Plaque on Broom bridge (Dublin) commemorating Hamilton's invention of quaternions
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.[1] 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.[2]
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[3]), and in applications to 3D computer graphics,[4] robotics[5][6] and computer vision.[7] Polynomials with coefficients given by (non-zero real norm) dual quaternions have also been used in the context of mechanical linkages design.[8][9]
^A.T. Yang, Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms, Ph.D thesis, Columbia University, 1963.
^Valverde, Alfredo; Tsiotras, Panagiotis (2018). "Dual Quaternion Framework for Modeling of Spacecraft-Mounted Multibody Robotic Systems". Frontiers in Robotics and AI. 5: 128. doi:10.3389/frobt.2018.00128. ISSN 2296-9144. PMC 7805728. PMID 33501006.
^Cite error: The named reference mccarthy was invoked but never defined (see the help page).
^Kenwright, Ben. "Dual-Quaternions: From Classical Mechanics to Computer Graphics and Beyond" (PDF). Retrieved December 24, 2022.
^Figueredo, L.F.C.; Adorno, B.V.; Ishihara, J.Y.; Borges, G.A. (2013). "Robust kinematic control of manipulator robots using dual quaternion representation". 2013 IEEE International Conference on Robotics and Automation. pp. 1949–1955. doi:10.1109/ICRA.2013.6630836. ISBN 978-1-4673-5643-5. S2CID 531000.
^Vilhena Adorno, Bruno (2017). Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra — Part I: Fundamentals.
^A. Torsello, E. Rodolà and A. Albarelli, Multiview Registration via Graph Diffusion of Dual Quaternions, Proc. of the XXIV IEEE Conference on Computer Vision and Pattern Recognition, pp. 2441-2448, June 2011.
^Li, Zijia; Schröcker, Hans-Peter; Scharler, Daniel F. (2022-09-07). "A Complete Characterization of Bounded Motion Polynomials Admitting a Factorization with Linear Factors". arXiv:2209.02306 [math.RA].
^Huczala, D.; Siegele, J.; Thimm, D.; Pfurner, M.; Schröcker, H.-P. (2024). Rational Linkages: From Poses to 3D-printed Prototypes. Advances in Robot Kinematics 2024. arXiv:2403.00558.
mathematics, the dualquaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they...
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton...
subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford...
Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three...
of the dualquaternion algebra to 2D geometry. At this present time, the article is focused on a 4-dimensional subalgebra of the dualquaternions which...
axis appears in the dualquaternion formulation of a spatial displacement D = ([A], d). The dualquaternion is constructed from the dual vector S = (S, V)...
of space science experimental payloads. Hybrid Global Finite-Time Dual-Quaternion Observer and Controller for Velocity-Free Spacecraft Pose Tracking...
worked with are dual numbers, dualquaternions, and split-biquaternions, all being associative algebras over R. Study's work with dual numbers and line...
Split-biquaternions when the coefficients are split-complex numbers. Dualquaternions when the coefficients are dual numbers. This article is about the ordinary biquaternions...
dualer Quaternionen auf Kinematik", Annales Academiae Scientiarum Fennicae (1958), 1-13; Gesammelte Werke, volume 2, Applications of dualquaternions...
split-biquaternions, while H ⊗ N {\displaystyle H\otimes N} forms dualquaternions. The algebra of dualquaternions is used to express screw displacement, a common mapping...
{\textbf {q}}^{0}} denote a unit dualquaternion. A homogeneous dualquaternion may be written as a pair of quaternions, Q ^ = Q + ε Q 0 {\displaystyle...
In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were...
TriAx Weight Maps, where many other industry platforms were using DualQuaternion. This changed in Genesis 3 to allow Daz 3D figures to be more compatible...
dynamics. This is in part because of the relationship between screws and dualquaternions which have been used to interpolate rigid-body motions. Based on screw...
representing a rotation as numbers in a computer, some people prefer the quaternion representation or the axis+angle representation, because they avoid the...
TriAx Weight Maps, where many other industry platforms were using DualQuaternion. This changed in Genesis 3, released in 2015, to allow Daz 3D figures...
analysis of movement in machines: Burmester theory Clifford algebra Dualquaternion Euler's rotation theorem Gear ratio Ideal machine Instantaneous center...
University of Tasmania, Hobart, Tasmania. He was also a proponent of dualquaternions, which he termed "octonions" or "Clifford biquaternions". McAulay was...
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε, where a and...
of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dualquaternions do not...
algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero...
unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates...
September 2010). "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product" (PDF). International Journal of the Physical...
group PSL(2, Z). Ring homographies have been used in quaternion analysis, and with dualquaternions to facilitate screw theory. The conformal group of spacetime...
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