Wing-pitch modulation in maneuvering fruit flies is explained by an interplay between aerodynamics and a torsional spring

Tsevi Beatus and Itai Cohen

Phys. Rev. E 92, 022712 – Published 14 August 2015

Abstract

While the wing kinematics of many flapping insects have been well characterized, understanding the underlying sensory, neural, and physiological mechanisms that determine these kinematics is still a challenge. Two main difficulties in understanding the physiological mechanisms arise from the complexity of the interaction between a flapping wing and its own unsteady flow, as well as the intricate mechanics of the insect wing hinge, which is among the most complicated joints in the animal kingdom. These difficulties call for the application of reduced-order approaches. Here this strategy is used to model the torques exerted by the wing hinge along the wing-pitch axis of maneuvering fruit flies as a damped torsional spring with elastic and damping coefficients as well as a rest angle. Furthermore, we model the air flows using simplified quasistatic aerodynamics. Our findings suggest that flies take advantage of the passive coupling between aerodynamics and the damped torsional spring to indirectly control their wing-pitch kinematics by modulating the spring parameters. The damped torsional-spring model explains the changes measured in wing-pitch kinematics during roll correction maneuvers through modulation of the spring damping and elastic coefficients. These results, in conjunction with the previous literature, indicate that flies can accurately control their wing-pitch kinematics on a sub-wing-beat time scale by modulating all three effective spring parameters on longer time scales.

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Received 16 April 2015
Revised 10 July 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022712

Authors & Affiliations

Tsevi Beatus and Itai Cohen

Department of Physics, Cornell University, Ithaca, New York 14853, USA

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Vol. 92, Iss. 2 — August 2015

