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Clarke Coordinates Are Generalized Improved State Parametrization for Continuum Robots

Reinhard M. Grassmann, Jessica Burgner-Kahrs

TL;DR

The paper addresses fragmentation in state parameterizations for continuum robots by proposing Clarke coordinates as a generalized, unifying framework. It derives and relates existing improved parameterizations to the generalized Clarke transformation matrix $\boldsymbol{M}_\mathcal{P}$ and its right-inverse, showing how $\overline{\boldsymbol{\rho}} = \boldsymbol{M}_\mathcal{P}\boldsymbol{\rho}$ subsumes prior approaches. Key contributions include expressing the Dian et al. ($n=3$), Della Santina et al. ($n=4$), and Allen et al. ($n=3,4$) parameterizations as Clarke coordinates, clarifying actuation versus displacement, and generalizing to non-constant curvature while addressing numerical singularities. The unified framework extends to arbitrary joint numbers and layouts, enabling systematic knowledge transfer across soft and continuum robotics subfields and broadening design and analysis capabilities.

Abstract

In this letter, we demonstrate that previously proposed improved state parameterizations for soft and continuum robots are specific cases of Clarke coordinates. By explicitly deriving these improved parameterizations from a generalized Clarke transformation matrix, we unify various approaches into one comprehensive mathematical framework. This unified representation provides clarity regarding their relationships and generalizes them beyond existing constraints, including arbitrary joint numbers, joint distributions, and underlying modeling assumptions. This unification consolidates prior insights and establishes Clarke coordinates as a foundational tool, enabling systematic knowledge transfer across different subfields within soft and continuum robotics.

Clarke Coordinates Are Generalized Improved State Parametrization for Continuum Robots

TL;DR

The paper addresses fragmentation in state parameterizations for continuum robots by proposing Clarke coordinates as a generalized, unifying framework. It derives and relates existing improved parameterizations to the generalized Clarke transformation matrix and its right-inverse, showing how subsumes prior approaches. Key contributions include expressing the Dian et al. (), Della Santina et al. (), and Allen et al. () parameterizations as Clarke coordinates, clarifying actuation versus displacement, and generalizing to non-constant curvature while addressing numerical singularities. The unified framework extends to arbitrary joint numbers and layouts, enabling systematic knowledge transfer across soft and continuum robotics subfields and broadening design and analysis capabilities.

Abstract

In this letter, we demonstrate that previously proposed improved state parameterizations for soft and continuum robots are specific cases of Clarke coordinates. By explicitly deriving these improved parameterizations from a generalized Clarke transformation matrix, we unify various approaches into one comprehensive mathematical framework. This unified representation provides clarity regarding their relationships and generalizes them beyond existing constraints, including arbitrary joint numbers, joint distributions, and underlying modeling assumptions. This unification consolidates prior insights and establishes Clarke coordinates as a foundational tool, enabling systematic knowledge transfer across different subfields within soft and continuum robotics.

Paper Structure

This paper contains 18 sections, 32 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Clarke Transform
  3. Displacement-Actuated Continuum Robot
  4. Intuition Using Generic Clarke Transformation Matrix
  5. Generalized Clarke transformation and its Inverse
  6. Clarke Transform and the Two-Dimensional Manifold
  7. Relation to Improved State Parameterizations
  8. Absolute Length or Displacement as Actuation?
  9. Relation to Dian et al.
  10. Relation to Della Santina et al.
  11. Relation to Allen et al.
  12. Note On the Constant Curvature Assumption
  13. Generalization to Non-Constant Curvature
  14. Numerical Singularities in Constant Curvature Kinematics
  15. Future Directions
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Displacement-actuated continuum robot. It has $n = 4$ joints, where $i\textsuperscript{th}$ joint location is defined by $\psi_i = 2\pi\left(i - 1\right)/n$ and $d_i = d$.
  • Figure 2: Transformation between joint space and arc space. Segment length $l$ is omitted since only a non-extensible displacement-actuated continuum robot is considered. Therefore, $l$ is one of the kinematic parameters and not an arc parameter.