Figure 10.

Optimal lateral undulation gait. (a–d) Instances at different times of a snake characterized by the identified optimal gait. (e) Evolution of the fitness function f=v^fwd_max as a function of the number of generations produced by CMA-ES. Solid blue, solid red and dashed black lines represent, respectively, the evolution of f corresponding to the best solution, the best solution within the current generation, and the mean generation value. (f) Scaled forward (red) and lateral (blue) centre of mass velocities versus normalized time. (g) Gait envelope over one oscillation period T_m. Red lines represent head and tail displacement in time. Settings: Froude number Fr=0.1, length L=1 m, radius r= 0.025 m, density ρ=10³ kg m⁻³, T_m=1 s, Young’s modulus E=10⁷ Pa, shear modulus G=2E/3 Pa, shear/stretch matrix $\widehat{\mathbf{\text{S}}}=\text{diag}(4G\hat{A}/3,4G\hat{A}/3,E\hat{A})$ N, bend/twist matrix $\widehat{\mathbf{\text{B}}}=\text{diag}(E{I}_1,E{I}_2,G{I}_3)\hspace{0.17em}{\mathrm{Nm}}^2$, dissipation constant γ= 5 kg (ms)⁻¹, gravity g=9.81 m s⁻², friction coefficient ratios ${\mu }_{\mathrm{k}}^f:{\mu }_{\mathrm{k}}^b:{\mu }_{\mathrm{k}}^r=1:1.5:2$ and ${\mu }_{\mathrm{s}}^f:{\mu }_{\mathrm{s}}^b:{\mu }_{\mathrm{s}}^r=1:1.5:2$ with ${\mu }_{\mathrm{s}}^f=2{\mu }_{\mathrm{k}}^f$, 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=50, timestep δt=2.5⋅10⁻⁵ T_m, wavelength λ_m=0.97L, phase shift ϕ_m=0, torque B-spline coefficients β_i=0,…,5={0,17.4,48.5,5.4,14.7,0} Nm, bounds maximum attainable torque |β|^max_i=0,…,5=50 Nm.