Low speed maneuvering flight of the rose-breasted cockatoo (Eolophus roseicapillus). II. Inertial and aerodynamic reorientation

Birds and other flapping fliers have long been noted for their agility in flight, especially in comparison with fixed wing aircraft. While much of the difference in apparent maneuverability is no doubt a product of the lower wing loading and therefore greater `intrinsic' maneuverability of biological fliers(Warrick et al., 1998), some may be due to means of reorientation available to flapping fliers but not to fixed wing aircraft. Changes in roll orientation and the resulting redirection of net aerodynamic force form the basis for changes in flight path or direction in many flying animals, including pigeons(Warrick and Dial, 1998),cockatoos (Hedrick and Biewener,2007b), bats (Aldridge,1987), houseflies (Wagner,1986) and fruit flies (Fry et al., 2003), as well as fixed wing aircraft. Therefore, mechanisms for changing roll orientation may strongly influence maneuvering performance. In addition to changes in orientation due to external forces such as aerodynamic torques acting on the wings, flapping fliers may also reorient via inertia, much as a cat does when dropped from a height (e.g. Frohlich, 1980). In flapping flight, right–left asymmetry in the arcs swept by the two wings leads to instantaneous changes in body orientation. When the moments of inertia of the wings and body do not vary through the stroke cycle, these inertial reorientations lead to no net change over the course of a complete wingbeat cycle. However, in vertebrate fliers with jointed wings capable of changes in moment of inertia about each axis throughout the stroke, net inertial reorientations may also occur and contribute to maneuvering performance.

Here, we use a blade-element aerodynamic model of force production to estimate the aerodynamic torques and resulting changes in orientation generated by turning cockatoos Eolophus roseicapillus. We also estimate the amount of inertial reorientation due to wing arc and moment of inertia asymmetries at different points in the wingbeat cycle. Based on our initial investigation (Hedrick and Biewener, 2007b), we hypothesized that the estimated change in orientation due to inertial effects would predominate during the wingbeat and that aerodynamic forces would have a greater influence on among-wingbeat changes in orientation.

Prior to the analysis presented in Hedrick and Biewener(Hedrick and Biewener, 2007b)and based on an earlier study of turning flight in pigeons(Warrick and Dial, 1998), we hypothesized that the pectoralis would be the key muscle for control of maneuvering flight. However, we then discovered that much of the link between wrist velocity and change in roll rested on inertial relationships that might have only a small effect over a complete wingbeat cycle. This suggested that other muscles might also play an important role in turning flight, as was reported in an earlier electromyographic and kinematic study of pigeons maneuvering through a slalom course (Dial and Gatesy, 1993). We therefore hypothesized that all four muscles examined, the pectoralis, supracoracoideus, biceps brachii and extensor metacarpi radialis, contribute to different factors related to aerodynamic and inertial reorientation. The biceps brachii influences wing rotation on the spanwise axis and therefore angle of attack(Dial and Gatesy, 1993),contributing to aerodynamic asymmetry. The pectoralis, the main source for muscular force and power during the downstroke in flapping flight, was previously shown to be important in inertial reorientation(Hedrick and Biewener, 2007b). The supracoracoideus, the main wing elevator and a key supinator during upstroke (Poore et al.,1997b), likely influences the arc swept by the wing and therefore both inertial and aerodynamic forces. Finally, the wrist extensor influences both wing area and distal wing moments of inertia, which are also likely to influence aerodynamic torque and inertial reorientation forces,respectively.

Aside from the differences noted in this section, the kinematic and electromyographic data used in this analysis were identical to those described in the companion paper (Hedrick and Biewener, 2007b). The methods of data reduction, kinematic frames of reference, and EMG processing were also identical except where otherwise noted.

In addition to the points digitized previously on the cockatoos Eolophus roseicapillus Viellot(Hedrick and Biewener, 2007b),we also digitized the trailing edge of the wing at the tips of the 4th primary and 1st secondary feathers. These points were digitized only at mid-downstroke when the wing was fully extended and the individual feathers were easily identified by counting in from the tip; the trailing edge points could not be accurately identified at other points in the wingbeat cycle. However, flight muscle and therefore aerodynamic forces are greatest at mid-downstroke(Hedrick et al., 2003), making mid-downstroke the most characteristic point in a wingbeat cycle.

