Optimizing Snake Locomotion on a Plane: A Variational and Analytical Framework for Friction and Normal Force Modulation

Abstract

Snake locomotion represents one of the most versatile and efficient forms of limbless movement across diverse terrains. This thesis develops an analytical framework to investigate the optimal body shapes and gaits that maximize locomotor efficiency and speed for elongated bodies interacting with frictional substrates. By assuming steady, periodic deformations and utilizing resistive force theory (RFT), we derive variational optimization conditions for minimizing the mechanical cost of transport (mCoT) and maximizing forward displacement. The results show that sawtooth-shaped body waveforms—characterized by alternating constant body angles—achieve optimal efficiency and velocity. In anisotropic environments, where transverse friction dominates, lateral undulation is both energetically favorable and effective for propulsion. However, in isotropic environments, lateral undulation fails to produce net movement due to symmetry constraints. We extend the framework to analytically derive two additional natural gaits: sinus lifting, which restores net movement in isotropic media by periodic lifting of body segments; and sidewinding, which achieves exceptionally high efficiency and displacement across both anisotropic and isotropic environments. Further generalization of the model demonstrates that the underlying optimization principles apply across a wide range of resistive environments, including granular media and low-Reynolds-number viscous fluids, highlighting a universal tendency toward sawtooth-like kinematics in efficient undulatory locomotion. This work provides a unified theoretical foundation for understanding the biomechanics of limbless locomotion and offers guiding principles for the design of efficient snake-like robotic systems.