Singularity analysis and representation of planar parallel manipulators

doi:10.1016/0921-8890(92)90001-F

Robotics and Autonomous Systems

Volume 10, Issue 4, 1992, Pages 209-224

https://doi.org/10.1016/0921-8890(92)90001-F Get rights and content

Abstract

In this paper, analytical expressions describing the singularity loci of simple planar three-degree-of-freedom parallel manipulators are derived. They are obtained using the analytical form of the Jacobian matrix, which is known to be singular when the manipulator is in a degenerate configuration. The expressions derived are then used to superimpose graphical representations of the manipulators' singularity loci and workspace. Both types of local degeneracies that can occur for parallel manipulators are studied. However, architecture singularities, which are rather trivial for the type of manipulators studied here, are assumed to be avoided by a proper choice of the manipulator's kinematic parameters. It is shown that, given certain assumptions on the geometry and for a given orientation of the end-effector, the singularity locus in the ( $x, y$ ) plane will always be a hyperbola. Moreover, when the passive revolute joints on the base and on the platform are respectively placed symmetrically, the hyperbola degenerates into a straight line. All these results are illustrated with examples of planar parallel manipulators. Some cases for which the singularities are located outside of the workspace are also presented. The analytic description of the singularity loci of manipulators in the Cartesian space can be used for analysis and optimum kinematic design. The expressions derived here have been integrated in a CAD package for parallel manipulators which provides a quick graphical representation of the singularity loci.

References (23)

J. Angeles et al.
The kinematic inversion of robot manipulators in the presence of singularities
ASME Journal of Dynamic Systems, Measurement, and Control
(1988)
J.E. Baker
Screw system algebra applied to special linkage configurations
Mechanism and Machine Theory
(1980)
C.M. Gosselin et al.
The optimum kinematic design of a planar three-degree-of-freedom parallel manipulator
ASME Jounral of Mechanisms, Transmissions and Automation in Design
(1988)
C.M. Gosselin et al.
Singularity analysis of closed-loop kinematic chains
IEEE Transactions on Robotics and Automation
(1990)
C.M. Gosselin
Determination of the worspace of 6-DOF parallel manipulators
ASME Journal of Mechanical Design
(1990)
C.M. Gosselin et al.
SIMPA: A graphic simulator for the CAD of parallel manipulators
C.M. Gosselin et al.
A global performance index for the kinematic optimization of robotic manipulators
ASME Journal of Mechanical Design
(1991)
K.H. Hunt
Special configurations of robot-arms via screw theory, Part 1: The Jacobian and its matrix cofactors
Robotica
(1986)
K.H. Hunt
Special configurations of robot-arms via screw theory, Part 2: Available end-effector displacements
Robotica
(1987)
A.C. Jones
Introduction to Algebratical Geometry
(1912)

C.A. Klein et al.

Dexterity measures for the design and control of kinematically redundant manipulators

The International Journal of Robotics Research

(1987)

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Singularity analysis and representation of planar parallel manipulators

Abstract

The kinematic inversion of robot manipulators in the presence of singularities

ASME Journal of Dynamic Systems, Measurement, and Control

Screw system algebra applied to special linkage configurations

Mechanism and Machine Theory

The optimum kinematic design of a planar three-degree-of-freedom parallel manipulator

ASME Jounral of Mechanisms, Transmissions and Automation in Design

Singularity analysis of closed-loop kinematic chains

IEEE Transactions on Robotics and Automation

Determination of the worspace of 6-DOF parallel manipulators

ASME Journal of Mechanical Design

SIMPA: A graphic simulator for the CAD of parallel manipulators

A global performance index for the kinematic optimization of robotic manipulators

ASME Journal of Mechanical Design

Special configurations of robot-arms via screw theory, Part 1: The Jacobian and its matrix cofactors

Robotica

Special configurations of robot-arms via screw theory, Part 2: Available end-effector displacements

Robotica

Introduction to Algebratical Geometry

Dexterity measures for the design and control of kinematically redundant manipulators

The International Journal of Robotics Research