Wasserstein Distributionally Robust Motion Control for Collision Avoidance Using Conditional Value-at-Risk

Abstract

In this article, a risk-aware motion control scheme is considered for mobile robots to avoid randomly moving obstacles when the true probability distribution of uncertainty is unknown. We propose a novel model-predictive control (MPC) method for limiting the risk of unsafety even when the true distribution of the obstacles’ movements deviates, within an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ambiguity set</i> , from the empirical distribution obtained using a limited amount of sample data. By choosing the ambiguity set as a statistical ball with its radius measured by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wasserstein metric</i> , we achieve a probabilistic guarantee of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">out-of-sample risk</i> , evaluated using new sample data generated independently of the training data. To resolve the infinite-dimensionality issue inherent in the distributionally robust MPC problem, we reformulate it as a finite-dimensional nonlinear program using modern distributionally robust optimization techniques based on the Kantorovich duality principle. To find a globally optimal solution in the case of affine dynamics and output equations, a spatial branch-and-bound algorithm is designed using McCormick relaxation. The performance of the proposed method is demonstrated and analyzed through simulation studies using nonlinear dynamic and kinematic vehicle models and a linearized quadrotor model. The simulation results indicate that, even when the sample size is small, the proposed method can successfully avoid randomly moving obstacles with a guarantee of out-of-sample risk, while its sample average approximation counterpart fails to do so.