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 in accordance with affordable velocity ranges. A key strategy is to exploit the underlying inner-outer-loop structure for the design of a new cascade controller for the class of EL systems. In particular, the outer-loop controller 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 conservation-of-energy property, a nonlinear velocity-tracking controller 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 refined QP algorithm with the feasible set reshaped by an appropriately chosen positive basis such that the feasibility is retained while the resulting outer-loop controller is locally Lipschitz. It is proved that the constraint-satisfaction problem is solvable as long as the ball obstacles satisfy a mild distance condition. The proposed design is validated by numerical simulation and an experiment based on a $2$-link planar manipulator.