Direct Optimal Control of Second Order Mechanical State Space Systems⁎

Abstract

In this paper, we derive a new numerical method for the solution of systems of second order ordinary differential equations, as well as the boundary value problems that arise in the optimal control of such systems. Systems of second order arise generally in mechanics, and are critical to problems in robotics, structural dynamics, and distributed parameter systems. Typically though, second order systems are transformed into systems of first order ordinary differential equations so that Runge-Kutta methods and allied methods can be invoked. The justification given for this is that first order system are more stable to solve numerically. The new algorithm, however, is shown to be stable, principally due to the proposed matrix-based approach, whereby the solution to the boundary value problems of optimal control are solved at all points simultaneously. The method is demonstrated by solving the optimal control problem of multiple pendulums of differing lengths on a single cart.