The analysed trials had a duration of 0.59 ± 0.09 s (mean ± s.d.), during which the moths rotated by 91 ± 23° at a whole-recording angular velocity of 159 ± 57° s−1. Maximum instantaneous yaw rates within each trial ranged from 723 to 306° s−1. Mean linear velocity during the trials was 0.37 ± 0.14 m s−1.
The moths exhibited a regular pattern of within-wingbeat changes in yaw rate, with the angular velocity at the end of downstroke greater than the angular velocity at the end of the following upstroke (a
). Considered across all wingbeats collected in this study (b
), the pattern was statistically significant (p
< 0.0001) and the ratio of the velocity at the end of upstroke to velocity at the end of the preceding downstroke was found to be 0.70 ± 0.06 (c
, model II regression slope ± s.e.), i.e. the moths accelerated in downstroke and decelerated in upstroke. To see whether this result was consistent with FCT damping of a passive upstroke following an active downstroke, we used the FCT model to predict the upstroke to downstroke velocity ratio. Since FCT shows that an exponential decay model predicts angular deceleration in yaw turns, the FCT half-life and event duration are all that is required to predict the amount of deceleration. We used the previously reported FCT half-life for M. sexta
(28.4 ms, Hedrick et al. 2009
) and the duration of upstroke, found to be 17.3 ± 2.2 ms (mean ± s.d.) for these data, resulting in a predicted ratio of:
Figure 2. (a) Yaw velocity through time for a single trial, with end of downstroke and upstroke points marked. (b) Aggregate data comparing end of downstroke and end of upstroke yaw velocities for all trials and wingbeats along with model II regression results. (more ...)
The predicted ratio is within the 95% CI for the ratio determined from the data, 0.59–0.82.
Wing kinematic measures associated with yaw turns were similar to those described for banked yaw turns in the forward flight of the hawkmoth species Agrius convolvuli
(Wang et al. 2008
). As in that study, the relative positions of the left and right wing at the end of the downstroke (i.e. at the time of ventral stroke reversal) were significant predictors of the mean yaw velocity for the wing stroke in question, with an R2
of 0.58 and p
< 0.0001 for the relationship between sweep angle (ϕ
) differences and velocity rate and an R2
of 0.50 and p
< 0.0001 for elevation angle (θ
) differences relating to yaw velocity (see the electronic supplementary material, figure S1). The correlations were such that yaw to the right was associated with the right wing reaching a greater elevation angle than the left wing at the end of the downstroke and also with the right wing ending with a lesser sweep angle than the left wing. Neither of these wing kinematic measurements was associated with body acceleration to a statistically significant (p
< 0.05) degree.
The effects of the measured kinematic pattern were examined by recreating them in a simulation of hawkmoth flight (Hedrick & Daniel 2006
). A flapping pattern which created a −12.5° right–left difference in wing elevation angle and 10° difference in wing sweep angle at the end of the downstroke, the approximate midpoints of the kinematic observations (see the electronic supplementary material, figure S1) resulted in rapid yaw acceleration to the right during the downstroke followed by deceleration during the upstroke (d
). Beginning from rest, the simulated moth using this biologically derived kinematic accelerated during the downstroke then decelerated during the upstroke, reaching a yaw velocity of 120° s−1
at the end of the first stroke, similar to that exhibited by real moths with similar kinematic deviations. However, these were not the only kinematics capable of producing yaw in the simulated moth. An alternative kinematic which modified the basic flapping motion only by increasing the mean spanwise rotation of the right wing by 10°, thus increasing wing angle of attack during the downstroke while decreasing it during the upstroke, was chosen for comparison to the biological case as the two kinematics reach similar peak yaw rates (d
). This alternative kinematic produced acceleration during both downstroke and upstroke and a yaw velocity of 320° s−1
after one stroke. Analysis of the simulation outputs revealed that the alternative kinematic reduced manoeuvring torque and damping by 59 and 64 per cent, respectively. Thus, continuous turn dynamics of the type hypothesized in b
, and characterized by reduced but less variable torque and reduced damping, are probably feasible in flapping flight. There is no evidence that moths turn in this way, but a recent report of fruitfly yaw turning showed that those animals yaw by increasing the spanwise rotation of one wing (Bergou et al. 2010
), the hypothetical kinematic which gave rise to more continuous turns in our simulations.