The Journal of Experimental Biology 206, 2803-2829 (2003)
Copyright © 2003 The Company of Biologists Limited
doi: 10.1242/jeb.00501
Dynamic flight stability in the desert locust Schistocerca gregaria
Graham K. Taylor* and
Adrian L. R. Thomas
Department of Zoology, Oxford University, Tinbergen Building, South
Parks Road, Oxford, OX1 3PS, UK
*
Author for correspondence (e-mail:
graham.taylor{at}zoo.ox.ac.uk)
Accepted 16 May 2003
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Summary
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Here we provide the first formal quantitative analysis of dynamic stability
in a flying animal. By measuring the longitudinal static stability derivatives
and mass distribution of desert locusts Schistocerca gregaria, we
find that their static stability and static control responses are insufficient
to provide asymptotic longitudinal dynamic stability unless they are sensitive
to pitch attitude (measured with respect to an inertial or earth-fixed frame)
as well as aerodynamic incidence (measured relative to the direction of
flight). We find no evidence for a `constant-lift reaction', previously
supposed to keep lift production constant over a range of body angles, and
show that such a reaction would be inconsequential because locusts can
potentially correct for pitch disturbances within a single wingbeat. The
static stability derivatives identify three natural longitudinal modes of
motion: one stable subsidence mode, one unstable divergence mode, and one
stable oscillatory mode (which is present with or without pitch attitude
control). The latter is identified with the short period mode of aircraft, and
shown to consist of rapid pitch oscillations with negligible changes in
forward speed. The frequency of the short period mode (approx. 10 Hz) is only
half the wingbeat frequency (approx. 22 Hz), so the mode would become coupled
with the flapping cycle without adequate damping. Pitch rate damping is shown
to be highly effective for this purpose — especially at the small scales
associated with insect flight — and may be essential in stabilising
locust flight. Although having a short period mode frequency close to the
wingbeat frequency risks coupling, it is essential for control inputs made at
the level of a single wingbeat to be effective. This is identified as a
general constraint on flight control in flying animals.
Key words: stability, control, flapping flight, desert locust, Schistocerca gregaria, insect, flight dynamics, modes of motion, equations of motion, frequency response, stabilising pitch reaction, constant-lift reaction, flight speed, body angle
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Introduction
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The insect flight literature still lacks a quantitative empirical analysis
of stability and control in terms of its flight mechanics. Many previous
studies have correlated changes in the aerodynamic forces or moments on
tethered insects with changes in wing kinematics during fictive manoeuvres
(for a review, see Taylor,
2001
), but the measured force systems have almost always been
incomplete in the sense of being insufficient to specify the direction,
magnitude and line of action of the resultant force. Some studies (e.g.
Weis-Fogh, 1956b;
Cloupeau et al., 1979) have
only measured the force in a single direction. Others (e.g.
Thüring, 1986;
Eggers et al., 1991) have
measured the turning moment about one or more axes, but have not measured the
forces normal to these axes (but for an exemplary exception, see
Blondeau, 1981). A few studies
(e.g. Wilkin, 1990) have
measured both the direction and magnitude of the resultant force but not
defined its line of action (but for a prescient exception, see
Hollick, 1940). Of these
studies, almost all have failed to record the centre of mass, which plays an
analogous role in flight dynamics to the fulcrum in a game of seesaw.
These limitations have made it impossible to predict even the initial
direction of a turn induced by a measured change in the forces, with the
result that we still lack any quantitative empirical understanding of the
stability of flying insects. An initial directional tendency to return to
equilibrium after a disturbance is called static stability, which qualifies
the fact that the dynamics of a system may prevent it from actually settling
back to equilibrium. This means that even if we could measure an insect's
initial turning tendency in response to a disturbance (i.e. measure its static
stability), this would still be insufficient to say anything about the more
interesting problem of dynamic stability without a formal theoretical
framework for analysing the flight dynamics. Analyses of static stability in
gliding animals (Thomas and Taylor,
2001; McCay, 2001)
and flapping flight (Taylor and Thomas,
2002) have been provided elsewhere. Here we analyse the
longitudinal flight dynamics of locusts empirically, providing the first
formal framework for analysing the dynamic stability of flying animals. Our
strategy parallels the engineering approach of measuring how the aerodynamic
forces and moments change with attitude and velocity in a wind tunnel, in
order to define the parameters of the linearized equations of motion. Writing
these equations enables us to use techniques of eigenvalue and eigenvector
analysis to provide the first formal description of dynamic stability in a
flying animal.
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A formal framework for analysing
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Animal flight stability
The framework that we use is founded upon the linearized equations of rigid
body motion, which are widely used in aircraft flight dynamics (e.g.
Etkin and Reid, 1996;
Boiffier, 1998) and have
recently been used in approximate (Taylor
and Thomas, 2002) or reduced form
(Deng et al., 2002;
Schenato et al., 2002) in
theoretical considerations of animal flight dynamics. The framework is
developed fully in the Appendix, but can be treated as a `black box' if
required (Fig. 1). Its single
most important assumption is that the animal has only the 6 degrees of freedom
of a rigid flying body: 3 in translation and 3 in rotation. This means
dropping the wings' degrees of freedom relative to the body from the explicit
formulation, so that although in reality the centre of mass moves and the
forces, moments and moments of inertia change through every wingbeat, all are
assumed to average out to make a constant contribution for a given flight
configuration. Conceptually, the aerodynamic forces resulting from the
flapping motion of each wing are collapsed into a single force vector, which
may vary with speed and attitude to reflect changes in the wing kinematics
with respect to the body as well as with respect to the air. These
approximations are all reasonable if the wings beat fast enough not to excite
the natural oscillatory modes of the system
(Taylor and Thomas, 2002). In
addition, the rigid body equations of motion contain no gyroscopic terms,
which is reasonable because the combined mass of the oscillating wings is
small (<4% of total body mass).
