Robust trajectory optimization combining Gaussian mixture models with stochastic collocation

This paper presents a method for extending the generalized polynomial chaos method with stochastic collocation, so that expansion coefficients for further continuous distributions other than the standard distributions of the generalized polynomial chaos can be calculated. This is achieved by using a Gaussian mixture model to approximate the desired arbitrary continuous input distribution. This incorporation of the Gaussian mixture model basically yields a repetitive solution of only Hermite-chaos problems. The developed method was applied to a time-optimal trajectory optimization problem for a fighter aircraft using the MATLAB^®-based toolbox, FALCON.m [1]. Within the generalized polynomial chaos stochastic collocation framework only the repetitive solution of standard, deterministic trajectory optimization problems at Gaussian quadrature nodes were required. This makes the stochastic collocation approach very efficient for carrying out robust trajectory optimizations. The developed framework was tested against a Jacobi-chaos problem, a multi-modal Gaussian distribution, and a Weibull distribution. The latter two were validated by comparing them to Latin-Hypercube-Sampling.