Robust adaptive control Lyapunov-barrier function for non-collocated control and safety of underactuated robotic systems

Quadratic program-based control Lyapunov-control barrier function (QP-CLBF) is a recently developed control scheme that balances stability and safety in an optimal fashion. However, direct application of such controllers to uncertain nonlinear systems with model uncertainties and disturbances could potentially degrade the performance of closed-loop systems and violate/restrict safety-critical constraints. This article presents a novel robust QP-based adaptive approach for non-collocated control of a class of underactuated robotic systems with a semi-strict-feedback form through which exponential disturbance-to-error (eDE) stability of all system solutions is ensured. We begin by developing a backstepping design technique based on which a neural network-based adaptive control is designed to approximate unknown nonlinear functions. To compensate for uncanceled uncertainties, including modeling approximation error, chained error effects between coordinates, and disturbances, virtual inputs are designed whose gains are evolved by projection-based adaptation mechanisms. To construct a three-term control law, including feedforward, adaptive, and optimal terms, a QP optimization problem is synthesized to compute a family of optimal stabilizing signals while encoding time-varying robust CLF (TVRCLF) and CBF (TVRCBF), and control bounds as constraints. In contrast with existing QP-CLBF, our control technique with the proposed TVRCLF and TVRCBF significantly improves control objectives and strictly guarantees safety by automatically compensating for unknown uncertainties without the need for knowing their bounds a priori. The eDE stability of all system errors is proven using Lyapunov stability arguments for both collocated and non-collocated coordinates. Simulations and comparisons to a baseline QP-CLBF-based feedback linearization (QP-CLBF/FL) on a single-link flexible-joint robot verify the benefits of the proposed control approach.