Higher-order kinematic analysis and output port synthesis of compliant four-bar linkages via epicyclic pseudo-rigid models

This work presents a higher-order kinematic formulation for planar compliant mechanisms based on epicyclic pseudo-rigid models of flexible hinges. A compliant four-bar linkage is transformed into an equivalent epicyclic pseudo-rigid four-bar linkage (E-PR4B), for which position, velocity, acceleration, and jerk equations are derived analytically. The formulation determines instantaneous poles, Bresse circles, polodes, the cubic of stationary curvature, and the Ball point, extending classical rigid-body kinematic tools to compliant systems. These invariants are used in an output-port synthesis procedure to identify coupler points providing prescribed motion characteristics, focusing on locally straight-line trajectories. The methodology is assessed through two case studies differing in scale, hinge topology, and material, and validated against nonlinear flexible multibody models. Overall agreement is obtained for the kinematic invariants and synthesized trajectories: maximum pointwise trajectory deviations remain below 5% of each mechanism’s characteristic length over the investigated ranges, although a larger Ball-point location error is observed in the first case study. The framework provides a geometry-based route to the analysis and synthesis of compliant mechanisms with higher-order motion requirements.