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Research Papers

Synthesis and Singularity Analysis of N-$UU$ Parallel Wrists: A Symmetric Space Approach

[+] Author and Article Information
Yuanqing Wu

Department of Industrial Engineering,
University of Bologna,
Via Risorgimento 2,
Bologna 40136, Italy
e-mail: yuanqing.wu@unibo.it

Marco Carricato

Department of Industrial Engineering,
University of Bologna,
Via Risorgimento 2,
Bologna 40136, Italy
e-mail: marco.carricato@unibo.it

1Corresponding author.

Manuscript received January 2, 2017; final manuscript received August 3, 2017; published online August 24, 2017. Assoc. Editor: Jian S. Dai.

J. Mechanisms Robotics 9(5), 051013 (Aug 24, 2017) (11 pages) Paper No: JMR-17-1002; doi: 10.1115/1.4037547 History: Received January 02, 2017; Revised August 03, 2017

Abstract

We report some recent advances in kinematics and singularity analysis of the mirror-symmetric $N$-$UU$ parallel wrists using symmetric space theory. We show that both the finite displacement and infinitesimal singularity kinematics of a $N$-$UU$ wrist are governed by the mirror symmetry property and half-angle property of the underlying motion manifold, which is a symmetric submanifold of the special Euclidean group SE(3). Our result is stronger than and may be considered a closure of Hunt's argument for instantaneous mirror symmetry in his pioneering exposition of constant velocity shaft couplings. Moreover, we show that the wrist can, to some extent, be treated as a spherical mechanism, even though dependent translation exists, and the singularity-free workspace of a $N$-$UU$ wrist may be analytically derived. This leads to a straightforward optimal design for maximal singularity-free workspace.

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Figures

Fig. 1

Schematic of a general 3-UU PM: (a) components of the PM, (b) synthesis condition of the PM: the two U joints in each leg are mirror-symmetric about the xy-plane, and the revolute axes of all proximal (or distal) U joints in all legs intersect at a point s+ (or s), (c) geometry of the first leg, and (d) geometry of the U joints in the first leg

Fig. 2

Collapsing of a UU leg to a symmetric spherical chain: (a) schematic, (b) instantaneous symmetric movement (θ˙ij+=θ˙ij−=θ˙ij, i=1,…,N,j=1,2), and (c) screw system of the corresponding LTS m: a planar pencil of zero-pitch screws

Fig. 3

(a) Displacement kinematics of the 3-UU PM (leg 1 is hidden for clarity), (b) constraint wrenches (ζ11, ζ21, ζ31 and ζ2) and actuation wrenches (ζa1 and ζa2) of leg i at initial configuration, and (c) twists (arrows emanated from O_), constraint wrenches and actuation wrenches of the 3-UU PM

Fig. 4

A configuration of leg singularity of a UU leg

Fig. 5

A configuration of active constraint singularity for a 3- UU PM (α=90deg,β=0deg,γ=40deg)

Fig. 6

A configuration of passive constraint singularity for a 3-UU PM (α=90deg,β=0deg,γ=20deg)

Fig. 7

Active constraint singularity loci of a N - UU PM and its differential loci (γ=30deg; coordinates are scaled up to match the true tilt angle 2ψ): (a) N=3 (ϕ0=120deg) and (b) N=4 (ϕ0=90deg)

Fig. 8

Passive constraint singularity locus of a 3-UU PM (ϕ0=120deg, γ=20deg; coordinates are scaled up to match the true tilt angle 2ψ). (a) branch 1; (b) branch 2; (c) branches 3–5; (d) branches 6–8; (e) all branches: (a) +++, (b) − − −, (c) −++ / +−+ / ++−, (d) + − −/− + −/− − +, and (e) all branches

Fig. 9

Distribution of 2ψmax versus γ for a 3-UU PM using a tilt angle margin of 2ψmargin=20deg

Fig. 10

Distribution of 2ψmax versus γ for a 3-UU PM using a singularity margin of (a) imargin=0.1 and (b) imargin=0.2

Fig. 11

Distribution of 2ψmax versus γ for a 4-UU PM using a singularity margin of (a) margin=0.25 and (b) imargin=0.5

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