Bundling and Tumbling in Bacterial-inspired Bi-flagellated Soft Robots for Attitude Adjustment

TL;DR

This work addresses attitude control for a bacterial-inspired bi-flagellated soft robot operating in viscous fluids. It introduces a multi-physics computational framework that couples Discrete Elastic Rods (DER) for flagellum elasticity, Regularized Stokeslet Segments (RSS) for hydrodynamics, and an Implicit Contact Model (IMC) for realistic contact interactions, with quaternion-based rotation dynamics to predict yaw and pitch adjustments. The authors validate the framework experimentally in glycerin and demonstrate bundling/tumbling phenomena, stabilization of attitude angles, and an Attitude Map showing how actuator speeds $(\omega_1,\omega_2)$ define attainable orientations, complemented by parametric insights via dimensionless groups. The results offer design guidance for flagellated robots in confined viscous environments and provide a foundation for precise, reorientation-enabled mobility at micro- and macroscale scales.

Abstract

We create a mechanism inspired by bacterial swimmers, featuring two flexible flagella with individual control over rotation speed and direction in viscous fluid environments. Using readily available materials, we design and fabricate silicone-based helical flagella. To simulate the robot's motion, we develop a physics-based computational tool, drawing inspiration from computer graphics. The framework incorporates the Discrete Elastic Rod method, modeling the flagella as Kirchhoff's elastic rods, and couples it with the Regularized Stokeslet Segments method for hydrodynamics, along with the Implicit Contact Model to handle contact. This approach effectively captures polymorphic phenomena like bundling and tumbling. Our study reveals how these emergent behaviors affect the robot's attitude angles, demonstrating its ability to self-reorient in both simulations and experiments. We anticipate that this framework will enhance our understanding of the directional change capabilities of flagellated robots, potentially stimulating further exploration on microscopic robot mobility.

Figures (6)

• Figure 1: Attitude adjustment of bi-flagellated robot in the viscous fluid (glycerin). (a) Rotation in the horizontal plane (pointing opposite to gravity direction). (b) Rotation in the vertical plane (pointing out to the paper).
• Figure 2: Robot schematic and attitude representation. (a) The bi-flagellated robot comprises an assembly base and two identical soft flagella. Each flagellum with length $l$, cross-sectional radius $r$, helix pitch $\lambda$, and helix radius $R$ is actuated by a miniature brushed DC motor. The robot is affixed to a ball joint to reorient. (b) Yaw angle $\psi$ (b.1), Pitch angle $\theta$ (b.2), and Roll angle $\phi$ (b.3) signifies the angular displacement around the vertical axis $z_B$, lateral axis $y_B$, and longitudinal axis $x_B$, respectively.
• Figure 3: Fabrication of a soft flagellum. (a) Step 1: blending VPS with iron powder to precisely calibrate the density of the silicone composite. (b) Step 2: using a syringe to inject the mixture into the PVC tube preformed by a mold. (c) Step 3: slitting the tube to extract the silicone elastomer.
• Figure 4: Computation framework of bi-flagellated robot. (a) Discretization scheme. Each soft flagellum is discretized into $\mathbf{N}$ nodes (denoted as $\mathbf{x_i}$) and $\mathbf{N-1}$ edges (denoted as $\mathbf{e^i}$), and connected to the base node $\mathbf{x_N}$. (b) Representation of material frame ([$\mathbf{m_{1}^{i-1}}, \mathbf{m_{2}^{i-1}}, \mathbf{t_{i-1}}$] and [$\mathbf{m_{1}^{i}}, \mathbf{m_{2}^{i}}, \mathbf{t_{i}}$]) and reference frame ([$\mathbf{d_{1}^{i-1}}, \mathbf{d_{2}^{i-1}}, \mathbf{t_{i-1}}$] and [$\mathbf{d_{1}^{i}}, \mathbf{d_{2}^{i}}, \mathbf{t_{i}}$]). The signed angle from $\mathbf{d_1^i}$ to $\mathbf{m_1^i}$ is $\mathbf{\theta^i}$ and twist at node $\mathbf{x_i}$ is $\mathbf{\tau_i} = \mathbf{\theta^i}-\mathbf{\theta^{i-1}}$. (c) Regularized Stokeslets Segments. This method establishes a connection between the velocity $\mathbf{\dot{x}_m}$ at a point $\mathbf{x_m}$ and the forces acting on the nodes $\mathbf{x_i}$ and $\mathbf{x_{i+1}}$. (d) Implicit contact model. The artificial contact energy is defined by the minimal distance $\Delta_{\mathbf{ij}}$ between two edges $\mathbf{e^i}$ and $\mathbf{e^j}$.
• Figure 5: Time evolution of reaction forces and attitude angles under dual actuation of $\omega_1 = 50$ rpm and $\omega_2 = 10$ rpm. (a) Examination of Flagellum 1 (Top) and Flagellum 2 (Bottom) reveals that the vertical force component, $F_z$, rapidly stabilizes after an initial transition, while the horizontal forces, $F_x$ and $F_y$, exhibit sinusoidal behavior. For a right-handed helix, the counter-clockwise rotation ($\omega > 0$) generates an downward propulsion force. (b) Yaw angle $\psi$ and pitch angle $\theta$ eventually stabilize, reaching steady-state values of $\psi_{\text{ss}} = 60.76 ^\circ$ and $\theta_{\text{ss}} = -3.29 ^\circ$ in simulation, and $\psi_{\text{ss}} = 67.95 ^\circ$ and $\theta_{\text{ss}} = -4.96 ^\circ$ in experiment.