Clifford Algebra Exponentials and Planar Linkage Synthesis Equations

[+] Author and Article Information
Alba Perez

Robotics and Automation Laboratory, University of California, Irvine, Irvine, CA 92697maperez@uci.edu

J. Michael McCarthy

Robotics and Automation Laboratory, University of California, Irvine, Irvine, CA 92697jmmccart@uci.edu

J. Mech. Des 127(5), 931-940 (Nov 01, 2004) (10 pages) doi:10.1115/1.1904047 History: Received June 24, 2004; Revised November 01, 2004

This paper uses the exponential defined on a Clifford algebra of planar projective space to show that the “standard-form” design equations used for planar linkage synthesis are obtained directly from the relative kinematics equations of the chain. The relative kinematics equations of a serial chain appear in the matrix exponential formulation of the kinematics equations for a robot. We show that formulating these same equations using a Clifford algebra yields design equations that include the joint variables in a way that is convenient for algebraic manipulation. The result is a single formulation that yields the design equations for planar 2R dyads, 3R triads, and nR single degree-of-freedom coupled serial chains and facilitates the algebraic solution of these equations including the inverse kinematics of the chain. These results link the basic equations of planar linkage design to standard techniques in robotics.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

A planar 2R serial chain

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Figure 2

Local frames for a serial chain

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Figure 3

A planar 3R chain in the reference configuration

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Figure 4

Two positions of a planar 2R chain

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Figure 5

The two 2R design candidates reaching each of the five specified task positions

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Figure 6

RRR design reaching positions 1, 2, 3, and 4

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Figure 7

RRR design reaching positions 5, 6, and 7



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