Unscented Trajectory Optimization

TL;DR

This paper reintroduces unscented trajectory optimization as a special case of tychastic optimal control, providing a systematic method for generating optimal trajectories under uncertainty without complex stochastic calculus.

Contribution

It presents a novel instantiation of tychastic optimal control using the unscented transform, simplifying stochastic trajectory optimization and enabling effective management of uncertainties.

Findings

Effective risk reduction demonstrated in numerical examples

Rapid formulation and computation of optimal trajectories

Management of trajectory dispersions with nonlinear transformations

Abstract

In a nutshell, unscented trajectory optimization is the generation of optimal trajectories through the use of an unscented transform. Although unscented trajectory optimization was introduced by the authors about a decade ago, it is reintroduced in this paper as a special instantiation of tychastic optimal control theory. Tychastic optimal control theory (from \textit{Tyche}, the Greek goddess of chance) avoids the use of a Brownian motion and the resulting It\^{o} calculus even though it uses random variables across the entire spectrum of a problem formulation. This approach circumvents the enormous technical and numerical challenges associated with stochastic trajectory optimization. Furthermore, it is shown how a tychastic optimal control problem that involves nonlinear transformations of the expectation operator can be quickly instantiated using an unscented transform. These nonlinear transformations are particularly useful in managing trajectory dispersions be it associated with path constraints or targeted values of final-time conditions. This paper also presents a systematic and rapid process for formulating and computing the most desirable tychastic trajectory using an unscented transform. Numerical examples are used to illustrate how unscented trajectory optimization may be used for risk reduction and mission recovery caused by uncertainties and failures.