Research Papers

Application of Hyper-Dual Numbers to Multibody Kinematics

[+] Author and Article Information
Avraham Cohen

Robotics Laboratory,
Department of Mechanical Engineering,
Technion–Israel Institute of Technology,
Technion City, Haifa 32000, Israel
e-mail: avico@tx.technion.ac.il

Moshe Shoham

Robotics Laboratory,
Department of Mechanical Engineering,
Technion–Israel Institute of Technology,
Technion City, Haifa 32000, Israel
e-mail: shoham@technion.ac.il

Manuscript received January 30, 2015; final manuscript received May 1, 2015; published online August 18, 2015. Assoc. Editor: J. M. Selig.

J. Mechanisms Robotics 8(1), 011015 (Aug 18, 2015) (4 pages) Paper No: JMR-15-1019; doi: 10.1115/1.4030588 History: Received January 30, 2015

Hyper-dual numbers (HDNs) are applied in this paper to multibody kinematics. First, the hyper-dual angle that encompasses a body's position, orientation, as well as its velocity, is defined as an element of the hyper-dual transformation matrix. Then, the “automatic differentiation” feature of the dual numbers is used to obtain the second derivative of a body pose. The body's velocity and acceleration are obtained from the elements of the hyper-dual transformation matrix by algebraic manipulations only, with no need for further time derivatives of the body pose. A robot manipulator is presented as an exemplary application of HDNs to multibody kinematics.

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Clifford, W. K. , 1873, “Preliminary Sketch of Bi-Quaternions,” Proc. London Math. Soc., 4, pp. 381–395.
Kotelnikov, A. P. , 1895, Screw Calculus and Some Applications to Geometry and Mechanics, Annals of the Imperial University of Kazan, Kazan, Russia.
Keler, M. L. , 1973, “Kinematics and Statics Including Friction in Single-Loop Mechanisms by Screw Calculus and Dual Vectors,” ASME J. Eng. Ind., 95(2), pp. 471–480. [CrossRef]
Yang, A. T. , and Freudenstein, F. , 1964, “Application of Dual Number Quaterian Algebra to Analysis of Spatial Mechanisms,” ASME J. Appl. Mech., 31(2), pp. 300–308. [CrossRef]
Dimentberg, F. M. , 1965, The Screw Calculus and Its Application in Mechanics, Nauka, Moscow (Clearinghouse for Federal and Scientific Technical Information).
Pennestrì, E. , and Stefanelli, R. , 2007, “Linear Algebra and Numerical Algorithms Using Dual Numbers,” Multibody Syst. Dyn., 18(3), pp. 323–344. [CrossRef]
Veldkamp, G. R. , 1976, “On the Use of Dual Numbers, Vectors and Matrices in Instantaneous, Spatial Kinematics,” Mech. Mach. Theory, 11(2), pp. 141–156. [CrossRef]
Spall, R. E. , and Yu, W. , 2013, “Imbedded Dual-Number Automatic Differentiation for Computational Fluid Dynamics Sensitivity Analysis,” ASME J. Fluids Eng., 135(1), p. 014501. [CrossRef]
Yu, W. , and Blair, M. , 2013, “DNAD, a Simple Tool for Automatic Differentiation of Fortran Codes Using Dual Numbers,” Comput. Phys. Commun., 184(5), pp. 1446–1452. [CrossRef]
Gu, Y. L. , and Loh, N. K. , 1985, “Dynamic Model for Industrial Robots Based on a Compact Lagrangian Formulation,” 24th IEEE Conference Decision and Control, Fort Laudedale, FL, Dec. 11–13, pp. 1497–1501.
Gu, Y. L. , and Luh, J. Y. S. , 1987, “Dual Number Transformation and Its Applications to Robotics,” IEEE Trans. Rob. Autom., 3(6), pp. 615–623. [CrossRef]
Fike, J. A. , Jongsma, S. , Alonso, J. J. , and van der Weida, E. , 2011, “Optimization With Gradient and Hessian Information Calculated Using Hyper-Dual Numbers,” AIAA Paper No. 2011-3807.
Fike, J. A. , 2009, “Numerically Exact Derivative Calculations Using Hyper-Dual Numbers,” 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design, Stanford University, Stanford, CA, June 18.
Fike, J. A. , and Alonso, J. J. , 2011, “The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations,” AIAA Paper No. 2011-886.
Fike, J. A. , and Alonso, J. J. , 2011, “Automatic Differentiation Through the Use of Hyper-Dual Numbers for Second Derivatives,” Recent Advances in Algorithmic Differentiation (Lecture Notes in Computational Science and Engineering, Vol. 87), Springer-Verlag, Berlin, pp. 163–173.
Penunuri, F. , Mendoza, O. , Peon-Escalante, R. , Villanueva, C. , and Cruz-Villar, C. A. , 2013, “A Dual Number Approach for Numerical Calculation of Velocity and Acceleration in the Spherical 4R Mechanism,” e-print arXiv:1301.1409.
Study, E. , 1903, Die Geometrie der Dynamen, Verlag Teubner, Leipzig, Germany, p. 437.
Pradeep, A. K. , Yoder, P. J. , and Mukundan, R. , 1989, “On the Use of Dual-Matrix Exponentials in Robotic Kinematics,” Int. J. Rob. Res., 8(5), pp. 57–66. [CrossRef]
Jen, F. H. , and Shoham, M. , 1993, “On Rotations and Translations With Application to Robot Manipulators,” Adv. Rob., 8(2), pp. 203–229. [CrossRef]
Whitney, D. E. , 1972, “The Mathematics of Coordinated Control of Prosthetic Arms and Manipulators,” ASME J. Dyn. Syst. Meas. Control, 94(4), pp. 303–309. [CrossRef]
Brodsky, V. , and Shoham, M. , 1998, “Derivation of Dual Forces in Robot Manipulators,” Mech. Mach. Theory, 33(8), pp. 1241–1248. [CrossRef]


Grahic Jump Location
Fig. 1

Two cylindrical, four degrees-of-freedom robot




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