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Fig. 1. Black box representation of the rigid body equations of motion underpinning
the quantitative framework of this analysis. The integral sign represents a
bank of single integrators.
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Provided there exists a longitudinal plane of symmetry, pure longitudinal
or symmetric motions are possible. This means that we can consider
longitudinal dynamic stability in isolation from lateral dynamic stability, so
we need only consider three of the animal's six degrees of freedom in the
present analysis (Fig. 2). The
rigid body equations of motion are intrinsically non-linear, but may be
linearized by approximating the body's motion as a series of small
disturbances from a steady, symmetric reference flight condition, and
retaining only the linear terms in the Taylor series expansion. This yields a
set of time (t) dependent equations, summarised by the expression:
 | (1) |
for symmetric longitudinal motion (subscript `sym'). Here
xsym(t) is the longitudinal state vector [u w
q 
]T, containing the longitudinal state variables
(Fig. 2): forward (u)
and dorsoventral (w) components of velocity along the x- and
z-axes, the angular pitch rate at the centre of mass (q),
and the pitch angle between the x-axis and the horizontal ().
Fsym is the longitudinal system matrix, containing partial
derivatives of the longitudinal forces and moments with respect to the
longitudinal state variables: the stability derivatives.
Csym is the control system matrix, and has as many rows and
columns as the number of symmetric control inputs available to populate the
control state vector csym. Since the system matrices
Fsym and Csym contain only (real) constant
numbers, Equation 1 is linear time-invariant with respect to disturbances from
the equilibrium condition about which the equations are linearized.
Although the usual practice in the aircraft literature is to place
aerodynamic effects arising from pilot and automatic control in
Csym and to reserve Fsym for passive
aerodynamic effects, it is sometimes helpful to view automatic control as
augmenting the stability derivatives in Fsym
(Etkin and Reid, 1996). We
must use the latter approach here, because it is impossible to isolate the
passive aerodynamic stability of a flapping insect without abolishing all of
the control inputs that provide the feedback necessary for stimulating normal
forward flight. The stability derivatives then conflate passive aerodynamic
effects with aerodynamic effects arising from `automatic' changes in muscle
firing due to changes in body angle, wind speed, etc. The longitudinal system
matrix Fsym containing these derivatives is written:
 | (2) |
where the terms (Xu, Zu,
Mu), (Xw, Zw,
Mw), (Xq, Zq,
Mq) denote partial derivatives of the forward (X)
and dorso-ventral (Z) components of force and the pitching moment
(M) about the centre of mass, with respect to the motion variables
u, w and q, measured during tethered flight. The terms
m and Iyy denote the reference body mass and
pitching moment of inertia; ue, we,
e denote values of the longitudinal state variables at
equilibrium; g is the acceleration due to gravity
(g=9.81 m s-2). We have dropped partial derivatives
with respect to accelerations and angular accelerations, but Equation 2 is
otherwise rather general. Similar equations can be found in any textbook on
flight dynamics, whether for fixed-wing aircraft (e.g.
Nelson, 1989;
Etkin and Reid, 1996;
Cook, 1997), helicopters (e.g.
Padfield, 1996), airships
(e.g. Khoury and Gillett,
1999), or spacecraft (e.g.
Bryson, 1994). The differences
between animals and aircraft are instead manifested in the stability
derivatives determining the system's response to perturbation.
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Materials and methods
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The partial derivatives in the system matrix Fsym
(Equation 2) can be empirically defined by measuring the changes in the forces
and moments induced by imposed changes in velocity or orientation away from an
equilibrium flight condition. This can only be done under open-loop
conditions, where the insect is immobilised such that changes in the stroke
cycle do not affect the external stimuli that the insect receives. We must
therefore assume that the insect responds similarly to (instantaneously)
identical perturbations from equilibrium in both the open-loop conditions of
tethered flight and the closed-loop conditions of free flight. The slopes of
linear regressions fitted to the open-loop data can then be used to provide
direct estimates of the stability derivatives for populating the system
matrix. This is the general strategy used to measure the stability derivatives
in wind tunnel studies on aircraft, helicopters, airships, submarines and
spacecraft. Here we extend it to flying animals.
Animals
Three adult male gregarious phase desert locusts Schistocerca
gregaria Forskål were drawn from a population that had been reared
in crowded laboratory culture in Oxford University's Zoology Department for 13
years. The small sample size was imposed by the difficulty of finding
individuals that would fly reliably for the full 2–3 h required for each
experiment, but is identical to that of previous studies (e.g.
Zarnack and Wortmann, 1989)
and is justified on the basis of the complexity of the analysis, which must be
performed separately for each individual. The individuals are called `R'
(Red), `G' (Green), and `B' (Blue), and the data are colour coded in
subsequent figures. Relaxed selection and inbreeding depression are expected
to have lowered the mean flight performance of the population, so individuals
were chosen on the basis of flight ability. Locusts were only picked if their
wings and appendages were in perfect condition. Each individual was also
checked for dynamically stable free flight performance by releasing it from an
indoor balcony prior to the experiment.
Tethering
Each locust was rigidly tethered to a 6-component aerodynamic force balance
(I-666, FFA Aeronautical Research Institute of Sweden) in a low speed, low
turbulence, open-circuit wind tunnel (Fig.
