Analysis of Pursuit-Evasion on a Sphere

核心概念

Investigating equilibrium strategies in pursuit-evasion games on a sphere.

摘要

The content delves into the pursuit-evasion game dynamics on a sphere, exploring the equilibrium intercept point and Apollonius domain. It discusses the relationship between agents' motions, geodesics, and optimal strategies. The paper presents theoretical derivations, applications to multiple pursuers scenarios, and target guarding games. Key concepts include Hamilton-Jacobi-Isaacs equation, Apollonius circle/domain, and geodesic paths.
I. Introduction and Background:

Classical works on planar differential pursuit-evasion games.
Study of pursuit-evasion game on a sphere.
Relation of equilibrium intercept point to Apollonius domain.
II. Problem Description:

Two agents restricted to motion on a spherical surface.
Pursuer (P) and slower evader (E) defined by positions and velocities.
Control variables for P and E in pursuit-evasion game.
III. Pursuit-Evasion Equilibrium Strategies:

Derivation of angular distance rate between P and E.
Equilibrium strategies for P and E based on Hamilton-Jacobi-Isaacs equation.
Special cases when α = 0 or π analyzed.
IV. Apollonius Domain and Intercept Point:

Definition of Apollonius domain A on the sphere.
Relationship between relative position α, intercept points, and boundary of A.
V. Results:

Illustration of Apollonius domain for different scenarios.
Application to two-pursuer scenarios with cooperative strategies.
VI. Conclusion:

Summary of findings in pursuit-evasion games on a sphere.
Future research directions in multi-agent scenarios on spherical geometry.

統計資料

This paper is based at AFRL Control Science Center supported by AFOSR LRIR 24RQCOR002.

引述

"The goal of P is to minimize J, i.e., time to capture."
"Equilibrium strategies are optimal for minimizing capture time."
"Apollonius domain exhibits similar properties to the planar case."

從以下內容提煉的關鍵洞見

Pursuit-Evasion on a Sphere and When It Can Be Considered Flat

by Dejan Miluti... 於 arxiv.org 03-25-2024

https://arxiv.org/pdf/2403.15188.pdf

深入探究

How does the concept of Apollonius domain differ from traditional Euclidean space

In traditional Euclidean space, the Apollonius circle is a set of points equidistant from two fixed points. However, in the context of pursuit-evasion games on a sphere, the concept extends to what is known as the Apollonius domain. This domain represents all points on the sphere that an evader can reach before being captured by one or more pursuers. The boundary of this domain includes intercept points where capture may occur and plays a crucial role in determining optimal strategies for both pursuers and evaders.

What implications do these findings have for real-world applications beyond theoretical pursuits

The findings regarding the Apollonius domain have significant implications for real-world applications beyond theoretical pursuits. One practical application lies in security and surveillance scenarios where multiple agents need to track or intercept a target moving on a spherical surface. By understanding how equilibrium strategies operate within the Apollonius domain, security systems can be optimized to efficiently track targets while minimizing capture time.
Another application could be in robotics and autonomous systems operating in dynamic environments. By incorporating these equilibrium strategies into their decision-making processes, robots can adapt their movements based on changing parameters such as speed ratios between pursuers and evaders or variations in target locations.
Furthermore, these findings could also have implications in sports analytics for tracking players' movements on curved surfaces like basketball courts or soccer fields during gameplay analysis.

How can these equilibrium strategies be adapted for dynamic environments with changing parameters

Adapting equilibrium strategies for dynamic environments with changing parameters involves continuously updating control variables based on real-time data feedback. In pursuit-evasion scenarios where conditions are not static, agents must adjust their actions to account for evolving circumstances.
One approach could involve implementing adaptive algorithms that consider varying speeds of agents or changes in initial positions during gameplay. These algorithms would dynamically calculate optimal paths based on current information to ensure efficient pursuit or evasion tactics.
Additionally, machine learning techniques could be employed to analyze patterns from past interactions and predict future movements of both pursuers and evaders. By leveraging predictive models trained on historical data, agents can anticipate opponents' behaviors and adjust their strategies accordingly to maintain competitive advantages even as parameters shift over time.