# Coordinated Target Assignment and Intercept for

## Comments

## Transcription

Coordinated Target Assignment and Intercept for

Proceedings of the 2002 IEEE International Conference on Robotics & Automation Washington, DC • May 2002 Coordinated Target Assignment and Intercept for Unmanned Air Vehicles Randal W. Beard∗ Timothy W. McLain† Abstract is feasible for each UAV on the team. Third, for each UAV with velocity constraints v ∈ [Vmin , Vmax ], determine a path (specified via waypoints), such that if the UAV were to fly along straight-line paths, it could complete the path in the specified TOT, while satisfying the given velocity constraints. Fourth, transform each waypoint path into a feasible trajectory for the UAV. By feasible we mean that the turning rate constraints, and the velocity constraints are not violated along the trajectory. Also, the trajectory should have the same TOT as the path specified by waypoints. In this paper we offer a complete solution to this problem. Our solution is derived via a systematic hierarchical approach to multiple vehicle systems. Many of the subproblems outlined above have previously been addressed in the literature. The assignment problem is a well known optimization problem and is similar to problems addressed in [1]. Our approach is based on satisficing decision theory [2]. The simultaneous rendezvous problem has been addressed in [3, 4]. A similar approach for simultaneous target intercept will be used in this paper. The problem of planning waypoint paths through a cluttered environment has been addressed in a number of works including [5, 6, 7]. The trajectory generation problem has been addressed in [8, 9, 10] to mention just a few. In this paper we will derive a novel trajectory generator that allows realtime generation of the trajectories such that the pathlength of the trajectory is equal to the corresponding Voronoi path. This paper presents an end-to-end solution to the battlefield scenario where M unmanned air vehicles are assigned to strike N known targets, in the presence of dynamic threats. The problem is decomposed into the subproblems of (1) cooperative target assignment, (2) coordinated UAV intercept, (3) path planning, and (4) feasible trajectory generation. The design technique is based on a hierarchical approach to coordinated control. Detailed simulation results are presented. 1 Introduction Consider a battlefield scenario where a group of M unmanned air vehicles (UAVs) are required to drop munitions on N known target locations. In addition, on the battlefield there are a number of known antiaircraft facilities. Suppose that each of the targets is equipped with anti-aircraft facilities with radars that detect the location of attacking aircraft, but that these facilities are only capable of firing on one aircraft at a time. Therefore to maximize the probability of success, it is desirable to have multiple UAVs arrive on the boundary of each target’s radar detection region simultaneously. The rationale is that simultaneous arrival of multiple UAVs on the detection boundary will confuse the software or operators of the anti-aircraft facility, enabling a non-empty subset of the attacking UAVs to drop their munitions. The problem can therefore be decomposed into several sub-problems. First, given M UAVs with N targets, assign each vehicle to a target, such that each target has, if possible, multiple UAVs assigned to it, and such that preference is given to high priority targets. Second, for each team of UAVs assigned to a single target determine an estimated time-over-target (TOT) that ensures simultaneous intercept and that 2 System Architecture A detailed schematic of the system architecture for a single UAV is shown in Figure 1. At the lowest level of the architecture is the controlled UAV which we assume to be equipped with trajectory tracking controllers. In Figure 1, the target manager, path planner, and intercept manager work together to generate waypoint paths for the UAV. The path planner generates k best paths from the specified UAV to a specific target. In addition, the path planner returns information about each path, namely the estimated fuel expenditure and the estimated threat exposure. Both the target manager and the intercept manager make calls to the path ∗ Assistant Professor, Electrical