3) designed specifically for insect flight work (G. K. Taylor and
A. L. R. Thomas, manuscript in preparation). During steady flight the locusts
made no attempt to grasp the tether, which consisted of a 0.5 mm sheet
aluminium platform set upon a brass M2 screw and cemented with cyanoacrylate
adhesive to the fused sternal sclerites forming the plastron of the
pterothorax. This arrangement ensured that the angle of the force balance back
from the vertical equated with the body angle (
b), defined
by Weis-Fogh (1956a) as the
angle between the oncoming wind and the plastron.
Force measurements
The force balance was sting-mounted to a rotary stage (Edmund Industrial
Optics, Barrington, NJ, USA) providing repeatable pitch adjustment
0°
b14°±0.5°. Force transduction
was by foil strain gauges in full Wheatstone bridge configuration, with 5 mA
constant current excitation provided by a 6-channel bridge amplifier (2210A
Signal Conditioning Amplifier/2250A Rack Adapter, Vishay Measurements Group,
Raleigh, NC, USA). Each output was filtered online with a 1 kHz low-pass
hardware filter (4th order Chebyshev), sampled at 10 kHz using a
16-bit analogue-to-digital converter (Maclab 8s, ADInstruments, Pty Ltd.,
Castle Hill, NSW, Australia), and recorded using 12 bits in Chart 3.6/s (AD
instruments 1998) on an Apple 9600 PowerMac.
Flight conditions
Standard conditions of 29±1°C temperature and 35±5%
relative humidity were maintained throughout the experiments. To give reliable
flight performance, we used diffuse overhead low-level (20 lux) red lighting
(Weis-Fogh, 1956a) provided by
light from a red-filtered 250 W slide projector. Blackout cloth prevented
light entering the sides of the tunnel, and the room was kept in darkness to
minimise extraneous visual input. The locusts initially attempted to escape
from the tether, displaying pronounced deflections of the abdomen that are
normally associated with avoidance manoeuvres (for a review, see
Taylor, 2001). Measurements
were made only when the locust had settled into flying in the complete flight
posture, defined by Weis-Fogh
(1956a) as when "the
antennae are stretched obliquely forwards, the forelegs are drawn up, the
middle legs and the hind femora are stretched backwards along the abdomen, the
hind tibiae are drawn up against the shallow groove on the underside of the
femora, and the abdomen points straight backwards in continuation of the
pterothorax" (p. 463). The transition to this posture was
accompanied by a switch to a very regular and pulse-like lift trace, with no
unbalanced side forces and roll or yaw moments. Once adopted, the complete
flight posture tended to be assumed for the duration of the experiments, which
lasted between 2 and 3 h, including a further 10 min for the locust to settle
into steady flight at the reference speed (Uref=3.50 m
s-1) and reference body angle (b,ref=7°):
values previously found to stimulate tethered flight with lift balancing body
weight (Weis-Fogh,
1956a,b;
Zarnack and Wortmann, 1989;
Wortmann and Zarnack,
1993).
Flight experiments
Each experiment comprised two consecutive measurement series: an angle
series, in which b was varied whilst tunnel speed U
was fixed at Uref, and a speed series, in which U
was varied whilst b was fixed at b,ref.
The ranges for U and b were within those observed
in cruising and climbing natural free flight for Schistocerca
gregaria (Waloff, 1972)
and Locusta migratoria L. (Baker
et al., 1981). The angle series comprised 14 pairs of force
measurements lasting 13 s each, made first at a perturbed angle
0°b14° and then at b,ref
within the same minute. We alternated between perturbed angles higher and
lower than b,ref to avoid correlating any systematic effect
of time with b. For each pair of measurements, we subtracted
the forces measured at b,ref from those measured at the
perturbed body angle to remove the temporal variation in flight performance
that has beset previous studies (e.g.
Zarnack and Wortmann, 1989).
We then added the mean of the 14 measurements at b,ref to
reconstruct an absolute value of force production. The speed series comprised
eight measurements made over a range of speeds (2.00U5.00 m
s-1), beginning and ending with Uref. Each
measurement lasted approximately 13 s, and we alternated between speeds higher
and lower than Uref to avoid correlating any systematic
effect of time with U. This procedure is important because it ensures
that temporal variation in flight performance cannot introduce systematic bias
into our estimates of the stability derivatives. The speed series measurements
were necessarily unpaired to avoid completely fatiguing the locust.
Conversion of balance output to dimensionless force—moment
data
Each force-balance recording was trimmed in Chart to contain an integer
number of complete stroke cycles (typically around 250). Subsequent analysis
was performed using custom-written programmes in Matlab 5.2.1 (1998; The
Mathworks Inc., Natick, MA, USA) on an Apple G4 PowerMac. A number of
extraneous factors affect balance output: amplifier zeroes drift with
temperature, and bridge resistance changes with balance orientation.
Corrections were applied to remove these effects, and will be described in
detail elsewhere (G. K. Taylor and A. L. R. Thomas, in preparation). The
corrected balance output was converted to force—moment units using a
static calibration analysed as a General Linear Model (GLM), in which we
retained significant terms up to third order in any one channel plus all
significant second order interactions (P0.05; G. K. Taylor and A.
L. R. Thomas, in preparation). We then took the mean of each
force—moment measurement, subtracted the gravitational forces and
moments due to the locust and its tether, and resolved the forces at the
centre of mass. Drag on the tether was estimated using a standard empirical
formula for a cylinder (White,
1974), but was sufficiently small to be ignored. For convenience
and ease of comparison with the existing literature, we resolved the resultant
aerodynamic force on the locust into an upward component (`lift') and a
forward component (called `thrust—drag' to indicate that it is
equivalent to thrust minus drag, although the two were never separated). The
forces were normalised by reference body weight, which is consistent with the
form of Equation 2 and allows the dimensionless forces to be compared directly
with each other. The pitching moment about the centre of mass was made
dimensionless by dividing through by the product of reference body weight and
length. These dimensionless quantities are henceforth referred to as `relative
lift' (Lr), `relative thrust—drag'
(Tr) and `relative pitching moment'
(Mr), following the notation of Weis Fogh
(1956a).
