Approaching Dual Quaternions From Matrix Algebra

Adv. Appl. Clifford Algebras

2016

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Abstract. A 4D rotation can be decomposed into a left-and a rightisoclinic rotation. This decomposition, known as Cayley's factorization of 4D rotations, can be performed using Elfrinkhof-Rosen method. In this paper, we present a more straightforward alternative approach using the corresponding orthogonal subspaces, for which orthogonal bases can be defined. This yields easy formulations, both in the space of 4×4 real orthogonal matrices representing 4D rotations and in the Clifford algebra C4,0,0.Cayley's factorization has many important applications. It can be used to easily transform rotations represented using matrix algebra to different Clifford algebras. As a practical application of the proposed method, it is shown how Cayley's factorization can be used to efficiently compute the screw parameters of 3D rigid-body transformations.Mathematics Subject Classification (2010). Primary 99Z99; Secondary 15A66.

“…In [8], the following mapping between 3D transformations in homogeneous coordinates and a subset of 4D rotation matrices was proposed:…”

Section: A Useful Mappingmentioning

confidence: 99%

Section: A Useful Mappingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Cayley’s Factorization of 4D Rotations and Applications

Pérez-Gracia

Adv. Appl. Clifford Algebras

2016

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“…which are known as left-and right-isoclinic rotation matrices, respectively (see [16] for details on Cayley's factorization). Now, it can observed that…”

Section: A B and Rotations In Rmentioning

confidence: 99%

A bilinear formulation for the motion planning of non-holonomic parallel orienting platforms

Grosch

2013 IEEE/RSJ International Conference on Intelligent Robots and Systems

2013

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Abstract-This paper deals with the motion planning problem for parallel orienting platforms with one non-holonomic joint and two prismatic actuators which can maneuver to reach any three-degree-of-freedom pose of the moving platform. Since any system with two inputs and up to four generalized coordinates can always be transformed into chained form, this path planning problem can be solved using well-established procedures. Nevertheless, the use of these procedures requires a good understanding of Lie algebraic methods whose technicalities have proven a challenge to many practitioners who are not familiar with them. As an alternative, we show how by (a) properly locating the actuators, and (b) representing the platform orientation using Euler parameters, the studied path planning problem admits a closed-form solution whose derivation requires no other tools than ordinary linear algebra.

On closed‐form solutions to the 4D nearest rotation matrix problem

Sarabandi

Math Methods in App Sciences

2022

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In this paper, we address the problem of restoring the orthogonality of a numerically noisy 4D rotation matrix by finding its nearest (in Frobenius norm) correct rotation matrix. This problem can be straightforwardly solved using the Singular Value Decomposition (SVD). Nevertheless, to avoid numerical methods, we present two new closed‐form methods. One relies on the direct minimization of the mentioned Frobenius norm, and the other on the passage to double quaternion representation. A comparison of these two methods with respect to the SVD reveals that the method based on a double quaternion representation is superior in all aspects.