Efficient sliding locomotion of three-link bodies with inertia

Abstract

Many previous studies of sliding locomotion have assumed that body inertia is negligible. Here we optimize the kinematics of a three-link body for efficient locomotion and include among the kinematic parameters the temporal period of locomotion, or equivalently, the body inertia. The optimal inertia is non-negligible when the coefficient of friction for sliding transverse to the body axis is small. Inertia is also significant in a few cases with relatively large coefficients of friction for transverse and backward sliding, and here the optimal motions are less sensitive to the inertia parameter. The optimal motions seem to converge as the number of frequencies used is increased from one to four. For some of the optimal motions with significant inertia we find dramatic reductions in efficiency when the inertia parameter is decreased to zero. For the motions that are optimal with zero inertia, the efficiency decreases more gradually when we raise the inertia to moderate and large values. I. INTRODUCTIONThis work addresses the physics of terrestrial locomotion, part of a larger field of interdisciplinary studies of the locomotion of organisms, robots, and vehicles, often bio-inspired [1][2][3][4][5][6][7]. Aerial and aquatic locomotion problems are often dominated by the interaction between a locomoting body (or bodies) and the surrounding fluid, and may involve complicated fluid dynamics [4,7]. Terrestrial locomotion is usually dominated by local contact forces involving friction [3,5,6,8,9], which may also be complicated to characterize. Here we study sliding locomotion inspired by biological and robotic snakes [3,5,6,[10][11][12]. As with several recent models [6,[13][14][15][16][17][18][19][20][21], we use a local Coulomb friction force model for the interaction between the body and the surface that it slides across. The problem is similar to a larger body of work that has used resistive force theory to approximate fluid forces on swimming bodies in the viscousdominated (zero-Reynolds-number or Stokes) regime [2,[22][23][24][25]. For the sliding locomotion problem here we adopt the three-link body that has been used Purcell and many others; it is one of the simplest bodies that can locomote at zero Reynolds number, by performing time-irreversible (e.g. undulatory) motions [2,23,[26][27][28]. Unlike the Stokes swimmers, for sliding locomotion the body's inertia can play an important role, though it has mostly been neglected for simplicity [16][17][18]29]. Neglecting inertia allows one to analyze this system and the three-link swimmer using a geometrical-mechanics framework [19,[30][31][32][33].The body's inertia can be neglected for motions with small accelerations, which is a reasonable approximation for