Morphometric measurements
Morphometric measurements are given in
Table 1. Each locust was
weighed at the beginning and end of the experiment, and was then frozen in a
sealed container at -40°C. Reference body mass was defined as the mass of
the locust at the mean time of force measurement, assuming linear loss of mass
with time. Reference body length was measured from the frons to the tip of the
abdomen using a pair of electronic callipers (Absolute Digimatic, Mitutoyo
Corporation, Kawasaki, Kanagawa, Japan); reference wing length was measured
from wing base to wing tip.
Centre of mass measurements
The locusts were later defrosted and fixed in the complete flight posture
using tiny amounts of cyanoacrylate adhesive. The centre of mass was
determined using a model aircraft propeller balancer (Precision Magnetic
Balancer, Top-Flite, Hobbico Inc., Champaign, Il, USA), which uses powerful
magnets to levitate a steel shaft under almost frictionless conditions.
Balanced stays were used to hang the locust from the shaft so that its centre
of mass always hung vertically below the shaft axis. Plumb lines were hung on
either side of the locust for sighting purposes, and were aligned in the
viewfinder of a Canon XL1 Camcorder. We were later able to overlay
left—right pairs of camcorder images and use the intersection of the
plumblines to determine the position of the centre of mass
(Fig. 4). Measurements agreed
to better than ±1 mmbetween the three locusts (approximately 2% of body
length). The wings comprise <4% of total body mass, so the centre of mass
is expected to vary little through the wingbeat.
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Fig. 4. Centre of mass measurements. Dorsal, lateral and ventral views of a locust,
showing the average position of the centre of mass (x). The line
drawings were traced from images generated by placing locust `G' directly onto
a flatbed scanner, and are scaled accordingly. Plumb lines passing through the
centre of mass are shown as coloured lines, with the colours denoting the
locust from which they were obtained (red `R', green `G', blue `B'). The
intersection of the plumb lines corresponds to the position of the centre of
mass; dotted lines join the most forward and rearward plumb line intersections
on the dorsal and ventral views to give an indication of measurement
error.
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Calculation of pitching moment of inertia
The body's pitching moment of inertia was approximated by sectioning the
frozen body of each locust into eight transverse sections of equal thickness
(Fig. 5B). The mass of each
section (expressed as a percentage of total frozen body mass) was then
averaged across the three locusts (Fig.
5A). The lateral projected area of each section was measured in
NIH Image 1.62 and expressed as a percentage of the total lateral projected
area of the locust. Each section was then approximated as a rectangle of
equivalent width and area, located with its centre of area vertically
coincident with the centre of area of the section
(Fig. 5C). The relative density
(
i) of each rectangle i (mean percentage mass over
mean percentage area) is represented in
Fig. 5C by the density of
shading. The dimensionless contribution of each rectangle to
Iyy is then:
 | (3) |
evaluated over the area of the rectangle, where
and
are dimensionless coordinates in the
body axes, normalised by body length. The total pitching moment of inertia is
the sum of the eight contributions multiplied by
mbl4/a for each locust, where
mb is the reference body mass less the mass of the wings,
l is the reference body length, and a is the lateral
projected area of the locust.
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Fig. 5. Pitching moment of inertia (Iyy) measurements. (A)
Stacked bar graphs showing the masses of eight transverse sections of the body
corresponding to the sections illustrated in (B) below. The different colours
distinguish data from the three locusts according to the scheme `R' (red), `G'
(green), `B' (blue). The mass of each section is expressed as a proportion of
the total frozen body mass and the y-axis is scaled such that the
total area of the bar graphs is 100%. (B) Line drawing showing the position of
the eight sections. (C) Block model of a locust. The area of each of the
blocks corresponds to the projected area of each section in B and the density
of the shading corresponds to the mass per unit area of each section. The
white crosses mark the positions of the wing roots. The red cross marks the
position of the centre of mass. (D) Bar graph illustrating the percentage
contribution of each body section to the total pitching moment of inertia.
Note that this bar graph mirrors the bar graph in A above, indicating that
percentage contributions to the total pitching moment of inertia are inversely
correlated with percentage contributions to the total body mass.
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The wings' contribution to Iyy is complex, but may be
modelled by representing each wing as a point mass (mw),
giving:
 | (4) |
for each wing, where 
is the period of a half stroke, and where x
and z are the instantaneous coordinates of the wing's centre of mass
in the body axes. These are closely approximated as:
 | (5) |
and
 | (6) |
where r is the distance from the centre of mass of the wing to the
wing root, b is the angle of the stroke plane back from the vertical,
and 
is the positional angle of the wings, defined as the angle between
the long axis of the wing and the z-axis (all angles in radians).
Equations 5 and 6 are exact if the centre of mass lies on the long axis of the
wing. The centre of mass was determined by balancing the frozen wings on a pin
(the hindwings had to be dried first to spread the vannal fan), and was
located approximately one third of the way out on both the forewings and the
hindwings.
