Abstract
Quaternions and dual quaternions are interesting elements which are being used to robot kinematics over five decades. They arise from Clifford algebras as many isomorphisms. In this chapter we offer representations to points, vectors, lines, screws and planes in dual quaternions coordinates, allowing a huge possibilities to solve problems, especially robot kinematics. No Clifford algebra is necessary, we will use only quaternions units. The displacement of the given elements are found in terms of dual quaternions algebra. For all these elements we must define the right dual quaternions conjugation and operations to handle with. Also, the principle of transference now is not sufficient, as we will explain into the chapter. Examples are presented to show the applicability of our results.
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References
Agrawal, O.P.: Hamilton operators and dual-number-quaternions in spatial kinematics. Mech. Mach. Theor. 22(6), 569–575 (1987)
Bayro-Corrochano, E., Falcón, L.E.: Geometric algebra of points, lines, planes and spheres for computer vision and robotics. Robotica 23(6), 1469–8668 (2005). doi: 10.1017/S0263574705001657
Christoph, M.H., Yang, w.: In: LI, Z., Sit, W. (eds.) Compliant motion constraints. Singapore, World Scientific (2003)
Dooley, J.R., McCarthy, J.M.: On the geometric analysis of optimum trajectories for cooperating robots using dual quaternion coordinates. In: IEEE Transactions on Robotics and Automation, pp. 1031–1036 (1993)
Funda, J., Paul, R.P.: A computational analysis os screw transformations in robotics. IEEE Trans. Robot. Autom. 6(3), 382–388 (1990)
Lounesto, P.: Clifford Algebras and Spinors, 2nd edn. Cambridge University Press, Cambridge (2001)
Martfnez, J.M.R.: The principle of transference: history, statement and proof. Mech. Mach. Theor. 28(1), 165–177 (1993)
Radavelli, L., Simoni, R., Pieri, E.R.D., Martins, D.: A comparative study of the kinematics of robot manipulators by Denavit- Hartenberg and dual quaternion. Mecnica Computacional XXXI, 2833–2848 (2012)
Radavelli, L.: Análise cinemática direta de robôs manipuladores via álgebra de clifford e quatérnios. Dissertação, UFSC (2013)
Sahul, S., Biswall, B.B., Subudhi, B.: A novel method for representing robot kinematics using quaternion theory. In: IEEE Sponsored Conference on Computational Intelligence, Control and Computer Vision in Robotics and Automation (2008)
Selig, J.M.: Clifford algebra of points, lines and planes. Robotica 18, 545–546 (2000a)
Selig, J.M.: Geometric Fundamentals of Robotics. Springer, New York (2000b)
Shoham, M., Ben-Horin, P.: Application of grassmann-cayley algebra to geometrical interpretation of parallel robot singularities. Int. J. Rob. Res. 28, 127–141 (2009)
Vince, J.: Geometric Algebra for Computer Graphics. Springer, London (2008)
Wang, J-Y., Liang, H-Z., Sun, Z-W., Wu, S-N., Zhang, S-H.: Relative motion coupled control based on dual quaternion. Aerosp. Sci. Technol. 25(1), 102–113 (2013)
Wang, X., Han, D., Yu, C., Zheng, Z.: The geometric structure of unit dual quaternion with application in kinematic control. J. Math. Anal. Appl. 389(2), 1352–1364 (2012)
Wang, X., Yu, C.: Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics. Syst. Control Lett. 62(3), 225–233 (2013). doi: http://dx.doi.org/10.1016/j.sysconle.2012.11.019
Woo, L., Freudenstein, F.: Application of line geometry to theoretical kinematics and the kinematic analysis of mechanical systems. Mechanisms 5, 417–460 (1970)
Yang, A.T., Freudenstein, F.: Application of dual-number quaternions to the analysis of the spatial mechanism. AMSE Trans. J. Appl. Mech. 86, 300–308 (1964)
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Radavelli, L.A., De Pieri, E.R., Martins, D., Simoni, R. (2014). Points, Lines, Screws and Planes in Dual Quaternions Kinematics. In: Lenarčič, J., Khatib, O. (eds) Advances in Robot Kinematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06698-1_30
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DOI: https://doi.org/10.1007/978-3-319-06698-1_30
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