### Data reporting

No statistical methods were used to predetermine sample sizes. The experiments were not randomized and the investigators were not blinded to allocation during experiments and outcome evaluation.

### Material

Adult feather beetles (*P. cakes* (Coleoptera: Ptiliidae)) was collected in Cát Tiên National Park, Vietnam, in November 2017. The beetles were collected and delivered to the laboratory along with the substrate for their safety. High-speed video recordings were made the same day within a few hours of collection.

### Morphology and morphometry

The material for morphological studies was fixed in alcoholic Bouin solution or in 70% ethanol. The wing structure was examined using a scanning electron microscope (SEM Jeol JSM-6380 and FEI Inspect F50), after dehydration of the samples and drying at critical points, followed by gold atomization. A confocal microscope (CLSM Olympus FV10i-O) and a transmitted light microscope (Olympus BX43) were also used, for which the samples were clarified and microscopic slides were made^{26} (More info). Measurements were taken from digital photographs in Autodesk AutoCAD software for ten replications (unless otherwise noted). Body weights and weights of specific body parts were calculated on the basis of three-dimensional reconstructions (Supplementary Information).

### Wing mass and moments of inertia

The volume of the leaf stalk and the membranous part (leaf) of the wing were measured using CLSM image-based geometric models. Uniform cuticle density 1,200 kg m^{−3} was assumed^{27}. The wing mass was obtained by summing the contributions from leaf stalk, blade and setae. To calculate the mass of setae, we first estimated their linear density (0.96 μg m^{−1}) using a three-dimensional model^{25} and multiply it by the length. The leaf stalk and the leaf on the wing model have a constant thickness without veins. A possible range of membrane thickness was assumed based on measurements in *T. telengai* (Hymenoptera: Trichogrammatidae, body length 0.45 mm), *O. atoms* (Coleoptera: Corylophidae, body length 0.8 mm) and *L. atom* (Coleoptera: Hydraenidae, body length 1.1 mm), of 0.5 µm thick histological sections obtained by diamond knife cutting using a Leica microtome, after fixation and embedding in araldite. These values are the minimum thicknesses measured in each species. The measurements were performed with an Olympus BX43 microscope. The measurement error for linear dimensions is of the order of 1% for the voltage and chord direction and 10% for the thickness. Sd for wing cuticle density^{25} is about 100 kg m^{−3}. This indicates that the overall root sum error in the wing mass calculation is around 13%. To evaluate the moments of inertia, the surface density of the membranous parts and linear density of the bristles were calculated. The moments of inertia of the individual setae were calculated using the formula for a thin rod at an angle and the parallel axis theorem. The moments of inertia of the membranous parts were calculated by means of a two-dimensional quadrature rule with the discretization step of 50 μm.

### High speed recording

The flight of the beetles was recorded in closed 20 × 20 × 20 mm chambers, specially made of 1.0 mm thick microscopic slides and 0.15 mm coverslips at a natural illumination level in visible light. There were 20-30 insects in the flight chamber during the recording. For temperature stabilization, the flight chamber was cooled by an outside air fan. The ambient temperature measured with a digital thermocouple was 22-24 ° C; the temperature in the flight chamber was 22–26 ° C.

High-speed video recordings were made using two synchronized Evercam 4000 cameras (Evercam) with a frequency of 3,845 FPS and a shutter speed of 20 μs in infrared light (850 nm LED). The high-speed cameras were mounted on optical rails exactly perpendicular to each other and both located 0 ° from the horizon. Two IR LED lights were placed opposite the cameras and one light above the flight chamber. A graphical representation of the experimental setup can be found in the previous study^{2}.

### Measurement of kinematics

For analysis, 13 recordings were selected. For four of them (PP2, PP4, PP5 and PP12), we reconstructed the kinematics of body parts in four kinematic cycles for each and performed CFD calculations because the flight of these samples was particularly similar to conventional hovering: relatively slow normal flight with horizontal speed 0.057 ± 0.014 ms^{−1} (hereafter mean ± sd) and 0.039 ± 0.031 ms^{−1} vertical velocity (PP2, PP4, PP5 and PP12). In CFD analysis with the membranous wing model, we chose kinematics of PP2 that do not cross the wings while patting. This case is practical for comparing the performance of brushed wings with replacement diaphragm wings because it guarantees that the latter do not cross each other. The circumference of the membrane is formed by lines connecting the tips of the bristles (see previous study^{25} for more information). The descriptions of kinematics and aerodynamics as well as the illustrations refer to results obtained for individual PP2. For the results obtained for other samples, see Supplementary information and extended data fig. 2, 4-6.

The average wingbeat frequency was calculated as the mean of the wingbeat frequency in all recordings. In each shot, the number of shots was counted in several complete kinematic cycles, 104 cycles in total.

