Safety-Critical Control of Euler-Lagrange Systems Subject to Position and Velocity Constraints

This paper presents a study on safety-critical control problems for Euler-Lagrange (EL) systems subject to multiple safety constraints. Specifically, we focus on position constraints characterized by a set of ball obstacles and velocity constraints that adhere to feasible velocity ranges. Our novel contribution lies in a new cascade design of safety-critical controllers, which feature an inner-outer-loop structure. We have developed an outer-loop controller based on quadratic programming (QP) to handle position constraints and generate velocity reference signals that conform to velocity limitations. By fully utilizing the energy-conservation property, we have designed a nonlinear velocity-tracking controller to form the inner loop. However, a significant challenge is posed by the non-Lipschitz continuity of the standard QP algorithm when there are multiple constraints. To address this issue, we propose a refined QP algorithm with the feasible set reshaped by an appropriately chosen positive basis to ensure that feasibility is maintained while the resulting outer-loop controller is locally Lipschitz. We have proven that safety-critical control can be achieved as long as the ball obstacles satisfy a mild distance condition. Finally, we have validated our proposed design via numerical simulations of safety-critical control of a 3-link planar manipulator.