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A General Lie-Group Framework for Continuum Soft Robot Modeling

Lingxiao Xun, Benoît Rosa, Jérôme Szewczyk, Brahim Tamadazte

Abstract

This paper introduces a general Lie group framework for modeling continuum soft robots, employing Cosserat rod theory combined with cumulative parameterization on the Lie group SE(3). This novel approach addresses limitations present in current strain-based and configuration-based methods by providing geometric local control and eliminating unit quaternion constraints. The paper derives unified analytical expressions for kinematics, statics, and dynamics, including recursive Jacobian computations and an energy-conserving integrator suitable for real-time simulation and control. Additionally, the framework is extended to handle complex robotic structures, including segmented, branched, nested, and rigid-soft composite configurations, facilitating a modular and unified modeling strategy. The effectiveness, generality, and computational efficiency of the proposed methodology are demonstrated through various scenarios, including large-deformation rods, concentric tube robots, parallel robots, cable-driven robots, and articulated fingers. This work enhances modeling flexibility and numerical performance, providing an improved toolset for designing, simulating, and controlling soft robotic systems.

A General Lie-Group Framework for Continuum Soft Robot Modeling

Abstract

This paper introduces a general Lie group framework for modeling continuum soft robots, employing Cosserat rod theory combined with cumulative parameterization on the Lie group SE(3). This novel approach addresses limitations present in current strain-based and configuration-based methods by providing geometric local control and eliminating unit quaternion constraints. The paper derives unified analytical expressions for kinematics, statics, and dynamics, including recursive Jacobian computations and an energy-conserving integrator suitable for real-time simulation and control. Additionally, the framework is extended to handle complex robotic structures, including segmented, branched, nested, and rigid-soft composite configurations, facilitating a modular and unified modeling strategy. The effectiveness, generality, and computational efficiency of the proposed methodology are demonstrated through various scenarios, including large-deformation rods, concentric tube robots, parallel robots, cable-driven robots, and articulated fingers. This work enhances modeling flexibility and numerical performance, providing an improved toolset for designing, simulating, and controlling soft robotic systems.
Paper Structure (50 sections, 6 theorems, 137 equations, 30 figures, 9 tables)

This paper contains 50 sections, 6 theorems, 137 equations, 30 figures, 9 tables.

Table of Contents

  1. Introduction
  2. State of the art
  3. Contributions and outline
  4. Cosserat Rod Theory
  5. Continuous formulation
  6. Strain-based parameterization
  7. Configuration-based parameterization
  8. Challenges and motivation
  9. Cumulative Parameterization
  10. Standard parameterization
  11. Cumulative parameterization
  12. Cumulative parameterization on Lie group
  13. B-spline approximation
  14. Hermite interpolation
  15. Kinematics
  16. ...and 35 more sections

Key Result

Theorem 1

Consider two configurations $g_a,g_b\in\mathcal{G}$ related by a Lie-algebra increment $g_b = g_a \boxplus x = g_a \,\tau(x), \;x\in\mathfrak g.$ Let be the corresponding infinitesimal variations, their variation holds the following relationship: with $\delta \zeta_a={g_a^{-1}}\delta g_a\in \mathfrak{g}$ and $\delta \zeta_b={g_b^{-1}}\delta g_b\in \mathfrak{g}$. The operator $\mathrm{Ad}_g: \mat

Figures (30)

  • Figure 1: Schematic diagram of the soft manipulator.
  • Figure 2: Bspline and Hermite curve on $SE(3)$ manifold.
  • Figure 3: B-spline basis functions.
  • Figure 4: Cumulative basis functions.
  • Figure 5: The variation of the kinematic Jacobian matrix concerning the arc-length coordinate $s$. The left figure shows the Jacobian matrix obtained using the Piecewise Linear Strain (PLS) li2023piecewise parameterization method. In contrast, the right figure shows the Jacobian matrix obtained using the proposed Lie group B-spline parameterization method.
  • ...and 25 more figures

Theorems & Definitions (19)

  • Definition 1: Homogeneous transformation matrix
  • Definition 2: Strain and velocity of Cosserat rod
  • Definition 3: Cumulative Parameterization
  • Definition 4: Retraction map
  • Definition 5: Local operator
  • Definition 6: Cumulative parameterization of Lie group
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 7: Left-trivialized tangent
  • ...and 9 more