Trajectory Optimization Under Stochastic Dynamics Leveraging Maximum Mean Discrepancy

This paper addresses sampling-based trajectory optimization for risk-aware navigation under stochastic dynamics. Typically such approaches operate by computing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\tilde{N}$</tex-math></inline-formula> perturbed rollouts around the nominal dynamics to estimate the collision risk associated with a sequence of control commands. We consider a setting where it is expensive to estimate risk using perturbed rollouts, for example, due to expensive collision-checks. We put forward two key contributions. First, we develop an algorithm that distills the statistical information from a larger set of rollouts to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reduced-set</i> with sample size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N&lt; &lt; \tilde{N}$</tex-math></inline-formula>. Consequently, we estimate collision risk using just <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> rollouts instead of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\tilde{N}$</tex-math></inline-formula>. Second, we formulate a novel surrogate for the collision risk that can leverage the distilled statistical information contained in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reduced-set</i>. We formalize both algorithmic contributions using distribution embedding in Reproducing Kernel Hilbert Space (RKHS) and Maximum Mean Discrepancy (MMD). We perform extensive benchmarking to demonstrate that our MMD-based approach leads to safer trajectories at low sample regime than existing baselines using Conditional Value-at Risk (CVaR) based collision risk estimate.