A Robust Saddle-Point Optimization Method for Adaptive Motion Control in Robotic Applications

Abstract

In this paper, a robust saddle-point optimization (RSPO) method for the adaptive motion control of robotic manipulators under severe model uncertainties and external disturbances is developed. Here, each robotic entity is assumed to be a controlled dynamical system, which interacts with neighboring entities by means of a communication network, while the cooperative objective – the formation acquisition, consensus, and leader tracking – is formulated as a penalty of deviations from a certain configuration manifold in terms of coupled quadratic-in-control cost functionals. As a result, the multi-robot optimization problem is stated as an N-person differential game, and by utilizing the calculus of variations and Pontryagin's principle, obtain the necessary conditions for a cooperative Nash equilibrium in terms of a coupled set of Hamilton-Jacobi-Isaacs partial differential equations, for which a verification theorem is proved. In the case of a linear-quadratic problem statement, the closed-form solution of the problem is derived as a feedback strategy using coupled Riccati differential equations. The existence and uniqueness of the cooperative equilibrium are shown under the condition of smallness that depends on the graph structure of the weights associated with the interaction among the agents, and the asymptotic stability of the formation manifold is also shown through a Lyapunov function that is the sum of the value functions of each agent. A numerical example of a robotic network composed of six agents is provided for showing how the formation error and the aggregated cost can converge according to the synthesized approach.