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Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space

Published online by Cambridge University Press:  28 April 2015

Davood Naderi ,
Mehdi Tale-Masouleh  and
Payam Varshovi-Jaghargh
Davood Naderi*
Affiliation:
Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, Iran
Mehdi Tale-Masouleh
Affiliation:
Human and Robot Interaction Laboratory, Department of Mechatronics, Faculty of New Science and Technology, University of Tehran, Tehran, Iran
Payam Varshovi-Jaghargh
Affiliation:
Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, Iran
*
*Corresponding author. E-mail: d_naderi@basu.ac.ir
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Summary

In this paper, the forward kinematic analysis of 3-degree-of-freedom planar parallel robots with identical limb structures is presented. The proposed algorithm is based on Study's kinematic mapping (E. Study, “von den Bewegungen und Umlegungen,” Math. Ann.39, 441–565 (1891)), resultant method, and the Gröbner basis in seven-dimensional kinematic space. The obtained solution in seven-dimensional kinematic space of the forward kinematic problem is mapped into three-dimensional Euclidean space. An alternative solution of the forward kinematic problem is obtained using resultant method in three-dimensional Euclidean space, and the result is compared with the obtained mapping result from seven-dimensional kinematic space. Both approaches lead to the same maximum number of solutions: 2, 6, 6, 6, 2, 2, 2, 6, 2, and 2 for the forward kinematic problem of planar parallel robots; 3-RPR, 3-RPR, 3-RRR, 3-RRR, 3-RRP, 3-RPP, 3-RPP, 3-PRR, 3-PRR, and 3-PRP, respectively.

Keywords

Forward kinematic problem3-DoF planar parallel robotSeven-dimensional kinematic spaceGröbner basisStudy's parametersResultant method
Type
Articles
Information
Robotica , Volume 34 , Issue 11 , November 2016 , pp. 2610 - 2628
Copyright
Copyright © Cambridge University Press 2015 

