Quaternions and the rotation of a rigid body

Arribas, M.; Elipe, A.; Palacios, M.

doi:10.1007/s10569-006-9037-6

Cited by 44 publications

(25 citation statements)

References 13 publications

(20 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This product is often referred to as the Grassman product [3,23]. In vector representation, the product of Quaternions p and q is given as Quaternion t:…”

Section: Multiplication Of Quaternionsmentioning

confidence: 99%

A novel Quaternion integration approach for describing the behaviour of non-spherical particles

Zhao

Wachem

2013

Acta Mech

View full text Add to dashboard Cite

There are three main frameworks to describe the orientation and rotation of non-spherical particles: Euler angles, rotation matrices and unit Quaternions. Of these methods, the latter seems the most attractive for describing the behaviour of non-spherical particles. However, there are a number of drawbacks when using unit Quaternions: the necessity of applying rotation matrices in conjunction to facilitate the transformation from body space to world space, and the algorithm integrating the Quaternion should inherently conserve the length of the Quaternion. Both drawbacks are addressed in this paper. The present paper derives a new framework to transform vectors and tensors by unit Quaternions, and the requirement of explicitly using rotation matrices is removed altogether. This means that the algorithm derived in this paper can describe the rotation of a non-spherical particle with four parameters only. Moreover, this paper introduces a novel corrector-predictor method to integrate unit Quaternions, which inherently conserves the length of the Quaternion. The novel framework and method are compared to a number of other methods put forward in the literature. All the integration methods are discussed, scrutinised and compared to each other by comparing the results of four test cases, involving a single falling particle, nine falling and interacting particles, a 2D prescribed torque on a sphere and a 3D prescribed torque on a non-spherical particle. Moreover, a convergence study is presented, comparing the rate of convergence of the various methods. All the test cases show a significant improvement of the new framework put forward in this paper over existing algorithms. Moreover, the new method requires less computational memory and fewer operations, due to the complete omission of the rotation matrix in the algorithm.

show abstract

“…This product is often referred to as the Grassman product [3,23]. In vector representation, the product of Quaternions p and q is given as Quaternion t:…”

Section: Multiplication Of Quaternionsmentioning

confidence: 99%

A novel Quaternion integration approach for describing the behaviour of non-spherical particles

Zhao

Wachem

2013

Acta Mech

View full text Add to dashboard Cite

show abstract

“…Recently, the Space Mechanics Group of the University of Zaragoza (Spain) took advantage of the elegance of the quaternion language in various applications in orbital and rigid-body dynamics, see, e.g., Arribas et al (2006).…”

Section: Introductionmentioning

confidence: 99%

Quaternions for regularizing Celestial Mechanics: the right way

Waldvogel

2008

Celest Mech Dyn Astr

View full text Add to dashboard Cite

Quaternions have been found to be the ideal tool for describing and developing the theory of spatial regularization in Celestial Mechanics. This article corroborates the above statement. Beginning with a summary of quaternion algebra, we will describe the regularization procedure and its consequences in an elegant way. Also, an alternative derivation of the theory of Kepler motion based on regularization will be given. Furthermore, we will consider the regularization of the spatial restricted three-body problem, i.e. the spatial generalization of the Birkhoff transformation. Finally, the perturbed Kepler motion will be described in terms of regularized variables.

show abstract

“…This is in contrast to previous works by Maciejewski [9] and Morton [10] (see also Arribas et al [11] and Borri et al [12,Section 5.2]), which require the introduction of an undetermined inertia term in order to set up the quaternionbased Hamiltonian formulation.…”

Section: Introductionmentioning

confidence: 79%

Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration

Betsch

Siebert

2009

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYIn the present paper unit quaternions are used to describe the rotational motion of a rigid body. The unitlength constraint is enforced explicitly by means of an algebraic constraint. Correspondingly, the equations of motion assume the form of differential-algebraic equations (DAEs). A new route to the derivation of the mass matrix associated with the quaternion formulation is presented. In contrast to previous works, the newly proposed approach yields a non-singular mass matrix. Consequently, the passage to the Hamiltonian framework is made possible without the need to introduce undetermined inertia terms. The Hamiltonian form of the DAEs along with the notion of a discrete derivative make possible the design of a new quaternion-based energy-momentum scheme. Two numerical examples demonstrate the performance of the newly developed method. In this connection, comparison is made with a quaternion-based variational integrator, a director-based energy-momentum scheme, and a momentum conserving scheme relying on the discretization of the classical Euler's equations.

show abstract

Quaternions and the rotation of a rigid body

Cited by 44 publications

References 13 publications

A novel Quaternion integration approach for describing the behaviour of non-spherical particles

A novel Quaternion integration approach for describing the behaviour of non-spherical particles

Quaternions for regularizing Celestial Mechanics: the right way

Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration

Contact Info

Product

Resources

About