Study of segmental scale synthesis algorithms for Procrustes curves in linkage mechanisms

Abstract

Accurate dimensional synthesis of four-bar linkages for reproducing complex closed paths (e.g. lemniscates) remains challenging due to the inherent phase misalignment and starting point ambiguity in trajectory comparison. Conventional Euclidean distance metrics fail to address these issues, while Procrustes analysis alone cannot resolve the cyclic starting point correspondence. To bridge this gap, this study proposes a novel integrated framework that synergistically combines Procrustes analysis with a curve segmentation algorithm. The primary objective is to achieve high-fidelity path synthesis by simultaneously ensuring optimal spatial alignment and correct temporal point matching. Within this framework, a curvature-sensitive mechanism first identifies the optimal starting point by minimizing a global Procrustes error index. Subsequently, an adaptive curve segmentation algorithm establishes precise point correspondence, effectively decoupling the problems of pose normalization and phase synchronization. A kinematic model is established, and a minimum-error objective function incorporating trajectory deviation and Grashof constraints is formulated. The proposed method is rigorously validated by synthesizing three distinct closed trajectories (a lemniscate, a droplet, and an ellipse) using three disparate metaheuristic optimizers (Multiverse Optimizer, Genetic Algorithm, and Artificial Hummingbird Algorithm). The consistent high-precision results across all algorithms demonstrate the robustness, generality, and optimizer-independence of the “Procrustes-curve segmentation” framework. This work not only provides a reliable new methodology for the path synthesis of linkage mechanisms but also establishes a theoretical basis for trajectory segmentation based on shape similarity.