as wheeled and/or other terrestrial or ground-based vehicles). These control systems and methods may apply to collision-free navigation among heterogeneous groups of decision-making agents. The following paragraphs and discussion highlights several contributions provided by the inventors including: (1) arbitrary kinematic model and dynamic constraints; (2) convex and non-convex, centralized and distributed algorithms for collision avoidance; (3) a novel extension to truly cooperative distributed collision avoidance where a prediction over future velocities of the neighboring agents can be taken into account in the computation of the velocity partition; and (4) extensive experimental evaluation of the proposed algorithms including different robot kinematics (e.g., small differentially driven robots, car-like robots, robotic wheelchairs, and boats). Although the following description generally focuses on vehicles/agents navigating in 2D space, the control methods can be used for aerial vehicles navigating in 3D space.

[0190] Throughout the following paragraphs, x denotes scalars, x vectors, M matrices, and X sets. The Minkowsky sum of two sets is denoted by X⊕y and x=∥x∥ denotes the Euclidean norm of vector x. The super index *^k indicates the value at time t^k and {tilde over (t)}=t−t^k the appropriate relative time. Subindex *_i indicates agent i, and relative vectors are denoted by x_ij=x_i−x_j. For ease of exposition, no distinction is made between estimated and real states.

[0191] The control problem can be stated as a group of n agents freely moving in 2D space for which shared control is required or desired. For each agent i, p_i∈³ denotes its position, v_i={dot over (p)}_i its velocity, and a_i={umlaut over (p)}_i its acceleration. The shape of each agent is modeled by an enveloping circle of radius r_i, centered at the agent's position p_i, and denoted by D(p_i,r_i). One goal of the control method is to compute for each agent, at high frequency (e.g., 10 Hz), a local motion that respects the kinematics and dynamic constraints of the ego agent and that is collision free with respect to neighboring agents for a short time horizon (typically, 2 to 5 seconds or the like), and the problem may be labeled simply as “collision avoidance.” As discussed above, centralized and distributed scenarios can be considered and addressed with local motion planning using the same terminology (e.g., global trajectory, preferred velocity, local trajectory, candidate local trajectory, candidate trajectory, and optimal reference trajectory) as discussed with reference to FIGS. 13A and 13B.

[0192] With regard to definition of candidate local trajectories, each candidate is that of an omnidirectional agent moving at a constant velocity, u, and starting at the current location, p^k, of the agent:
p_ref(t)=p^k+u(t−t^k),t∈[t^k,∞)  Eq. (41)

[0193] A trajectory tracking controller, respecting the kinematics and dynamic constraints of the agent, p({tilde over (t)})=ƒ(z^k,u,{tilde over (t)}) is considered that is continuous in the initial state z^k=[p^k,{dot over (p)}^k,{umlaut over (p)}^k, . . . ] of the agent and converging to the candidate reference trajectory. This defines the candidate local trajectories given by p({tilde over (t)}) and a bijective mapping from candidate reference velocities u.

[0194] One focus of the inventors' work is on local motion planning using a preferred velocity, ū, computed by a guidance system in order to track a global trajectory. An optimal reference trajectory can be computed such that its associated local trajectory is collision free. This provides a parameterization of local motions given by u and allows for an efficient optimization in candidate reference velocity space, (³), to achieve collision-free motions. FIG. 26 illustrates with functional block or data flow diagrams 2610 and 2620 local motion planning according to the present teaching that uses a distributed approach and using a centralized approach, respectively.

[0195] Velocity prediction involves consideration that in each time instance a probability distribution over the future velocities of the ego agent and its neighbors is available or, due to the structure of the problem, it can be relaxed to a utility over the possible future velocities that represents their individual likelihood. This distribution can be computed in an environment around the agent's current velocity. Given a candidate reference velocity, ui, for an agent, i, its utility is defined by a probability distribution:
Utility,φ_i:²→[0,1]u_iφ_i(u_i)  Eq. (42)
such that ∫_ui∈2φ_i(u_i)=1. A value close to 1 indicates that the agent is very likely to choose that reference velocity, while a value close to 0 indicates that it is very unlikely.

[0196] In multi-agent applications where several robots/agents/vehicles are controlled, a prediction over the future velocities of the neighboring agents may not be suitable. In that case, a cost can be defined:
Cost,χ_i:²→[0,∞)u_iχ_i(u_i)  Eq. (43)
which is then transformed into a utility distribution with a mapping [0,∞)→[0,1]. To maintain the linear characteristics for low cost, one can employ the following (but, note, other mappings are possible and may be used):
φ_i,aux(u_i)=max(0,1−χ_i(u_i)/K₀)  Eq. (44)
where K₀ is the cut-off cost, and this can be normalized to

[0197] $\begin{array}{cc}{\varphi }_i\left(u_i\right)=\frac{{\varphi }_{i,\mathrm{aux}}\left(u_i\right)}{{\int }_{u_i\in R^2}{\varphi }_{i,\mathrm{aux}}\left(u_i\right)}& \mathrm{Eq}.\left(45\right)\end{array}$
In the following paragraphs, three cases are discussed: (1) a simplified prediction; (2) utility computed from a motion prediction; and (3) utility computed from a cost distribution.

[0198] Using a simplified prediction may be useful in some implementations of a controller or control system. Simplifying assumptions can be made with respect to the neighboring agents such as the following: (1) static agent to provide a zero-order approximation (φ_j(u_j)=1, for u_j=0; φ_j(u_j)=0 otherwise); (2) constant velocity for a first-order approximation (φ_j(u_j)=1, for u_j=v_j; φ_j(u_j)=0); and (3) constant preferred velocity to provide a first-order approximation (φ_j(u_j)=1, for u_j=ū_j; φ_j(u_j)=0). In these formulations of the velocity predictions, ū_j is the preferred velocity of agent j.