Aside from the EMG plug base and the trailing edge points described above,the cockatoos were digitized only at the end of downstroke, mid-upstroke, the start of downstroke, and the seven frames surrounding mid-downstroke. These four points in the wingbeat cycle were identified visually from the video sequences; we include a series of still images showing typical body and wing posture at each stage (Fig. 1).

We used a blade-element model to estimate the aerodynamic forces and torques generated by the cockatoos' wings at mid-downstroke. This mid-downstroke estimate was then used as a basis for estimating the force and torque impulses delivered during the entire wingbeat. While blade-element models have a long history in biological flight research (e.g. Osborne, 1951; Ellington, 1984), recent advances in measurement of the aerodynamic coefficients appropriate for rotating wings enhance their applicability to kinematic studies of animal flight (Sane, 2003). While the majority of these examples are from insect flight and at lower Reynolds numbers (∼100 to ∼2000) than those characteristic of the cockatoos(∼26 000), Usherwood and Ellington(Usherwood and Ellington,2002b) found high force coefficients for revolving quail wings,also at a Reynolds number of ∼26 000.

The direction of aerodynamic force acting on each wing segment was determined by assuming that the net aerodynamic force on a wing was directed orthogonal to the upper surface of the wing(Usherwood and Ellington,2002a). A coefficient of net force was estimated for each segment from empirical measurement of the force coefficients on a revolving quail wing(Usherwood and Ellington,2002b) and an estimate of the angle of attack of each segment(Eqn 3). Although the wings of the cockatoos studied here have a greater aspect ratio than the quail wings studied by Usherwood and Ellington(Usherwood and Ellington,2002b), variation in wing shape parameters had little influence on the force coefficients measured from revolving wings(Usherwood and Ellington,2002b).

The wing surface orientation, for use with the normal forces assumption and determination of angle of incidence, was computed from the X_b and Z_b components of the normal vector to the plane defined by the positions of the wrist, tip of the 4th primary and tip of the 1st secondary, giving a spanwise rotation angle,γ, for the entire wing (Fig. 2B).

General results characteristic of the entire turning flight were computed as the inter-individual mean of the six cockatoos. Regression and partial correlation tests against kinematic data were computed from the standardized wingbeats from each of the cockatoos in each of the turn directions, resulting in N of ∼66 (six cockatoos, six wingbeats when turning left, five wingbeats when turning right). Regression and partial correlation tests against EMG data were computed from the standardized EMG differences,resulting in N of ∼30 in each comparison. All computations were performed in MATLAB 7.0 (Mathworks Inc., Natick, Massachusetts).

We found that the estimated net aerodynamic torque at mid-downstroke was a significant predictor of roll acceleration (r²=0.55, P<0.00001, Fig. 3). This made it a more informative predictor than right–left asymmetry in wrist velocity, the basic kinematic parameter most closely related to roll acceleration, which predicted roll acceleration with an r²of 0.34 (Hedrick and Biewener,2007b).

The estimated mean upward force was 2.47±0.58 N (N=6). This corresponds to 88±17% (N=6) of the force required to support the bird in the air. Vertical force also varied through the turn, reaching a local minimum at the 0th wingbeat, the middle wingbeat of the turn(Fig. 4).

Centripetal acceleration varied widely among the wingbeats that made up the turn, but was on average 8.37±1.95 m s^–2(N=6). Centripetal acceleration was greatest during the 0th wingbeat of the turn, with an inter-individual mean of 10.86±2.79 m s^–2 (N=6). Not surprisingly, the inward aerodynamic force providing centripetal acceleration also varied widely among wingbeats,and reached a maximum at the 0th wingbeat(Fig. 5). The estimated mean inward force generated during each wingbeat was somewhat less than the product of centripetal acceleration and body mass for each cockatoo(Fig. 5). On average, the estimated inward force accounted for 79±29% (N=6) of the centripetal force required to produce the observed change in heading.

