基於橢圓傅立葉描述子之平面與球面四連桿機構運動合成

Abstract

在機構設計領域，連桿機構的尺度合成對於精確控制機械運動至關重要。本研究將聚焦於平面與球面四連桿機構的尺度合成。傳統的傅立葉描述子 (Fourier Descriptors, FD) 的複數坐標在描述空間曲線時存在限制。因此，我們引入橢圓傅立葉描述子 (Elliptical Fourier Descriptors, EFD) 作為新的形狀表示法，以提高描述的準確性和通用性。EFD通過本研究改良後的正規化係數取得幾何不變性，使用傅立葉功率分析自動選擇諧波數，並可擴展至空間曲線。本研究採用圖集與差分進化法進行數值最佳化，對機構尺度進行最佳化設計。通過各種耦桿曲線和合成條件的測試，我們實現了更精確的路徑生成與剛體導引。以平面與球面四連桿機構為例，多種案例分析表明，EFD能有效解決開放曲線與空間曲線的描述問題，驗證了方法的有效性。研究創新點在於首次將EFD應用於機構合成領域，解決了FD在描述複雜曲線時的局限性。此外，本研究還提出了一種新的開放曲線描述方法，其精確性和效率相較於過往方法有顯著提升。

In mechanism design, dimensional synthesis of linkages is crucial for precisely controlling mechanical motion. This study focuses on the dimensional synthesis of planar and spherical four-bar linkages. The complex coordinates of conventional Fourier Descriptors (FD) have limitations in describing spatial curves. Therefore, we introduce Elliptical Fourier Descriptors (EFD) as a new shape representation to enhance accuracy and generality. EFD achieves geometric invariance through improved normalization coefficients in this study, automatically selects the number of harmonics using Fourier power analysis, and can be extended to spatial curves. Numerical optimization of linkage dimensions is conducted using atlas-based and differential evolution methods. Through tests on various coupler curves and synthesis conditions, we verified path generation and rigid body guidance can be achieved efficiently and precisely through EFD. Using planar and spherical four-bar linkages as examples, multiple case studies demonstrate that EFD effectively addresses the description issues of open and spatial curves, validating the method’s effectiveness. The innovation of this research lies in the first application of EFD in the field of linkage synthesis, overcoming the limitations of FD in describing complex curves. Additionally, a new method for describing open curves is proposed in this study, which significantly improves accuracy and efficiency compared to previous methods.