The Clifford Algebra and the Optimization of Robot Design

Ahlers, Shawn G.; McCarthy, John Michael

doi:10.1007/978-1-4612-0159-5_12

Cited by 7 publications

(5 citation statements)

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“…The two parts of G behave independently, and this allows combinations of transforms to be formed and manipulated. In particular, given control poses which are double quaternions, it is possible to form Bézier and B-spline combinations of the individual parts separately to form a changing transform and hence a motion (Etzel andMcCarthy 1999, Ahlers andMcCarthy 2001).…”

Section: Discussionmentioning

confidence: 99%

“…The growth in the complexity of computer games has seen a renewed interest in the use of quaternions to represent rotations (Leeney 2009, Fang et al 1998, Shoemake 1985. These have been extended to form double and dual quaternions which can be used to describe motions (Wu and You 2010, Jin and Ge 2010, Purwar and Ge 2005, Ahlers and McCarthy 2001, and handle kinematic control problems (Sariyildiz and Temeltas 2012, Wang et al 2012, Akyar 2008.…”

Section: Introductionmentioning

confidence: 99%

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Using geometric algebra to represent and interpolate tool poses

Cripps

Mullineux

2015

International Journal of Computer Integrated Manufacturing

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In conventional computer aided manufacturing techniques, a tool path is sent to the machine tool controller as a sequence of precision points. This means that errors are introduced when the controller interpolates between the points to recover the intended path. For 4-and 5-axis machining, a sequence of precision poses (positions and orientations) is needed. The use of geometric algebra in representing poses is discussed. The form of algebra used can represent rigidbody transforms exactly and can interpolate, in a natural way, between two or more poses using spherical linear interpolation (slerp). Sequences of poses from free-form surfaces are investigated. Tool motions are generated and compared with tool paths derived from purely positional information. It is found that the motion paths more naturally follow the original surface so that the interpolation errors are smaller.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Using geometric algebra to represent and interpolate tool poses

Cripps

Mullineux

2015

International Journal of Computer Integrated Manufacturing

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“…The use of double quaternions for handling transforms and motions [15,18] depends upon the idea of representing a transform as a pair of quaternions which are regarded as commuting. This corresponds to representing the transform by a 4 × 4 orthogonal matrix which is then factorized as a pair of commuting factors.…”

Section: Introductionmentioning

confidence: 99%

“…15 For some applications, matrix exponentials need to be formed 16 which perhaps adds to the complication since this involves a move into Clifford (geometric) algebra. 17 The use of double quaternions for handling transforms and motions 15,18 depends upon the idea of representing a transform as a pair of quaternions which are regarded as commuting. This corresponds to representing the transform by a 4 Â 4 orthogonal matrix which is then factorized as a pair of commuting factors.…”

Section: Introductionmentioning

confidence: 99%

Characterizing isoclinic matrices and the Cayley factorization

Hunt

Mullineux

Cripps

et al. 2016

Proceedings of the Institution of Mechanical Engineers, Part C:

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Quaternions, particularly the double and dual forms, are important for the representation rotations and more general rigid-body motions. The Cayley factorization allows a real orthogonal 4 × 4 matrix to be expressed as the product of two isoclinic matrices and this is a key part of the underlying theory and a useful tool in applications. An isoclinic matrix is defined in terms of its representation of a rotation in four-dimensional space. This paper looks at characterizing such a matrix as the sum of a skew symmetric matrix and a scalar multiple of the identity whose product with its own transpose is diagonal. This removes the need to deal with its geometric properties and provides a means for showing the existence of the Cayley factorization.

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“…The area of kinematics to a large extent considers and solves compliance problems, for instance when investigating linkages, or, more generally, when designing and analyzing machinery [6,7]. Other relevant research is done in robotics [1], and in some areas of geometric constructions [5]. To some extent, the compliance problem overlaps with the construction problem, as seen in [4], where a system of equations is attacked by considering the residual compliant motion of geometric primitives when restricting to a subset of the given constraints.…”

Section: Introductionmentioning

confidence: 99%