Evaluation of invariant cost functions for the optimal control of robotic manipulators

In the framework of Pontryagin’s Maximum Principle, the cost function choice is fundamental because it has a direct impact on the system Hamiltonian. This choice naturally affects the resulting controls and the cost function should therefore be chosen appropriately, in accordance with the desired system behavior. We deal with robotic systems for which the governing equations and optimality conditions are commonly affected by numerical stiffness. As such, computing the optimal controls can be difficult for such nonlinear systems. We apply Pontryagin’s Maximum Principle to the optimal control of a robotic system and compare the impact of four different cost functions on a commercial ODE solver that determines the optimal solution. The focus is on numerical stability and overall computing time. Two of these selected cost functions are typically found in the literature. We propose alternative forms of these cost functions where the robot mass tensor is embedded. Numerical trials show that the mass tensor acts as a stabilizing factor that leads to: lower link motion amplitudes, improved numerical stability as well as decreased CPU computing times when solving the optimal control of a robotic manipulator. These cost functions can be beneficial for motion planning and optimal control of robotic manipulators.