Geometric mechanics reveals optimal coordination of limb and body waves during centipede locomotion

Running epimorphic centipedes display a characteristic body undulation. Elucidating the role of body-leg coordination is difficult because of the many degrees of freedom in the flexible body and numerous limbs. We decomposed the centipede locomotion into body undulation and leg stepping waves, and used the mathematical framework of geometric mechanics to study body-leg coordination. We characterized the body-leg coordination by the phase offset between two waves. We recorded high-speed video and tracked the motion of a centipede (Scolopendra polymorpha, 5 trials) running on a treadmill. Our theory predicted that the speed (in body lengths per cycle, BLC) was maximized when the body undulation (1.6 spatial waves) had a phase of 2.50 rad leading that of the leg wave (2 spatial waves). The theoretical prediction of centipede speed was 0.40 BLC, 0.10 BLC and 0.25 BLC with best, worst and no body undulation respectively. Experiments revealed that the centipede moved at 0.41±0.05 BLC with the phase offset 2.51±0.71 rad, in good agreement with our theoretical prediction. Our preliminary results suggest that the geometric approach captures centipede locomotion dynamics without including inertial effects.