# A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators

International Journal of Applied Mathematics and Computer Science (2013)

- Volume: 23, Issue: 2, page 373-382
- ISSN: 1641-876X

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topIgnacy Dulęba, and Michał Opałka. "A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators." International Journal of Applied Mathematics and Computer Science 23.2 (2013): 373-382. <http://eudml.org/doc/257109>.

@article{IgnacyDulęba2013,

abstract = {The objective of this paper is to present and make a comparative study of several inverse kinematics methods for serial manipulators, based on the Jacobian matrix. Besides the well-known Jacobian transpose and Jacobian pseudo-inverse methods, three others, borrowed from numerical analysis, are presented. Among them, two approximation methods avoid the explicit manipulability matrix inversion, while the third one is a slightly modified version of the Levenberg-Marquardt method (mLM). Their comparison is based on the evaluation of a short distance approaching the goal point and on their computational complexity. As the reference method, the Jacobian pseudo-inverse is utilized. Simulation results reveal that the modified Levenberg-Marquardt method is promising, while the first order approximation method is reliable and requires mild computational costs. Some hints are formulated concerning the application of Jacobian-based methods in practice.},

author = {Ignacy Dulęba, Michał Opałka},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {serial manipulator; inverse kinematics; Jacobian methods; comparison; applications},

language = {eng},

number = {2},

pages = {373-382},

title = {A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators},

url = {http://eudml.org/doc/257109},

volume = {23},

year = {2013},

}

TY - JOUR

AU - Ignacy Dulęba

AU - Michał Opałka

TI - A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators

JO - International Journal of Applied Mathematics and Computer Science

PY - 2013

VL - 23

IS - 2

SP - 373

EP - 382

AB - The objective of this paper is to present and make a comparative study of several inverse kinematics methods for serial manipulators, based on the Jacobian matrix. Besides the well-known Jacobian transpose and Jacobian pseudo-inverse methods, three others, borrowed from numerical analysis, are presented. Among them, two approximation methods avoid the explicit manipulability matrix inversion, while the third one is a slightly modified version of the Levenberg-Marquardt method (mLM). Their comparison is based on the evaluation of a short distance approaching the goal point and on their computational complexity. As the reference method, the Jacobian pseudo-inverse is utilized. Simulation results reveal that the modified Levenberg-Marquardt method is promising, while the first order approximation method is reliable and requires mild computational costs. Some hints are formulated concerning the application of Jacobian-based methods in practice.

LA - eng

KW - serial manipulator; inverse kinematics; Jacobian methods; comparison; applications

UR - http://eudml.org/doc/257109

ER -

## References

top- Ben-Isreal, A. and Cohen, D. (1966). On iterative computation of generalized inverses and associated projections, SIAM Journal on Numerical Analysis 3(3): 410-419. Zbl0143.37402
- Ben-Isreal, A. and Greville, T. (2003). Generalized Inverses: Theory and Applications, CMS Books in Mathematics, 2nd Edn., Springer, New York, NY. Zbl1026.15004
- Chiacchio, P. and Siciliano, B. (1989). A closed-loop Jacobian transpose scheme for solving the inverse kinematics of nonredundant and redundant wrists, Journal of Robotic Systems 6(5): 601-630. Zbl0683.70029
- D'Souza, A., Vijaykumar, S. and Schaal, S. (2001). Learning inverse kinematics, International Conference on Intelligent Robots and Systems, Maui, HI, USA, pp. 298-303.
- Dulęba, I. and Jagodziński, J. (2011). Motion representations for the Lafferriere-Sussmann algorithm for nilpotent control systems, International Journal of Applied Mathematics and Computer Science 21(3): 525-534, DOI: 10.2478/v10006-011-0041-y. Zbl1234.93042
- Dulęba, I. and Sasiadek, J. (2002). Modified Jacobian method of transversal passing through the smallest deficiency singularities for robot manipulators, Robotica 20(4): 405-415.
- Golub, G. and Van Loan, C. (1996). Matrix Computations, 3rd Edn., Johns Hopkins, Baltimore, MD. Zbl0865.65009
- Horn, R. and Johnson, C. (1986). Matrix Analysis, Cambridge University Press, New York, NY.
- Hunek, W. and Latawiec, K.J. (2011). A study on new right/left inverses of nonsquare polynomial matrices, International Journal of Applied Mathematics and Computer Science 21(2): 331-348, DOI: 10.2478/v10006-011-0025-y. Zbl1282.93144
- Lee, C. (1982). Robot arm kinematics, dynamics, and control, Computer 15(12): 62-80.
- Levenberg, K. (1944). A method for the solution of certain problems in least squares, Quarterly of Applied Mathematics 2: 164-168. Zbl0063.03501
- Maciejewski, A. and Klein, C. (1989). The singular value decomposition: Computation and applications to robotics, International Journal of Robotics Research 8(6): 63-79.
- Marquardt, D. (1963). An algorithm for least-squares estimation of nonlinear parameters, SIAM Journal on Applied Mathematics 11(2): 431-441. Zbl0112.10505
- Nakamura, Y. (1991). Advanced Robotics: Redundancy and Optimization, Addison Wesley, New York, NY.
- Nearchou, A. (1998). Solving the inverse kinematics problem of redundant robots operating in complex environments via a modified genetic algorithm, Mechanism and Machine Theory 33(3): 273-292. Zbl1045.70505
- Tchoń, K. and Dulęba, I. (1993). On inverting singular kinematics and geodesic trajectory generation for robot manipulators, Journal of Intelligent and Robotic Systems 8(3): 325-359.
- Tchoń, K., Dulęba, I., Muszyński, R., Mazur, A. and Hossa, R. (2000). Manipulators and Mobile Robots: Models, Motion Planning, Control, PLJ, Warsaw, (in Polish).
- Tchoń, K., Karpińska, J. and Janiak, M. (2009). Approximation of Jacobian inverse kinematics algorithms, International Journal of Applied Mathematics and Computer Science 19(4): 519-531, DOI: 10.2478/v10006-009-0041-3. Zbl1300.93123
- Tejomurtula, S. and Kak, S. (1999). Inverse kinematics in robotics using neural networks, Information Sciences 116(2-4): 147-164. Zbl0942.70006

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