Lateral contact yields longitudinal cohesion in active undulatory systems

Many animals and robots move using undulatory motion of their bodies. When the bodies are in close proximity undulatory motion can lead to novel collective behavior such as gait synchronization, spatial reconfiguration, and clustering. Here we study the role of contact interactions between model undulatory swimmers: three-link robots in experiment and multilink swimmers in simulation. The undulatory gait of each swimmer is generated through a time-dependent sinusoidal-like waveform which has a fixed phase offset, $\varphi$. By varying the phase relationship between neighboring swimmers we seek to study how contact forces and planar configurations are governed by the phase difference between neighboring swimmers. We find that undulatory actuation in close proximity drives neighboring swimmers into planar equilibrium configurations that depend on the actuation phase difference. We propose a model for stable planar configurations of nearest-neighbor undulatory swimmers which we call the gait compatibility condition, which is the set of planar and phase configurations in which no collisions occur. Robotic experiments with two, three, and four swimmers exhibit good agreement with the compatibility model. To study the contact forces and the time-averaged equilibrium between undulatory systems we perform simulations. To probe the interaction potential between undulatory swimmers we apply a small force to each swimmer longitudinally to separate them from the compatible configuration and we measure their steady-state displacement. These studies reveal that undulatory swimmers in close proximity exhibit attractive longitudinal interaction forces that drive the swimmers from incompatible to compatible configurations. This system of undulatory swimmers provides new insight into active-matter systems which move through body undulation. In addition to the importance of velocity and orientation coherence in active-matter swarms, we demonstrate that undulatory phase coherence is also important for generating stable, cohesive group configurations.

Figure 1

Motivation and overview of gait compatibility among undulatory swimmers. (a) Large groups of swimmers experience contact interactions. (b) Contact interactions among pairs of undulatory swimmers confined to a lateral distance $d$ require planar ($\mathrm{\Delta }x,\mathrm{\Delta }y$) reconfiguration when there is a gait phase difference $\mathrm{\Delta }\varphi$.

Figure 2

Overview of three-link robots. (a) Geometry of the three-link system. Links are length, $l=17$ cm, height, $h=5$ cm, and width, $w=2.5$ cm. (b) The joint angles ${\beta }_1$ and ${\beta }_2$ are controlled through position-commanded servos. (c) We studied groups of robots in a narrow channel of variable width $d$.

Figure 3

Gait compatibility in undulatory swimmer pairs. (a) Image of two undulatory robot swimmers in experiment and illustration of longitudinal motion from contact interactions. Illustrations are traced from experiment images. (b) Steady-state longitudinal separation versus phase difference between robot pairs with $\xi =1$. Black diamonds are experimental results; green dots are simulation results. Solid lines are compatibility predictions from equation (2). (c) The right column shows the simulation results with different $\xi$ at three ranges. Experiment results with $\xi =[0,\frac{3}{4},\frac{3}{2}]$ are included accordingly. The solid lines are the compatibility prediction.

Figure 4

Contact compatibility criteria. (a) An overview of the phase-range ($\mathrm{\Delta }\varphi$) for compatible sinusoidal curves that are separated by a $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ displacement. (b) The three-dimensional representation of the compatibility criteria represented by equation (5). The shaded region above the solid blue curves are allowable configurations of gait compatibility. As the lateral separation distance increases (vertical axis, $\frac{\mathrm{\Delta }y}{2A}$) the range of compatible phases increases. (c) We show cross sections of the compatibility condition at three different lateral separations: $\frac{\mathrm{\Delta }y}{2A}=[0.2,0.5,0.9]$ from top to bottom. When $\frac{\mathrm{\Delta }y}{2A}>1$ any combination of $\mathrm{\Delta }\varphi$ and $\mathrm{\Delta }x$ will be in compatibility. (d) Dots indicate the steady-state longitudinal separation between two three-link swimmers in simulation with $\mathrm{\Delta }\varphi =0$ as a function of wall width. The shaded area is drawn to guide the eye and represents the growing compatibility region as lateral spacing is allowed to increase.

