Safety in the face of uncertainty is a key challenge in robotics. In this work, we propose a real-time capable framework to generate safe and task-efficient robot trajectories for stochastic control problems. For that, we first formulate the problem as a chance-constrained optimisation problem, in which the probability of the controlled system to violate a safety constraint is constrained to be below a user-defined threshold. To solve the chance-constrained optimisation problem, we propose a Monte--Carlo approximation relying on samples of the uncertainty to estimate the probability of violating a safety constraint given a controller. We use this approximation in the motion planner VP-STO to solve the sampled-based problem. Consequently, we refer to our adapted approach as CC-VPSTO, which stands for Chance-Constrained VP-STO. We address the crucial issue concerning the Monte--Carlo approximation: given a predetermined number of uncertainty samples, we propose several ways to define the sample-based problem such that it is a reliable over-approximation of the original problem, i.e. any solution to the sample-based problem adheres to the original chance-constrained problem with high confidence. The strengths of our approach lie in i) its generality, as it does not require any specific assumptions on the underlying uncertainty distribution, the dynamics of the system, the cost function, and for some of the proposed sample-based approximations, on the form of inequality constraints; and ii) its applicability to MPC-settings. We demonstrate the validity and efficiency of our approach on both simulation and real-world robot experiments. For additional material, please visit https://sites.google.com/oxfordrobotics.institute/cc-vpsto.

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