Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-23T12:17:51.704Z Has data issue: false hasContentIssue false
  • English
  • Français

A geometric approach for singularity analysis of 3-DOF planar parallel manipulators using Grassmann–Cayley algebra

Published online by Cambridge University Press:  17 August 2015

Kefei Wen ,
TaeWon Seo  and
Jeh Won Lee
Kefei Wen
Affiliation:
School of Mechanical Engineering, Yeungnam University, Gyeongsan 712–749, Republic of Korea
TaeWon Seo*
Affiliation:
School of Mechanical Engineering, Yeungnam University, Gyeongsan 712–749, Republic of Korea
Jeh Won Lee*
Affiliation:
School of Mechanical Engineering, Yeungnam University, Gyeongsan 712–749, Republic of Korea
*
*Corresponding author. E-mail: taewon_seo@yu.ac.kr, jwlee@yu.ac.kr
*Corresponding author. E-mail: taewon_seo@yu.ac.kr, jwlee@yu.ac.kr
Get access Rights & Permissions [Opens in a new window]

Summary

Singular configurations of parallel manipulators (PMs) are special poses in which the manipulators cannot maintain their inherent infinite rigidity. These configurations are very important because they prevent the manipulator from being controlled properly, or the manipulator could be damaged. A geometric approach is introduced to identify singular conditions of planar parallel manipulators (PPMs) in this paper. The approach is based on screw theory, Grassmann–Cayley Algebra (GCA), and the static Jacobian matrix. The static Jacobian can be obtained more easily than the kinematic ones in PPMs. The Jacobian is expressed and analyzed by the join and meet operations of GCA. The singular configurations can be divided into three classes. This approach is applied to ten types of common PPMs consisting of three identical legs with one actuated joint and two passive joints.

