## SUMMARY

The reconfigurable, flapping wings of birds allow for both inertial and
aerodynamic modes of reorientation. We found evidence that both these modes
play important roles in the low speed turning flight of the rose-breasted
cockatoo *Eolophus roseicapillus*. Using three-dimensional kinematics
recorded from six cockatoos making a 90° turn in a flight corridor, we
developed predictions of inertial and aerodynamic reorientation from estimates
of wing moments of inertia and flapping arcs, and a blade-element aerodynamic
model. The blade-element model successfully predicted weight support
(predicted was 88±17% of observed, *N*=6) and centripetal force
(predicted was 79±29% of observed, *N*=6) for the maneuvering
cockatoos and provided a reasonable estimate of mechanical power. The
estimated torque from the model was a significant predictor of roll
acceleration (*r*^{2}=0.55, *P*<0.00001), but greatly
overestimated roll magnitude when applied with no roll damping.
Non-dimensional roll damping coefficients of approximately –1.5,
2–6 times greater than those typical of airplane flight dynamics
(approximately –0.45), were required to bring our estimates of
reorientation due to aerodynamic torque back into conjunction with the
measured changes in orientation. Our estimates of inertial reorientation were
statistically significant predictors of the measured reorientation within
wingbeats (*r*^{2} from 0.2 to 0.37, *P*<0.0005).
Components of both our inertial reorientation and aerodynamic torque estimates
correlated, significantly, with asymmetries in the activation profile of four
flight muscles: the pectoralis, supracoracoideus, biceps brachii and extensor
metacarpi radialis (*r*^{2} from 0.27 to 0.45,
*P*<0.005). Thus, avian flight maneuvers rely on production of
asymmetries throughout the flight apparatus rather than in a specific set of
control or turning muscles.

## Introduction

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.

## Materials and methods

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.

### Wing and body kinematics

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).

### A blade-element model of aerodynamic force and torque

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.

In the blade-element approach used here, we first divided the wings of four
cockatoos into 1 cm wide strips, measuring the area of each strip and
combining the results from the four birds into a single standard wing [table 3
in the companion paper (Hedrick and
Biewener, 2007b)]. The velocity of the *i*th segment
(*V*_{i}) at mid-downstroke was calculated as:
(1)
where *d*_{i} is the distance from the shoulder to the midpoint
of the *i*th segment of the standard wing,
is
the velocity of the wrist in the body coordinate system,
*d*_{wrist} the distance from the shoulder to the wrist segment
in the standard wing, and
the velocity
of the EMG plug attachment site in the global coordinate system
(Fig. 2). Note that the
blade-element velocity calculation, Eqn
1, assumes still air and does not incorporate any estimate of the
effect of induced velocity on overall flow direction or magnitude. However, we
compensated for this by using aerodynamic force coefficients derived from data
that relate angles of attack, also computed assuming still air, to forces
measured with fully developed induced velocity (see below).

The position of each wing segment in the body coordinate system, , was estimated as: (2) where is the position vector of the wrist in the body coordinate system.

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).

This spanwise rotation angle was combined with an estimate of the flow
velocity for each segment,
, computed by
applying the roll, pitch and yaw rotations to the
, the result of
Eqn 1. The chordwise components
of the estimated flow were combined with spanwise wing rotation angle to
compute the wing's angle of attack, α_{i}
(Fig. 2B):
(3)
where the zero in the [cosγ,0,sinγ] orientation vector enforces
computation of the angle of attack from chordwise flow only, assuming that
chordwise flow is in the *X*_{b}*Z*_{b} plane.
This assumption is reasonable at mid-downstroke, when the wings are extended
along the *Y*_{b} axis, but not at other points within the
wingbeat cycle. The angles of attack (α_{i}) were converted to
coefficients of resultant force (*C*_{r,i}) using the following
equation, generated by a polynomial fit to the data in
(Usherwood and Ellington,
2002b) for real and model quail wings in steady state rotation:
(4)
where α is the angle of attack in rad. The range of α in the data
used to create the fit extend from –0.4 to 1.25 rad. These high force
coefficients derived from measurement of revolving wings allow estimation of
the forces due to three-dimensional and unsteady aerodynamic effects and are
thus appropriate for slow, flapping flight
(Sane, 2003).

The results of these equations were combined to provide a net estimated
force vector for the wing as a whole:
(5)
where ρ is air density and *s* is wing area, as well as an
estimated torque vector
for the
wing:
(6)
where is position
vector in the body coordinate system.

To extend
from mid-downstroke to a mean force for the entire stroke, we made the
following assumptions: (1) aerodynamic force is generated only during the
downstroke and (2) aerodynamic force in downstroke varies as a half-sine,
peaking at mid-downstroke. These assumptions are consistent with measurements
of pectoralis force production (Dial and
Biewener, 1993; Biewener et
al., 1998; Hedrick et al.,
2003) and the time course of wing pressure distribution
(Usherwood et al., 2005).
Under these assumptions, the average force produced by the bird during the
entire stroke is:
(7)
where *t*_{ds} is the downstroke duration, and
*t*_{ws} is the whole stroke duration. We extended the
estimated torque,
, to
estimated mean torque,
,
using an identical approach.

