Trajectory optimization under uncertainty based on polynomial chaos expansion

A general procedure of trajectory optimization under uncertainty, which considers probabilistic uncertainties from both initial state and system parameter under both path and boundary constraints, is presented in this paper. With the proposed method, based on the robust design theory, the original stochastic trajectory optimization problem is transformed into an equivalent deterministic one in the expanded higher-dimensional state space by the polynomial chaos expansion method. Quantification of the stochastic cost, boundary and path constraints in terms of polynomial chaos expansion is described in detail in a straightforward way. Through the application of the proposed procedure to two examples of optimal trajectory generation, it is observed that the obtained optimal solutions are evidently less sensitive to uncertainties and more reliable compared to that of the deterministic optimization, which demonstrates the effectiveness of the proposed method.