Continuous Safety-Critical Control of Mobile Robots With Set-Valued Feedback in Body-Fixed Coordinate Frame

This paper investigates a safety-critical control problem for a mobile robot navigating cluttered environments. Obstacles are represented as a closed set, and a direction-distance function is used to model the obstacle measurement process. Based on real-time obstacle measurements, a safety-critical controller is expected to maintain a safe distance between the robot and obstacles while ensuring that the robot's velocity, considered as the control input, follows a given velocity command signal as closely as possible. This paper contributes a regularization method for the obstacle-measurement model in the body-fixed frame and a systematic design of a Lipschitz continuous safety-critical controller based on the regularized obstacle-measurement model and quadratic programming (QP). Specifically, a novel Lipschitz-truncated regularization term is used to replace the conventional quadratic regularization in Moreau-Yosida regularization. It is guaranteed that, for each local minimum distance given by the original direction-distance function and its corresponding direction, there exists a neighborhood around that direction where the value of the regularized direction-distance function does not exceed the local minimum distance. This property is crucial for constructing a feasible set of control input that ensures safety in all directions using only a finite number of constraints. The QP framework is then employed to integrate these safety constraints with nominal velocity commands. A specific positive basis is employed to meet the linear independence constraint qualification, ensuring the Lipschitz continuity of the QP-based safety-critical controller. The effectiveness of the proposed design is demonstrated through numerical simulations and physical experiments.