cs.ROApr 10, 2026

On the Terminology and Geometric Aspects of Redundant Parallel Manipulators

Authors: Andreas Mueller

Organizations: Chair of Mechanics and Robotics, University Duisburg-Essen, Lotharstrasse 1, 47057 Duisburg, Germany

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Abstract

Parallel kinematics machines (PKM) can exhibit kinematic as well as actuation redundancy. While the meaning of kinematic redundancy has been clarified already for serial manipulators, actuation redundancy, that is only possible in PKM, is differently classified in the literature. In this paper a consistent terminology for general redundant PKM is proposed. A kinematic model is introduced with the configuration space (c-space) as central part. The notion of kinematic redundancy is recalled for PKM. C-space, output, and input singularities are distinguished. The significance of the c-space geometry is emphasized, and it is pointed out geometrically that input singularities can be avoided by redundant actuation schemes. In order to distinguish different actuation schemes of PKM a non-linear control system is introduced whose dynamics evolves on the c-space. The degree of actuation (DOA) is introduced as the number of independent control vector fields, and PKM are classified as full-actuated and underactuated. Relating this DOA to the degree of freedom (DOF) allows to classify the actuation redundancy.

Figures & tables

Figure 1: Geometric interpretation of the PKM control system. The operation modes refer to input space I\mathcal{I}
Figure 2: 5-bar mechanism a), and its redundant extension b) the R R/2 R RR PKM.
Figure 3: Manipulability measure 1/κ1/\kappa of a) 5-bar mechanism and b) R R/2 R RR PKM.
Figure 4: q1q^{1} - q2q^{2} - q4q^{4} c-space section of the 5-bar mechanism.
Figure 5: A 2 DOF manipulator in different actuation modes.
Figure 6: 3URU DYMO PKM in its planar operation mode a), c-space singularity b), and lockup mode c).
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