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Research Papers

Higher-Order Analysis of Kinematic Singularities of Lower Pair Linkages and Serial Manipulators

[+] Author and Article Information
Andreas Müller

Institute of Robotics,
Johannes Kepler University,
Linz 4040, Austria
e-mail: a.mueller@jku.at

Manuscript received December 27, 2016; final manuscript received October 26, 2017; published online December 22, 2017. Assoc. Editor: Philippe Wenger.

J. Mechanisms Robotics 10(1), 011008 (Dec 22, 2017) (13 pages) Paper No: JMR-16-1390; doi: 10.1115/1.4038528 History: Received December 27, 2016; Revised October 26, 2017

Kinematic singularities of linkages are configurations where the differential mobility changes. Constraint singularities are critical points of the constraint mapping defining the loop closure constraints. Configuration space (c-space) singularities are points where the c-space ceases to be a smooth manifold. These singularity types are not identical and can neither be distinguished nor identified by simply investigating the rank deficiency of the constraint Jacobian (linear dependence of joint screws). C-space singularities are reflected by the c-space geometry. In a previous work, a kinematic tangent cone was introduced as an approximation of the c-space, defined as the set of tangents to smooth curves in c-space. Identification of kinematic singularities amounts to analyze the local geometry of the set of critical points. As a computational means, a kinematic tangent cone to the set of critical points is introduced in terms of Jacobian minors. Closed form expressions for the derivatives of the minors in terms of Lie brackets of joint screws are presented. A computational method is introduced to determine a polynomial system defining the kinematic tangent cone. The paper complements the recently proposed mobility analysis using the tangent cone to the c-space. This allows for identifying c-space and kinematic singularities as long as the solution set of the constraints is a real variety. The introduced approach is directly applicable to the higher-order analysis of forward kinematic singularities of serial manipulators. This is briefly addressed in the paper.

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Figures

Grahic Jump Location
Fig. 1

Decision tree for classifying a point q ∈V as regular or singular, where r=rank J(q). It is presumed that the zero set of the loop constraints is real at q.

Grahic Jump Location
Fig. 2

6R linkage in a rank 4 singularity belonging to L41

Grahic Jump Location
Fig. 3

Underconstrained 7R linkage in the rank 3 singularity q0

Grahic Jump Location
Fig. 4

(a) Planar linkage in the reference configuration where its c-space has a cusp singularity. Bodies are numbered with I,…,VIII and joints with 1,…,10. (b) Topological graph and the γ = 3 FCs.

Grahic Jump Location
Fig. 5

Redundant serial 7R manipulator (KUKA LWR) in the singular configuration q0 with rank 3

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