Analysis of a distributed model of leg coordination - I. Individual coordination mechanisms

Using tools from discrete dynamical systems theory, we begin a systematic a nalysis of a distributed model of leg coordination with both biological and robotic applications. In this paper, we clarify the role of individual coo rdination mechanisms by studying a system of two leg oscillators coupled in one direction by each of the three major mechanisms that have been describ ed for the stick insect Carausius morosus. For each mechanism, we derive an alytical return maps, and analyze the behavior of these return maps under i teration in order to determine the asymptotic phase relationship between th e two legs. We also derive asymptotic relative phase densities for each mec hanism and compare these densities to those obtained from numerical simulat ions of the model. Our analysis demonstrates that, although each of these m echanisms can individually compress a range of initial conditions into a na rrow band of relative phase, this asymptotic relative phase relationship is , in general, only neutrally stable. We also show that the nonlinear depend ence of relative phase on walking speed along the body in the full hexapod model can be explained by our analysis. Finally, we provide detailed parame ter charts of the range of behavior that each mechanism can produce as coup ling strength and walking speed are varied.