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Research Papers

# Experimental Analysis of Rigid Body Motion. A Vector Method to Determine Finite and Infinitesimal Displacements From Point Coordinates

[+] Author and Article Information
Álvaro Page

Instituto de Biomecánica de Valencia, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spainafpage@ibv.upv.es

Helios de Rosario, Carlos Atienza

Instituto de Biomecánica de Valencia, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain

Vicente Mata

Departamento de Ingeniería Mecánica y de Materiales, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain

J. Mech. Des 131(3), 031005 (Feb 05, 2009) (8 pages) doi:10.1115/1.3066468 History: Received August 02, 2007; Revised November 10, 2008; Published February 05, 2009

## Abstract

This paper presents a vector method for measuring rigid body motion from marker coordinates, including both finite and infinitesimal displacement analyses. The common approach to solving the finite displacement problem involves the determination of a rotation matrix, which leads to a nonlinear problem. On the contrary, infinitesimal displacement analysis is a linear problem that can be easily solved. In this paper we take advantage of the linearity of infinitesimal displacement analysis to formulate the equations of finite displacements as a generalization of Rodrigues’ formula when more than three points are used. First, for solving the velocity problem, we propose a simple method based on a mechanical analogy that uses the equations that relate linear and angular momenta to linear and angular velocities, respectively. This approach leads to explicit linear expressions for infinitesimal displacement analysis. These linear equations can be generalized for the study of finite displacements by using an intermediate body whose points are the midpoint of each pair of homologous points at the initial and final positions. This kind of transformation turns the field of finite displacements into a skew-symmetric field that satisfies the same equations obtained for the velocity analysis. Then, simple closed-form expressions for the angular displacement, translation, and position of finite screw axis are presented. Finally, we analyze the relationship between finite and infinitesimal displacements, and propose vector closed-form expressions based on derivatives or integrals, respectively. These equations allow us to make one of both analyses and to obtain the other by means of integration or differentiation. An experiment is presented in order to demonstrate the usefulness of this method. The results show that the use of a set of markers with redundant information $(n>3)$ allows a good accuracy of measurement of kinematic variables.

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## Figures

Figure 1

(a) The field of finite displacements is not skew-symmetric. (b) The field of virtual displacements, defined on the virtual intermediate body, is skew-symmetrical. The associate instantaneous axis of rotation (ISAv) agrees with the finite axis of rotation (FSA).

Figure 2

The virtual body is similar to that of the rigid one in the midrotated position, though shrinked around the rotation axis by the factor cos(θ/2)

Figure 3

Composition of the finite rotation Ω and the infinitesimal angle displacement dΦ=wdt/2. The infinitesimal rotation dΦ′ is the same dΦ but rotated −Ω, from location 2′ to location 1′.

Figure 4

Table used in the analysis. The table rolls on the laboratory floor without sliding.

Figure 5

Pencil of lines corresponding to measured ISA. The arrows represent the angular velocities w. We have represented only a selection of axes in order to clarify the image.

Figure 6

Markers set at reference position (black circles), at a given position (black squares), and virtual body (gray circles). The pencil of gray lines is the FSA. Note that FSA only remains on the floor at the reference position.

Figure 7

Components of angular velocity w. The lines represent the measured values, and the markers represent those calculated by derivation from measured Ω.

Figure 8

Components of vector θu. The lines represent the measured values, and the markers represent the calculated ones by integration from measured w.

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