### Estimating upward aerodynamic force

To verify the accuracy of forces and torques estimated by the above
equations, we also used the blade-element model to estimate the upward
component of aerodynamic force for comparison with the bird's weight. For
non-accelerating flight, the upward aerodynamic force produced over an integer
number of wingbeats should equal the bird's weight. As before, we assumed that
aerodynamic force varied as a half sine wave during downstroke, peaking at
mid-downstroke, the point at which we estimated force from the kinematic data
using Eqn 5. We also assumed that
forces on the wing were oriented normal to the upward surface, and corrected
for force orientation due to the measured spanwise rotation, γ, and
changes in wing elevation (or dihedral) angle, ϕ, through the stroke.
Elevation angle was assumed to vary from 45° above horizontal in the body
coordinate system to 45° below. Finally, we assumed that forces were
generated only during downstroke (Hedrick
et al., 2004). These assumptions were expressed as:
(8)
where *F*_{z,est} is the upward component of
, β is the
roll angle at mid-downstroke (Fig.
2), and 4√2/3π is the average of the curve
*y*=sin(*x*)sin[(*x*/2)+(π/4)] for *x* from 0
to π, the combination of the assumed sinusoidal variation of force and
elevation angle through the downstroke. Note that
Eqn 8 is for one wing only; we
therefore summed the mean forces from the left and right wings to obtain the
total net upward aerodynamic force.

## Estimating inward aerodynamic force and mechanical power

The mean inward, i.e. centripetal aerodynamic force was estimated with a
similar process, substituting only sine β for cosine β in
Eqn 8. Rather than overcoming
gravity, the inward force generated by the wings must provide the centripetal
force required to change heading. The mean centripetal force for a wingbeat
was estimated from the data as follows:
(9)
where *M* is body mass, *r* is turn radius, and *ū*
is the bird's average flight speed during the turn. Turn radius was computed
as:
(10)
where ΔΨ is the change in heading or flight path during the
stroke.

The mechanical power *P* required to generate the aerodynamic forces
was estimated as:
(11)
where λ is the arc swept by the wing during downstroke. The estimated
power was calculated by combining our estimate of aerodynamic force
(Eqn 5) with the distance moved
by each wing segment during downstroke and the duration of the entire
wingbeat. The estimate formulated in Eqn
11 assumes that the wing flaps with a constant angular velocity
during downstroke, an assumption supported by *in vivo* measurements of
muscle length change in the avian pectoralis during flight (e.g.
Askew et al., 2001;
Hedrick et al., 2003). These
studies note both an initial rapid shortening and later slow shortening phase
of muscle contraction during downstroke, but the overall muscle lengthening–
shortening cycle was best described as a sawtooth, rather than
sinusoidal wave (Askew and Marsh,
2001). Alternatively, assuming that wing angular velocity varies
sinusoidally during the stroke would increase the estimated mechanical power
by a factor of π^{2}/8, or approximately 23%.

### Inertial reorientation

As was shown in the companion paper
(Hedrick and Biewener, 2007b),
both instantaneous and net inertial changes to orientation are likely in avian
maneuvering flight. The degree of both these effects is related to the moments
of inertia of the wing through time and the arc swept by the wing over the
course of the wingbeat cycle. Here we show how measurements of wing position
at four different points within the wingbeat cycle were used to estimate the
magnitude of these inertial effects. First, for each wing at each of the four
positions (see Fig. 1) we
computed the wing moment of inertia with respect to the *X*_{b}
or roll axis for rotation about its shoulder joint:
(12)
where *M*_{i} is the mass of the *i*th wing section and
*d*_{y,i} is its distance from the shoulder joint in the
*Y*_{b}*Z*_{b} plane. Distances from the
shoulder were computed for each section by evenly distributing the proximal
wing sections along the shoulder–wrist segment and the distal wing
sections along the wrist–tip segment. We also used a similar formula to
compute an *I*_{wing,o}, the moment of inertia for rotation
about the opposite shoulder joint. We then estimated the wing moment of
inertia during a stroke interval as the average of the wing moment at the
interval end points. As shown in Appendix 1 of the companion paper
(Hedrick and Biewener, 2007b),
these were combined with the change in elevation angle through which each wing
moved during the specified stroke interval to give an estimate of inertial
body roll during that interval:
(13)
where Δϕ_{wing} is the elevation angle change for the wing
during the interval of interest and an r or l prefix in the subscript
indicates the right or left wing. The predicted change in body roll over a
complete wingbeat was estimated as the sum of Δβ for the start of
downstroke to mid-downstroke, mid-downstroke to end of downstroke, end of
downstroke to mid-upstroke, and mid-upstroke to end of upstroke sequence.

### Statistics

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).

## Results

### Aerodynamic force, torque and power estimates

We found that the estimated net aerodynamic torque at mid-downstroke was a
significant predictor of roll acceleration (*r*^{2}=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 *r*^{2}
of 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).