Aerodynamic power varied slightly among wingbeats in the turn and among individual birds. The overall mean estimated aerodynamic power was 13.48±4.23 W (N=6), which corresponds to a pectoralis mass-specific power of 233.2±65.8 W kg^–1(N=6). Power was greatest at the middle wingbeat of the turn and tended to decline by the last wingbeat as the birds approached the landing perch (Fig. 6).

The predicted inertial reorientations for each phase of the wingbeat cycle were significantly related to measured reorientation during that phase(Fig. 7). However, the degree of correlation between the inertial predictions and actual reorientation was moderate and varied among phases, ranging from r²=0.37 for the mid-upstroke to start of downstroke phase to r²=0.20 for the end of downstroke to mid-upstroke phase. The slopes and intercepts of the linear regression lines averaged 1.19° and –1.07°,respectively. Thus, our predictions of change in orientation over these subsections of the wingbeat cycle were of similar magnitude to the measured changes. When summed over an entire wingbeat, the predicted inertial changes in body roll were significantly but moderately (r²=0.19)related with the measured change in roll(Fig. 8). As before, the regression slope and intercept were near one and zero, respectively.

Although we found no individual muscle activation parameters that explained or were correlated with either the net aerodynamic torque or inertial reorientation summed over an entire wingbeat, we found many relationships between muscle activation measurements and different components of our aerodynamic and inertial reorientation estimates. These relationships are summarized in Table 1 and noted below. The outside wing–inside wing difference in spanwise rotation angle (γ) was significantly related to the outside–inside difference in pectoralis EMG intensity (r²=0.331, P<0.005, F=12.9) such that greater EMG intensity was associated with upward rotation of the trailing edge. The outside–inside difference in the supracoracoideus rectified impulse was also correlated with spanwise rotation (r²=0.247, P=0.01, F=8.2) such that greater supracoracoideus EMG was associated with downward rotation of the trailing edge, an increase in the spanwise rotation angle. Biceps activation duration was correlated with spanwise rotation angle at mid-downstroke (r²=0.273, P=0.015, F=9.4, Fig. 9), with greater activation duration associated with greater spanwise rotation(trailing edge down, supination). The square of wing velocity in the world coordinate system was correlated with pectoralis time to mid-burst, an expression of muscle activation delay (r²=0.433, P<0.0001, F=22.1) such that increased time to mid-burst was associated with increased wing velocity at mid-downstroke. Finally, we found that increased pectoralis activation intensity was correlated with greater aerodynamic force coefficients (r²=0.277, P<0.005, F=9.6, Fig. 9).

We did not find a significant correlation between any individual muscle activation parameter and inertial reorientation over a complete stroke. However, inertial reorientation during discrete portions of the wingbeat was often related to one or several muscle activation parameters. Inertial reorientation from mid-upstroke to the start of downstroke was significantly correlated to the duration of supracoracoideus activation(r²=0.446, P<0.0005, F=20.2, Fig. 9), with longer activation durations corresponding to inertial reorientation toward the same side.

Three different muscles were significantly related to inertial reorientation of roll from the start to the middle of downstroke. A larger biceps EMG impulse was associated with inertial roll to the same side(r²=0.379, P<0.001, F=15.3, Fig. 9). The mean spike amplitude of the wrist extensor was inversely correlated with inertial reorientation (r²=0.420, P<0.0005, F=18.1). Finally, the pectoralis mean spike amplitude was also inversely correlated with inertial reorientation in early downstroke(r²=0.322, P<0.005, F=11.4). Inertial reorientation from the middle to the end of downstroke was positively correlated with only one EMG parameter, the duration of pectoralis activation(r²=0.292, P<0.005, F=10.3).

