A Modular Network for Legged Locomotion
Physica D. 115 (1998) 56-72.
We show that this difficulty with four-oscillator networks is unavoidable, but that the problems can be overcome by using a larger network. Specifically, we show that the standard gaits of a quadruped, including walk, trot and pace, cannot all be realized by a single four-cell network without introducing unwanted conjugacies between trot and pace --- conjugacies that imply a dynamic equivalence between these gaits that seems inconsistent with observations. In this sense a single network with four cells cannot model the CPG of a quadruped. We also introduce a single eight-cell network that can model all of the primary gaits of quadrupeds without these unwanted can model all of the primary gaits of quadrupeds without these unwanted conjugacies. Moreover, this network is modular in that it naturally generalizes to provide models of gaits in hexapods, centipedes, and millipedes. The analysis of models for many-legged animals shows that wave-like motions, similar to those obtained by Kopell and Ermentrout, can be expected. However, our network leads to a prediction that the wavelength of the wave motion will divide twice the length of the animal. Indeed, we reproduce illustrations of wave-like motions in centipedes where the animal is approximately one and a half wavelengths long --- motions that are consistent with this prediction. We discuss the implications of these results for the development of modular control networks for adaptive legged robots.