Skip to main content Accessibility help
×
Home

Singularity analysis of the H4 robot using Grassmann–Cayley algebra

  • Semaan Amine (a1), Stéphane Caro (a1), Philippe Wenger (a1) and Daniel Kanaan (a1)

Summary

This paper extends a recently proposed singularity analysis method to lower-mobility parallel manipulators having an articulated nacelle. Using screw theory, a twist graph is introduced in order to simplify the constraint analysis of such manipulators. Then, a wrench graph is obtained in order to represent some points at infinity on the Plücker lines of the Jacobian matrix. Using Grassmann–Cayley algebra, the rank deficiency of the Jacobian matrix amounts to the vanishing condition of the superbracket. Accordingly, the parallel singularities are expressed in three different forms involving superbrackets, meet and join operators, and vector cross and dot products, respectively. The approach is explained through the singularity analysis of the H4 robot. All the parallel singularity conditions of this robot are enumerated and the motions associated with these singularities are characterized.

Copyright

Corresponding author

*Corresponding author. E-mail: stephane.caro@irccyn.ec-nantes.fr

References

Hide All
1.Merlet, J. P., “Singular configurations of parallel manipulators and Grassmann geometry,” Int. J. Robot. Res. 8 (5), 4556 (1989).
2.Angeles, J., Caro, S. and Morozov, A. W. K., “The design and prototyping of an innovative Schonflies motion generator,” IMechE Part C, J. Mech. Eng. Sci., Special Issue: Kinematics, Kinematic Geometry and their applications 220 (C7), 935944 (July 2006).
3.Merlet, J. P., “A formal–numerical approach for robust in-workspace singularity detection,” IEEE Trans. Robot. 23 (3), 393402 (June 2007).
4.Li, Q. and Huang, Z., “Mobility Analysis of Lower-Mobility Parallel Manipulators Based on Screw Theory,” Proceedings of the IEEE International Conference on Robotics and Automation, Taipei, Taiwan (Sep. 14–19, 2003) pp. 11791184.
5.Fang, Y. and Tsai, L. W., “Structure synthesis of a class of 4-DoF and 5-DoF parallel manipulators with identical limb structures,” Int. J. Robot. Res. 21 (9), 799810 (2002).
6.Zlatanov, D., Fenton, R. G. and Benhabib, B., “Singularity Analysis of Mechanisms and Robots via a Velocity–Equation Model of the Instantaneous Kinematics,” Proceedings of the IEEE International Conference on Robotics and Automation, San Diego, CA, USA (May 8–13, 1994) pp. 986991.
7.Ball, R. S., A Treatise on the Theory of Screws (Cambridge University Press, Cambridge, MA, 1900).
8.Waldron, K. J., ‘The Mobility of Linkages’. Ph.D. Thesis (Stanford, CA: Stanford University, 1969).
9.Hunt, K. H., Kinematic Geometry of Mechanisms (Clarendon Press, Oxford, 1978).
10.Kong, X. and Gosselin, C., Type Synthesis of Parallel Mechanisms (Springer, Heidelberg, 2007) vol. 33.
11.Amine, S., Kanaan, D., Caro, S. and Wenger, P., “Constraint and Singularity Analysis of Lower-Mobility Parallel Manipulators with Parallelogram Joints”. Proceedings of the ASME 2010 International Design Engineering Technical Conferences, no. 28483 in DETC2010, Montreal, Quebec, Canada (Aug. 15–18, 2010).
12.Joshi, S. A. and Tsai, L. W., “Jacobian analysis of limited-DOF parallel manipulators,” ASME J. Mech. Des. 124 (2), 254258 (June 2002).
13.Zlatanov, D., Bonev, I. and Gosselin, C. M., “Constraint Singularities of Parallel Mechanisms,” Proceedings of the IEEE International Conference on Robotics and Automation, Washington, DC, USA (May 11–15, 2002) pp. 496502.
