Continuous safety-critical control of Euler–Lagrange systems subject to multiple obstacles and velocity constraints

This paper studies the safety-critical control problem for Euler–Lagrange (EL) systems subject to multiple ball obstacles and velocity constraints. A key strategy is to exploit the underlying cascade structure of EL systems to design a new safety-critical controller featuring an inner–outer-loop structure. In particular, the outer-loop control law is developed based on quadratic programming (QP) to avoid ball obstacles and generate velocity reference signals fulfilling the velocity limitation. Taking full advantage of the energy conservation property, a nonlinear velocity-tracking control law is designed to form the inner loop. One major difficulty is caused by the possible non-Lipschitz continuity of the standard QP algorithm when there are multiple constraints. To solve this problem, we propose a new feasible-set reshaping technique such that the refined QP algorithm with the reshaped feasible set admits a Lipschitz continuity property. Additionally, inspired by small-gain analysis, we construct a max-type Lyapunov-like function to integrate the safety constraints and the velocity-tracking error, and prove the achievement of the safety-critical control objective. The effectiveness of the proposed design is validated through numerical simulations and experiments on a 2-link planar manipulator.