Research Papers

A Robust Solution of the Spatial Burmester Problem

[+] Author and Article Information
Shaoping Bai

 Department of Mechanical and Manufacturing Engineering, Aalborg University, Denmark, Aalborg 9220shb@m-tech.aau.dk

Jorge Angeles

 Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 2K6, Canadaangeles@cim.mcgill.ca

That is, if abstraction is made of the translation of the rigid body when formulating the problem stated in Section 2, then points Rj , for j = 1,…,m, coincide with point R0 , and the problem at hand becomes one of spherical synthesis.

Here, of course, the antipodal solutions of those lying in the square of side-length π are excluded.

Every m×n matrix with mn can be factored into a m×m orthogonal matrix Q and an upper-triangular matrix R [38].

J. Mechanisms Robotics 4(3), 031003 (May 22, 2012) (10 pages) doi:10.1115/1.4006658 History: Received July 21, 2011; Revised April 07, 2012; Published May 21, 2012; Online May 22, 2012

The spatial Burmester problem is studied in this work, focusing on the synthesis of CCCC and RCCC linkages for rigid-body guidance, where R stands for revolute, C for cylindrical pair. The synthesis equations for CC and RC dyads are formulated using dual algebra. The formulation is developed in such a way that it leads to a robust solution, based on a semigraphical approach, which produces all the real solutions to the problem of CC-dyad synthesis for five given poses. This eases the equation-solving process by filtering out the complex solutions, while allowing for the handling of the special cases of none or infinitely many solutions. The synthesis procedure is illustrated with examples for four and five given poses.

Copyright © 2012 by American Society of Mechanical Engineers
Topics: Linkages , Equations
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Figure 1

The RCCC linkage

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Figure 5

A CCCC linkage for five given poses

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Figure 8

The dual angle between two lines

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Figure 2

The spherical 4R linkage

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Figure 3

A CC spatial dyad, which becomes an RC dyad if the sliding sj vanishes

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Figure 4

Contour plots for (a) angles φb and θb , and (b) angles φa and θa

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Figure 6

Centerpoint and circlepoint spherical curves

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Figure 7

Congruences of (a) the fixed axis; and (b) the moving axis




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