Efficient sliding locomotion of three-link bodies

We study the efficiency of sliding locomotion for three-link bodies with prescribed joint angle motions. The bodies move with no inertia, under dry (Coulomb) friction that is anisotropic (different in the directions normal and tangent to the links) and directional (different in the forward and backward tangent directions). Friction coefficient space can be partitioned into several regions, each with distinct types of efficient kinematics. These include kinematics resembling lateral undulation with very anisotropic friction, small-amplitude reciprocal kinematics, very large-amplitude kinematics near isotropic friction, and kinematics that are very asymmetric about the flat state. In the two-parameter shape space, zero net rotation for elliptical trajectories occurs mainly with bilateral or antipodal symmetry. These symmetric subspaces have about the same peak efficiency as the full space but with much smaller dimension. Adding the second or third harmonics greatly increases the numbers of local optimal for efficiency, but only modestly increases the peak efficiency. Random ensembles with higher harmonics have efficiency distributions that peak near a certain nonzero value and decay rapidly up to the maximum efficiency. A stochastic optimization algorithm is developed to compute optima with higher harmonics. These are simple closed curves, sharpened versions of the elliptical optima in most cases, and achieve much higher efficiencies mainly for small normal friction. With a linear (viscous) resistance law, the optimal trajectories are similar in much of the friction coefficient space, and relative efficiencies are much lower except with very large normal friction.

Figure 1

Left: Classification of local optima across friction coefficient space, presented in [21]. The triangles, crosses, and circles mark locations where optima that are direct, standing, or retrograde waves were found, respectively. The solid lines mark interfaces between regions containing a distinct type of wave optimum, while the dashed lines delineate a region with both standing- and retrograde-wave optima. Right: Three sequences of snapshots of locally optimal motions giving examples of direct, standing, and retrograde waves. These occur at particular friction coefficient ratios, listed above the snapshots and marked with green, red, and blue symbols in the panel at left. The three sequences of snapshots are given over one period of motion, and they are displaced vertically to enhance visibility, but the actual net displacement is horizontal.

Figure 2

(a) Schematic diagram of a three-link body with changes in angles $\mathrm{\Delta }{\theta }_1$ (here positive) and $\mathrm{\Delta }{\theta }_2$ (here negative) between the links. The body is parametrized by arc length $s$ (nondimensionalized by body length) at an instant in time. The tangent angle and the unit vectors tangent and normal to the curve at a point are labeled. Vectors representing forward, backward, and normal velocities are shown with the corresponding friction coefficients ${\mu }_f, {\mu }_b$, and ${\mu }_n$. (b) Examples of body shapes in the ($\mathrm{\Delta }{\theta }_1,\mathrm{\Delta }{\theta }_2$)-plane. Shapes that do not self-intersect are shown in black; a few shapes at the threshold of self-intersection are shown in red.

Figure 3

(a) Examples of elliptical trajectories in the region of non-self-intersecting configurations (inside the black polygonal outline). Examples of body configurations at the boundary of the region are shown at the upper right. The gray ellipse has center $A_{10},A_{20}$ and shape given by $\{A_{11},A_{21},B_{11},B_{21}\}$. (b) $(\mathrm{\Delta }{\theta }_1\left(t\right), \mathrm{\Delta }{\theta }_2\left(t\right))$ for a three-link body, symmetric about the line $\mathrm{\Delta }{\theta }_1=-\mathrm{\Delta }{\theta }_2$. $A_0$ is the average of $\mathrm{\Delta }{\theta }_1$ over the ellipse and $\sqrt{2}{A}_1$ and $\sqrt{2}\left|B_1\right|$ are the semimajor and semiminor axes of the ellipse. The sign of $B_1$ gives the direction in which the path is traversed.

Figure 4

Cluster classification of the best [panel (a)] and the second best [panel (b)] local optima in efficiency for elliptical trajectories across a grid of (${\mu }_n/{\mu }_f,{\mu }_b/{\mu }_f$) values. The set of 192 local optima are used to define 10 clusters based on proximity in ($A_0,A_1,B_1$) space. At each (${\mu }_n/{\mu }_f,{\mu }_b/{\mu }_f$) pair, the number and color of the square denotes the cluster to which it belongs. (c) The elliptical trajectory of the optimum closest to the centroid of each cluster, with color corresponding to that cluster. The cluster number of each ellipse is located along the right side of the panel, at the minimum vertical position of the corresponding ellipse. Each ellipse corresponds to a square in panel (a) that is outlined in black or purple. (d) For each ellipse in (c), snapshots of the body motion at five instants spaced 1/4 period apart, starting from the thin colored line, proceeding from light to dark gray, and ending with the thick colored line. The friction coefficient ratios for each motion are labeled, with the abbreviations ${\mu }_{n/f}$ and ${\mu }_{b/f}$ in place of ${\mu }_n/{\mu }_f$ and ${\mu }_b/{\mu }_f$.