We found that both aerodynamic and inertial mechanisms contribute to the changes in roll orientation that underlie low speed turning in the rose-breasted cockatoo. Predicted within-wingbeat inertial roll reorientations were significantly related to measured reorientations(Fig. 7). Estimated aerodynamic torque at mid-downstroke was a good predictor of among-wingbeat roll acceleration (Fig. 3). Furthermore, the estimated mean upward and inward aerodynamic forces were good matches to body weight (Fig. 4)and centripetal force (Fig. 5),respectively. Finally, several of the factors that contributed to the estimates of both aerodynamic torque and inertial reorientation were correlated with specific muscle activation measurements, providing some insight into how the neuromuscular system manages changes in heading and orientation in avian flight. These EMG results support an integrated model of neuromuscular control of maneuvering, where all aspects of the flight apparatus are modulated.

As noted above, inertial reorientation was a major factor in determining within-wingbeat changes in roll. In each of the four phases of the wingbeat cycle, our estimates of inertial reorientation were significantly related to the measured change in roll (Fig. 7). The correlation was not particularly strong in some cases,especially from the end of downstroke to mid-upstroke; this may be due to the poor temporal resolution of our wing moment of inertia estimates. Because we measured moment of inertia at only four instants throughout the wingbeat cycle, we used an average of the two instants defining an interval to characterize wing moment of inertia for the entire interval. This simplification, along with changes in orientation due to non-inertial factors(i.e. aerodynamic forces), likely accounts for much of the error in our inertial predictions.

Within-wingbeat inertial changes in orientation may play an important role in maneuvering flight, especially in generating changes in heading smaller than those studied here. Because the aerodynamic force experienced by the wings varies widely over the course of a wingbeat, inertial reorientation that leads to a change in roll at mid-downstroke would cause a lateral aerodynamic force, a lateral acceleration and a change in heading. This change would come with the added advantage that the net change in roll would be slight to non-existent, leaving the bird well oriented for steady flight or another slight change in direction during the next wingbeat.

Although inertial roll reorientation within a wingbeat may be sufficient for some maneuvers, the cockatoos studied here rolled into the turn over the course of several wingbeats [see fig. 9 in the companion paper(Hedrick and Biewener,2007b)]. Inertial reorientation can also contribute to these changes, although the rate of change in orientation will be less than that within a wingbeat. As for the within-wingbeat data, our estimate of inertial reorientation during a complete wingbeat was a significant although not especially strong correlate to the measured change in orientation(Fig. 8). As before, errors may be the result of the small number of moment of inertia measurements along with the accumulation of changes in orientation due to aerodynamic effects. Despite the weak correlation, the magnitude of the estimated net inertial changes to orientation (>10° per wingbeat) demonstrates that inertial reorientation cannot be ignored, even among wingbeats. The magnitude of estimated net inertial reorientation exceeds our initial estimates of∼2.0° (Hedrick and Biewener,2007b) because the cockatoos simultaneously modulated both wing inertia and angular velocity, and because downstroke and upstroke arcs did not always match one another. For instance, a bird might sweep its wing through a 90° arc in downstroke but only a 60° arc in upstroke. This clearly has consequences for the subsequent downstroke, but leads to a larger net inertial roll reorientation in the interim.

Net aerodynamic torque, estimated from a blade-element analysis of the kinematic data, was a significant predictor of among-wingbeat roll acceleration (Fig. 3). Additionally, our extension of the model to estimate both mean upward and inward force for an entire wingbeat resulted in values close to the measured quantities. Specifically, the mean upward force from the blade-element analysis was 89% of body weight and the mean inward force was 72% of that required for centripetal acceleration during the turn. Finally, the mean aerodynamic power computed from the model, 238 W kg^–1, was within the range of pectoralis power measured from cockatiels flying in a wind tunnel (Tobalske et al., 2003)and blue-breasted quail in ascending flight(Askew et al., 2001). The match between the blade-element results and the measured flight forces, body weight and centripetal acceleration, demonstrates that the large aerodynamic coefficients, occasionally exceeding 2.0, drawn from measurement of the forces on a revolving quail wing (Usherwood and Ellington, 2002b), likely reflect the actual force coefficients experienced by a bird wing in slow, flapping flight. Indeed, use of more moderate force coefficients such as those measured from bird wings held fixed in a wind tunnel (e.g. Withers, 1981) would leave the cockatoos severely deficient in mean upward and inward force, a result analogous to Ellington's proof by contradiction for the existence of unsteady aerodynamic effects in insect flight (Ellington,1984). Here we find that low speed avian flight requires aerodynamic force coefficients greater than those obtained from bird wings positioned in a steady flow, suggesting that time-varying aerodynamic effects such as delayed stall are important in slow avian flight.

