Controllability and Displacement Analysis of a Three-Link Elastic Microswimmer: A Geometric Control Approach

TL;DR

This work analyzes a planar Purcell-type three-link elastic microswimmer with passive torsional springs, controlled by the resting angles of the springs. It combines linear and weakly nonlinear analyses with geometric control theory to characterize motion and controllability: linearization yields purely periodic angular/shape motion with no net translation, while a weakly nonlinear step reveals net translation along the central-link orientation; using Lie-bracket-based geometric control, the system is shown to be small-time locally controllable near equilibrium, and displacement estimates are derived for piecewise-constant controls. Numerical simulations corroborate the analytical predictions, showing that oscillatory controls can achieve motion in all directions near equilibrium and that piecewise-constant approximations closely track continuous controls over time. The results establish foundational controllability and displacement planning for elastic microswimmers, enabling trajectory design and subsequent optimization for micro-robotic applications.

Abstract

This study investigates the dynamics and controllability of a Purcell three-link microswimmer equipped with passive elastic torsional coils at its joints. By controlling the spontaneous curvature, we analyse the swimmers motion using both linear and weakly nonlinear approaches. Linear analysis reveals steady harmonic solutions for small-amplitude controls but does not predict any net displacement, whereas weakly nonlinear analysis predicts translation along the orientation of the central link. Using geometric control theory, we prove that the system is small time locally controllable near equilibrium and derive displacement estimates for periodic piecewise constant controls, which are validated through numerical simulations. These findings indicate that oscillatory controls can enable motion in all directions near equilibrium. This work offers foundational insights into the controllability of elastic microswimmers, paving the way for advanced motion planning and control strategies.

Figures (11)

• Figure 1: Schematic representation of an elastic $3$-links microswimmer.
• Figure 2: (Left) Plots of $\bar{v}$ versus frequency $\omega$, are shown for $\varepsilon=0.1, 0.3, 0.5, 0.7$. Both theoretical results (solid lines) and numerical results (dotted lines) are included. (Right) The relative error of the mean velocity, $\mathcal{E}_{\bar{v}}$, quantifies the discrepancy between the theoretical and the numerical results.
• Figure 3: Trajectories in the $x-y$ plane for different values of $\epsilon$. Solid blue lines represent the numerical solutions of the fully nonlinear equations of motion, while dashed red lines represent solutions obtained from the weak nonlinear analysis.
• Figure 4: Orientation angle $\vartheta$ as a function of time (left) and parametric plot of the shape angles in the $\alpha_{-1}- \alpha_{+1}$ plane (right), are shown for several values of the amplitude control gaits $\epsilon$. Solid blue lines represent the numerical solutions of the full nonlinear equations of motion, while dashed red lines represent solutions obtained from linear analysis
• Figure 5: Continuous inputs with $\epsilon=0.1$ and their approximation with piecewise constant functions.

Theorems & Definitions (8)

• Theorem 1
• Remark 1
• Definition 1: Equilibrium Point
• Definition 2: Small-Time Local Controllability
• Definition 3: Lie Algebra Rank Condition
• Definition 4: Iterated Lie Brackets
• Definition 5: Sussmann Condition
• Theorem 2