Construction of Barrier in a Fishing Game With Point Capture

Multiplayer Reach-Avoid Differential Games in 3D Space Inspired by Harris' Hawks' Cooperative Hunting Tactics

This paper investigates a multiplayer reach-avoid differential game in 3-dimensional (3D) space, which involves multiple pursuers, multiple evaders, and a designated target region. The evaders aim to reach the target region, while the pursuers attempt to guard the target region by capturing the evaders. This class of research holds significant practical value. However, the complexity of the problem escalates substantially with the growing number of players, rendering its solution extremely challenging. In this paper, the multiplayer game is divided into many subgames considering the cooperation among pursuers, reducing the computational burden, and obtaining numerically tractable strategies for players. First, the Apollonius sphere, a fundamental geometric tool for analyzing the 3D differential game, is formulated, and its properties are proved. Based on this, the optimal interception point for the pursuer to capture the evader is derived and the winning conditions for the pursuer and evader are established. Then, based on the Apollonius sphere, the optimal state feedback strategies of players are designed, and simultaneously, the optimal one-to-one pairings are obtained. Meanwhile, the Value function of the multiplayer reach-avoid differential game is explicitly given and is proved to satisfy Hamilton-Jacobi-Isaacs (HJI) equation. Moreover, the matching algorithm for the case with pursuers outnumbered evaders is provided through constructing a weighted bipartite graph, and the cooperative tactics for multiple pursuers are proposed, inspired by the Harris' Hawks intelligent cooperative hunting tactics. Finally, numerical simulations are conducted to illustrate the effectiveness of the theoretical results for both cases where the number of adversary players is equal and unequal between the 2 groups.

Adaptive multi-player pursuit-evasion games with unknown general quadratic objectives

In this paper, an adaptive parameter estimation scheme is investigated for a class of multi-player pursuit-evasion games subject to unknown objectives with linear feedback and cross terms in control strategy. In virtue of the adaptive method, each player updates controller strategies to estimate opponent's control parameters before getting the Nash equilibrium solution. Moreover, the sufficient condition is provided to guarantee stability of the proposed adaptive procedure. Meanwhile, the stability behavior of closed-loop Nash strategy with adaptive estimator is guaranteed by the contraction mapping method. Finally, a simulation of an all-against-one game with one escaper and three pursuers is carried out to demonstrate the effectiveness of the proposed methodology.

Guarding a Subspace in High-Dimensional Space With Two Defenders and One Attacker

This article considers a subspace guarding game in high-dimensional space which consists of a play subspace and a target subspace. Two faster defenders as a team cooperate to protect the target subspace by capturing an attacker which strives to enter the target subspace from the play subspace without being captured. A closed-form solution is provided from the perspectives of kind and degree. Contributions of the work include the use of the attack subspace (AS) method to construct the barrier, by which the game winner can be perfectly predicted before the game starts. In addition to this inclusion, with the priori information about the game result, a critical payoff function is designed when the defenders can win the game. Then, the optimal strategy for each player is explicitly reformulated as a saddle-point equilibrium. Finally, we apply these theoretical results to two half-space and half-plane guarding games in 3-D space and 2-D plane, respectively. Since the entire achieved developments are analytical, they require a little memory without the computational burden and allow for real-time updates, beyond the capacity of the traditional Hamilton-Jacobi-Isaacs method. It is worth noting that this is the first time in the current work to consider the target guarding games for arbitrary high-dimensional space and in a fully analytical form.

Cooperative Pursuit With Multi-Pursuer and One Faster Free-Moving Evader

This article addresses a multi-pursuer single-evader pursuit-evasion game where the free-moving evader moves faster than the pursuers. Most of the existing works impose constraints on the faster evader, such as limited moving area and moving direction. When the faster evader is allowed to move freely without any constraint, the main issues are how to form an encirclement to trap the evader into the capture domain, how to balance between forming an encirclement and approaching the faster evader, and what conditions make the capture possible. In this article, a distributed pursuit algorithm is proposed to enable pursuers to form an encirclement and approach the faster evader. An algorithm that balances between forming an encirclement and approaching the faster evader is proposed. Moreover, sufficient capture conditions are derived based on the initial spatial distribution and the speed ratios of the pursuers and the evader. Simulation and experimental results on ground robots validate the effectiveness and practicability of the proposed method.

Kinematic Characterization of a Target-Defense Problem With an Interception and Expelling Strategy

This paper considers a target-defense problem of one defender and one adversary (or called intruder), in which the defender tries to approach a desired interception-position between the intruder and the target to intercept the intruder and expel it from the target. The defender adopts this interception and expelling strategy since the defender is assumed to not capture or destroy the intruder. An expelling-decay exponent is introduced to characterize the expelling-decay rate on the intruder. The system is nonlinear, with distinct physical meaning and rich kinematics. The main concern in this paper is the analysis of the kinematic properties. First, two motion patterns of the system are characterized with respect to different values of the system parameters. Then, the stability and the transition condition of the two motion patterns are provided. Finally, the optimal interception of the defender is provided, which interestingly coincides with the transition condition for the two motion patterns. The interpretations for the physical meaning of the optimal interception are also provided.

Reach-Avoid Games With Two Defenders and One Attacker: An Analytical Approach

This paper considers a reach-avoid game on a rectangular domain with two defenders and one attacker. The attacker aims to reach a specified edge of the game domain boundary, while the defenders strive to prevent that by capturing the attacker. First, we are concerned with the barrier, which is the boundary of the reach-avoid set, splitting the state space into two disjoint parts: 1) defender dominance region (DDR) and 2) attacker dominance region (ADR). For the initial states lying in the DDR, there exists a strategy for the defenders to intercept the attacker regardless of the attacker's best effort, while for the initial states lying in the ADR, the attacker can always find a successful attack strategy. We propose an attack region method to construct the barrier analytically by employing Voronoi diagram and Apollonius circle for two kinds of speed ratios. Then, by taking practical payoff functions into considerations, we present optimal strategies for the players when their initial states lie in their winning regions, and show that the ADR is divided into several parts corresponding to different strategies for the players. Numerical approaches, which suffer from inherent inaccuracy, have already been utilized for multiplayer reach-avoid games, but computational complexity complicates solving such games and consequently hinders efficient on-line applications. However, this method can obtain the exact formulation of the barrier and is applicable for real-time updates.