An explicit analytical model for passive twist and systematic optimization of membrane flapping wings.

The hovering performance of membrane flapping-wing micro air vehicles (FWMAVs) is governed by numerous coupled parameters, making rapid forward optimization difficult via traditional empirical trials or localized fluid-structure interaction (FSI) analysis. This paper proposes a systematic design framework based on a decoupled parametric configuration and an explicit twist formula. First, a parametric wing design method is developed, in which the wing planform and dominant vein constraints are reduced to a small set of geometric variables, thereby simplifying the design space while preserving their main effects on passive twist. Leveraging this parametric modeling, the study successfully derived an explicit analytical formula reflecting the quantitative relationship between spanwise twist and veins by performing high-fidelity FSI simulations on only a minimal set of key parameter points. This facilitates a transition from costly physical simulations to efficient mathematical expressions. Subsequently, this formula was integrated into the quasi-steady blade element method alongside electromechanical constraints, such as motor torque saturation, to construct a fully coupled system optimization model. The optimized wing, identified via global search, achieved an 11% efficiency improvement and a lift-to-weight ratio exceeding 1.3, as validated by bench and guide-rail flight tests. This research transforms complex empirical design into deterministic analytical optimization, providing an efficient design tool for high-payload, high-efficiency FWMAVs.