A minimax Optimal Control Framework for Nonlinear Robotic Systems using Hamilton-jacobi formulation

Abstract

A minimax optimal control framework for a wide range of nonlinear robotic systems with bounded exogenous disturbances and unmodeled dynamics is formulated in this paper. The synthesis problem is formulated as a two-player zero-sum, differential game with the performance functional to be minimised by the control input over an infinite horizon and to be maximised by the disturbance input, which is adversarial. It is shown using arguments of dynamic programming that the value function of the game is the unique viscosity solution of a stationary Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). Initial nonrigorous ideas of admissible policies, the Hamiltonian structure that goes with them, and regularity of the value function are developed, followed by a proof of a verification theorem that connects a solution of the HJI equation to the existence of a saddle-point minimax feedback pair. Existence and uniqueness of the value function is proved using a monotone-operator argument, and convergence of an associated policy-iteration scheme to the optimal value is proved. Three corollaries are drawn: The framework is reduced to the classical Hamilton–Jacobi–Bellman equation in the disturbance-free limit, a closed-form saddle-point feedback law for control-affine robotic dynamics under the quadratic running cost is produced, and a guaranteed closed-loop L2-gain bound is obtained together with a Lyapunov stability certificate. The theory is demonstrated with a two-link rigid robotic manipulator with torque bounded actuator and bounded joint-torque disturbances, and using a numerically implemented policy-iteration algorithm compared with a linear-quadratic baseline regulator. The results provide formal guarantees of robustness for nonlinear robotic control in worst case uncertainty and are meant to be used as a blueprint for further development of minimax dynamic programming for more classes of Lagrangian and networked robotic systems.