Figure 5

Relative efficiencies of the global (a) and second-best optima (b) among elliptical trajectories with bilateral symmetry.

Figure 6

Relative efficiency ($\lambda /{\lambda }_{ub}$) vs net rotation ($|\mathrm{\Delta }{\theta }_0|$, in radians) for elliptical trajectories that are bilaterally symmetric (blue dots), antipodally symmetric (green dots), or reciprocal (red dots). Values for other trajectories are shown by gray dots. Each panel shows data at a given pair of friction coefficient ratios, labeled along the top and left of the figure.

Figure 7

Examples of the effect of adding higher harmonics to elliptical trajectories. The trajectories are given by (20). In (a), (c), (e), and (g), we have $A_1=0.5$ and $B_1=1$; in (b), (d), (g), and (h), we have $A_1=1$ and $B_1=0.5$. To these ellipses we add just one additional nonzero mode, setting either $A_2$ [in (a) and (b)], $B_2$ in [(c) and (d)], $A_3$ [(e) and (f)], or $B_3$ [(g) and (h)] to 0.2 (light blue lines) or 0.4 (dark blue lines).

Figure 8

The numbers of local optima of efficiency at various friction coefficient ratios in the space of $\{A_0,A_1,B_1\}$ describing bilaterally symmetric ellipses [panel (a)], the larger space of $\{A_0,A_1,B_1,A_2,B_2\}$ with second harmonics added [panel (b)], and the space of $\{A_0,A_1,B_1,A_3,B_3\}$ with third harmonics added [panel (c)].

Figure 9

The factor of improvement in maximum relative efficiency when the space of modes is enlarged from (a) $\{A_1,B_1\}$ to $\{A_0,A_1,B_1\}$; (b) $\{A_0,A_1,B_1\}$ to $\{A_0,A_1,B_1,A_2,B_2\}$; (c) $\{A_0,A_1,B_1\}$ to $\{A_0,A_1,B_1,A_3,B_3\}$. The modes corresponding to these coefficients are listed in Eqs. (20). The factor is plotted at various friction coefficient values shown on the axes.

Figure 10

Probability densities of relative efficiency, estimated from histogram data for various friction coefficient ratios (labeled at top and left). Each color corresponds to bilaterally symmetric trajectories with a given maximum harmonic $k$, labeled at left, resulting in $2k+1$ modes. Each curve corresponds to a different random ensemble of about ${10}^6$ trajectories.

Figure 11

Efficiency-maximizing trajectories with bilateral symmetry, with different maximum harmonics $k$—3 (red), 5 (green), 7 (light blue), and 9 (purple)—corresponding to $2k+1$ modes in each case. The trajectories are plotted in the region of nonintersecting trajectories, plotted at various friction coefficient ratios labeled at the bottom and left. The trajectories are computed with the stochastic optimization algorithm described in the text.

Figure 12

Efficiency-maximizing trajectories with antipodal symmetry, with different maximum harmonics $k$—3 (red), 5 (green), 7 (light blue), and 9 (purple)—corresponding to $2k+2$ modes in each case. The trajectories are plotted in the region of nonintersecting trajectories, plotted at various friction coefficient ratios labeled at the bottom and left. The trajectories are computed with the stochastic optimization algorithm described in the text.

Figure 13

(a) The factors by which the efficiencies of the bilaterally symmetric optima exceed those of the antipodally symmetric optima. (b) The relative efficiencies of the bilaterally symmetric optima.

Figure 14

(a) Trajectories (with bilateral or antipodal symmetry) that maximize relative efficiency, with different maximum harmonics $k$—3 (red), 5 (green), 7 (light blue), and 9 (purple)—when the resistance law is linear in velocity. (b) The relative efficiencies corresponding to the motions in panel (a).

Figure 15

The left column [panels (a), (c), and (e)] gives the values of the kinematic parameters $A_0, A_1$, and $B_1$, respectively, for the global optimizers of efficiency among elliptical trajectories with bilateral symmetry. The right column [panels (b), (d), and (f)] gives the same values for the second best local optimizers. The parameters are defined in (18) and shown in Fig. 3.