New Computational Framework for Trajectory Optimization of Higher-Order Dynamic Systems

Using tools from systems theory, it is now well known that classes of linear and nonlinear dynamic systems in firstorder form can be alternatively written in higher-order form, i.e., 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. In this paper, we address the question of trajectory optimization of higher-order systems with general nonlinear constraints. 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 firstorder forms. The program implements both direct and indirect methods and uses collocation in conjunction with a nonlinear programming solver. infancy. However, a few studies in the literature address this issue applied to limited classes of problems. For example, a direct method was used for PVTOL systems to compute the optimal solution, where the inequality constraints were not considered [3]. In recent work, Agrawal and coworkers have used the indirect method to compute the optimal solution in the absence of inequality constraints. In these studies, the system equations were explicitly embedded into the cost functional. The results from higher-order variational theory were used to find the optimality conditions [4]. This approach was applied successfully to linear systems ([5], [6]) and feedback linearizable nonlinear systems [7]. The purpose of this paper is to extend the optimality theory applicable to higher-order systems, without converting the system equations to the first-order form. Using the resulting optimality conditions, direct and indirect computational algorithms are developed and implemented. The same direct or indirect algorithm is used to solve the firstorder and the higher-order optimization problems, thereby, allowing to benchmark the computation requirements in a systematic manner.