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Spectral variational integrators for trajectory optimization under parametric uncertainties and stochastic disturbances

We propose a novel methodology for stochastic trajectory optimization which is based on merging the theory of spectral expansions with Differential Dynamic Programming. Specifically, we employ polynomial chaos expansions to handle parametric uncertainties and utilize the Karhunen-Loéve transformation to represent stochastic forces. This allows us to build a generic framework and avoid relying on restrictive assumptions regarding the form of stochasticity. In addition, Differential Dynamic Programming provides an iterative algorithm for finding optimal controls which attains scalability and, under mild assumptions, fast convergence. The obtained method is capable of controlling the distribution of the system's trajectory and can be used in both planning and control. Last but not least, we formulate a variational integration scheme for uncertain systems that are represented by spectral expansions. Simulated examples validate the applicability and numerical efficiency of the proposed approach.