L₁ Adaptive Control Barrier Functions for Nonlinear Underactuated Systems

Abstract

This paper presents a novel integration of adaptive control and control barrier functions that offers tracking stability as well as safety-critical constraints for nonlinear underactuated systems in the presence of model uncertainty. The proposed method is based on L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> adaptive control with a nonlinear closed-loop reference model based on control barrier functions. For underactuated systems, adaptation based on the control output does not imply adaptation in the system state. Therefore, to guarantee adaptive constraint enforcement which depends on the entire state, we introduced a modified reference state to represent the zero dynamics or internal states of the real system inside the reference model. We evaluate our proposed control design for the problem of dynamic walking of an underactuated bipedal robot subject to safety-critical constraints of foot placements on stepping stones under significant model uncertainty. We present numerical results on RABBIT, a five-link planar bipedal robot carrying a large unknown load on its torso. Our proposed controller is able to demonstrate walking while strictly enforcing the above constraints with an unknown load of up to 30 Kg (94% of the robot mass).