[0199] In other cases, the velocity prediction may be provided using a utility given a motion prediction. As a step towards accounting for uncertainties and predictions, one may consider the case of having a prediction of the future motion of the neighboring agent, i, given by a Gaussian Process (GP) over its future positions and velocities. The utility of each reference velocity can then be obtained by computing the posterior probability given the Gaussian Process, with different levels of complexity and using the following: (1) reference velocity (φ_j(u_j)=P(u_j,GP)); (2) candidate reference trajectory (φ_j(u_j)=P(p_k^k+u_j(t−t^k),GP(t))); and (3) candidate local trajectory (φ_j(u_j)=P(ƒ(z_j^k,u_j,t),GP(t))).

[0200] In still other cases, the velocity prediction may be provided using a utility given by a cost from the environment. The environment characteristics can be taken into account when constructing the weights over the reference velocities, u_j. This approach then provides information about the maneuverability of the neighboring agents and their motion preferences and can later be incorporated in the local motion planning step.

[0201] As an example, the cost, χ_j(u_j), can be given by the sum of several terms, which may include the following and should be saturated: (1) deviation from current velocity, where the agent is expected to continue with its current velocity (K₁∥u_j−v_j∥²); (2) deviation from preferred velocity, where velocities close to its preferred velocity incur lower cost (K₂∥u_j−ū_j∥²); (3) cost to go, determined as being proportional to the distance from the desired goal to the expected position after the time horizon of the local planner and as may be determined with a search algorithm or from a pre-computed distance transform map, and, with K₃ being a design constant, the term can be normalized to [0,K₃v_maxτ] by K₅max (0, cost2go(p_j^k+u_jτ)−C₀) Eq. (46) where C₀=max (0, cost2go(p_j^k)−v_maxτ); (4) static obstacles, where a high cost, K₄→∞, is added for reference trajectories in collision with a static obstacle such that p_j^k+u_j{tilde over (t)}∈O_r for some {tilde over (t)}∈[0,τ]; (5) congestion, where a cost, K₅<<K₄, can be added for reference trajectories in collision with another agent for {tilde over (t)}∈[0,τ]; for this weight, the other agents can be considered static or moving at constant speed, and the intuition behind this is that an agent would prefer to move away from other neighbors but, since they are mobile, the cost of a potential collision is not too high; note, this term penalizes movement towards potentially congested areas and may only be used for guidance as hard collision avoidance constraints are included in the local motion planning framework; and (6) motion velocity field, where the case is considered where a motion velocity field of the environment is available given by a velocity, V(p), for each position p, and this is the case, for example, in multi-lane navigation, with intuition indicating that velocities against the flow field are unlikely; this could be encoded by K₅∫₀^τ∥V(p_j^k+u_jt)−u_j∥dt for areas where the velocity field is defined and 0 in free areas (but alternative formulations are possible).

[0202] The different cost teams represent the belief the agent has over the environment and their relative weights are design parameters. Costs 1 and 4 to 6 can be obtained by all agents with common observations. On the other hand, costs 2 and 3 are agent specific and cannot be observed, although an estimation may be available and used. These terms are given as an example and can be modified or simplified. One contribution of this description is that of incorporating the velocity prediction in the planning.

[0203] Now, turning to determination of preferred velocity, in each time step, each agent, computes a preferred velocity, ū_i, which would be its desired reference velocity in the absence of other agents. It can be obtained to achieve different objectives. A first objective may be goal directed. In this case, the preferred velocity, ū_i, can be computed from the “cost to the goal” map (e.g., obtained by an F1 expansion as discussed above). Its orientation is that of the shortest path to the goal from the current position of the agent, and its magnitude equals the preferred speed, V>0, of the agent reduced linearly in the neighborhood of the goal.

[0204] A second objective may be trajectory following. In this case, the preferred velocity, ū_i, is given by a trajectory tracking controller for omnidirectional agents, such as a controller in position with feed forward velocity, such as:
ū=K_p(q_i−p_i)+{dot over (q)}_i  Eq. (47)
where q_i is the ideal position on the guidance trajectory at the current time and K_p is a design constant. A third objective may be semi-autonomous driving. In this case, the preferred velocity, ū_i, can be specified by a driver, with the preferred velocity, ū_i, being proportional to the driving wheel or joystick position (in the relative frame of the vehicle).

[0205] In practice, the preferred velocity, ū_i, often will be corrected with a repulsive velocity, _i, that pushes the vehicle away from both static and dynamic obstacles. Collision-free motion is guaranteed by the constraints that are directly included in the local planning framework. Optimal solutions that minimize deviation with respect to the preferred velocity do produce solutions where agents come arbitrarily close to each other. This behavior can be unsafe in some real world scenarios with uncertainties in sensing and kinematic model. The repulsive velocity is, thus, included to help maintain a minimum distance to other agents and walls, and it moves the ego-agent (controlled vehicle) away when it is too close.

[0206] A linear function weighting the distance to other vehicles can be used that is adapted to specify the maximum velocity allowed towards the obstacles or other vehicles as shown by:

[0207] $\begin{array}{cc}{\stackrel{.}{u}}_i=u_{io}^{\mathrm{rep}}{n}_{i,o}+{\sum }_{jp_{\mathrm{ij}}<D}{u}_{ij}^{\mathrm{rep}}\frac{p_{\mathrm{ij}}}{p_{\mathrm{ij}}}& \mathrm{Eq}.\left(48\right)\\ \mathrm{where}& \\ u_{io}^{\mathrm{rep}}=\max \left(V_r\frac{\stackrel{\_}{D}-d\left(p_i,O_{{\tau }_i}\right)}{\stackrel{\_}{D}}-\stackrel{\_}{V},{\stackrel{\_}{u}}_i·n_{i,o}\right)& \mathrm{Eq}.\left(49\right)\\ u_{ij}^{\mathrm{rep}}=\max \left(V_r\frac{D-p_{\mathrm{ij}}}{D-r_i-r_j}-\stackrel{\_}{V},{\stackrel{\_}{u}}_i·\frac{p_{\mathrm{ij}}}{p_{\mathrm{ij}}}\right)& \mathrm{Eq}.\left(50\right)\end{array}$
where V_r is the maximum repulsive velocity, D and D are the maximum distance from which a repulsive force is added, d(p_i,O) is the distance to the closest static obstacle within the O_ri map and n_i,O is the normal vector from it. In order to avoid corner issues, not only the closest point it considered but also an average over the closest points. The norm of _i is then limited to ∥_i∥≦V_r. For ease of notation, consider the following ū:=ū+.

