  We analyze two- and three-link planar snake-like locomotion and optimize the
motion for efficiency. The locomoting system consists of two or three identical
inextensible links connected via hinge joints, and the angles between the links
are actuated as prescribed periodic functions of time. An essential feature of
snake locomotion is frictional anisotropy: the forward, backward and transverse
coefficients of friction are different. The dynamics are studied analytically
and numerically for small and large amplitudes of the internal angles.
Efficiency is defined as the ratio between distance traveled and the energy
expended within one period, i.e. the inverse of the cost of locomotion. The
optimal set of coefficients of friction to maximize efficiency consists of a
large backward coefficient of friction and a small transverse coefficient of
friction, compared to the forward coefficient of friction. For the two-link
case with a symmetrical motion, efficiency is maximized when the internal angle
amplitude is approximately $\pi/2$, for transverse coefficient sufficiently
large. For the three-link case, the efficiency-maximizing paths are triangles
in the parameter space of internal angles.
