Dynamic flight stability in the desert locust Schistocerca gregaria
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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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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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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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) |
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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) |
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Materials and methods |
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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 23 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.
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Force measurements
The force balance was sting-mounted to a rotary stage (Edmund Industrial
Optics, Barrington, NJ, USA) providing repeatable pitch adjustment
0°b
14°±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°
b
14° 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 (2.00
U
5.00 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 forcemoment
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 forcemoment 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
forcemoment 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 `thrustdrag' 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 thrustdrag'
(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.
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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
leftright 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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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) |
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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) |
![]() | (5) |
![]() | (6) |
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:
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Results |
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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 thrustdrag 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.
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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. 6AC), 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. 6GI), 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.
6AF). 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. 6AF). 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. 6AF; 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 9198% of the
within-individual variation in Lr for the paired analysis
(Fig. 6DF). 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. 6GI) 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 thrustdrag
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 thrustdrag decreased linearly with b over
the range of disturbances studied (Fig.
7DF). 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 thrustdrag
(Fig. 7AC). 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.
7GI) 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 (P
0.004), 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.
7GI): 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.
8AF), 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 thrustdrag,
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. 8GI)
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. 8GI), 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 (P
0.002), 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.
8GI): it is clear that the deviation from linearity is
small over the range of speeds used.
![]() |
Analysis of results |
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Whereas we have so far presented force data resolved into vertical lift and
horizontal thrustdrag 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) |
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) |
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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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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) |
![]() | (11) |
![]() | (12) |
![]() | (13) |
![]() | (14) |
![]() | (15) |
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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) |
![]() | (17) |
![]() | (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) |
![]() | (20) |
![]() | (21) |
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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) |
![]() | (23) |
|
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) |
![]() | (25) |
Further information on the oscillatory mode may be gathered directly from
the relevant eigenvectors, which are simply values of x(0) satisfying
Equation 17 for each of the roots in Table
8. The components of the eigenvectors are displayed in
Table 9, in polar form: since
eigenvectors are unique in direction but not in magnitude we have scaled them
to make =1. Note that a pitch perturbation of 1
rad(approximately 57°) would be substantial indeed, so the scale is chosen
only for convenience. Scaling the eigenvectors in this way aids in comparison
of the three locusts, but since the values of the various state variables of a
single locust are only significant relative to each other, some further
normalisation of the variables is required for ease of comparison. We have
already shown that
w
ue
b (Equation
14), so normalising
w by ue as we have
done in Table 9 converts
w to radian measure (i.e. the same units as
).
Normalising
u by ue similarly brings
u into line with the dimensionless form of
w.
We can see immediately from Table
9 that changes in the forward velocity component
(
u) are negligible in the oscillatory mode. As a result,
oscillatory changes in pitch (
) manifest themselves as
oscillations in angle of attack (
b) of almost
identical phase and amplitude to the pitch oscillations themselves. This
combination of characters seems to identify this oscillatory mode of locusts
with the short period pitching mode of aircraft, which is essentially a rapid
pitch oscillation with negligible change in forward velocity (e.g.
Nelson, 1989
;
Etkin and Reid, 1996
;
Cook, 1997
).
|
The subsidence and divergence modes
The non-oscillatory nature of the subsidence and divergence modes is at
first sight puzzling. In most aircraft, the longitudinal equations of motion
have two pairs of complex conjugate roots, which means that most aircraft
display two oscillatory modes of motion: the short period pitching motion just
mentioned, and a second, lightly damped motion of much longer period called
the phugoid. The latter is classically described as an interchange between
kinetic and potential energy, manifested as a steady rise and fall in altitude
(Lanchester, 1908). This is
accompanied by slow changes in pitch, which result in the angle of attack
remaining substantially constant throughout the motion. Assuming it is correct
to identify the oscillatory mode found in locusts with the short period mode
of aircraft, then the results of this analysis show the complex phugoid root
of aircraft appears to split into two real roots in locusts.
The lack of any clear relationship between lift and flight speed in our
results (see also Zarnack and Wortmann,
1989) goes a long way towards explaining this difference. The
apparent independence of speed and lift is most clearly manifested in the
speed derivative Zu, which is statistically
non-significant (P>0.05) in locusts `G' and `B'. This speed
derivative plays a critical role in the phugoid motion of aircraft, because
potential energy lost as the aircraft loses altitude is converted into kinetic
energy with a concomitant increase in speed. This increase in speed increases
the lift and causes the aircraft to gain altitude, so it is the speed
derivative Zu that is responsible for the oscillatory
behaviour of the phugoid. Given the lack of any clear relationship between
lift and flight speed in locusts, it is not surprising to find that they
exhibit no analogous oscillatory mode.
