Limbless undulatory propulsion on land

Abstract

We analyze the lateral undulatory motion of a natural or artificial snake or other slender organism that "swims" on land by propagating retrograde flexural waves. The governing equations for the planar lateral undulation of a thin filament that interacts frictionally with its environment lead to an incomplete system. Closures accounting for the forces generated by the internal muscles and the interaction of the filament with its environment lead to a nonlinear boundary value problem, which we solve using a combination of analytical and numerical methods. We find that the primary determinant of the shape of the organism is its interaction with the external environment, whereas the speed of the organism is determined primarily by the internal muscular forces, consistent with prior qualitative observations. Our model also allows us to pose and solve a variety of optimization problems such as those associated with maximum speed and mechanical efficiency, thus defining the performance envelope of this mode of locomotion.

Fig. 1.

The model. (a) A schematic view of lateral undulation (after ref. 1). The snake is moving at +x direction with a velocity v_x. Solid dots indicate the location of inflection points. The shaded areas qualitatively describe the pattern and amplitude of muscle activity (4, 5). The dotted lines show the track left behind. For undulations without lateral slip, the flexural waves are stationary relative to the ground, and thus the pattern of muscle activity is stationary relative to the lab frame. For undulations with lateral slip, the dashed arrows show the slip velocity in the lab frame U and its components along the tangent and normal. (b) A small segment of the organism shows the internal forces and moments at a cross-section. (c) A visco-elastic model of the same segment contains three parallel elements, a passive elastic element (spring), a passive viscous element (dashpot), and an active muscular element.

Fig. 2.

Comparison of analytical and numerical solutions. (a) The maximum tangent angle θ_max obtained by numerically solving (Eq. 9) is well approximated by θ₀, the amplitude of θ obtained from the linearized Eqs. 11–13. Different circles correspond to the values of Pr for the shapes shown in Fig. 3b. Here, Be = 0.4, Vi = 0.04, although Be and Vi do not affect the normalized shape of the snake (i.e., θ_max) (see Fig. 3a). (b) The dimensionless active moment Mo, obtained from the numerical solution of Eq. 9, is a linear function of the dimensionless passive viscosity Vi (left and bottom axis, Pr = 0.1, Be = 0.4). We also fit Mo using k₁ + k₂ Vi and plot k₁/Mo₀ (triangles), k₂/(2π)⁴ θ₀ (circles) (top and right axis). k₁/Mo₀ and k₂/(2π)⁴ θ₀ are ≈1 when Pr > 0.2, which validates Eq. 14. The scaled elastic bending stiffness Be used here is 0.4, but the exact value does not affect Mo (see text).

Fig. 3.

The effects of exogenous and endogenous dynamics on the normalized shape of a snake. (a) The effects of scaled passive elasticity Be and viscosity Vi on the shape of the snake normalized by the arc-length in a period. The solid line shows one period, whereas the dashed line shows the track, which is the same as the solid line, left behind. The observed collapse of all the shapes shows that the passive elasticity and viscosity do not affect the normalized shape. Here, Pr = 0.18. (b) The effects of exogenous dynamics, i.e., the substrate resistance Pr, on the normalized shape of the snake; we see that the normalized amplitude is large when Pr is small and small when Pr is large. Here, Be = 0.4, Vi = 0.04. (c) The effects of two extreme forms of the active moment on the normalized shape, with M_a1 = m_a1 sin(2πs) (solid), M_a2 = m_2a tanh(3 sin(2πs)) (dashed and dotted) (Inset). We see that the normalized shape is not strongly affected by the form of the active moment, for a range of values of Be and Vi. Here, Pr = 0.1.

Fig. 4.

The energetics and efficiency of lateral undulatory locomotion. (a) The mechanical efficiency χ as a function of Pr (top right axes) and Vi (bottom left axes), obtained by solving Eq. 9 numerically (solid line) is well approximated by Eq. 22 (dash-dot line). (b) The optimum arclength l_opt (red curve) and the maximum projected velocity v_xmax (rest curves) as a function of Pr. The environment temperatures are 15°C (V′ = 3.0/s) and 25°C (V′ = 8.4/s). The maximum of l_opt is 1 m, the length of the snake. For all the calculations, c = 4, δ = 0.2, and Be = 0.4.