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Research Papers

# Analytic and Geometric Technique for the Singularity Analysis of Multi-Degree-of-Freedom Spherical Mechanisms

[+] Author and Article Information
Raffaele Di Gregorio

Department of Engineering,
University of Ferrara,
Via Saragat, 1,
FERRARA 44122, Italy
e-mail: rdigregorio@ing.unife.it

Following [3,4], the connectivity between two links of a mechanism is the dof number of the relative motion between the two links. Hereafter, the phrase “local connectivity” will denote the dof number of the instantaneous relative motion at a given mechanism configuration.

It is worth noting that, since the output link belongs to the mechanism, its local connectivity, gq, must respect the inequality: gq ≤ min(f, nq), where f is the dof number of the unconstrained rigid body in the space the mechanism's links move in.

Superposition principle states that “In a linear and homogeneous system, say

$Ax=By,$
IF (x1,y1) and (x2,y2) are both solutions, THEN (k1x1+k2x2, k1y1+k2y2) where k1 and k2 are any two constant coefficients is solution, too.” [9].

It is worth noting that relationship (1) defines ujk and ωjk but a sign (i.e., {ujk, ωjk} and {-ujk, -ωjk} yield the same angular velocity ωjk)

It is worth stressing that Eq. (2) yields a type-I singularity iff d is equal to zero, a type-II singularity iff c is equal to zero and, accordingly, a type-III singularity iff c and d are both equal to zero.

Hereafter, mechanism's geometric variables that are not input variables will be called “secondary variables” (e.g., nonactuated joints variables).

Hereafter, the notation (P, a) denotes an oriented line passing through point P and with the direction of the unit vector a.

In fact, in this case, since link 4 is at a stationary configuration, if link 5 is locked link 6 cannot move (i.e., this is not a type-II singularity); vice versa, if link 6 is locked link 5 cannot move (i.e., this is not a type-I singularity).

In fact, in this case, since link 4 is at a stationary configuration, if link 3+5 is locked link 6 cannot move (i.e., this is not a type-II singularity); vice versa, if link 6 is locked link 3+5 cannot move (i.e., this is not a type-I singularity).

1The right subscript m × n denotes the size of the matrix. In this case, the number of rows, m, is equal to the number of velocity constraints, and the number of column, n, is equal to the total number of joint variables. In a mechanism, n is always greater than m since a number of independent constraints equal to n would lead to zero mobility (i.e., to a structure).

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received September 5, 2013; final manuscript received August 30, 2014; published online December 4, 2014. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 7(3), 031008 (Aug 01, 2015) (9 pages) Paper No: JMR-13-1175; doi: 10.1115/1.4028625 History: Received September 05, 2013; Revised August 30, 2014; Online December 04, 2014

## Abstract

Mechanisms' instantaneous kinematics is modeled by linear and homogeneous mappings whose coefficient matrices are also meaningful to understand their static behavior through the virtual work principle. The analysis of these models is a mandatory step during design. The superposition principle can be used for building and studying linear and homogeneous models. Here, multi-degree-of-freedom (multi-DOF) spherical mechanisms are considered. Their instantaneous-kinematics model is written by exploiting instantaneous-pole-axes' (IPA) properties and the superposition principle. Then, this general model is analyzed and an exhaustive analytic and geometric technique to identify all their singular configurations is deduced. Eventually, the effectiveness of the deduced technique is shown with two relevant case studies.

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## Figures

Fig. 1

General scheme: O is the center of the spherical motion; with reference to the angular velocity of the output link, u and ω are its unit vector and signed magnitude, respectively, in the actual case, whereas ui and ωi, for i = 1,…,g, are its unit vector and signed magnitude, respectively, in the ith single-DOF mechanism

Fig. 2

Spherical 6R single-loop linkage: notations

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