A robust technique for determining the wing kinematics, body position and attitude of a free-flight dragonfly is described. The new method is based on a projected comb-fringe technique combined with the natural landmarks on a dragonfly, allowing us to establish the local body-centered coordinate system with high accuracy, and to measure the body attitude at any instant. The kinematic parameters, including wingbeat frequency, flapping angle, angle of attack, torsional angle and camber deformation, required no assumptions to be made with respect to wing geometry, deformability (except the assumption of rigid leading edges) or bilateral wing symmetry. Two typical flight behaviors,forward flight and turning maneuvers, of dragonflies Polycanthagyna melanictera Selys were measured and analyzed.

Details of the kinematics of free flight are very important to understanding insect flight mechanics. Important data for aerodynamic analysis and modeling include flight trajectory, body attitude and wing kinematics for individuals flying over a diverse array of behaviors, such as hovering,climbing and turning.

A number of recent studies have focused on the kinematics of hovering and forward flight, using a variety of techniques. Azuma and Watanabe(1988) changed the velocity of the wind tunnel in their measurements. Dudley and Ellington(1990) calculated angles of attack in the free forward flight of bumblebees. Willmott and Ellington(1997) employed a variable-speed wind tunnel associated with the optomotor response to investigate wing and body kinematics during free forward flight of a hawkmoth over a range of speeds from hovering to 5 m s-1. Wakeling and Ellington (1997) filmed the free flights of dragonflies and damselflies flying over the pond in the greenhouse at the University of Cambridge. The individuals were not restrained by either tethers or wind tunnels, but were free to vary the velocity and acceleration and could perform any flight action. In their analyses of forward flight, the stroke plane was constructed based on the assumption of bilateral wing symmetry, and variations in roll, yaw and pitch angles of the body through each flapping cycle were neglected. To date no detailed information on wing orientation or shape during free flight has been acquired.

All kinds of flight behaviors are important for studying the aerodynamics and the control of flight. In turning maneuvers, the wings move asymmetrically, and the change in attitude is obvious even during one flapping cycle. We have also found that dragonflies exhibit substantial chordwise deformation and changes in camber during free flight, which might be important for aerodynamic models of flight performance.

To study turning maneuvers involving obvious changing of the insect attitude, the description of wing kinematics should be based on a local body-centered coordinate system, together with the body attitude and flight trajectory. We have developed a method utilizing a Projected, Comb-Fringe technique combined with the Landmarks procedure (PCFL), in which a comb-fringe pattern with high intensity and sharpness was projected onto the transparent wing of a dragonfly in free flight. Images of the wings with distorted fringes were then recorded by a high-speed camera. Based on the distorted fringe pattern and the natural landmarks on the dragonfly wings, we reconstructed wing shape and established the body-centered coordinate system. This method allowed us to derive kinematic parameters without assumptions of rigid chords or kinematic symmetry, except for the assumption of rigid leading edges. The instantaneous attitude of the body was also measured simultaneously. We measured dragonflies in two flight behaviors: forward flight and turning maneuvers, and compared the kinematics results obtained for each of them.

Dragonflies of the species Polycanthagyna melanictera Selys were caught at Tsinghua University, Beijing, China, during the period of July-August 2001. The kinematic parameters together with the flight trajectory and attitude were measured for several dragonflies in free flight. The mean body mass was 0.29±0.02 g, and the forewing and hindwing lengths were 44 mm and 40 mm, respectively. The mid-span chord lengths for forewing and hindwing were 10 mm and 15 mm, respectively. The individuals were flying within an area were recorded by a high-speed camera. Eight individuals were recorded with no more than ten sequences for each individual. These records were grouped according to flight behaviors, and two typical flight behaviors,forward flight and turning maneuvers, were selected for analysis.

Global coordinate system and PCFL system

Fig. 1 shows the PCFL system based on the global coordinate system (OXYZ). A dragonfly was induced by a fluorescent lamp (pilot light) to fly across the experimental region. The flight was captured by a high-speed camera HCCD (Dalsa D256). Frame capture speed was 955 frames s-1 at a resolution of 256×256 pixels;exposure time was 1/4000 s. The experimental background was illuminated by an incandescent lamp with an output spectrum close to the response peak of the camera, thereby allowing the outline of the wing to be easily identified.

Fig. 1.

PCFL system based on the global coordinate system (OXYZ). The measurement and reference fringes come from points W1 and W2, respectively. The global coordinate is defined as follows: the XOY plane is at the lowest position; the Z-axis is the optical axis of the high-speed camera; the X-axis is perpendicular to the fringes and the Y-axis is parallel to the fringes. A calibration plane (CP) perpendicular to the optical axis of the high speed camera (HCCD)was used to calibrate the system. PC, personal computer; FPP, fringe pattern projector.

Fig. 1.

PCFL system based on the global coordinate system (OXYZ). The measurement and reference fringes come from points W1 and W2, respectively. The global coordinate is defined as follows: the XOY plane is at the lowest position; the Z-axis is the optical axis of the high-speed camera; the X-axis is perpendicular to the fringes and the Y-axis is parallel to the fringes. A calibration plane (CP) perpendicular to the optical axis of the high speed camera (HCCD)was used to calibrate the system. PC, personal computer; FPP, fringe pattern projector.

A fringe pattern projector (FPP) with an interbeam fringe angle of 0.77° was used to project a fringe pattern onto the wing. Wing deformation was calculated by using the resulting distorted fringe pattern observed from the camera view. Given the camera position, the location of the FPP, and the angle of the measurement fringe, the three-dimensional position of any point on the distorted fringe could be calculated by using spatial analytic geometry(Zeng et al., 1996, 2000; Song et al., 2001). The angle of the measurement fringe was determined using a reference fringe, which was thicker and brighter than the measurement fringes. The reference fringe was generated by using a semiconductor laser together with a cylindrical lens. The rank order of measurement fringes was counted from the reference fringe,allowing determination of the angle of each measurement fringe. Although the reference fringe was not visible during some frames while the dragonfly was flying through the experimental region, the fringe order was maintained, as the fringe shift caused by wing flapping was much smaller than the distance between fringes. Constancy of fringe order could therefore be used to identify individual fringes by consecutive counting.

