Opposite Treatment on Null Space: A Unified Control Framework for a Class of Underactuated Robotic Systems With Null Space Avoidance

Design of feedback control for underactuated robotic systems (URSs) is often complex and nonintuitive due to the inherent incapability of generating arbitrary velocities or accelerations at each moment. Researchers have designed various controllers; however, there are few unified intuitive feedback stabilization controllers for a wide range of URSs. In this article, we propose a real-time and unified control framework for a class of URSs with three states and two inputs based on the intuition of avoiding null space. To control diverse URSs, unlike transforming them to canonical forms, we approach them from an inverse perspective based on a Jacobian-like mapping matrix and disclose a fundamental issue, null space attraction, which fails all Jacobian-based standard kinematics control. An opposite treatment on null space, avoiding null space, is applied to oppose null space attraction, allowing the standard kinematics controllers to keep control authority. Overall, the proposed control framework consisting of a standard kinematics controller and a null space avoidance controller can fundamentally solve point-to-point stabilization for the class of driftless URSs and also render <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta $ </tex-math></inline-formula> -neighborhood stabilization for URSs with a nonvanishing and constant drift. Both the convergence properties are rigorously analyzed. Simulations and applications are conducted to demonstrate the generality, effectiveness, and robustness of our controllers.