A Predictive Algorithm for Rate Control of Mechanisms Near Singularities

Abstract

A new kinematic algorithm for the control of mechanisms is described that produces smoother and more accurate control near kinematic singularities than the usual damped least-squares method. The new method predicts the occurence of singularities, evaluates the singular terms of the inverse kinematic function us ing a nonsingular expression, and smoothly interpolates between the singular and nonsingular forms. The desingularized expres sion makes use of the rates of change of the singular vectors and values of the Jacobian matrix; formulas for these derivatives are presented. Simulations of the control of two- and six-joint serial manipulators moving through and near singular configurations are given to illustrate the method.

Get full access to this article

View all access options for this article.

References

Chen, Y.C. , Seng, J. , and O'Neil, K. 1997 (Albuquerque, NM). Lowest-order rate control of mechanisms near singularities. Proc. of the 1997 IEEE Conf. on Robot. and Automat . Piscataway, NJ: IEEE, 3597-3602.

Chiaverini, S. 1993. Estimate of the two smallest singular values of the Jacobian matrix: Application to damped least-squares inverse kinematics. J. Roboti. Sys. 10(8):991-1008.

Kieffer, J. 1994. Differential analysis of bifurcations and isolated singulaities for robots and mechanisms. IEEE Trans. Robot. Automat . 10(1):1-10.

Kirćanski, M. , and Borić, M. 1993 (Atlanta, GA). Symbolical singular value decomposition for a 7-DOF manipulator and its application to robot control. Proc. Int. Conf. on Robot. and Autom., vol. 3. Los Alamitos, CA: IEEE, pp.895-900.

Klein, C. , Chu-Jenq, C. , and Ahmed, S. 1995. A new formulation of the extended Jacobian method and its use in mapping algorithmic singularities for kinematically redundant manipulators. IEEE Trans. Robot. Automat . 11(2):50-55.

Lloyd, J. 1996 (Minneapolis, MN). Using Puiseux series to control nonredundant robots at singularities. Proc. Int. Conf. on Robot. and Automat . IEEE, pp. 1877-1882.

Maciejewski, A.A. 1987. The analysis and control of robotic manipulators operating at or near kinematically singular configurations. Ph.D. thesis Department of Electrical Engineering, Ohio State University.

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

Nakamura, Y. , and Hanafusa, H. 1986. Inverse kinematic solutions with singularity robustness for robot manipulator control. J. Dyn. Sys. Meas. Contro l 109:163-171.

10.

Narasimhan, S. , and Kumar, V. 1994 (Minneapolis, MN). A second order analysis of manipulator kinematics in singular configurations. Proc. 23rd Biennial Mechanisms Conf., pp.477-484.

11.

Nenchev, D.N. , Tsumaki, Y. , Uchiyama, M. 1996 (Minneapolis, MN). Adjoint Jacobian closed-loop kinematic control of robots. Proc. Int. Conf. Robot. Automat ., pp. 1235-1240.

12.

O'Neil, K. , and Chen, Y.C. In preparation. Critical points of the singular value decomposition .

13.

O'Neil, K. , Chen, Y.C. , Seng, J. 1997. Removing singularities of resolved motion rate control of mechanisms, including self-motion. IEEE Trans. Robot. Automat. 13(5):741-751.

14.

Park, J. , Chung, W.K. , and Youm, Y. 1995. Specification and control of motion for kinematically redundant manipulators. Proc. Int. Conf. on Intell. Robots and Sys. IEEE, pp. 89-93.

15.

Pohl, E.D. , and Lipkin, H. 1991. A new method of robotic rate control. Proc. Int. Conf. on Robotics and Aut. IEEE, pp. 1708-1713.

16.

Sciavicco, L. , and Siciliano, B. 1988. A solution algorithm to the inverse kinematic problem for redundant manipulators. IEEE J. of Robot. Automat . RA-4(4):303-310.

17.

Singh, S.K. 1993. Motion planning and control of nonredundant manipulators at singularities. Proc. Int. Conf. Robot. and Automat. IEEE, pp. 487-492.

18.

Tchoń, K. , and Dulęba, I. 1993. On inverting singular kinematics and geodesic trajectory generation for robot manipulators. J. Intell. Robot. Sys. 8:325-359.

19.

Wampler, C.W. II. 1986. Manipulator inverse kinematic solutions based on vector formulations and damped least-squares methods. IEEE Trans. Sys. Man Cybernetics SMC-16:93-101.