Three-dimensional (3-D) coordinates of points on the distorted fringes were calculated by spatial analytic geometry(Song et al., 2001). Interpolation was used to calculate the spatial position of points of interest not on the fringes. We reconstructed 3-D wing shape throughout the flapping cycle by digitizing the fringe coordinates frame-by-frame from the recorded fringe sequence. The body position, attitude and kinematic parameters of wing were based on the 3-D reconstruction.

Local body-centered coordinate, flight trajectory and attitude

The body position and attitude of an insect in free flight should be described on the basis of a global coordinate system in order to obtain the flight trajectory. By contrast, the kinematic parameters should be described on the basis of the local body-centered coordinate system. Uncertainties in body position and attitude bring errors in local body-centered coordinate system construction, and therefore uncertainties in the wing kinematics description.

We defined the local body-centered coordinate relative to the body position and attitude in the global coordinate system using the four wingbases. Because the wingbases cannot be identified in captured images, the wingbase coordinates were constructed using identifiable landmarks on the wings. Fig. 2A shows the natural landmarks near the wing tip (Q). Because the leading edge of the wing bends little during flapping motion, it can be considered a rigid bar. Given the 3-D coordinates of two arbitrary points on leading edge projected by fringes, the equation of the leading edge (the line joining wingbase E and landmark Q) can be obtained. Then, the coordinate of Q can be calculated based on the image of Q and the leading edge equation. According to the distance S between E and Q,which is constant for a given dragonfly wing and can be measured in advance,the coordinate of each wingbase can be calculated based on spatial analytic geometry. Then the local body-centered coordinate system is constructed based on the four wingbases, its origin KF being the midpoint of the two fore-wingbases (see Fig. 2B).

Fig. 2.

(A) Landmarks on forewing and hindwing of a dragonfly; (B) definition of the local body-centered coordinate system and yaw, pitch and roll angles. Arrows define the positive rotation directions. The local body-centered coordinate system O′X′Y′Z′ is based on four wingbases (B1, B2, B3 and B4), where points B1 and B2 are wingbases of the forewing, and B3 and B4 those of the hindwing. The midpoint KF of B1 and B2 is the origin of the local body-centered coordinate. The Y′-axis is along the line KFKH that joins points KF and the midpoint of B3 and B4, KH. The plane X′O′Y′ is constructed by points KF,KH and B1. The Z′-axis is perpendicular to the X′O′Y′ plane and upward.

Fig. 2.

(A) Landmarks on forewing and hindwing of a dragonfly; (B) definition of the local body-centered coordinate system and yaw, pitch and roll angles. Arrows define the positive rotation directions. The local body-centered coordinate system O′X′Y′Z′ is based on four wingbases (B1, B2, B3 and B4), where points B1 and B2 are wingbases of the forewing, and B3 and B4 those of the hindwing. The midpoint KF of B1 and B2 is the origin of the local body-centered coordinate. The Y′-axis is along the line KFKH that joins points KF and the midpoint of B3 and B4, KH. The plane X′O′Y′ is constructed by points KF,KH and B1. The Z′-axis is perpendicular to the X′O′Y′ plane and upward.

A dragonfly in flight has six degrees of freedom (d.f.) for its movement. It can translate in three dimensions in space and rotate around its center of mass, which is approximated by the origin of the body-centered coordinate system. The flight trajectory (or the variation in body position of a free flight dragonfly) is defined by the movement of the origin of the local body-centered coordinate system, and measured in the global coordinates. Body attitude is defined by the orientation of a body (or vectors of three axes of the body-centered coordinate) with respect to a reference orientation, and described by three angles: yaw angle τ, pitch angle ϕ and roll angleψ (Schilstra, 1999). The reference orientation is defined when the body axis of a dragonfly is along the X-axis and the symmetry plane of its body is in XOZ plane in the global coordinate system. The three angles can be deduced from the local body-centered coordinate system O′X′Y′Z′ and the reference orientation. For the reference orientation, the body-centered coordinate system was rotated in the following sequence: first, around its Z′-axis with angleτ; second, around its X′-axis with angle ϕ; and finally, around its Y′-axis with angle ψ. To describe the attitude change of free flight visually, we used the common aeronautical descriptions where the angles τ, ϕ and ψ are positive as they are turning clockwise, respectively. The arrows in Fig. 2B show the direction definitions.

Because insects can fly within the same trajectory but with different body attitudes, it is important to describe flight behavior by combining the flight trajectory together with the aeronautical descriptions of roll, pitch and yaw angles.

Stroke plane and kinematic parameters

The kinematic parameters, including wingbeat frequency, flapping angles,angles of attack, torsional angle and camber deformation of the fore- and hindwings are relevant to the aerodynamic analysis of insect flight. We assumed that the wing's leading edge did not bend during its stroke in calculating the kinematic parameters. The assumption is robust for almost all flight behaviors. The leading edge equation, the arch κ and the chord vector MN of each wing at any flapping angle can be calculated based on the 3-D reconstruction of the wings.

The plane in which the wings oscillate relative to the dragonfly's body is called the stroke plane, which is defined by three points: the wing base, and the wingtip at the maximum and minimum angular positions in a flapping cycle. The kinematic parameters, including the flapping angle φ, angle of attackα, torsional angle ρ and camber deformation ζ, are described based on the stroke plane. Fig. 3 shows the parameters of the left forewing; those of other three wings are defined by the same way.

Fig. 3.

