Analytic Non-Gaussian Confidence Boundary Method for Chance-Constrained Trajectory Control

TL;DR

This paper introduces a non-Gaussian confidence boundary method for chance-constrained control, improving accuracy in highly nonlinear systems where Gaussian assumptions fail.

Contribution

It develops a perturbation-based technique using higher-order moments to better approximate confidence boundaries in non-Gaussian distributions.

Findings

Outperforms Gaussian-based methods in non-Gaussian scenarios

Effective for 'banana-shaped' distributions in orbital mechanics

Applied successfully to spacecraft maneuver control

Abstract

Standard chance constrained control algorithms typically rely on the assumption that uncertainties in vehicle states obey Gaussian statistics. Highly nonlinear systems tend to disrupt Gaussianity, challenging standard chance-constrained control methods. This paper develops a non-Gaussian confidence boundary parameterization technique for such cases where the problem departs appreciably from the Gaussian assumption. The approach is to consider the true confidence boundary as a perturbation of the one predicted from covariance, deriving perturbed boundary geometry from computed higher-order statistical moments. Applying this technique to so-called "banana-shaped distributions" (found e.g. in orbital mechanics problems) enables a simple parameterization of the confidence boundary using the skew and kurtosis tensors. The method is then applied to an impulsive stochastic spacecraft maneuver targeting problem in two-body dynamics. An algorithmic implementation outperforms a standard linear covariance-based approach in computing control parameters satisfying certain probabilistic bounds on the non-Gaussian distribution.