Although the two real roots identify modes that are non-oscillatory, it is
nevertheless possible to gain some impression of the timescales on which they
operate by specifying the time to halve or double (Equation 25). This gives:
thalf,R=0.28 s, thalf,G=0.20 s,
thalf,B=0.18 s, for the subsidence mode, and
tdouble,R=0.14 s, tdouble,G=0.12 s,
tdouble,B=0.12 s for the divergence mode. Clearly, the
timescales of the non-oscillatory modes are similar to the timescale of the
oscillatory mode, and we may expect to find some interaction between the
various modes. Further information is again given by the eigenvectors, which
indicate that the non-oscillatory modes involve significant interactions
between all of the state variables (Table
9). In the divergence mode an increase in pitch angle
(>0) is accompanied by a decrease in forward velocity
(
u<0), which leads to a stall following a nose-up
disturbance and a nosedive following a nose-down disturbance. This is not
dissimilar to a divergent motion that appears to limit the length of glides in
Locusta migratoria (Baker and
Cooter, 1979
). These are begun at a positive pitch angle
(typically 1520°), but end at a negative pitch (as low as
-30°). Stable flight is only recovered when flapping is reinitiated. This
rotation takes place over about 0.25 s, which identifies a divergent mode
operating on a similarly short timescale to the mode we have identified.
In both the subsidence and divergence modes, changes in pitch rate
(q) are small relative to changes in pitch attitude
(
), whereas changes in forward velocity (
u) are
comparatively large. This implies significant coupling of speed and pitch, as
already shown by the strong negative relationship found between pitching
moment and tunnel speed. This kind of coupling does not occur in fixed wing
aircraft (in which the speed derivative Mu is close to
zero at subsonic speeds), but does occur in helicopters as a result of the
positive pitching moments induced as the rotor `flaps' back with increasing
speed (e.g. Padfield, 1996
).
This leads to mildly unstable longitudinal dynamics in helicopters, and it
therefore seems likely that the speed derivative Mu may be
contributing to the instability observed in the divergence mode we have
identified. This is confirmed by the root locus plots of
Fig. 11AC, which
illustrate the effect of systematically reducing the value of
Mu to zero for each of the locusts. In each case, the
positive real root becomes less positive (i.e. less unstable) as
Mu
, and in the case of locust `R' the root is
negative (i.e. stable) by the time Mu=0. In fact, for
Mu
0, locust `R' is completely stable.
|
One surprising outcome of the eigenvector analysis is that the signs of
u and
w are switched in the subsidence mode of
locust `B' compared to the other two locusts
(Table 9). We have already
remarked, on the basis of the regression statistics, that the system matrix
for locust `B' is the least reliable. Comparison of the system matrices
indicates that the incidence derivative Xw is considerably
smaller in locust `B' than in either of the other two locusts, and is actually
non-significant in locust `B' (P>0.05). Substituting the
corresponding value from either of the other two locusts renders the
subsidence mode eigenvector qualitatively identical to those of the other
locusts. We are therefore reasonably confident that locusts `R' and `G'
correctly identify the nature of the subsidence mode.
Error analysis
How reliable are these results? There are two components to answering this
question. The first concerns the sensitivity of the results to measurement
error; the second (see Discussion) concerns the effect of the various
assumptions made in the course of arriving at the static system matrices. To
investigate the effects of measurement error, we used the 95% confidence
intervals for the partial derivatives in
Table 6 to parameterize each of
the six static stability derivatives as a normally distributed variable. We
then ran a Monte Carlo simulation in Matlab in which the stability derivatives
were varied randomly together. For the purposes of this analysis, it was
assumed that the equilibrium flight speed ue, equilibrium
body angle b,e, reference body mass m and pitching
moment of inertia Iyy are all measured without error. This
assumption is not meant literally, but is reasonable to make because errors in
b,e, m and Iyy will only modify
the same six matrix elements that we are already varying, and variation in
ue between individuals provides a wide bracket upon the
effects of errors in ue.
We ran the Monte Carlo simulation 5000 times for each locust, recalculating
the eigenvalues after each iteration. The results are plotted in Argand
diagram format in Fig.