Wing kinematics. The figure illustrates the parameters of the left forewing; those of the other three wings are defined by the same way. (A)Definition of flapping angle φ. Point B1 represents the wingbase of the left forewing, point T1 the wingtip of that. The line B1-T1 denotes the leading edge of the wing. The stroke plane ζ is defined by three points:the wingbase B1, and the wingtip at the maximum and minimum angular positions(T1H and T1L) in a flapping cycle based on the local body-centered coordinate system. (Note that the stroke plane of the left wing is generally different from that of the right wing without any assumption of kinematic symmetry, i.e. the stroke planes with respect to the left and right wing are not in one plane.) The line L1 represents the intersection of the plane X′—O′—Y′ and the stroke plane. The flapping angle φ is defined as the angular position of the wing in the stroke plane, measured from dorsal reversal (start of downstroke) to ventral reversal (start of upstroke). φ=0 for the leading edge in the plane X′—O′—Y′. (B) Definition of the angle of attack, the torsional angle and camber deformation. The planeΩ, which is normal to the leading edge and at a spanwise position of 50%of the total wing length, is defined as the mid-span chord plane. The intersection arch of the wing surface and the plane Ω is the mid-span arch κ. The mid-span chord vector MN is corresponding to the archκ, whose direction definition is from the wing trailing edge(M) to leading edge (N). The line L is the intersection of the plane Ω and the stroke plane Π, which denotes the tangent of the wing's trajectory. The line L′ is the perpendicular of the line L. The angle of attack α is defined as the included angle between the vector MN and the line L and the torsional angle ρ is defined as the included angle between the vector MN and the line L′. Note that α and ρ are complementary angles. When MNΠL, ρ=0; α=+90°(if downstroke) or α=-90° (if upstroke). The two-dimensional coordinate system (o′, x′, y′) is established in the plane Ω with the x-direction from N to M,the y-direction from the lower surface to the upper surface of the wing and the origin in the leading edge. The camber deformation [UNK] is defined as the ratio of the maximum arch rise Hmax to the mid-span chord length ζMN|.

Fig. 3.

Wing kinematics. The figure illustrates the parameters of the left forewing; those of the other three wings are defined by the same way. (A)Definition of flapping angle φ. Point B1 represents the wingbase of the left forewing, point T1 the wingtip of that. The line B1-T1 denotes the leading edge of the wing. The stroke plane ζ is defined by three points:the wingbase B1, and the wingtip at the maximum and minimum angular positions(T1H and T1L) in a flapping cycle based on the local body-centered coordinate system. (Note that the stroke plane of the left wing is generally different from that of the right wing without any assumption of kinematic symmetry, i.e. the stroke planes with respect to the left and right wing are not in one plane.) The line L1 represents the intersection of the plane X′—O′—Y′ and the stroke plane. The flapping angle φ is defined as the angular position of the wing in the stroke plane, measured from dorsal reversal (start of downstroke) to ventral reversal (start of upstroke). φ=0 for the leading edge in the plane X′—O′—Y′. (B) Definition of the angle of attack, the torsional angle and camber deformation. The planeΩ, which is normal to the leading edge and at a spanwise position of 50%of the total wing length, is defined as the mid-span chord plane. The intersection arch of the wing surface and the plane Ω is the mid-span arch κ. The mid-span chord vector MN is corresponding to the archκ, whose direction definition is from the wing trailing edge(M) to leading edge (N). The line L is the intersection of the plane Ω and the stroke plane Π, which denotes the tangent of the wing's trajectory. The line L′ is the perpendicular of the line L. The angle of attack α is defined as the included angle between the vector MN and the line L and the torsional angle ρ is defined as the included angle between the vector MN and the line L′. Note that α and ρ are complementary angles. When MNΠL, ρ=0; α=+90°(if downstroke) or α=-90° (if upstroke). The two-dimensional coordinate system (o′, x′, y′) is established in the plane Ω with the x-direction from N to M,the y-direction from the lower surface to the upper surface of the wing and the origin in the leading edge. The camber deformation [UNK] is defined as the ratio of the maximum arch rise Hmax to the mid-span chord length ζMN|.

After calculating flapping angle φ and torsional angle ρ from the distorted fringe, we described the wing kinematics by fitting the anglesφ and ρ using a Fourier series:
\[\ {\varphi},{\rho}={\mu}0+{{\sum}_{n=1}^{k}}A_{\mathrm{n}}\mathrm{cos}(n{\bar{{\omega}}}t+{\delta}_{\mathrm{n}}),\]
1
where μ0, An and δn are the Fourier coefficients,
\({\bar{{\omega}}}\)
is the fundamental frequency and n is the harmonic number, k=1 for flapping angle and k=3 for torsional angle (see Azuma et al., 1985). The uncertainties (error bars) are expressed by the standard deviation S:
\[\ S=\sqrt{\frac{1}{M-P}{{\sum}_{i=1}^{M}}(y_{\mathrm{i}}-f_{\mathrm{i}})^{2}},\]
2
where M is the total number of data points within the wingbeat, yi the measured values, fi the fitted values, and P the total number of adjustable parameters used in the fitting, for flapping angle P=4 and for torsional angle P=8.

Uncertainty in measurement

To estimate the uncertainty of the measurement system, we measured a half-round surface with a radius of 19.00 mm, which was placed on a horizontal plane normal to the optical axis of the high speed camera. The profile of the object was measured from the distorted fringe. Using this real curve as a reference, the fitted curve from the measured results had a standard deviation of 0.10 mm. The accuracy is mainly limited by the pixel resolution of the image. If the error of any point on the distorted fringe is 0.1 mm, the error in wingbase is 0.14 mm, according to the equation of leading edge.

Because the 3-D coordinate of wingbase E can be deduced according to only two arbitrary fringes on one wing, the number of the calculated coordinate of E is

\(C_{\mathrm{N}}^{2}\)
(N is the total number of fringes on the wing). The final coordinate of wingbase is the optimization of these coordinates according to a least-squares method. The relationship of the four wingbases is nearly invariable, and is used to estimate the effectivity of the local body-centered coordinate system.

Applicability of the PCFL method

The PCFL method is based on the assumption of rigid leading edges, which is applicable for most insects. We can measure the kinematic parameters for a wide range of insect wing size, from a small Drosophila wing to a large magpie wing, by adjusting the magnification of optical system and the interbeam angle of fringes. For the system with only one camera and one FPP,the flight measurement is limited, because the fringes may not project on a certain wing or leading and trailing edges may be confluent in the camera view field when the insect is at a particular place, especially for the insect wings that oscillate through large angles. Moreover, the FPP or camera view of a wing may be obstructed by the body or other wings when the insect turns. On such occasions, two cameras and two FPPs, mounted in the measurement system with different angles, should be used.

