Abstract
Singularity analysis of fully-parallel manipulators (FPMs) produced a wide literature that tried to overcome the difficulty of algebraically calculating the determinant of general FPM’s Jacobian. An early work of this author addressed this problem by using Laplace expansion, and proposed an analytic expression of general FPM’s singularity locus which contains ten terms easy to compute and geometrically interpret. Such an expression is exploited here to classify the singularity loci of all the m-n FPM architectures.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
S and P stand for spherical pair and prismatic pair, respectively, and the underscore indicates the actuated pair.
- 2.
The instantaneous input-output relationship of the general FPM relates platform’s twist (output) to leg-lengths’ rates (input). It is a linear and homogeneous mapping which contains two 6×6 matrices, one, here referred to as left-Jacobian, multiplies platform’s twist and the other multiplies the 6-tuple collecting leg-lengths’ rates [4].
- 3.
In the platform (base), a multiple spherical pair with multiplicity greater than three allows the redistribution of a single transmitted force along more than three directions, what brings to an indeterminate static problem with an infinite number of solutions.
- 4.
The 6-tuple \(\$_{i} = ({\mathbf{a}}_{i}^{T},\mathbf{b}_{i}^{T})^{T}\) is the screw of the ith leg axis, and identifies the location of this axis in the space.
References
Conconi, M., Carricato, M.: A new assessment of singularities of parallel kinematic chains. IEEE Trans. Robot. 25(4), 757–770 (2009)
Innocenti, C., Parenti-Castelli, V.: Exhaustive enumeration of fully-parallel kinematic chains. In: Proc. of the 1994 ASME Int. Winter Annual Meeting, Chicago, USA. DSC-Vol. 55-2, pp. 1135–1141 (1994)
St-Onge, B.M., Gosselin, C.M.: Singularity analysis and representation of the general Gough–Stewart platform. Int. J. Robot. Res. 19(3), 271–288 (2000)
Di Gregorio, R.: Singularity-locus expression of a class of parallel mechanisms. Robotica 20, 323–328 (2002)
Li, H., Gosselin, C.M., Richard, M.J., St-Onge, B.M.: Analytic form of the six-dimensional singularity locus of the general Gough–Stewart platform. ASME J. Mech. Des. 128(1), 279–287 (2006)
Ben-Horin, P., Shoham, M.: Singularity analysis of a class of parallel robots based on Grassmann–Cayley algebra. Mech. Mach. Theory 41(8), 958–970 (2006)
Eberly, D.: The Laplace expansion theorem: computing the determinants and inverses of matrices. Geometric Tools, LLC (2008). http://www.geometrictools.com/Documentation/LaplaceExpansionTheorem.pdf
Alberich-Carramiñana, M., Thomas, F., Torras, C.: Flagged parallel manipulators. IEEE Trans. Robot. 23(5), 1013–1023 (2007)
Faugère, J.C., Lazard, D.: Combinatorial classes of parallel manipulators. Mech. Mach. Theory 30(6), 765–776 (1995)
Di Gregorio, R.: Analytic formulation of the 6-3 fully-parallel manipulator’s singularity determination. Robotica 19, 663–667 (2001)
Di Gregorio, R.: Singularity locus of 6-4 fully-parallel manipulators. In: J. Lenarčič, M.M. Stanišić (eds.) Advances in Robot Kinematics: Motion in Man and Machine, pp. 437–445. Springer (2010)
Acknowledgements
This work has been developed at the Laboratory of Advanced Mechanics (MECH-LAV) of Ferrara Technopole, supported by UNIFE funds and by Regione Emilia Romagna (District Councillorship for Productive Assets, Economic Development, Telematic Plan) POR-FESR 2007–2013, Attività I.1.1.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Di Gregorio, R. (2012). Classification of the Singularity Loci of m-n Fully-Parallel Manipulators. In: Lenarcic, J., Husty, M. (eds) Latest Advances in Robot Kinematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4620-6_6
Download citation
DOI: https://doi.org/10.1007/978-94-007-4620-6_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-4619-0
Online ISBN: 978-94-007-4620-6
eBook Packages: EngineeringEngineering (R0)