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References

1. Merlet, J.-P., Parallel Robots (Springer, New York, NY, 2006).Google Scholar
2. Stewart, D., “A platform with six degrees of freedom,” Proc. Inst. Mech. Eng. 180, 371–386 (1965).Google Scholar
3. Huang, T., Whitehouse, D. J. and Wang, J., “The local dexterity, optimal architecture and design criteria of parallel machine tools,” CIRP Ann. Manuf. Technol. 47, 346–350 (1998).Google Scholar
4. Cleary, K. and Brooks, T., “Kinematic Analysis of a Novel 6-DOF Parallel Manipulator,” Proceedings of the 1993 IEEE International Conference on Robotics and Automation, Atlanta, 02–06 May 1993 (1993) pp. 708–713.Google Scholar
5. Agrawal, S. K., Desmier, G. and Li, S., “Fabrication and analysis of a novel 3-DOF parallel wrist mechanism,” J. Mech. Des. 117, 343 (1995).CrossRefGoogle Scholar
6. Gosselin, C. M. and St-Pierre, É., “Development and experimentation of a fast 3-DOF camera-orienting device,” Int. J. Robot. Res. 16, 619–630 (1997).Google Scholar
7. Albus, J., Bostelman, R. and Dagalakis, N., “The NIST robocrane,” J. Robot. Syst. 10, 709–724 (1993).CrossRefGoogle Scholar
8. Arai, T., Stoughton, R., Homma, K., Adachi, H., Nakamura, T. and Nakashima, K., “Development of a Parallel Link Manipulator,” Proceedings of the Fifth International Conference on Advanced Robotics, 1991: Robots in Unstructured Environments (91 ICAR), Pisa, Italy, 19–22 June 1991, (1991) pp. 839–844.Google Scholar
9. Brandt, G., Zimolong, A., Carrat, L., Merloz, P., Staudte, H.-W., Lavallee, S., Radermacher, K. and Rau, G., “CRIGOS: A compact robot for image-guided orthopedic surgery,” IEEE Trans. Inf. Technol. Biomed. 3, 252–260 (1999).Google Scholar
10. Liu, X.-J., Wang, J., Gao, F. and Wang, L.-P., “On the Design of 6-DOF Parallel Micro-Motion Manipulators,” Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems, Maui, HI, 29 Oct–03 Nov 2001, (2001) pp. 343–348.Google Scholar
11. Li, Y.-M. and Xu, Q., “Design and Analysis of a New 3-DOF Compliant Parallel Positioning Platform for Nanomanipulation,” Proceedings of the 5th IEEE Conference on Nanotechnology, 2005,” 11–15 July 2005, (2005) pp. 861–864.Google Scholar
12. Siciliano, B. and Khatib, O., Springer Handbook of Robotics (Springer, Berlin, Germany, 2008).Google Scholar
13. Kong, X. and Gosselin, C. M., “Type synthesis of 3-DOF spherical parallel manipulators based on screw theory,” J. Mech. Des. 126, 101 (2004).Google Scholar
14. Kong, X. and Gosselin, C. M., “Type synthesis of 4-DOF SP-equivalent parallel manipulators: A virtual chain approach,” Mech. Mach. Theory, 41, 1306–1319 (2006).Google Scholar
15. Innocenti, C., “Forward kinematics in polynomial form of the general Stewart platform,” J. Mech. Des. 123, 254 (2001).Google Scholar
16. Angeles, J., Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms (Springer, New York, NY, 2007).Google Scholar
17. Lee, T.-Y. and Shim, J.-K., “Forward kinematics of the general 6–6 Stewart platform using algebraic elimination,” Mech. Mach. Theory, 36, 1073–1085 (2001).Google Scholar
18. Study, E., “von den Bewegungen und Umlegungen,” Math. Ann. 39, 441–565 (1891).Google Scholar
19. Kalker-Kalkman, C., “An implementation of Buchbergers' algorithm with applications to robotics,” Mech. Mach. Theory 28, 523–537 (1993).Google Scholar
20. Husty, M. L., “An algorithm for solving the direct kinematics of general Stewart–Gough platforms,” Mech. Mach. Theory, 31, 365–379 (1996).Google Scholar
21. Wampler, C. W., “Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates,” Mech. Mach. Theory 31, 331–337 (1996).Google Scholar
22. Hayes, M. and Husty, M., “On the kinematic constraint surfaces of general three-legged planar robot platforms,” Mech. Mach. Theory 38, 379–394 (2003).Google Scholar
23. Brunnthaler, K., Schröcker, H.-P. and Husty, M., “A New Method for the Synthesis of Bennett Mechanisms,” Proceedings of CK2005, International Workshop on Computational Kinematics, Cassino, May 4–6 (2005).Google Scholar
24. Hayes, M., Zsombor-Murray, P. and Chen, C., “Unified kinematic analysis of general planar parallel manipulators,” J. Mech. Des. 126, 866–874 (2004).Google Scholar
25. Husty, M. L., Pfurner, M. and Schröcker, H.-P., “A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator,” Mech. Mach. Theory 42, 66–81 (2007).Google Scholar
26. Walter, D. R., Husty, M. L. and Pfurner, M., “The SNU 3-UPU Parallel Robot from a Theoretical Viewpoint,” Proceedings of the Second International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators (2008) pp. 1–8.Google Scholar
27. Masouleh, M. T., Gosselin, C., Husty, M. and Walter, D. R., “Forward kinematic problem of 5-RPUR parallel mechanisms (3T2R) with identical limb structures,” Mech. Mach. Theory 46, 945–959 (2011).Google Scholar
28. Gosselin, C. M. and Merlet, J.-P., “The direct kinematics of planar parallel manipulators: Special architectures and number of solutions,” Mech. Mach. Theory 29, 1083–1097 (1994).Google Scholar
29. Merlet, J.-P., “Direct Kinematics of Planar Parallel Manipulators,” Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, MN, 22–28 Apr 1996, (1996) pp. 3744–3749.Google Scholar
30. Kong, X. and Gosselin, C. M., Type Synthesis of Parallel Mechanisms (Springer, New York, NY, 2007).Google Scholar
31. Husty, M. L. and Schröcker, H.-P., Algebraic Geometry and Kinematics, Nonlinear Computational Geometry (Springer, New York, NY, 2010) pp. 85–107.Google Scholar
32. Craig, J. J., Introduction to Robotics: Mechanics and Control (Prentice Hall; 3 edition, August 6, 2004).Google Scholar
33. Cox, D. A., Little, J. B. and O'Shea, D., Using Algebraic Geometry (Springer, New York, NY, 1998).CrossRefGoogle Scholar
34. Bates, D. J., Hauenstein, J. D., Sommese, A. J. and Wampler, C. W. II, “Software for Numerical Algebraic Geometry: A Paradigm and Progress Towards Its Implementation,” In: Software for Algebraic Geometry, Editors: Stillman, Michael E., Takayama, Nobuki, Verschelde, Jan (Eds.), (Springer, New York, NY, 2008) pp. 1–14.Google Scholar
35. Wampler, C. W., “Bezout number calculations for multi-homogeneous polynomial systems,” Appl. Math. Comput. 51, 143–157 (1992).Google Scholar