12AC, which show sufficient overlap for us to be confident
that all three locusts share the same underlying flight dynamics. The plots
show that the complex conjugate roots (represented by the clouds of points for
which 0) are stable (i.e. (n<0) in 100% of cases for
locusts `R' and `G', and are stable in 99.9% of cases for locust `B'. The
conclusion that the static stability derivatives imply the existence of a
single, damped oscillatory mode of motion is therefore robust within the
bounds of experimental error. The results also show that there are two real
roots - 1 positive and 1 negative in 100% of cases. The conclusion
that the static stability derivatives of locusts are insufficient to confer
complete dynamic stability is therefore also robust within the bounds of
experimental error.
|
![]() |
Discussion |
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Assuming the locusts were flying acceptably close to equilibrium, the measured static stability derivatives are sufficient to confer static stability with respect to speed and pitch, but insufficient to confer complete dynamic stability. Since all of the locusts were capable of stable free flight, the analysis we have presented so far must therefore be deficient in some way. There are five major ways in which the analysis could be deficient, and these may be ranked in hierarchical fashion according to the point at which the analysis fails. It makes best sense to work backwards that is to consider deficiencies entering at a later stage of the analysis first.
1. The static stability derivatives that we have measured may be
somehow unrepresentative of natural free flight
The experimental conditions used are not a literal open-loop version of the
conditions experienced by a free-flying locust. Most significantly, the
locusts lacked any optic flow to simulate steady progression through the
environment visually. We do not consider this to be a likely source of the
observed instability, because locusts commonly fly at altitude or in twilight,
when there is little or no optic flow. However, it is possible that the
locusts would have perceived some of the nearby objects as fixed, which is
quite different to there being no optic flow and might produce unusual flight
behaviour. It is also possible that the locusts were attempting take-off from
the force balance, and were therefore intentionally inducing a divergence
mode, but this also seems unlikely given that the complete flight posture they
were using is not adopted during take-off.
2. It is erroneous to conflate w and in
determining the stability derivatives if the locusts possess a static sense of
pitch attitude
In an aircraft, passive stability arises through changes in the flow around
the wings and body. This means that wind tunnel measurements of the passive
aerodynamic stability derivatives of a model aircraft can be made just as well
with the model upside down as with the model the right way up, provided the
aerodynamic incidence is the same (e.g.
Barlow et al., 1999). It
follows that the forces and moments generated by an automatic control system
could also be characterised with the model inverted if the control settings
were linked only to aerodynamic instruments. This equivalence would of course
cease to apply if the controls were linked to instruments sensing pitch
attitude with respect to the ground. Framed in this way, the difference
between pitch attitude and aerodynamic incidence control seems rather obvious.
What may be less obvious is the corollary that nervous control mechanisms
linked to changes in pitch attitude (measured relative to an inertial frame)
are not dynamically equivalent to nervous control mechanisms linked to
aerodynamic incidence or body angle (measured relative to the direction of
flight).
Consider the phugoid motion of an aircraft. In this mode, large changes in
pitch attitude are accompanied by relatively insignificant changes in
aerodynamic incidence. Since the static pitch stability built into most
aircraft derives from the derivative M, which in
turn derives from the incidence derivative Mw (Equation
15), static pitch stability has rather little effect upon the damping of the
phugoid. By contrast, the short period mode does involve significant changes
in incidence, and is therefore heavily damped in any aircraft in which
M
is large and negative. An automatic control
system linked to aerodynamic incidence is therefore only effective at
providing damping of the short period mode of motion. On the other hand, since
the phugoid and short period modes both involve large changes in pitch
attitude, both modes of motion are effectively damped by a control mechanism
linked to pitch attitude. In fact, just such mechanism is usually present in
aircraft autopilots, in which it is linked to the vertical gyro (e.g.
Etkin and Reid, 1996
).
In the analysis we have presented so far, we have not distinguished between
changes in the forces and moments due to aerodynamic incidence and changes in
the forces and moments due to pitch attitude. Pitch attitude is measured with
respect to inertial space, which for the purposes of this analysis is fixed
with respect to both gravity and the horizon. It follows that visual or
gravimetric input could provide the necessary reference for pitch attitude
control in flying animals. Separating pitch attitude control from aerodynamic
incidence control experimentally would therefore require a more complex
arrangement in which the airstream could be rotated relative to an Earth-fixed
visual frame of reference. A much simpler alternative is to simulate the
effect of introducing pitch attitude control mathematically by introducing a
new derivative M into the system matrix.
