Hybrid under-actuated control for autonomous dynamic balance of a running electrical bicycle including motor dynamics and in the presence of huge uncertainty

Hybrid under-actuated control for the autonomous dynamic balance of a running electrical bicycle including frictional torque and motor dynamics is developed, which includes two control inputs: steering and pendulum voltages, and three system outputs: steering, lean and pendulum angles. Due to the under-actuated feature, two novel reference signals using three system outputs are designed so that the number of control inputs and sliding surfaces is the same. According to the input-output data of each subsystem, two scaling factors for each subsystem are first employed to normalize the sliding surface and its derivative. Based on the concept of if-then rule, an appropriate rule table for the ith subsystem is obtained. The output scaling factor based on Lyapunov stability is also determined. It is called fuzzy decentralized sliding mode under-actuated control (FDSMUC). Because the uncertainties of a running electrical bicycle system, caused by different ground conditions, gusts of wind, and interactions among subsystems, are often huge, an extra compensation of learning uncertainty is plunged into FDSMUC to enhance the system performance. We call it fuzzy decentralized sliding mode adaptive under-actuated control (FDSMAUC). To avoid the unnecessary transience caused by uncertainties and preserve the dynamic balance, the combination of FDSMUC and FDSMAUC with a transition (i.e., Hybrid FDSMUC) is suggested. The stability of the closed-loop system using the proposed control is verified by Lyapunov stability theorem. Finally, the compared simulations validate the efficiency among the FDSMUC, FDSMAUC and Hybrid FDSMUC.