Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Prying Pedestrian Surveillance-Evasion: Minimum-Time Evasion from an Agile Pursuer

Philipp Braun, Timothy L. Molloy, Iman Shames

TL;DR

The paper introduces a surveillance-evasion differential game with an agile pursuer and a turn-limited evader, analyzed in both game-of-kind and game-of-degree formulations. It develops a rigorous Hamiltonian-based framework to derive Nash-equilibrium strategies, revealing a speed-ratio dependent structure with singular arcs and UP/NUP/BUP partitions, and demonstrates a concrete minimum-time-circle-escape specialization via a Dubins-car model. The results show that when the pursuer is slower ($v_p<v_e$), the evader can escape, while a faster pursuer ($v_p>v_e$) guarantees surveillance, with the degree game producing richer, region-specific strategies. The findings have practical implications for designing surveillance and evasion strategies in agile-vehicle contexts and lay groundwork for extensions to multiple pursuers and obstacles, supported by analytic and illustrative examples in inertial and evader-centered coordinates.

Abstract

A new surveillance-evasion differential game is posed and solved in which an agile pursuer (the prying pedestrian) seeks to remain within a given surveillance range of a less agile evader that aims to escape. In contrast to previous surveillance-evasion games, the pursuer is agile in the sense of being able to instantaneously change the direction of its velocity vector, whilst the evader is constrained to have a finite maximum turn rate. Both the game of kind concerned with conditions under which the evader can escape, and the game of degree concerned with the evader seeking to minimize the escape time whilst the pursuer seeks to maximize it, are considered. The game-of-degree solution is surprisingly complex compared to solutions to analogous pursuit-evasion games with an agile pursuer since it exhibits dependence on the ratio of the pursuer's speed to the evader's speed. It is, however, surprisingly simple compared to solutions to classic surveillance-evasion games with a turn-limited pursuer.

Prying Pedestrian Surveillance-Evasion: Minimum-Time Evasion from an Agile Pursuer

TL;DR

The paper introduces a surveillance-evasion differential game with an agile pursuer and a turn-limited evader, analyzed in both game-of-kind and game-of-degree formulations. It develops a rigorous Hamiltonian-based framework to derive Nash-equilibrium strategies, revealing a speed-ratio dependent structure with singular arcs and UP/NUP/BUP partitions, and demonstrates a concrete minimum-time-circle-escape specialization via a Dubins-car model. The results show that when the pursuer is slower (), the evader can escape, while a faster pursuer () guarantees surveillance, with the degree game producing richer, region-specific strategies. The findings have practical implications for designing surveillance and evasion strategies in agile-vehicle contexts and lay groundwork for extensions to multiple pursuers and obstacles, supported by analytic and illustrative examples in inertial and evader-centered coordinates.

Abstract

A new surveillance-evasion differential game is posed and solved in which an agile pursuer (the prying pedestrian) seeks to remain within a given surveillance range of a less agile evader that aims to escape. In contrast to previous surveillance-evasion games, the pursuer is agile in the sense of being able to instantaneously change the direction of its velocity vector, whilst the evader is constrained to have a finite maximum turn rate. Both the game of kind concerned with conditions under which the evader can escape, and the game of degree concerned with the evader seeking to minimize the escape time whilst the pursuer seeks to maximize it, are considered. The game-of-degree solution is surprisingly complex compared to solutions to analogous pursuit-evasion games with an agile pursuer since it exhibits dependence on the ratio of the pursuer's speed to the evader's speed. It is, however, surprisingly simple compared to solutions to classic surveillance-evasion games with a turn-limited pursuer.

Paper Structure

This paper contains 19 sections, 9 theorems, 100 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Evader-Centric Coordinate System
  4. Game of Kind
  5. Game of Degree
  6. Game of Degree Solution
  7. Preparatory Results and Definitions
  8. Candidate Optimal Strategies, the Hamiltonian and the Adjoint Equation
  9. Candidate Optimal Solutions Terminating on Usable Part with $x\neq 0$
  10. Trajectories Converging to $y$-axis in Finite Time
  11. Extension of Solution \ref{['eq:sol_xP2']} for $y\geq 0$
  12. Illustrative Analysis of Solution of Game of Degree
  13. Specialization to Minimum-Time Escape from a Circular Region for a Dubins Car
  14. Game Solutions in Inertial Coordinate Frame
  15. Optimal Value Function Compared to Pure Pursuit
  16. ...and 4 more sections

Key Result

Lemma 1

Consider Problem prob:game_of_degree. Under optimal play (of the evader and the pursuer), the game of degree eq:problem_constraints can only end on the usable part defined through the set Conversely, under optimal play, the game cannot end on the nonusable part and the boundary of the usable part given by respectively. $\lrcorner$

Figures (11)

  • Figure 1: Coordinate system attached to the evader.
  • Figure 2: Illustration of the decrease condition in \ref{['eq:decrease_game_kind']}. On the left, $\xi$ and $f(\xi,u_e,u_p)$ point in opposite directions and thus $f(\xi,u_e,u_p)^\top \xi < 0$. On the right, $\xi$ and $f(\xi,u_e,u_p)$ point in the same direction and thus $f(\xi,u_e,u_p)^\top \xi > 0$.
  • Figure 3: Trajectories corresponding to solutions of the game of degree for different parameter selections. On the left, the parameters $v_p=1$, $v_e=2$, $\rho=1$ and $\omega_e=1$ are used. On the right, $v_e$ is replaced by $v_e=1.5$.
  • Figure 4: Solutions of the game of degree emanating from the usable part in backward time characterized through Theorem \ref{['thm:solutions_usable_part']} for the parameters $v_p=1$, $v_e=2$, $\rho=1$ and $\omega_e=2$.
  • Figure 5: Trajectories characterized through Theorem \ref{['thm:opt_strat_game2']} using the parameter selection $\omega_e=2$, $v_p=1$, $\rho=1$ and $v_e=1.5$ (left) and $v_e=2$ (right), respectively.
  • ...and 6 more figures

Theorems & Definitions (20)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1: Usable and Nonusable parts
  • Remark 4
  • Remark 5
  • Lemma 2
  • Corollary 1: Symmetry with respect to $y$-axis
  • Lemma 3
  • Definition 1: Regular Points & Parts, Lewin2012
  • ...and 10 more