Iterative Synthesis of Initial Kinematic Chains with Rotational Pairs Based on Stepwise Minimization of Geometric Deviations

This paper addresses the problem of synthesizing four-link initial kinematic chains with rotational pairs for designing flat lever mechanisms with specified motion laws of the input and output links. The proposed method formulates the synthesis problem as an optimization process involving three moving planes Q, Q1 and Q2. Each plane performs distinct motions: Q is fixed, Q1 rotates around a point A, and Q2 undergoes plane-parallel motion. The goal is to determine the optimal position of point A∈Q, B∈Q1 and C∈Q2 such that the distance between points B and C remains nearly constant across multiple configurations. The synthesis problem is reduced to a sequence of linear systems of equations corresponding to three iterative minimization stages: Determining the fixed-point A and the radius R for given points B and C; Determining the point B and radius R for a given point C; Determining the point C and radius R for a given point B. The algorithm uses a hierarchical optimization process to iteratively refine the positions of A, B, and C until convergence is achieved. The convergence criteria are defined by the small changes in the positions of these points and the radius R between iterations. The proposed method avoids solving a highly nonlinear system directly by decomposing the problem into simpler linear systems, ensuring computational efficiency and robustness. The solution provides the desired positions of the points, forming an open kinematic chain ABCD with three degrees of freedom. The approach is applicable to the synthesis of complex planar mechanisms with unrestricted parameters, making it a valuable tool for mechanism design.