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A Constraint Embedding Approach for Dynamics Modeling of Parallel Kinematic Manipulators with Hybrid Limbs

Andreas Mueller

TL;DR

This paper introduces a local constraint-embedding approach to model dynamics of parallel kinematic manipulators with hybrid limbs. By treating each limb as a separate sub-system and solving intra-limb loop constraints numerically, the method yields a modular, scalable dynamic model that can be assembled into a full PKM model in task space. A nested Newton-Raphson scheme (outer-loop IK for limbs and inner-loop loop-constraint solves) enables robust handling of ill-conditioned or redundant loops, with optional compound iteration when constraints are well-behaved. The framework is demonstrated on the IRSBot-2, showing accurate inverse kinematics and inverse dynamics, with clear pathways for parallel computation and $O(n)$-time formulations. Overall, the approach provides a practical, high-fidelity dynamic modeling method for PKM with complex, hybrid limbs, facilitating advanced control and simulation tasks.

Abstract

Parallel kinematic manipulators (PKM) are characterized by closed kinematic loops, due to the parallel arrangement of limbs but also due to the existence of kinematic loops within the limbs. Moreover, many PKM are built with limbs constructed by serially combining kinematic loops. Such limbs are called hybrid, which form a particular class of complex limbs. Design and model-based control requires accurate dynamic PKM models desirably without model simplifications. Dynamics modeling then necessitates kinematic relations of all members of the PKM, in contrast to the standard kinematics modeling of PKM, where only the forward and inverse kinematics solution for the manipulator (relating input and output motions) are computed. This becomes more involved for PKM with hybrid limbs. In this paper a modular modeling approach is employed, where limbs are treated separately, and the individual dynamic equations of motions (EOM) are subsequently assembled to the overall model. Key to the kinematic modeling is the constraint resolution for the individual loops within the limbs. This local constraint resolution is a special case of the general \emph{constraint embedding} technique. The proposed method finally allows for a systematic modeling of general PKM. The method is demonstrated for the IRSBot-2, where each limb comprises two independent loops.

A Constraint Embedding Approach for Dynamics Modeling of Parallel Kinematic Manipulators with Hybrid Limbs

TL;DR

This paper introduces a local constraint-embedding approach to model dynamics of parallel kinematic manipulators with hybrid limbs. By treating each limb as a separate sub-system and solving intra-limb loop constraints numerically, the method yields a modular, scalable dynamic model that can be assembled into a full PKM model in task space. A nested Newton-Raphson scheme (outer-loop IK for limbs and inner-loop loop-constraint solves) enables robust handling of ill-conditioned or redundant loops, with optional compound iteration when constraints are well-behaved. The framework is demonstrated on the IRSBot-2, showing accurate inverse kinematics and inverse dynamics, with clear pathways for parallel computation and -time formulations. Overall, the approach provides a practical, high-fidelity dynamic modeling method for PKM with complex, hybrid limbs, facilitating advanced control and simulation tasks.

Abstract

Parallel kinematic manipulators (PKM) are characterized by closed kinematic loops, due to the parallel arrangement of limbs but also due to the existence of kinematic loops within the limbs. Moreover, many PKM are built with limbs constructed by serially combining kinematic loops. Such limbs are called hybrid, which form a particular class of complex limbs. Design and model-based control requires accurate dynamic PKM models desirably without model simplifications. Dynamics modeling then necessitates kinematic relations of all members of the PKM, in contrast to the standard kinematics modeling of PKM, where only the forward and inverse kinematics solution for the manipulator (relating input and output motions) are computed. This becomes more involved for PKM with hybrid limbs. In this paper a modular modeling approach is employed, where limbs are treated separately, and the individual dynamic equations of motions (EOM) are subsequently assembled to the overall model. Key to the kinematic modeling is the constraint resolution for the individual loops within the limbs. This local constraint resolution is a special case of the general \emph{constraint embedding} technique. The proposed method finally allows for a systematic modeling of general PKM. The method is demonstrated for the IRSBot-2, where each limb comprises two independent loops.

Paper Structure

This paper contains 42 sections, 57 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Graph Representation of PKM Topology
  3. Kinematics of a Limb
  4. Kinematics of Tree-Topology Limbs
  5. Task Space Velocity
  6. Loop Constraints within Hybrid Limbs
  7. Constraint Resolution
  8. Geometric Constraints:
  9. Velocity and Acceleration Constraints:
  10. Forward Kinematics of Separated Limb
  11. Inverse Kinematics
  12. Inverse Kinematics of PKM
  13. Numerical Constraint Embedding
  14. Nested Solution Scheme
  15. Inverse Kinematics of Limb ---Outer-Loop
  16. ...and 27 more sections

Figures (22)

  • Figure 1: a) Drawing of the 2-DOF IRSBot-2, whose EE performs pure translations in the 1-3-plane of $\mathcal{F}_{0}$. b) Separated limb comprising two loops with platform frame $\mathcal{F}_{\mathrm{P}}$ assigned.
  • Figure 2: a) Topological graph $\Gamma$ of the IRSBot-2. Indicated are the FCs $\Lambda _{1\left( l\right) }$ and $\Lambda _{2\left( l\right) }$ of the two limbs $l=1,2$. b) Subgraph $\Gamma _{\left( l\right) }$ of $\Gamma$ representing limb $l=1,2$ of the IRSBot-2. c) A ground-directed spanning tree $\vec{G}_{\left( l\right) }$ on $\Gamma _{\left( l\right) }$.
  • Figure 3: A possible tree $G$ on the topological graph $\Gamma$ of the IRSBot-2 in fig. \ref{['figIRSBotGraph']}a).
  • Figure 4: Kinematic topology of the system for which EOM are derived in case of the IRSBot-2.
  • Figure 5: Geometry of the IRSBot-Model -- Front view of right limb. $\varphi _{0}$ and $\psi _{0}$ describe the orientation of the proximal and distal limb, and $h_{0}$ is the platform hight in the reference configuration $\mathbold{\vartheta}=\mathbf{0}$, respectively.
  • ...and 17 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6