Frequency-Domain-Based Function Generation Synthesis for Planar Multi-Loop Mechanisms

Abstract This paper presents a novel frequency-domain-based approach for function generation synthesis in planar multi-loop mechanisms featuring two independent kinematic loops. To our knowledge, this is the first extension of frequency-domainbased synthesis to planar multi-loop mechanisms. Leveraging the Orthogonality of Trigonometric Functions, the system of equations is derived using a frequency-domain physical interpretation, and reformulated into an optimization problem that balances constraint accuracy and numerical stability. This formulation enables the direct acquisition of multiple, defect-free solutions. Unlike time-domain methods, the proposed approach greatly reduces circuit and branch defects without requiring complex singularity or defect analysis. In contrast to existing frequency-domain methods limited to four-bar mechanisms with full-cycle crank inputs, it supports both revolute and prismatic joints and allows motion profiles such as trapezoidal or S-curve planning. The method also avoids explicit trajectory specification for each loop by treating the Fourier coefficients of the shared passive joint as unknowns, while using linear constraints and coefficient symmetry to reduce complexity. Its effectiveness is demonstrated through two novel multi-loop finger mechanisms, the Watt II mechanism, and a comparison with a time-domain method, highlighting its practicality and broad applicability.