Figure 5

(a)–(d) Gait compatibility simulations of two swimmers with 3, 5, 7, or 9 links from left to right. As the link number increases the longitudinal separation versus phase difference (red circles) show good agreement with the compatibility prediction (solid-lines). (e) Bottom figures i–v show example configurations of two nine-link swimmers. Red circles represent swimmer pairs that reached the compatibility condition (ii, iv), blue diamonds represent swimmer pairs that separate longitudinally with each other (i, v), green squares represent swimmer pairs that are jammed with each other (iii).

Figure 6

Gait compatibility in larger robot groups. (a) Image of a four-robot experiment at the beginning (top) and in the middle of the experiment (bottom). (b) Steady-state longitudinal separation versus phase difference groups experiments with three (red squares) and four robots (blue circles). Solid lines are compatibility predictions from Eq. (2).

Figure 7

Lateral density is influenced by phase variance in undulatory groups. (a) A representation of a group of undulatory swimmers separated laterally (the vertical direction) in tangent contact with phase variation. (b) A heatmap of time-averaged absolute value of joint error induced through collisions within a swimmer group for varying normalized lateral density, $\tilde{\rho }$ ($y$ axis) and the range of phase variation ($x$ axis). Blue circles indicate the measured joint error threshold below which contact typically does not occur in the group from simulations. Error bars represent the standard deviation of the mean taken over five periods. (c) A heatmap of time-averaged magnitude of contact forces induced through collisions within a swimmer group. Blue circles indicate the baseline contact force threshold between a swimmer and the wall. Error bars represent the standard deviation of the mean taken over five-periods. In both panels (b) and (c): The red curve is a Monte Carlo estimate based on the compatibility equation. The dashed green curve is the calculation from the math model (10).

Figure 8

Compatible configurations minimize the contact between swimmers. (a) We envision that longitudinal dynamics are governed by a potential-energy landscape dependent on the phase difference between swimmers. Swimmers initialized with $\mathrm{\Delta }\varphi =0$ and different longitudinal positions ($\mathrm{\Delta }{x}_0/\lambda$) evolve to one of three compatible configurations depending on initial position. All initial positions $|\mathrm{\Delta }{x}_0/\lambda |\le 0.3$ evolve to $\mathrm{\Delta }{x}_f/\lambda =0$. (b) Heatmap represents the distance traveled from initial condition to compatibility, $\mathrm{\Delta }{x}_f-\mathrm{\Delta }{x}_0$ for three-link swimmers. (c) Heatmap represents the distance traveled from initial condition to compatibility, $\mathrm{\Delta }{x}_f-\mathrm{\Delta }{x}_0$ for sinusoidal swimmers. The solid lines are the compatibility lines. The dashed lines separate regions of attraction between the middle and outer compatibility lines.

Figure 9

Cohesive longitudinal interactions depend on confinement. (a) Magnitude of contact force between two swimmers during five periods of oscillation, with initial conditions $\mathrm{\Delta }x=0$ and $\mathrm{\Delta }\varphi$ varied from $-\pi$ to $\pi$. The contact-interaction force grows with compatibility detuning. Dots represent mean contact force and error bars are standard deviation. (b) The cohesive magnitude of the compatible configuration was measured by applying equal and opposite perturbation forces $\delta F$. The separation distance from compatibility, $\delta x$, is measured. (c) Force-displacement relationship for nine confinement distances of wall widths (0.10 to 0.18 m in 0.01 m increments). Error bars represent the standard deviation of $\delta x$ at applied $\delta F$. (d) Escape time for wall widths (0.12 to 0.18 m in 0.01 m increments). The black curve represents the escape time without walls. Error bars represent the standard deviation of escape time. (e) Effective interaction spring constant as a function of wall width. Error bars represent estimated spring stiffness with 95% confidence bounds. (f) Offset distance ($\delta x_0$) as a function of wall width. Error bars represent estimated $\delta x_0$ intersection with 95% confidence bounds.