Introduction

The dynamic flight stability of insects is of great significance in biomechanics research and plays an important role in the development of small autonomous aircraft. In recent years, remarkable progress has been made in the study of the dynamic flight stability of insects.

Taylor and Thomas performed the first quantitative analysis of the longitudinal stability of desert locusts using the averaged model and linear analysis1. Based on that, Sun and Xiong2 studied the longitudinal dynamic flight stability of a hovering bumblebee. Since the insect’s flight is symmetric about the longitudinal plane, stability in the longitudinal and lateral planes was studied separately. They found that for the longitudinal motion, there existed three natural modes: one unstable oscillatory mode, one stable fast subsidence mode, and one stable slow subsidence mode. Follow-up research on hoverfly, cranefly, dronefly, and hawkmoth revealed the same conclusion3,4. Researchers also investigated the lateral stability of insects. Sun and his colleagues5,6 studied the lateral dynamic flight stability of several hovering model insects (dronefly and hoverfly). They found that the lateral motion of the insects had three natural modes: an unstable divergence mode, a stable oscillatory mode, and a stable subsidence mode. Faruque and Sean Humbert7,8 and Cheng and Deng9 also studied the dynamic flight stability in several hovering insects. Their results all showed that hovering flight in the insects mentioned above was dynamically unstable due to the unstable modes of the longitudinal and lateral motions.

The researchers also studied the dynamic stability of forward-flight insects. Xiong and Sun10 studied a bumblebee’s longitudinal dynamic stability and found that the flight became more and more unstable as the flight speed increased. Meanwhile, the lateral motion tends to be neutral or weakly stable at medium and high flight speeds11. Zhu et al.12 analyzed the dynamic stability of a dronefly in the full range of flight speeds. They found that increasing speed led to greater instability, indicating that the strong instability may limit the flight speed of insects. Furthermore, the researchers analyzed the insect stability control by adding control terms into the linearized average model3,13,14,15,16. Wu and Sun17 and Noest and Wang18 found that the Floquet analysis could better deal with the problem of the large effect of wing-body coupling. Liang and Sun19 conducted direct numerical simulations to analyze the nonlinear flight dynamics of hovering model insects under large disturbances. Results showed that the insects were unable to return to their equilibrium state, confirming that the insect flight is inherently unstable.

All the above studies were performed on insects with normal wings. However, wing wear is an inevitable consequence of demanding flight activities, which alters the shape, size, and structural integrity of the wing and even results in increased mortality. Mountcastle20 et al. tested the flight maneuvering performance of bumblebees by designing flight paths with dynamic obstacles and found that symmetrical wing damage reduced their maximum acceleration. Vance and Roberts21 studied the effect of wing damage on the hovering flight ability of bees under different gas density environments. They found that with the increase of asymmetric mutilation, the hovering ability of bees in low-density gas decreased. The acceleration and maneuverability of other insects with damaged wings (butterfly22 and dragonfly23 were also found to be reduced in either evasive or predation flight.

Most of the above studies indirectly estimated the effects of wing damage on energy consumption and maneuverability by measuring the maximum lift, flight performance, and the ability to avoid obstacles in complex environments. However, when insect wings are damaged, the flight stability would change first, which subsequently affects flight performance, such as maneuverability. Therefore, how wing damage affects its flight dynamic stability needs to be investigated. It not only deepens the understanding of their agile flight compensation strategies but also offers valuable guidance for controlling flapping-wing aerial vehicles. As far as the authors know, the dynamic flight stability of insects with wing damage has not been reported. In addition, previous studies on insect stability have shown that the lateral and longitudinal motions of insects can be decoupled due to wing symmetry. However, when the wing is damaged unilaterally, whether the stability of longitudinal motion and lateral motion can be solved by decoupling is still unknown. Based on our previous work on kinematic compensation strategies of damaged insects24, this paper employed experimentally measured wing kinematics to investigate the effect of unilateral wing damage on insect flight stability. This paper was recognized as follows: Section Methods detailed the stability analysis method of insects’ flight and gave the flight data. Then, Sect. Results and discussion showed the stability results of the normal wing insect and the unilateral wing damage insect. Section Conclusion summarized the results of this paper.

Methods

Equations of motion

As Taylor and Thomas, Sun and colleagues1,2, we used the averaged model: the insect is treated as a rigid body of 6 degrees of freedom, and the wingbeat-cycle-average forces and moments represent the time-varying forces and moments of the flapping wings. Figure 1 shows the coordinate systems of insect’s motion. Let ogxgygzg be the ground coordinate frame, oxyz be a non-inertial frame fixed to the body. The origin o is at the mass center of the insect. When hovering, the x- and y-axes are horizontal, the x-axis points forward of the flight, and the y-axis points to the right of the insect.