[0208] Turning now to consideration of motion constraints, for an agent, i, the candidate local trajectories can be given by a tracking controller, p_i(t)=ƒ(z_i^k,u_i,{tilde over (t)}), towards a constant velocity reference trajectory parameterized by u. The set of feasible local trajectories given the current state is defined by:
_i(z_i^k)={u_i∈²,such that the trajectory ƒ(z_i^k,u_i,{tilde over (t)}) respects all dynamic constraints}  Eq. (51)
And, in the following discussion, the super-index, *^k, is dropped to simplify the notation.

[0209] With regard to a first constraint regarding dynamic restrictions (“Constraint 1”), in order to guarantee collision-free motions, it should be guaranteed that each agent remains within a maximum tracking error, ∈_i, of its reference trajectory. For a maximum tracking error ∈_i and a current state, z_i=[p_i,{dot over (p)}_i,{umlaut over (p)}_i], the set of reference velocities u_i that can be achieved with position error below ∈_i is given by R_i:=R(z_i,∈_i) and then:
R_i:={u_i∈_i(z_i)|∥(p_i+{tilde over (t)}u_i)−ƒ(z_i,u_i,{tilde over (t)})∥≦∈_i,∀{tilde over (t)}>0}  Eq. (52)

[0210] If a local trajectory that is continuous in acceleration is used (such as one generated by one of the trajectory controllers taught herein), the set of reachable reference velocities, R_i, depends on {dot over (p)}_i, {umlaut over (p)}_i, and ∈_i. A mapping, γ, from the initial state, {dot over (p)}_i, {umlaut over (p)}_i, and a reference velocity, u_i, to maximum tracking error is precomputed and stored in a look up table as:

[0211] $\begin{array}{cc}\gamma \left(u_i,{\stackrel{.}{p}}_i,{\ddot{p}}_i\right)=\begin{array}{c}\max \\ \tilde{t}>0\end{array}\left(p_i+\tilde{t}{u}_i\right)-f\left(z_i,u_i,\tilde{t}\right)& \mathrm{Eq}.\left(53\right)\end{array}$
A bounding box given by four linear constraints of the form H_l={u_i|n_lu_i≦b_l} is computed such that R_i ⊂ ∪_l=1⁴H_l. These linear constraints can then be included in the algorithms to reduce the search space.

[0212] Although some examples of agents, vehicles, or robots have been provided earlier, it may be useful at this point in the description to provide several additional examples of typical robot models used by the inventors to implement and/or test their control methods (and controllers and control systems). Below, five different robot kinematics and dynamics models are described that have been implemented and tested with this framework, and these cover most of the typical robot platforms presently in use.

[0213] First, one may model a robot, vehicle, or agent as a simply unicycle that may take the form of a small, differently-drive robot. In this case, the local trajectories can be defined by two sections including a circumference arc followed by a straight line path at a constant speed, u_i. The angular velocity over the arc was, in one model, defined such that the correct orientation was achieved within a fixed time, T, and the linear velocity minimized the tracking error of the reference trajectory. A closed form solution was obtained for the set R_i, but, in this simple formulation, the robots had no constraints in acceleration and the linear and angular velocities presented a discontinuity.

[0214] Second, one may model the controlled agent as a bicycle. For example, this modeling may be applied to car-like robots with bicycle kinematics. A trajectory controller can be employed to track the referenced trajectory given by a constant speed, u_i, and was obtained by applying full-state linearization via dynamic feedback to the non-linear system. Additional constraints such as maximum steering angle and velocity and linear velocity and acceleration were added as saturation limits. This framework was useful as it guaranteed continuity in both linear velocity and steering angle.

[0215] Third, a point mass v^th-order integrator may be used to model the controlled vehicle. A controller for point mass v^th-order integrator dynamics can and has been described, which is continuous up to the v_th derivative of the initial state. For v>0, the state feedback control law was given by:
p^(v+1)({tilde over (t)})=−a_vp^(v)− . . . −a₂{umlaut over (p)}+a₁(u−{dot over (p)})+a₀(p^k+u{tilde over (t)}−p)  Eq. (54)
In particular, for v=2 (continuity in initial acceleration), the trajectory is given by:
p({tilde over (t)})=p^k+u{tilde over (t)}+(({dot over (p)}^k−u+(ω₁−ω₀)F){tilde over (t)}−F)e^{−ω0{tilde over (t)}}  Eq. (55)
with F=({umlaut over (p)}^k+2ω₀({dot over (p)}^k−v∞))/(ω₀−ω₁)² and ω₀, ω₁ being design parameters related to the decay of the initial velocity and acceleration.

