A Differential Game of Collision Coordination Between Two Robots.

This paper explores a differential game involving two robots whose movements are described by linear differential equations with integral energy constraints, where the first robot possesses twice the energy of the resource of the second. We propose a novel multi-stage control strategy enabling the robots to execute position swaps while ensuring collision avoidance across three distinct phases, maintaining safe separation throughout. By employing time-specific control functions, we achieve precise coordination, culminating in a planned convergence at a shared location at predetermined terminal time. The admissibility of control strategies under the given constraints is rigorously verified and also, the timing sequence to achieve collision avoidance until critical endpoint is mathematically demonstrated. This work advances differential game theory by introducing a structured, multi-stage approach to balancing collision-free navigation and intentional terminal convergence.