Time-Optimal Point Stabilization Control for WIP Vehicles Using Quasi-Convex Optimization and B-Spline Adaptive Interpolation Techniques

This paper addresses a time-optimal point stabilization control for inherent unstable wheeled inverted pendulum (WIP) vehicles using quasi-convex optimization and B-spline adaptive interpolation techniques. First, to handle the difficulty caused by the inherent unstable characteristic, a state feedback control law is introduced to the state-space model deduced with the kinematic coupling relationship between the longitudinal motion and tilt angle of vehicle body. Then, by system discretization, a standard quasi-convex optimization problem is formulated to plan a time-optimal trajectory with various constraints being taken into account, and the optimization problem can be solved by a bisection method combined with solving a series of convex feasibility problems. Next, to obtain the analytic expression of the discrete optimal trajectory, a B-spline interpolation algorithm with adaptive curve refinement is presented based on feature points recognition. Furthermore, as the time-optimal trajectories with analytic expressions, including displacement, velocity, and acceleration trajectories, have been obtained, a proportional-integral-derivative (PID) tracking controller is applied to control the vehicle to track the optimal trajectories and, thus, the time-optimal point stabilization control for the WIP vehicle can be realized. Finally, simulation results of a numerical example is presented to validate the feasibility and effectiveness of the proposed method.