On the other hand, in order to ensure the measurement accuracy of the method, the insect should be sufficiently large in the camera view field, that it only needs to be kept in view for a short time. In our experiment, the proportion of dragonfly in the view field was selected carefully to ensure that the dragonfly was monitored for at least for one stroke period. We are now designing a tracking measurement system, including video tracking and fringe pattern tracking, to allow us to record the image of dragonfly continuously over more stroke cycles.

All analyses were based on the data from the 3-D reconstruction of the four wings of the flight dragonfly. One digital image from a forward flight sequence and its 3-D reconstruction are shown in Fig. 4.

Fig. 4.

Three-dimensional (3-D) reconstruction of the four wings of the flight dragonfly. (A) An original digital image of a dragonfly, and its 3-D reconstruction observed from the same perspective (overhead view). (B) The 3-D reconstruction observed from different viewpoints, represented by two parameters, azimuth or horizontal rotation (as AZ) and vertical elevation (as EL), both in degrees. AZ is positive as the viewpoint is rotated clockwise,while positive values of elevation correspond to movement above the object. Notice that the three orthogonal short arrows represent the body-centered coordinate system.

Fig. 4.

Three-dimensional (3-D) reconstruction of the four wings of the flight dragonfly. (A) An original digital image of a dragonfly, and its 3-D reconstruction observed from the same perspective (overhead view). (B) The 3-D reconstruction observed from different viewpoints, represented by two parameters, azimuth or horizontal rotation (as AZ) and vertical elevation (as EL), both in degrees. AZ is positive as the viewpoint is rotated clockwise,while positive values of elevation correspond to movement above the object. Notice that the three orthogonal short arrows represent the body-centered coordinate system.

Flight trajectory and attitude

The 3-D flight trajectories of the two flight behaviors, forward flight and turning maneuvers, are described by each trajectory of the origin of the body-centered coordinate system (see Fig. 5). Note that the forward flight is not a proper rectilinear forward flight, for it contains alternating pitching turns and large pitch velocities and accelerations (see Figs 5, 6). Because the body pitch of a dragonfly is nearly a constant during hovering(Azuma et al., 1985), we assume that it should also be a constant during proper rectilinear forward flight.

Fig. 5.

Flight trajectories of two typical flight behaviors. The flight trajectories of forward flight and turning maneuvers are expressed by lines with open squares and open circles, respectively, and the flight trajectories projected onto the plane XOY by the solid and dotted lines,respectively. The interval of the vertical projection line is 1/955 s.

Fig. 5.

Flight trajectories of two typical flight behaviors. The flight trajectories of forward flight and turning maneuvers are expressed by lines with open squares and open circles, respectively, and the flight trajectories projected onto the plane XOY by the solid and dotted lines,respectively. The interval of the vertical projection line is 1/955 s.

Fig. 6.

Attitude of two flight behaviors: (A,C,E,G,I) for forward flight;(B,D,F,H,J) for turning maneuvers. (A,B) The flight trajectory, velocity and acceleration. The position of the dragonfly is drawn at an interval of 1/955 s. The solid and dotted lines are with respect to the velocity v and acceleration a vectors, starting from the body position, denoted by crosses (the vector arrow is neglected). In forward flight, the maximum and minimum velocities are 1.71 m s-1 and 1.22 m s-1, and the maximum and minimum accelerations are 129.8 m s-2 and 16.7 m s-2; and in turning maneuvers, these are 2.01 m s-1 and 1.26 m s-1, 101.5 m s-2 and 39.1 m s-2,respectively. (C,D) Flight trajectories projected on the X, Y and Z axes. The filled circles, measured data. The curves are fitted by cubic regression functions as follows. For forward flight,

\(\begin{array}{l}{\ }{\ }X=10^{-3}{\times}(-0.3t^{3}+16.6t^{2}-52.3t-10832.6),\\{\ }{\ }Y=10^{-3}{\times}(-9.4t^{2}-1049.8t+27657.5)\\\mathrm{and}Z=10^{-3}{\times}(-1.3t^{3}+72.7t^{2}-1373.6t-16116.6);\end{array}\)
and for turning maneuvers,
\(\begin{array}{l}{\ }{\ }X=10^{-3}{\times}(0.1t^{3}-19.3t^{2}+441.8t-6669),\\{\ }{\ }Y=10^{-3}{\times}(-1.1t^{2}+44.1t^{2}-1885.4t+26309.8)\\\mathrm{and}Z=10^{-3}{\times}(0.4t^{3}-34.2t^{2}+868.2t+11029.6),\end{array}\)
where the time t is ms, and coordinates X, Y and Zin mm. (E,F) Attitude of the dragonfly expressed by orientation angles; filled square, filled circle and filled triangle denote roll, pitch and yaw,respectively. To calculate the angular velocity and acceleration, these raw data are also fitted using cubic regression functions, denoted by solid,broken and dotted lines for roll, pitch and yaw, respectively. For forward flight,
\(\begin{array}{l}\mathrm{Roll}=0.0026902t^{3}-0.14277t^{2}+2.3014t-22.2541,\\{\ }{\ }\mathrm{Pitch}=-0.0081571t^{3}+0.37106t^{2}-2.736t-2.5483\\\mathrm{and}{\ }\mathrm{Yaw}=0.0022405t^{3}-0.11112t^{2}+0.86824t+88.9312;\end{array}\)
and for turning maneuvers,
\(\begin{array}{l}\mathrm{Roll}=-0.0082091t^{3}+0.38716t^{2}-3.4528t+8.9258,\\{\ }{\ }\mathrm{Pitch}=0.0017947t^{3}-0.069399t^{2}+1.9321t+21.963\\\mathrm{and}{\ }\mathrm{Yaw}=-0.0012337t^{3}+0.082754t^{2}+0.37377t+61.5567,\end{array}\)
where the time t is ms, and attitude angle is degrees. (G,H) Angular velocities; (I,J) angular accelerations.

