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Research Papers

Invariant Description of Rigid Body Motion Trajectories

[+] Author and Article Information
Joris De Schutter1

Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300, P.O. Box 02420, 3001 Leuven-Heverlee, Belgiumjoris.deschutter@mech.kuleuven.be

The description is unique if signed curvature and torsion are defined.

Making use of ω̇×ω̇=0,a×(b×c)=b(ac)c(ab) and (ω×ω̇).ω=0.

The same occurs in curvature theory, where only positive generalized curvatures are defined. For example for a 3D space curve, only positive values of curvature are defined and the normal direction of the Frenet frame flips over 180 deg accordingly.

In a practical experiment the twist data are generated from velocity and position measurement data of at least three noncollinear points on the rigid body, for example, using one of the methods compared in Ref. 22. Obviously, for noiseless data these methods yield the same results as the procedure used in the numerical example.

1

Corresponding author.

J. Mechanisms Robotics 2(1), 011004 (Nov 19, 2009) (9 pages) doi:10.1115/1.4000524 History: Received December 27, 2008; Revised July 18, 2009; Published November 19, 2009

This paper presents a minimal invariant coordinate-free description of rigid body motion trajectories. Based on a motion model for the instantaneous screw axis, a time-based coordinate-free description consisting of six scalar functions of time is defined. Analytical formulas are presented to obtain these functions from the pose or twist coordinates of a motion trajectory. The time-based functions are then stripped from their temporal information yielding five independent geometric functions together with a scalar motion profile. The geometric functions are shown to be invariant with respect to time scale, linear and angular scale, motion profile, reference frame, and reference point on the rigid body used to express the translational components of the motion. An algorithm is given to reconstruct a coordinate representation of a motion trajectory from its coordinate-free description. A numerical example illustrates the validity of the approach.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

The instantaneous screw axis at three consecutive time instants

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Figure 2

Decomposition of Δp⊥

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Figure 3

Pose Coordinates of Motions I and II: given coordinates for motions I and II (dashed); open-loop reconstructed coordinates for motions I and II (dotted); the dashed and dotted lines coincide

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Figure 4

Dimensionless Degree of Advancement for Motions I (dashed) and II (solid)

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Figure 5

Time-based Invariants for Motions I (dashed) and II (solid)

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Figure 6

Dimensionless Geometric Invariants for Motions I (dashed) and II (dotted)

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