Abstract
The left and right dual Moore–Penrose generalized inverses are the subject of this paper. It is shown that, contrary to the real case, these inverses are not unique, those with minimum Frobenius norm being obtained. Their application in kinematic synthesis is discussed. It is shown that, in the case of function-generating RCCC linkages, the left dual generalized inverse leads to a linkage that meets the prescribed input-output relations with both a least-square error and a minimum size. The study concludes with the synthesis of a linkage that approximates a homokinetic transmission between shafts with skew, orthogonal axes.
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Notes
- 1.
Actually, the authors do not stress the difference between the right and the left generalized inverses; they represent both with the same symbol, \((\; \cdot\; )^{+}\).
- 2.
If e and e o denote the primal and dual parts of \(\hat{\mathbf{e}}\), then \(\|\hat{\mathbf {e}}\|^{2} = \|\mathbf{e}\|^{2} +\varepsilon2\mathbf{e}^{T} \mathbf{e}_{o}\).
- 3.
The new variable u i is introduced with the purpose of avoiding double subscripts.
- 4.
This high number was used with the purpose of bringing the optimum design error e 0 as close as possible to the structural error, which measures the actual deviation of the synthesized output angle from its prescribed value, as per the results reported in [14].
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Acknowledgements
The support provided by a James McGill Professorship is dutifully acknowledged. Partial support received from the NSERC (Canada’s Natural Sciences and Engineering Research Council) Discovery Grant 4532-2008 is equally acknowledged.
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Angeles, J. (2012). The Dual Generalized Inverses and Their Applications in Kinematic Synthesis. In: Lenarcic, J., Husty, M. (eds) Latest Advances in Robot Kinematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4620-6_1
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DOI: https://doi.org/10.1007/978-94-007-4620-6_1
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