Fig. 1
Fig. 1
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Sketches of the reference frames and definitions of the state variables.

The variables that define the longitudinal motion are the component of velocity along x-axis (denoted as u) and z-axis (denoted as w), the angular velocity around the y-axis (denoted as q), and the pitch angle between the x-axis and the horizontal (θ). The variables that define the lateral motion are the component of velocity along y-axis (denoted as v), the angular velocity around the x-axis (denoted as p), the angular velocity around the z-axis (denoted as r), the roll angle (γ), and the yaw angle (ψ). As for the forces, X is defined as the x-component of the aerodynamic force acting on the insect. Y and Z are the corresponding y- and z-components, respectively. L is the x-component of the aerodynamic moment, and M and N are the corresponding y- and z-components, respectively.

The motion equations of insects are the same as those of airplanes or helicopters. which are listed as follows25:

$$X - mg\sin \theta =m(\dot {u}+wq - vr)$$
(1)
$$Y+mg\cos \theta \sin \gamma =m(\dot {v}+ru - pw)$$
(2)
$$Z{\text{+}}mg\cos \theta \cos \gamma =m(\dot {w} - qu+pv)$$
(3)
$$L={I_x}\dot {p} - {I_{xz}}\dot {r} - {I_{xz}}pq - ({I_y} - {I_z})qr$$
(4)
$$M={I_y}\dot {q}+{I_{xz}}({p^2} - {r^2})+({I_x} - {I_z})rq$$
(5)
$$N= - {I_{xz}}\dot {p}+{I_z}\dot{r} - ({I_x} - {I_y})pq+{I_{xz}}qr$$
(6)
$$\dot {\gamma }=p+\tan \theta (q\sin \gamma +r\cos \gamma )$$
(7)
$$\dot {\theta }=q\cos \gamma - r\sin \gamma$$
(8)
$$\dot {\psi }=\frac{1}{{\cos \theta }}(q\sin \gamma +r\cos \gamma )$$
(9)

where m is insect mass; g is the gravity constant; Ix, Iy and Iz are the moments of inertia about the x-, y-, and z-axis, respectively, and Ixz, Ixy, and Iyz are the corresponding products of inertia. Since the insect body is symmetrical about the oxz plane, the values of Ixy and Iyz are equal to zero; “∙” indicates time (t) derivative. Then the motion equations are linearized by approximating the body’s motion as a series of small disturbances from a steady, symmetric reference flight condition. Variables can be expressed by u = ue + δu, X = Xe + δX et al., among which “e” denotes the equilibrium flight condition and the symbol “δ” represents a small disturbance quantity. Note that at equilibrium flight, ue=we=qe = θe = ve=pe=re = γe = ψe = 0. As for the forces, Ze=-mg, the other forces and moments are equal to 0. Thus, at equilibrium hovering flight, the linearized equations of motions are:

$$\delta \dot {u}=\delta X/m - g\delta \theta$$
(10)
$$\delta \dot {v}=\delta Y/m+g\delta \gamma$$
(11)
$$\delta \dot {w}=\delta Z/m$$
(12)
$$\delta \dot {p}=({I_z}\delta L+{I_{xz}}\delta N)/({I_x}{I_z} - I_{{xz}}^{2})$$
(13)
$$\delta \dot {q}=\delta M/{I_y}$$
(14)
$$\delta \dot {r}=({I_{xz}}\delta L+{I_x}\delta N)/({I_x}{I_z} - I_{{xz}}^{2})$$
(15)
$$\delta \dot {\theta }=\delta q$$
(16)
$$\delta \dot {\gamma }=\delta p$$
(17)
$$\delta \dot {\psi }=\delta r$$
(18)

