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Research Papers

# The Synthesis of Spherical Motion Generators in the Presence of an Incomplete Set of Attitudes

[+] Author and Article Information
Khalid Al-Widyan

Mechatronics Engineering Department,
Hashemite University,
Zarqa 13115, Jordan
e-mail: alwidyan@hu.edu.jo

Jorge Angeles

Department of Mechanical Engineering
and Centre for Intelligent Machines,
McGill University,
e-mail: angeles@cim.mcgill.ca

A real function is said to be analytic if it possesses derivatives of all orders and agrees with its Talyor series in a neighborhood of every point—Weisstein, Eric W., “Real Analytic Function.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/RealAnalyticFunction.html

The CPM Ej of vector $ej∈IR3$ is defined, for any vector $v∈IR3$, as $Ej=CPM(ej)≡(∂(ej×v)/∂v)⇔ej×v=Ejv$.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received October 25, 2012; final manuscript received July 22, 2013; published online April 15, 2014. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 6(3), 031008 (Apr 15, 2014) (8 pages) Paper No: JMR-12-1174; doi: 10.1115/1.4025297 History: Received October 25, 2012; Revised July 22, 2013

## Abstract

Proposed in this paper is a general methodology applicable to the synthesis of spherical motion generators in the presence of an incomplete set of finitely separated attitudes. The spherical rigid-body guidance problem in the realm of four-bar linkage synthesis can be solved exactly for up to five prescribed attitudes of the coupler link, and hence, any number of attitudes smaller than five is considered incomplete in this paper. The attitudes completing the set are determined to produce a linkage whose performance is robust against variations in the unprescribed attitudes. Robustness is needed in this context to overcome the presence of uncertainty due to the selection of the unspecified attitudes, that many a time are specified implicitly by the designer upon choosing, for example, the location of the fixed joints of the dyads. A theoretical framework for model-based robust engineering design is thus, recalled, and a methodology for the robust synthesis of spherical four-bar linkages is laid down. An example is included here to concretize the concepts and illustrate the application of the proposed methodology.

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Topics: Linkages , Design , Generators

## Figures

Fig. 1

Fig. 2

A spherical RR dyad at a reference and at the jth attitude

Fig. 3

The synthesized linkage at the pick attitude

Fig. 4

The synthesized linkage at the place attitude

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