and Computer Engineering, BYU, Provo, Utah 84602, [email protected], corresponding author. † Associate Professor, Mechanical Engineering, BYU, Provo, Utah 84602, [email protected] ‡ Assistant Professor, Computer Science, BYU, Provo, Utah 84602, [email protected] 0-7803-7272-7/02/$17.00 © 2002 IEEE Michael Goodrich‡ 2581 1 Communication Manager Target Manager Target Target & Possible paths Team Intercept Manager from the initial vehicle location to the target location are derived from a k-best paths graph search [7] of a Voronoi diagram that is constructed from the known threat, vehicle, and target locations [12]. Figure 2 shows a Voronoi diagram created for a set of specified threats, UAV location, and target location. Best path Path Planner 30 target 20 Target 3 3 10 Way−points Flag Trajectory Generator Desired Trajectory Trajectory Controlled UAV X (North) 1,2,4,5 0 1,2,4,5 1,2,4 2,3 −10 5 1,5 4 −20 start Figure 1: System Architecture for a Single UAV. −30 −10 10 20 Y (East) 30 40 50 Figure 2: Threat-based Voronoi diagram. The threats are represented by dots. planner. The role of the target manager is to assign each UAV a target. The role of the intercept manager is to ensure that when multiple UAVs are assigned to the same target by the target manager, that they arrive on the radar detection boundary of the target simultaneously. The design of the path planner, target manager, and intercept manager, are discussed in Section 2.1, Section 2.2, and Section 2.3 respectively. The trajectory generator in Figure 1 receives a set of waypoints, which specify the desired path of the UAV. The waypoints received by the trajectory generator are not parameterized in time, they are simply a set of X, Y , and h coordinates. The role of the trajectory generator is to generate a time trajectory for the UAV that is feasible within its dynamic constraints. The design of the trajectory generator is discussed in Section 2.4. The communication manager shown in Figure 1 facilitates communication between different UAVs. Each UAV implements a separate target manager, intercept manager, and path planner. Therefore the decisions reached by these functional blocks must be synchronized among the different UAVs. The primary role of the communication manager is to ensure this synchronization. The design of communication managers for multiple cooperating autonomous vehicles has been discussed in [11]. 2.1 0 Each edge of the Voronoi diagram is assigned two costs: a threat cost, specified as Ã ! N X 1 1 1 + 4 + 4 Jthreat,i = Li , d41/6,i,j d1/2,i,j d5/6,i,j j=1 (1) where N is the total number of threats and d1/2,i,j is the distance from the 1/2 point on the ith edge to the jth threat, and a length cost, specified as Jlength,i = Li . The total cost for traveling along an edge comes from a weighted sum of the threat and length costs: Ji = kJlength,i + (1 − k)Jthreat,i , (2) where k ∈ [0, 1]. With the cost determined for each of the Voronoi edges, the Voronoi diagram is searched to find the set of lowest-cost candidate paths using a variation of Eppstein’s k-best paths algorithm [7]. In Figure 2, the five best paths to the target are shown. A great advantage of the Voronoi diagram approach is that it reduces the path-planning problem from an infinite-dimensional search, to a finite-dimensional graph search. This important abstraction makes the path-planning problem feasible in near-real-time. Path Planner 2.2 The Path Planner shown in Figure 1 is used by both the Target Manager and the Intercept Manager. The output of the Path Planner is a set of waypoints and a commanded velocity for each vehicle. Paths Target Manager Assume that there are M targets, N UAVs, and P threats. The task is to assign each vehicle to a target 2582 2 such that the overall group cost is mitigated, while maximizing the number of targets that are destroyed. In assigning targets to UAVs, there are four objectives. First, minimize the group path length to the targets, second, minimize group threat exposure, third, maximize the number of vehicles attacking each target (to maximize survivability), and fourth, maximize the number of targets attacked. We refer to these objectives as the ShortPath, AvoidThreats, MaxForce, and MaxSpread objectives respectively. The ShortPath and AvoidThreats objectives are myopic