The six force and moment disturbances are further expanded as Taylor’s series in terms of the disturbance state variables:

$$\delta X={X_u}\delta u+{X_v}\delta v+{X_w}\delta w+{X_p}\delta p+{X_q}\delta q+{X_r}\delta r$$
(19)
$$\delta Y={Y_u}\delta u+{Y_v}\delta v+{Y_w}\delta w+{Y_p}\delta p+{Y_q}\delta q+{Y_r}\delta r$$
(20)
$$\delta Z={Z_u}\delta u+{Z_v}\delta v+{Z_w}\delta w+{Z_p}\delta p+{Z_q}\delta q+{Z_r}\delta r$$
(21)
$$\delta L={L_u}\delta u+{L_v}\delta v+{L_w}\delta w+{L_p}\delta p+{L_q}\delta q+{L_r}\delta r$$
(22)
$$\delta M={M_u}\delta u+{M_v}\delta v+{M_w}\delta w+{M_p}\delta p+{M_q}\delta q+{M_r}\delta r$$
(23)
$$\delta N={N_u}\delta u+{N_v}\delta v+{N_w}\delta w+{N_p}\delta p+{N_q}\delta q+{N_r}\delta r$$
(24)

where Xu, Yv, etc. are the aerodynamic derivatives. In the former studies, the lateral dynamic stability and longitudinal dynamic stability were usually decoupled since insects’ motion had a longitudinal plane of symmetry1,2,3,26. In this study, however, the lateral disturbance may produce relatively large longitudinal forces and moments change due to the asymmetry of the wings after damage. Thus, the lateral and longitudinal equations should be considered at the same time. With Eqs. (10–18) can be written as:

$$\left[ {\begin{array}{*{20}{c}} {\delta {{\dot {u}}^+}} \\ {\delta {{\dot {w}}^+}} \\ {\delta {{\dot {q}}^+}} \\ {\delta {{\dot {\theta }}^+}} \\ {\delta {{\dot {v}}^+}} \\ {\delta {{\dot {p}}^+}} \\ {\delta {{\dot {r}}^+}} \\ {\delta {{\dot {\gamma }}^+}} \end{array}} \right]={\bf{A^+}}\left[ {\begin{array}{*{20}{c}} {\delta {u^+}} \\ {\delta {w^+}} \\ {\delta {q^+}} \\ {\delta {\theta ^+}} \\ {\delta {v^+}} \\ {\delta {p^+}} \\ {\delta {r^+}} \\ {\delta {\gamma ^+}} \end{array}} \right]$$
(25)

where A+ is the system matrix,

$$\bf{A^+}=\left[ {\begin{array}{*{20}{c}} {\bf{A_1}^{+}}&{\bf{B^+}} \\ {\bf{C^+}}&{\bf{A_2}^{+}} \end{array}} \right]$$
(26)
$$\bf{A_1}^{+}=\left[ {\begin{array}{*{20}{c}} {\frac{{X_{u}^{+}}}{{{m^+}}}}&{\frac{{X_{w}^{+}}}{{{m^+}}}}&{\frac{{X_{q}^{+}}}{{{m^+}}}}&{ - {g^+}} \\ {\frac{{Z_{u}^{+}}}{{{m^+}}}}&{\frac{{Z_{w}^{+}}}{{{m^+}}}}&{\frac{{Z_{q}^{+}}}{{{m^+}}}}&0 \\ {\frac{{M_{u}^{+}}}{{I_{y}^{+}}}}&{\frac{{M_{w}^{+}}}{{I_{y}^{+}}}}&{\frac{{M_{q}^{+}}}{{I_{y}^{+}}}}&0 \\ 0&0&1&0 \end{array}} \right]$$
(27)
$$\bf{B^+}=\left[ {\begin{array}{*{20}{c}} {\frac{{X_{v}^{+}}}{{{m^+}}}}&{\frac{{X_{p}^{+}}}{{{m^+}}}}&{\frac{{X_{r}^{+}}}{{{m^+}}}}&0 \\ {\frac{{Z_{v}^{+}}}{{{m^+}}}}&{\frac{{Z_{p}^{+}}}{{{m^+}}}}&{\frac{{Z_{r}^{+}}}{{{m^+}}}}&0 \\ {\frac{{M_{v}^{+}}}{{I_{y}^{+}}}}&{\frac{{M_{p}^{+}}}{{I_{y}^{+}}}}&{\frac{{M_{r}^{+}}}{{I_{y}^{+}}}}&0 \\ 0&0&0&0 \end{array}} \right]$$
(28)
$$\bf{C^+}=\left[ {\begin{array}{*{20}{c}} {\frac{{Y_{u}^{+}}}{{{m^+}}}}&{\frac{{Y_{w}^{+}}}{{{m^+}}}}&{\frac{{Y_{q}^{+}}}{{{m^+}}}}&0 \\ {\frac{{I_{z}^{+}L_{u}^{+}+I_{{xz}}^{+}N_{u}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&{\frac{{I_{z}^{+}L_{w}^{+}+I_{{xz}}^{+}N_{w}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&{\frac{{I_{z}^{+}L_{q}^{+}+I_{{xz}}^{+}N_{q}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&0 \\ {\frac{{I_{{xz}}^{+}L_{u}^{+}+I_{x}^{+}N_{u}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&{\frac{{I_{{xz}}^{+}L_{w}^{+}+I_{x}^{+}N_{w}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&{\frac{{I_{{xz}}^{+}L_{q}^{+}+I_{x}^{+}N_{q}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&0 \\ 0&0&0&0 \end{array}} \right]$$
(29)
$$\bf{A_2}^{+}=\left[ {\begin{array}{*{20}{c}} {\frac{{Y_{v}^{+}}}{{{m^+}}}}&{\frac{{Y_{p}^{+}}}{{{m^+}}}}&{\frac{{Y_{r}^{+}}}{{{m^+}}}}&{{g^+}} \\ {\frac{{I_{z}^{+}L_{v}^{+}+I_{{xz}}^{+}N_{v}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&{\frac{{I_{z}^{+}L_{p}^{+}+I_{{xz}}^{+}N_{p}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&{\frac{{I_{z}^{+}L_{r}^{+}+I_{{xz}}^{+}N_{r}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&0 \\ {\frac{{I_{{xz}}^{+}L_{v}^{+}+I_{x}^{+}N_{v}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&{\frac{{I_{{xz}}^{+}L_{p}^{+}+I_{x}^{+}N_{p}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&{\frac{{I_{{xz}}^{+}L_{r}^{+}+I_{x}^{+}N_{r}^{+}}}{{I_{x}^{+}I_{z}^{+} - I_{{xz}}^{{+2}}}}}&0 \\ 0&1&0&0 \end{array}} \right]$$
(30)

