Chance constraints transcription and failure risk estimation for stochastic trajectory optimisation

TL;DR

This paper introduces two new general-purpose methods for transcribing multi-dimensional Gaussian chance constraints in stochastic trajectory optimization, reducing conservatism and computational complexity.

Contribution

It presents the spectral radius and refined first-order methods, along with a high-dimensional risk estimation technique, improving robustness and efficiency in uncertain trajectory planning.

Findings

Spectral radius method reduces conservatism in multi-dimensional constraints.

Refined first-order method achieves tighter bounds with linear complexity.

Risk estimation method provides accurate failure probabilities with limited conservatism in high dimensions.

Abstract

Stochastic trajectory optimisation under uncertainty requires robust constraint satisfaction through chance constraints. However, existing transcription methods remain limited to scalar constraints or highly specific structures while introducing substantial conservatism. This work presents two general-purpose transcription methods for multi-dimensional Gaussian chance constraints for trajectory optimisation problems under uncertainty. The spectral radius method extends existing methods to arbitrary multi-dimensional constraints with reduced conservatism. The refined first-order method achieves superior tightness with linear complexity. In addition, a d-th order risk estimation methodology provides conservative failure probability estimates with limited conservatism in high dimensions in quadratic complexity. Applied to an optimal control with uncertainties setting, the first-order transcription achieves near-optimal fuel consumption while maintaining the failure risk below the target. The spectral radius method incurs approximately 0.7 kg additional fuel consumption due to excessive conservatism and a 51% increase in computational time due to its cubic complexity. High-dimensional tests show that the proposed risk estimation method provides accurate risk estimates, while previously developed methods exhibit exponential growth in conservatism with respect to constraint dimension.