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.
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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.
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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.
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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
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