Keywords

SingularityPlanar parallel manipulatorsGrassmann–Cayley AlgebraScrew theoryPlücker coordinates
Type
Articles
Information
Robotica , Volume 35 , Issue 3 , March 2017 , pp. 511 - 520
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Hunt, K. H., “Structural kinematics of in-parallel actuated robot arms,” J. Mech. Transm.-T. ASME, 105 (4), 705–712 (1983).Google Scholar
2. Merlet, J. P., Parallel Robot (Dordrecht, The Netherlands, Springer, 2nd ed. 2006).Google Scholar
3. Dash, A. K., Chen, I., Yeo, S. H. and Yang, G., “Singularity-Free path Planning of Parallel Manipulators using Clustering Algorithm and Line Geometry,” Proceedings of the 2003 IEEE International Conference on Robotics and Automation, Taipei, Taiwan (Sept. 14–19, 2003).Google Scholar
4. Chung, Y. H., Choo, J. H. and Lee, J. W., “SenSation: A New 2-DOF Parallel Mechanism for Haptic Device,” Proceedings 2nd Workshop on Computational Kinematics, Seoul, South Korea (May 20–22, 2001) pp. 45–56.Google Scholar
5. Liu, G. F., Wu, Y. L., Wu, X. Z., Kuen, Y. Y. and Li, Z. X., “Analysis and Control of Redundant Parallel Manipulators,” Proceedings of the 2001 IEEE International Conference on Robotics and Automation, Seoul, Korea (May 21–26, 2001).Google Scholar
6. Saglia, J. A., Dai, J. S. and Caldwell, D. G., “Geometry and kinematic analysis of a redundantly actuated parallel mechanism that eliminates singularities and improves dexterity,” J. Mech. Des. 130 (12), 124501 (2008).CrossRefGoogle Scholar
7. Ebrahimi, I., Carretero, J. A. and Boudreau, R., “3-PRRR redundant planar parallel manipulator: Inverse displacement, workspace, and singularity analysis,” Mech. Mach. Theory 42 (8), 1007–1016 (2007).Google Scholar
8. Gosselin, C. and Angeles, J., “Singularity analysis of closed-loop kinematic chains,” IEEE Trans. Robot. Autom. 6 (3), 281–290 (1990).CrossRefGoogle Scholar
9. Bonev, I. A., Zlatanov, D. and Gosselin, C. M., “Singularity analysis of 3-DOF planar parallel mechanisms via screw theory,” J. Mech. Des. 125, 573–581 (Sept. 2003).Google Scholar
10. Davidson, J. K. and Hunt, K. H., Robots and Screw Theory (Oxford University Press, Oxford, 2004).Google Scholar
11. Wolf, A., Ottaviano, E., Shoham, M. and Ceccarelli, M., “Application of line geometry and linear complex approximation to singularity analysis of the 3-DOF CaPaMan parallel manipulator,” Mech. Mach. Theory 39, 75–95 (2004).Google Scholar
12. Degani, A. and Wolf, A., “Graphical Singularity Analysis of Planar Parallel Manipulators,” Proceedings of the 2006 IEEE International Conference on Robotics and Automation, Orlando, Florida (May, 2006).Google Scholar
13. Huang, Z. and Li, Q. C., “General methodology for type synthesis of symmetrical lower-mobility parallel manipulators and several novel manipulators,” Int. J. Robot. Res. 21 (2), 131–145 (2002).CrossRefGoogle Scholar
14. Kong, X. and Gosselin, C. M., “Type synthesis of 3-DOF spherical parallel manipulators based on screw theory,” J. Mech. Des. 126 (1), (2004).Google Scholar
15. Yu, J., Li, S., Su, H. and Culpepper, M. L., “Screw theory based methodology for the determinisitic type synthesis of flexure mechanisms,” J. Mech. Robot. 3 (3), 031008 (2011).CrossRefGoogle Scholar
16. Ben-Horin, P. and Shoham, M., “Singularity analysis of a class of parallel robots based on Grassmann–Cayley algebra,” Mech. Mach. Theory 41 (8), 958–970 (2006).Google Scholar
17. Ben-Horin, P. and Shoham, M., “Singularity condition of six-degree-of-freedom three-legged parallel robots based on Grassmann–Cayley algebra,” IEEE Trans. Robot. 22 (4), 577–590 (2006).CrossRefGoogle Scholar
18. Staffetti, E., “Kinestatic analysis of robot manipulators using the Grassmann–Cayley algebra,” IEEE Trans. Robot. Autom. 20 (2), 200–210 (2004).Google Scholar
19. Amine, S., Masouleh, M. T., Caro, S., Wenger, P. and Gosselin, C., “Singularity analysis of 3T2R parallel mechanisms using Grassmann–Cayley algebra and Grassmann geometry,” Mech. Mach. Theory 52, 326–340 (2012).Google Scholar
20. Kanaan, D., Wenger, P., Caro, S. and Chablat, D., “Singularity analysis of lower-mobility parallel manipulators using Grassmann-Cayley Algebra,” IEEE Trans. Robot. 25 (5), 995–1004 (2009).Google Scholar
21. Huang, Z., Zhao, Y. S. and Zhao, T. S., Advanced Spatial Mechanism (Beijing, P. R. China, Higher Education Press, 2006), ch, 1.Google Scholar
22. White, N. L., “Grassmann–Cayley Algebra and Robotics Applications,” In: Handbook of Geometric Computing (Corrochano, E. B., ed.) (Berlin Heidelberg, Springer-Verlag, 2005) pp. 629–656.Google Scholar
23. Staffetti, E., “Kinestatic analysis of robot manipulators using the Grassmann–Cayley algebra,” IEEE Trans. Robot. Autom. 20, 200–210 (Apr. 2004).Google Scholar
24. Doubilet, P., Rota, G.-C. and Stein, J., “On the foundations of combinatorial theory: IX combinatorial methods in invariant theory,” Stud. Appl. Math. 53 (3), 185–216 (1974).Google Scholar
25. Ben-Horin, P. and Shoham, M., “Application of Grassmann–Cayley algebra to geometrical interpretation of parallel robot singularities,” Int. J. Robot. Res. 28 (1), 127–141 (2009).Google Scholar
26. Choi, J. H., Seo, T. W. and Lee, J. W., “Singularity analysis of a planar parallel mechanism with revolute joints based on a geometric approach,” Int. J. Precis. Eng. Manuf. 14 (8), 1369–1375 (2013).Google Scholar
27. White, N. L., “The bracket of 2-extensors,” Congressus Numerantium, 40, 419–428 (1983).Google Scholar
28. White, N. L. and McMillan, T., Cayley Factorization, IMA Preprint Series # 371, University of Minenesota, 206 Church St, S.E., Minneapolis, MN, USA (1987).Google Scholar