Figure 1

Three-link swimmers. (a) The robot resting on a granular medium (GM), a bed of plastic spheres. The lit masts at the ends of the links are markers for tracking the robot when its body is submerged. The rubber skin on the front section has been removed to display the mechanical structure. Each link is $5.4\times 2.8\times 14.7{\mathrm{cm}}^3$. (b) Analytical three-link model with a body frame corresponding to a weighted average of the link positions and orientations. (c) Force relations for a rod dragged in GM (adapted from Ref. 6) (solid) and a long slender rod dragged in a low Re fluid (dashed). Scaling is chosen such that the maximum values of $F_{\parallel }$ are equal for the GM and low Re systems.Reuse & Permissions

Figure 2

The components of $\mathbf{\xi }$ for a particular initial shape (inset), with $\alpha =(0.85,-1.14)$ radians; velocities are given in terms of body-lengths ($\xi {\text{}}_x$ and $\xi {\text{}}_y$) or radians ($\xi {\text{}}_{\theta }$) per second. The components of the equilibrium $\mathbf{\xi }$ (shown here for a body frame attached to the middle link) are almost linear functions of the shape velocity, and so can be characterized by the gradient vectors of the best-fit planes (black arrows). The color map shows the agreement between the equilibrium $\mathbf{\xi }$ and the fitted plane, with light regions exhibiting the least error.Reuse & Permissions

Figure 3

(a) The local connection vector fields [Eq. (1)] ${\mathbf{A}}^x$, ${\mathbf{A}}^y$, ${\mathbf{A}}^{\theta }$. The gray (red) path through the shape space denotes a circle gait, and the smaller gray circle identifies the gradients taken from Fig. 2. (b) Sequence of shapes and displacements along the gait shown above. The vertical red line is a reference position for displacement. (c) Along the gait, the $\mathbf{\xi }$ and net displacement predicted by the linear model (thick red line), the DEM simulation (dotted line), and robot experiments (thin black line) for a representative experiment at frequency 0.5 Hz. Lengths and angles are given in terms of body-lengths and radians, respectively. Time is in fractions of a gait cycle.Reuse & Permissions

Figure 4

Fundamental geometric diagrams, the constraint curvature functions (CCFs) for the (a) granular three-link swimmer and (b) low Re swimmer of equivalent dimensions (bottom). Each swimmer’s body frame is optimized to a weighted average of the three link frames; see Ref. 28. The units of the field values are $(\mathrm{body}\mathrm{\text{−}}\mathrm{lengths})/(\mathrm{joint}\mathrm{\text{−}}\mathrm{radians}{)}^2$ for $x^b$ and $y^b$ and $(\mathrm{orientation}\mathrm{\text{−}}\mathrm{radians})/(\mathrm{joint}\mathrm{\text{−}}\mathrm{radians}{)}^2$ for ${\theta }^b$. Values on the $x^b$ and $y^b$ fields have been multiplied by 100.Reuse & Permissions

Figure 5

Predictions of movement for the three-link swimmer. (a) CCF estimate of displacement compared to experimental and DEM results as a function of stroke amplitude (radius of the circle traced in the shape space). Dashed horizontal lines indicate displacements for the butterfly gait in DEM and resistive force theory. The inset shows the circle gait overlaid on the $x^b$ CCF, along with a graphically optimized “butterfly” gait (dashed path). (b) CCF estimate of net rotation for figure-eight gait of different stroke amplitudes, the radius of one circle of the figure-eight gait overlaid on the ${\theta }^b$ CCF in the inset. Frequency was 0.5 Hz in (a) and 0.17 Hz in (b).Reuse & Permissions