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First published online November 1, 2006
Journal of Experimental Biology 209, 4533-4545 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02552
Response characteristics of visual altitude control system in Bombus terrestris
Department of Aeronautics and Astronautics, the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
* Author for correspondence (e-mail: tanaka{at}kawachi.rcast.u-tokyo.ac.jp)
Accepted 8 September 2006
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The dynamic control characteristics of the bumblebees were analyzed on the basis of the frequency response data. First, we showed that the measured system possesses a substantial stability margin. This means that the control system has substantial damping characteristics, and was suitable for stable flight control. In addition, our results showed that the measured bumblebee system possesses superior steady state and quick-response characteristics in comparison with a human pilot-vehicle system. Such excellence in both the steady state and transient characteristics (i.e. damping and quick response characteristics) provide the evidence that bumblebees can effectively control their flight with stability and maneuverability.
Key words: bumblebee, Bombus terrestris, flight, frequency response, visual oscillation, altitude control, vertical force, Bode plot, transfer function, dynamic stability
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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) and correlations
between wing kinematics and aerodynamic force generation have been extensively
studied (Lehmann and Dickinson,
1997; Lehmann and Dickinson,
1998; Sane, 2003),
the dynamic flight performance has not been studied thoroughly.
Studies on dynamic flight stability of the desert locust Schistocerca
gregaria (Taylor and Thomas,
2003) provided the first quantitative analysis of dynamic
stability of a flying animal. Taylor and Thomas measured the longitudinal
static stability derivatives and mass distribution of the desert locusts, and
solved the longitudinal equations of motion by utilizing a classical
linearized framework of aircraft flight analyses. In the same framework, Sun
and Xiong studied the longitudinal dynamic flight stability of a hovering
bumblebee (Sun and Xiong,
2005). They computed the aerodynamic derivatives by means of
computational fluid dynamics, and solved the equations of motion by eigenvalue
and eigenvector analyses. Both studies succeeded in identifying three natural
modes of the longitudinal flight: one unstable oscillatory mode, one stable
fast subsidence mode, and one stable slow subsidence mode. Neither of them,
however, succeeded in explaining the flight stability fully, because of the
unstable oscillatory mode.
In the present study, we suggest another approach for the dynamic flight
control analysis. Instead of solving the equations of motion to identify the
natural modes, we focused on visual altitude control in the flight of a
bumblebee Bombus terrestris, and on its dynamic control performance.
We can evaluate such performances on the basis of a transfer function, which
is a mathematical representation of the relation between the input and output
of a linear time-invariant system (Franklin
et al., 2002). We utilized the frequency response method to obtain
the transfer function for the bumblebee system. The input of the frequency
response measurements was a visual oscillation in the vertical direction,
which elicited vertical flight modulation from a tethered bumblebee. The
output was the vertical force variation of the bee, measured by using a load
cell. The oscillation frequency of the input was varied at 0.9, 1.8, 3.6 and
7.4 Hz. We summarized the frequency response characteristics in terms of
amplitude and phase differences, and showed them on a Bode plot. The transfer
function of the visual altitude control system was identified from the plot.
Our results revealed that the measured control system possessed a substantial
stability margin, equivalent to that of a human pilot-vehicle system
(McRuer and Graham, 1964). In
addition, the bumblebee system was revealed to be superior to the human
pilot-vehicle system in terms of both steady state and quick-response
characteristics. These results provide the evidence that bumblebees can
effectively control their flight with stability and maneuverability.
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In preparation for each experiment, we captured a bee in a Petri dish from the colony, and cooled the dish with ice for approximately 30 min. While the bee was anesthetized at that cold temperature, an iron wire 2.5 cm longx0.7 mm diameter (1 mN weight) was glued to the bee near the characteristic yellow line on its thorax. Although all the bees recovered from the anesthesia within a few minutes, some were inactive for nearly 1 h. We used the bees for experiments after they seemed to be fully active, usually within 2 h.