If locusts do use pitch attitude control, the derivative that we have thus
far characterised as M will include elements due to
both M
and Mw. Assuming these
combine additively (which follows from the linearization of this analysis), we
may therefore partition M
into components due to
aerodynamic incidence and pitch angle. The root locus plots of
Fig. 13 show all of the
possible additive combinations of M
and
Mw that are compatible with the values of
M
measured for each of the three locusts. The
damping of the short period mode decreases a little as the ratio of
Mw to M
tends towards zero, but
the mode remains stable. By contrast, the positive real root part of the
unstable divergence mode becomes strikingly less positive (i.e. less unstable)
as the ratio of Mw to M
tends
toward zero. In the case of locust `G' there comes a point at which the real
parts of all the roots are negative (i.e. completely stable). Beyond this
point the two real roots converge and split into a complex conjugate pair of
long period (1.72 s) that resembles the phugoid mode of aircraft. The two real
roots of locusts `R' and `B' also converge if the pitch attitude damping is
made sufficiently large, although this is not compatible with the measured
values of M
. This analysis therefore indicates that
pitch attitude control could be critical in stabilising locust flight.
|
Pitch attitude control has received little attention in the locust flight
literature, but there is some evidence to suggest that locusts are sensitive
to pitch attitude. For example, locusts are known to pitch their heads in
response to pitching of an artificial horizon
(Taylor, 1981a), and whilst
nothing direct is known about the accompanying changes in aerodynamic force
production, head movements are known to be closely associated with turning
about the roll and yaw axes (e.g. Goodman,
1965
; Camhi, 1970
;
Cooter, 1979
; Taylor,
1981a
,b
;
Robert and Rowell,
1992a
,b
).
It was not possible to make sufficiently accurate measurements from the video
recordings to determine whether the locusts displayed similar head movements
in this study. Diffuse overhead illumination such as we used in this study
ought, however, to be sufficient to elicit a pitch attitude response
(Taylor, 1981b
;
contra Zarnack and Möhl,
1977
), because the pitching movements of the head are mediated by
an ocellar response (Taylor,
1981a
). This is presumably based upon changes in the relative
stimulation of the median ocellus and lateral ocelli
(Rowell, 1988
), but it is
likely that the compound eyes might also play some role in detecting pitch
disturbances. It is therefore possible that brighter lighting conditions than
we used would have elicited a stronger response to changes in pitch.
3. The dropped rate derivatives may be important
Since we were only able to measure static stability derivatives, rate
derivatives were dropped at an early stage of the analysis. Dropping the rate
derivatives does not in general lead to a loss of stability in aircraft. This
is clearly shown by Table 10, which compares the characteristic roots of the complete and static system
matrices for a Navion NA-154 light aircraft (calculated from data in
Nelson, 1989). The real parts
of the roots are negative in both cases, indicating that on a simple yes/no
assessment, the stability of the system is unchanged. In general, the static
system matrix gives a surprisingly good first approximation to the behaviour
of the system: only the damping of the short period mode is significantly
affected by dropping the rate derivatives.
|
Physically, the reason why only the damping of the short period mode is strongly affected by the rate derivatives is that significant linear and angular accelerations are only experienced during rapid motions. Even a high degree of rate damping will have little effect upon a slow motion like the phugoid, so pitch rate damping should not greatly affect damping of the unstable divergence mode in locusts. This prediction is borne out by the root locus plots of Fig. 14, which illustrate the effects of introducing pitch rate damping into the system matrix by increasing the value of the most important rate derivative, Mq, from zero. Increasing Mq has practically no effect upon the non-oscillatory modes, so we are reasonably confident that dropping the rate derivatives has not generated the instability that we have identified. By contrast, the short period oscillatory mode becomes much more heavily damped as Mq is increased. Eventually, critical damping is reached and the complex conjugate roots split into two negative real roots. This suggests that pitch rate damping could be important in damping out the short period mode in locusts, as even a moderate degree of pitch rate damping is sufficient to suppress the oscillations completely.
|
In quantitative terms, the short period mode becomes non-oscillatory when
pitch rate damping is introduced such that the ratio
Mq:M is 0.035 in locust `R',
0.021 in locust `G', and 0.040 in locust `B'. For a Boeing 747 cruising at 12
200 m, this ratio is approximately 0.242
(Etkin and Reid, 1996
).
Physically, the effects of a given degree of pitch rate damping (as measured
by the ratio Mq:M
) are more
pronounced at smaller size because of the larger angular accelerations
associated with the intrinsically shorter timescales of the characteristic
modes of motion. Some degree of pitch rate damping of the short period mode
may therefore be essential in stabilising locust flight, and possibly insect
flight in general. This is because the natural frequency of the short period
mode is only half that of the wingbeat, which makes it likely that the
flapping cycle would couple with the short period mode if the latter were not
adequately damped.