Despite our success in extending the blade-element estimates from mid-downstroke to whole stroke mean aerodynamic force and power, a similar approach did not give reasonable results for average torque, and thus angular acceleration, during the complete wing stroke. In fact, applying the approach outlined in Eqn 7 for estimating the mean roll acceleration from the instantaneous torque resulted in roll accelerations tenfold greater than those measured. However, our success of applying similar assumptions to our estimates of mean force and power suggests that the core estimates of aerodynamic force at mid-downstroke are not in error, but that the assumptions used to extend the estimate to a mean for the entire stroke were not appropriate.

Our main assumption was that the torque asymmetry measured at mid-downstroke was a good proxy for the degree of torque asymmetry throughout the downstroke. However, when closely examining one of the primary factors used in estimating roll torque, the square of wrist speed in the global coordinate system, we found that differences at mid-downstroke were not characteristic of differences during the entire downstroke(Fig. 10). Instead, the relationships at mid-downstroke were reversed later in the downstroke. This change may be the result of roll damping in avian flight. Roll damping occurs when roll to one direction creates a torque opposing the roll. Consider the case shown in Fig. 10B. From early to mid-downstroke, the cockatoo generates a greater roll torque on the right wing, leading to body roll to the left. As the bird rolls, the left wing moves downward while the right wing moves upward, increasing the velocity of the left wing relative to the right and leading to the conditions seen at the end of downstroke in Fig. 10B. The increased relative velocity of the left wing leads to a counter-roll torque to the right; the right and left wings do not act mechanically (or aerodynamically) independently.

A high degree of roll damping is typical of airplane flight, in which roll velocity rapidly declines to zero once the applied torque ceases(Nelson, 1997). Unlike the velocity-based mechanism proposed above, the mediating factor for fixed-wing aircraft is the angle of attack asymmetry created by rotational velocity. This effect may add to the velocity changes shown in Fig. 10B, but our data were insufficient to measure angle of attack throughout an entire wingbeat cycle. The different sources of roll damping in fixed-wing and flapping flight suggest that the degree of damping may also vary substantially.

This analysis of C_K, the normalized damping coefficient, cannot distinguish between an active counter-torque generated by asymmetric motion wing during upstroke or late downstroke and passive damping that would occur with symmetric wing kinematics. However, as noted above,rolling motion will enhance velocity on the outside wing during upstroke so a passive damping mechanism is plausible. The experimentally derived values for C_K are similar to those predicted by a simple model of roll damping in low speed flapping flight(Hedrick and Biewener, 2007a). Additionally, the flexible nature of the cockatoo's wings and body may enhance roll damping (Sneyd et al.,1982; Krus,1997).

To assess the importance of aerodynamic torque to overall patterns of roll reorientation, we used the damped roll equation in combination with initial roll velocity, estimated mid-downstroke torque, and our assumption of sinusoidal torque during downstroke, to predict change in roll over an entire wingbeat. The resulting prediction was significantly correlated with the measured change in roll (r²=0.213, P<0.0005, F=14.6, Fig. 11). The modest correlation is likely indicative of both errors in the initial estimate of net aerodynamic torque at mid-downstroke and variability in the damping coefficient K, both within and between wingbeats. Net inertial reorientation (Fig. 8) may also play a role, although estimated inertial reorientation was not a significant predictor of overall change in roll, either independent of estimated aerodynamic reorientation or when included as a co-predictor.