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Images

Figure 1
Roll correction maneuver following a midair impulsive perturbation. (a) Superimposed images from three orthogonal cameras of the fly during the maneuver. The 3D-rendered fly represents the measured kinematics. The perturbation location [red (dark gray) line] is shown on the fly's center-of-mass trajectory in green (light gray). In the second snapshot the fly is deflected $45^{\circ}$ to its right with respect to its preperturbation roll angle. (b) Definition of body Euler angles with respect to the laboratory frame. ${\hat{x}}_{b}$ is the long body axis. (c) Definition of the body frame $({\hat{x}}_{b}, {\hat{y}}_{b}, {\hat{z}}_{b})$ and wing Euler angles, measured in the body frame with respect to the stroke plane shown in shaded blue (dark gray). The wing-pitch angle $ψ$ describes the rotation of the wing about its vein, or leading edge; vector is shown for the right wing by a curved arrow. A pitch angle of $ψ = 0^{\circ}$ implies that the wing is parallel to the wing stroke plane with the leading edge ahead of the wing surface. (d) Body Euler angles versus time. Perturbation was applied between 0 and 5 ms in yellow (black vertical lines). White and gray stripes represent forward and back strokes, respectively. Roll was measured manually at the middle of each half-stroke and smoothed by a spline (dashed black line). Measurement errors are smaller than the symbols size. The snapshots above the plot show top side views of nine consecutive wing strokes of the maneuver, taken when the wings are at their forwardmost position. The perturbation wing beat is numbered 0.
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Figure 2
Wing kinematics. (a) The stroke angle $ϕ$ for both wings as a function of time in $ms$ . Throughout the entire figure, red (light gray) indicates the right wing and blue (dark gray) indicates the left wing data. The magnetic pulse perturbation is indicated by a yellow strip between $t = 0$ –5 ms. White and gray stripes indicate forward and back stroke, respectively. (b) The peak-to-peak amplitude of the stroke angle $Φ$ for each half-stroke. (c) The wing-pitch angles $ψ$ . (d) Wing elevation angles $θ$ . (e) The wing-pitch angle $ψ$ as a function of the stroke angle $ϕ$ for both wings, plotted separately for six wing beats during the roll correction maneuvers. The data in each plot start when the wings are at their backmost position [black squares with red or blue (light or dark gray) outline], and time propagates clockwise as shown by the curved arrow on the leftmost plot. Inset images show top-view snapshots of the fly taken when the wings are in their forwardmost position (minimum $ϕ$ ) during each wing beat. The wing-beat numbering is the same as in Fig. 1.
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Figure 3
Lissajous-Bowditch plots of $ψ$ as a function of $ϕ$ (a) before and (b) during each of the 10 measured correction maneuvers. Each loop on the $(ϕ, ψ)$ plane corresponds to one wing-beat cycle of one wing. Red (light gray) curves correspond the the “bottom” wing in each maneuver, which is the wing that increases its stroke amplitude. Blue (dark gray) curves correspond to the “top” wing, which reduces its stroke amplitude. A square symbol on each curve indicates the point of maximum $ψ$ in the front half-stroke. A circle on each curve indicates the point of minimum (frontmost) $ϕ$ . (a) Before the maneuver the curves for the “top” and “bottom” wings are indistinguishable ( $p$ values indicated on the plot). (b) During the maneuver the maximum $ψ$ for the “top” wing is $169^{\circ} \pm 10^{\circ}$ ( $mean \pm standard$ deviation) and its value for the “bottom” wing is $144^{\circ} \pm 8^{\circ}$ ( $p$ value of $8.5 \times 10^{- 6}$ ). The frontmost $ϕ$ values during the maneuver are $40^{\circ} \pm 9^{\circ}$ for the “top” wing and $9^{\circ} \pm 9^{\circ}$ for the “bottom” wing, corresponding to an increase of the stroke amplitude ( $p$ value of $5.2 \times 10^{- 7}$ ).
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Figure 4
The three components of the aerodynamic torque generated by the two wings during wing beat 3 of the roll maneuver plotted as a function of time in $ms$ (solid black lines). [(a)–(c)] The torque components shown are along the $(x, y, z)$ axes of the body frame of reference [defined in Fig. 1]. The torques were calculated based on the full wing and body kinematics and using a quasi-steady-state aerodynamic force model. Plotted in dashed red (light gray) lines are the aerodynamic torque generated by the same wing-beat kinematics but with $ψ (t)$ taken from a nonmaneuvering wing beat (0). Solving the Euler equation of motion for the body using the torque calculated for these two cases and the experimental initial conditions showed that the measured kinematics reduced the body roll velocity from $2710^{\circ} s^{- 1}$ to $- 870^{\circ} s^{- 1}$ (a slowdown of $3580^{\circ} s^{- 1}$ ), while the “mixed” kinematics reduces the body roll velocity to $360^{\circ} s^{- 1}$ (a slowdown of $2350^{\circ} s^{- 1}$ ).
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Figure 5
Torsional spring model for wing pitch. [(a) and (b)] A fly scheme with the spring torque indicated by red arrows. The wing frame of reference is shown in (b). [(c) and (d)] The operation of the spring (black spiral) on a moving wing (thick black line). For wing with low air speed (c) Red (light gray) line indicates the aerodynamic force that twists the torsional spring such that in steady state (when $\dot{ψ} = 0$ ) the spring torque and aerodynamic torques are balanced. (d) When the wing is moving faster, the aerodynamic force is stronger and so is the torque it generates about the wing base. In steady state the torsional spring is twisted further than in (c) such that the two torques balance at a shallower wing-pitch angle.