where “+” denotes dimensionless value. Equation (25) is non-dimensionalized using c, U and tw as the reference length, velocity and time, respectively (c, mean chord length; U, mean flapping velocity defined as U = 2Φfr2; tw, the period of the wingbeat cycle defined as tw =1/f; Φ is the wing flapping amplitude; f is the wing flapping frequency, r2 is radius of the second moment of wing area). m+=m/0.5ρUStw (S is the area of two wings, ρ is the air density), Ix+= Ix/0.5ρU2Sctw2, Iy+= Iy/0.5ρU2Sctw2, Iz+= Iz/0.5ρU2Sctw2, Ixz+=Ixz/0.5ρU2Sctw2, and g+=gtw/U; δu+ = δu/U, δv+ = δv/U, δw+ = δw/U, δp+ = δptw, δq+ = δqtw and δr+ = δrtw. X+=X/0.5ρU2S, Y+=Y/0.5ρU2S, Z+=Z/0.5ρU2S, L+=L/0.5ρU2Sc, M+=M/0.5ρU2Sc and N+=N/0.5ρU2Sc; t+=t/tw.

Similar to Sun and Xiong2, we used the method of eigenvalue and eigenvector analysis for the study. In this method, the central elements of the solution for the dynamic stability problem are the eigenvalues and eigenvectors of A+. A real eigenvalue and the corresponding eigenvector (or a conjugate pair of complex eigenvalues and the corresponding eigenvector pair) represent a simple motion called the natural mode of motion of the system. The free motion of the flying body after an initial deviation from its reference flight is a linear combination of the natural modes of motion. A positive (or negative) real eigenvalue, λ, will result in an unstable divergent (or stable subsidence) mode. The time to double (or to half) the starting value is given by 0.693/|λ|. A pair of complex conjugate eigenvalues, e.g., λ1,2 =\(\:{n}\pm\omega{i}\), will result in oscillatory time variation of the disturbance quantities. The period of the oscillatory motion is 2π/\(\:\omega\) and the time to double or half the oscillatory amplitude is 0.693/|\(\:n\)|. With the stability derivatives (Xu+, Yv+, Lv+, etc.) and morphological parameters (m, Ix, etc.) known, the system matrix A+ in Eq. (26) is determined.

Insect model and flight data

The insect models were the same as those in our previous studies, in which the dronefly of Eristalis tenax was selected to study the wing kinematics of damaged insects24. The wing cut ratio was chosen with reference to previous studies on naturally and artificially damaged wings27,28,29, which have shown that most insects with approximately 20% wing area loss were still capable of performing normal flight behavior, such as foraging and avoiding, while the flight performance declined when the damage exceeded 20%. Thus, a damaged insect with a nearly 20% wing area loss was considered representative. Similar to most studies on artificially damaged insects, a straight cut was introduced by trimming the distal part of the wing using scissors. Figure 2a,d show the planforms of the normal and damaged wings, with the removed area being about 18% of a single wing’s area.

