Application of Dual-Number Quaternion Algebra to the Analysis of Spatial Mechanisms

A line symmetric motion is the motion obtained by reflecting a rigid body in the successive generator lines of a ruled surface. In this work we review the dual quaternion approach to rigid body displacements, in particular the representation of the group SE(3) by the Study quadric. Then some classical work on reflections in lines or half-turns is reviewed. Next two new characterisations of line symmetric motions are presented. These are used to study a number of examples one of which is a novel line symmetric motion given by a rational degree five curve in the Study quadric. The rest of the paper investigates the connection between sets of half-turns and linear subspaces of the Study quadric. Line symmetric motions produced by some degenerate ruled surfaces are shown to be restricted to certain 2-planes in the Study quadric. Reflections in the lines of a linear line complex lie in the intersection of a the Study-quadric with a 4-plane.

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“…However, they were kept alive in Kinematics, notable by Dimentberg [6], Freudenstein and his student Yang, [25] and [26].…”

Section: Dual Quaternions and The Study Quadricmentioning

confidence: 99%

Half-turns and line symmetric motions

Selig

Husty

2011

Mechanism and Machine Theory

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“…Dual quaternions have been used to describe rigid transformations in robotics (Yang and Freudenstein 1964;McCarthy 1990;Bottema and Roth 1979), computer graphics (Kavan et al 2007(Kavan et al , 2008) and other fields. They offer a simple way to describe arbitrary geometric relations and provide a mathematical background that allows combining transformations by simple multiplication of the corresponding dual quaternions.…”

Section: Dual Quaternion Mmcsmentioning

confidence: 99%

Universally manipulable body models—dual quaternion representations in layered and dynamic MMCs

Schilling

2011

Auton Robot

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Surprisingly complex tasks can be solved using a behaviour-based, reactive control system, i.e., a system that operates without an explicit internal representation of the environment and the own body. Nevertheless, application of internal representations has gained interest in recent years because such internal representations can be used to solve problems of perception and motor control (sensor fusion, inverse modeling) and may in addition be applied to higher cognitive functions as are the ability to plan ahead. To endow such a system with the ability to find new behavioural solutions to a given problem in a broad range of possibilities, the internal representation must be universally manipulable, i.e. the model should be able to simulate all movements that are physically possible for the body given. Using recurrent neural networks, models showing this faculty have been proposed being based on the principle of mean of multiple computation (MMC). The extension of this approach to three dimensions requires the introduction of a joint angle representation which allows for computation of mean values. Here we use dual quaternions that are singularity-free and unambiguous which allow for shortest path interpolation. In addition, it has been shown that dual quaternions are the most efficient and most compact form for representing rigid transformations. The model can easily be adapted to bodies of

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“…The spatial RCCC linkage is a single-input-double-output four-bar linkage whose input-output equations can be written in the form of the foregoing Freudenstein equations [15]:…”

Section: Robust Function-generation Synthesis For the Spatial Rccc LImentioning

confidence: 99%

Robustness to algorithmic singularities and sensitivity in computational kinematics

Gracia

Ángeles

2011

Proceedings of the Institution of Mechanical Engineers, Part C:

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A robust approach to computational kinematics intended to cope with algorithmic singularities is introduced in this paper. The approach is based on the reduction of the original system of equations to a subsystem of bivariateequations, as opposed to the multivariate-polynomial reduction leading to the characteristic univariate-polynomial. The effectiveness of the approach is illustrated for the exact function-generation synthesis of planar, spherical, and spatial four-bar linkages. Some numerical examples are provided for the case of the spherical four-bar function generator with six precision points to show the benefits of the proposed method with respect to methods reported in the literature.

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Application of Dual-Number Quaternion Algebra to the Analysis of Spatial Mechanisms

Cited by 259 publications

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Half-turns and line symmetric motions

Half-turns and line symmetric motions

Universally manipulable body models—dual quaternion representations in layered and dynamic MMCs

Robustness to algorithmic singularities and sensitivity in computational kinematics

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