Stimulus presentation
The family of bees has highly developed eye optics
(Spaethe and Chittka, 2003)
that are sensitive to image motion
(Srinivasan et al., 1999). We
utilized this characteristic, and elicited vertical flight modulation from the
bumblebees by means of vertical image motion. This technique has mainly been
used with flies for some time (e.g.
Götz and Wehrhahn,
1984).
We prepared a flight arena, as shown in Fig. 1A, for the purpose of stimulus presentation. This arena shaped a hexagonal cylinder, which is 19.2 cm high, and each side of whose hexagon is 9.6 cm wide. Fig. 1B shows the magnified view of one wall of the arena. The inside wall contained 16 rows of 5.5 mm-diameter LEDs (DU-256N-64C, Azuma Electric, Tokyo, Japan, LED color: orange, wavelength: 610 nm) in the horizontal direction, and 32 in the vertical direction. Horizontal stripe patterns were displayed by lighting alternate 8 rows of LEDs, that is, the period of the stripes was 88 mm. Those stripes were visually oscillated in the vertical direction by using a computer control.
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During the experiments, we placed a tethered bumblebee in the middle of the
arena, and showed it the visual oscillation. We could verify that the bee was
responding to the visual stimulus because the wingbeat sound varied
continuously according to the visual oscillation. In other words, the
amplitude of the sound oscillated at the same frequency as the stripe motion.
Although some bees stopped flapping their wings within a few seconds after the
stimulus was given, other bees remained responsive. In the analysis, we used
data from the responsive bees. The oscillation frequency of the stripes
(
) was varied at 0.9, 1.8, 3.6 and 7.4 Hz, while the amplitude was kept
constant at 33 mm (i.e. 6 dots of LEDs).
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). The step response characteristics are usually
approximated as a first order or a second order transfer function. We combined
these two approximation methods, and expressed the transfer function
M(s) as follows:
 | (1) |
Here, T is the time constant, n is the resonance
frequency, and 
is the attenuation coefficient. These parameters were
determined through a trial and error process: T=0.016,
n=62x2
, and =0.001. Finally, we identified
the following transfer function for the force measurement system:
 | (2) |
The step response characteristics of M(s) were
superimposed on the measured step response data in
Fig. 5, showing reasonable
agreement. The dynamic properties in a frequency domain were also identified
from Eqn 2. The frequency response characteristics are, in general, defined as
the magnitude and phase differences between the sinusoidal input and output.
We can obtain these values at each input frequency by replacing s
with j in Eqn 2, where j is the imaginary unit and
is the input frequency (Franklin et
al., 2002). The frequency response characteristics of
M(s), calculated by using MATLAB (Mathworks, Natick, MA,
USA), are shown in Fig. 6. The
style of Fig. 6 is called a
`Bode plot', showing the magnitude and phase differences against logarithmic
angular frequency. The frequencies used for the measurements (0.9, 1.8, 3.6
and 7.4 Hz) are equal to 5.6, 11, 22 and 46 rad s-1, respectively
(Fig. 6).
Fig. 6A shows the magnitude,
given as the gain attenuation (Ga) in decibels, according
to the following equation:
 | (3) |
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a) at each in
Table 1. These results indicate
that the measurement data obtained by using this force measurement system will
include an artifact, which may not be negligible, especially in the high
frequency domain. In the analysis of the frequency response of the bumblebees,
the values in Table 1 were used
to compensate the raw measurement data.
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F0) gradually lagged behind that of
zv with increasing .
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as the compensated force
response of the bee, and obtained its frequency response characteristics
(
and
)
according to Table 1 and the
following equations:
 | (4) |
 | (5) |
We plotted
and
against a logarithmic frequency axis in
Fig. 8.
Fig. 8A reveals that most of
the
data are distributed between 0.1 and 0.5 at each , and the mean values
are approximately 0.3 in common. The bumblebees did not change
distinctly with respect to . In contrast,
clearly decreases with increasing
(Fig. 8B). When is 0.9
and 1.8 Hz (i.e. 5.6 and 11 rad s-1),
is positive in all the data. Thus, the phase of
is always earlier than
that of zv. When is around 3.6 Hz (i.e. 22 rad
s-1), the mean value of
is approximately 0°, meaning that the phases of
and zv
are almost synchronized. When is 7.4 Hz (i.e. 46 rad s-1),
F0 is negative in all the data, i.e. the phase of
lagged behind that of
zv.