Fig. 6.

Attitude of two flight behaviors: (A,C,E,G,I) for forward flight;(B,D,F,H,J) for turning maneuvers. (A,B) The flight trajectory, velocity and acceleration. The position of the dragonfly is drawn at an interval of 1/955 s. The solid and dotted lines are with respect to the velocity v and acceleration a vectors, starting from the body position, denoted by crosses (the vector arrow is neglected). In forward flight, the maximum and minimum velocities are 1.71 m s-1 and 1.22 m s-1, and the maximum and minimum accelerations are 129.8 m s-2 and 16.7 m s-2; and in turning maneuvers, these are 2.01 m s-1 and 1.26 m s-1, 101.5 m s-2 and 39.1 m s-2,respectively. (C,D) Flight trajectories projected on the X, Y and Z axes. The filled circles, measured data. The curves are fitted by cubic regression functions as follows. For forward flight,

\(\begin{array}{l}{\ }{\ }X=10^{-3}{\times}(-0.3t^{3}+16.6t^{2}-52.3t-10832.6),\\{\ }{\ }Y=10^{-3}{\times}(-9.4t^{2}-1049.8t+27657.5)\\\mathrm{and}Z=10^{-3}{\times}(-1.3t^{3}+72.7t^{2}-1373.6t-16116.6);\end{array}\)
and for turning maneuvers,
\(\begin{array}{l}{\ }{\ }X=10^{-3}{\times}(0.1t^{3}-19.3t^{2}+441.8t-6669),\\{\ }{\ }Y=10^{-3}{\times}(-1.1t^{2}+44.1t^{2}-1885.4t+26309.8)\\\mathrm{and}Z=10^{-3}{\times}(0.4t^{3}-34.2t^{2}+868.2t+11029.6),\end{array}\)
where the time t is ms, and coordinates X, Y and Zin mm. (E,F) Attitude of the dragonfly expressed by orientation angles; filled square, filled circle and filled triangle denote roll, pitch and yaw,respectively. To calculate the angular velocity and acceleration, these raw data are also fitted using cubic regression functions, denoted by solid,broken and dotted lines for roll, pitch and yaw, respectively. For forward flight,
\(\begin{array}{l}\mathrm{Roll}=0.0026902t^{3}-0.14277t^{2}+2.3014t-22.2541,\\{\ }{\ }\mathrm{Pitch}=-0.0081571t^{3}+0.37106t^{2}-2.736t-2.5483\\\mathrm{and}{\ }\mathrm{Yaw}=0.0022405t^{3}-0.11112t^{2}+0.86824t+88.9312;\end{array}\)
and for turning maneuvers,
\(\begin{array}{l}\mathrm{Roll}=-0.0082091t^{3}+0.38716t^{2}-3.4528t+8.9258,\\{\ }{\ }\mathrm{Pitch}=0.0017947t^{3}-0.069399t^{2}+1.9321t+21.963\\\mathrm{and}{\ }\mathrm{Yaw}=-0.0012337t^{3}+0.082754t^{2}+0.37377t+61.5567,\end{array}\)
where the time t is ms, and attitude angle is degrees. (G,H) Angular velocities; (I,J) angular accelerations.

The detailed differences between the two flight behaviors are difficult to distinguish from the flight trajectories alone. However, the difference between the two flight behaviors becomes obvious on considering the velocities, accelerations and attitude of the dragonfly (see Fig. 6). When the dragonfly is in forward flight, its acceleration at each position is nearly normal to the horizontal plane, so that the body moves in the vertical plane containing the body axis; in contrast when it performs turning maneuvers, its acceleration at each position is nearly parallel to the horizontal plane, which must induce the body axis to deviate from the vertical plane, resulting in a turn.

The difference between forward and turning maneuvers can also be found from the variations in the roll, pitch and yaw angles. During forward flight, only the pitch angle has a distinct fluctuation, while the roll and yaw angles are almost constant, implying flight in a vertical plane. During turning maneuvers, the roll, pitch and yaw angles are increasing by almost the same slope, implying that the dragonfly turns right with the body inclined to the right and the head elevated. Within the limits of the camera view field, the turning was not completed before the dragonfly flew out of the fringe pattern.

Kinematics of flight

Based on the 3-D reconstruction of the dragonfly wings and the interpolation algorithm, the time-dependent variation in flapping angleφ, torsional angle ρ and the camber deformation | for each wing were obtained (see Fig. 7). The wingbeat frequency was 33.4 Hz and 35.0 Hz with respect to forward flight and turning maneuvers, respectively.

Fig. 7.

Wing kinematics for two flight behaviors: (A,C,E) for forward flight;(B,D,F) for turning maneuvers. The middle of the error bar is the actual data point. (A,B) Flapping angle. The smooth curves are fitted by the Fourier series as Equation 1, with the parameter k=1. The value of the error bar is the standard error of fitting expressed as equation 2, with the parameter P=4. The flapping angles of right and left hindwing and right and left forewing are represented by the symbols φrhlh, φrf and φlf,respectively. For forward flight,