Electrophysiological evidence suggests that locusts are sensitive to pitch
rate (Möhl and Zarnack,
1977), although it is not clear which sensory system provides the
input for this response. Pitch rate sense has only been discussed in any
detail for flies (Diptera). In flies, angular velocity is detected about all
three orthogonal axes by the highly non-orthogonal halteres (Nalbach,
1993
,
1994
;
Nalbach and Hengstenberg,
1994
). The stimulus for this response is the Coriolis force
experienced by the beating halteres during turning
(Nalbach, 1993
).
Interestingly, the non-orthogonality of the halteres results in enhanced
sensitivity for pitch compared to roll
(Nalbach, 1994
), and it is
possible that this relates to the importance of pitch rate damping in
providing adequate damping of the short period mode in the face of excitation
by the flapping cycle.
Coupling of the short period mode with the flapping cycle could lead to
catastrophic flight handling problems for the locust, which begs the question
of why it should want to have a mode of motion so close to the flapping cycle
in frequency. Much the same problem arises in aircraft, where the frequency of
the short period mode is usually similar to the human pilot's own response
time (Cook, 1997). Adequate
damping of the mode is therefore essential in avoiding pilot-induced
oscillations. Nevertheless, for the aircraft to display good handling
qualities, it is essential for the short period mode to have a frequency close
to the pilot's response time. At low forcing frequencies, all of the state
variables show a clear sinusoidal response to a sinusoidal control input.
Although control inputs are rarely purely sinusoidal in aircraft, an arbitrary
waveform can be represented as a summation of sinusoids of varying amplitude
and phase. The same principle applies in flapping flight, because control
inputs will generally be harmonics of the wingbeat frequency. As the forcing
frequency is increased, inertia will prevent an aircraft from responding
quickly enough to follow the input, so the response of the state variables
will lag behind the control input. At the same time, the amplitude of the
sinusoidal response will diminish, eventually becoming imperceptible at a
frequency corresponding to the bandwidth of the aircraft's frequency response.
Effectively, the aircraft acts like a low-pass filter to the control inputs
applied to it. The same will be true of flying animals, and this is clearly of
relevance to understanding the evolution of their neuromuscular control
systems.
A generalised plot of an aircraft's frequency response looks something like the pitch attitude gain response illustrated in Fig. 15, with clear peaks in the response of the aircraft to pilot inputs made at the frequencies of the natural modes of motion. At forcing frequencies close to the phugoid frequency, the aircraft behaves as an amplifier and relatively small control inputs lead to large amplitude responses in the state variables. The gain is usually smaller at the frequency of the short period mode because the short period mode is more heavily damped than the phugoid. It follows that the pitch attitude bandwidth frequency is usually only a little higher than the frequency of the short period mode. This means that whilst it would be undesirable for an oscillatory mode of motion to coincide exactly with the flapping cycle, its frequency should be only a little less than the wingbeat frequency (and certainly of the same order of magnitude) if control inputs made at the level of a single wingbeat are to be effective in controlling the animal's flight.
|
4. A linear model may be inadequate to describe this system
"It is almost invariably true to say that behind a linear system
is a non-linear one to which it is intended to be an approximation"
(Power and Simpson, 1978).
With this principle in mind we note that the issue we must address here is not
whether locust flight is a linear phenomenon, which it certainly is not, but
whether the linearized approximations are sufficient to capture the essentials
of the system's behaviour. It is reassuring that the aerodynamic forces and
moments are well modelled as linear functions of speed and angle of attack
perturbations when one or other of these variables is held constant and the
other is varied. In a rigid aerodynamic system this would be sufficient to
imply that changes in speed and incidence should also combine additively in
their effects. This will not necessarily hold true in a controlled system,
however, because there is no guarantee that a locust will respond in the same
way to a given pitch disturbance at different speeds.
Such interactions would clearly violate the assumption of linearity and
would greatly complicate the required analysis if the non-linear effects were
significant. Interaction effects could be either stabilising or destabilising
in their effects, so it is possible that their neglect could explain the
instability of the linear model. For example, since an increase in is
accompanied by an increase in u and a decrease in w in the
stable subsidence mode, but by opposite changes in u and w
in the unstable divergence mode, it might be possible to stabilise the latter
by building into the system appropriate interactions of the control responses
to u, w and
. Other forms of non-linearity such as aeroelastic
hysteresis or control hysteresis are likely to promote instability, so
neglecting them here is unlikely to have been the cause of instability in the
system.
