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Research Papers: Design of Mechanisms and Robotic Systems

Performing Nonsingular Transitions Between Assembly Modes in Analytic Parallel Manipulators by Enclosing Quadruple Solutions

[+] Author and Article Information
Adrián Peidró

Systems Engineering
and Automation Department,
Miguel Hernández University,
Elche 03202, Spain
e-mail: apeidro@umh.es

José María Marín

Department of Mechanical Engineering
and Energy,
Miguel Hernández University,
Elche 03202, Spain
e-mail: jmarin@umh.es

Arturo Gil

Systems Engineering
and Automation Department,
Miguel Hernández University,
Elche 03202, Spain
e-mail: arturo.gil@umh.es

Óscar Reinoso

Systems Engineering
and Automation Department,
Miguel Hernández University,
Elche 03202, Spain
e-mail: o.reinoso@umh.es

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 4, 2015; final manuscript received August 25, 2015; published online October 15, 2015. Assoc. Editor: Charles Kim.

J. Mech. Des 137(12), 122302 (Oct 15, 2015) (11 pages) Paper No: MD-15-1070; doi: 10.1115/1.4031653 History: Received February 04, 2015; Revised August 25, 2015

This paper analyzes the multiplicity of the solutions to forward kinematics of two classes of analytic robots: 2RPR-PR robots with a passive leg and 3-RPR robots with nonsimilar flat platform and base. Since their characteristic polynomials cannot have more than two valid roots, one may think that triple solutions, and hence nonsingular transitions between different assembly modes, are impossible for them. However, the authors show that the forward kinematic problems of these robots always admit quadruple solutions and obtain analytically the loci of points of the joint space where these solutions occur. Then, it is shown that performing trajectories in the joint space that enclose these points can produce nonsingular transitions, demonstrating that it is possible to design simple analytic parallel robots with two and three degrees-of-freedom (DOF) and the ability to execute these transitions.

Copyright © 2015 by ASME
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References

Figures

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Fig. 1

An analytic 2RPR-PR parallel robot

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Fig. 2

Zeros of P(ψ) in (−1, 1) with different multiplicities: (a) simple zero, (b) double zero (singularity), and (c) triple zero (cusp point)

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Fig. 3

Simple (left) and double (right) zeros of P(ψ) at ψ = −1

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Fig. 4

(a) Joint space of an analytic 2RPR-PR robot, without cusps or α-curves. (b) Polar plots of the evolution of the six solutions σi (i = 1,…,6) to forward kinematics along the circular trajectory abcdefgha of the joint space.

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Fig. 5

Rectangular (left) and polar (right) representations of ϕ

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Fig. 6

Polar plots of the solutions to forward kinematics when approaching the point λπ along tπ (left). Solutions at λπ (right).

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Fig. 7

Polar plots of the solutions to forward kinematics when approaching the point λ0 along t0 (left). Real quadruple solution at λ0 (right).

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Fig. 8

Trajectory of the solution σ4 in the reduced configuration space

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Fig. 9

Configuration of the analytic 2RPR-PR robot for the solution σ4 along the circular trajectory abcdefgha of Fig. 4(a)

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Fig. 10

The analytic 3-RPR robot with flat base and platform

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Fig. 11

(a) Joint space of the analytic 3-RPR robot, with the singularity locus and the γ-curves. Dotted lines indicate lines hidden by the singularity surface. (b) Polar diagrams with the evolution of the six solutions σi (i = 1,…,6) to forward kinematics along the trajectory t0 that encircles the curve γ0−.

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Fig. 12

Evolution of the configuration of the 3-RPR robot for the solution σ1 along the nonsingular assembly mode changing trajectory t0

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Fig. 13

(a) Slice of the joint space of Fig. 11(a) at ρ2 = 1. (b) Polar plots with the evolution of the six solutions σi (i = 1,…,6) to forward kinematics along the circle centered at d, shown in Fig. 13(a).

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Fig. 14

Trajectory of the solution σ1 in the reduced configuration space of the 3-RPR robot, for the trajectory of Eq. (48) and ρ2 = 1

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Fig. 15

Trajectory of the solution σ1 in the output space of the 3-RPR robot, for the trajectory of Eq. (48) and ρ2 = 1

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Fig. 16

Conics C1 (continuous line) and C2 (dashed line) for different values of 0 < ψ < 1. For ψ = 0.999, C1 and C2 almost intersect, as shown in the zoomed areas.

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