14.Ben-Horin, P. and Shoham, M., “Singularity analysis of parallel robots based on Grassmann–Cayley algebra,” Proceedings of the International Workshop on Computational Kinematics, Cassino, Italy (May 4–6, 2005).
15.Ben-Horin, P. and Shoham, M., “Singularity analysis of a class of parallel robots based on Grassmann–Cayley algebra,” Mech. Mach. Theory 41 (8), 958970 (2006).
16.Ben-Horin, P. and Shoham, M., “Application of Grassmann–Cayley algebra to geometrical interpretation of parallel robot singularities,” Int. J. Robot. Res. 28 (1), 127141 (2009).
17.Kanaan, D., Wenger, P., Caro, S. and Chablat, D., “Singularity analysis of lower-mobility parallel manipulators using Grassmann–Cayley algebra,” IEEE Trans. Robot. 25, 9951004 (2009).
18.Mohamed, M. and Duffy, J., “A direct determination of the instantaneous kinematics of fully parallel robot manipulators,” ASME J. Mech. Transm. Autom. Des. 107 (2), 226229 (1985).
19.White, N. L., “The bracket ring of a combinatorial geometry. I,” Trans. Am. Math. Soc. 202, 7995 (1975).
20.White, N. L., “The bracket of 2-extensors,” Congressus Numerantium 40, 419428 (1983).
21.White, N. L., Grassmann–Cayley Algebra and Robotics Applications (Handbook of Geometric Computing, 2005) vol. VIII. Springer, Berlin Heidelberg.
22.McMillan, T., Invariants of Antisymmetric Tensors Ph.D. Thesis (University of Florida, Gainesville, 1990).
23.Pierrot, F. and Company, O., “H4: A New Family of 4-dof Parallel Robots,” Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Atlanta, Georgia, USA (Sep. 19–22, 1999) pp. 508513.
24.Pierrot, F., Marquet, F., Company, O. and Gil, T.. “H4 Parallel Robot: Modeling, Design and Preliminary Experiments,” Proceedings of the IEEE International Conference on Robotics and Automation, Seoul, Korea (May 21–26, 2001) vol. 4, pp. 32563261.
25.Gregorio, R. D., “Determination of singularities in Delta-like manipulators,” Int. J. Robot. Res. 23 (1), 8996 (January 2004).
26.Zhao, T. S., Dai, J. S. and Huang, Z., “Geometric analysis of overconstrained parallel manipulators with three and four degrees of freedom,” JSME Int. J. Ser. C: Mech. Syst. Mach. Elem. Manuf. 45 (3), 730740 (2002).
27.Angeles, J., “The qualitative synthesis of parallel manipulators,” ASME J. Mech. Des. 126 (4), 617624 (July 2004).
28.Caro, S., Khan, W. A., Pasini, D. and Angeles, J., “The rule-based conceptual design of the architecture of serial Schönflies-motion generators,” Mech. Mach. Theory 45 (2), 251260 (2010).
29.Wolf, A. and Shoham, M., “Investigation of parallel manipulators using linear complex approximation,” ASME J. Mech. Des. 125, 564572 (2003).
30.Krut, S., Contribution L'étude des Robots Parallèles Légers, 3T-1R et 3T-2R, à Forts Débattements Angulaires Ph.D. Thesis (Montpellier, France: Université Montpellier II, 2003).
31.Wu, J., Yin, Z. and Xiong, Y., “Singularity analysis of a novel 4-dof parallel manipulator H4,” Int. J. Adv. Manuf. Technol. 29, 794802 (2006).
32.Amine, S., Kanaan, D., Caro, S. and Wenger, P., “Singularity Analysis of Lower-Mobility Parallel Robots with an Articulated Nacelle,” In: Advances in Robot Kinematics: Motion in Man and Machine 2010, Part 5, (Springer, Berlin, 2010) pp. 273282.
33.Nabat, V., Rodriguez, M. de la O, Company, O., Krut, S. and Pierrot, F.. “Par4: Very High Speed Parallel Robot for Pick-and-Place”. Proceedings of the Intelligent Robots and Systems, IEEE/RSJ International Conference, Edmonton, Alberta, Canada (Aug. 2005) pp. 553558.

Keywords

Related content

Powered by UNSILO

Singularity analysis of the H4 robot using Grassmann–Cayley algebra

  • Semaan Amine (a1), Stéphane Caro (a1), Philippe Wenger (a1) and Daniel Kanaan (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.