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) are inconsistent. The
dimension of the output is in Newtons (or nondimensionalized), which is a
force, whereas that of the input is in mm (a length). When the force
measurements are performed under open-loop conditions, the physical values of
the input and output can be freely selected. In actual control systems,
however, the output is fed back to the input. In order to estimate the direct
correlation between the input and output, it is preferable for both the
dimensions to be identical. Most of the advanced researches on control
engineering have therefore used homogeneous dimension analyses (e.g.
Hess and Siwakosit, 2001). In
our analysis, we hypothesized that the vertical acceleration was proportional
to the measured vertical force, and then calculated the hypothetical vertical
position of the bumblebees by integrating the acceleration twice. The equation
of motion of a bumblebee in the vertical direction is as follows:
 | (6) |
 | (7) |
where m is the mass of a bumblebee, zb is the
hypothetical vertical position of the bee,
is the non-dimensional
vertical force, and g is the acceleration of gravity. Because
the mean value of is
generally not equal to 1, direct solution of Eqn 7 involves acceleration
motion, which is unnecessary for the frequency response analysis. To avoid
this problem, we added an appropriate constant to the right side of Eqn 7, and
solved the differential equation. As a result, the input parameter was the
vertical position of the visual stripes, zv, and the
output was the hypothetical vertical position of the bee,
zb. Fig. 9
shows typical results of the correlation between zv and
zb at each . The lines in red represent
zb, and the bold broken lines in green represent
zv. We defined Azb and
Azv as amplitudes of zb and
zv, respectively, and as phase difference between
zb and zv. As shown in
Fig. 9A-D,
Azb is clearly attenuated, and that
decreases with increasing .
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We summarized all the results on a Bode plot
(Fig. 10).
Fig. 10A shows the gain of the
system (G), calculated by the following equation:
 | (8) |
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). The mean values of G and at each are
shown in Table 2. These values
were used for identifying the transfer function of the visual altitude control
system. Because the sets of G and were obtained at only four
frequencies, the number of representable expressions of the transfer function
were infinite. Therefore, we focused on finding the simplest expression by
means of the following two steps.
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First, we hypothesized that the transfer function B(s)
was represented as a product of a linear part and a non-linear exponential
part:
 | (9) |
This hypothesis is based on an earlier study
(McRuer and Graham, 1964), in
which a transfer function of human systems was simply approximated as a style
of Eqn 9. The exponential part means a time delay of the system, mostly due to
transport delays and high frequency neuromuscular lags in animals. In insect
motion, effective time delay 
e is approximately a few dozen ms
(Azuma, 1992;
Höltje and Hustert, 2003;
Ridgel et al., 2001). We
hypothesized that e for the bumblebees was 0.02 s. The
frequency response characteristics of e-es are
remarkable in that the gain is 0 dB throughout the frequency domain, whereas
the phase lag enlarges with increasing frequency. We showed the gain
(Ge) and the phase (e) of
e-eS at each measured in
Table 3. Here, we benefit from
using the Bode plot, in which the gain is expressed on a logarithmic scale,
and the phase is expressed on a linear scale. In this case, the frequency
response characteristics (i.e. gain and phase difference) of an arbitrary
transfer function,
T(s)=T1(s)T2(s)...Tn(s),
are calculated as the summation of those of
T1(s), T2(s),...,
and Tn(s). For example, the gain and phase of
B(s) can be calculated as follows:
 | (10) |
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and
 | (11) |
where G0 and 0 are the gain and phase
of B0(s), respectively. We can, therefore,
identify the frequency response characteristics of B0
(s) by subtracting those of e-es from
those of B(s). We showed the resultant values of
G0 and 0 in
Table 4.
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Next, we determined the expression of B0(s).