The relationships between various EMG parameters and different factors important to aerodynamic and inertial reorientation, presented above and summarized in Table 1,generally agree with those uncovered in prior studies or evident from the simplified model of inertial reorientation presented in the companion paper(Hedrick and Biewener, 2007b). The association between pectoralis EMG intensity and spanwise rotation angle(γ) was such that greater EMG intensities correlated with upward rotation of the trailing edge of the wing, as might be expected given the position of the pectoralis insertion on the cranial margin of the humerus. Similarly, the relationship we found between increases in the supracoracoideus EMG rectified impulse and downward rotation of the trailing edge of the wing may be explained by the demonstration(Poore et al., 1997a) that the supracoracoideus imparts a substantial (supinating) rotation to the humerus. Persistence of this torque into early downstroke might influence wing orientation through the downstroke by changing the initial position of the wing before the pectoralis begins contracting. The correlation between increased biceps activation duration and increased downward (supinating)rotation of the trailing edge at mid-downstroke was consistent with that reported by Dial and Gatesy (Dial and Gatesy, 1993).

Unlike the results described above, the positive correlation between pectoralis time to mid-burst and the square of wing velocity in the world coordinate system was somewhat counter-intuitive. We expected muscle activation parameters to relate most strongly to motion in the body frame of reference. However, the observed correlation may be explained by early activation of the pectoralis causing roll to the opposite side, which then appears as greater world coordinate system wing velocity on the side with the muscle activation delay. Finally, the association between greater pectoralis activation intensity and greater aerodynamic force coefficients was due to an association between greater pectoralis activation and greater downward-to-forward velocity ratio for the wing at mid-downstroke. This greater downward velocity influenced the angle of attack in opposition to the previously mentioned pectoralis influence on wing rotation, leading to a greater aerodynamic force coefficient.

Like the majority of the EMG to aerodynamic reorientation results described above, the relationships between the various EMG parameters and different aspects of inertial reorientation typically allowed simple interpretation. The positive correlation between inertial reorientation from mid-upstroke to the start of downstroke and the duration of supracoracoideus activation is consistent with the expected effect of a larger upstroke arc. For example, a longer activation of the right supracoracoideus might increase the elevation of the right wing relative to the left, leading to simultaneous inertial rotation toward the right. The correlation between increased biceps EMG impulse and inertial roll in the direction of the wing with elevated impulse is potentially the result of the biceps activity reducing the wing moment of inertia during downstroke. Likewise, the inverse correlation between wrist extensor mean spike amplitude and inertial roll could be the result of the wrist extensor increasing the wing moment of inertia. Finally, the inverse relationship between pectoralis mean spike amplitude and inertial roll in early downstroke is consistent with the expected inertial effects of a more rapid downstroke.

The positive relationship between pectoralis activation duration and inertial reorientation was initially more difficult to understand. We expected that greater activation duration would lead to a greater downward arc of the wing and therefore cause inertial reorientation toward the opposite side,leading to an inverse correlation. However, upon examination of the arcs, we found that greater pectoralis EMG duration was associated with net upward motion of the wrist during the final portion of downstroke. This may be due to the storage and release of energy in the supracoracoideus tendon at the end of upstroke (Hedrick et al.,2004), as greater energy storage could lead to a more rapid upstroke and therefore the observed net upward motion of the wrist. Alternatively, greater pectoralis activation was also associated inertial reorientation to the opposite side earlier in the downstroke (see above). Therefore, the observed reorientation toward the same side in the later half of downstroke may simply be a reaction to decelerating the faster moving wing. This relationship means that greater pectoralis activation in downstroke is initially correlated with inertial roll to the opposite side, then subsequently to the near side, for a small net effect over the entire downstroke.