\(\begin{array}{l}{\varphi}_{\mathrm{rh}}=0.80+24.19\mathrm{cos}(0.21t+73.34),\\{\varphi}_{\mathrm{lh}}=-1.95+22.27\mathrm{cos}(0.21t+65.47),\\{\varphi}_{\mathrm{rf}}=0.61+20.70\mathrm{cos}(0.21t-31.28)\\\mathrm{and}{\ }{\varphi}_{\mathrm{lf}}=1.96+25.31\mathrm{cos}(0.21t-40.11);\end{array}\)
and for turning maneuvers,
\(\begin{array}{l}{\varphi}_{\mathrm{rh}}=-0.20+16.67\mathrm{cos}(0.22t+168.15),\\{\varphi}_{\mathrm{lh}}=0.78+23.94\mathrm{cos}(0.22t+166.56),\\{\varphi}_{\mathrm{rf}}=1.45+24.08\mathrm{cos}(0.22t+75.44)\\\mathrm{and}{\ }{\varphi}_{\mathrm{lf}}=-9.15+31.91\mathrm{cos}(0.22t+57.14),\end{array}\)
where the time t is ms, and flapping angle in degrees. (C,D)Torsional angle at the 50% spanwise positions. The smooth curves are fitted by the Fourier series as Equation 1 with k=3. The value of the error bar is the standard error of fitting expressed as Equation 2 with P=8. The torsional angles of right and left hindwing and right and left forewing are represented by the symbols ρrh, ρlhrf and ρlf, respectively. For forward flight,
\(\begin{array}{l}{\rho}_{\mathrm{rh}}=86.04+43.92\mathrm{cos}(0.21t-30.95)+13.75\mathrm{cos}(2{\times}0.21t+78.12)+0.86\mathrm{cos}(3{\times}0.21t-4.85),\\{\rho}_{\mathrm{lh}}=79.83+36.88\mathrm{cos}(0.21t-19.62)+14.26\mathrm{cos}(2{\times}0.21t+72.21)+1.12\mathrm{cos}(3{\times}0.21t-11.75),\\{\rho}_{\mathrm{rf}}=97.50+34.40\mathrm{cos}(0.21t+214.63)+6.26\mathrm{cos}(2{\times}0.21t+38.25)+2.68\mathrm{cos}(3{\times}0.21t-31.94),\\{\rho}_{\mathrm{lf}}=105.55+36.74\mathrm{cos}(0.21t+215.70)+1.96\mathrm{cos}(2{\times}0.21t+163.17)+3.16\mathrm{cos}(3{\times}0.21t+14.45),\end{array}\)
and for turning maneuvers,
\(\begin{array}{l}{\rho}_{\mathrm{rh}}=82.07+21.47\mathrm{cos}(0.22t+60.28)+5.79\mathrm{cos}(2{\times}0.22t+95.23)+5.51\mathrm{cos}(3{\times}0.22t-17.60),\\{\rho}_{\mathrm{lh}}=84.45+27.60\mathrm{cos}(0.22t+67.86)+6.67\mathrm{cos}(2{\times}0.22t+93.49)+4.71\mathrm{cos}(3{\times}0.22t-11.93),\\{\rho}_{\mathrm{rf}}=72.08+51.15\mathrm{cos}(0.22t-5.75)+13.90\mathrm{cos}(2{\times}0.22t+121.72)+10.80\mathrm{cos}(3{\times}0.22t+6.77),\\{\rho}_{\mathrm{lf}}=75.00+46.98\mathrm{cos}(0.22t-25.89)+19.65\mathrm{cos}(2{\times}0.22t+62.20)+3.29\mathrm{cos}(3{\times}0.22t+64.46),\end{array}\)
where the time t is ms, and torsional angle in degrees. (E,F) Camber deformation (the ratio of the maximum arch rise to the mid-span chord length). The smooth curves are fitted by B-spline function. The value of the error bar is determined from the ratio of measurement error of any point on the distorted fringe to the mid-span chord length.

Fig. 7.

Wing kinematics for two flight behaviors: (A,C,E) for forward flight;(B,D,F) for turning maneuvers. The middle of the error bar is the actual data point. (A,B) Flapping angle. The smooth curves are fitted by the Fourier series as Equation 1, with the parameter k=1. The value of the error bar is the standard error of fitting expressed as equation 2, with the parameter P=4. The flapping angles of right and left hindwing and right and left forewing are represented by the symbols φrhlh, φrf and φlf,respectively. For forward flight,

\(\begin{array}{l}{\varphi}_{\mathrm{rh}}=0.80+24.19\mathrm{cos}(0.21t+73.34),\\{\varphi}_{\mathrm{lh}}=-1.95+22.27\mathrm{cos}(0.21t+65.47),\\{\varphi}_{\mathrm{rf}}=0.61+20.70\mathrm{cos}(0.21t-31.28)\\\mathrm{and}{\ }{\varphi}_{\mathrm{lf}}=1.96+25.31\mathrm{cos}(0.21t-40.11);\end{array}\)
and for turning maneuvers,
\(\begin{array}{l}{\varphi}_{\mathrm{rh}}=-0.20+16.67\mathrm{cos}(0.22t+168.15),\\{\varphi}_{\mathrm{lh}}=0.78+23.94\mathrm{cos}(0.22t+166.56),\\{\varphi}_{\mathrm{rf}}=1.45+24.08\mathrm{cos}(0.22t+75.44)\\\mathrm{and}{\ }{\varphi}_{\mathrm{lf}}=-9.15+31.91\mathrm{cos}(0.22t+57.14),\end{array}\)
where the time t is ms, and flapping angle in degrees. (C,D)Torsional angle at the 50% spanwise positions. The smooth curves are fitted by the Fourier series as Equation 1 with k=3. The value of the error bar is the standard error of fitting expressed as Equation 2 with P=8. The torsional angles of right and left hindwing and right and left forewing are represented by the symbols ρrh, ρlhrf and ρlf, respectively. For forward flight,
\(\begin{array}{l}{\rho}_{\mathrm{rh}}=86.04+43.92\mathrm{cos}(0.21t-30.95)+13.75\mathrm{cos}(2{\times}0.21t+78.12)+0.86\mathrm{cos}(3{\times}0.21t-4.85),\\{\rho}_{\mathrm{lh}}=79.83+36.88\mathrm{cos}(0.21t-19.62)+14.26\mathrm{cos}(2{\times}0.21t+72.21)+1.12\mathrm{cos}(3{\times}0.21t-11.75),\\{\rho}_{\mathrm{rf}}=97.50+34.40\mathrm{cos}(0.21t+214.63)+6.26\mathrm{cos}(2{\times}0.21t+38.25)+2.68\mathrm{cos}(3{\times}0.21t-31.94),\\{\rho}_{\mathrm{lf}}=105.55+36.74\mathrm{cos}(0.21t+215.70)+1.96\mathrm{cos}(2{\times}0.21t+163.17)+3.16\mathrm{cos}(3{\times}0.21t+14.45),\end{array}\)
and for turning maneuvers,
\(\begin{array}{l}{\rho}_{\mathrm{rh}}=82.07+21.47\mathrm{cos}(0.22t+60.28)+5.79\mathrm{cos}(2{\times}0.22t+95.23)+5.51\mathrm{cos}(3{\times}0.22t-17.60),\\{\rho}_{\mathrm{lh}}=84.45+27.60\mathrm{cos}(0.22t+67.86)+6.67\mathrm{cos}(2{\times}0.22t+93.49)+4.71\mathrm{cos}(3{\times}0.22t-11.93),\\{\rho}_{\mathrm{rf}}=72.08+51.15\mathrm{cos}(0.22t-5.75)+13.90\mathrm{cos}(2{\times}0.22t+121.72)+10.80\mathrm{cos}(3{\times}0.22t+6.77),\\{\rho}_{\mathrm{lf}}=75.00+46.98\mathrm{cos}(0.22t-25.89)+19.65\mathrm{cos}(2{\times}0.22t+62.20)+3.29\mathrm{cos}(3{\times}0.22t+64.46),\end{array}\)
where the time t is ms, and torsional angle in degrees. (E,F) Camber deformation (the ratio of the maximum arch rise to the mid-span chord length). The smooth curves are fitted by B-spline function. The value of the error bar is determined from the ratio of measurement error of any point on the distorted fringe to the mid-span chord length.

