Pseudospectral methods for optimal motion planning of differentially flat systems

This paper presents some preliminary results on combining two new ideas from nonlinear control theory and dynamic optimization. We show that the computational framework facilitated by pseudospectral (PS) methods applies quite naturally and easily to the Lie-Backlund equivalence of nonlinear controlled dynamical systems. The optimal motion planning problem for differentially flat systems is equivalent to a classic Bolza problem of the calculus-of-variations. In this paper, we exploit the notion that derivatives of flat outputs given in terms of Lagrange interpolating functions can be quickly and easily computed using PS differentiation matrices. The application of this method to the crane control problem demonstrates how flatness may be readily exploited. In the case of partial dynamic inversion, or systems with non-zero defects, differential constraints are satisfied at optimal node points whose optimality criterion is some error norm. Integral cost functionals are handled by Gauss-type quadrature rules. Numerical experiments suggest that PS methods are superior to other methods that exploit full or partial dynamic inversion; however, a number of problems inherent to utilizing flat outputs to real-time trajectory optimization remain open.