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Talk Abstract

Legged Locomotion from the Symmetry Viewpoint

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

1/4 3/4font0 1/2back left right

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 2*n* 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.