[0216] Fourth, the robot may be modeled as a unicycle with dynamic constraints. For example, as shown in FIG. 2, a robotic wheelchair may be modeled as a unicycle. The kinematic model of a differentially-driven agent (unicycle) was considered:
[ p,{dot over (δ)}]^T=[cos δ, sin δ,0]^Tv₁+[0,0,1]^Tv₂  Eq. (56)
with p being the point in between both tractor wheels, δ being the orientation of the vehicle, and v₁, v₂ being the forward and rotational speed of the platform, respectively. The relevant variables are shown in FIG. 2. For this non-linear system, p is not fully controllable. Any reference point, p, to the front of the vehicle axles is fully controllable and linearization via static feedback becomes feasible. Consider p= p+[cos δ, sin δ]^TΔ, where Δ>0 defines the maneuverability of the vehicle. The following relations then hold:
{dot over (p)}=ξ,v₁=ξ·[cos δ, sin δ]^T,v₂=[−sin δ, cos δ]^T/Δ  Eq. (57)
and the system is then controllable about p with holonomic velocity inputs given by ξ∈² and a local trajectory may be applied at this point. A circle with a radius, r, is considered as the enveloping shape of the vehicle. Equation (55) defines the local motion of the vehicle's holonomic point. Additional constraints can be added such as maximum linear and angular velocity (|v₁|≦v_1,max, |v₂|≦v_2,max) and maximum linear and angular accelerations (|{dot over (v)}₁|≦{dot over (v)}_1,max, |{dot over (v)}₂|≦{dot over (v)}_2,max).

[0217] Fifth, the vehicle may be modeled as an omnidirectional boat with dynamic constraints. The methods taught herein have also been applied to control a team of omnidirectional boats controlled by three rotating propellers. An LQR controller in position in velocity was employed to track the reference trajectory. Additional saturation limits were included that accurately model the boat dynamics.

[0218] In the control method, static obstacles are typically given by a grid map, O₀, of occupied space. The dilated obstacle map of the positions, p, where a disk, D(p,r), is in collision with a static obstacle is denoted by O_r. Its complement defines the positions for which the agent is not in collision. A constraint (“Constraint 2” in some of this description) can be stated regarding collision avoidance with static obstacles. For agent i≦m this constraint is given by the reference velocities such that the new positions are not in collision with the enlarged obstacle map, p_i+u_i{tilde over (t)} ∉ O_ri+∈i, for all {tilde over (t)}∈[0,τ].

[0219] The constraint can be kept in its original grid-based representation or convexified. Grid-based forms can be used due to the mapping between velocities and future positions. This provides a vehicle with full navigation capability including being able to move close to the static obstacles and being able to go inside small openings. Particularly, given a known grid map, O₀, the enlarged map O_ri+∈i can be precomputed for several values of ∈_i. With regard to convexification, for simplicity and lower computational cost, the static obstacles within a neighborhood of the robot may be approximated by a union of linear constraints forming a convex polygon. This is the case, of example, for a polygon where the closest side is taken as the linear constraint, but, in complex environments, this can be suboptimal. Depending on the algorithm employed either the grid-based representation or its convex approximation can be employed.

[0220] With regard to pair-wise collision avoidance, the following discussion teaches the constraint for inter-agent avoidance (or avoidance of dynamic obstacles), and it may be based on work on velocity obstacles, reciprocal velocity obstacles, and/or their convex counterpart. The constraint is computed for agents of radius {circumflex over (r)}_i=r_i+∈_i following the reference trajectory. First, the constraint is described in its relative velocity space formulation, which can be employed in a centralized optimization. Then, three different approaches to obtain a distributed version are described.

[0221] Regarding a velocity obstacle and relative velocity space, the inter-agent collision avoidance constraint can be obtained for the reference trajectory following or using work on velocity obstacles. The velocity obstacle is formed by the relative reference velocities that lead to a collision between the two agents within the given time horizon τ. This constraint (“Constraint 3” or a constraint to provide inter-agent collision avoidance) may be stated: for every pair of neighboring agents i≦m, j∈[i+1,n], the constraint is given be the candidate reference velocities such that ∥p_i−p_j+(u_i−u_j){tilde over (t)}∥≧{circumflex over (r)}_i+j={circumflex over (r)}_i+{circumflex over (r)}_j, for all {tilde over (t)}∈[0,τ]. This constraint can be rewritten as u_i−u_j ∉ VO_ij^τ=∪_{{tilde over (t)}=0}^τ((D(p_j,{circumflex over (r)})) ⊕ D(p_i,{circumflex over (r)}_i)/{tilde over (t)}), representing a truncated cone, which is only computed if the distance between the two agents is below a threshold (p_ij<K_d). Since this constraint is non-convex (free space is the complement of VO_ij^τ), it should be linearized to incorporate it into a convex optimization. Alternatively, for homogeneous teams of robots, the control obstacle (or C^N) can be employed as a substitute of the velocity obstacle. Then, C^N-CO considers all interacting agents or robots as linear systems with order N dynamics. After linearization of the C^N-CO, the method described herein proceeds analogously.

[0222] An approximation can be made using linear constraints. Particularly, the non-convex constraint ²\VO_ij^τ (inter-agent collision-free reference velocities) can be approximated by three linear constraints of the form n_ij^l·u_ij−b_ij^l, with l∈{1,2,3} and given by:

[0223] $\begin{array}{cc}\left[\begin{array}{c}\cos \left({\gamma }^{+}\right)\\ \sin \left({\gamma }^{+}\right)\end{array}\right]u_{\mathrm{ij}}\le 0,-\frac{p_{\mathrm{ij}}}{p_{\mathrm{ij}}}·u_{\mathrm{ij}}\le \frac{p_{\mathrm{ij}}-r_{i+j}}{\tau },\left[\begin{array}{c}\cos \left({\gamma }^{-}\right)\\ \sin ({\gamma }^{-}\end{array}\right]u_{\mathrm{ij}}\le 0& \mathrm{Eq}.\left(58\right)\end{array}$
where γ⁺=α+β, γ⁻=α−β, α=a tan 2(−p_ij) and β=a cos(r_i+j/p_ij). The first and last constraints represent avoidance to the right and to the left, respectively, and the middle constraint represents a head-on maneuver, which remains collision free up to {tilde over (t)}=τ.

