Symbolic Singular Value Decomposition for a PUMA Robot and Its Application to a Robot Operation Near Singularities

Abstract

Singular value decomposition has received considerable atten tion in robot control during the past several years because of the fact that it may be used in solving the problem of robot operation in singular configurations. However, its main draw back is its high computational complexity when compared with that of the Gaussian elimination or analytical inverse kinematic solution. In this article we have derived symbolic expressions for matrices of the decomposition for both a 2-DOF planar manipulator and a 6-DOF PUMA robot. This reduces the com putational burden by an order of magnitude and gives us more insight into the nature of the singularities. The symbolic ex pressions have been applied to obtain the damped least-squares solution for joint velocities. Some modifications of the damped least-squares solution are proposed, and the simulation results are included. The method can be applied to other robots when position and orientation problems can be separated.

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References

Dubey, R. , and Luh, J.Y.S. 1988. Redundant robot control using task based performance measures . J. Robot. Sys. 5(5):409-432.

Featherstone, R. 1983. Position and velocity transformations between robot end-effector coordinates and joint angles. Int. J. Robot. Res. 2(2):35-45.

Fijany, A. , and Bejczy, A.K. 1988 (April 24-29, Philadelphia). Efficient Jacobian inversion for the control of simple robot manipulators . Proc. 1988 IEEE Int. Conf. Robotics and Automation . Washington, DC: IEEE Computer Society Press, pp. 999-1007.

Golub, G.H. , and Van Loan, Ch. F. 1989. Matrix Computations. Baltimore: The John Hopkins University Press.

Klein, Ch. A. , and Blaho, B.E. 1987. Dexterity measures for the design and control of kinematically redundant manipulators. Int. J. Robot. Res. 6(2):72-83.

Khatib, O. 1987. A unified approach for motion and force control of robot manipulators: the operational space formulation. IEEE J. Robot. Automation RA-3:43-53.

KirΛanski, M. , and BoriΛ M. 1992 (October 6-9, Barcelona). Symbolical singular value decomposition for PUMA robot and its application to robot control , Proc. of Int. Symp. on Industrial Robots ISIR'92 , pp. 255-260.

Lawson, C.L. , and Hanson, R.J. 1974. Solving Least-Squares Problems. Englewood Cliffs, NJ: Prentice Hall.

Lee, S. , and Lee, J.M. 1990 (May 13-18, Cincinnati). Multiple task point control of a redundant manipulator. Proc. 1990 IEEE Int. Conf. Robotics and Automation. Los Alamitos: IEEE Computer Society Press, pp. 988-993.

10.

Maciejewski, A.A. , and Klein, Ch. A. 1988. Numerical filtering for the operation of robotic manipulators through kinematically singular configurations . J. Robot. Sys. 5(6):527-552.

11.

Maciejewski, A.A. , and Klein, Ch. A. 1989. The singular value decomposition: computation and applications to robotics. Int. J. Robot. Res. 8(6):63-79.

12.

Nakamura, Y. , and Hanafusa, H. 1986. Inverse kinematic solutions with singularity robustness for robot manipulator control. ASME J. Dynam. Sys. Measurement Control 108:163-171.

13.

Renaud, M. 1981 (October, Tokyo). Geometric and kinematic models of a robot manipulator: calculation of the Jacobian matrix and its inverse. Proc. of 11th Int. Symp. on Industrial Robots

14.

Shin, K.G. , and Lee, C.P. 1985. (December, Ft. Lauderdale). Compliant control of robotic manipulators with resolved acceleration. Proc. 24th IEEE Conf. Decision and Control, pp. 350-357.

15.

StanišiΛ, M. 1988. Coordinate systems and the inverse velocity problem of manipulators with spherical wrists. Int. J. Robot. Res. 7(2):62-64.

16.

StokiΛ, D. 1991. Constrained motion control of manipulation robots—a contribution. Robotica 9:157-163.

17.

Tarn, T.J. , Bejczy, A.K. , Ganguly, S. , Ramadorai, A.K. , and Marth, G.T. 1991 (April 9-11, Sacramento). Experimental evaluation of the nonlinear feedback robot controller. Proc. 1991 IEEE Int. Conf. Robotics and Automation. Los Alamitos: IEEE Computer Society Press, pp. 1638-1644.

18.

Wampler, Ch. W. 1986. Manipulator inverse kinematic solutions based on vector formulations and damped least-squares methods. IEEE Trans. Sys. Man Cybernet. SMC-16(1):93-101.

19.

Wolfram, S. 1988. Mathematica: A System for Doing Mathematics by Computer . New York: Addison-Wesley Publishing Company, Inc.