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Research Papers

# Geometric Construction-Based Realization of Planar Elastic Behaviors With Parallel and Serial Manipulators

[+] 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.

Manuscript received December 14, 2016; final manuscript received May 18, 2017; published online August 4, 2017. Assoc. Editor: Robert J. Wood.

J. Mechanisms Robotics 9(5), 051006 (Aug 04, 2017) (10 pages) Paper No: JMR-16-1373; doi: 10.1115/1.4037019 History: Received December 14, 2016; Revised May 18, 2017

## Abstract

This paper addresses the passive realization of any selected planar elastic behavior with a parallel or a serial manipulator. Sets of necessary and sufficient conditions for a mechanism to passively realize an elastic behavior are presented. These conditions completely decouple the requirements on component elastic properties from the requirements on mechanism kinematics. The restrictions on the set of elastic behaviors that can be realized with a mechanism are described in terms of acceptable locations of realizable elastic behavior centers. Parallel–serial mechanism pairs that realize identical elastic behaviors (dual elastic mechanisms) are described. New construction-based synthesis procedures for planar elastic behaviors are developed. Using these procedures, one can select the geometry of each elastic component from a restricted space of kinematically allowable candidates. With each selection, the space is further restricted until the desired elastic behavior is achieved.

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## Figures

Fig. 1

Planar parallel mechanism with three line springs. The three spring axes typically form a triangle ABC.

Fig. 2

Force and resulting motion of a three-spring parallel mechanism: (a) a force along one spring axis results in a rotation about the opposite vertex of the triangle and (b) a force passing through one vertex results in a twist with an instantaneous center located on the line along the opposite side of the triangle

Fig. 3

Location of center of elastic behavior: (a) if T(q) moves from C→B along the finite segment CB as q varies 0→1, there must be a q̂ such that f(q̂) passes points A and T(q̂) and (b) the center must be in the shaded area

Fig. 4

Location of stiffness center associated with a parallel and serial mechanism: (a) for a parallel mechanism, the center must lie within the triangle formed by the three spring axes and (b) for a serial mechanism, the center must lie within the triangle formed by the three joints

Fig. 5

Dual elastic mechanisms in parallel and serial construction. The triangle formed by the three spring axes in the parallel mechanism is coincident with the triangle formed by the three joints in the serial mechanism.

Fig. 7

Realization of a planar stiffness with a parallel mechanism. The first spring axis w1 can be arbitrarily selected. The second spring can be selected from the pencil of lines passing through point T1. The third spring axis is determined by the line passing through the instantaneous centers of twists t1 and t2, T1 and T2.

Fig. 9

Realization of a planar compliance with a serial mechanism. The location of the first joint J1 can be arbitrarily selected. The second joint can be selected from any point on the line along wrench w1. The third joint is determined by the intersection of the two lines along wrenches w1 and w2.

Fig. 11

Synthesis of planar stiffness with a parallel mechanism. The line of action for each spring is identified based on its geometry. A parallel mechanism can be constructed with three springs along the three spring wrenches w1, w2, and w3.

Fig. 12

Synthesis of planar compliance with a serial mechanism. The location for each joint is identified based on its geometry. A serial mechanism can be constructed with three joints located at J1, J2, and J3.

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