[0224] Turning now to the selection of a linear constraint, ²\VO_ij^τ is linearized by selecting one of the three linear constraints. Sensible choices include the following: (1) fixed side for avoidance: if agents are moving towards each other (v_ij·p_ij<0) avoid on a predefined side, on the left (l=1) or on the right (l=3) and if the agents are not moving towards each other the constraint perpendicular to the apex of the cone (l=2) is selected to maximize maneuverability; (2) maximum constraint satisfaction for the current relative velocity,

[0225] $\begin{array}{c}\arg \min \\ l\end{array}\left(n_{\mathrm{ij}}^l·\left(v_i-v_j\right)-b_{\mathrm{ij}}^l\right):$
this selection maximizes the feasible area of the optimization when taking into account the motion continuity constraints; and (3) maximum constrain satisfaction for the preferred velocity,

[0226] $\begin{array}{c}\arg \min \\ l\end{array}\left(n_{\mathrm{ij}}^l·\left({\stackrel{\_}{u}}_i-{\stackrel{\_}{u}}_j\right)-b_{\mathrm{ij}}^l\right)$
if centralized and

[0227] $\begin{array}{c}\arg \min \\ l\end{array}\left(n_{\mathrm{ij}}^l·\left({\stackrel{\_}{u}}_i-v_j\right)-b_{\mathrm{ij}}^l\right)$
if distributed: this selection may provide faster progress towards the goal position but the optimization may become quickly infeasible if the agent greatly deviates from its preferred trajectory. Option 1 provides the best coordination results as it incorporates a social rule, Option 2 maximizes the feasible area of the optimization, and Option 3 provides (or may provide) faster convergence to the ideal trajectory. Without loss of generality, consider n_ij·u_ij≦b_ij the selected linearization of ²\VO_ij^τ, which can be included in a centralized convex optimization.

[0228] Pair-wise collision avoidance may also include avoidance of a velocity obstacle in the distributed case. The new reference velocity of the neighboring agent, u_j, is unknown. One of the following assumptions can be made regarding this velocity: (1) static agent: the assumption u_j=0 is made, and the constraint is then given by u_i∈²\VO_ij^τ and its linearization by n_ij·u_i≦b_ij, with this approach being prone to collisions in highly dynamic environments; and (2) constant velocity: the assumption u_j=v_j is made, and the constraint is then given by u_i∈²\VO_ij^τ ⊕ v_j and its linearization by n_ij·u_i≦b_ij+n_ij·v_j, with this approach typically not considering the case where the other agent is also decision-making.

[0229] Pair-wise collision avoidance may further include velocity obstacles (or a distributed reciprocal). In the following discussion, the assumption is made that all agents compute collision-free motions following identical algorithms. One attempt towards taking into account decision making can be considered the work on the reciprocal velocity obstacle, where the velocity obstacle was slightly translated and the assumption was made that both agents share the avoidance effort with a known weight. To avoid reciprocal dances, to obtain stronger guarantees, and to coin the problem as a convex optimization, the idea was later extended to optimal reciprocal collision avoidance (ORCA).

[0230] The goal with regard to the velocity obstacle is to obtain two sets F_i|j ⊂² and F_j|i ⊂² such that for every velocity u_i∈F_i|j and for every velocity u_j∈F_j|i the relative velocity is collision free, thus:

[0231] $\begin{array}{cc}\begin{array}{c}{u}_i\in {ℱ}_{ij}\\ u_j\in {ℱ}_{ji}\end{array}\}\Rightarrow u_i-u_j\in {ℝ}^2\text{}{\mathrm{VO}}_{\mathrm{ij}}^{\tau }& \mathrm{Eq}.\left(59\right)\end{array}$
equivalent to F_i|j ⊕ (−F_j|i) ⊂²\VO_ij^τ. Therefore, agent i can freely select a velocity within F_i|j and agent j can freely select a velocity within F_j|i.

[0232] An avoidance topology can be fixed by first linearizing the constraint ²\VO_ij^τ. In this case, the partition is reduced to obtaining a value b_i|j such that:

[0233] $\begin{array}{cc}\begin{array}{c}{n}_{\mathrm{ij}}·u_i\le b_{ij}\\ -n_{\mathrm{ij}}·u_j\le b_{\mathrm{ij}}-b_{ij}\end{array}\}\Rightarrow n_{\mathrm{ij}}·u_{\mathrm{ij}}\le b_{\mathrm{ij}}& \mathrm{Eq}.\left(60\right)\end{array}$
where the implication holds for any b_i|j∈. For reciprocal collision avoidance, b_i|j can be defined such that if the minimum required change in relative velocity is denoted by Δu_ij then the minimum required change in velocity for agent i is Δu_i=λ_i|jΔu_ij, with λ_i|j being a constant to indicate how the avoidance effort is shared between both agents. For agent j, it is then assumed Δu_j=−(1−λ_i|j)Δu_ij. The implication of Equation (60) then holds, and, for agent i, the linear constraint is computed from:

[0234] $\begin{array}{cc}{n}_{\mathrm{ij}}·u_{\mathrm{ij}}=n_{\mathrm{ij}}·\left(v_i-v_j+\Delta u_{\mathrm{ij}}\right)=n_{\mathrm{ij}}·\left(v_i-v_j+\frac{\Delta u_i}{{\lambda }_{ij}}\right)=n_{\mathrm{ij}}·\left(v_i-v_j+\left(u_i-v_i\right)/{\lambda }_{ij}\right)\le b_{\mathrm{ij}}& \mathrm{Eq}.\left(61\right)\end{array}$
which implies:
n_ij·u_i≦b_i|j=λ_i|jb_ij+n_ij·((1−λ_i|j)v_i+λ_i|jv_j)  Eq. (62)
Although decision making is encoded, the parameter λ_i|j defining the collaboration effort is fixed and known, typically considering a known partition between agents that equally cooperate, λ_i|j=0.5, and dynamic obstacles, λ_i|j=1. FIG. 27 illustrates with the functional block system or data flow diagram 2700 exemplary computation of collision constraints.