The hovering flights of insects were filmed using three orthogonally aligned synchronized high-speed cameras. The dronefly was allowed to fly freely inside an enclosed space built with translucent acrylic plates. When the insect was hovering, the synchronized cameras were manually triggered, and the images represented the intersecting field of view. A total of six intact insects performing hovering flights were recorded. After that, three of them were artificially cut in the wings and their hovering flights were filmed. For each damaged insect, 1–2 flight trials were recorded, and stable hovering segments were selected for kinematic reconstruction. More details of the high-speed filming can be found in24. After filming, the insects were killed and the total mass (m) was measured. Then the shapes of wings and bodies were scanned separately, as shown in Fig. 2a–c. The body and wing motions were extracted in MATLAB, with the body represented by a line and the wings by their outlines. Their positions and orientations were determined by adjusting the models to best match the images in three views simultaneously. Six wingbeat cycles were digitized for one insect, and each Euler angle was phase-averaged to obtain more representative kinematic data. In the wing kinematics results of damaged insects24, the flapping patterns of six different intact individuals were highly consistent. After damage, changes in wing kinematics, such as increased wingbeat frequency and larger stroke amplitude on the damaged side, were also consistent across all three unilateral damaged insects. Therefore, DF1 in24 was considered representative and selected to study the dynamic stability.

Fig. 2
Fig. 2
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Intact wing and damaged wing (a); scanned body photos (b,c); Intact wing and damaged wing models (d); morphological parameters of body (e,f)24.

As defined in former studies24, the wing kinematics is described as three Euler angles in the stroke plane reference frame (OsXsYsZs) (the stroke plane XsOsYs was identified using Ellington’s method25, Fig. 3), i.e., stroke angle (ϕs), pitch angle (ψs), and deviation angle (θs). In this frame, the origin is at the wing root, and XsZs plane is parallel to the sagittal plane of the insect body with Zs-axis perpendicular to the stroke plane and Xs-axis pointing backward. The inclination angle of the stroke plane about the horizontal plane was named the stroke plane angle and denoted as β. The changes in three Euler angles within one flapping cycle were given in Fig. 4, which were obtained from the experimental measurements, i.e., high-speed filming and wing kinematic reconstruction method mentioned above.

Fig. 3
Fig. 3
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Definitions of the angles that determine the wing kinematics.

Fig. 4
Fig. 4
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Variations of the Euler angles of wing motion in one cycle.

To specify matrix A+, morphological data, m, Ix, Iy, Iz and Ixz, etc., and aerodynamic derivatives, Xu+, Xv+ and Xq+, etc., are needed. The majority of the required morphological parameters were provided by Ref24,30. The available morphological parameters are the wing length (R, wing length of damaged wing is defined as Rdamaged); mean chord length of the wing (c); radius of the second moment of wing area (r2); total mass of the insect (m); the radial distance between the wing-root and the center of mass of insect (h1); distance from wing base axis to center of mass (l1); the distance between the wing roots (lr); body length (lb). The definitions for each parameter were shown in Fig. 2d–f. DF-N and DF-U were used to represent the normal wing dronefly and the unilateral wing damage dronefly, respectively. The morphological parameters data of DF-N and DF-U are listed in Table 1 (wing data of DF-U listed in the table are from the damaged wing). Note that wing damage only changed the wing morphological parameters (R, c and r2), but not the morphological parameters of the insect body (m, h1, l1, lb, lr, and χ0). Additionally, the relative position of the wing root to the center of mass did not change after wing damage, so the center of mass was also unchanged between DF-N and DF-U. In addition to the body and wing shape parameters, the body postures of insects were also measured. Table 2 lists the averaged wing kinematics and body posture parameters of DF-N and DF-U in the hovering equilibrium state: Φ is the flapping amplitude, Φ = ϕsmax-ϕsmin, where the ϕsmax and ϕsmin are the maximum and minimum values of ϕs; β is the stroke plane angle; χ is the angle between the stroke plane and the body axis; the Reynolds number Re is defined as Re = cU/ν, where U is the mean flapping velocity defined above and ν is the kinematic viscosity of the air; \(\bar{\phi}\)s is the mean stroke angle of the wing, \(\bar{\phi}\)s=(ϕsmax-ϕsmin)/2; the definitions of other values can be found above. The Ix, Iy, Iz and Ixz values of the dronefly were estimated using the same method as that in Zhang and Sun5 and listed in Table 3. In brief, the side and dorsal images of the body were captured. Assumptions were made that the cross sections of the body were ellipses and the body density was uniform. With that, the center of mass and the moments and products of inertia can be estimated. It is worth noting that the moment of inertia of DF-N was used to calculate the eigenvalues for both DF-N and DF-U. This was because the wing weight was less than 1% of the whole weight of the dronefly, and the weight of the wing tip region accounted for an even smaller portion of the weight, so changes in the moment of inertia due to the area loss in the wing tip region were ignored.