In forward flight, time-dependent variation in wing flapping angles indicated remarkable right—left symmetry of the flapping angles, though the flight had up-and-down fluctuation. The phase difference between the fore and hind pairs of wings, by which the fore pair followed the hind pair, was approximately 100°. And for both forewings and hindwings, the flapping amplitudes were approximately ±30°. In turning maneuvers, the wingbeat motion was not right—left symmetrical for the forewing pair,but was nearly symmetrical for the hindwing pair. The flapping amplitude of the inner wing (the right forewing) was not as distinct as that of the outboard wing (the left forewing). A similar phenomenon was also seen with the hind pair.

The right—left symmetry of the torsional angles in both fore and hind pairs showed little differences between two flight behaviors. Note that the variation in torsional angles of forewing in turning maneuvers was larger than that in forward flight, while that of hindwing in turning maneuvers was smaller than that in forward flight.

In forward flight, the negative camber deformation in both fore and hind pairs during their upstrokes was of short duration, while in turning maneuvers, the negative camber deformation only occurred in the hind pair, but the duration was longer than that in forward flight.

Unsteady aerodynamics of forward flight

Unsteady aerodynamics of forward flight was further studied numerically by solving the unsteady flow about a single airfoil as a two-dimensional model of the right forewing in the forward flight mode. The computation was conducted using an in-house Navier—Stokes solver developed by Liu (see Liu and Kawachi, 1998). As illustrated in Fig. 8A, the geometric model is a 10 mm long, flat-plate airfoil of the same length as the mid-span chord length (defined as c) and thickness approximately 2%of the chord length. The kinematic model consists of two motions, a translational component (plunging), defined as the movement of the mid-span chord corresponding to the flapping angles(Fig. 7A), and a rotational one(pitching) described by the torsional angles(Fig. 7C). The camber deformation is described by a time-varying mean line of the airfoil, with the maximum ordinate occurring constantly at the one-quarter chordwise position throughout a complete beating cycle (Fig. 8A). The shape of the mean line is analytically expressed as two parabolic arcs tangent at the position of maximum mean-line ordinate. The mid-span chord length (c=10 mm) is defined as the reference length and the maximum flight speed (U0=1.71 m s-1) as the reference speed. Hence, the Reynolds number was calculated to be approximately 1140 and the reduced frequency, K=2πfc/2U0 is 0.6136. Computed lift and drag forces are nondimensionalized, as in the study by Liu and Kawachi(1998).

Fig. 8.

A computational fluid dynamic study of unsteady aerodynamics of a two-dimensional (2-D) model of the right forewing in forward flight. (A) A systematic diagram of the 2-D computational model. The geometric model is a flat-plate airfoil with the same length as the mid-span chord length and a thickness approximately 2% of the chord length. The angle of stroke planeγ is defined as the included angle of the stroke plane and the plane X′O′Z′ plane of the body-centered coordinate system. The kinematic model consists of a translational motion and a rotational motion. (B) Plots of time-varying force coefficients in a complete beating cycle with and without camber deformation. The force coefficients Cy and Cz represent the time-variations of drag and lift, respectively. LPCD, large positive camber deformation; NCD, negative camber deformation. (C,D) Flow patterns at zero flapping angle during downstroke with (C) and without (D) camber deformation. Velocity vectors and pressure contours are drawn to visualize the flow patterns about the airfoil; and blue in the color map denotes low pressure and red is high pressure.

Fig. 8.

A computational fluid dynamic study of unsteady aerodynamics of a two-dimensional (2-D) model of the right forewing in forward flight. (A) A systematic diagram of the 2-D computational model. The geometric model is a flat-plate airfoil with the same length as the mid-span chord length and a thickness approximately 2% of the chord length. The angle of stroke planeγ is defined as the included angle of the stroke plane and the plane X′O′Z′ plane of the body-centered coordinate system. The kinematic model consists of a translational motion and a rotational motion. (B) Plots of time-varying force coefficients in a complete beating cycle with and without camber deformation. The force coefficients Cy and Cz represent the time-variations of drag and lift, respectively. LPCD, large positive camber deformation; NCD, negative camber deformation. (C,D) Flow patterns at zero flapping angle during downstroke with (C) and without (D) camber deformation. Velocity vectors and pressure contours are drawn to visualize the flow patterns about the airfoil; and blue in the color map denotes low pressure and red is high pressure.

