Dual Quaternions Robotics: A) The 3R Planar Manipulator
DOI:
https://doi.org/10.12970/2308-8354.2018.06.02Keywords:
Climate change, disease transmission, host, pathogen, vector.Abstract
Abstract Kinematics analysis studies the relative motions, such as, first of all, the displacement in space of the end effector of a given robot, and thus its velocity and acceleration, associated with the links of the given robot that is usually designed so that it can position its end-effector with a three degree-of-freedom of translation and three degree-of-freedom of orientation within its workspace. This chapter presents mainly, on the light of both main concepts; the first being the screw motion or/ and dual quaternions kinematics while the second concerns the classical ‘Denavit and Hartenberg parameters method’ the direct kinematics of a planar manipulator.We must emphasize the fact that the use of Matlab software and quaternions and / or dual quaternions in the processing of 3D rotations and/or screw movements is and will always be the most efficient, fast and accurate first choice. Dual quaternion direct kinematics method could be generalised, in the future, to more complicated spatial and/ or industrial robots as well as to articulated and multibody systems.References
[1] Hamilton WR. On quaternions; or on a new system of imaginaries in algebra. London, Edinburgh, and Dublin.
[2] McDonald J. Teaching Quaternions is not Complex. Comp Graphics Forum 2010; 29(8): 2447-2455. https://doi.org/10.1111/j.1467-8659.2010.01756.x
[3] Chou JCK, Kamel M. Quaternions approach to solve the kinematic equation of rotation, of a sensor mounted rob. manip. In: Proceedings of the IEEE Int. Conf. Rob.s and automation (ICRA), Philadelphia 1988; pp. 656-662.
[4] Gouasmi M, Ouali M, Brahim F. Rob. Kin. using dual quat. Int Jourl of Rob and Autom 2012; 1(1): 13-30.
[5] Clifford WK. Preliminary sketch of bi-quaternions. Proceedings of the London Mathematical 1882.
[6] Perez A. Dual Quaternion Synthesis of Constrained Robotic Systems, Ph.D. Dissertation, Department of Mechanical and Aerospace Engineering, University of California, Irvine 2003.
[7] Perez A, McCarthy JM. Dual Qua. Synth. of Constr. Rob. Syst. Journal of Mechanical Design 2003; in press.
[8] Kavan L, Collins S, Žára J, O’Sullivan C. Geometric skinning with approximate dual quaternion blending. ACM Transactions on Graphics (TOG) 2008; 27(4): 105. https://doi.org/10.1145/1409625.1409627
[9] Ivo FZ, Ivo H. Spher. skin. with dual quat and Q. Tangents. ACM SIGGRAPH Talks, 2011; 27: 4503.
[10] Selig J. Rat. Int. of r-b. m. Adv. in the Theory of Control, Sign. and Syst. with Phys. Mod. 2011; 213-224.
[11] Vasilakis A, Fudos I. Skeleton-based rigid skinning for character animation, in Proc. of the Fourth International Conference on Computer Graphics Theory and Applications, 2009; pp. 302-308.
[12] Kuang Y, Mao A, Li G, Xiong Y. A strategy of real-time animation of clothed body movement, in Multimedia Technology (ICMT), 2011 International Conference on 2011; pp. 4793-4797.
[13] Pham HL, Perdereau V, Adorno BV, Fraisse P. “osition and orientation control of robot manipulators using dual quaternion feedback, in Intelligent Robots and Systems (IROS), 2010 IEEE/RS J Int Conf 2010; pp. 658-663.
[14] Schilling M. Univer. manip body models - dual quaternion rep. in lay. and dyn. MMCs. Autonomous Robots, 2011.
[15] Ge Q, Varshney A, Menon JP, Chang CF. Double quaternions for motion interpolation, in Proceedings of the ASME Design Engineering Technical Conference 1998.
[16] Lin Y, Wang H, Chiang Y. Estim. of real. orientation using dual. quat, Sys. Sci. and, 2010; 2: 413-416.
[17] Perez A, McCarthy JM. Dual quat synthesis of constr. rob. systs, Jou. of Mech. Des 2004; 126: 425. https://doi.org/10.1115/1.1737378
[18] van den Bergen G. Dual Numbers: Simple Math, Easy C++ Coding, and Lots of Tricks, GDC Europe, 2009. [Online]. Available: www.gdcvault.com/play/10103/Dual-NumbersSimple-Math-Easy.
[19] Amanpreet, Singh, Ashish, Singla. Kinematic Modeling of Robotic. Manip. The Nat. Acad of Sciences 2016.
[20] Chasles M. Note sur les propriétés générales du système de deux corps semblables entr'eux. Bulletin des Sciences Mathématiques, Astronomiques, Physiques et Chimiques (in French). 1830; 14: 321-326.
[21] Louis Poinsot. Théorie nouvelle de la rotation des corps, Paris, Bachelier, 1851; p. 170.
[22] Whittaker ET. A Treatise on Analytical Dynamics of Particles and Rigid Bodies, 1904; p. 4, at Google Books
[23] Ball RS. The Theory of Screws. Cambridge, U.K., Cambridge Univ. Press, 1900.
[24] Murray RM, Li Z, Sastry SS. A Math. Intro. to Robot Manip. Boca. Raton, FL: CRC Press, 1993.
[25] Denavit J, Hartenberg RS. A Kin. Not. for Low-pair Mech.s Based on Matr. ASME Jour. of App. Mechs 1955; 22: 215-221.
[26] Selig JM. Introductory robotics. Prentice hall international (UK) Ltd, 1992.
[27] Selig JM. Geometrical fundamentals of Robotics, Springer, second edition, 2004.
[28] Selig JM. Lie groups and Lie algebras in robotics. Course report, south bank university, London.
[29] Yang AT, Freudenstein F. App. of Dual-Num. Quat. Alg. to the Ana of Spa. Mec.” ASME Jour. of Ap. Mec., 1964; pp. 300-308.
[30] Bottema O, Roth B. Theoretical Kinematics, Dover Publications, New York 1979.
[31] McCarthy JM. Introduction to Theoretical Kinematics. The MIT. Press, Cambridge, MA. 1990.
[32] Ge QJ, Ravani B. Geom. Cons. of Bezier Motions. ASME Jour of Mech. Des 1994; 116: 749-755.
[33] Perez A, McCarthy JM. Dimen.Synth.of Spa RR rob. Advan. in Rob. Kin., Piran-Portoroz, Slovenia 2000.
[34] Marsden JE, Ratiu TS. Introduction to Mechanics and Symmetry. Springer Verlag, New York, NY, sec.ed. 1999. https://doi.org/10.1007/978-0-387-21792-5
[35] Oliveira VM. Estudo e controle de robôs bracejadores subatuados. Ph.D. thesis, Escola de Engenharia, Departamento de Engenharia Elétrica, Universidade Federal do Rio Grande do Sul 2008.
[36] Spong MW, Vidvasagar M. Robot Dynamics and Control 2004.
[37] Klasing K. Parallelized sampling-based path planning for tree structured rigid robots, Master’s thesis, Institute of Automatic Control Engineering, Technische Universität München 2009.
[38] Khalil W, Dombre E. Modélisation, Identification et Commande des robots, Hermès 2002.