[0235] At this point in the description, it may be useful to provide more explanation of cooperative pair-wise collision avoidance. Reciprocal methods are a first step towards modeling inter-agent decision making, but they consider all agents equal or a known cooperation effort. Velocity predictions and external factors such as static obstacles were not taken into account. In contrast, the following provides a method to incorporate a velocity prediction or utility function over future velocities in the selection of the sets F_i|j and F_j|i of the velocity partition for distributed collision avoidance. This produces a truly cooperative partition that takes into account the state of each agent and can improve vehicle navigation. A cooperation measure is defined as a function of the utility functions φ_i(u_i) and φ_j(u_j) of both interacting agents and the partition is found such that the cooperation measure is maximized.

[0236] With regard to the cooperation measure, to bound the problem, one can consider two disks, D_i and D_j, that are centered at the current velocity of each agent and of known radius, such as τa_i^max representing the velocities that can be achieved within the time horizon. For each linearization option, l∈{1,2,3}, of the collision-free relative velocities ²\VO_ij^τ and for each value of b_i|j defining the partition, the cooperative measure Y(l,b_i|j) is given by: (1) Y(l) which can favor a certain avoidance topology, for example to encode preferences for avoidance to the right; and (2) Y(i,l,b_i|j) which is a function of the utility of the collision-free velocities within the partition F_i|j of agent i (analogously for agent j). One can then define each agent's term by:

[0237] $\begin{array}{cc}Y\left(i,l,b_{ij}\right)=\frac{{\int }_{u_i\in D_i\bigcap {ℱ}_{ij}}{\varphi }_i\left(u_i\right)d{u}_i}{{\int }_{u_i\in D_i}{\varphi }_i\left(u_i\right)d{u}_i}& \mathrm{Eq}.\left(63\right)\end{array}$
with F_i|j and F_j|i defined by Equation (60). Then, two alternative cooperation measures are proposed in the form of a joint value and a min-max value. The joint value is given by:
Y₁(l,b_i|j)=Y(l)+Y(i,l,b_i|j)+(j,l,b_i|j)  Eq. (64)
The min-max value is given by:
Y₂(l,b_i|j)=Y(l)+min(Y(i,l,b_i|j),Y(j,l,b_i|j))  Eq. (65)
Note that, for each agent and fixed l, Y(i,l,b_i|j) is a monotonically increasing/decreasing function of b_i|j taking values from 0 to 1.

[0238] With regard to an optimal partition and linearization, for a cooperative measure Y_r(l,b_i|j), the optimal partition and linearization are then given by:

[0239] $\begin{array}{cc}\left(l^{*},b_{ij}^{*}\right)={\arg }_{l,b_{ij}}^{\max }{\rm Y}\left(l,b_{ij}\right)& \mathrm{Eq}.\left(66\right)\end{array}$
obtained by first computing the value Y_r(l,b_i|j) for each l∈{1,2,3} and then selecting the maximum among the three. The reciprocal partition discussed earlier is obtained, for λ=0.5, by employing the cooperation measure:
Y_r(l,b_i|j)=max(n_ij^l·v_i−b_i|j−n_ij^l·v_i−(b_ij−b_i|j))  Eq. (67)

[0240] The following discussion teaches another exemplary algorithm or method for performing local motion planning (e.g., as may be implemented with the shared control system 100 of FIG. 1). In each control loop, an optimal reference velocity, u*_i, is computed by solving an optimization in candidate reference velocity space. The preferred velocity, ū_i, of the ego-agent is given, together with the current position and velocity of neighboring agents. The following describes a distributed optimization and a centralized optimization. The optimization cost, which is described in detail below, is denoted by C(u_i) and C(u_1:m).

[0241] For a first optimization technique (“Optimization 1” or distributed optimization), each agent, i, independently solves an optimization where its optimal reference velocity, u*_i, is computed as follows, with the position p_j and velocity v_j of neighboring agents being known:

[0242] $u_i^{*}={\arg }_{u_i}^{\min }C\left(u_i\right)$ $s.t.u_i\le u_{\max }$ $u_i\in R_i\mathrm{respects}\mathrm{motion}\mathrm{constraints}$ $u_i\mathrm{collision}\text{-}\mathrm{free}w.r.t.\mathrm{static}\mathrm{obstacles}$ $u_i\in {ℱ}_{i|j}\mathrm{collision}\text{-}\mathrm{free}w.r.t.\mathrm{neighboring}\mathrm{agents}$

[0243] For a second optimization technique or algorithm (“Optimization 2” or centralized optimization), a single optimization is solved where the optimal reference velocities of all vehicles, u*_1:m=[u*₁, . . . , u*_m], are jointly computed as follows:

[0244] $u_{1:m}^{*}=\arg \begin{array}{c}\min \\ u_{1:m}\end{array}C\left(u_{i:m}\right)$ $s.t.u_i\le u_{\max },\forall i\le m$ $u_i\in R_i\mathrm{respects}\mathrm{motion}\mathrm{constraints}\forall i\le m$ $u_i\mathrm{collision}\text{-}\mathrm{free}w.r.t\mathrm{static}\mathrm{obstacles}\forall i\le m$ $u_i\mathrm{collision}\text{-}\mathrm{free}w.r.t\mathrm{neighboring}\mathrm{agents}\forall i\le m$
Note that in the centralized optimization the partitions, F_i|j, for distributed avoidance are not required or utilized.

