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Defending a Static Target Point with a Slow Defender

Goutam Das, Michael Dorothy, Zachary I. Bell, Daigo Shishika

TL;DR

This paper studies a target-defense game played between a slow defender and a fast attacker, and provides the barrier surface that divides the state space into attacker-win and defender-win regions, and presents corresponding strategies that guarantee win for each region.

Abstract

This paper studies a target-defense game played between a slow defender and a fast attacker. The attacker wins the game if it reaches the target while avoiding the defender's capture disk. The defender wins the game by preventing the attacker from reaching the target, which includes reaching the target and containing it in the capture disk. Depending on the initial condition, the attacker must circumnavigate the defender's capture disk, resulting in a constrained trajectory. This condition produces three phases of the game, which we analyze to solve for the game of kind. We provide the barrier surface that divides the state space into attacker-win and defender win regions, and present the corresponding strategies that guarantee win for each region. Numerical experiments demonstrate the theoretical results as well as the efficacy of the proposed strategies.

Defending a Static Target Point with a Slow Defender

TL;DR

This paper studies a target-defense game played between a slow defender and a fast attacker, and provides the barrier surface that divides the state space into attacker-win and defender-win regions, and presents corresponding strategies that guarantee win for each region.

Abstract

This paper studies a target-defense game played between a slow defender and a fast attacker. The attacker wins the game if it reaches the target while avoiding the defender's capture disk. The defender wins the game by preventing the attacker from reaching the target, which includes reaching the target and containing it in the capture disk. Depending on the initial condition, the attacker must circumnavigate the defender's capture disk, resulting in a constrained trajectory. This condition produces three phases of the game, which we analyze to solve for the game of kind. We provide the barrier surface that divides the state space into attacker-win and defender win regions, and present the corresponding strategies that guarantee win for each region. Numerical experiments demonstrate the theoretical results as well as the efficacy of the proposed strategies.
Paper Structure (18 sections, 7 theorems, 25 equations, 12 figures)

This paper contains 18 sections, 7 theorems, 25 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Dominance Regions
  4. Defender's Dominance Region
  5. Attacker's Dominance Region
  6. The Blocking Game
  7. Dynamics of Phase-II
  8. Phase-III / Termination of Phase-II
  9. Defender Winning Terminal Condition
  10. Attacker Winning Terminal Condition
  11. Phase-II: Optimal Constrained Trajectories
  12. Phase-I: Optimally Entering Phase-II
  13. Barrier and Security Strategies
  14. Barrier Surface
  15. Security Strategies
  16. ...and 3 more sections

Key Result

Lemma 1

The CO for the slower defender and the faster attacker is described by the following equations: where $l$ is the distance of a point on CO from the attacker: for $\bar{\phi} \in [-\bar{\phi}_S, \bar{\phi}_S]$, and $\lambda$ is the line-of-sight (LOS) angle,

Figures (12)

  • Figure 1: Illustration of the target-defense game with a slower-defender.
  • Figure 2: Defender's dominance region, $\mathcal{D}_D$ is shown as a Cartesian Oval (CO) and the attacker's dominance region, $\mathcal{D}_A$ is shown as a shaded region outside the CO.
  • Figure 3: The defender and the attacker in $(\rho_D,\theta)$ space in Phase-II.
  • Figure 4: Defender-win terminal conditions: defender reaches the target (left), and attacker is blocked while the defender has angular advantage (right).
  • Figure 5: Terminal surfaces of the Phase-II indicating the end of game trivially by either defender or attacker win.
  • ...and 7 more figures

Theorems & Definitions (18)

  • Lemma 1: Lemma 2 in garcia2021cooperative_containment
  • Remark 1
  • Remark 2
  • Lemma 2
  • proof
  • Theorem 1: Defender Win Condition
  • proof
  • Lemma 3
  • proof
  • Remark 3
  • ...and 8 more