Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Optimal Evading Strategies and Task Allocation in Multi-Pursuer Single-Evader Problems Venkata Ramana Makkapati and Panagiotis Tsiotras Dynamics and Control Systems Laboratory Department of Aerospace Engineering Georgia Institute of Technology 12 th July, 2018 12 th July, 2018 Ramana ISDG 2018 1 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Outline Motivation and problem statement Optimal evading strategies Active/Redundant pursuers Simulations 12 th July, 2018 Ramana ISDG 2018 2 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Motivation Airspace security Regulate the traffic and usage of UAVs Figure 1: DroneHunter a a https://fortemtech.com/ 12 th July, 2018 Ramana ISDG 2018 3 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations A scenario Assume: Territory n agents (pursuers) E m guarding a territory P n P 1 m adversaries (evaders, ... ... typically m ≤ n ) E 1 P 2 P 5 Pursuers want to capture the evaders E 2 P 3 Pursuers are faster than P 4 the evaders 12 th July, 2018 Ramana ISDG 2018 4 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Some questions! Relevant Questions: What is the shortest time-to-capture, while evaders will try to postpone capture indefinitely? Which pursuer(s) should go after which evader(s)? A multi-pursuer multi-evader game! 12 th July, 2018 Ramana ISDG 2018 5 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Approach Divide and Conquer Territory E m Solve m multi-pursuer P n P 1 single-evader games ... ... E 1 Pursuers follow simple P 2 P 5 navigation laws: Pure E 2 Pursuit (PP) or Constant P 3 P 4 Bearing (CB) staregies 12 th July, 2018 Ramana ISDG 2018 6 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Problem Set Up i 2 i 2 Inertial Inertial Frame Frame P n P n Q i ... P 1 ... P 1 Collision triangle for P i θ Ε θ Ε E E θ i r i ϕ i ϕ i r i P i P i P 2 P 2 ... ... i 1 i 1 O O (a) CB (b) PP Figure 2: Schematics of the proposed pursuit-evasion problems. Identical pursuers, pursuers faster than evader. 12 th July, 2018 Ramana ISDG 2018 7 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Regions of Non-Degeneracy 1 (a) CB (b) PP Figure 3: Regions of non-degeneracy 1 Makkapati et al., Pursuit-Evasion Problems Involving Two Pursuers and One Evader , AIAA GNC Conference, Kissimmee, FL, 2018 12 th July, 2018 Ramana ISDG 2018 8 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Two Pursuers - CB (Previous work) (a) A degenerate case (b) A non-degenerate case Figure 4: Trajectories of the players for optimal control inputs in Scenario 1: black - evader, blue - P 1 , red - P 2 . 12 th July, 2018 Ramana ISDG 2018 9 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Two Pursuers - PP (Previous work) Optimal Suboptimal Optimal Suboptimal (a) Trajectories (b) Time variation of difference in relative distances Figure 5: Performance of the optimal and suboptimal strategies for a non-degenerate case in Scenario 2: black - evader, blue - P 1 , red - P 2 . 12 th July, 2018 Ramana ISDG 2018 10 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Optimal Evading Strategies In both CB and PP cases: Proposition The time-optimal evading strategy is dependent only on the initial positions of those pursuers that (simultaneously) capture the evader . Let’s call them the “influential” pursuers! 12 th July, 2018 Ramana ISDG 2018 11 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Some Issues In both cases No analytical expression for the optimal strategy of the evader Hard to identify the influential pursuers - no theoretical backing! 12 th July, 2018 Ramana ISDG 2018 12 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations What If? The pursuers don’t know the evader’s strategy 12 th July, 2018 Ramana ISDG 2018 13 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Capturing Pursuer Set Definition Given the initial positions of the players (at t = 0) in an MPSE problem and assuming that the pursuers follow either a CB or a PP strategy, for a given evading strategy, the capturing pursuer set P is the set of pursuers that are in the capture zone of the evader at the time of capture ( t c ) . 12 th July, 2018 Ramana ISDG 2018 14 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Active/Redundant Pursuers At time 0 ≤ t < t c Definition If there exists an evading strategy for which pursuer P i is in P , then P i is an active pursuer . Definition If there exists no evading strategy for which pursuer P i is in P , then P i is a redundant pursuer 12 th July, 2018 Ramana ISDG 2018 15 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Apollonius Curves 1 0.5 0 -0.5 -1 0 0.5 1 1.5 2 2.5 Figure 6: The locus of capture points for a non-maneuvering evader in the cases CB and PP. Simulation parameters: u = 1, v = 0 . 6. 12 th July, 2018 Ramana ISDG 2018 16 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Apollonius Boundary 5 5 0 0 -5 -5 -5 0 5 -5 0 5 (a) CB (b) PP Figure 7: Apollonius boundaries for CB and PP cases (Simulation parameters: u = 1, v = 0 . 6) 12 th July, 2018 Ramana ISDG 2018 17 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations A Formal Definition Definition The Apollonius boundary is the set of points � n n � � � B = { X ∈ A i | M ( E , X ) ∩ A i = { X }} , where i =1 i =1 M ( E , X ) denotes the set of points on the line segment with endpoints E (position of the evader) and X . 12 th July, 2018 Ramana ISDG 2018 18 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations A Conjecture for the CB case Conjecture Pursuer P i is active if B ∩ A i � = ∅ , and is redundant otherwise. 5 0 -5 -5 0 5 12 th July, 2018 Ramana ISDG 2018 19 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Lemma 1 Pursuer P i is the only active pursuer if and only if � n � � A i ∩ A j = ∅ , (1) j =1 , j � = i � n � � M ( E , T i ) ∩ A j = ∅ , (2) j =1 , j � = i T i is the closest point to the evader on the Apollonius circle A i 12 th July, 2018 Ramana ISDG 2018 20 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Lemma 2 Assumption: A i intersects one or more of the other Apollonius circles. P i is an active pursuer if and only if there exists at least one X ∈ I i such that: � n � � M ( E , X ) ∩ A j = { X } , (3) j =1 I i is the set of intersections points between A i and the rest of the Apollonius circles. 12 th July, 2018 Ramana ISDG 2018 21 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Algorithm to Identify Pursuer Status 12 th July, 2018 Ramana ISDG 2018 22 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Simulations 20 20 15 15 10 10 5 5 0 0 0 5 10 15 20 0 5 10 15 20 (a) Initial Apollonius circles (b) Trajectories Figure 8: CB case 12 th July, 2018 Ramana ISDG 2018 23 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Simulations 20 20 15 15 10 10 5 5 0 0 0 5 10 15 20 0 5 10 15 20 (a) Initial Apollonius curves (b) Trajectories (refined) Figure 9: PP case 12 th July, 2018 Ramana ISDG 2018 24 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations An extension to multi-evaders case 1 Find the set of all active pursuers for each evader 2 Check if each active pursuer is assigned to a single evader 3 Break the tie by assigning the closest evader 4 Obtain the set of unassigned pursuers 5 Add the unassigned pursuers to the current assignment, and recheck active pursuers 6 Repeat steps (3)-(5) until (2) is satisfied 12 th July, 2018 Ramana ISDG 2018 25 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Simulations 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 (a) t = 0 (b) t = 1.3 (c) t = 2.5 Figure 10: CB case 12 th July, 2018 Ramana ISDG 2018 26 / 27
Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations Future Work Estimate when the assignment can change to avoid unnecessary calculations Account for turn-radius constraints on the players 12 th July, 2018 Ramana ISDG 2018 27 / 27
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