Dynamics of Parallel Manipulators with Hybrid Complex Limbs -- Modular Modeling and Parallel Computing

TL;DR

The paper addresses the challenge of modeling the dynamics of PKM with complex, multi-loop limbs by introducing a modular approach based on constraint embedding and Lie-group formalisms. It develops a tree-topology based forward and inverse kinematics framework, combines cut-joint and cut-body loop-closure formulations, and derives task-space EOM that modularly aggregate limb contributions for efficient computation. A key contribution is the explicit demonstration of a parallel computation scheme that exploits PKM topology to distribute kinematic and dynamic evaluations across limbs, validated on Delta-type robots such as the 3RR[2RR]R Delta and IRSBot-2. The work provides a practical, scalable method for accurate PKM dynamics, enabling time-integration, inverse dynamics, and potential extension to continuum parallel robots, with rigorous formulation and numerical validation against commercial tools.

Abstract

Parallel manipulators, also called parallel kinematics machines (PKM), enable robotic solutions for highly dynamic handling and machining applications. The safe and accurate design and control necessitates high-fidelity dynamics models. Such modeling approaches have already been presented for PKM with simple limbs (i.e. each limb is a serial kinematic chain). A systematic modeling approach for PKM with complex limbs (i.e. limbs that possess kinematic loops) was not yet proposed despite the fact that many successful PKM comprise complex limbs. This paper presents a systematic modular approach to the kinematics and dynamics modeling of PKM with complex limbs that are built as serial arrangement of closed loops. The latter are referred to as hybrid limbs, and can be found in almost all PKM with complex limbs, such as the Delta robot. The proposed method generalizes the formulation for PKM with simple limbs by means of local resolution of loop constraints, which is known as constraint embedding in multibody dynamics. The constituent elements of the method are the kinematic and dynamic equations of motions (EOM), and the inverse kinematics solution of the limbs, i.e. the relation of platform motion and the motion of the limbs. While the approach is conceptually independent of the used kinematics and dynamics formulation, a Lie group formulation is employed for deriving the EOM. The frame invariance of the Lie group formulation is used for devising a modular modeling method where the EOM of a representative limb are used to derived the EOM of the limbs of a particular PKM. The PKM topology is exploited in a parallel computation scheme that shall allow for computationally efficient distributed evaluation of the overall EOM of the PKM. Finally, the method is applied to the IRSBot-2 and a 3\underline{R}R[2RR]R Delta robot, which is presented in detail.

Figures (16)

• Figure 1: a) Drawing of a 3RR[2RR]R Delta robot. b) Representative limb of this Delta robot
• Figure 2: a) Topological graph of the 3RR[2RR]R Delta. b) Subgraphs $\Gamma _{\left( l\right) }$ representing the $L=3$ limbs of this Delta.
• Figure 3: a) Topological graph of the 3R[2US] Delta robot. b) Subgraphs $\Gamma _{\left( l\right) }$ representing the $L=3$ limbs of this design.
• Figure 4: a) Drawing of the IRSBot-2 presented in Germain2013 (courtesy of Sébastien Briot, Laboratoire des Sciences du Numérique de Nantes). b) Representative limb comprising two loops.
• Figure 5: a) Topological graph of the IRSBot-2. b) Subgraphs $\Gamma _{\left( l\right) }$ representing the $L=2$ limbs. The hybrid topology of the limbs is clearly visible as $\Lambda _{1\left( l\right) }$ and $\Lambda _{2\left( l\right) }$ are arranged in series.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Example 1: 3RR[2RR]R Delta
• Example 2: 3R[2US] Delta
• Example 3: IRSBot-2
• Definition 3
• Example 4: 3RR[2RR]R Delta --cont.
• Example 5: 3R[2US] Delta --cont
• Example 6: IRSBot-2 --cont.
• Example 7: 3RR[2RR]R Delta --cont.