The airfoil is placed in a body-centered inertial frame of reference that undergoes the plunging motion, and pitches about a fixed axis at the one-quarter chordwise position. Inflow conditions of the forward flight are realized by defining a velocity vector at upstream, U0, in which two velocity components form an angle identical to the stroke plane inclination of approximately 15°, namely γ=15°, constantly during a complete beating cycle. Simulations were undertaken till the periodicity of force coefficients was clearly captured with a constant fluctuation, typically more than three cycles for the given reduced frequency;the results at the fourth cycle were used in the evaluation of the force-related quantities and in the flow visualization.

Fig. 8B illustrates comparison of the time variations of the force coefficients, Cy (drag) and Cz (lift), with and without the camber deformation. Their relationship with the vertical force and the thrust force can be given as:
\[\ \begin{array}{c}C_{\mathrm{thrust}}=-C_{\mathrm{y}}\mathrm{cos}{\gamma}-C_{\mathrm{z}}\mathrm{sin}{\gamma}\\C_{\mathrm{vertical}}=-C_{\mathrm{y}}\mathrm{sin}{\gamma}-C_{\mathrm{z}}\mathrm{cos}{\gamma},\end{array}\]
3
where the terms Cthrust and Cverticaldenote the thrust force coefficient and the vertical force coefficient,respectively. Note that negative drag —Cycontributes to the force generation of both vertical force and thrust, whereas positive lift Cz contributes to the vertical force as well as to the drag force. The flexible airfoil with prescribed, time-varying camber deformation shows obvious discrepancy in the force generation compared with the rigid airfoil, particularly at the middle downstroke where the large positive camber deformations (LPCD as illustrated in Fig. 8B) are observed of approximately 9-10% of the chordwise length (see Fig. 7E). Fig. 8C,D) shows comparison of the velocity vectors and pressure contours about the airfoils with and without the camber deformation, which supports such discrepancy in the force generation. Note that the negative camber deformation (NCD in Fig. 8B) seems merely to lead to a slight difference at upstroke.

Overall patterns of flapping angles, angles of attack and camber deformation throughout the flapping cycle can be explained partly by aerodynamic forces. In fact, Ennos showed that inertial forces may be more important, as well as the constraints of the musculo—skeletal system. When the flapping angle increases (upstroke), the angle of attack and the torsional angle are negative, but when the flapping angle decreases(downstroke), they are both positive. This pattern is necessary for generating the higher lift force during the downstroke and thrust during the upstroke. Near the maximum and minimum angular positions, the torsional angle is approximately zero and the angle of attack ±90°, indicating that the wing performs pronation at the maximum angular position and supination at the minimum angular position. Several mechanical models with this flapping pattern were used to study the unsteady aerodynamic force in insect flight(see Dickinson et al., 1999;Sane and Dickinson, 2001).

The camber deformation may play an important role in lift and drag coefficients. Positive camber deformation of the hindwing during the downstroke generates the vertical force, whereas negative camber deformation of the wing during upstroke generates thrust force. In contrast to some previous reports that the camber deformation has a rather minor effect on aerodynamic forces at high angles of attack (see Dickinson and Gutz, 1993; Sunada et al., 1993), we notice that it may be important for aerodynamic models of flight performance. Our present computational fluid dynamics study of the unsteady aerodynamics of the right forewing undergoing forward flight indicates that the airfoil with time-varying camber deformation very likely plays a role in delaying the development and shedding of the leading-edge vortex (see Fig. 8C,D), and hence enhances the delay of the dynamic stall. At the same moment with a zero flapping angle,the vortex over the upper surface of the airfoil with the camber deformation clearly is smaller and closely attached, leading to a much lower negative pressure region on the upper surface (Fig. 8C,D). Hence, at downstroke the airfoil with the camber deformation obviously generates greater forces in both thrust and lift than the rigid airfoil (Fig. 8B). At upstroke, however, it is very interesting to notice that this discrepancy is remarkably reduced (Fig. 8B). On the other hand, the value of wing camber deformation under steady state conditions is clear. In the range of Reynolds' number within which insects operate (Re<4×104) an arched plate gives higher lift coefficient values than a flat plate(Hertel, 1966). Though any non-steady benefits are less apparent, active alteration of camber deformation by the raising and lowering of flaps may be important in maneuvering and in the maintenance of stability. Wootton(1981) mentioned that Agrion (Odonata) asymmetrically lowers the cubitoanal region of fore and hindwings during the downstroke when turning; and Pringle(1961) described how the hindwing of bees might similarly be depressed in the control of rolling movements. During insect flapping, it was reported that the wings themselves deform semi-automatically, optimizing aerodynamic forces(Wootton, 2000). Alexander(2000) remarked,`Previously, we thought of insect wings as stiff, flat plates — now we know that some bend and twist in flight in ways that must have large aerodynamic effects.' However, the effect of the wing camber deformation on unsteady aerodynamic forces is still not clear, particularly in the case of large angles of attack; more extensive studies need to be done.

The flight characteristics of the forewings reveal many differences between forward flight and turning maneuvers, while differences in the hindwings are rare. All show that the freely flying dragonfly's hindwings play an important role in providing a stable lift force, and the forewings are very important for controlling flight behavior, such as turning, climbing and so on.

Further understanding of wingbeat kinematics and flight attitude would require a detailed analysis of simultaneous aerodynamic and inertial forces on the wing. Using a PCFL method, flapping angles, angles of attack and camber deformation of insect wings can be determined for unrestrained free flight. No assumptions concerning wing geometry or deformability are necessary, and wing contours can be determined with high accuracy. Moreover, the body attitude at any instant can be determined simultaneously, which is important for analyzing flight control in the insect. These experimental results show that there is considerable camber deformation during free flight of dragonflies. The wingbeat kinematic parameters of the dragonfly studied here are broadly similar to those reported by previous works for similar-sized dragonflies (see Azuma and Watanabe, 1988; Wakeling and Ellington, 1997),but to date no detailed information on wing profile and flight attitude during free flight has been acquired. This study thus demonstrates the feasibility of obtaining detailed information on wing geometry during the flight of insects.

The authors wish to thank Professor Robert Dudley for his kind advice. This work was supported by National Natural Science Foundation of China 59925514 and The Research Fund for the Doctoral Program of Higher Education.

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