The Bode plot is useful again in finding the simplest expression. In
Fig. 10, the slope of the gain
curve is observed to be approximately -40 dB/decade, which should be identical
to that of B0(s). In general, the gain slope
becomes steeper by -20 dB/decade per a power of 1/s in a Bode plot
(Franklin et al., 2002). This
indicates that 1/s2 is dominant in
B0(s). Therefore, the simplest expression of
B0(s), around the measured frequency domain, is
represented as follows:
 | (12) |
Each coefficient (a, b and c) in Eqn 12 was calculated by
a MATLAB program, using an algorithm of identifying continuous-time filter
parameters from frequency response data. The resultant
B0(s) was:
 | (13) |
Consequently, the transfer function of the visual altitude control system
in the bumblebees is represented as:
 | (14) |
 | (15) |
In Fig. 11, the frequency
responses of B0(s) and B(s)
[i.e. B0(j) and
B(j), respectively] were shown with the measurement
data. The agreement between B(s) and the measurement data is
observed to be reasonable in both G and .
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). We
measured the temporal transitions of F0 at each (0.9, 1.8,
3.6 and 7.4 Hz) (Fig. 8).
Because the raw measurement data included an artifact due to the dynamics of
the force measurement system, we compensated the data for the influence of the
artifact, and obtained corrected force response of the bees
().
The frequency responses have been studied for other insects. Sherman and
Dickinson (Sherman and Dickinson,
2003) measured the frequency response of fruit flies
Drosophila melanogaster by mechanically and visually oscillating the
tethered flies about the pitch, roll and yaw axes, and recorded the changes in
wingbeat amplitude and wingbeat frequency. They characterized the dynamics of
the visual and mechanosensory systems, and revealed that both feedback systems
were composed of band-pass filters of different frequency characteristics. In
a subsequent study, they also successfully identified the contribution of each
sensory modality (Sherman and Dickinson,
2004).
In our study, we focused on the variations in amplitude and phase of the
frequency response for visual altitude control. Our results showed that the
amplitude
() was
almost constant, and that the phase
()
gradually lagged with increasing
(Fig. 8). Next, we
solved the equation of motion (Eqn 7) to obtain variations in hypothetical
vertical position of the bee. For the purpose of adjusting the dimensions of
the input and output, we defined the input parameter as the vertical position
of the visual stripes (zv) and the output as the
hypothetical vertical position of the bumblebee (zb).
Fig. 9 shows that the amplitude
of zb is clearly attenuated, and that the phase of
zb lagged with increasing . We summarized all the
results on a Bode plot (Fig.
10). The simplest transfer function, representing the frequency
response characteristics of the bumblebees, was obtained as Eqn 15.
Influence of compensations for the load cell dynamics
We measured the dynamic response of the bumblebees by using a load cell.
The raw measurement data, however, included an artifact due to the load cell
dynamics, which should be compensated for in the analysis. We identified these
dynamic properties by measuring the step response of the force measurement
system (Figs 3,
4). We represented the step
response characteristics by using a transfer function, M(s)
(Eqn 1), and estimated the gain attenuation and phase lags in the frequency
domain, on the basis of M(s)
(Fig. 6 and
Table 1). These dynamic
properties were compensated for in the analysis of the vertical force data of
the bees, according to Eqn 4, 5. As shown in
Fig. 6 and
Table 1, the influence of the
artifact enlarges with increasing frequency. In other words, the data
corrections we performed may have importance in the high frequency domain.
Here, we discuss how the compensation affected our frequency response
analysis.
First, we focus on the gain control analysis. The maximum of the gain
attenuation due to the load cell dynamics is 1.7 dB at =46 rad
s-1 (Table 1). This
is much smaller than the gain variation through (from 7.8 to -29.6 dB)
(Table 2). We notify that the
correction values in gain included in the position response data are also
equal to Ga. The compensations performed for the gain
results, therefore, had little influence on our analysis. On the other hand,
phase lags caused by the load cell dynamics are not negligible. When
is 22 or 46 rad s-1, the phase lags due to the artifact are more
than 10% of the lags in the response of the bees (Tables
1,
2). The influence of the
compensations was more significant for the phase analysis, rather than for the
gain analysis.
