Dual quaternion

In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε2 = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra. In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigid transformation has six real degrees of freedom, three for translations and three for rotations, dual quaternions obeying two algebraic constraints are used in this application. Since unit quaternions are subject to two algebraic constraints, unit quaternions are standard to represent rigid transformations. Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics (see McCarthy), and in applications to 3D computer graphics, robotics and computer vision. Polynomials with coefficients given by (non-zero real norm) dual quaternions have also been used in the context of mechanical linkages design. W. R. Hamilton introduced quaternions in 1843, and by 1873 W. K. Clifford obtained a broad generalization of these numbers that he called biquaternions, which is an example of what is now called a Clifford algebra. In 1898 Alexander McAulay used Ω with Ω2 = 0 to generate the dual quaternion algebra. However, his terminology of "octonions" did not stick as today's octonions are another algebra. In Russia, Aleksandr Kotelnikov developed dual vectors and dual quaternions for use in the study of mechanics.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (10)

Related concepts (8)

Related courses (1)

Related lectures (13)

MATH-113: Algebraic structures

Le but de ce cours est d'introduire et d'étudier les notions de base de l'algèbre abstraite.

Screw theory

Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics). Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axes of spatial movement and the lines of action of forces.

Biquaternion

In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers.

William Kingdon Clifford

William Kingdon Clifford (4 May 1845 - 3 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics, geometry, and computing.

Geometric Algebra for Optimal Control With Applications in Manipulation Tasks

Sylvain Calinon, Tobias Löw

Many problems in robotics are fundamentally problems of geometry, which have led to an increased research effort in geometric methods for robotics in recent years. The results were algorithms using the various frameworks of screw theory, Lie algebra, and d ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2023

The accurate, robust and efficient transfer of the deformation gradient tensor between meshes of different resolution is crucial in cardiac electromechanics simulations. This paper presents a novel method that combines rescaled localized Radial Basis Funct ...

Lausanne2023

We consider multiagent decision making where each agent optimizes its convex cost function subject to individual and coupling constraints. The constraint sets are compact convex subsets of a Euclidean space. To learn Nash equilibria, we propose a novel dis ...

2019

Fourier Expansion and Quaternion Algebra MATH-603: Subconvexity, Periods and Equidistribution

Discusses Fourier expansion, quaternion algebra, and Whittaker model in automorphic forms.

Quaternion Algebra and Hermite Space MATH-603: Subconvexity, Periods and Equidistribution

Covers quaternion algebra, Waldspurger formula, and Hermite space in relation to L-functions and inner products.

Quaternion Algebra: Properties and Applications MATH-603: Subconvexity, Periods and Equidistribution

Covers the properties and applications of quaternion algebra, including norm calculations and distribution in different spaces.