Optimizing snake locomotion in the plane

Silas Alben

School of Mathematics

University of Michigan




Snake locomotion has recently drawn interest from biologists, engineers and applied mathematicians. Snakes propel themselves by a variety of gaits including slithering, sidewinding, concertina motion and rectilinear progression. We develop a numerical scheme to determine which planar snake motions are optimal for locomotory efficiency, across a wide range of frictional parameter space. For a large coefficient of transverse friction, we show that retrograde traveling waves are optimal. We give an asymptotic analysis showing that the optimal wave amplitude decays as the -1/4 power of the coefficient of transverse friction. This result agrees well with the numerical optima. At the other extreme, zero coefficient of transverse friction, we propose a triangular direct wave which is optimal. Between these two extremes, a variety of complex, locally optimal, motions are found. Some of these can be classified as standing waves (or ratcheting motions).