In the present study of rose-breasted cockatoos making 90° turns, we found that asymmetric pectoralis activation was involved in several aspects of both inertial and aerodynamic reorientation(Table 1 and above), although not to a degree sufficient to associate it with our overall predictions of net aerodynamic torque or inertial reorientation. As predicted in our initial examination of turning in cockatoos in the companion paper(Hedrick and Biewener, 2007b),all other flight muscles examined in this study were associated with at least one component of our estimates of inertial and aerodynamic reorientation. The extensive involvement of the main flight power muscles, the pectoralis and supracoracoideus, as well as the intrinsic wing muscles, demonstrates that flight control and maneuvering in birds is not managed by a discrete set of muscles but is, instead, an integrated system with effects distributed throughout the flight muscles. While it is likely that some of these muscles are more important to maneuvering than others, the experiments conducted here were insufficient to rank muscles in order of importance. Wing muscle denervation experiments performed on pigeons demonstrated that the birds were not capable of maneuvers such as landing and take-off without the use of their intrinsic wing muscles (Dial,1992). Thus, we expect that the cockatoos would not be capable of turning without use of their intrinsic wing muscles. However, they may also not be capable of maneuvering without asymmetrically activating the pectoralis and supracoracoideus muscles. In summary, our results best support an integrated model of the neuromuscular control of maneuvering flight, where many components of the flight apparatus are modulated during maneuvering. This is in contrast to insect flight systems, where accessory muscles modulate the effects of flight motor muscles (e.g. Tu and Dickinson, 1994; Balint and Dickinson, 2004). The integrated control model also highlights differences between human engineered flight systems, with their limited set of actuators capable of generating variation in aerodynamic forces, and the vast range of aerodynamic and inertial reorientation possibilities open to animals with reconfigurable flapping wings.

This study points out a number of interesting avenues for further research. The possibility that birds may, in some circumstances, maneuver with primarily inertial rather than aerodynamic reorientations remains interesting and might occur in smaller amplitude, slalom-type turns where inertial reorientation over the course of a single wingbeat leads to sufficient change in heading. Additionally, assessing the importance of different morphological factors,such as wing area and wing moment of inertia, to maneuvering flight requires an integrated model that includes both aerodynamic and inertial reorientations along with velocity based damping. Finally, our understanding of the neural control of flapping flight would benefit from studies recording muscle activation asymmetry over turns of different radii as well as perturbed turns where the bird's anticipated flight path was interrupted, forcing the bird to perform a rapid and unexpected maneuver.

List of abbreviations and symbols

 
• b

  wing span

 
• C_r

  resultant force coefficient

 
• C_K

  roll damping coefficient

 
• d

  distance

 
• d_wrist

  distance from the wing root to the wrist

 
• \({\vec{F}}\)

  force vector

 
• \(\overline{{\vec{F}}}\)

  mean force vector for a complete wingbeat

 
• F_z

  upward force

 
• F_c

  centripetal force

 
• I

  moment of inertia

 
• K

  roll damping constant

 
• M

  mass

 
• P

  power

 
• Q

  dynamic pressure

 
• r

  turn radius

 
• \({\vec{r}}_{\mathrm{b}}\)

  position vector in the body coordinate system

 
• s

  wing area

 
• t_ds

  duration of downstroke

 
• t_ws

  duration of whole stroke

 
• u

  flight speed

 
• \({\vec{V}}\)

  velocity vector in the global coordinate system

 
• \({\vec{V}}_{\mathrm{b}}\)

  velocity vector in the body coordinate system

 
• α

  angle of attack

 
• β

  roll angle

 
• \({\dot{{\beta}}}\)

  roll velocity

 
• \({\ddot{{\beta}}}\)

  roll acceleration

 
• γ

  wing spanwise axis rotation angle

 
• λ

  arc swept by the wing during a half-stroke

 
• ρ

  air density

 
• τ

  roll moment

 
• \({\vec{{\tau}}}\)

  torque vector

 
• \(\overline{{\vec{{\tau}}}}\)

  mean torque vector for a complete wingbeat

 
• ϕ

  wing elevation angle

 
• Ψ

  heading angle

We wish to thank the late Dr Russell Baudinette for facilitating this work at the University of Adelaide. Jayne Skinner and Craig McGowan also contributed enormously to the experiments; the work could not have been completed without their assistance. The manuscript was greatly improved by comments from Sanjay Sane and two anonymous referees. This project was funded by NSF IBN-0090265 to A.A.B.