Dual quaternion

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]

  1. ^ A.T. Yang, Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms, Ph.D thesis, Columbia University, 1963.
  2. ^ 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.
  3. ^ Cite error: The named reference mccarthy was invoked but never defined (see the help page).
  4. ^ Kenwright, Ben. "Dual-Quaternions: From Classical Mechanics to Computer Graphics and Beyond" (PDF). Retrieved December 24, 2022.
  5. ^ 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.
  6. ^ Vilhena Adorno, Bruno (2017). Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra — Part I: Fundamentals.
  7. ^ 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.
  8. ^ 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].
  9. ^ 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.