Sufficient conditions for the Lipschitz continuity of QP-based multi-objective control of humanoid robots

In this paper we analyze the continuity properties of feedback controllers that are formulated as state-dependent quadratic programs (QP), with specific application to motion control for humanoid robots. With a desire to achieve multiple simultaneous goals in locomotion and manipulation, we develop a generalized QP-based control law through the use of multiple control Lyapunov functions (CLFs). Motivated by simulation studies showing cases where QP-based control loses Lipschitz continuity, the main result of this paper is a set of sufficient conditions under which such continuity properties will hold. This result provides conditions under which any number of tasks encoded as CLFs can be simultaneously exponentially stabilized. Finally, these results are demonstrated in a simulation of a simple humanoid robot climbing a vertical ladder.