A Differential Game-Theoretic Variational Approach for Cooperative Control of Autonomous Robotic Networks

Abstract

This paper presents a differential game-theoretic variational approach to cooperative control of autonomous robotic networks. The cooperative task is formulated as a coupled quadratic-in-control cost functionals for each robotic agent, that penalize the deviation from a defined relative-configuration manifold, and the robotic agents are modeled as controlled dynamical systems coupled with each other via a defined communication graph, in which the formation acquisition, consensus and leader tracking problem are embedded. The multi-agent optimization is formulated as an N-player differential game and necessary conditions for a cooperative Nash equilibrium are obtained by a variational treatment by the calculus of variations and Pontryagin's minimum principle. These conditions are then described by a coupled system of Hamilton-Jacobi-Isaacs partial differential equations and a verification theorem is derived. In the linear-quadratic case of the problem, closed-form feedback strategies are derived from a set of coupled Riccati differential equations, and a fully distributed control law is constructed that only uses locally available information from a neighbouring set of nodes. Under a smallness condition on the inter-agent coupling weights which depends on the graph, the existence and uniqueness of the cooperative equilibrium are derived, and by using a Lyapunov function constructed from the sum of the individual value functions, the asymptotic stability of the formation manifold is proved, with an extension to jointly connected switching topologies. The numerical example demonstrates how the formation error and the decay of the aggregate running cost are converging under the synthesized strategy with a 6-agent robotic network. The results offer a solid variational link between differential game theory and distributed cooperative control of robotic networks.