[0245] In the following, four algorithms are described to compute the optimal reference velocities. First, a distributed, convex optimization may be used. Optimization 1 is solved with quadratic cost C(u_i) and linearized constraints. Computational complexity is very low and scalability is very good, but the solution space is greatly reduced due to the linearization of the constraints and the partition of the reference velocity space. Second, a centralized, convex optimization may be used to compute the optimal reference velocities. In this case, Optimization 2 is solved with quadratic cost, C(u_i:m), and linearized constraints. Computation complexity is low and scalability is relatively good, but the solution space is partially reduced due to the linearization of the constraints. Third, a centralized, non-convex optimization may be utilized. With this algorithm, Optimization 2 is solved with quadratic cost, C(u_i:m), and non-convex constraints. Computational complexity is high and scalability is poor, but the complete solution space is explored. Fourth, a distributed, non-convex search may be performed within the convex region to compute the optimal reference velocities. With the use of this algorithm, Optimization 1 is solved with quadratic cost, C(u_i:m), or arbitrary utilities φ_i(u_i), linearized inter-agent collision avoidance constraints, non-convex motion constraints, and static obstacle avoidance constraints. Computational complexity is low and stability is good, but the solution space is reduced due to the linearization of the collision-avoidance constraints and the partition of the reference velocity space to guarantee safety in the distributed case.

[0246] These algorithms provide either collision-free reference inputs that respect all the dynamic constraints (u*_i if distributed or u*_1:m if centralized) or an infeasible flag. If the optimization is infeasible, the time horizon might be decreased or a different linearization might be selected. If the problem remains infeasible, agent i (distributed) or agents 1:m (centralized) decelerate on their last feasible local trajectory (obtained at time t_ƒ) following the time remap given by:
γ(t)=−t_ƒ²/(2τ)+(1+t_ƒ/τ)t−t²/(2τ),t∈[t_ƒ,t_ƒ+τ],γ(t)=t_ƒ+τ/2,t>t_ƒ+τ  Eq. (68)
The re-parameterization of the trajectories guarantees that the vehicles stop within the time horizon of the local planner, unless the optimization becomes feasible in a later time step.

[0247] Turning first to distributed convex optimization, the optimization cost, C(u_i) is defined by a quadratic function given by two terms. The first one is a regularizing term penalizing changes in velocity, and the second one minimizes the distance to the preferred velocity. The optimization cost may be determined as:
C(u_i):=K_o∥u_i−v_i∥²+∥u_i−ū_i∥²  Eq. (69)
where K_o is a design constant.

[0248] Typically, pedestrians prefer to maintain a constant velocity in order to minimize energy. Hence, preference was given to turning instead of changing speed. Furthermore, penalizing changes in speed leads to a reduction of deadlock situations. This idea can be formalized as an elliptical cost, higher in the direction parallel to ū_i and lower perpendicular to it. One can then let Λ<0 represent the relative weight and let γ_i represent the orientation of ū_i. Then, the relative weighting is defined by:

[0249] $L=\left[\begin{array}{cc}\Lambda & 0\\ 0& 1\end{array}\right]\mathrm{and}{D}_i=\left[\begin{array}{cc}\cos {\gamma }_i& \sin {\gamma }_i\\ -\sin {\gamma }_i& \cos {\gamma }_i\end{array}\right]$
And the optimization cost is redefined as:
C(u_i):=K_o∥u_i−v_i∥²+(u_i−ū_i)^TD_i^TLD_i(u_i−ū_i)  Eq. (70)
For each neighboring agent j, a collision-avoidance constraint is computed, linearized, and partitioned (u_i∈F_i|j).

[0250] The constraints for avoidance of static obstacles are also linearized, and the set, R_i, defining the motion constraints is approximated by a convex polygon (formed by a union of linear constraints). If R_i is given by two convex regions, only one is selected such as either the closest one to the previous reference velocity or the closest one to the preferred reference velocity. Note that at high speeds, R_i is formed by a single region. The set of all linear constraints may be denoted by C_lin. Then, Optimization 1 may be formulated as a quadratic program with linear constraints as:

[0251] $\begin{array}{cc}{u}_i^{*}=\arg \begin{array}{c}\min \\ u_i\end{array}C\left(u_i\right)s.t.u_i\le u_{\max }{n}_s·u_i\le b_s\forall s\in C_{\mathrm{lin}}& \mathrm{Eq}.\left(71\right)\end{array}$

[0252] Turning now to centralized convex optimization, the optimization cost, C(u_1:m), is defined by a quadratic function given by a weighted sum of the cost functions, C(u_i), of each agent as:
C(u_1:m):=Σ_i=1^mω_iC(u_i)  Eq. (72)
where ω_i represents the weight of each individual agent in the avoidance effort and can be used to define characteristics such as shy versus aggressive behavior.

[0253] For each neighboring pair of neighboring agents, i and j, a collision-avoidance constraint is computed and linearized (e.g., as described above). The set of linear constraints for inter-agent collision avoidance (one for each pair of neighboring agents) is denoted by C_rel. The constraints for avoidance of static obstacles are also linearized and the sets R_i are approximated by a convex polygon. The set of these linear constraints for agent i is denoted by C_i,abs. Then, Optimization 2 can be formulated as a quadratic program with linear constraints as:

[0254] $\begin{array}{cc}{u}_{1\text{:}m}^{*}={\arg }_{u_{1\text{:}m}}^{\min }C\left(u_{i\text{:}m}\right)s.t.u_i\le u_{\max },\forall i\le m{n}_{\mathrm{ij}}·\left(u_i-u_j\right)\le b_{\mathrm{ij}},\forall i,j\mathrm{neighboring}\mathrm{agents}{n}_s·u_i\le b_s,\forall i\in \left[1,m\right],\forall s\in {𝒞}_{i,\mathrm{abs}}& \mathrm{Eq}.73\end{array}$

[0255] Turning now to centralized non-convex optimization, to coin the problem as a convex optimization, the inter-agent collision avoidance constraints were linearized, which led to a sub-optimal solution due to the reduction of the solution space (an avoidance topology was fixed). To obtain the global optimum, a non-convex problem should be solved. The solution may be obtained via sampling, which may use a large amount of samples in this high dimensional space with many constraints. Instead, the inventors propose that one of the two following algorithms may be utilized to provide shared control of vehicles.

