Continuous approximate synthesis of planar function-generators minimising the design error

It has been observed in the literature that as the cardinality of the prescribed discrete input–output data set increases, the corresponding four-bar linkages that minimise the Euclidean norm of the design and structural errors tend to converge to the same linkage. The important implication is that minimising the Euclidean norm, or any p-norm, of the structural error, which leads to a nonlinear least-squares problem requiring iterative solutions, can be accomplished implicitly by minimising that of the design error, which leads to a linear least-squares problem that can be solved directly. Apropos, the goal of this paper is to take the first step towards proving that as the cardinality of the data set tends towards infinity the observation is indeed true. In this paper we will integrate the synthesis equations in the range between minimum and maximum input values, thereby reposing the discrete approximate synthesis problem as a continuous one. Moreover, we will prove that a lower bound of the Euclidean norm, and indeed of any p-norm, of the design error for planar RRRR function-generating linkages exists and is attained with continuous approximate synthesis.