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Research Papers

Incremental Forward Kinematics of Wire-Suspended Parallel Mechanical System (A Many-Worlds Interpretation Approach)

[+] Author and Article Information
Kazuyuki Hanahara

Associate Professor
Department of Systems Engineering,
Graduate School of System Informatics,
Kobe University,
Nada, Kobe 657-8501, Japan
e-mail: hanahara@cs.kobe-u.ac.jp

Manuscript received September 26, 2013; final manuscript received March 14, 2015; published online April 15, 2015. Assoc. Editor: Xianmin Zhang.

J. Mechanisms Robotics 7(4), 041021 (Nov 01, 2015) (16 pages) Paper No: JMR-13-1191; doi: 10.1115/1.4030163 History: Received September 26, 2013; Revised March 14, 2015; Online April 15, 2015

In this paper, we discuss an incremental forward kinematics of wire-suspended mechanical systems. In order to deal with a mechanical system of this type, we have to take into account the characteristic property of a wire, in that it cannot take any compressive force. We give a general formulation and its solution of the incremental forward kinematics problem taking account of the slack condition of a wire. We also consider the influence of displacement of the mass center position of the suspended platform on the kinematic behavior of the system. An interval arithmetic approach is proposed to deal with the uncertainty in the kinematic parameters. One of the important points is that the number of possible solutions of the formulated incremental kinematics problem is often more than one. We introduce a kind of parallelism, referred to as the “many-worlds interpretation” taken from the quantum mechanics theory, to this problem and offer an approach to deal with plural possible kinematic states simultaneously. The developed approach is based on basic equations in general form and is applicable to various wire-suspended mechanical systems. The feasibility of the proposed incremental kinematics approach is demonstrated by example calculations of three-, six-, and eight-wire systems. On the basis of the example forward kinematics calculation results, we conclude the following. The influence of displacement of mass center position of the platform is not insignificant. The number of possible kinematic states becomes large in the case of the neighborhood of singular configuration. In spite of an incremental kinematics based on linearization, the required computational cost of the proposed parallelism approach is considerable; it is, however, demonstrated that the proposed approach is still fairly practical from the viewpoint of computation. The developed approach cannot deal with drastic change in kinematic configuration in a single incremental step, since it is based on linearization of the kinematic relation; however, the approach can distinguish such discontinuity.

FIGURES IN THIS ARTICLE
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Copyright © 2015 by ASME
Topics: Kinematics , Wire , Tension
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References

Figures

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Fig. 1

Parallel wire mechanism of various types: (a) three-wire system, (b) six-wire system, and (c) eight-wire system

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Fig. 2

Singular configuration example: 2D three-wire system (tension and slack wires are shown in solid and broken lines, respectively). (a) Initial neutral configuration. (b) Central wire shortened under slightly displaced mass center: (b1) left-shifted, (b2) lower-shifted, and (b3) right-shifted.

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Fig. 3

Example forward kinematic motion of three-wire system: (a) initial state (with wire numbering), (b) wire 3 lengthened 1 m from (a), and (c) wire 3 lengthened 1 m from (b)

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Fig. 4

Some possible tension patterns for initial singular configuration (only 6 of 19 patterns are shown). (a) Six taut-wire pattern (with wire numbering), (b) five taut-wire pattern, (c) four taut-wire pattern, (d) four taut-wire pattern, (e) three taut-wire pattern, and (f) two taut-wire pattern.

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Fig. 5

Possible kinematic states at initial configuration with interval of ±0.01m. Example additional patterns due to the interval are shown in (b) and (c). (a) Superimpose of 25 patterns, (b) four taut-wire pattern, and (c) three taut-wire pattern.

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Fig. 6

Two example kinematic motions from initial configuration shown in Fig. 5. (a) Example kinematic motion 1: (a1) mass center displaced right at 0.4 m, (a2) mass center displaced right at 0.5 m, and (a3) mass center displaced left at 0.5 m. (b) Example kinematic motion 2: (b1) central two wires shortened 0.3 m, (b2) mass center displaced right at 0.5 m, and (b3) mass center displaced left at 0.5 m (fail to attain).

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Fig. 7

Example kinematic motion of eight-wire system: (a) initial configuration (superimpose of 78 patterns), (b) mass center displaced 0.3 m rightward, backward, and downward, respectively, and (c) lower four wires lengthened 1 m

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Fig. 8

Influence of unification parameters αUx and αUθ on possible kinematic states (attaining configuration Fig. 6(b1) from initial configuration Fig. 5(a) in 100 incremental steps). (a) Intermediate states: case αUx = αUθ = 0.4: (a1) 10th step (20 possible kinematic states), (a2) 20th step (27 possible kinematic states), and (a3) 30th step (21 possible kinematic states). (b) Intermediate states: case αUx = αUθ = 0.1: (b1) 10th step (76 possible kinematic states), (b2) 20th step (39 possible kinematic states), and (b3) 30th step (21 possible kinematic states).

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Fig. 9

Influence of unification parameters αUx and αUθ on possible kinematic states—2 (attaining configuration Fig. 6(b1) from initial configuration corresponding to Fig. 5(a)). (a) Intermediate states: case αUx = αUθ = 0.01: (a0) initial configuration (3852 possible kinematic states), (a1) 10th step (5337 possible kinematic states), (a2) 20th step (525 possible kinematic states), (a3) 30th step (134 possible kinematic states), (a4) 50th step (130 possible kinematic states), and (a5) 70th step (29 possible kinematic states).

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Fig. 10

Conceptual illustration of wire-suspended platform system

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