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Kinetostatic Analysis for 6RUS Parallel Continuum Robot using Cosserat Rod Theory

Vinayvivian Rodrigues, Bingbin Yu, Christoph Stoeffler, Shivesh Kumar

TL;DR

This work models a 6-RUS PCR using Cosserat-rod theory to derive forward and inverse kinetostatic formulations for a closed-loop parallel continuum robot. A boundary-value problem with distal and proximal conditions is solved via a shooting method, coupling six elastic rods to the end-effector and base universal joint. The approach yields quantitative insights into axial stiffness ($k_z \approx 989.75\ \mathrm{N/m}$ for $F \in [10,300]\ \mathrm{N}$), maximum end-effector compression ($\approx 267\ \mathrm{N}$), rotational range, trajectory tracking accuracy (FK error ~ $1\times 10^{-7}$), and a reachable workspace mapped through IK sampling. The results validate the boundary-conditions formulation and demonstrate a practical pathway for design, control, and workspace analysis of PCRs, with open-source implementation for reproducibility and further development.

Abstract

Parallel Continuum Robots (PCR) are closed-loop mechanisms but use elastic kinematic links connected in parallel between the end-effector (EE) and the base platform. PCRs are actuated primarily through large deflections of the interconnected elastic links unlike by rigid joints in rigid parallel mechanisms. In this paper, Cosserat rod theory-based forward and inverse kinetostatic models of 6RUS PCR are proposed. A set of simulations are performed to analyze the proposed PCR structure which includes maneuverability in 3-dimensional space through trajectory following, deformation effects due to the planar rotation of the EE platform, and axial stiffness evaluation at the EE.

Kinetostatic Analysis for 6RUS Parallel Continuum Robot using Cosserat Rod Theory

TL;DR

This work models a 6-RUS PCR using Cosserat-rod theory to derive forward and inverse kinetostatic formulations for a closed-loop parallel continuum robot. A boundary-value problem with distal and proximal conditions is solved via a shooting method, coupling six elastic rods to the end-effector and base universal joint. The approach yields quantitative insights into axial stiffness ( for ), maximum end-effector compression (), rotational range, trajectory tracking accuracy (FK error ~ ), and a reachable workspace mapped through IK sampling. The results validate the boundary-conditions formulation and demonstrate a practical pathway for design, control, and workspace analysis of PCRs, with open-source implementation for reproducibility and further development.

Abstract

Parallel Continuum Robots (PCR) are closed-loop mechanisms but use elastic kinematic links connected in parallel between the end-effector (EE) and the base platform. PCRs are actuated primarily through large deflections of the interconnected elastic links unlike by rigid joints in rigid parallel mechanisms. In this paper, Cosserat rod theory-based forward and inverse kinetostatic models of 6RUS PCR are proposed. A set of simulations are performed to analyze the proposed PCR structure which includes maneuverability in 3-dimensional space through trajectory following, deformation effects due to the planar rotation of the EE platform, and axial stiffness evaluation at the EE.
Paper Structure (17 sections, 5 equations, 6 figures)

This paper contains 17 sections, 5 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Organization:
  3. Mathematical Background
  4. Kinetostatic Model of 6-RUS PCR
  5. Boundary Conditions
  6. Distal joint boundary condition ($s_i=l_i$)
  7. Proximal joint boundary condition ($s_i=0$):
  8. Inverse and Forward Kinetostatic formualtion
  9. Inverse Kinetostatic (IK) model:
  10. Forward Kinetostatic (FK) model:
  11. Shooting Method Implementation
  12. Results
  13. Analysis for compressing forces at the EE:
  14. Analysis of rotation range of the EE:
  15. Trajectory following of the EE under constant load:
  16. ...and 2 more sections

Figures (6)

  • Figure 1: Schematic explaining the kinematics for the proposed 6-RUS PCR.
  • Figure 2: Internal forces $\textbf{n} \in \mathbb{R}^3$ and internal moments $\textbf{m} \in \mathbb{R}^3$ acting on an arbitrary section of the rod. These forces and moments are expressed in the global frame H(s).
  • Figure 3: A shooting method overview to evaluate the boundary value problem John2015.
  • Figure 4: (a) IK solution describing the deformation with 150 N acting at the EE. (b) IK solution for a rotated EE platform by $+60^\circ$ about the $z$-axis.
  • Figure 5: (a) Helical trajectory comparison under 5 N load. (b) Euclidean distance between the reference EE position and the FK solution.
  • ...and 1 more figures