Figure 19.

Isotropic versus anisotropic friction driven locomotion. (a) Gait envelope computed over the 10th muscular activation cycle in the case of isotropic friction. The blue triangle denotes the location of the snake’s centre of mass at time t=0, reported as reference. (b) Lateral (blue) and forward (red) velocities as functions of time normalized by the activation period T_m in the case of isotropic friction. (c) Gait envelope computed over the 10th muscular activation cycle in the case of anisotropic friction. The blue triangle denotes the location of the snake’s centre of mass at time t=0, reported as reference. (d) Lateral (blue) and forward (red) velocities as functions of time normalized by the activation period T_m in the case of anisotropic friction. Settings: length L=0.5 m, radius r= 0.025 m, mass m=1 kg, Young’s modulus E=10⁷ Pa, shear modulus G=2E/3 Pa, shear/stretch matrix S=10⁵×1 N, bend/twist matrix B=diag(EI₁,EI₂,GI₃) Nm², dissipation constant γ= 10⁻¹ kg (ms)⁻¹, gravity g=9.81 m s⁻², static μ^f_s=0.2, ${\mu }_{\mathrm{s}}^{\mathrm{r}}={\mu }_{\mathrm{s}}^{\mathrm{f}}$, ${\mu }_{\mathrm{s}}^{\mathrm{b}}={\mu }_{\mathrm{s}}^{\mathrm{f}}$ and kinetic μ^f_k=0.1, ${\mu }_{\mathrm{k}}^{\mathrm{r}}={\mu }_{\mathrm{k}}^{\mathrm{f}}$, ${\mu }_{\mathrm{k}}^{\mathrm{b}}={\mu }_{\mathrm{k}}^{\mathrm{f}}$ friction coefficients in the isotropic case, static μ^f_s=0.2, ${\mu }_{\mathrm{s}}^{\mathrm{r}}=2{\mu }_{\mathrm{s}}^{\mathrm{f}}$, ${\mu }_{\mathrm{s}}^{\mathrm{b}}=20{\mu }_{\mathrm{s}}^{\mathrm{f}}$ and kinetic μ^f_k= 0.1, ${\mu }_{\mathrm{k}}^{\mathrm{r}}=2{\mu }_{\mathrm{k}}^{\mathrm{f}}$, ${\mu }_{\mathrm{k}}^{\mathrm{b}}=20{\mu }_{\mathrm{k}}^{\mathrm{f}}$ friction coefficients in the anisotropic case, friction threshold velocity $v_{ϵ}={10}^{-8}\hspace{0.17em}\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$, ground stiffness and viscous dissipation k_w=1 kg s⁻² and γ_w=10⁻⁶ kg s⁻¹, discretization elements n=100, timestep δt= 10⁻⁵ s, muscular activation period T_m=1 s, wavelength ${\mathrm{\lambda }}_m=\mathrm{\infty }$, phase shift ϕ_m=0, torque B-spline coefficients β_i=0,…,5={0,10,15,15,10,0} Nm.