Asymmetry-Induced Chiral Dynamics in Coupled Self-Propelled Robots: Spinning and Circular Motion

Technical Summary: Asymmetry-Induced Chiral Dynamics in Coupled Self-Propelled Robots

Problem Statement
Microscopic biological swimmers, such as E. coli, Chlamydomonas reinhardtii, and spermatozoa, exhibit diverse motility strategies including run-and-tumble (RnT) motion, circular trajectories, and spinning. These behaviors are often driven by structural asymmetries (e.g., off-center flagellar attachment or morphological asymmetry) that couple propulsion forces to net torques. While previous minimal robotic models (e.g., Paramanick et al.) successfully replicated RnT motion using two self-propelled robots connected by a rigid rod, a key question remains: can controlled mechanical asymmetry in a system with elastic coupling generate circular motion and other distinct dynamical regimes? Furthermore, the functional role of elastic compliance (as opposed to rigid constraints) in switching between motility behaviors is not fully explored. This paper investigates the dynamics of a spring-coupled pair of self-propelled active Brownian robots with asymmetric propulsion directions to address these gaps.

Methodology
The authors develop a theoretical model extending the rigid-rod framework of Paramanick et al. by replacing the rigid connection with a Hookean spring of stiffness $k$. The system consists of two active Brownian robots, each characterized by:

• Self-propulsion: A constant active force $F_a$ acting along an orientation vector $n_i$.
• Geometric Asymmetry: A pivot offset $\delta$ (distance between the robot center and the spring attachment point) and a propulsion angle $\alpha_i$ (the angle between the propulsion direction and the line joining the pivot to the center).
• Coupling: The robots are connected by a spring with rest length $l$ and stiffness $k$.
• Noise: Rotational diffusion $D_r$ is included to model stochastic reorientation.

The equations of motion are derived in the overdamped limit, neglecting inertial terms. The model accounts for anisotropic translational friction (different coefficients parallel and perpendicular to the robot's orientation). The study utilizes numerical simulations to analyze the system's behavior across the parameter space of propulsion angles ($\alpha_1, \alpha_2$), spring stiffness ($K$), and rotational noise ($D_r$). Key observables include the average spin angular velocity ($\langle \omega_s \rangle$), orbital angular velocity ($\omega_c$), average centroid speed ($\langle v_c \rangle$), and orientational autocorrelation functions ($C_0(\tau)$ and $C(\tau)$).

Key Contributions and Results

1. Emergence of Three Dynamical Regimes:
   The system exhibits three distinct regimes determined primarily by the geometric configuration ($\alpha_1, \alpha_2$):

   • Run-and-Tumble (RnT): Occurs in symmetric configurations ($\alpha_1 = \alpha_2$). The system alternates between straight runs and reorientation events, similar to the rigid-rod model.
   • Spinning Motion: Occurs when $\alpha_1 + \alpha_2 = 360^\circ$ (propulsion directions are anti-aligned). In this regime, the net active force vanishes, and the geometric torque induces rapid in-place rotation with negligible translation. The chirality (clockwise vs. counter-clockwise) is determined by the sign of $\alpha_1 - \alpha_2$.
   • Circular Motion: Occurs in asymmetric configurations where $\alpha_1 + \alpha_2 \neq 360^\circ$ and $\alpha_1 \neq \alpha_2$. Here, a non-zero net active force coexists with a net torque, driving the system's centroid along sustained curved trajectories. The radius of curvature and chirality are controlled by the propulsion angles.

2. Role of Geometric Asymmetry and Fixed Points:
   The authors analyze the deterministic flow in the $(\theta_+, \theta_-)$ plane (where $\theta_\pm$ are combinations of robot orientations). They identify stable fixed points corresponding to spinning and circular states.

   • For $\alpha_1 + \alpha_2 = 360^\circ$, stable fixed points correspond to perfectly anti-aligned robots, resulting in pure spinning.
   • For $\alpha_1 + \alpha_2 \neq 360^\circ$, the stable fixed points involve non-perfectly anti-aligned robots, generating a net force that causes the centroid to drift, resulting in circular motion.
   • The study identifies small parameter regions where the chirality of motion reverses relative to the sign of $\alpha_1 - \alpha_2$ due to the presence of alternative stable fixed points.

3. Effect of Rotational Noise ($D_r$):
   Increasing rotational noise progressively randomizes the propulsion orientations, weakening the geometric torque.

   • Spinning: As $D_r$ increases, $\langle \omega_s \rangle$ decreases. High noise drives the system away from the stable spinning fixed point toward the run state. Configurations with larger $\alpha_1$ (near $90^\circ$) are more robust against noise than those with smaller $\alpha_1$.
   • Circular Motion: Circular motion persists only below a threshold noise level ($D_r \approx 0.15 \, \text{rad}^2 \text{s}^{-1}$). Above this threshold, orbital angular velocity $\omega_c$ drops to zero, and the motion becomes diffusive.

4. Effect of Spring Stiffness ($K$):
   The spring stiffness acts as a tuning parameter for mechanical compliance.

   • Both $\langle \omega_s \rangle$ and $\omega_c$ increase monotonically with $K$ and saturate as $K \to \infty$, recovering the rigid-rod limit.
   • At low $K$ (high compliance), the spring cannot sustain the necessary constraint forces to transmit torque effectively, suppressing the dynamical response.
   • This finding suggests that elastic compliance can continuously interpolate between suppressed dynamics and fully developed collective motion.

5. Mean Square Displacement (MSD):

   • Circular Regime: MSD shows ballistic growth at short times, followed by pronounced oscillations at intermediate times (signature of orbital confinement), and suppressed long-time diffusion.
   • Spinning Regime: MSD shows ballistic growth followed by a smooth crossover to diffusion without oscillations, reflecting the absence of orbital confinement.

Significance and Claims
The paper claims to demonstrate that geometric asymmetry and elastic compliance are sufficient to generate the full spectrum of collective motility behaviors observed in biological microswimmers—from straight runs and tumbles to chiral circular trajectories and spinning—without requiring hydrodynamic interactions, chemical gradients, or active sensing.

The authors position their work as a bridge between artificial active systems and biological microswimmers. By showing that a simple mechanical design (two robots coupled by a spring with asymmetric propulsion) can reproduce complex behaviors like the chirality switching in Chlamydomonas reinhardtii (which modulates flagellar beating to change circling handedness), the study establishes a minimal mechanical design principle. This principle highlights how structural asymmetries, coupled with mechanical constraints, can produce diverse motility behaviors through purely mechanical interactions. The results suggest that the elastic coupling found in biological systems (e.g., the distal striated fiber in Chlamydomonas or the bacterial flagellar hook) may play a functional role in tuning motility regimes, a hypothesis supported by the model's ability to replicate these transitions via stiffness tuning.