Optimal Input Waveform for an Indirectly Controlled Limit Cycle Walker

Abstract

Precisely manipulating the center of mass (CoM) of the underactuated locomotion robot can't be easily achieved by common control mechanisms which apply only joint torques. A novel and indirect method has been recently introduced using an active wobbling mass attached to limit cycle walkers. The next important issue is to design an optimal control input to reduce the forcing energy. In this paper, we use combined rimless wheels as a simplified example to apply our method, which is based on the theory of phase oscillators. First, we introduce the typical modeling and control of this underactuated robot. Second, we obtain the phase response curve by numerically applying perturbations at different phases of the walker's gait interval and calculating the deviations from the unperturbed. Third, we analytically derive an optimal forcing waveform for the wobbling mass to entrain the combined rimless wheel based on the phase response curve. As an ecological extension, an ideal forcing waveform for m: 1 entrainment was further generated. Finally, the proposed method was evaluated by locking range of the Arnold tongues. The results show that the optimal forcing waveform we derived achieves the best performance for 1: 1 entrainment among all the candidates. One of the strongest advantages of our method is the easiness of its implementation, prompting its applicability to a wide variety of locomotion systems.