The mechanics of slithering locomotion David L. Hua,b,1, Jasmine Nirodya, Terri Scotta, and Michael J. Shelleya aApplied Mathematics Laboratory, Courant Institute of Mathematical Sciences, New York University, New York, NY 10003; and bDepartments of Mechanical Engineering and Biology, Georgia Institute of Technology, Atlanta, GA 30332 Edited by Charles S. Peskin, New York University, New York, NY, and approved April 22, 2009 (received for review December 12, 2008) In this experimental and theoretical study, we investigate the the sum of frictional forces per unit length, ffric, and internal slithering of snakes on flat surfaces. Previous studies of slithering forces, fint, generated by the snakes. That is, have rested on the assumption that snakes slither by pushing laterally against rocks and branches. In this study, we develop a ␯X¨ ϭ ffric ϩ fint , [1] theoretical model for slithering locomotion by observing snake ¨ motion kinematics and experimentally measuring the friction co- where X is the acceleration at each point along the snake’s body. efficients of snakeskin. Our predictions of body speed show good To avoid complex issues of muscle and tissue modeling, fint is agreement with observations, demonstrating that snake propul- determined to be that necessary for the model snake to produce sion on flat ground, and possibly in general, relies critically on the the observed shape kinematics. We use a sliding friction law in k frictional anisotropy of their scales. We have also highlighted the which the friction force per unit length is f ϭϪ␮k␯guˆ, where ␮ importance of weight distribution in lateral undulation, previously is the sliding friction coefficient, uˆ ϭ u/u is the velocity difficult to visualize and hence assumed uniform. The ability to direction, and ␯g is the normal force per unit length applied by redistribute weight, clearly of importance when appendages are the snake to the ground. We assume that the sliding friction airborne in limbed locomotion, has a much broader generality, as coefficients ␮k are equal (or at least proportional) to the ␮k ␮k ␮k ϭ shown by its role in improving limbless locomotion. corresponding static ones we have measured [( f , t , b) ␣(␮f, ␮t, ␮b)]. Applied to incorporate our measured frictional friction ͉ locomotion ͉ snake anisotropies, we model ffric simply as a weighted average of independent frictional responses to motions in the forwards (sˆ), Ϫ SCIENCES imbless creatures are slender and flexible, enabling them to backwards ( sˆ), and transverse directions (nˆ). That is, use methods of locomotion that are fundamentally different APPLIED PHYSICAL L f ϭϪ␯g͑␮k͑uˆ ⅐ nˆ͒nˆ ϩ ͓␮kH͑uˆ ⅐ sˆ͒ ϩ ␮k͑1 Ϫ H͑uˆ ⅐ sˆ͔͒͒ from the more commonly studied flying, swimming, walking, and fric t f b running used by similarly sized limbed or finned organisms. ⅐ ͑uˆ ⅐ sˆ͒sˆ͒, [2] These methods can be as efficient as legged locomotion (1) and moreover are particularly versatile when moving over uneven where H ϭ 1[1 ϩ sgn(x)] is the Heaviside step function used 2 terrain or through narrow crevices, for which the possession of to distinguish the components in the sˆand Ϫsˆdirections. Eqs. 1 and limbs would be an impediment (2, 3). Limbless invertebrates 2 can be made dimensionless by scaling lengths on the snake body SCIENCES such as slugs propel themselves by generating lubrication forces length L and time on the snake’s period of undulation ␶, together APPLIED BIOLOGICAL with their mucus-covered bodies; earthworms move by ratchet- with the force redefinition f ϭ ␮␯g˜f, where ␮ is a characteristic ing: propulsion is achieved by engaging their hairs in the ground sliding friction coefficient, taken to be 0.2. Eq. 1 then becomes as they elongate and shorten their bodies (4, 5). Terrestrial snakes propel themselves by using a variety of techniques, FrX¨ ϭ ˜ffric ϩ ˜fint, [3] including slithering by lateral undulation of the body, rectilinear progression by unilateral contraction/extension of their belly, where Fr is defined below. We prescribe the body shape in terms concertina-like motion by folding the body as the pleats of an of body curvature ␬(s, t), which is a quantity without reference to accordion, and sidewinding motion by throwing the body into a absolute position or orientation in the plane. The force balance, Eq. series of helices. This report will focus on lateral undulation, 3, is used to find the dynamics of the snake center of mass X៮ and whose utility to locomotion by snakes has been previously mean orientation ␪៮ through the requirement the total internal force described on the basis of push points: Snakes slither by driving and torque generated by the snake to execute its specified shape their flanks laterally against neighboring rocks and branches dynamics are zero. Those constraints, ͐1 ˜f (s)ds ϭ 0 and͐1(X(s, t) Ϫ ៮ 0 int ទ ទ 0 found along the ground (6–11). This key assumption has in- X) ϫ fint(s)ds ϭ 0, generate equations for X and ␪. The resultant formed numerous theoretical analyses (12–17) and facilitated system of ordinary differential equations are easily evolved the design of snake robots for search-and-rescue operations. numerically, with the only parameters being the Froude number Previous investigators (7, 9, 18, 19) have suggested that the Fr, often used in modeling terrestrial locomotion (23), and the frictional anisotropy of the snake’s belly scales might play a role friction coefficients ␮f,b,t. A natural definition for a mechanical in locomotion over flat surfaces. The details of this friction- efficiency is the ratio of the power to drag a straight snake to the based process, however, remain to be understood; consequently, active power mediated by friction on a slithering snake, ␩ ϭ snake robots have been generally built to slither over flat ៮ ͐1 ˜ ⅐˙ Uavg/[ 0 ffric Xds]avg, where the subscript avg denotes a time average surfaces by using passive wheels fixed to the body that resist over a period. lateral motion (18, 20–22). In this report, we present a theory for A critical parameter that arises in our theory is the so-called how snakes slither, or how wheelless snake robots can be ‘‘Froude number,’’ Fr ϭ (L/␶ 2)/␮g, which measures the relative designed to slither, on relatively featureless terrain, such as sand importance of inertial to frictional forces (which both scale with or bare rock, which do not provide obvious push points. Model Author contributions: D.L.H., J.N., T.S., and M.J.S. performed research; D.L.H. analyzed We model snakes (Fig. 1 A and B) as inextensible 1-dimensional data; and D.L.H. and M.J.S. wrote the paper. curves X(s,t) ϭ (x(s,t),y(s,t)) of fixed length L and uniform mass The authors declare no conflict of interest. per unit length ␯ (Fig. 1C). Here, s is the curve arc length This article is a PNAS Direct Submission. measured from the head, and t is time. The snakes’ motion arises 1To whom correspondence should be addressed at: Georgia Institute of Technology, ATTN: through balancing body inertia at each point along the curve with David Hu, 801 Ferst Drive, MRDC 1308, Atlanta, GA 30332-0405. E-mail: [email protected]. www.pnas.org͞cgi͞doi͞10.1073͞pnas.0812533106 PNAS ͉ June 23, 2009 ͉ vol. 106 ͉ no. 25 ͉ 10081–10085 Downloaded by guest on September 27, 2021 Fig. 1. Frictional anisotropy of snakeskin. (A and B) One of the milk snakes used in our experiments (A) and its ventral scutes (B), whose orientation allows them to interlock with ground asperities. (Scale bars, 1 cm.) (C) Schematic diagram for our theoretical model, where X៮ denotes the snake’s center of mass, ␪៮ its mean orientation, and sˆand nˆthe tangent and normal vectors to the body, taken toward the head. (D) The experimental apparatus, an inclined plane, used to measure the static friction coefficient ␮ of unconscious snakes. (E) The relation between the static friction coefficient ␮ and angle ␪៮ with respect to the direction of motion, for straight unconscious snakes. Filled symbols indicate measurements on cloth; open symbols, on smooth fiberboard; solid curve derived from theory (Eq. 2)by using ␮f ϭ 0.11, ␮t ϭ 0.19, and ␮b ϭ 0.14. Error bars indicate the standard deviation of measurement. mass). From our observations, we estimate that frictional forces 0.14). The values of these coefficients are slightly less than those are an order of magnitude greater than inertial forces (i.e., Fr Ϸ measured elsewhere (7) for dead snakes on various surfaces (␮f and 0.1, using ␮ ϭ 0.2; see Fig. 1), which gives the important result ␮b between 0.24 and 1.30). We also observe that while the snakes that, unlike most other terrestrial organisms of this size, body were recovering from unconsciousness, they began twitching their inertia is not central to slithering locomotion. Motion instead scales individually. This level of control is consistent with their arises by the interaction of surface friction and internal body neuromuscular anatomy (2), which shows that each scale has its own forces. Even the red racer (24), one of the world’s fastest snakes, muscular attachment. has Fr Ϸ 1 Ϫ 1.5(L ϭ 60 Ϫ 130 cm, U ϭ 130 cm/s), indicating The snake’s frictional anisotropy with the surface is critical to that its body inertia is not predominant. In the zero Fr limit, its motion, at least on horizontal surfaces (angle of inclination forces are transduced directly to velocities rather than acceler- ␾ ϭ 0). Fig. 2aЈ shows time-lapse photographs of a snake ations, which is also a feature of the low Reynolds number unsuccessfully attempting to slither forward when placed on a locomotion of microorganisms through a fluid (25).