Helical Locomotion in a Granular Medium

The physical mechanisms that bring about the propulsion of a rotating helix in a granular medium are considered. A propulsive motion along the axis of the rotating helix is induced by both symmetry breaking due to the helical shape, and the anisotropic frictional forces undergone by all segments of the helix in the medium. Helix dynamics is studied as a function of helix rotation speed and its geometrical parameters. The effect of the granular pressure and the applied external load were also investigated. A theoretical model is developed based on the anisotropic frictional force experienced by a slender body moving in a granular material, to account for the translation speed of the helix. A good agreement with experimental data is obtained, which allows for predicting the helix design to propel optimally within granular media. These results pave the way for the development of an efficient sand robot operating according to this mode of locomotion.

Figure 1

Sketch of the experimental setup. The left inset presents the helix and vector definitions used in the model.

Figure 2

(a) Horizontal position $x(t)$ of the helix as a function of time for different $\omega$, a constant load ($L=1.0\mathrm{N}$) and at a constant depth ($h=75\mathrm{mm}$). (b) $U$ along the $x$ direction as a function $R\omega$, for a constant load ($L=1.0\mathrm{N}$) and at a constant depth ($h=75\mathrm{mm}$). The inset shows $U$ as a function of $h$, for $L=1.5\mathrm{N}$ and $\omega =10.4\mathrm{rad}/\mathrm{s}$. (c) Normalized helix velocity $U/R\omega$ as a function of $L$ for a constant $\omega$ ($\omega =10.4\mathrm{rad}/\mathrm{S}$). Blue triangles, green diamonds, yellow squares, orange circle, and red triangles correspond to experimental data for $h=45$, 55, 65, 77, and 90 mm, respectively. Solid lines correspond to Eq. (5) taken from the model results. (d) Black dots show the critical load $L_c$, above which the helix moves as a function of $h$. Experimental data are interpolated by a linear trend (black solid line). The intersection of this linear fit with the $y$ axis provides an estimate of the internal resistance $F_f$ of the system. This solid line separates the domains of phase space, indicating if helix motion is or is not propulsive. Open triangles indicate the critical load, which is measured when friction is reduced by means of air bearings.

Figure 3

(a) $U/R\omega$ vs $\phi$ for ${\mathcal{L}}_x=58$, $R=5.0$, $d=3.0$, and $h=50\mathrm{mm}$. To compensate for head’s drag, a load $L=1.2\mathrm{N}$ is applied. The solid line shows the best fit to the experimental data with the relation ${\tilde{U}}_m=(C_n-C_t)/(C_t\tan \phi +C_n/\tan \phi )$ and $C_n/C_t=1.6$. Inset: 3D printed helices of varying $\phi$. (b) Helical robot for motion in a granular medium. Black bar is 20 mm.