Fig 1.
(A) We take the theoretical optic flow vector () to be the opposite of the 3-D speed vector (
) experienced by the hoverfly. We calculated a horizontal (
) and a vertical (
) component of this theoretical optic flow vector in the hoverfly’s reference frame (Rfly) depending on the estimated pitch orientation. (B) The force produced by the hoverflies’ flapping wings (
) is assumed to be orthogonally oriented with respect to the body pitch orientation [11]. Moving forward is then achieved by pitching down from the head and moving backward, or braking, by pitching up from the head. Lift force (
) corresponds to the vertical component of
in the inertial reference frame and thrust force (
) to the horizontal component. As depicted in [35], we assumed a pure active control of the pitch torque which is seen to occur during a fraction of the wingstroke, about half of a wing beat period (i.e., about 2ms for an hoverfly, see [3]).
Fig 2.
(A) Control of the pitch rate () in response to horizontal optic flow component (ωxRfly). The measured ωxRfly is compared with an input reference set-point,
, and the error (ϵω) is then sent to a Proportional-Derivative controller, which delivers a reference pitch rate,
. We then used a second loop mimicking the halteres, to measure and adjust the pitch rate,
, to this set-point. The pitch θp obtained by integration (by definition) is then used to calculate the orientation of the thrust and the speed vector in the inertial reference frame (see B). The norm of the force produced by flapping wings is assumed to correspond to a second order transfer function with a zero based on the data. The two components of the optic flow vector in the fly reference frame, ωxRfly and ωzRfly, are then calculated geometrically (see B). ωxRfly is used to close the pitch control loop. (B) Details of the pitch rate control based on horizontal optic flow component.
Fig 3.
Setup and stereo reconstruction.
(A) The setup consists of a 40x40x40cm3 box illuminated from above with a halogen light. Hoverflies were filmed in the box with a fast camera through a two-way mirror at a rate of 1600 frames per second in full resolution (1280 by 800 pixels). The mirror was tilted at an angle of 45° to make the hoverfly see a uniform white wall (i.e. the reflection of the white light-diffusing ceiling). A manual switch was used to trigger the camera and simultaneously turn off the power of the electromagnet, thus releasing the resting fly and causing it to fall. A set of mirrors was used to split the images, giving two different views of the experimental box. (B) Example of a split image obtained with the set of mirror with the two trajectories superimposed. The two trajectories were used to obtained a 3-D reconstruction with the MATLAB stereo vision toolbox.
Fig 4.
Measured 3D trajectories and wingbeat triggering times.
(A) Hoverflies’ XY trajectories in all the trials in which flight was initiated before 150ms. Trajectories were stopped at the moment were animals overpass 0cm.s−1 vertical speed. Green lines correspond to stabilized flights, blue one to non stabilized flights. Red arrows indicate the direction between the start and the end of stabilized flight. Red dotted lines give the outlines of the experimental box. The high-speed camera is positioned approximately in coordinate (0,-100). The values on the X-axis are width coordinates and those on the Y-axis are depth coordinate. (B) Height of the hoverflies’ flight during the trials in which flight was initiated before 150ms. Green lines correspond to stabilized flights, blue lines to non stabilized flights. (C) Histogram of the times to initiate wingbeat after the onset of the fall observed during the experiments. Distribution has been fitted with both normal (red) and lognormal (green) rules distribution giving mean ΔWB respectively equal to 103.121ms and 103.11ms.
Fig 5.
Model’s responses to a free fall (1).
(A) Averaged simulated pitch orientation versus time in the 3 initiation time groups: 75 to 100ms (green line, top panel), 100 to 125ms (blue line, middle panel) and 125 to 150ms (red line, bottom panel). Gray lines are the experimental individual timelines. All the data were synchronized with the reference wingbeat triggering time: tWB = 0. (B) Averaged simulated optic flows, (solid line) and
(dashed line), versus time in the 3 initiation time group: 75 to 100ms (green line, top panel), 100 to 125ms (blue line, middle panel) and 125 to 150ms (red line, bottom panel). Gray lines show experimental data of
(solid line, dark gray) and
(dashed line, light gray).
Fig 6.
Model simulations in free fall (2).
(A) Simulated acceleration produced by the flapping wings with time (green line) and median from thrust data ± interquantile deviation (dark lines). Time zero corresponds to wingbeat initiation. (B) Averaged simulated flight height versus time corresponding to wingbeat initiation time ranging from 75 to 100ms (green lines), 100 to 125ms (blue lines) and 125 to 150ms (red lines). Vertical bars show the wingbeat triggering times with the same color code. Grey shadowed lines are the 44 experimental time courses of the flight height. (C) Averaged heave speeds observed experimentally (solid lines) and in the simulation (dotted lines) in the three wingbeat initiation group: 75 to 100ms (green lines), 100 to 125ms (blue lines) and 125 to 150ms (red lines). (D) Averaged surge speeds observed experimentally (solid lines) and in the simulation (dotted lines) in the three wingbeat initiation group: 75 to 100ms (green lines), 100 to 125ms (blue lines) and 125 to 150ms (red lines).
Fig 7.
Block diagram of the improved model including a flapping wing force feedback control based on the vertical optic flow (green) and a pitch rate closed-loop control based on the absolute pitch measurement (purple).
(A) Diagram of the model including a feed-back control loop based on ωZRfly accounting for the force produced by flapping wings and a feed-back control loop based on θp accounting for the control of . (B) Averaged (large solid line) and 10 samples (thin solid line) height measured during 150 simulations of free fall conducted with the new model. Model simulations results are presented in color (wingbeat triggering time of 75-100ms are shown in green, 100-125ms in blue and 125-150ms in red) and experimental data appears in grey. Inset shows the percentage of cumulated crash during 150 model simulations lasting 3 seconds. (C) Averaged heave (top panel) and surge speed (bottom panel) measured during the 150 simulations of free fall conducted with the new model. Model simulations are presented in dotted lines and experimental data in solid lines. Results are split in the three wingbeat initiation groups: 75-100ms are shown in green, 100-125ms in blue and 125-150ms in red.
Fig 8.
Experimental versus improved model responses in terms of pitch control dynamics.
(A) Left panels give experimental (grey: individuals; dark: mean) and the simulated (color shaded: 10 individual samples; color plain: mean) pitch orientation versus time corresponding to the 3 groups of initiation time: 75-100ms are shown in green, 100-125ms in blue and 125-150ms in red. (B) Right panels represent the experimental (grey: individuals; dark: mean) and the simulated (shadowed: individuals; dark: mean) optic flows (: dotted lines,
: solid lines) versus time corresponding to the 3 groups of initiation time: 75-100ms are shown in green, 100-125ms in blue and 125-150ms in red.