A Minimax Differential Game Approach for Multi-Agent Robotic Coordination Under Uncertainty

Abstract

Coordination of networked robots under uncertain environments demands a robust control law that is not compromised by the worst possible admissible disturbance but is optimized only under nominal conditions. In this paper, a minimax differential-game approach is developed for the formation coordination problem of multi-agent robotic systems affected by the presence of uncertainty in their models, exogenous disturbances, input constraints, and collision avoidance restrictions. The robot group is considered as a minimizing player while aggregated uncertainty as a maximizing player. Graph-based cost functional depends on the formation error, the velocity difference, the control effort, the terminal error, and a penalty term for unsafe separation. Dynamic programming leads to the Hamilton–Jacobi–Isaacs equation whose solution produces the optimal control and disturbance strategies. Under compact convex action spaces, Lipschitz dynamics, and the Isaacs condition, the equality of the lower and upper game values is demonstrated. Further proofs lead to uniqueness of the viscosity solution, input-to-state practical stability of the coordination error, induced disturbance attenuation performance, collision-free set forward invariance, and asymptotic convergence of the projected extragradient saddle solver. Numerical examples with six robots examine nominal, stochastic, parametric, impulsive, and mixed uncertain conditions. The designed controller exhibits the lowest coordination error and maintains the specified separation constraint for all experiments and is proven to have a monotonically decreasing saddle residual. Minimax theory provides the proper theoretical framework for robust robotic coordination as well as a computationally feasible methodology, and the engineering problem is not allowed to dominate the mathematics.