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Figure 6
Calculated wing-pitch kinematics from the torsional-spring model plotted in the $(ϕ, ψ)$ plane (solid black lines) along with the smoothed measured kinematics of the left and right wings (circles). Fit results for for a wing beat before the correction maneuver (0) are shown in (a) for the left wing and in (b) for the right wing. Similarly, fit results for a maneuvering wing beat (3) are plotted in (c) for the left wing and in (d) for the right wing. Each plot [(a)–(d)] includes also the values of the fitted spring parameters. The mean RMSE values for the fits in (a)–(d) were $5 . 7^{\circ}$ , $5 . 2^{\circ}$ , $6 . 2^{\circ}$ , and $6 . 1^{\circ}$ , respectively. The torsional spring model captures the salient features of the measured $ψ$ kinematics. The dashed gray curves in (c) and (d) correspond the the calculated $ψ$ kinematics resulting from the wing and body kinematics of the maneuvering wing beat combined with the fitted spring for the nonmaneuvering wing beat in (a) and (b), respectively.
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Figure 7
The calculated torque exerted by the spring during the same wing beats shown in Fig. 6. The torques were calculated using Eq. (1) with the fitted spring parameters and the measured wing kinematics. The total spring torque $τ_{s}$ is plotted in dashed thick green (light gray) lines, the elastic torque term $- k (ψ (t) - ψ_{0})$ is plotted in black, and the damping torque term $- c \dot{ψ}$ is plotted in red (darker gray). The wing beats shown are a nonmaneuvering wing beat for the left (a) and right (b) wing, as well as a manuevering wing beat for the left (c) and right (d) wing. Black arrows indicate a time when the torque in (d) is more negative than in (a)–(c) as a result of the damping term after the front wing flip.
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Figure 8
Intuition for the individual effect of each spring parameter on the $ψ (ϕ)$ curve. We calculated the wing-pitch kinematics of the nonmaneuvering wing beat in Fig. 6 and as a reference used the spring with $k = 50 pN \times m / \deg$ , $c = 25 fN \times m / (\deg s^{- 1})$ , and $ψ_{0} = 0^{\circ}$ (with respect to the vertical). The curves resulting from the reference spring are plotted in solid black lines. (a) Wing-pitch curves for increasing $k$ . Panels (b) and (c) show similar curves for increasing $c$ and $ψ_{0}$ , respectively. Color bars show values of each parameter.
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Figure 9
[(a)–(c)] the fitted spring parameters $k$ , $c$ , and $ψ_{0}$ , as a function of time in wing beats for the maneuver analyzed in Figs. 1–4 and 6. The time $t = 0$ corresponds to wing beat in which the fly was perturbed. Data are shown for the right wing in red (light gray) and for the left wing in blue (dark gray). Error bars correspond to the fit's confidence intervals (CI). [(d)–(f)] The same fit results for $k$ , $c$ , and $ψ_{0}$ , plotted as a function of the wing-stroke amplitude for the corresponding wing beat of each wing. The CI are the same as in (a)–(c).
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Figure 10
[(a)–(c)] The differences in the fitted spring parameters $Δ k$ , $Δ c$ , and $Δ ψ_{0}$ calculated with respect to their values for the corresponding wing prior to each maneuver. The differences are plotted as a function of the wing-stroke amplitude $Φ$ . The first six wing beats for each of the 10 maneuvers analyzed are shown, constituting the active part of each maneuver [21]. Red (light gray) circles indicate data for the “bottom” wing in each roll maneuver, which is the wing that flapped with increased stroke amplitude to exert roll correcting torque. Blue (dark gray) squares indicate data for the “top” wing in each maneuver. A dashed black line on each plot in (a)–(c) indicates the mean stroke amplitude before the maneuver $Φ_{mean} = 156^{\circ} \pm 2 . 5^{\circ}$ ( $mean \pm standard$ error for $n = 20$ ). Wing beats with $Φ > Φ_{mean}$ . The data for $Δ k$ and $Δ c$ are correlated with $Φ$ , with correlation coefficients of $\sim 0.5 \pm 0.15$ (value $\pm 95 %$ confidence interval) and $p$ values of $p = 3.6 \times 10^{- 10}$ for $Δ k$ and $p = 8.8 \times 10^{- 8}$ for $Δ c$ . [(d)–(f)] The mean values of $Δ k$ , $Δ c$ , and $Δ ψ_{0}$ , below and above the $Φ = 165^{\circ}$ threshold. Black error bars indicate the standard error of each mean. The $p$ values in each plot are the results of a t test comparing the data below and above the threshold.
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Figure 11
[(a)–(c)] The fitted spring parameters $k$ , $c$ , and $ψ_{0}$ as a function of time in wing beats for a fly hovering for 22 wing beats. Data are shown for the right [red (light gray)] and left [blue (dark gray)] wing and the error bars correspond to the fit's confidence intervals (CI). The mean and standard deviation values for the spring parameters of the left ( $L$ ) and right ( $R$ ) wings are $k_{L} = 47 \pm 2.6 pN \times m / \deg$ , $c_{L} = 29 \pm 1.6 fN \times m / (\deg s^{- 1})$ , $ψ_{0}, L = - 10 \pm 3 . 5^{\circ}$ , $k_{R} = 54 \pm 2.3 pN \times m / \deg$ , $c_{R} = 24 \pm 2 fN \times m / (\deg s^{- 1})$ , and $ψ_{0}, R = - 4 \pm 3 . 5^{\circ}$ . [(d)–(f)] The same fit results for $k$ , $c$ , and $ψ_{0}$ , plotted as a function of the wing-stroke amplitude for the corresponding wing beat of each wing. The CI are the same as in (a)–(c). The mean stroke amplitude of both wings was $151.4 \pm 2^{\circ}$ ( $mean \pm standard$ deviation, $n = 44$ ).
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