Mathematical Modeling and Vision-Guided Triple-Loop Control of an Underactuated Bicycle Robot

This paper presents a mathematical modeling-based vision-guided triple-loop control method for lane tracking of an underactuated bicycle robot. To describe the coupling between lateral balance and path tracking, a reaction-wheel-based inverted-pendulum model is established using the Lagrange formulation. Based on the linearized dynamics, the transfer function between the flywheel rotational speed and the motor torque is derived, providing a mathematical basis for designing the gain-scheduled triple-loop PID controller. To generate continuous control inputs under practical visual disturbances, an improved Hough transform, a near-field multi-layer sliding window detector, and a multi-scenario finite-state-machine strategy are incorporated for lateral deviation estimation and path reconstruction. A cascaded smoothing filter is further introduced to reduce high-frequency command fluctuations and improve the closed-loop control response. Real-vehicle experiments on an STM32F407-based underactuated bicycle robot demonstrate that the proposed framework achieves stable dynamic balance and robust lane tracking. Compared with a conventional Hough-transform and sliding window method, the lateral RMSE is reduced by 40.2%, 39.85%, and 32.35% in straight, left-turn, and right-turn scenarios, respectively.