### “Iterative Solutions to the Inverse Geometric Problem for Manipulators with no Closed Form Solution”

**Authors:**Pål J. From and Jan T. Gravdahl,

**Affiliation:**NTNU, Department of Engineering Cybernetics

**Reference:**2008, Vol 29, No 3, pp. 77-92.

**Keywords:**Robotics, Kinematics, Inverse Kinematics

**Abstract:**A set of new iterative solutions to the inverse geometric problem is presented. The approach is general and does not depend on intersecting axes or calculation of the Jacobian. The solution can be applied to any manipulator and is well suited for manipulators for which convergence is poor for conventional Jacobian-based iterative algorithms. For kinematically redundant manipulators, weights can be applied to each joint to introduce stiffness and for collision avoidance. The algorithm uses the unit quaternion to represent the position of each joint and calculates analytically the optimal position of the joint when only the respective joint is considered. This sub-problem is computationally very efficient due to the analytical solution. Several algorithms based on the solution of this sub-problem are presented. For difficult problems, for which the initial condition is far from a solution or the geometry of the manipulator makes the solution hard to reach, it is shown that the algorithm finds a solution fairly close to the solution in only a few iterations.

PDF (275 Kb) DOI: 10.4173/mic.2008.3.1

**References:**

[1] Ahuactzin, J. M. Gupka, K. K. (1999). The kinematic roadmap: A motion planning based global approach for inverse kinematics of redundant robots, IEEE Trans. on Robotics and Automation, 15.

[2] Bronshtein, I. N., Semendyayev, K. A., Musiol, G., Muehlig, H. (2003). Handbook of Mathematics, Springer.

[3] Corke, P. (1996). A robotics toolbox for MATLAB, IEEE Robotics and Automation Magazine, .1:24-32 doi:10.1109/100.486658

[4] From, P. J. (2006). Modelling and Optimal Trajectory Planner for Industrial Spray Paint Robots, Master's thesis, NTNU.

[5] From, P. J. Gravdahl, J. T. (2007). General solutions to kinematic and functional redundancy, Proc. 46th IEEE Conf. on Decision and Control.

[6] Grudic, G. Z. Lawrence, P. D. (1993). Iterative inverse kinematics with manipulator configuration, IEEE Transactions on Robotics and Automation, 9, no. 4:476-483 doi:10.1109/70.246059

[7] Hanson, A. J. (2006). Visualizing Quaternions, Morgan Kaufmann.

[8] Johnson, M. P. (1995). Exploiting Quaternions to Support Expressive Interactive Character Motion, Ph.D. thesis, MIT.

[9] Khalil, W. Dombre, E. (2002). Modeling, Identification and Control of Robots, Hermes Penton.

[10] Kuipers, J. B. (2002). Quaternions and Rotation Sequences, Princeton University Press.

[11] Lin, Q. Burdick, J. W. (2000). Objective and frame-invariant kinematic metric functions for rigid bodies, International Journal of Robotics Research, 19.

[12] Luenberger, D. G. (2003). Linear and Nonlinear Programming, Kluwer Academic Publishers.

[13] Perdereau, V., Passi, C., Drouin, M. (2002). Real-time control of redundant robotic manipulators for mobile obstacle avoidance, Robotics and Autonomous Systems, 41 doi:10.1016/S0921-8890(02)00274-9

[14] Wang, L.-C. T. Chen, C. C. (1991). A combined optimization method for solving the inverse kinematics problem of mechanical manipulators, IEEE Trans. on Robotics and Automation. 7, no. 4 doi:10.1109/70.86079

[15] Wen, J. T.-Y. Kreutz-Delgado, K. (1991). The attitude control problem, IEEE Transactions on Automatic Control, 30 no. 10.

[16] Yuan, J. S. C. (1988). Closed-loop manipulator control using quaternion feedback, IEEE Journal of Robotics Automation, no. 4 doi:10.1109/56.809

**BibTeX:**

@article{MIC-2008-3-1,

title={{Iterative Solutions to the Inverse Geometric Problem for Manipulators with no Closed Form Solution}},

author={From, Pål J. and Gravdahl, Jan T.},

journal={Modeling, Identification and Control},

volume={29},

number={3},

pages={77--92},

year={2008},

doi={10.4173/mic.2008.3.1},

publisher={Norwegian Society of Automatic Control}

};