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TECHNICAL PAPERS

# On the Effect of Dual Weights in Computer Aided Design of Rational Motions

[+] Author and Article Information
Anurag Purwar

Department of Mechanical Engineering, State University of New York, Stony Brook, NY 11794-2300purwar@design.eng.sunysb.edu

Qiaode Jeffrey Ge1

Department of Mechanical Engineering, State University of New York, Stony Brook, NY 11794-2300Qiaode.Ge@stonybrook.edu

1

Corresponding author.

J. Mech. Des 127(5), 967-972 (Jan 27, 2005) (6 pages) doi:10.1115/1.1906263 History: Received April 09, 2004; Revised January 27, 2005

## Abstract

In recent years, it has become well known that rational Bézier and B-spline curves in the space of dual quaternions correspond to rational Bézier and B-spline motions. However, the influence of weights of these dual quaternion curves on the resulting rational motions has been largely unexplored. In this paper, we present a thorough mathematical exposition on the influence of dual-number weights associated with dual quaternions for rational motion design. By deriving the explicit equations for the point trajectories of the resulting motion, we show that the effect of real weights on the resulting motion is similar to that of a rational Bézier curve and how the change in dual part of a dual-number weight affects the translational component of the motion. We also show that a rational Bézier motion can be reparameterized in a manner similar to a rational Bézier curve. Several examples are presented to illustrate the effects of the weights on rational motions.

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## Figures

Figure 1

(a) Screw motion with unit real weights ŵi=1+ε0; i=0,1. (b) Screw motion with nonunit real weights ŵ0=1+ε0, ŵ1=2+ε0.

Figure 2

(a) A rational Bézier motion of degree 6 with unit real weights, ŵi=1+ε0; i=0,…,3. (b) A rational Bézier motion with nonunit real weights ŵi=1+ε0; i=0,3 and ŵi=4+ε0; i=1,2.

Figure 3

(a) A rational Bézier motion of degree 6 ŵi=1+ε1; i=0,…,3. (b) A reparameterized rational Bézier motion ŵi=λi+ελi; λ=2 and i=0,…,3.

Figure 4

The effect of dual weights on screw interpolation.

Figure 5

The effect of dual weights on a rational Bézier motion.

Figure 6

(a) Key frames and the trajectory. (b) C2 spline interpolating motion for key-frames of (a).

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