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On the Certification of the Kinematics of 3-DOF Spherical Parallel Manipulators

Alexandre Lê, Guillaume Rance, Fabrice Rouillier, Damien Chablat

TL;DR

This work tackles robust kinematic certification for non-redundant $3$-DOF Spherical Parallel Manipulators with unlimited rolling under fabrication/measurement uncertainties. It contributes two complementary certification pipelines: a symbolic discriminant-variety method for the inverse geometric model (IGM) and a semi-numerical Kantorovich-based path-tracking approach for the forward geometric model (FGM), enabled by polynomial reformulation via tangent-half-angle substitutions and interval/ball arithmetic for uncertainty propagation. Key results include a Type-1 singularity-free prescribed workspace $m{W}^\star$, computed joint-stops, and a validated forward-path tracking strategy with guaranteed convergence. The framework provides a rigorous route to certify safe operation of SPMs and is extendable to other parallel robots, potentially integrating GCI-based design optimization.

Abstract

This paper aims to study a specific kind of parallel robot: Spherical Parallel Manipulators (SPM) that are capable of unlimited rolling. A focus is made on the kinematics of such mechanisms, especially taking into account uncertainties (e.g. on conception & fabrication parameters, measures) and their propagations. Such considerations are crucial if we want to control our robot correctly without any undesirable behavior in its workspace (e.g. effects of singularities). In this paper, we will consider two different approaches to study the kinematics and the singularities of the robot of interest: symbolic and semi-numerical. By doing so, we can compute a singularity-free zone in the work- and joint spaces, considering given uncertainties on the parameters. In this zone, we can use any control law to inertially stabilize the upper platform of the robot.

On the Certification of the Kinematics of 3-DOF Spherical Parallel Manipulators

TL;DR

This work tackles robust kinematic certification for non-redundant -DOF Spherical Parallel Manipulators with unlimited rolling under fabrication/measurement uncertainties. It contributes two complementary certification pipelines: a symbolic discriminant-variety method for the inverse geometric model (IGM) and a semi-numerical Kantorovich-based path-tracking approach for the forward geometric model (FGM), enabled by polynomial reformulation via tangent-half-angle substitutions and interval/ball arithmetic for uncertainty propagation. Key results include a Type-1 singularity-free prescribed workspace , computed joint-stops, and a validated forward-path tracking strategy with guaranteed convergence. The framework provides a rigorous route to certify safe operation of SPMs and is extendable to other parallel robots, potentially integrating GCI-based design optimization.

Abstract

This paper aims to study a specific kind of parallel robot: Spherical Parallel Manipulators (SPM) that are capable of unlimited rolling. A focus is made on the kinematics of such mechanisms, especially taking into account uncertainties (e.g. on conception & fabrication parameters, measures) and their propagations. Such considerations are crucial if we want to control our robot correctly without any undesirable behavior in its workspace (e.g. effects of singularities). In this paper, we will consider two different approaches to study the kinematics and the singularities of the robot of interest: symbolic and semi-numerical. By doing so, we can compute a singularity-free zone in the work- and joint spaces, considering given uncertainties on the parameters. In this zone, we can use any control law to inertially stabilize the upper platform of the robot.
Paper Structure (23 sections, 1 theorem, 15 equations, 10 figures, 1 table)

This paper contains 23 sections, 1 theorem, 15 equations, 10 figures, 1 table.

Table of Contents

  1. Context of the study
  2. Introduction
  3. Generalities on parallel robots
  4. Generalities on SPMs
  5. Modeling of a non-redundant 3-DOF SPM
  6. Description
  7. Geometric and first order kinematic models
  8. Implementation of the geometric model
  9. Requirements and strategy
  10. Uncertainty analysis
  11. Propagation of uncertainty on the fabrication parameters
  12. On the coaxiality of the input shafts
  13. Certifying the Inverse Geometric Model of 3-DOF SPMs
  14. Workspace analysis
  15. Joint space analysis
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\bm{f}:\mathcal{D}\subseteq\mathbb{R}^n \to\mathbb{R}^n$ a function of class $\mathcal{C}^2$. Let $\bm{x}_0$ be a point and $\overline{\bm{U}}\left(\bm{x}_0\right)$ its neighborhood defined by $\overline{\bm{U}}\left(\bm{x}_0\right)\triangleq\left\{\bm{x}\in \mathcal{D}\;\text{s.t.}\;\left\lVer then there is a unique solution of $\bm{f}(\bm{x})=\bm{0}$ in $\overline{\bm{U}}\left(\bm{x}_0\righ

Figures (10)

  • Figure 1: General structure of a parallel robot
  • Figure 2: Examples of non-redundant SPMs (3-RRR)
  • Figure 3: Illustration of a typical SPM with conception paramaters (red $+$ green), local frames (dark blue)
  • Figure 4: Principle of the geometric model
  • Figure 5: Certification by avoiding the discriminant variety $\mathcal{W}_D$ w.r.t. the projection onto the paramater space
  • ...and 5 more figures

Theorems & Definitions (9)

  • Remark 2
  • Definition 1: Discriminant Variety
  • Definition 2: Ball interval
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1: Newton-Kantorovich
  • Remark 6
  • Definition 3: Interval