Approximation of Jacobian inverse kinematics algorithms
This paper addresses the synthesis problem of Jacobian inverse kinematics algorithms for stationary manipulators and mobile robots. Special attention is paid to the design of extended Jacobian algorithms that approximate the Jacobian pseudoinverse algorithm. Two approaches to the approximation problem are developed: one relies on variational calculus, the other is differential geometric. Example designs of the extended Jacobian inverse kinematics algorithm for 3DOF manipulators as well as for the unicycle mobile robot illustrate the theoretical concepts.
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Baillieul J. (1985). Kinematic programming alternatives for redundant manipulators Proceedings of the 1985 IEEE International Conference on Robotics and Automation St. Louis LO USA pp. 722-728.
Chitour Y. and and Sussmann H. J. (1998). Motion planning using the continuation method in J. Baillieul S. S. Sastry and H. J. Sussmann (Eds) Essays on Mathematical Robotics Springer-Verlag New York NY pp. 91-125.
Gelfand I. M. and Fomin S. V. (1963). Calculus of Variations Prentice-Hall Englewood Cliffs NJ.
Janiak M. and Tchoń K. (2008). Extended Jacobian inverse kinematics and approximation of distributions in J. Lenarcic and Ph. Wenger (Eds) Advances in Robot Kinematics Springer Science+Business Media Berlin pp. 137-146.
Klein Ch. A. and Huang C. (1983). Review of the pseudoinverse control for use with kinematically redundant manipulators IEEE Transactions on Systems Man and Cybernetics13(3): 245-250.
Klein Ch. A. Chu-Jenq C. and Ahmed Sh. (1995). A new formulation of the extended Jacobian method and its use in mapping algorithmic singularities for kinematically redundant manipulators IEEE Transactions on Robotics and Automation11(1): 50-55.
Roberts R. G. and Maciejewski A. A. (1992). Nearest optimal repeatable control strategies for kinematically redundant manipulators IEEE Transactions on Robotics and Automation8(3): 327-337.
Roberts R. G. and Maciejewski A. A. (1993). Repeatable generalized inverse control strategies for kinematically redundant manipulators IEEE Transactions on Automatic Control38(5): 689-699.
Roberts R. G. and Maciejewski A. A. (1993). Singularities stable surfaces and the repeatable behavior of kinematically redundant manipulators International Journal of Robotics Research13(1): 207-213.
Shamir T. and Yomdin Y. (1988). Repeatability of redundant manipulators: Mathematical solution of the problem IEEE Transactions on Automatic Control33(11): 1004-1009.
Sluis W. M. Banaszuk A. Hauser J. and Murray R. M. (1996). A homotopy algorithm for approximating geometric distributions by integrable systems Systems & Control Letters27(5): 285-291.
Tchoń K. (2002). Repeatability of inverse kinematics algorithms for mobile manipulators IEEE Transactions on Automatic Control47(8): 1376-1380.
Tchoń K. (2007). Continuation method in robotics Proceedings of the 7th Conference on Computer Methods and Systems Cracow Poland pp. 17-24.
Tchoń K. (2008). Optimal extended Jacobian inverse kinematics algorithms for robotic manipulators IEEE Transactions on Robotics28(6): 1440-1445.
Tchoń K. and Jakubiak J. (2006). Extended Jacobian inverse kinematics algorithm for non-holonomic mobile robots International Journal of Control79(8): 895-909.