Cooperative Pursuit-Evasion Games with a Flat Sphere Condition

In planar pursuit-evasion differential games considering a faster pursuer and a slower evader, the interception points resulting from equilibrium strategies lie on the Apollonius circle. This property is instrumental for leveraging geometric approaches for solving multiple pursuit-evasion scenarios in the plane. Here, we study a pursuit-evasion differential game on a sphere and generalize the planar Apollonius circle to the spherical domain. For the differential game, we provide equilibrium strategies for all initial conditions, including a special case when the pursuer and evader are on opposite sides of the sphere where a dilemma occurs. In contrast to planar scenarios, on the sphere we find that the interception point from the equilibrium strategies can leave the Apollonius domain boundary. We present a condition to ensure the intercept point remains on the boundary of the Apollonius domain. This condition allows for generalizing planar pursuit-evasion strategies to the sphere, and we show how these results are applied by analyzing the scenarios of target guarding and two-pursuer, single evader differential games on the sphere.

DOI: 10.1109/OJCSYS.2025.3604640