Reach-avoid games for players with damped double integrator dynamics

TL;DR

This work analyzes a reach-avoid differential game between two damped double integrator players, where an attacker aims to reach a static target while a faster defender attempts interception. It develops a geometric-dynamics framework based on isochrones to define the attacker’s dominance region (ADR) and introduces three ADR types, with a novel third type arising from damped dynamics and requiring a multiple reachable region (MRR) strategy. The authors derive necessary optimality conditions using the Maximum Principle via a Hamiltonian $H$, and provide algorithms to compute circumscribing/inscribing times of isochrones, revealing that optimal strategies depend on the ADR type and the location on the boundary $\mathcal{L}$ where $t_{Aj}=t_{Dk}$. Simulations confirm the theory, showing when strategy I is optimal and when entering $\mathcal{R}_{III}$ necessitates alternative strategies like MRR; they also discuss a special rest-start case where the ADR boundary reduces to an Apollonius circle. The findings offer a rigorous, geometry-driven approach to defender-winning scenarios in damped-dynamics pursuit-evasion and suggest directions for handling non-convex ADRs and multiple minimal points in future work.

Abstract

This paper studies a reach-avoid game of two damped double integrator players. An attacker aims to reach a static target, while a faster defender tries to protect the target by intercepting the attacker before it reaches the target. In scenarios where the defender succeeds, the defender aims to maximize the attacker's final distance from the target, while the attacker aims to minimize it. This work focuses on determining the equilibrium strategy in the defender-winning scenarios. The optimal state feedback strategy is obtained by a differential game approach combining geometric analysis. We construct a multiple reachable region to analyse the damped double integrator player's motion under optimal strategy. Building on this, a new type of the attacker's dominance region is introduced for the first time. It is shown that different strategies are required when the terminal point lies in distinct areas of the attacker's dominance region. Then, a necessary condition is derived for the proposed strategy to be optimal using differential game approach. Furthermore, a case where both players start at rest is discussed, and some useful properties about the dominance region and the optimal strategy are presented. Simulations are conducted to show the effectiveness of the proposed strategy.

Figures (11)

• Figure 2: An illustration of isochrones and region $\mathcal{M}_i$ of a player. The isochrones are marked by colored circles, and the region $\mathcal{M}_i$ is shaded in Fig \ref{['fig:region']}. The player has three strategies to reach a point $\mathbf{x}\in\mathcal{M}_i$, with corresponding isochrones shown in Fig \ref{['fig:reaching-time-iso']}. For points outside $\mathcal{M}_i$, the player has one strategy to reach. On the boundary of the area, denoted as $\mathcal{B}_I^i,\mathcal{B}_{II}^i$, the player has two strategies.
• Figure 3: An illustration of the reaching times of player $i$.
• Figure 4: Illustrations of dominance regions. The yellow shaded regions indicate the attacker's dominance regions (ADR). Blue and red circles represent isochrones of the defender and attacker, and inscribe circle and circumscribe circles are displayed in bold. The intersections of isochrones form the black curve $\mathcal{L}$. Boxes at the bottom of each figure are schematics of the tangential orders, with intermediate states above the arrows. The red and blue circles in schematics represent attacker's and defender's isochrones. Three different tangential orders are given.
• Figure 5: Illustrations for the third type of the attacker's dominance region
• Figure 6: Illustrations about attacker's dominance region. The attacker can reach $\mathcal{R}_I$ without being caught by the defender, while it might be caught before reaching $\mathcal{R}_{II}$. To reach $\mathcal{R}_{III}$, the attacker has to change strategy, such as adopting a smaller acceleration.

Theorems & Definitions (18)

• Remark 1
• Lemma 2
• Lemma 3
• proof
• Definition 3: ADR
• Remark 2
• Lemma 4
• proof
• Definition 4
• Theorem 2