# Analytical Forward Kinematics to the 3 DOF Congruent Spherical Parallel Robot Manipulator

### Ping Ji, Hongtao Wu

#### Abstract

This paper studies the kinematics of a special three degree-of-freedom (3 DOF) spherical parallel robot manipulator, where the two pyramids are exactly the same and so it is commonly called the 3 DOF congruent spherical parallel platform. Due to this special structure, the movement of the mobile pyramid can be regarded as the rotation of a rigid body from its base posture to its current status. By use of this special property, the forward kinematics of the parallel robot manipulator is obtained in this paper, and the final solution is a univariate quartic equation, which can be solved analytically without numerical iterations. A numerical example is provided to illustrate the method.

#### References

- Angeles, J., 1997. Fundamentals of Robotic Mechanical Systems, Springer-Verlag, New York.
- Bonev, I., Zlatanov, D. and Gosselin, C. M.; 2003. Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory, Transactions of the ASME, Journal of Mechanical Design, Vol. 125, pp. 573-581.
- Bruyninckx, H., 1998. Closed-form forward position kinematics for a (3-1-1-1)2 fully parallel manipulator, IEEE Transactions on Robotics and Automation, Vol. 14, No. 2, pp. 326-328.
- Gosselin, C. M., Sefrioui, J. and Richard, M. J., 1994a. On the direct kinematics of spherical three-degree-offreedom parallel manipulators with coplanar platform, Transactions of the ASME, Journal of Mechanical Design, Vol. 116, pp. 587-593.
- Gosselin, C. M., Sefrioui, J. and Richard, M. J., 1994b. On the direct kinematics of spherical three-degree-offreedom parallel manipulators of general architecture, Transactions of the ASME, Journal of Mechanical Design, Vol. 116, pp. 594-598.
- Guo, S., Wang, C., Qu, H., and Fang, Y., 2012. A novel 4- RRCR parallel mechanism based on screw theory and its kinematics analysis, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, DOI: 10.1177/0954406212469774.
- Huang, Z. and Yao, Y. L., 1999. A new closed-from kinematics of the generalized 3-DOF spherical parallel manipulator, Robotica, Vol. 17, pp. 475-485.
- Innocenti, C. and Parenti-Castelli, V., 1993. Echelon form solution of direct kinematics for the general fullyparallel spherical wrist, Mechanism and Machine Theory, Vol. 28, No. 4, pp. 553-561.
- Innocenti, C., 1998. Forward kinematics of a 6-6 fullyparallel manipulator with congruent base and platform, Advances in Robot Kinematics: Analysis and Control, Edited by J. Lenarcic and M. L. Husty, Kluwer Academic Publishers, pp. 137-146.
- Leguay-Durand, S. and Reboulet, C., 1997. Optimal design of a redundant spherical parallel manipulator, Robotica, Vol. 15, pp. 399-405.
- Li, Y. and Xu, Q., 2007. Kinematic analysis of a 3-PRS parallel manipulator, Robotics and ComputerIntegrated Manufacturing, Vo. 23, pp. 395-408.
- Mavroidis, C., 1997. Method to determine the displacement Screws Related to the direct kinematics of parallel mechanisms, Proceedings of the 5th Applied Mechanisms and Robotics Conference, Cincinnati, October 12-15.
- Mavroidis, C., 1998. Completely specified displacements of a rigid body and their application in the direct kinematics of in-parallel mechanisms, Proceedings of 1998 ASME Design Engineering Technical Conference, Atlanta, GA, September 13-16, Paper No. DETC98/MECH-5891.
- Wohlhart, K., 1994. Displacement analysis of the general spherical Stewart platform, Mechanism and Machine Theory, Vol. 29, No. 4, pp. 581-589.
- Zhang, Y., Crane III, C. D. and Duffy, J., 1998. Determination of the unique orientation of two bodies connected by a ball-and socket joint form four measured displacement, Journal of Robotic Systems, Vol. 15, No. 5, pp. 299-308.

#### Paper Citation

#### in Harvard Style

Ji P. and Wu H. (2013). **Analytical Forward Kinematics to the 3 DOF Congruent Spherical Parallel Robot Manipulator** . In *Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,* ISBN 978-989-8565-71-6, pages 111-115. DOI: 10.5220/0004399601110115

#### in Bibtex Style

@conference{icinco13,

author={Ping Ji and Hongtao Wu},

title={Analytical Forward Kinematics to the 3 DOF Congruent Spherical Parallel Robot Manipulator},

booktitle={Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,},

year={2013},

pages={111-115},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004399601110115},

isbn={978-989-8565-71-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics - Volume 2: ICINCO,

TI - Analytical Forward Kinematics to the 3 DOF Congruent Spherical Parallel Robot Manipulator

SN - 978-989-8565-71-6

AU - Ji P.

AU - Wu H.

PY - 2013

SP - 111

EP - 115

DO - 10.5220/0004399601110115