Weis-Fogh (1956a) showed
that the wingbeat of tethered locusts approximates simple harmonic motion if
upstroke and downstroke are treated separately. Baker
(1979) has shown that the
wingbeat is even more closely sinusoidal in free flight. We therefore have:
 | (7) |
for each half stroke, where 0t, 
is the
total stroke excursion, and 
is the mean
positional angle of the wings. The kinematic parameters of the `standard'
stroke defined by Weis-Fogh
(1956a) are given in
Table 2 and were used to
evaluate the wings' mean contribution to Iyy. Equation 5
was evaluated numerically for the fore- and hindwings of each locust in Maple
6 (Maplesoft Waterloo, ON, Canada) on an Apple G4 PowerMac. Compared to
gliding flight, flapping approximately doubles the wings' mean contribution to
Iyy, by increasing the average moment arm about the
pitching axis. The mean contribution of the wings was nevertheless small,
always averaging less than 3.5% of Iyy
(Table 3).
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Results
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Analysis of variance for the force measurements
For ease of comparison with the literature, the results of the force
measurements are described as relative lift, thrust—drag and pitching
moment, before conversion to the form used in the longitudinal system matrix
(Equation 2). Figs 6,
7,
8 plot the angle series and
speed series data, with Model I least-squares linear regressions (see, for
example, Sokal and Rohlf,
1995) fitted to the data for each individual (called `individual
regressions'). We combined the data for the three locusts in various General
Linear Model (GLM) analyses to test the overall significance of the results
for each of the dimensionless forces and moments, including
individual in the model as a random effect and U or
b as a covariate (see
Table 4 for models used). In
all but one case (and then only in the unpaired analysis), relationships that
were significant in the individual regressions also attained overall
significance in the GLM, affirming that the individual regressions do not lead
us to overestimate the significance of the results. Post hoc residual
analysis indicated that the assumptions of normality of error and homogeneity
of variance were fulfilled.
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Fig. 7. Graphs of relative thrust—drag Tr against
b for the unpaired (A—C) and paired (D—F)
analyses of the angle series data, and against U (G—I) for the
speed series data. For further explanation, see legend to
Fig. 6. Individual regressions
of Tr against U2 (G—I) are shown
as black dotted lines where the individual regression and the corresponding
GLM treating U2 as the covariate were both significant; it
is clear that the deviation from linearity is small over the range of speeds
used.
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Fig. 8. Graphs of relative pitching moment Mr against
b for the unpaired (A—C) and paired (D—F)
analyses of the angle series data, and against U (G—I) for the
speed series data. For further explanation, see legend to
Fig. 6. Individual regressions
of Mr against U2 (G—I) are shown
as black dotted lines where the individual regression and the corresponding
GLM treating U2 as the covariate were both significant; it
is clear that the deviation from linearity is small over the range of speeds
used.
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For subsonic fixed-wing aircraft, the aerodynamic forces and moments vary
quadratically with flight speed and linearly with angle of attack only up to
the stall point. Linear approximations can therefore only be used to model the
effects of small changes in speed and pitch, as in the small disturbance
formulation of the rigid body equations of motion used here. It is not clear
in advance whether we should expect the aerodynamic forces and moments to vary
similarly in locusts, which appear to use unsteady aerodynamics
(Cloupeau et al., 1979;
Wilkin, 1990) and may also
vary their wing kinematics with changes in speed and attitude. Nevertheless,
for the range of disturbances studied, the forces and moments were all well
modelled as linear functions of U and b. Although
relative thrust—drag and pitching moment both varied significantly with
U2 (P<0.0001) when U2 was
treated as the covariate, no significant quadratic term could be found in any
hierarchical GLM of the form
Lr=individual+U+U*U in which
U was treated as the covariate. There is therefore no evidence for
any significant quadratic effect over and above a linear dependency of the
aerodynamic forces and moments on U.
The term individual was significant in all but one of the GLM
analyses (Table 4), because
locust `G' produced significantly less force in proportion to its body mass
than either of the other locusts. The interaction term
individual*b was significant for the GLM of the
paired data for Lr (P<0.001) and
Mr (P=0.025), and only just non-significant for
the paired data for Tr (P=0.058), indicating that
the slopes of the dimensionless forces and moments against b
do differ significantly between individuals. The interaction
individual*b was not significant in any of the
unpaired analyses, suggesting that the paired analyses are more sensitive and
better at picking up subtle differences between individuals. On average, the
paired analyses explain 33% more of the total variation than the unpaired
analyses, as can be seen from the dramatic decrease in scatter between the
graphs of the unpaired and paired analyses (Figs
6,
7,
8).
The tighter fit of the data from the paired analysis is a good indication
that the locusts' flight performance varied through time. This is shown
directly in Fig. 9, which plots
Lr against time for the angle series; equivalent graphs
for Tr and Mr are of similar form.
Locusts `R' and `B' produced decreasing amounts of lift over time, but,
surprisingly, locust `G' actually increased the lift it produced. In all three
cases, temporal lift variation (indicated by the range of the reference
measurements) is of the same order of magnitude as pitch-dependent lift
variation (indicated by the range of scatter about the line joining the
reference measurements in Fig.
9):temporal variation in aerodynamic force production cannot be
ignored (contra Weis-Fogh,
1956a,1956b).
Mismatches between the reference levels of force production measured in the
consecutive angle and speed series experiments are presumably the result of
temporal variation in flight performance.
Relative lift
All three locusts produced lift in excess of body weight at certain times
during the angle series but below body weight at others
(Fig. 6A—C), so each must
have supported their body weight exactly at some point in time. The mean lift
generated by locusts `R' and `B' was not significantly different from body
weight over the angle series as a whole. None of the locusts ever produced
lift in excess of body weight during the speed series
(Fig. 6G—I), but this is
not surprising in light of the general decline in lift production through time
observed in locusts `R' and `B' (Fig.
