Closed form solution for direct and inverse kinematics of a US-RS-RPS 2-DOF parallel robot

Document Type : Article

Authors

Department of Mechanical Engineering, Universidad del Norte, Km.5 Via Puerto Colombia, Barranquilla, Colombia

Abstract

Parallel mechanisms with reduced degree of freedom (DOF) have grown in importance
for industry and researchers as they o er a simpler architecture and lower manufactur-
ing/operating costs with great performance. In this paper, a two degree of freedom
parallel robot is proposed and analyzed. The robot with a xed base, a moving platform
and three legs achieve translational and rotational motion through actuation on prismatic
and revolute joints, and can be applied on pick and place applications, vehicle simulators
among others. By making use of homogeneous transformation matrices and Sylvesters
dialytic elimination method a closed form solution for direct kinematics is obtained for
all possible assembly modes. Inverse kinematics was solved in closed form as well. This
greatly decreases computational time and proposed approach is optimal. A case study
was done to validate the solutions found and compared with a CAD model to corroborate
results. Finally, a workspace calculation was made for di erent geometrical parameters
of the robot.

Keywords

Main Subjects


References
1. Fassi, I. and Wiens, G.J. Multiaxis machining: PKMs
and traditional machining centers", Journal of Manufacturing
Processes, 2, pp. 1-14 (2000).
2. Taghirad, H.D., Parallel Robots: Mechanics and Control,
CRC Press (2013).
3. Merlet, J.-P., Parallel Robots, Springer, Netherlands
(2006).
4. Merlet, J.-P. Direct kinematics of parallel manipulators",
IEEE Transactions on Robotics and Automation,
9, pp. 842-846 (1993).
5. Bonev, I., Ilian A., and Gosselin, C.M. Analytical determination
of the workspace of symmetrical spherical
parallel mechanisms", IEEE Transactions on Robotics,
22, pp. 1011-1017 (2006).
6. Sadjadian, H., Member, H.T., and Fatehi, A. Neural
networks approaches for computing the forward kinematics
of a redundant parallel manipulator", International
Journal of Computer, Electrical, Automation,
Control and Information Engineering, 2, pp. 1664-1671
(2008).
7. Sadjadian, H. and Taghirad, H. Comparison of di erent
methods for computing the forward kinematics of a
redundant parallel manipulator", Journal of Intelligent
and Robotic Systems: Theory and Applications, 44, pp.
225-246 (2005).
8. Jamwal, P.K., Xie, S.Q., Tsoi, Y.H., and Aw, K.C.
Forward kinematics modelling of a parallel ankle
rehabilitation robot using modi ed fuzzy inference",
J. Sanjuan et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2144{2154 2153
Mechanism and Machine Theory, 45, pp. 1537-1554
(2010).
9. Uchida, T. and McPhee, J. Using Grobner bases to
generate ecient kinematic solutions for the dynamic
simulation of multi-loop mechanisms", Mechanism and
Machine Theory, 52, pp. 144-157 (2012).
10. Abbasnejad, G., Daniali, H.M., and Fathi, A. Closed
form solution for direct kinematics of a 4PUS 1PS
parallel manipulator", Scientia Iranica, 19, pp. 320-
326 (2012).
11. Lu, Y., Shi, Y., and Hu, B. Solving reachable
workspace of some parallel manipulators by computeraided
design variation geometry", Proceedings of the
Institution of Mechanical Engineers, 222, pp. 1773-
1782 (2008).
12. Enferadi, J. and Shahi, A. On the position analysis
of a new spherical parallel robot with orientation
applications", Robotics and Computer-Integrated Manufacturing,
37, pp. 151-161 (2016).
13. Varedi, S.M., Daniali, H.M., and Ganji, D.D. Kinematics
of an o set 3-UPU translational parallel manipulator
by the homotopy continuation method",
Nonlinear Analysis: Real World Applications, 10, pp.
1767-1774 (2009).
14. Huang, X.H.X. and He, G.H.G. New and ecient
method for the direct kinematic solution of the general
planar Stewart platform", IEEE International Conference
on Automation and Logistics, pp. 1979-1983
(2009).
15. Dhingra, A.K., Almadi, A.N., and Kohli, D. A
Grobner-Sylvester hybrid method for closed-form displacement
analysis of mechanisms", Journal of Mechanical
Design, 122, pp. 431-438 (2000).
16. Dash, A.K., Chen, I.M., Yeo, S.H., and Yang, G.
Workspace generation and planning singularity-free
path for parallel manipulators", Mechanism and Machine
Theory, 40, pp. 776-805 (2005).
17. Dunlop, G.R. and Jones, T.P. Position analysis
of a two DOF parallel mechanism the Canterbury
tracker", Mechanism and Machine Theory, 34, pp.
599-614 (1999).
18. Zhang, C. and Zhang, L. Kinematics analysis and
workspace investigation of a novel 2-DOF parallel
manipulator applied in vehicle driving simulator",
Robotics and Computer-Integrated Manufacturing, 29,
pp. 113-120 (2013).
19. Cammarata, A. Optimized design of a largeworkspace
2-DOF parallel robot for solar tracking
systems", Mechanism and Machine Theory, 83, pp.
175-186 (2015).
20. Chaparro-Rico, B. and Castillo-Castaneda, E. Design
of a 2DOF parallel mechanism to assist therapies for
knee rehabilitation", Ingeniera e Investigacion, 36,
pp. 98-104 (2016).
21. Kucuk, S. and Bingul, Z. Inverse kinematics solutions
for industrial robot manipulators with o set wrists",
Applied Mathematical Modelling, 38, pp. 1983-1999
(2014).
22. Zhao, J.-S., Zhou, K., and Feng, Z.-J. A theory of
degrees of freedom for mechanisms", Mechanism and
Machine Theory, 39, pp. 621-643 (2004).
23. Zhao, J.-S., Feng, Z.-J., and Dong, J.-X. Computation
of the con guration degree of freedom of a spatial
parallel mechanism by using reciprocal screw theory",
Mechanism and Machine Theory, 41, pp. 1486-1504
(2006).
24. Shmakov, S.L. A universal method of solving quartic
equations", International Journal of Pure and Applied
Mathematics, 71, pp. 251-259 (2011).
25. Lu, Y. Using CAD functionalities for the kinematics
analysis of spatial parallel manipulators with 3-, 4-,
5-, 6-linearly driven limbs", Mechanism and Machine
Theory, 39, pp. 41-60 (2004).
26. Lukanin, V. Inverse Kinematics, Forward Kinematics
and working space determination of 3DOF parallel
manipulator with SPR Joint Structure", Periodica
Polytechnica, 49, pp. 39-61 (2005).