Table 1 Morphological parameters of normal wings dronefly (DF-N) and unilateral wing-damaged dronefly (DF-U).
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Table 2 Averaged kinematic data of the wing and body posture.
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Table 3 Moments and products of inertia of the DF-N (Ib represents the moments of inertia concerning only the body).
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Calculation of aerodynamic derivatives

The stability derivatives are determined using the same CFD methods as those used by Sun et al. in the study of longitudinal and lateral motions of insects2,5,10,11. When hovering, the body’s speed was approximately equal to zero, so the aerodynamic force on the body was ignored, and only the flows around the wings were needed31,32. Therefore, in the numerical simulations, the body was neglected; only a two-wing model was constructed, and the flows around both wings were computed simultaneously. The in-house code used to solve the Navier–Stokes equations was the same as that described by Sun and Tang33,34,35. To obtain the stability derivatives, as in the earlier studies10,36, the hovering equilibrium flights were taken as the reference conditions. The longitudinal and lateral stability derivatives were obtained by varying six state variables separately. For example, a u-series means that u is varied whilst v, w, p, q and r were fixed at the reference values (i.e., v = w = p = q = r = 0). Let Δu, ΔX+, ΔL+, etc. denote the differences between u, X+, L+, etc., and their corresponding values at equilibrium flights. A stability derivative is defined as the partial derivative obtained by fitting the tangent near the equilibrium state.

Results and discussion

The stability derivatives

Figure 5 illustrates the variations in ΔX+, ΔZ+, ΔM+, ΔY+, ΔL+, and ΔN+ for DF-N under different u, w, q, v, p, and r disturbances. Since the flight stability of a normal insect can be decoupled, only the forces and moments change corresponding to disturbances within the same plane were given. Figure 6 displays the ΔX+, ΔZ+, ΔM+, ΔY+, ΔL+, and ΔN+ for DF-U under different u, w, q, v, p, and r disturbances, including the longitudinal (lateral) forces and moments changes corresponding to lateral (longitudinal) disturbances. The linearization of forces and moments for small disturbances was validated, as ΔX+, ΔZ+, and other parameters showed good linearity with Δu+, Δw+, and other state variables. Data in Figs. 5 and 6 were then fitted with straight lines to obtain the stability derivatives (Xu+, Xv+, etc.).

Fig. 5
Fig. 5
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The u-series, w-series, q-series, v-series, p-series, and r-series force and moment data for the dronefly with normal wing (DF-N).

Fig. 6
Fig. 6
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The u-series, w-series, q-series, v-series, p-series, and r-series force and moment data for the dronefly with unilateral wing damage (DF-U).

Table 4 Comparison of non-dimensional aerodynamic derivatives of DF-N and DF-U.
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Table 4 compares the stability derivatives of DF-N and DF-U. Since the longitudinal and lateral motion of DF-N can be decoupled, only the longitudinal derivatives produced by longitudinal disturbances were given, and so were the lateral derivatives. The results of the DF-N are similar to those of the previous studies4. The magnitudes of Xu+, Mu+, and Zw+ are much larger than those of the other derivatives. Xu+ and Zw+ are negative, showing that translational motions in the horizontal and vertical directions generate damping aerodynamic forces. However, Mu+ is positive, indicating that horizontal translation would cause a destabilizing pitch moment. Among the lateral stability derivatives, Yv+, Lv+, Lp+, and Nr+ had larger magnitudes compared to the others, making them the main contributors to flight stability. Notably, only Lv+ had a positive sign, indicating a destabilizing influence.