9). The decline in Lr might also indicate some
form of physiological compensation for the loss of body mass, although the
percentage changes in body mass are relatively small (<6%).
Relative lift increased linearly with b over the range of
disturbances studied (Fig.
6A—F). This increase was highly significant
(P<0.001) in the GLM analyses using both the paired and unpaired
methods (Table 4), and was just
as highly significant in the individual regressions
(Fig. 6A—F). Importantly,
the upper and lower confidence limits for the slopes of the lines of
Lr against b remain positive even when
the confidence interval for the correction for balance orientation is added to
the confidence interval for the regression slope
(Fig. 6A—F; see figure
legend). Errors in correcting for balance orientation are therefore small
enough not to affect any of the qualitative conclusions above, and we can be
confident that the positive relationship between Lr and
b is both real and consistent across the three individuals.
In addition, the r2 values of the individual regressions
were very high, with b explaining 91–98% of the
within-individual variation in Lr for the paired analysis
(Fig. 6D—F). The GLM
Lr=individual+b explained a
similar proportion of the total variation in the paired analysis
(R2=0.91), and explained only slightly less of the total
variation than the equivalent GLM including the highly significant
(P<0.001; Table 4)
interaction term individual*b
(R2=0.94). This implies that such individual differences
in the underlying slopes as may exist are small in the context of the overall
variation in Lr. In the GLM analyses, b
explains 84% of the total variation, based on the sequential sums of
squares.
Relative lift increased just significantly (P=0.047) with
U in the GLM Lr=individual+U
(Table 4), but inspection of
the individual regressions (Fig.
6G—I) shows that only locust `R' shows any significant
effect. Neither locust `G' nor locust `B' offered any evidence of
Lr varying linearly with U, but the data are too
widely scattered to discount the possibility that some form of relationship
exists. The same was true for individual regressions of Lr
against U2, which only showed a significant effect of
U2 for locust `R' (P=0.039), and even then the
corresponding GLM treating U2 as a covariate failed to
attain overall significance.
Relative thrust—drag
Locust `R' always produced a net thrust, as did locust `B' except at the
highest tunnel speed (Fig. 7).
Only locust `G' alternated between producing net thrust and net drag
(Fig. 7). The preferred flight
speeds of locusts `R' and `B' are therefore likely to be higher than for
locust `G', consistent with their larger overall size (some 30% greater by
body mass).
Relative thrust—drag decreased linearly with b over
the range of disturbances studied (Fig.
7D—F). The negative slope of the GLM
Tr=individual+b was just
significant (P=0.024) for the unpaired analysis
(Table 4), but since only the
individual regression for locust `G' showed any significant effect
(P=0.026), the unpaired analysis offers no strong evidence for a
general effect of body angle on thrust—drag
(Fig. 7A—C). On the other
hand, the GLM Tr=individual+b
revealed a highly significant (P<0.001) negative relationship
between Tr and b in the paired analysis
(Table 4), and in this case the
negative slopes of the individual regressions were highly significant
(P<0.001) for locusts `R' and `G'
(Fig. 7D,E) and just
significant (P=0.048) for locust `B'
(Fig. 7F). In the case of
locust `B', the confidence interval widened just enough to include zero when
error in correcting for balance orientation was taken into account
(Fig. 7F). Nevertheless, the
negative relationship between Tr and b
that was weakly detected by the unpaired analysis is clearly revealed by the
more sensitive paired analysis, and we can be reasonably confident that this
relationship is both real and consistent across the three individuals.
Although the GLM
Tr=individual+b explained an
extremely high proportion of the total variation in the paired analysis
(R2=0.96), this largely reflects the wide variation in
Tr between individuals, which tends to swamp the variation
due to b in the pooled analysis. In fact, the term
individual explains over 92% of the total variation in the GLM,
whereas b explains only 4%. Under these circumstances, the
r2 values for the individual regressions (average
r2=0.55) give a better indication of the importance of
b in explaining variation in Tr —
at least within individuals.
A highly significant negative relationship was found between
Tr and U (P<0.001) in the GLM
Tr=individual+U
(Table 4). The significance
levels of the individual regressions (Fig.
7G—I) were also very high (P0.002), and all
showed a negative relationship between Tr and U.
We are therefore confident that this relationship is both real and consistent
across individuals, which is reassuring because a negative relationship
between Tr and U is necessary to provide static
stability with respect to flight speed. Individual regressions of
Tr against U2 were also highly
significant (P0.004), as was the corresponding GLM treating
U2 as the covariate (P<0.001), so the
individual regressions of Tr against
U2 are plotted for comparison with the linearized response
to small perturbations (Fig.
7G—I): it is clear that the deviation from linearity is
small over the range of speeds used. Although the total proportion of the
variation explained by the GLM
Tr=individual+U was very high
(R2=0.95), U explained only 23% of the total
variation in Tr when fitted after individual in
the model (note that for the speed series analysis, the sequential and
adjusted sums of squares generally differ). Once again, the
r2 values of the individual regressions (average
r2=0.91) give a better indication of the importance of
U in explaining variation in Tr.
Relative pitching moment
Locust `R' consistently produced a nose-up pitching moment
(Fig. 8), and is therefore
unlikely to have experienced pitch equilibrium at any point in the
experiments. Locusts `G' and `B' both produced nose-up and nose-down pitching
moments at various moments in time (Fig.
8), so both must have experienced pitch equilibrium at some
point.
The GLM Mr=individual+b for
the unpaired analysis was the only GLM in which the slope was not quite
significant (P=0.065), but the more sensitive paired analysis was
able to resolve the underlying relationship, with a highly significant
(P<0.001) negative relationship found in the corresponding GLM
(Table 4). A negative
relationship between Mr and b was found
in all of the individual regressions (Fig.
