Safety-Critical Control of a Class of Underactuated Systems: A Case Study on Wheeled Inverted Pendulum

This paper studies the safety-critical control of a class of underactuated systems, with a focus on a wheeled inverted pendulum as a case study. A safety-critical controller is desired to simultaneously keep the pendulum upright, restrict the position of the wheel within a desired range, and ensure that the rotational angular velocity of the wheel tracks a nominal reference signal as closely as possible. We first reformulate the safety constraints as saturated constraints on the state variables of a cascade connection of an integrator and a feedforward system, thereby mitigating conflicts among the safety constraints. Then, to ensure the satisfaction of the constraints, we propose a safety-critical controller that consists of a safety-critical virtual controller based on control barrier functions (CBFs) and a tracking controller designed using forwarding methods. Nonlinear small-gain techniques are employed to ensure the stable integration of the two controllers and the achievement of the safety-critical control objectives.