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Vol: 59(73) No: 2 / December 2014 Regularization of the task Jacobian of the differential inverse orientation problem of serial revolute joint manipulators Dániel András Drexler Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Magyar tudósok krt. 2., 1117, Budapest, phone: (361) 463-4027, e-mail: drexler@iit.bme.hu István Harmati Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Magyar tudósok krt. 2., 1117, Budapest, e-mail: harmati@iit.bme.hu Keywords: differential inverse kinematics, inverse orientation, singularity, Lie algebra, Jacobian regularization, singular configurations Abstract Solution of the inverse orientation problem of robot manipulators is encumbered with singularities. The solution of the differential inverse orientation problem requires the inverse of the task Jacobian, however in singularities the inverse does not exist. The differential inverse orientation problem is transformed into a new representation called the spherical representation, and then the resulting transformed task Jacobian is regularized using a methodology developed to regularize the inverse positioning problem. The algorithm is illustrated on the inverse orientation problem of an Euler wrist. References [1] R. M. Murray, S. S. Sastry, and L. Zexiang, A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994. [2] B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo, Robotics - Modelling, Planning and Control. Springer, 2009. [3] D. A. Drexler and I. Harmati, “Optimization issues on the motion planning of kinematically redundant manipulators,” in Optimization in computer engineering - Theory and applications, M. Z. Á, Ed. Scientific Research Publishing, 2011, pp. 81 – 98. [4] D. H. Gottlieb, “Topology and the robot arm,” Acta Applicande Mathematica, vol. 11, pp. 117–121, February 1988. [5] Y. Nakamura and H. Hanafusa, “Inverse kinematic solutions with singularity robustness for robot manipulator control,” Journal of Dynamic Systems, Measurement, and Control, vol. 108, no. 3, pp. 163–171, 1986. [6] D. A. Drexler and I. Harmati, “Joint constrained differential inverse kinematics algorithm for serial manipulators,” Periodica Polytechnica - Electrical Engineering and Computer Science, vol. 56, pp. 95–104, 2012. [7] S. Chiaverini, “Singularity-robust task-priority redundancy resolution for real-time kinematic control of robot manipulators,” IEEE Transactions on Robotics and Automation, vol. 13, no. 3, pp. 398–410, 1997. [8] P. Donelan and A. Müller, “Singularities of regional manipulators revisited,” in Advances in Robot Kinematics: Motion in Man and Machine, J. Lenarcic and M. M. Stanisic, Eds. Springer Netherlands, 2010, pp. 509–519. [9] F. Caccavale, S. Chiaverini, and B. Siciliano, “Second-order kinematic control of robot manipulators with jacobian damped least-squares inverse: Theory and experiments,” IEEE/ASME Transactions on Mechatronics, vol. 2, no. 3, pp. 188–194, 1997. [10] J. Tan, N. Xi, and Y. Wang, “A singularity-free motion control algorithm for robot manipulators–a hybrid system approach,” Automatica, vol. 40, no. 7, pp. 1239 – 1245, 2004. [11] T. Sugihara, “Solvability-unconcerned inverse kinematics by the Levenberg–Marquardt method,” IEEE Transactions on Robotics, vol. 27, no. 5, pp. 984–991, 2011. [12] J. Kim, G. Marani, W. K. Chung, and J. Yuh, “A general singularity avoidance framework for robot manipulators: task reconstruction method,” in Proceedings of 2004 IEEE International Conference on Robotics and Automation (ICRA04), 2004, vol. 5, pp. 4809–4814. [13] D. Oetomo and M. H. A. Jr, “Singularity robust algorithm in serial manipulators,” Robotics and Computer-Integrated Manufacturing, vol. 25, no. 1, pp. 122–134, 2009. [14] D. A. Drexler and I. Harmati, “Regularized Jacobian for the differential inverse positioning problem of serial revolute joint manipulators,” in Proceedings of the IEEE 11th International Symposium on Intelligent Systems and Informatics (SISY2013), Subotica, Serbia, 2013. [15] D. A. Drexler and I. Harmati, “Regularization of the differential inverse orientation problem of serial revolute joint manipulators,” in Proceedings of the 9th IEEE International Symposium on Applied Computational Intelligence and Informatics (SACI2014), Timisoara, Romania, 2014. [16] M. Postnikov, Lectures in Geometry, Semester V, Lie Groups and Lie Algebras. MIR Publishers Moscow, 1986. [17] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory. Springer Verlag, New York, 1972. |