8A—F), although the slope for locust `B' just failed to
attain significance (P=0.088). The slopes for locusts `R' and `G'
were both highly significant (P=0.007, P<0.001,
respectively), although the confidence interval for locust `R' widened just
enough to include zero when error in correcting for balance orientation was
taking into account (Fig. 8D).
Nevertheless, the consistency with which a negative slope was found gives us
confidence in the generality of the relationship, which is reassuring because
a negative relationship between Mr and b
is essential for static pitch stability. The GLM
Mr=individual+b explained a
very high proportion of the total variation in the paired analysis
(R2=0.94), but as in the analysis of thrust—drag,
b itself explained a relatively small proportion (4%) of
this variation, owing to the much smaller relative force production of locust
`G' as compared to locusts `R' and `B'. Here again, the r2
values of the individual regressions give a better indication of the
proportion of the variation in Mr explained by
b (average r2=0.44). As in the analysis
of lift, the proportion of the variation in Mr explained
by the GLM Mr=individual+b was
only 1% less than the proportion explained by the corresponding GLM including
the significant interaction (P=0.025) term. Such individual
differences in the underlying slopes as may exist are therefore likely to be
small in the context of the overall variation in Mr.
All of the individual regressions of Mr against
U (Fig. 8G—I)
had a highly significant (P0.001) negative slope, and explained a
high proportion of the within-individual variation in Mr
(r2=0.90 on average). This was mirrored by the high
significance (P<0.001) of the GLM
Mr=individual+U
(Table 4) and the very high
proportion of the total variation explained by the model
(R2=0.94), although U itself explained only 31%
of the total variation in Mr when fitted after
individual in the model. Incorporating the 95% confidence interval
for the correction for U did not widen the combined confidence
intervals for the slopes of the individual regressions to include zero
(Fig. 8G—I), and we are
therefore confident that this negative relationship between
Mr and U is both real and consistent across the
three individuals. Individual regressions of Mr against
U2 were also highly significant (P0.002), as
was the corresponding GLM treating U2 as the covariate
(P<0.001), so the individual regressions of Mr
against U2 are plotted for comparison with the linearized
response to small perturbations (Fig.
8G—I): it is clear that the deviation from linearity is
small over the range of speeds used.
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Analysis of results
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Populating the system matrix
The fact that Lr, Tr and
Mr are so well modelled as linear functions of small
perturbations to b and U substantially validates
the linearization of the equations within the range of disturbances studied,
and also suggests that the slopes of the regressions will give a good estimate
of the slopes of the underlying relationships at equilibrium. Because
b was held constant whilst U was varied, and
vice versa, the slopes of these functions already define partial
derivatives like those in the longitudinal system matrix
Fsym (Equation 2), with the important caveat that because
flight equilibrium was never achieved for the combinations of
b and U we used, we must assume that the same
slopes would apply for small disturbances from equilibrium. This assumption
allows us to calculate all of the stability derivatives in Equation 2, except
for the pitch rate derivatives (q-derivatives), and conveniently
preserves the linear time-invariance of the equations. Our general strategy
will be to analyse the system matrix assuming that the pitch rate derivatives
are all zero, and then to investigate the effect of assigning a range of
realistic non-zero values to them. This means that the empirically-defined
longitudinal system matrices contain only the static stability derivatives
(i.e. Xu, Zu, Mu,
Xw, Zw, Mw), and
we will term them `static system matrices' to highlight this important
qualification.
Whereas we have so far presented force data resolved into vertical lift and
horizontal thrust—drag components, the stability derivatives in
Fsym are resolved into X and Z components
fixed with respect to the body. The corresponding axes are usually defined
such that the x-axis is aligned with the direction of flight at
equilibrium, which means that we=e=0.
The axes are then referred to as stability axes. With these simplifications,
we may write the equation of motion for nonmanoeuvring flight with
correctional control enabled as:
 | (8) |
where the 4x4 matrix is the static system matrix. Equation 8 explicitly
requires that we specify the equilibrium flight speed (ue)
and implicitly requires that we specify the equilibrium body angle
(b,e) in order to define the axes in which the forces are to
be resolved. Unfortunately, it is not possible to determine whether a locust
was flying at equilibrium in advance of analysing the flight data. Not
surprisingly, none of the locusts ever flew in perfect equilibrium (i.e. lift
balancing body weight, thrust balancing drag, and no net pitching moment), but
the reference measurements taken during the angle series experiments provide
reliable benchmarks from which to solve for the equilibrium flight condition,
about which the equations of motion are linearized.
Since the individual locusts differed significantly in their flight
performance, we will calculate the equilibrium flight conditions and stability
derivatives separately for each individual. In general, we have:
 | (9a) |
 | (9b) |
 | (9c) |
where Lr,ref, Tr,ref and
Mr,ref denote the mean levels of relative force production
at the reference speed (Uref=3.50 m s-1) and
body angle (b,ref=7°). Solving for the equilibrium
flight conditions means solving Equation 9a—c for
Lr=1 and Tr=Mr=0.
Unfortunately, since we have three dependent variables (lift,
thrust—drag and pitching moment), but only two independent variables
(speed and body angle), it is only possible to solve the equations such that
two of the three equilibrium conditions are satisfied. Pitch disequilibrium
and thrust—drag disequilibrium are closely linked in our dataset, so we
will solve Equation 9a—c for Lr=1, letting either
Tr=0 or Mr=0. This gives two
pseudo-equilibria, which each provide an estimate of the equilibrium tunnel
speed (Ue), and may be averaged to give the unique
estimate of ue that we require (in practice the two
estimates always differed by less than 13% of their mean). The solution for
b,e is already unique if we assume, as shown above, that
lift is independent of flight speed (i.e.
