Optimal Evading Strategies and Task Allocation in Multi-Pursuer - - PowerPoint PPT Presentation

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

12th July, 2018

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Outline

Motivation and problem statement Optimal evading strategies Active/Redundant pursuers Simulations

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Motivation

Airspace security Regulate the traffic and usage of UAVs

Figure 1: DroneHunter a

ahttps://fortemtech.com/

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

A scenario

Assume: n agents (pursuers) guarding a territory m adversaries (evaders, typically m ≤ n) Pursuers want to capture the evaders Pursuers are faster than the evaders

P1 P2 E1 P3 Pn ... P4 E2 Em ... P5 Territory

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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!

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Approach

Divide and Conquer Solve m multi-pursuer single-evader games Pursuers follow simple navigation laws: Pure Pursuit (PP) or Constant Bearing (CB) staregies

P1 P2 E1 P3 Pn ... P4 E2 Em ... P5 Territory

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Problem Set Up

O Inertial Frame i1 i2 P1 E ri θΕ ϕi θi Qi Collision triangle for Pi P2 Pi Pn ... ...

(a) CB

O Inertial Frame i1 i2 P1 E ri θΕ ϕi P2 Pi Pn ... ...

(b) PP

Figure 2: Schematics of the proposed pursuit-evasion problems.

Identical pursuers, pursuers faster than evader.

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Regions of Non-Degeneracy1

(a) CB (b) PP

Figure 3: Regions of non-degeneracy

1Makkapati et al., Pursuit-Evasion Problems Involving Two Pursuers and One Evader, AIAA GNC

Conference, Kissimmee, FL, 2018 Ramana ISDG 2018 12th July, 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 - P1, red - P2.

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Two Pursuers - PP (Previous work)

Optimal Suboptimal

(a) Trajectories

Optimal Suboptimal

(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 - P1, red

• P2.

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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!

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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!

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

What If?

The pursuers don’t know the evader’s strategy

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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 (tc).

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Active/Redundant Pursuers

At time 0 ≤ t < tc Definition If there exists an evading strategy for which pursuer Pi is in P, then Pi is an active pursuer. Definition If there exists no evading strategy for which pursuer Pi is in P, then Pi is a redundant pursuer

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Apollonius Curves

0.5 1 1.5 2 2.5

• 1
• 0.5

0.5 1

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.

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Apollonius Boundary

• 5

5

• 5

5

(a) CB

• 5

5

• 5

5

(b) PP

Figure 7: Apollonius boundaries for CB and PP cases (Simulation parameters: u = 1, v = 0.6)

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

A Formal Definition

Definition The Apollonius boundary is the set of points B = {X ∈

n

• i=1

Ai | M(E, X) ∩ n

• i=1

Ai

• = {X}}, where

M(E, X) denotes the set of points on the line segment with endpoints E (position of the evader) and X.

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

A Conjecture for the CB case

Conjecture Pursuer Pi is active if B ∩ Ai = ∅, and is redundant otherwise.

• 5

5

• 5

5

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Lemma 1

Pursuer Pi is the only active pursuer if and only if Ai ∩

• n
• j=1, j=i

Aj

• = ∅,

(1) M(E, Ti) ∩

• n
• j=1, j=i

Aj

• = ∅,

(2) Ti is the closest point to the evader on the Apollonius circle Ai

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Lemma 2

Assumption: Ai intersects one or more of the other Apollonius circles. Pi is an active pursuer if and only if there exists at least one X ∈ Ii such that: M(E, X) ∩ n

• j=1

Aj

• = {X},

(3) Ii is the set of intersections points between Ai and the rest of the Apollonius circles.

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Algorithm to Identify Pursuer Status

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Simulations

5 10 15 20 5 10 15 20

(a) Initial Apollonius circles

5 10 15 20 5 10 15 20

(b) Trajectories

Figure 8: CB case

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Simulations

5 10 15 20 5 10 15 20

(a) Initial Apollonius curves (refined)

5 10 15 20 5 10 15 20

(b) Trajectories

Figure 9: PP case

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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

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Motivation Optimal Evading Strategies Active/Redundant Pursuers Simulations

Simulations

2 4 6 8 10 2 4 6 8 10

(a) t = 0

2 4 6 8 10 2 4 6 8 10

(b) t = 1.3

2 4 6 8 10 2 4 6 8 10

(c) t = 2.5

Figure 10: CB case

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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

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