For the mathematical description of the kinematics of wings and elytra, we used Euler's angle system^{28.29} (Fig. 2b) based on picture-by-picture reconstruction of the location of the insect's body parts (wings, elytra and the body itself) performed in Autodesk 3Ds Max. Three-dimensional models of the body and elytra were obtained by confocal microscope image stacking, and the flat wing model was based on light microscopy photos of dissected wings. We used the rigid flat wing model for reconstruction of kinematics because the deformations of the wings are smaller (Supplementary information). First, we prepared frame sequences with four full kinematic cycles in each. The frames were then centered and cut pointwise between the bottoms of the wings and then placed as orthogonal protrusions. Virtual models of body parts were placed in a coordinate system with two image planes. We then manually changed the position and rotated body parts until their orthogonal projections were superimposed on the image planes. To calculate the Euler angles, a coordinate system was created (Fig. 2a). That *X0Y* Plane is a plane parallel to the plane of impact and intersects the bottom of the wing or elytron, which is located at the zero point. To determine the position of the plane of impact, we calculated the trend line of the main axis for the wingtip coordinates instead of the linear trend line.^{29}, because the wingtip orbit *P. cakes* forms a wide scatter plot. Impact deviation angle (*θ*) and position angle (*Phi*) was calculated from the base and apex coordinates. Pitch angle (*ψ*) is the angle between the plane of impact and the chord perpendicular to the line between base and top. Body pitch angle (*χ*) is the angle between the plane of impact and the longitudinal axis of the body, calculated as the line between the tip of the abdomen and the midpoint between the apical antennae. Pitch angle (*b*) of the plane of impact relative to the horizon was also measured.

For flight velocity analysis, we performed tracking the center of the body (midpoint between the outer edges of the head and abdomen) in Tracker (Open Source Physics) in both projections and calculated the instantaneous velocity and its vertical and horizontal components in each frame. The obtained velocity values were filtered by loose fitting in R (state package). The minimum distance between the wingtip tips under the bottom flap was also calculated.

### Computational fluid dynamics

Time intervals for low-speed flights of more than four wingspan were selected. The angles *Phi*, *θ* and *ψ* of left wing, right wing and elytra and body angle χ was interpolated on a uniform grid with time step size Δ*t*= 2.6 × 10^{−6} s. By solving numerically *Phi*(*t*) = 0 with respect to *t* , we identified four consecutive wingbeat cycles and calculated the average cycle period *T*and the wing beat frequency *f*= 1 /*T* . We then spline-interpolated the data for each of the four cycles on a grid that subdivides the time interval [0, *T*] with step Δ*t* , calculated phase averages, then calculated the average between the left and right wings. This gave the views shown in fig. 2c, d. Constant forward and upward / downward flight speeds were prescribed using the time average values of the loosely filtered time series.

The computational fluid dynamics analysis was performed using open-source Navier – Stokes solver WABBIT^{30}, which is based on the artificial compressibility method to force velocity pressure coupling, volume control method to model non-slip condition at the solid surfaces, and dynamic lattice adjustment using the wavelet coefficients as refining indicators. The flying insect was represented as a collection of five rigid fixed moving parts: the two elytra and the two wings move relative to the body, and the body oscillates about its lateral axis (Supplementary information). The kinematic protocol is described in Supplementary information and extended data. 2c. The computational domain is a 12*R* × 12*R* × 12*R* cube, where *R* is the wing length, with volume control used in combination with periodic external boundary conditions to enforce the desired long-range speed^{30}. The computational domain was decomposed into embedded Cartesian blocks, each containing 25 × 25 × 25 lattice points. The blocks were created, removed and redistributed among parallel calculation processes to ensure maximum level of refinement near the fixed limits and constant wavelet coefficient threshold otherwise during the simulations. The numerical simulations started from the quiet air state, continued for a time period of two wingbeat cycles with a coarse spatial lattice resolution of Δ*x*_{mine}= 0.00781*R* to allow the flow to evolve to its ultimate periodic state, the spatial discretization size was allowed to reduce to Δ*x*_{mine}= 0.00098*R*if the wing was brushed or to Δ*x*_{mine}= 0.00049*R*if it was membranous and the simulation was continued for another wingbeat period to obtain high resolution results. The air temperature in all cases was 25 ° C; its density was *ρ*= 1,197 kg m^{−3} and its kinematic viscosity was *ν*= 1.54 × 10^{−5} m^{2} s^{−1}; the artificial speed of sound was prescribed as *c*_{0}= 30.38*fR*, based on a previous experimental validation^{25}. The volume penalty and other case-specific parameter values are specified in Supplementary Information. The CFD simulation accuracy is discussed in Supplementary Information and Extended Data. 8.

### Degradation of the aerodynamic force of a wing for lifting and towing components

The resistance component of the total instantaneous aerodynamic force acting on the wing is defined as its projection on the direction of the wing velocity at the radius of oscillation. The lifting component is defined as a vector subtraction of the total force and the traction component. The total lift and traction vectors are projected vertically (*with*) direction to achieve the time courses shown in FIG. 3d.

### Reporting overview

Further information on research design is available in the Nature Research Reporting Summary attached to this paper.