[Paper Review] Trajectory Tracking Using Motion Primitives for the Purcell's Swimmer

Sudin Kadam, Kedar Joshi+3 more|arXiv (Cornell University)|Mar 4, 2017

Micro and Nano Robotics17 references4 citations

TL;DR

This paper presents a motion planning algorithm for a 3-link planar Purcell's swimmer using Lie algebraic notions to prove the existence and synthesize control profiles for motion primitives. By leveraging driftless control-affine system theory and vision-based open-loop control, the method enables trajectory tracking of arbitrary paths in SE(2), achieving precise tracking of straight and circular trajectories with experimental validation on a physical prototype.

ABSTRACT

Locomotion at low Reynolds numbers is a topic of growing interest, spurred by its various engineering and medical applications. This paper presents a novel prototype and a locomotion algorithm for the 3-link planar Purcell's swimmer based on Lie algebraic notions. The kinematic model based on Cox theory of the prototype swimmer is a driftless control-affine system. Using the existing strong controllability and related results, the existence of motion primitives is initially shown. The Lie algebra of the control vector fields is then used to synthesize control profiles to generate motions along the basis of the Lie algebra associated with the structure group of the system. An open loop control system with vision-based positioning is successfully implemented which allows tracking any given continuous trajectory of the position and orientation of the swimmer's base link. Alongside, the paper also provides a theoretical interpretation of the symmetry arguments presented in the existing literature to generate the control profiles of the swimmer.

Motivation & Objective

• To develop a motion planning algorithm for the 3-link planar Purcell’s swimmer based on controllability and motion primitives.
• To prove the existence of motion primitives using Lie algebraic structures of the control vector fields.
• To synthesize control inputs that enable tracking of arbitrary continuous trajectories in the Special Euclidean group SE(2).
• To implement and validate the control strategy experimentally using a vision-based open-loop system.
• To provide a generalizable framework applicable to other driftless control-affine systems in microswimming robotics.

Proposed method

• Model the swimmer’s kinematics using resistive force theory and Cox’s theory, resulting in a driftless control-affine system on S¹×S¹.
• Establish controllability via Lie algebra rank condition, proving existence of motion primitives using the Lie bracket structure of control vector fields.
• Synthesize control profiles by integrating flows of vector fields associated with the Lie algebra basis, enabling motion along desired group directions.
• Implement an open-loop control system with real-time vision-based feedback for position and orientation tracking of the base link.
• Use a hardware prototype with servo motors and a vision system to execute and validate the planned trajectories.
• Calibrate viscous drag coefficients experimentally to ensure accurate kinematic modeling.

Experimental results

Research questions

• RQ1 Can motion primitives be rigorously proven to exist for the 3-link Purcell’s swimmer using Lie algebraic control theory?
• RQ2 How can control profiles be synthesized to generate net motion along arbitrary directions in SE(2) using only the Lie algebra basis?
• RQ3 To what extent can open-loop control with vision feedback achieve accurate trajectory tracking for low-Reynolds-number swimmers?
• RQ4 How do symmetry-based gaits from prior work compare to the Lie algebra-based motion primitive approach in terms of trajectory generation?
• RQ5 Can the proposed method be generalized to other driftless control-affine systems beyond the Purcell swimmer?

Key findings

• The control vector fields g₁ and g₂ for the Purcell’s swimmer are complete, as their support on S¹×S¹ is compact, ensuring global existence of trajectories.
• The system is weakly controllable, enabling the synthesis of motion primitives that allow arbitrary trajectory tracking in SE(2).
• Experimental results show successful open-loop tracking of a 12 cm straight-line trajectory at 154°, with initial alignment taking ~260 seconds and translation ~720 seconds.
• A circular trajectory of radius 20 cm was approximated as a 10-sided polygon, requiring ~3 hours to complete with repeated theta and x-maneuvers.
• Tracking errors were observed due to servo motor jerks after prolonged use, non-zero cross-coupling in control inputs, and accumulation of open-loop noise.
• The method successfully reproduces symmetry-based gaits from prior work (e.g., [25]) while providing a systematic, non-empirical framework for gait design.