Efficient sliding locomotion with isotropic friction

Abstract

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. I. INTRODUCTIONSnake locomotion has attracted the interest of biologists and engineers for several decades [1][2][3][4][5][6]. Many locomoting animals use appendages such as legs, wings, or fins to exert a force on the substrate or surrounding fluid, and propel the rest of the body forward [7]. Snakes lack appendages, and thus it is less clear which parts of the snake body should exert propulsive forces, and at which instants during the motion, to move forward efficiently.A typical way to understand how organisms move is to study physical or computational models and compare their motions with those of the actual organisms [7][8][9][10][11]. One can take a step further and pose and solve optimization problems for the models. This can suggest locomotion strategies that are effective for man-made vehicles [12][13][14]. It can also help understand why organisms have evolved in particular ways under a multitude of constraints [15][16][17].Often what is optimized is a measure of the efficiency of locomotion. For example, one can maximize the average speed for a given time-averaged power expended by the organism. One can study the effects of physical parameters and constraints by varying them and studying how the optimal solutions change. Well-known examples are optimization studies of organisms moving in low-[18-24] and high-Reynolds-number fluid flows [25][26][27][28][29][30][31]. For locomotion in frictional (terrestrial or granular) media, frictional forces can result in distinctive modes of efficient (or optimal) locomotion [32,33].Snakes are limbless reptiles with elongated bodies, supported by a backbone with 100-500 bony segments (vertebrae) * Electronic address: alben@umich.edu arXiv:1902.03163v1 [physics.bio-ph] 8 Feb 2019 [34]. The vertebrae allow for high flexibility particularly in the lateral (side-to-side) direction, with less flexibility for vertical (dorso-ventral) bending or for torsion. Running along the backbone are muscles that attach to the sides of the vertebrae and cause bending. The snake bod...

Self propulsion via slipping: frictional resistive force theory for multi-legged locomotors

“…From a physics of locomotion perspective, this paper reveals that a unified framework can capture and explain undulatory swimming in highly damped environments with both intrinsic and acquired drag anisotropy. Drag anisotropy is believed to be the critical principle which enables effective undulatory swimming [18,36,37,49,50]. Prior work has typically considered the intrinsic property of an element translating through a flowable medium (e.g., viscous fluid [36] and granular media [16,51]) or a locomotors' surface structures [17,52] as the cause of drag anistropy and therefore effective undulatory swimming [49,51,[53][54][55].…”

“…More recently, studies have demonstrated effective undulatory swimming with no intrinsic drag anisotropy. These works modulated the magnitude of surface traction via either static friction [18,37] or periodic lifting and landing of body appendages [31] to produce undulatory locomotion. Here, we posit that effective undulatory swimming shares the same physical principles between these two scenarios (intrinsic drag anisotropy and friction modulation).…”

“…As documented in prior work on locomotion at low Reynolds number [18,36], the drag anisotropy of slender rods (higher reaction forces in the perpendicular than in the parallel direction) is the critical physical property enabling swimming in viscous flows. In terrestrial environments where drag force is typically assumed isotropic Coulomb friction, the direct implementation of undulatory motion would be ineffective [37].…”

Efficient bending and lifting patterns in snake locomotion

“…Many common biological snake motions such as serpentine locomotion have been modelled successfully by assuming planar motions with a simple (Coulomb) frictional model [18][19][20][21][22][23][24][25][26]. Serpentine and concertina-like motions were found to be optimally efficient among general time-periodic kinematics of three-link [27] and smooth bodies [28,29]. For certain body geometries and frictional anisotropies, other motions, beyond those observed biologically, were found to be optimal [30][31][32].…”

“…As in our previous planar locomotion studies [27][28][29][30][31][32], we study non-planar motions using an optimization framework. It is not feasible to describe the full range of locomotor behaviours across the space of geometrical and physical parameters.…”