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Research Papers

Geometric Approach to the Realization of Planar Elastic Behaviors With Mechanisms Having Four Elastic Components

[+] Author and Article Information
Shuguang Huang

Department of Mechanical Engineering,
Marquette University,
Milwaukee, WI 53201-1881
e-mail: huangs@marquette.edu

Joseph M. Schimmels

Department of Mechanical Engineering,
Marquette University,
Milwaukee, WI 53201-1881
e-mail: j.schimmels@marquette.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received July 26, 2017; final manuscript received January 22, 2018; published online April 18, 2018. Assoc. Editor: Guimin Chen.

J. Mechanisms Robotics 10(4), 041004 (Apr 18, 2018) (13 pages) Paper No: JMR-17-1227; doi: 10.1115/1.4039399 History: Received July 26, 2017; Revised January 22, 2018

This paper addresses the passive realization of any selected planar elastic behavior with redundant elastic manipulators. The class of manipulators considered are either serial mechanisms having four compliant joints or parallel mechanisms having four springs. Sets of necessary and sufficient conditions for mechanisms in this class to passively realize an elastic behavior are presented. The conditions are interpreted in terms of mechanism geometry. Similar conditions for nonredundant cases are highly restrictive. Redundancy yields a significantly larger space of realizable elastic behaviors. Construction-based synthesis procedures for planar elastic behaviors are also developed. In each, the selection of the mechanism geometry and the selection of joint/spring stiffnesses are completely decoupled. The procedures require that the geometry of each elastic component be selected from a restricted space of acceptable candidates.

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References

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Figures

Grahic Jump Location
Fig. 1

Simple planar compliant mechanisms: (a) serial mechanism. Each joint has joint compliance ci (b) parallel mechanism. Each spring has stiffness ki.

Grahic Jump Location
Fig. 2

Realization conditions for the case that l12 does not cross line l34 between J3 and J4: (a) A force w12 applied to the mechanism along line l12 must result in a twist t12 located on line l34 between joints J3 and J4 and (b) a force w34 applied to the mechanism along line l34 must result in a twist t34 located on line l12 between joints J1 and J2

Grahic Jump Location
Fig. 3

Realization conditions for the case in which l13 crosses line l24 between J2 and J4: (a) A force w13 applied on the mechanism along line l13 must result in a twist t13 located on line l24 outside the segment between J2 and J4 and (b) a force w24 applied to the mechanism along line l24 results in a twist t24 located on line l13 outside the segment between J1 and J3

Grahic Jump Location
Fig. 4

Realization conditions for four-spring parallel mechanisms. The wrench resulting from a twist tij whose center Tij is located at the intersection of two wrench axes wi and wj, must pass through the intersection of the other two wrench axes, Trs, and must lie in the interior of the shaded area that does not contain point Tij.

Grahic Jump Location
Fig. 5

Realization conditions for four-spring parallel mechanisms: (a) The wrench caused by a twist at T12 must be in the shaded area bounded by the two wrench axes w3 and w4 and (b) The wrench caused by a twist at T34 must be in the shaded area bounded by the two wrench axes w1 and w2

Grahic Jump Location
Fig. 6

Two wrenches satisfying reciprocal condition (51). Any wrench passing through T12 results in a twist located on line l12. Any wrench along line l12 results in a twist located at point T12. The center of the compliance must be in the interior of the shaded triangle.

Grahic Jump Location
Fig. 7

The locus of compliance centers of elastic behaviors realized with a four-joint serial and four-spring parallel mechanism. (a) For a four-joint serial mechanism, the center is inside the polygon formed by the union of all triangles formed by any three joints. (b) For a four-spring parallel mechanism, the center is inside the polygon formed by the union of all triangles formed by any three spring wrenches.

Grahic Jump Location
Fig. 9

A family of twists Tl centered on a line lt passing through the stiffness center Ck and the corresponding wrenches Wt: (a) the lines of action of wrenches resulting from w = Kt form a family of parallel lines, (b) the line of action of w and the center of t are on the opposite sides of line lc, (c) when the center of tCk, the line of action of w → ±, and (d) when the center of t → ±, the line of action of w approaches line lc

Grahic Jump Location
Fig. 8

A family of parallel wrenches Wp and the corresponding twists Tw: (a) the locus of t = Cw is a line lt passing through the compliance center Cc, (b) the line of action of wrench w and the center of twist t = Cw are separated by line lc, (c) when wlc from two sides, the center of t → ± along line lt, and (d) When w → ±, the center of t approaches Cc along line lt

Grahic Jump Location
Fig. 11

Synthesis of a compliance with a four-spring parallel mechanism based on geometry. The realization is achieved by selecting the spring axes of the four springs in the mechanism.

Grahic Jump Location
Fig. 10

Synthesis of a compliance with a four-joint serial mechanism based on geometry. The realization is achieved by selecting the locations of the four joints in the mechanism.

Grahic Jump Location
Fig. 13

Realization of a given stiffness with a four-spring parallel mechanism. In the process, the four axes of the springs (w1, w2, w3, w4) in the mechanism and the corresponding spring stiffnesses (k1, k2, k3, k4) are identified.

Grahic Jump Location
Fig. 12

Realization of a given compliance with a four-joint serial mechanism. In the process, the locations of the four joints (J1, J2, J3, J4) in the mechanism and the corresponding joint compliances (c1, c2, c3, c4) are identified.

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