[0256] First, one may utilize a mixed integer quadratic program. The optimization that is formulated in this way uses a program where three binary variables are added for each inter-agent collision avoidance constraint and that specifies which one of the three linear constraints is active. Another two binary variables are added for each R_i set that needs to be approximated by two convex regions. One can then let b be the vector of all binary variables, where each binary variable, β, is zero if the constraint is active (to be satisfied) and one otherwise.

[0257] The optimization cost of Equation (72) is kept, with an optional modification. Since an avoidance topology is not defined beforehand, a side preference for the collision avoidance can now be added by appropriately selecting a penalty for the binary variables. The three binary variables associated with each VO constraint, s∈C_rel, are denoted by β_s^l, l∈[1,3]. For example, the rule “prefer to avoid on the right” can be added by penalizing β_s^l=1 with a weight w. The vector of linear cost with respect to the binary variables is then given by:

[0258] $f_{\beta }=\underset{\underset{3C_{\mathrm{rel}}}{︸}}{[w00\dots w00}\underset{\underset{2I_2}{︸}}{00\dots 00]},$
where I₂ is the set of agents whose set R_i is approximated by two convex regions, I₁=[1,m]\I₂, and |X| the cardinality of set X. The set of linear constraints for static obstacle avoidance is denoted by C_i,obst and the set of linear constraints defining the polygonal approximation of R_i by C_i,dyn, and the sets verify C_i,abs=C_i,obst ∪ C_i,dyn.

[0259] Optimization 2, in this case, can then be formulated as a quadratic program with linear constraints as:

[0260] $\begin{array}{cc}{u}_{1:m}^{*}=\arg \begin{array}{c}\min \\ u_{1:m}b\end{array}\left({\mathrm{Cu}}_{i:m}\right)+f_{\beta }·bs.t.u_i\le u_{\max },\forall i\in \left[1,m\right]n_s^l·\left(u_i-u_j\right)-{\beta }_s^{l}M\le b_s^l,\forall s\in {𝒞}_{\mathrm{rel}},l\in \left[1,3\right]{\beta }_s^l+{\beta }_s^2+{\beta }_s^3=2,\forall s\in {𝒞}_{\mathrm{rel}}{n}_s·u_i\le b_s,\forall i\in \left[1,m\right],\forall s\in {𝒞}_{i,\mathrm{obst}}{n}_s·u_i\le b_s,\forall i\in I_1,\forall s\in {𝒞}_{i,\mathrm{dyn}}{n}_s·u_i-{\beta }_s^{l}M\le b_s,\forall i\in I_2,\forall s\in {𝒞}_{i,\mathrm{dyn}},l\in \left[1,2\right]{\beta }_s^1+{\beta }_s^2=1,\forall i\in I_2,\forall s\in {𝒞}_{i,\mathrm{dyn}}& \mathrm{Eq}.\left(74\right)\end{array}$
where M>>0 is a large enough constant and extra constraints have been added to impose that only one linear constraint is active for each original non-convex constraint. This optimization can be solved via branch-and bound, where the maximum number of explored nodes defines the ratio between optimality and computational time.

[0261] To reduce the computational time of the algorithm, an initial point may be specified. The solution of the convex optimization discussed herein may be well suited for this purpose since it typically returns a feasible (although not optimal) solution with very low computational time. Optimality of the solution increases with time as more nodes are explored. The algorithm can exit at any time to cope with run-time constraints, and, hence, this method features anytime properties.

[0262] Second, centralized, non-convex optimization may be formulated as message passing optimization. For example, this non-convex optimization can be solved using an improved version of the alternating direction method of multipliers (ADMM) algorithm known to those skilled in the art. This approach was naturally parallelizable and allowed for incorporating different cost functional with only minor adjustments, and binary constraints were no longer required and/or used. None the less, convergence to the global optimum was not guaranteed.

[0263] Turning now to distributed non-convex search within a convex region, convex (linear) and non-convex (grid-based) constraints are combined to explore a larger solution space while keeping the computational cost low. A convex cost, C(u_i), is given by Eq. 70. Additionally, a non-convex cost can be defined, for example, by {tilde over (C)}(u_i)=C(u_i)+χ_i(u_i). The set of convex constraints is formed by the inter-agent linearized collision avoidance constraints (i.e., set F_i|j), and the bounding box of the set R_i is denoted by C. For agent i, the set of non-convex constraints is formed by the static obstacle collision avoidance constraint and the set R_i accounting for the motion constraints and is denoted by {tilde over (C)}_i. Both non-convex constraints are given by grid representations of identical resolution.

[0264] To efficiently find a solution, the optimization is divided in two parts. First, a convex subproblem is solved resulting in u_i^c, and, second, this is followed by a search within the grid-based constraints restricted to the convex area defined by the linear constraints. The quadratic cost defined in Equation (70) is maintained, and the algorithm proceeds as follows:

[0265]

Input distributed: z1, ū1 and r1 + εi; pj, {dot over (p)}j and rj ∀j neighbor of 1. Consider
εj = ε1.
Result: ui*
Compute all constraints 1, 2 and 3
uic ← solution 2-dimensional convex optimization with quadratic cost
Eq. (70) and convex constraints C.
||Wave expansion from uic within convex area C.
Initialize sorted list L (increasing cost {tilde over (C)} (ui)) with uic.
while L ≠ Ø do
ui ← first point in L; L := L\ui; feasible := ‘true’;
if feasibleDynamics(ui)= ‘false’ or feasibleMap(ui)= ‘false’
then
L ←expandNeighbors(L, ui);
feasible := ‘false’;
end if
if feasible = ‘true’ then return ui* = ui end if.
end while;
return 0;