Multimodal motion and behavior switching of multistable ciliary walkers

Abstract

The collective motion of arrays of cilia - tiny, hairlike protrusions - drives the locomotion of numerous microorganisms, enabling multimodal motion and autonomous switching between gaits to navigate complex environments. To endow minimalist centimeter-scale robots with similarly rich dynamics, we introduce millimeter-scale flexible cilia that buckle under the robots weight, coupling multistability and actuation within a single physical mechanism. When placed on a vibrating surface, these ciliary walkers select their propulsion direction through the buckled states of their cilia, allowing multimodal motion and switching between modes in response to perturbations. We first show that bimodal walkers with left-right symmetric cilia can autonomously reverse direction upon encountering obstacles. Next, we demonstrate that walkers with isotropic cilia exhibit both translational and rotational motion and switch between them in response to environmental interactions. At increasing densities, swarms of such walkers collectively transition from predominantly spinning to translational motion. Finally, we show that the shape, placement and number of cilia controls the modes of motion of the walkers. Our results establish a rational, physically grounded strategy for designing minimalist soft robots where complex behaviors emerge from feedback between internal mechanical states and environmental interactions, laying the foundation for autonomous robotic collectives without the need for centralized control.

Figures (17)

• Figure 1: Bimodal motion of ciliary walkers with rectangular-section cilia. (A) A ciliary walker supported by an array of flexible cilia (inset); view from below a transparent glass plate. Each cilium is straight when unloaded (inset), and has a rectangular cross-section, enabling two distinct stable buckled states, i.e. the left- (B) and right-leaning (C) configurations (along and opposite to $e_{\parallel}$). (D) Superimposed snapshots of a walker translating on the vibrated plate, and (E) autonomously reversing direction upon collision with the boundary, exploiting the bistability of the buckled cilia.
• Figure 2: Different types of locomotion in a 1D track. (A) Time-lapse sequences (side view) of a bidirectional walker moving along a 1D track under three distinct regimes. (i) For $\Gamma = 1.97$, $W=12$ g, the walker exhibits steady motion until it reaches a boundary and then gets trapped. (ii) For $\Gamma = 2.45$, $W=12$ g, the walker reverses its motion upon collision with a boundary. (iii) For $\Gamma = 2.93$, $W=8$ g, the motion becomes erratic, allowing spontaneous reversal of motion. (B) Corresponding trajectories. (C) Probability distributions of the velocity $V_x$, color coded from light to dark blue as the vibration amplitude $\Gamma \in \{ 1.49, 1.97, 2.45 \}$ increases illustrating the transition from the trapped to the reversing regime. (D) When $\Gamma$ is increased from $1.97$ to $3.41$, the cruise velocity $v_0$ increases, and for strong driving the motion becomes erratic. (F-G) Corresponding PDFs for simulations of a single cilium (see inset of H and Supporting Information). (E and H) Second moment of the velocity $V_x$ as a function of $\Gamma$ for experiments (E) and for simulations of a single cilium (H).
• Figure 3: Multimodal motion of isotropic ciliary walkers. (A) A ciliary walker supported by an array of flexible cilia; view from below a transparent glass plate. Each cilium has a cylindrical cross-section, enabling multiple distinct buckled states at the level of the cilia array, i.e. translational (B) and rotational (D) configurations. (C-E) Superimposed snapshots of a walker performing translational (C) and rotational (E) motion on the vibrated plate.
• Figure 4: Transitions between locomotion modes for isotropic ciliary walkers in a 2D arena. (A) Probability distributions of the rotation rate and velocity, as obtained from experiments at fixed $f = 60$ Hz, $\Gamma = 5.05$, $W = 12$ g; in the $V_{\parallel}-V_{\perp}$ plane (i), and in the $|\boldsymbol{V}|-\omega$ plane (ii), overlaid with representative trajectories from (iii-iv); arrows indicate the direction of the transitions between locomotion modes. (iii-iv) Superimposed snapshots of a walker during collisions with a boundary, showing autonomous transitions from translational to rotational (iii) and from rotational to translational motion (iv); the colored markers indicate the orientation of the velocity $\boldsymbol{V}$ of the walker (inset colorbar), and collisions are marked with a star symbol. (B) Coordinates of the detected peaks from the experimental probability distributions as a function of $\Gamma$, as obtained from experiments at fixed $f = 40$ Hz, $W = 12$ g. (C) Same as (A), as obtained from simulations of a rigid active solid, with fixed $D = 0.2$, $\tau_n = 0.5$, $M = 1$. In the bottom snapshots, the arrows represent the active forces and are color coded according to their orientation (inset colorbar).
• Figure 5: Tunable locomotion behaviors in a 2D arena using different designs of cilia arrays. (i) Schematics of the different arrays of flexible cilia; A: $5 \times 5$ square array with bistable, rectangular-section cilia, all oriented along $\boldsymbol{\hat{e}}_{\parallel}$; B: circular line of bistable, rectangular-section cilia, all oriented along $\boldsymbol{\hat{e}}_{\theta}$; C: $10 \times 10$ square array with multistable, cylindrical-section cilia. (ii) Different arrays of cilia give rise to drastically different locomotion behaviors, as illustrated in the planes $V_{\parallel}-V_{\perp}$ and $|\boldsymbol{V}|-\omega$ (same conventions as Fig. \ref{['fig:4']}A-i,ii); as obtained from experiments at fixed $f = 40$ Hz, and $\Gamma = 2.45$,$2.45$, and $1.49$, respectively. Added weight on walkers in A and C was $8$ g.