Rank-deficiencies and ill-conditioning of the synthesis matrix in the optimization of function-generating linkages are often caused by an improper selection of the input-output data points given by the {ψi,ϕi}1m pairs, where ψ and ϕ denote the input and output values, respectively. Ten basic cases of rank-deficiencies in the synthesis matrix are discussed in this paper, the associated curves, termed singularity curves, being plotted in the ψ-ϕ plane. Measures to remedy the ill-conditioning that arises in the optimization procedure and means to find the best-conditioned synthesis matrices by minimizing their condition number are also proposed.

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