Abstract
A closed form solution of the inverse kinematics problem requires the derivation of a set of implicit determining equations for the joint variables. Depending on the sequence, in which joint variables are solved, different sets of equations will be obtained. The maximum polynomial degree among the equations in one set can be smaller than in other sets.
An approach for the symbolic computation of determining equations with minimum degree is presented. First a survey of the polynomial degrees of all equations in all possible solution sequences is obtained by application of the Buchberger Algorithm. This leads to the identification of optimal solution sequences for the joint variables. A basic technique is employed for actually calculating the equations in such a solution sequence.
The method can be used as well to examine arbitrary equation sub-system for extraneous roots. The presented strategies have been incorporated as components in an inverse kinematics expert system. This was used to verify that all closed form solutions for the general 6R robot must contain a determining equation of degree 16. An example of the application of the presented methods is given.
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References
Paul, R.P.: Robot manipulators: Mathematics, Programming, and Control. MIT Press, Cambridge, 1981.
Grzeschniok, F.:Ein Expertensystem zur Lösung der kinematischen Gleichung.Diploma Thesis, Technische Universität Berlin, 1987.
Kovacs, P.:Ermittlung von Triangulationen kleinsten Grades für die inverse kinematische Transformation.Robotersysteme 6 (1990), Springer Verlag, pp. 51–63.
Smith, D., Lipkin, H.: Analysis of fourth order Manipulator Kinematics using Conic Sections. Proc. IEEE Intl. Conf. Robotics and Autom., Cincinnati (1990) pp. 274–278.
Albala, H., Pessen, D.:Displacement Analysis of a Special Case of the 7R, Single-Loop Spatial Linkage. J. Mech., Transm. and Autom. in Design 105, No. 1 (1983) pp. 78–87.
Lee, H.J., Liang, C.G.:Displacement Analysis of the General Spatial 7-Link 7R Mechanism.Mech. Mach. Theory 23 (1988) pp. 219–226.
Herrera-Bendezu, L.G., Mu, E, Cain, J.T.:Symbolic Computation of Robot Manipulator Kinematics. Proc. IEEE Intl. Conf. Robotics and Autom., Philadelphia (1988) pp. 993–998.
Rieseler, H., Wahl, F.: Fast Symbolic Computation of the Inverse Kinematics of Robots. Proc. IEEE Intl. Conf. Robotics and Autom., Cincinnati (1990) pp. 86–91.
Hintenaus, P.: The Inverse Kinematics System (User’s Manual). Universität Linz, RISC Technical Report 87–18, 1987.
Heiß, H.:Die explizite Lösung für eine Klasse von Industrierobotern.Ph.D. Thesis, Technische Universität Berlin, 1985.
Faugère, J.C., Gianni, P., Lazard, D., Mora, F.:Efficient Computation of zero-dimensional Gröbner Bases by Change of Ordering. Preprint (1989).
Hofmann, C.M.:Geometric and Solid Modelling. Morgan Kaufmann Publishers, 1989.
Buchberger, B.: Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory. In: Multidimensional Systems Theory (Ed. N.K.Bose), Reidel, Dordrecht, 1985.
Buchberger, B.: Applications of Gröbner Bases in Non-Linear Computational Geometry. In: Trends in Computer Algebra (Ed. R.Janßen). Springer Lecture Notes 296 (1987) pp. 52–80.
Tsai, L.W., Morgan, A.P.: Solving the Kinematics of the most general six and five-degree-offreedom Manipulators by Continuation Methods. J. Mech., Transm. and Autom. in Design 107 (1985) pp. 48–57.
Van der Waerden, B.L.: Modern Algebra. 2 vols., Frederick Ungar Publ., New York, 1953.
Kovacs, P., Hommel, G.: Factorization and Decomposition in Kinematic Equation Systems. Technical Report 90–23, Technische Universität Berlin, 1990.
Kovacs, P., Hommel, G.: Reduced Equation Systems for the Inverse Kinematics Problem. Technical Report 90–20, Technische Universität Berlin, 1990.
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© 1991 Springer-Verlag Wien
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Kovács, P. (1991). Minimum Degree Solutions for the Inverse Kinematics Problem by Application of the Buchberger Algorithm. In: Stifter, S., Lenarčič, J. (eds) Advances in Robot Kinematics. Springer, Vienna. https://doi.org/10.1007/978-3-7091-4433-6_37
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DOI: https://doi.org/10.1007/978-3-7091-4433-6_37
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-82302-6
Online ISBN: 978-3-7091-4433-6
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