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Talk Abstract
Legged Locomotion from the Symmetry Viewpoint

Institute for Mathematics and Its Applications
Ian Stewart, University of Warwick

This talk is on joint work with Marty Golubitsky, Luciano Buono, and Jim Collins.

Legged locomotion involves many different gait patterns --- for example the horse can walk, trot, canter, or gallop. A key feature of these patterns is phase-locking: each leg moves periodically, with the same period for all legs, with a fixed pattern of phase relationships. For instance the walk gait has the phase pattern

               
	
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where the phases are stated as fractions of one complete period.

For many gaits (but not all) the phases are simple fractions of a full period. Moreover, many gaits exhibit spatio-temporal symmetries, and their phase patterns can be derived from those symmetries.

It is believed that the basic rhythms of legged gaits are set up by neural circuits known as Central Pattern Generators (CPGs). The above facts suggest that CPG architecture is symmetric, and that the phase patterns are `universal' patterns of symmetry-breaking oscillations.

Early work by Collins and Stewart (also Kelso and Schöner) showed that a small number of circuits can generate all the symmetric quadruped gaits. However, these authors did not identify a single circuit that could generate all quadruped gaits. Another difficulty was `spurious conjugacy': for instance any 4-oscillator network that can generate walk and pace can also generate trot, and trot and pace have identical dynamical characteristics.

These difficulties disappear if a suitable 8-oscillator network (schematic) is employed. This network has a simple architecture that relates sensibly to known physiology. A generalisation to 2n oscillators applies to n-legged animals. The circuit has a tidy `modular' structure that may make it easily applicable to legged robots. Moreover, the 8-oscillator circuit can exhibit `mixed mode' patterns corresponding to the rotary and transverse gallops, and to the canter. This appears to be the first model that includes a natural canter.

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