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Optimal Fast Entrainment Waveform for Indirectly Controlled Limit Cycle Walker Against External Disturbances

After occasional perturbation, it is crucial to spontaneously control the limit cycle walking so that it quickly returns to its closed orbit in phase space. Otherwise, its stability can not be sufficiently guaranteed if the speed of recovery is slow while successive perturbation is applied. The accumulated deviation may eventually drive the phase outside the basin of attraction, leading to failure of the walking. In this sense, a control law that quickly recovers the disturbed phase before encountering the following perturbations is indispensable. With this consideration, here we analytically derive an optimal fast entrainment waveform that maximizes the speed of phase recovery based on phase reduction theory. Our theoretical method is numerically evaluated using a limit cycle walker, which is indirectly controlled by the oscillation of a wobbling mass via entrainment effect. The obtained waveform is used as the desired trajectory of the wobbling motion. The simulation results show that the waveform we derived achieves the best performance among all candidates. Our method helps to enhance the stability of limit cycle walking.