objectives, whereas the MaxForce and MaxSpread objectives are team objectives. The proposed solution methodology is based on the satisficing paradigm [2] which is used to select a set of potential targets that are acceptable for each UAV. The set of targets that are selected are then used by each agent to negotiate an acceptable group assignment. The first step is to use the path planner described in the previous section to generate k best paths to each target. Associated with these k best paths is a median length cost, denoted as J¯length (Vi , Tj ) and a median threat exposure cost denoted as J¯threat (Vi , Tj ), where Vi is the ith UAV and Tj is the jth target. The median operator throws away targets that have only one good path but keeps targets that have at least k/2 reasonable paths. For an individual UAV, close targets are deemed acceptable, while targets with large threat exposure are deemed rejectable. Normalized measures of acceptability and rejectability can be defined as given by SVi = {Tj : µAi (Tj ) ≥ bi µRi (Tj )}, where bi is a selectivity index and can be used to ensure that the cardinality of SVi is greater than zero, but small enough to enable an efficient search. There are two competing objectives that affect how the team behavior is evaluated: the MaxForce objective prefers assigning every vehicle to the same target, and the MaxSpread objective prefers attacking as many targets as possible. These may be competing objectives. We will address each in turn. From the MaxForce perspective, the more vehicles assigned to a target the better. We can represent this objective by assigning a value to the team size for each assigned target. In other words, we can identify a number that encodes how much better a larger team is than a smaller team. We use a monotonically increasing function that increases dramatically between one and three team members, indicating that a minimally acceptable team consists of two members. The monotonic function we selected is the sigmoid UMaxForce (Gj ) = µRi (Tj ) = 1 , 1 + e−3(m(Gj )−2) where m(Gj ) = |{Vi : Vi → Gj }| ´ ´ ³ ³ maxTk J¯length (Vi , Tk ) − J¯length (Vi , Tj ) ´, ³ ´ ³ µAi (Tj ) = maxTk J¯length (Vi , Tk ) − minTk J¯length (Vi , Tk ) ³ (5) (3) ´ J¯threat (Vi , Tk ) − minTk J¯threat (Vi , Tj ) ³ ´. ³ ´ maxTk J¯threat (Vi , Tk ) − minTk J¯threat (Vi , Tk ) (4) The acceptability function µA assigns the closest target the highest measure of acceptability (equal to one), the most distant target the lowest measure of acceptability (equal to zero), and all other targets some assignment in the range [0, 1]. The rejectability function µR assigns the target with the most threat exposure the highest measure of rejectability (equal to one), the target with the least threat exposure the lowest measure of rejectability (equal to zero), and all other targets some assignment in the range [0, 1]. Notice that both of these functions can be computed in order kN M time. For each vehicle, we can identify the set of assignments that are individually satisficing. This set is 2583 3 is the number of vehicles assigned to target Gj . We can use the value of the team size, encoded in UMaxForce (Gj ), to estimate the value of an assignment. We encode the value of the assignment as Y JMaxForce = UMaxForce (Gj ). (6) Gj ∈{Gj :m(Gj )>0} This value takes the product of each team-size value over the set of assigned targets (the set of targets that have at least one vehicle attacking it). Using the product has a double effect. First, it lets targets with small team sizes dramatically reduce the value of the overall assignment. Second, it allows the preference for large team sizes to be tempered by the preference for having multiple targets attacked; i.e., it accounts for the MaxSpread objective. Since it is unlikely that all agents can feasibly attack the same target (it is unlikely that such an assignment is individually satisficing), the product avoids unbalanced team sizes (with, for example, five agents assigned to one target and only one to another target) from receiving high preference. Thus, multiple nearly equal-sized teams are preferred to multiple, differently-sized teams. For a scenario with five UAV’s and six targets, Figure 3 shows the targets assigned to each UAV and its associated path as a solid line. In addition, the satisficing paths for the UAV in the bottom left hand corner are plotted as dotted lines. Note the preference for nearby targets and targets that do not require passing too close to threats. 