It is also important to note that the small disturbance equations of motion
are only linear time-invariant with respect to disturbances from the
equilibrium condition about which the equations are linearized. Since flight
equilibrium was never attained experimentally, and since it was not possible
to calculate a unique condition in which all three conditions for flight
equilibrium were exactly satisfied simultaneously, the equations of motion are
not strictly linear time-invariant. In this case the entries in the system
matrix are (real) functions of time and it is no longer a sufficient condition
for asymptotic stability that the eigenvalues of the system matrix have
negative real parts for all time t0 (e.g.
Hahn, 1967
). Nevertheless, we
were able to calculate a mean pseudo-equilibrium for each locust that was
close to being an exact equilibrium (Table
5), and on the assumption that the empirical regression slopes
would have been the same had perturbations been made from this mean
pseudo-equilibrium instead of the reference flight condition, it is reasonable
to treat the equations of motion as linear time-invariant.
5. The rigid body approximation may be inappropriate
We have discussed the constraints upon treating periodic variables as
time-averaged constants in detail elsewhere
(Taylor and Thomas, 2002). In
general, this rigid body approximation only works well if the lowest frequency
of the periodic variables is at least an order of magnitude higher than the
highest frequency of the natural modes of motion. We previously reasoned
(Taylor and Thomas, 2002
) that
this could be expected to be true in animal flight, partly on the basis of
approximations to the natural modes of motion and partly on the basis of the
observation that resonance effects between the natural modes of motion and the
flapping cycle are rarely observed. By contrast, the analysis here identifies
a short period mode of motion whose frequency is only half that of the
wingbeat frequency, and we have already said that the flapping cycle is
therefore liable to excite the short period mode. How can we explain this
discrepancy?
The reduced-order approximation that we used to approximate the period of
the short period mode previously (Taylor
and Thomas, 2002) was:
![]() | (26) |
Specifically, because M scales as the product of
a force and a length, it scales with the fourth power of the linear
dimensions. On the other hand, because Iyy scales as the
product of mass and area, it scales as the fifth power of the linear
dimensions. Other things being equal, the period of the short period mode
should therefore scale as the square root of the linear dimensions. This is
why the short period mode is much closer in frequency to the stroke cycle than
we had previously predicted (Taylor and
Thomas, 2002
). It follows that the rigid body approximation is
only likely to be valid in flapping flight if the pitch rate damping is
sufficiently high that the flapping cycle does not normally excite the short
period mode. Equations of motion have recently been derived that include three
rotational degrees of freedom for each of two flapping wings
(Gebert et al., 2002
), so the
theoretical framework now exists for future experimental work to take account
of the wing motions.
General conclusions
Earlier work on the longitudinal flight mechanics of locusts has been
dominated by analysis of an influential paradigm: the so-called `constant-lift
reaction' (Wilson and Weis-Fogh,
1962), in which locusts are supposed to produce the same vertical
force across a 20° range of body angles (Weis-Fogh,
1956a
,b
;
Gewecke, 1975
). The
`constant-lift reaction' is supposed to result from a consistent and
well-known response to imposed changes in pitch, whereby forewing inclination
adjusts to compensate for the disturbance
(Gettrup and Wilson, 1964
;
Gettrup, 1966
;
Wortmann and Zarnack, 1993
;
Fischer and Kutsch, 2000
)
under active muscular control (Wilson and
Weis-Fogh, 1962
; Gettrup,
1966
; Zarnack and Möhl,
1977
). Because hindwing inclination remains unchanged with respect
to the body (Gettrup and Wilson,
1964
; Wortmann and Zarnack,
1993
), the lift on the hindwings would be expected to increase
with increasing body angle, but this is supposed to be compensated for by a
drop in flight speed (Gettrup and Wilson,
1964
), resulting in the maintenance of approximately constant
lift.
Remarkably, no study of the `constant-lift reaction' has ever measured the
pitching moments associated with changes in body angle, so it has not been
possible to say whether free-flying locusts would remain at perturbed angles
long enough for a `constant-lift reaction' to have any noticeable effect. The
high frequency of the stable short period mode indicates that angle of attack
disturbances would be corrected for rather quickly in stable free flight. A
`constant-lift reaction' would therefore be inconsequential because the locust
would correct its body angle within a single wingbeat, assuming critical
damping of the short period mode. The same changes in forewing kinematics that
have been implicated in the `constant-lift reaction' will tend to shift the
balance of lift between fore- and hindwings so as to generate a stabilising
pitching moment, and are probably responsible for the static pitch stability
of locusts (Wilson and Weis-Fogh,
1962; Gettrup and Wilson,
1964
; Taylor,
2001
). This may also explain why body angle disturbances are
corrected within a time period almost identical to that of a single wingbeat.
