M. Golubitsky, I. Stewart, P.L. Buono and J.J. Collins
A modular network for legged locomotion.
Physica D. 115 (1998) 56-72.
In this paper we use symmetry methods to study networks of coupled cells, which are models for central pattern generators (CPGs). In these models the cells obey identical systems of differential equations and the network specifies how cells are coupled. Previously, Collins and Stewart showed that the phase relations of many of the standard gaits of quadrupeds and hexapods can be obtained naturally via Hopf bifurcation in small networks. For example, the networks they used to study quadrupeds all had four cells, with the understanding that each cell determined the phase of the motion of one leg. However, in their work it seemed necessary to employ several different four-oscillator networks to obtain all of the standard quadrupedal gaits.
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 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.