For the dronefly with unilateral wing damage (DF-U, Table 4), the magnitude of Xu+ decreased by 25% compared to DF-N, but this didn’t affect the generation of damping forces in response to disturbances, and neither did Zq+. The sign of Xw+ changed from positive to negative, indicating that an upward vertical velocity disturbance would induce a downward damping force, which benefited the stable flight of DF-U. Changes in other stability derivatives were either minimal or their magnitudes were too small to influence the stability characteristics of the insect (e.g., Zu+). Furthermore, the lateral stability derivatives also showed limited variation. The main lateral stability derivatives only fluctuated in magnitudes, with no changes in the directions of the responses. The rest of the lateral derivatives of DF-U did not change significantly compared to the normal wing insect. Notably, the signs of Np+ and Yr+ changed: Np+ became positive and became a destabilizing factor, meaning that the rolling disturbances would induce a yawing moment that further amplified disturbances. On the contrary, the sign of Yr+ changed to negative but with a very small value, indicating little influence on the flight stability change.

For the longitudinal (lateral) derivatives produced by lateral (longitudinal) disturbances (Yu+, Xv+ et al.), it can be seen that they have relatively large magnitudes, especially Lu+, Lw+, Lq+. This suggests that longitudinal disturbances can induce a considerable rolling moment in the lateral plane. Consequently, whether the longitudinal and lateral stability can be decoupled becomes a key question, which will be examined in the following section.

The eigenvalues and eigenvectors

In this section, the eigenvalues of DF-N were first given (Table 5, decoupled results). To study whether the flight dynamics of the DF-U can be decoupled, the eigenvalues and eigenvectors of A+ were first obtained directly (see Table 6). Then, an assumption was made that the longitudinal motion and lateral motion can be decoupled. Under this assumption, the eigenvalues of the longitudinal system matrix A1+ (longitudinal) and the lateral system matrix A2+ (lateral) of DF-U were calculated, respectively (see Table 7). By comparing the coupled and decoupled results, the hypothesis can be tested.

Table 5 The eigenvalues of DF-N (decoupled, the eigenvalues of matrix A1+ and A2+).
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Table 6 The eigenvalues of DF-U (coupled, the eigenvalues of matrix A+).
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Table 7 The eigenvalues of DF-U (decoupled, the eigenvalues of matrix A1+ and A2+).
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It can be seen from the Tables 5 and 6 that both normal insect and unilateral wing damage insects have 6 natural modes, including two pairs of complex eigenvalues (λ1 and λ2 with positive real parts, λ6 and λ7 with negative real parts), three negative real eigenvalues (λ3, λ4, and λ5), and a positive real eigenvalue (λ8), which represent the natural modes of insect motion: an unstable oscillatory mode (mode 1) and a stable oscillatory modes (mode 5), two stable fast subsidence modes (mode 2 and mode 4), a stable slow subsidence mode (mode 3) and an unstable divergence mode (mode 6). Due to the presence of unstable modes (mode 1 and mode 6), the motions of both DF-N and DF-U are unstable.

Comparing the eigenvalues of DF-U in Tables 6 and 7, it can be seen that there were almost no differences between the coupled results and decoupled results, indicating that the lateral (longitudinal) force and moment caused by longitudinal (lateral) disturbance do not influence the natural modes of the system. Thus, lateral and longitudinal motion can still be considered separately when studying the stability of insects with unilateral wing damage. That is, the longitudinal motion of DF-U has an unstable oscillatory mode (mode 1), a stable fast subsidence mode (mode 2), and a stable slow subsidence mode (mode 3). The lateral motion has a stable subsidence mode (mode 4), a stable oscillatory mode (mode 5), and an unstable divergence mode (mode 6). Therefore, in the following analysis, the flight stability of the insects is compared separately in the longitudinal and lateral planes.

Table 8 Non-dimensional time constants of the natural modes of longitudinal motion.
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Table 9 Non-dimensional time constants of the natural modes of lateral motion.
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Table 10 Eigenvectors of the longitudinal system matrix A1+.
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Table 11 Eigenvectors of the lateral system matrix A2+.
Full size table