Given that more recent experiments
(Zarnack and Wortmann, 1989
;
Wortmann and Zarnack, 1993
)
contradict the observations upon which the `constant-lift reaction' is
premised, and are consistent with the force measurements in this study, it
seems that we may firmly reject the `constant-lift reaction' hypothesis.
The case of the `constant-lift reaction' is instructive because the hypothesis is predicated upon an oversimplification of the dynamics of flying bodies. As we have shown, all of the longitudinal state variables are coupled in some way, which means that the response to an apparently straightforward control input is far from simple. For example, in aircraft a step increase in the throttle results in an immediate increase in flight speed, as intuition would suggest. However, if the thrust line passes through the centre of mass, this increase in speed excites a damped phugoid oscillation, which converges towards a steady state in which the flight speed is unchanged from its initial value but in which the flight path is tilted up from the initial line of flight! We would strongly caution against making any conclusions about the dynamic effects of measured insect control responses without performing a dynamic analysis similar to that we have used here.
We have argued above that the frequency of the short period mode of motion will need to be close to the wingbeat frequency for an insect to be able to control its flight with control inputs made at the level of a single wingbeat. Similar considerations will apply in birds, but they are arguably less constrained than insects in this respect because the tail provides a control surface that can be operated at timescales not dictated by the wingbeat frequency. In insects, size-dependent scaling of the short period mode frequency will have important consequences for the frequencies at which the neuromuscular system must operate. Control inputs made at frequencies much higher than the short period mode frequency will be ineffective, and for maximum efficiency we would expect the control responses of the nervous system to be tuned to approximately this frequency. We therefore predict that for insects of similar morphology but differing size, the intrinsic timescales at which the neuromuscular control system should operate will scale as the square root of the insect's linear dimensions. The intrinsic frequencies of the neuromuscular control system should therefore scale as one over the square root of the linear dimensions.
The frequency of the short period mode is a function of the pitching moment of inertia. The effect on this of adding body mass is proportional to the square of the distance to the centre of mass from the point at which the mass is added. If locusts were compensating for this perfectly by putting less body mass towards the extremities then we would expect the bar graph of the percentage contributions of different body sections to the total pitching moment of inertia (Fig. 5D) to be flat. It is not: the bar graph of the percentage contributions to the total pitching moment of inertia (Fig. 5D) mirrors almost perfectly the bar graph of the percentage contributions of different body sections to total body mass in the three locusts (Fig. 5A), which is anything but flat. This indicates that locusts are under-compensating for the increasing cost of adding mass towards the extremities, and hints at strong functional constraints in body shape and mass distribution. For example, the head is by far the most costly section in terms of its contribution to the total pitching moment of inertia, even though it lies much closer to the centre of mass than many parts of the abdomen. This is presumably because the organs that it houses (eyes, brain, mouthparts) are some of the most important, their localisation in the body is crucial, and they are rather strictly limited in size to being above some useful minimum.
Because the frequency of the short period mode is a function of the mass distribution of the insect, body shape is crucial in determining an insect's frequency response. For example, the elongate body forms associated with more primitive groups of insect would tend to confer a narrower bandwidth and hence a poorer frequency response. This would constrain manoeuvrability, but would be preferable to having a wider bandwidth that would couple the short period mode and the flapping cycle in the absence of a highly evolved control system to provide pitch rate damping. Conversely, shortening of the abdomen tends to be associated with greater manoeuvrability: darter dragonflies and hoverflies are prominent examples of this. Although the conclusion that insects should reduce their moments of inertia to enhance manoeuvrability is old news, we have shown for the first time that mass distribution should be linked in predictable ways to wingbeat frequency, flight morphology and the response characteristics of the nervous system itself. Flight dynamics therefore provides a formal quantitative framework by which to unite the fields of flight mechanics and neurophysiology.
On the value of true open-loop studies
It has been argued previously
(Weis-Fogh, 1956a) that
allowing tethered insects to indirectly control wind tunnel speed simulates
flight more naturally than if the tunnel speed is held fixed. Whilst there is
certainly some logic in this argument, it must be stressed that the conditions
that this produces remain highly artificial unless body angle is also allowed
to vary freely. Even though body angle was allowed to vary in some of
Weis-Fogh's experiments (Weis-Fogh,
1956a
), the axis of rotation was not aligned with the centre of
mass, which would, in any case, have been affected by the mounting. Unless
there are clear reasons for doing otherwise, we think it preferable to vary
only one degree of freedom at a time, thereby preserving true open-loop
conditions, instead of using a hybrid system that closes some control loops
whilst leaving others open. In the present case, this means holding tunnel
speed fixed whilst body angle is varied, and vice versa.
