Efficient sliding locomotion with isotropic friction

Snakes' bodies are covered in scales that make it easier to slide in some directions than in others. This frictional anisotropy allows for sliding locomotion with an undulatory gait, one of the most common for snakes. Isotropic friction is a simpler situation (that arises with snake robots, for example) but is less understood. In this work we regularize a model for sliding locomotion to allow for static friction. We then propose a robust iterative numerical method to study the efficiency of a wide range of motions under isotropic Coulomb friction. We find that simple undulatory motions give little net locomotion in the isotropic regime. We compute general time-harmonic motions of three-link bodies and find three local optima for efficiency. The top two involve static friction to some extent. We then propose a class of smooth body motions that have similarities to concertina locomotion (including the involvement of static friction) and can achieve optimal efficiency for both isotropic and anisotropic friction.