Tables 8 and 9 gives the time constants tdouble, thalf, and T for the natural modes of longitudinal and lateral motion, respectively, non-dimensionalized by wingbeat period tw=1/n = 5.2·ms (see Sect. Equations of motion for formula). The eigenvectors of system matrix A1+ and A2+ expressed in polar form are given in Tables 10 and 11. Since the actual magnitude of an eigenvector is arbitrary, only its direction is unique, and we have scaled them to make δθ = 1 (δγ = 1). From Tables 8 and 10, it can be seen that in the longitudinal plane. For the unstable mode (mode 1), the period of the normal wing insect oscillation is about 80 times the wingbeat period. The initial value of the oscillation would double after 18 wingbeats (Table 8), before which the insect can respond to the disturbance and adjust its flight. Thus, for insect flight, the unstable mode can be considered as controllable2. In this mode, δq+ and δu+ are the main variables (δw+ is smaller than δq+ and δu+ by two orders of magnitude, see Table 10). Since δq+ and δu+ are in the same phase, the motion is forward with pitch-up motion and backwards with pitch-down motion. The fast subsidence mode (mode 2) also contains the horizontal (δu) and pitching (δq) rotation. Since their phases differ by 180°, the insect would pitch down and at the same time move forward (or pitch up and move backwards). For slow subsidence mode (mode 3), δw+ is the main variable, and this mode represents a relatively slowly descending (or ascending) motion. The time constants and eigenvectors of mode 1 and mode 2 of DF-U are similar to DF-N, except that the δw+ phase angle of mode 2 is more variable (Table 10). For mode 3, δw+, the main influence variable is much reduced compared to the normal wings, and the phase changed from 0° to 180°. This indicates that the amplitude of the slow descending mode of the DF-U is lower.

Next, we analyzed the lateral motion modes. In mode 4 of DF-N (stable fast subsidence mode), the magnitudes of δp+ and δr+ are approximately the same, whilst the phase angles differ by 180° (see Table 11). Since the angle between the x-axis and the long axis of the body is around 45° (χ = 56°), the resultant angular velocity of δp+ and δr+ is almost along the long axis of the body, i.e., in this mode, the insect rotates about the long axis of its body. For the stable oscillatory mode (mode 5, T = 108.9, thalf =8.45, see Table 9), since T is much larger than thalf, the disturbance dies out long before one complete oscillation cycle. In this mode, δv+, δp+, δγ are the main variables. The phase angle of δv+ and δγ exhibits a 150° difference (Table 11). Thus, the insect conducts slow horizontal side motion with its body rolled in the opposite direction. In mode 6 (unstable slow divergence), the disturbances doubled the initial values in about 11.7 wingbeats. Dominant variables δv+, δp+ and δγ have the same signs (Table 11), demonstrating that the insect conducts slow horizontal side motion with its body rolled in the same direction. For the DF-U, the oscillation mode period is nearly two times longer than that of DF-N, with little difference in thalf (see Table 9). For the unstable modes (mode 6), the period of DF-U is nearly 4 wingbeat periods longer than that of DF-N, suggesting that such dynamic differences may potentially influence its maneuverability.

Overall, for the unilateral damaged wing insect, despite the presence of unstable modes, its flight is still controllable. In addition, the flight modes in both the longitudinal and lateral planes exhibit little difference between the DF-U and the DF-N. These findings demonstrate the remarkable adaptability of insect flight stability.

Conclusion

This study investigated the flight stability of insects with unilaterally damaged wings. The wing model and flight data are the same as those in24. Numerical simulation was conducted to obtain the aerodynamic derivatives of both the normal wing insect (DF-N) and the unilateral wing damage insect (DF-U). Results showed that, in terms of dynamic derivatives, unilateral wing damage can affect Xu, Mq, Lv, and Nr, with the magnitudes of Xu, Lv, and Nr decreasing and Mq increasing. Additionally, dynamic derivatives with smaller magnitudes, such as Zu, Xw, Np, and Yr, changed directions, but with little impact on overall flight stability. Asymmetric wing motion resulted in relatively large longitudinal (lateral) derivatives produced by lateral (longitudinal) disturbances, especially a large rolling moment induced by longitudinal disturbances.

Tests show that the stability of the unilaterally damaged insects can still be decoupled and studied separately in the longitudinal and lateral directions. Modal analysis revealed that the stability modes of DF-N and DF-U were the same. That is, in the longitudinal plane: an unstable oscillatory mode, a stable fast subsidence mode, and a stable slow subsidence mode; in the lateral plane: a stable fast subsidence mode, a stable oscillatory mode, and an unstable slow divergence mode. Overall, though the coupling stability derivatives were not negligible, their influence on the dynamic stability modes was minimal. This indicated that the response to disturbances of damaged-wing insects remained unchanged. Insects with unilateral wing damage are still capable of maintaining the same flight stability as before the damage, showing the astonishing adaptability of insect flight. These findings provide theoretical guidance for the design of control strategies in bio-inspired flapping-wing vehicles.