Although there are some disadvantages to this scheme (for example, it may
be difficult to determine whether the animals are flying at their preferred
flight speed prior to analysing the data), using true open-loop conditions
removes many possible ambiguities in interpreting experimental results.
Results from free-flying locusts described above notwithstanding
(Baker et al., 1981), flight
speed and body angle do appear to be negatively correlated in many insects
(for a review, see Taylor,
2001
). The same is true of helicopters and fixed-wing aircraft.
The observed closed-loop changes in flight speed in Weis-Fogh's experiments
(Weis-Fogh
1956a
,b
)
are therefore of uncertain significance because body angle was not allowed to
co-vary naturally with flight speed.
A second advantage of using true open-loop conditions is that the changes in the flight forces measured then correspond directly to the partial derivatives, or stability derivatives, that are central to the dynamic analysis that we have made here. Although our tethered force measurements were made under open-loop conditions, the quantitative framework that we have provided offers a means of simulating the closed-loop dynamics of free flight. Provided there is no hysteresis in the system, such that open-loop measurements are equivalent to a snapshot of closed-loop conditions, then open-loop measurements of the forces and moments offer a tractable point of entry into the wonderful complexities of animal flight dynamics.
List of symbols
This list contains only those symbols that appear in the main body of the paper. Symbols appearing only in the Appendix are not included.
Stability derivatives
The following shorthand notation is used to denote a partial derivative with respect to the variable contained in the subscript.
Appendix
Development of the linearized equations of motion
In the present context, there is little to be gained from deriving the
equations of motion from first principles. The reader is referred to Etkin and
Reid (1996) and Boiffier
(1998
) for recent treatments of
the subject. At the scales we are considering, it will be satisfactory to
assume that gravitational acceleration (g) and air density are
constants and that the Earth's surface is flat and fixed in inertial space. In
addition, we will assume that there is no wind, and that the animal is a rigid
body with six degrees of freedom and perfect bilateral symmetry.
The nonlinear equations of motion of such a body may be written in the
state vector form:
![]() | (A1) |
![]() | (A2) |
Equation A1 may be expanded as a set of nine coupled nonlinear ordinary
differential equations incorporating the aerodynamic forces (X, Y, Z)
and moments (L, M, N) acting along or about the body axes (e.g.
Etkin and Reid, 1996;
Padfield, 1996
):
![]() | (A3a) |
![]() | (A3b) |
![]() | (A3c) |
![]() | (A4a) |
![]() | (A4b) |
![]() | (A4c) |
![]() | (A5a) |
![]() | (A5b) |
![]() | (A5c) |
Equations A3A5 offer little insight in their present form, but are
readily linearized using small perturbation theory, which assumes that the
animal's motion consists of small disturbances from a reference flight
condition of steady motion. In general we have:
![]() | (A6) |
![]() | (A7) |
![]() | (A8) |
If the following trigonometric approximations are also used:
![]() | (A9) |
![]() | (A10a) |
![]() | (A10b) |
![]() | (A10c) |
![]() | (A11a) |
![]() | (A11b) |
![]() | (A11c) |
![]() | (A12a) |
![]() | (A12b) |
![]() | (A12c) |
The next step of the linearization process requires the aerodynamic forces
and moments to be represented as analytical functions of the perturbed motion
variables and their derivatives. Taylor's theorem for analytical functions can
then be used to approximate each of the six forces and moments as an infinite
series about the reference condition, retaining only first order terms to give
an expansion of the form:
![]() | (A13) |
Partial derivatives of the aerodynamic forces and moments are referred to
as stability derivatives and we will hereon adopt the convention of writing
the stability derivatives in the form
X/
u=Xu. Since we have
assumed that the xz-plane is a plane of symmetry, changes in the
longitudinal variables u, w, q and
induce symmetrical changes
in the forces and moments. It follows that the lateral stability derivatives
with respect to the longitudinal motion variables must all be zero.
Reference values of the aerodynamic forces and moments can be replaced by
setting the disturbance quantities equal to zero in Equations A10A12,
yielding:
![]() | (A14) |
![]() | (A15) |
![]() | (A16) |
![]() |
Acknowledgments |
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References |
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