Lr/U=0). The pseudo-equilibria so
defined (Table 5) are
consistent with the speeds and body angles adopted by free-flying locusts in
the wild (Schistocerca gregaria:
Waloff, 1972; Locusta
migratoria: Baker et al.,
1981).
Having defined b,e for each of the locusts, we may
resolve the forces into X and Z components. Calculating the
stability derivatives directly from the regressions of Lr
and Tr against b and U
complicates interpretation of the regression model used, so we instead
resolved the forces into their X and Z components and fitted
Model I linear regressions to the data again. The slopes of these regressions
are given as partial derivatives in Table
6 and are shown enclosed in square brackets if the individual
regression slope was non-significant (P>0.05). It is immediately
clear that the stability derivatives are more reliable for locusts `R' and `G'
than for locust `B'. The significance of the regressions closely matches that
of the regressions of Lr and Tr; for
example, the non-significant Zu derivatives for locusts
`G' and `B' in Table 6 reflect
the absence of any significant relationship between Lr and
U for those individuals.
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Table 6. Dimensional static stability derivatives with respect to tunnel
speed U and body angle b for the
three locusts, resolved in the stability axes
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The partial derivatives of Table
6 must now be re-expressed as functions of the longitudinal
velocity components u and w in order to match the form of
the stability derivatives in Equation 8. For the symmetrical flight condition,
we have:
 | (10) |
and since w2 is well over an order of magnitude smaller
than u2, even at the most extreme body angles used in the
experiments, the first order approximation
 | (11) |
is perfectly acceptable at perturbed angles of attack. The
u-derivatives may therefore be expressed as:
 | (12) |
Similarly, we may express b as:
 | (13) |
which indicates that to a first-order approximation,
 | (14) |
where b is in radians. Hence, the w-derivatives are
simply:
 | (15) |
allowing us to calculate all six static stability derivatives for each locust
(Table 7), and so to populate
the system matrix in Equation 8.
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Table 7. Dimensional static stability derivatives with respect to forward speed
(u) and aerodynamic incidence (w) for the three locusts,
resolved in the stability axes
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Solution of the small disturbance equations
To yield any useful insight into locust flight we must use the system
matrices we have defined to solve Equation 8, which is of the general form:
 | (16) |
Solutions to this first-order differential equation are well known and are of
the general form:
 | (17) |
where eA is the matrix exponential (e.g.
Apostol, 1997). This is shown
by differentiating Equation 17 with respect to t:
 | (18) |
The exponential matrix etA is readily
calculated if the nxn matrix A can be
diagonalized (which is the case if its n eigenvalues are distinct),
in which case we have:
 | (19) |
where C is a non-singular matrix depending on A, and D is
an nxn diagonal matrix with the same eigenvalues
(
1,...,n) as A. Hence,
 | (20) |
and since
 | (21) |
it follows that the entries of the exponential matrix
etA are linear combinations of
et1,...,etn,
and therefore (Equation 17) the values of the state variables themselves are
also linear combinations of these modes.
Table 8 gives the eigenvalues
of the static system matrices for each of the locusts, calculated in
Matlab.
A positive real root will result in the exponential growth of each of the
disturbance quantities in Equations 22, so the modes of motion identified by
the positive real roots in Table
8 are dynamically unstable. On the other hand, the negative real
roots in Table 8 will result in
exponential decay of the disturbed quantities, so the modes of motion that
they identify are dynamically stable. The behaviour of a pair of complex
conjugate roots =n±i
is less straightforward,
but since the principle of linear superposition applies, the pair combines to
give:
 | (22) |
which can be expanded as:
 | (23) |
where the coefficients
A1,1=(a1,1+a1,2)
and
A1,2=i(a1,1—a1,2),
etc. are real numbers. Equation 23 describes an oscillatory motion of angular
frequency and period T=2/, so the complex
conjugate roots identify an oscillatory mode of motion. It is clear by
inspection of Equation 23 that this motion decays if the real part,
n, of the root is negative, but grows if n is positive. The
complex roots in Table 8 always
have negative real parts, so the mode of motion that they identify is
dynamically stable. The behaviours of the four types of general solution to
Equation 16 are indicated in Fig.
10A—D. The three modes of motion displayed by the locusts
correspond to the types of general solution illustrated in
Fig. 10AC. On the principle of
linear superposition, and in the absence of any control inputs beyond those
identified in the static system matrices, the natural flight behaviour of a
locust is describable as the sum of these three modes. Since one of these
modes is unstable, the model fails to explain the dynamic flight stability of
locusts completely. We will now consider each of the modes we have identified
in detail.
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Fig. 10. The four general types of solution to the longitudinal equations of motion.
(A) Monotonic subsidence (stable), corresponding to a negative real root. (B)
Monotonic divergence (unstable), corresponding to a positive real root. (C)
Damped oscillation (stable), corresponding to a complex conjugate pair of
roots with negative real parts. (D) Divergent oscillation (unstable),
corresponding to a complex conjugate pair of roots with positive real
parts.
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The damped oscillatory mode
The pair of complex conjugate roots
=n±i identify a damped oscillatory
mode of relatively short period (TR=0.10 s,
TG=0.06 s, TB=0.11 s). The damping of
the motion is defined by the damping ratio:
 | (24) |
which for the three locusts is: 
R=0.079,
G=0.071, B=0.095. Critical damping (i.e. the
transition from sinusoidal to exponential motion) occurs at 
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Summary
Introduction
