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Closeness to singularities of manipulators based on geometric average normalized volume spanned by weighted screws

Published online by Cambridge University Press:  09 June 2016

Wanghui Bu
Wanghui Bu*
Affiliation:
School of Mechanical Engineering, Tongji University, Shanghai 200092, P. R. China
*
*Corresponding author. E-mail: [email protected]
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Summary

In order to prevent robot manipulators from reaching singularities, the “distance” from the current configuration to a singular configuration should be measured. This paper presents a novel metric based on geometric average normalized volume spanned by weighted screws to measure closeness to singularities for both serial and parallel manipulators. The weighted screws are proposed to reinterpret the physical meaning of twists and wrenches, so the problem of inconsistent dimensions in the dot product of screws has been eliminated. Compared with other existing methods, the proposed metric can obtain an identical result for similar manipulators with different sizes. Furthermore, the metric is independent of the selection of base screws, which is very suitable for the overconstrained or lower mobility parallel manipulator whose base screws are not uniquely definite. Besides, the geometrical meaning of the metric is related to the dimensionless volume of a high dimensional polyhedron, and hence the metric is insensitive to screw magnitude.

Keywords

Robot manipulatorsSingularityMetricScrew theoryOverconstrained mechanisms
Type
Articles
Information
Robotica , Volume 35 , Issue 7 , July 2017 , pp. 1616 - 1626
Copyright
Copyright © Cambridge University Press 2016 

