###### Abstract
The analytical model accounting for only the translational mechanism was also adopted by [PERSON] and [PERSON] [8] to acquire the minimum power constrained by a given lift, and they concluded that the flapping angle with a triangular profile and the pitching angle with a rectangular profile consume the lowest power with the given wing geometry parameters. [PERSON] and [PERSON] [30] adopted a revised quasi-steady aerodynamic model to evaluate the wing performance of dynamically scaled fruit flies. Optimizations of wing geometry and kinematic parameters were conducted, revealing that the optimal flapping angle followed a harmonious profile, while the optimal pitching angle exhibited a rounded trapezoidal profile with a faster time scale of pitching reversal. Although previous studies have explored the optimization design of hovering wings to some extent using various aerodynamic models, the studies mentioned above either focused solely on optimizing kinematic parameters under a given shape or primarily investigated hovering fliers operating at relatively low Reynolds numbers. The wing was modeled as a single rigid body, and its flexibility or morphing capabilities were ignored. Compared to smaller hovering insects, most vertebrates, such as bats and hummingbirds, fly at Reynolds numbers beyond \(\sim\)10,000. However, flying creatures and robots that operate at higher Reynolds numbers have received less attention in research. Therefore, a broader parameter space, which includes shape and kinematic parameters, still holds significant potential for exploration. Wing performance can be significantly impacted by wing twisting. For example, [PERSON] et al. [31] conducted a parametric study on a three-dimensional passively twistable wing and reported that the vertical aerodynamic force and the aerodynamic power consumption for optimal flat and twisted wings are almost identical. However, another work [32] showed that the power loading of wings with torsion is higher than flat wings. The function of wing twist in biological hovering flappingflight is not yet fully understood, although twisting has been observed in vertebrate species. Therefore, disregarding spanwise twisting may lead to suboptimal optimization results when accounting for the highly coupled kinematic parameters. Considering that the rotational motion of the flapping wings of insects is not always actively controlled, the wings of FWMAVs are also designed to pitch passively. Some quasi-steady models [33; 34; 35] were developed previously to predict the passive pitching motion over an entire stroke. Structural dynamics modeling techniques have been utilized to model flexible wings and investigate the impacts of flexibility and inertia on wing performances. The wings were simplified as isotropic or anisotropic plates, reinforced with stiffeners, to study the interaction between passively twisted wings and air. Similarly, comparable studies are observed in the research on wind turbine blades [36; 37; 38]. Research on the optimization of passively twisted wings can be found in references [39; 40]. In this paper, we performed an optimization study to provide a pair of energy-efficient wings for a FWMAV in the initial design stage. Based on the findings of previously reported studies on wing twist effects, we recognized wing-spanwise twisting as a means to improve hovering flight efficiency. We specified a linear distribution of rotation angle along the wingspan on a hummingbird-scale model to introduce dynamic twisting. Accordingly, a set of optimal geometric and kinematic parameters was numerically determined. Next, we performed a sensitivity analysis of design parameters at the optimal point for the established wing model. That helped us identify some key parameters that may affect wing performance. The obtained results will assist in guiding the stiffness design and the deformation constraint of flexible wings widely used today. The rest of this paper is organized as follows.