### Inertial reorientation estimates

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 *r ^{2}*=0.37 for
the mid-upstroke to start of downstroke phase to

*r*

^{2}=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 (

*r*

^{2}=0.19) related with the measured change in roll (Fig. 8). As before, the regression slope and intercept were near one and zero, respectively.

### Muscle activation parameters: aerodynamic reorientation

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 (*r*^{2}=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 (*r*^{2}=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 (*r*^{2}=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 (*r*^{2}=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 (*r*^{2}=0.277,
*P*<0.005, *F*=9.6, Fig.
9).

### Muscle activation in relation to inertial reorientation

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
(*r*^{2}=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
(*r*^{2}=0.379, *P*<0.001, *F*=15.3,
Fig. 9). The mean spike
amplitude of the wrist extensor was inversely correlated with inertial
reorientation (*r*^{2}=0.420, *P*<0.0005,
*F*=18.1). Finally, the pectoralis mean spike amplitude was also
inversely correlated with inertial reorientation in early downstroke
(*r*^{2}=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
(*r*^{2}=0.292, *P*<0.005, *F*=10.3).

## Discussion

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.

### Inertial roll reorientation within a wingbeat

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.

### Inertial roll reorientation among wingbeats

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.

### Aerodynamic reorientation among wingbeats

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.

### Aerodynamic torque, roll rate and damping

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.

The degree of roll damping characteristic of these low speed turns can be
estimated from the torque-to-roll relationship shown in
Fig. 11. The change in roll
rate due to both applied torque and roll damping is given by:
(14)
where is roll rate, *K* is
the roll damping coefficient, *I*_{x} is the roll moment of
inertia, and τ is the roll moment due to wing asymmetry. We assume as
before that τ during downstroke is in the form of a half-sinusoid with a
peak at the estimated torque asymmetry, with τ equal to zero during
upstroke. The portion of
Eqn 14 provides the counter
torque that establishes the wrist velocity pattern shown in
Fig. 10B. The general
relationship between estimated roll moment and roll acceleration, averaged
over a whole wingbeat, was given in the regression equation from
Fig. 3:
(15)
This equation includes a constant term, an unlikely circumstance since it
implies that the cockatoos begin rolling with no wing asymmetry. This is
unlikely, though not impossible if, for instance, the data cable applied
torque to the bird. Given the unexpected appearance of the constant, we
computed *K* for a τ=0.5 Nm and three separate equations relating
and τ_{est}: (i)
Eqn 15, (ii)
Eqn 15 without the constant
term, and (iii) a linear regression with no constant fit through the data in
Fig. 3. In all cases we used
the *I*_{x} for a bird with wings flexed [table 2 in the
companion paper (Hedrick and Biewener,
2007b)]. These resulted in estimated *K* values (damping
coefficients) of –60.32, –99.65 and –158.66
s^{–1}, respectively. For comparison between different
individuals or species, *K* should be non-dimensionalized to
*C*_{K}, a normalized damping coefficient, as follows:
(16)
where *Q* is the dynamic pressure, *u* is flight speed,
*s* is wing area and *b* is wing span
(Nelson, 1997). Using the
average values for these constants [tables
1 and 2 in the companion paper
(Hedrick and Biewener, 2007b)]
resulted in an estimated *C*_{K} of –1.14, –1.87
and –2.99 rad^{–1} for the respective cases. These values
for *C*_{K} are 2–6 times greater than those typical of
human piloted aircraft (Roskam,
1995). Thus, roll velocity will decay even more quickly in
maneuvering birds than in airplanes. In fact, the time constants for these
*K* are less than a wingbeat, so moderate roll velocities established
at mid-downstroke will drop to near zero by the start of the next
mid-downstroke. The greater roll damping experienced by flapping cockatoos as
compared to fixed wing aircraft is likely due to the different sources of
damping in the two types of flight. In fixed wing flight, roll damping is due
to the largely linear relationship between angle of attack and aerodynamic
force. In flapping flight, roll damping is due to the exponential relationship
between the flow velocity over the wing and the resulting aerodynamic
force.

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 (*r*^{2}=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.

### Electromyogram correlations to aerodynamic and inertial reorientation

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.

### Multiple EMG modalities enable turning flight

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.

### Future work

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
- force vector
- 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
- position vector in the body coordinate system
- s
- wing area
*t*_{ds}- duration of downstroke
*t*_{ws}- duration of whole stroke
- u
- flight speed
- velocity vector in the global coordinate system
- velocity vector in the body coordinate system
- α
- angle of attack
- β
- roll angle
- roll velocity
- roll acceleration
- γ
- wing spanwise axis rotation angle
- λ
- arc swept by the wing during a half-stroke
- ρ
- air density
- τ
- roll moment
- torque vector
- mean torque vector for a complete wingbeat
- ϕ
- wing elevation angle
- Ψ
- heading angle

## ACKNOWLEDGEMENTS

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.

- © The Company of Biologists Limited 2007