Pursuit Winning Strategies for Reach-Avoid Games with Polygonal Obstacles

TL;DR

This work tackles multiplayer reach-avoid differential games in obstacle-rich planar environments, where multiple pursuers defend a convex goal against evaders trying to enter it. It develops a geometry-driven MOCG framework that fuses onsite and close-to-goal strategies, leveraging expanded Apollonius circles, convex goal-covering polygons, and Euclidean shortest paths with wavefronts to build three distinct pursuit-winning regions. A receding-horizon hierarchical task allocation (consisting of capture, enhanced, non-dominated, and closest matchings) assigns pursuer coalitions to evaders to guarantee an increasing lower bound on defeated evaders, without requiring state-space discretization. Simulations demonstrate robust performance across various obstacle layouts, pursuer/evader counts, and visibility conditions, confirming that the defeated set grows over time and illustrating practical applicability to security and robotics in cluttered environments.

Abstract

This paper studies a multiplayer reach-avoid differential game in the presence of general polygonal obstacles that block the players' motions. The pursuers cooperate to protect a convex region from the evaders who try to reach the region. We propose a multiplayer onsite and close-to-goal (MOCG) pursuit strategy that can tell and achieve an increasing lower bound on the number of guaranteed defeated evaders. This pursuit strategy fuses the subgame outcomes for multiple pursuers against one evader with hierarchical optimal task allocation in the receding-horizon manner. To determine the qualitative subgame outcomes that who is the game winner, we construct three pursuit winning regions and strategies under which the pursuers guarantee to win against the evader, regardless of the unknown evader strategy. First, we utilize the expanded Apollonius circles and propose the onsite pursuit winning that achieves the capture in finite time. Second, we introduce convex goal-covering polygons (GCPs) and propose the close-to-goal pursuit winning for the pursuers whose visibility region contains the whole protected region, and the goal-visible property will be preserved afterwards. Third, we employ Euclidean shortest paths (ESPs) and construct a pursuit winning region and strategy for the non-goal-visible pursuers, where the pursuers are firstly steered to positions with goal visibility along ESPs. In each horizon, the hierarchical optimal task allocation maximizes the number of defeated evaders and consists of four sequential matchings: capture, enhanced, non-dominated and closest matchings. Numerical examples are presented to illustrate the results.

Figures (10)

• Figure 1: Multiplayer reach-avoid differential games in the presence of black general polygonal obstacles, where the blue pursuers protect a green convex polygon $\Omega_{\rm goal}$ against the red evaders who start from the play region $\Omega_{\rm play}$ and aim to enter $\Omega_{\rm goal}$. The obstacles block the motion of all players.
• Figure 2: Onsite pursuit winning. $(a)$ one pursuer: if two regions $\mathbb{A}_{\delta}$ and $\mathcal{R}_{\textup{tri}}$ surrounded by the green expanded Apollonius circle and black triangle, respectively, are obstacle-free, then $P_i$ guarantees to capture $E_j$ in $\mathbb{A}_{\delta}$ in a finite time. $(b)$ pursuit coalition: if every pursuer in a coalition can capture an evader individually, then the evader will be captured in the intersection of all regions bounded by expanded Apollonius circles, i.e., $\cap_{i=1}^3 \mathbb{A}_{\delta,i}$.
• Figure 3: $(a)$ goal-visible point $\bm{x}$ with a pair $(\bm{x}_1, \bm{x}_2)$ of minimum-covering points: all points in $\Omega_{\rm goal}$ are visible to $\bm{x}$ and $\Omega_{\rm goal} \subset \mathcal{R}_{\textup{sec}}(\bm{x}, \bm{x}_1, \bm{x}_2)$. $(b)$ non-goal-visible point $\bm{x}$: at least one point in $\Omega_{\rm goal}$ is not visible to $\bm{x}$.
• Figure 4: Convex goal-covering polygon (GCP). The orange polygon $\mathcal{R}$ is a convex GCP for $\bm{x}$, as $i)$$\mathcal{R}$ is convex, $ii)$$\bm{x} \in \mathcal{R}$, $iii)$$\mathcal{R} \subset \Omega_{\rm free}$ and $\Omega_{\rm goal} \subset \mathcal{R}$. $(a)$ if $\bm{x}$ is at the boundary of $\mathcal{R}$, the direction range $D_{\mathcal{R}}(\bm{x})$ is the blue direction range; otherwise $(b)$$D_{\mathcal{R}}(\bm{x}) = [0, 2\pi)$.
• Figure 5: Goal-visible pursuit winning. If $P_i$ is goal-visible and the safe distance (i.e., the distance between the region $\mathbb{E}$ (red boundary) and $\Omega_{\rm goal}$) is non-negative, then $P_i$ guarantees to win against $E_j$, which also applies to a pursuit coalition. Let $\mathcal{R}_i$ be a (orange) convex GCP for $\bm{x}_{P_i}$ with the direction angle span $\phi$. The pursuit winning strategy, which ensures the consistent goal-visible property, is as follows. $(a)$ If the closest point $\bm{x}_I$ in $\mathbb{E}$ to $\Omega_{\rm goal}$ is in the angle span $\phi$, then $P_i$ moves towards $\bm{x}_I$. $(b)$ If $\bm{x}_I$ is not in $\phi$, then $P_i$ moves along one edge of $\mathcal{R}_i$. After a small time step, $P_i$ reaches $\bm{x}_{P_i}'$ which is goal-visible and has a convex GCP $\mathcal{R}_i'$ with the angle span $\phi'$. The evader and the closest point are updated to $\bm{x}_{E_j}'$ and $\bm{x}_I'$, respectively.

Theorems & Definitions (49)

• Definition 1: Pursuit winning
• Definition 2: Three pursuit winning regions
• Lemma 1: Pursuit strategy, MD-DM-DS-AVM:21
• Remark 1
• Remark 2
• Theorem 1: Onsite pursuit winning
• proof
• Lemma 2: Onsite pursuit winning for coalitions
• proof
• Remark 3