30 Resource Allocation Manager team and target assignments mission status 20 Cooperative Path Manager 2 subject to TOT∗ ∈ ∩ S STOT,1 X (North) km 10 n Choose TOT∗ to minimize Σ Ji ^ TOT∗ i=1 TOT,i ^ J1(TOT) UAV n 0 UAV 2 UAV 1 1 −10 3 Determine feasible TOT range: STOT,1 ^ Calculate coord. function: J1(TOT) J1 = f(ξ1,V1) −20 −30 −10 ^ J1 = h(TOT) TOT = g(ξ1,V1) 0 10 20 Y (East) km 30 40 Plan path (determine ξ1,V1) to match TOT∗ that minimizes J1(ξ1,V1) path description (to trajectory generator) ξ1,V1 50 Figure 4: Cooperative Path Planning Algorithm Figure 3: Target assignment for case study scenario. Solid lines connect each vehicle to its assigned target. The dotted lines represent satisficing paths for the UAV in the bottom left corner. 2.3 ordination function, is utilized. In this problem, the coordination function Jˆi models the cost to UAV i of achieving a particular TOT: Jˆi = h(T OT ). The coordination function is determined from the relations for T OTi and Ji . Based on the candidate paths determined from the Voronoi diagram and the feasible range of UAV air speeds, the feasible T OT range for each UAV on the team can also be determined. The set of feasible T OT ranges for UAV i is denoted by ST OT,i . Coordination function information for two UAVs composing a team is shown in Figure 5. Note that each candidate path produces a line segment in the plot. Intercept Manager Once teams have been formed and targets assigned to each team, it is necessary to plan the trajectories that each vehicle will take to its respective target. In military applications, such as the one presented here, the ability to prescribe and coordinate the timing of target interception is critically important. In the architecture developed here, the vehicles composing a team are required to strike their target simultaneously to maximize the element of surprise and enhance mission effectiveness. The challenge becomes one of determining the best time over target (TOT) specification for the team in light of the threat scenario and dynamic capabilities of the vehicles. The output of the intercept manager is a set of waypoints and a commanded velocity for each vehicle. The waypoints specified avoid the threats and allow for simultaneous arrival of the vehicles at the assigned target. The strategy for cooperatively planning UAV paths employed here is presented in detail in [4]. Figure 4 depicts graphically the steps of the cooperative path planning algorithm. Candidate paths to the target for UAV i are parameterized by the waypoints ξi and the UAV forward speed Vi . Time over target for each UAV is a function of the speed of the UAV along the selected path: T OTi = f (ξi , Vi ). Similarly, the cost (threat and fuel) to travel along any path to the target is a function of the path and the speed: Ji = g(ξi , Vi ). To choose the best TOT for the team in a cooperative manner, some information regarding the candidate paths for the UAVs must be exchanged. Rather than passing ξi and Vi around among all of the UAVs, a more efficient parameterization of information, called the co- 0.6 0.5 path cost 0.4 0.3 0.2 UAV 3 0.1 UAV 4 0 100 150 200 250 300 350 400 450 500 TOT (sec) Figure 5: Coordination Functions for Two UAVs Coordination function and feasible T OT range information is exchanged among vehicles to enable cooperative path planning. At the team level, the cooperative planning problem is simplified to finding T OT ∗ that minimizes the collective threat exposure of the 2584 4 team T OT ∗ = arg min n X Jˆi local reachability region i=1 subject to T OT ∗ ∈ ∩ST OT,i . From Figure 5 it can be seen that this optimization results in a team-optimal value of T OT ∗ = 160.5 sec. Time over target T OT is called the coordination variable. By requiring individual UAVs to match the team-optimal value of the coordination variable T OT ∗ , cooperation is ensured. We note that this optimization problem has been greatly simplified by searching over the coordination variable instead of the original path parameters (ξi , Vi ) of the system. For example, in the coordinated strike problem, the coordination variable is one-dimensional (T OT ), whereas the original path parameterization is of much higher dimension. Once T OT ∗ has been determined, each UAV must determine which of the candidate paths ξi to take and the appropriate flight speed Vi so that its own threat exposure Ji = f (ξi , Vi ) is minimized and the T OT ∗ value is matched. In implementation, the best path for a specified T OT was determined when the coordination function was computed. At this point, it is simply a matter of looking up the trajectory waypoints in memory and computing the appropriate velocity. Waypoint and velocity information is then supplied to the trajectory generator on the UAV. 2.4 u_1 = − c u_1 = + c x y Figure 6: Local reachability region of the trajectory generator. Ri = Vid /c. Note that as the desired velocity increases the minimum turning radius increases. Consider the problem of switching from one straight-line segment to another in minimum time at a constant velocity. If the trajectory is not constrained to pass through the waypoint connecting the two lines, then a time-optimal trajectory is shown as a dashed line in Figure 7. Trajectory Generator Given a set of waypoints and the desired velocity Vid , the task of the Trajectory Generator (TG) is to generate time-parameterized trajectories that are feasible within the dynamic constraints of the vehicle. The essential idea is to use a nonlinear filter that has a mathematical structure similar to the kinematics of the vehicle to generate trajectories that smooth through the waypoints in real-time, while the vehicle traverses the trajectory. The UAV dynamics given in [13] suggest the following structure for the Trajectory Generator for constant altitude maneuvers: waypoint u_1 = −c u_1 = + c u_1 = −c u_1 = + c Ẋid = Vid cos ψid Ẏid = Vid sin ψid ψ̇id = u1 ḣdi = 0, (7) Figure 7: Time-optimal transitions between path segments. The dashed line is not constrained to go through the waypoint. The solid line is constrained to pass through the waypoint. where u1 is constrained by the dynamic capability of the UAVs, namely the heading rate constraint −c ≤ u1 ≤ c. Note that if u1 = c, then the trajectory generator given in Equation (7) traces out a righthanded circle as shown in Figure 6. Similarly if u1 = −c then the TG traces out a left-handed circle. As shown in Figure 6, the local reachability region of the TG is bounded by these two circles. The radius of the circles defining the local reachability region is given by If (Xid , Yid , ψid , Vid ) is the configuration of the TG, then the center of the two circles that bound the reach- 2585 5 ability region are given by zright = µ Xid Yid ¶ zleft = µ Xid Yid ¶ + Ã + Ã Vid d c sin(ψi ) Vid d c cos(ψi ) − made, the UAVs composing each team must coordinate to find a feasible team time-over-target (T OT ). The target assignment problem was solved using the satisficing and social welfare paradigms. The T OT coordination problem was solved by encapsulating the essential myopic information in coordination functions that were then used to optimize the team T OT . The path planning problem was solved via a Voronoi diagram and Eppstein’s k-best paths algorithm. The trajectory generation problem was solved via a novel real-time nonlinear filter that explicitly accounts for the dynamic constraints of the vehicle. ! Vid d c sin(ψi ) d V − ci cos(ψid ) ! Since the boundaries of the reachability region is given by circles with known center and radius, finding the intersection of the reachability region with lines and circles can be done in a computationally efficient way. A simple, real-time algorithm for trajectory generation that either guarantees minimum-time transition between paths, or minimum-time transitions that are constrained to go through the waypoint, is therefore suggested by Figure 7, and is analyzed in detail in [14]. One of the disadvantages of both minimum time transitions and transitions constrained to go through the waypoint, is that the trajectories will have a different path length than the original Voronoi path. Since the Voronoi path is used for determining intercept times, we would like to transition between path segments, such that the path length of the feasible trajectory is equal to the path length of the original Voronoi path. From Figure 7 it is obvious that the path length of a minimum