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References

1. Selig, J. M., Geometric Fundamentals of Robotics (Springer, New York, US, 2004).Google Scholar
2. Murray, R. M., Li, Z. and Sastry, S. S., A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, US, 1994).Google Scholar
3. Müller, A., “On the manifold property of the set of singularities of kinematic mappings: Genericity conditions,” ASME J. Mech. Robot. 4 (1), 011006, 1–9 (2012).Google Scholar
4. Jiang, Q. and Gosselin, C. M., “Determination of the maximal singularity-free orientation workspace for the gough-stewart platform,” Mech. Mach. Theory 44 (6), 1281–1293 (2009).CrossRefGoogle Scholar
5. Kong, X., “Forward displacement analysis and singularity analysis of a special 2-DOF 5R spherical parallel manipulator,” ASME J. Mech. Robot. 3 (2), 024501, 1–6 (2011).Google Scholar
6. Liu, G. F., Lou, Y. J. and Li, Z. X., “Singularities of parallel manipulators: A geometric treatment,” IEEE Trans. Robot. Autom. 19 (4), 579–594 (2003).Google Scholar
7. Park, F. C. and Kim, J. W., “Singularity analysis of closed kinematic chains,” ASME J. Mech. Des. 121 (1), 32–38 (1999).Google Scholar
8. Collins, C. L. and McCarthy, J. M., “The quartic singularity surface of planar platforms in the clifford algebra of the projective plane,” Mech. Mach. Theory 33 (7), 931–944 (1998).CrossRefGoogle Scholar
9. Zlatanov, D., Bonev, I. A. and Gosselin, C. M., “Constraint Singularities of Parallel Mechanisms,” IEEE International Conference on Robotics and Automation, New York, USA (2002) pp. 496–502.Google Scholar
10. Qin, Y., Dai, J. S. and Gogu, G., “Multi-furcation in a derivative queer-square mechanism,” Mech. Mach. Theory 81 (11), 36–53 (2014).Google Scholar
11. Parsa, S. S., Boudreau, R. and Carretero, J. A., “Reconfigurable mass parameters to cross direct kinematic singularities in parallel manipulators,” Mech. Mach. Theory 85 (3), 53–63 (2015).CrossRefGoogle Scholar
12. Schadlbauer, J., Walter, D. R. and Husty, M. L., “The 3-RPS parallel manipulator from an agebraic viewpoint,” Mech. Mach. Theory 75 (5), 161–176 (2014).Google Scholar
13. Zlatanov, D., Fenton, R. G. and Benhabib, B., “Identification and classification of the singular configurations of mechanisms,” Mech. Mach. Theory 33 (6), 743–760 (1998).Google Scholar
14. Conconi, M. and Carricato, M., “A new assessment of singularities of parallel kinematic chains,” IEEE Trans. Robot. 25 (4), 757–770 (2009).Google Scholar
15. Blanc, D. and Shvalb, N., “Generic singular configurations of linages,” Topology Appl. 159, 877–890 (2012).Google Scholar
16. Yoshikawa, T., “Manipulability of robotic mechanisms,” Int. J. Robot. Res. 4 (2), 3–9 (1985).Google Scholar
17. Merlet, J. P., “Jacobian, manipulability, condition number, and accuracy of parallel robots,” ASME J. Mech. Des. 128 (1), 199–206 (2006).CrossRefGoogle Scholar
18. Doty, K. L., Melchiorri, C., Schwartz, E. M. and Bonivento, C., “Robot manipulability,” IEEE Trans. Robot. Autom. 11 (3), 462–468 (1995).CrossRefGoogle Scholar
19. Hubert, J. and Merlet, J. P., “Static of parallel manipulators and closeness to singularity,” ASME J. Mech. Robot. 1 (1), 011011, 1–6 (2009).CrossRefGoogle Scholar
20. Lee, J., Duffy, J. and Keler, M., “The optimum quality index for the stability of in-parallel planar platform devices,” ASME J. Mech. Des. 121 (3), 15–20 (1999).CrossRefGoogle Scholar
21. Lipkin, H. and Duffy, J., “The elliptic polarity of screws,” ASME J. Mech. Des. 107 (9), 377–386 (1985).Google Scholar
22. Voglewede, P. A. and Ebert-Uphoff, I., “Overarching framework for measuring closeness to singularities of parallel manipulators,” IEEE Trans. Robot. 21 (6), 1037–1045 (2005).CrossRefGoogle Scholar
23. Hartley, D. M. and Kerr, D. R., “Invariant measures of the closeness to linear dependence of six lines or screws,” IMechE Part C: J. Mech. Eng. Sci. 215 (10), 1145–1151 (2001).Google Scholar
24. Kerr, D. R. and Hartley, D. M., “Invariant measures of closeness to linear dependency of screw systems,” IMechE Part C: J. Mech. Eng. Sci. 220 (7), 1033–1043 (2006).Google Scholar
25. Liu, X. J., Wu, C. and Wang, J., “A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators,” ASME J. Mech. Robot. 4 (4), 041001, 1–10 (2012).Google Scholar
26. Liu, X. J., Chen, X. and Nahon, M., “Motion/force constrainability analysis of lower-mobility parallel manipulators,” ASME J. Mech. Robot. 6, 031006, 1–9 (2014).CrossRefGoogle Scholar
27. McCarthy, J. M., “Discussion: ‘The elliptic polarity of screws’,” ASME J. Mech. Des. 107 (9), 386–387 (1985).Google Scholar
28. Huang, T., Wang, M., Yang, S., Sun, T., Chetwynd, D. G. and Xie, F., “Force/motion transmissibility analysis of six degree of freedom parallel mechanisms,” ASME J. Mech. Robot. 6, 031010, 1–5 (2014).Google Scholar
29. Bu, W., “Closeness to singularities of robotic manipulators measured by characteristic angles,” Robotica (2014). DOI: 10.1017/S0263574714002823.CrossRefGoogle Scholar
30. Angeles, J., “The design of isotropic manipulator architectures in the presence of redundancies,” Int. J. Robot. Res. 11 (3), 196–201 (1992).Google Scholar
31. Angeles, J., “Is there a characteristic length of a rigid-body displacement?,” Mech. Mach. Theory 41 (8), 884–896 (2006).Google Scholar
32. Joshi, S. A. and Tsai, L. W., “Jacobian analysis of limited-DOF parallel manipulators,” ASME J. Mech. Des. 124 (6), 254–258 (2002).Google Scholar
33. Hong, M. B. and Choi, Y. J., “Formulation of unique form of screw based jacobian for lower mobility parallel manipulators,” ASME J. Mech. Robot. 3 (1), 011002, 1–6 (2011).Google Scholar
34. Zhao, J. S., Feng, Z. J. and Dong, J. X., “Computation of the configuration degree of freedom of a spatial parallel mechanism by using reciprocal screw theory,” Mech. Mach. Theory 41 (12), 1486–1504 (2006).Google Scholar
35. Park, F. C., “Distance metrics on the rigid-body motions with applications to mechanism design,” ASME J. Mech. Des. 117 (1), 48–54 (1995).Google Scholar