New computational framework for trajectory optimization of higher-order dynamic systems

By the use of tools from systems theory, it is now well known that classes of linear and nonlinear dynamic systems in first-order form can he alternatively written in higher-order form, that is, as sets of higher-order differential equations. Input-state linearization is one of the popular tools to achieve such a transformation. For mechanical systems, the equations of motion naturally have a second-order form. For real-time planning and control, a higher-order form offers a number of advantages compared to the first-order form. The question of trajectory optimization of higher-order systems with general nonlinear constraints is addressed. First, we develop the optimality conditions directly using their higher-order form. These conditions are then used to develop computational approaches. A general purpose program has been developed to benchmark computations between problems posed in alternate higher-order and first-order forms. The program implements both direct and indirect methods and uses collocation in conjunction with a nonlinear programming solver.