time trajectory will be less than the path length of the Voronoi path, and that the path length of the constrained trajectory will be greater than the path length of the Voronoi path. Let κ ∈ [0, 1] parameterize the location of the center of the turning radius circle along the bisector of the angle shown in Figure 7. The algorithm suggested by Figure 7 can be modified with a preliminary step to find κ such that the path lengths are equal [14]. The above algorithms are easily modified to account for short path segments that are smaller than the radius Ri . To account for those cases, the trajectory generator must look ahead several path segments. There are several advantages to our approach. First, it couples nicely with the Voronoi search algorithms described in the previous section. Second, the approach has low computational overhead. In fact, trajectories are generated in real-time, as the vehicles move. Finally, it can be shown using optimal control techniques that the approach minimizes the time that the vehicle deviates from the Voronoi path. Acknowledgments This work was partially funded by AFOSR award no. F49620-01-1-0091. The technical guidance of Dr. Phil Chandler at AFRL is gratefully acknowledged. The authors would like to thank Mr. Erik Anderson for assistance with developing the trajectory generator and Mr. Kevin Judd for assistance with developing the path planner. References [1] R. E. Burkard, “Selected topics in assignment problems,” tech. rep., Institute of Mathematics B, Technical University Graz, Movember 1999. To appear in Discrete Applied Mathematics. [2] M. A. Goodrich, “A satisficing approach to assigning vehicles to targets,” in Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, 2001. (to appear). [3] T. McLain and R. Beard, “Cooperative rendezvous of multiple unmanned air vehicles,” in Proceedings of the AIAA Guidance, Navigation and Control Conference, (Denver, CO), August 2000. Paper no. AIAA-2000-4369. [4] T. McLain, P. Chandler, S. Rasmussen, and M. Pachter, “Cooperative control of UAV rendezvous,” in Proc. of the ACC, June 2001. [5] F. Gentili and F. Martinelli, “Robot group formations: a dynamic programming approach for a shortest path computation,” in Proceedings of the IEEE International Conference on Robotics and Automation, (San Francisco), pp. 3152–3157, April 2000. [6] P. Chandler, S. Rasumussen, and M. Pachter, “UAV cooperative path planning,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, (Denver, CO), August 2000. AIAA Paper No. AIAA-2000-4370. [7] D. Eppstein, “Finding the k shortest paths,” SIAM Journal of Computing, vol. 28, no. 2, pp. 652–673, 1999. [8] M. J. Van Nieuwstadt and R. M. Murray, “Real time trajectory generation for differentially flat systems,” International Journal of Robust and Nonlinear Control, vol. 8, no. 11, pp. 995–1020, 1998. [9] M. B. Milam, K. Mushambi, and R. M. Murray, “A computational approach to real-time trajectory generation for constrained mechanical systems,” in IEEE Conference on Decision and Control, (Sydney, Australia), 2000. [10] N. Faiz, S. K. Agrawal, and R. M. Murray, “Trajectory planning of differentially flat systems with dynamics and inequalities,” AIAA Journal of Guidance, Control and Dynamics, vol. 24, pp. 219–227, March–April 2001. [11] F. Giulietti, L. Pollini, and M. Innocenti, “Autonomous formation flight,” IEEE Control Systems Magazine, vol. 20, pp. 34–44, December 2000. 3 [12] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications. Springer-Verlag, 2000. Conclusions and Discussion [13] A. W. Proud, M. Pachter, and J. J. D’Azzo, “Close formation flight control,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, (Portland, OR), pp. 1231–1246, American Institute of Aeronautics and Astronautics, August 1999. Paper no. AIAA-99-4207. In this paper we have developed a complete system architecture for the target assignment and coordinated intercept problem. Coordination between UAVs take place at two levels. At the highest level, the UAVs must negotiate a target assignment vector that assigns UAVs to targets. Once a target assignment has been [14] R. W. B. Erik P. Anderson, “Constrained extremal trajectories and uav path planning,” in Proceedings of the IEEE International Conference on Robotics and Automation, (Washington D.C.), May 2002. (submitted). 2586 6