As a result of performing the compensations for the measurement data, the phase lags decreased whereas the gain was almost unchanged. It is naturally desirable for the stable control to have smaller phase lags. This indicates that the data with the compensations show more stable control characteristics than the data without the compensations. We discuss the dynamic stability in the visual altitude control system of the bumblebees in the subsequent section (`Meaning of the obtained transfer function'), on the basis of the gain and phase characteristics. We define an indicator of the dynamic stability, `stability margin' in that section. We will see with ease that the smaller phase lags contribute to the larger stability margin.
Visual perception and flight control in the bumblebees
We utilized the sensitivity for image motion in bumblebees, and elicited
flight modulation in the vertical direction. Recent studies have revealed that
flying insects use cues derived from optic flow for navigational purposes
(e.g. Srinivasan et al.,
1996). For example, bees flying through a tunnel maintain
equidistance from the flanking walls by balancing the apparent speeds of the
images of the walls (Kirchner,
1989; Srinivasan et al.,
1996). Bees landing on a horizontal surface hold constant the
image velocity of the surface as they approach it
(Srinivasan et al., 2000;
Srinivasan et al., 1996). A
large number of studies have focused on modeling the visual motion detection
mechanisms of insects, which have been summarized
(Srinivasan et al., 1999).
Neumann and Bülthoff showed successful simulations of visual flight
control by using these models (Neumann and
Bülthoff, 2000; Neumann
and Bülthoff, 2001;
Neumann and Bülthoff,
2002).
The motion-sensitive mechanism in an insect measures the angular velocity
of the moving image (Srinivasan et al.,
1996). Therefore, the bumblebees used in our experiments most
certainly responded to the variation in the angular velocity of the stripes,
rather than the position of the stripes. This indicates that the transfer
function (Eqn 15), obtained in the positional dimension, is also likely
dependent on the angular velocity. In the process of finding Eqn 15, we
focused on the amplitude and phase of the hypothetical position of the bee
with respect to those of the stripe position. Even though
Azv (amplitude of the visual oscillation) is
given as a constant, the angular velocity perceived by the bee varies
according to the distance between the bee and the display. This implies that
the resultant transfer function, B(s), is applicable within
the framework of the present experimental conditions. However, the applicable
range is likely rather extensive, because the amplitude of the response is
dealt with in a logarithmic scale.
We analyzed the bumblebee's control system in the dimension of position,
for the purpose of evaluating its control characteristics from the viewpoint
of human-related control systems. Human beings can perceive structure and
position of a stationary object with high accuracy. Therefore, human beings
are most likely to guide their movement by using position control
(Rushton et al., 1998),
although they also seem to use the optic flow as well
(Warren et al., 2001). On the
other hand, insects have inferior spatial acuity in comparison with human
beings (Horridge, 1977). The
insects achieve the same tasks only by using the optic flow. Insects and human
beings, therefore, respond to different elements of image motion; insects
primarily respond to the velocity, whereas human beings primarily respond to
the position. However, we can compare their frequency response characteristics
in the same framework of analysis. This is because the available transfer
functions for the frequency response are identical, whether they are
calculated in the dimension of velocity or position, as far as both the
sinusoidal input and output have the same dimension. In addition, the purpose
of controlling their motion is the same: reliable guidance to a destination.
We can therefore estimate the control performance of the bumblebee system by
using Eqn 15, in the same manner as the control analysis for human systems.
Below, we discuss the control stability of the bumblebee system, and
differences between the bumblebee and a human pilot-vehicle system, on the
basis of the resultant transfer function (Eqn 15).
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;
McRuer, 1973). The inside loop
including K1 is called the `stabilization control loop'.
SAS (Stability Augmentation System) and CAS (Control Augmentation System) are
examples of such loops. This K1 loop reflexively controls
quick responses. The outside loop including K2 is called
the `guidance and navigation control loop'. A typical example of this loop is
the automatic flight control system consisting of TMS (Thrust Management
System) and FMS (Flight Management System). The K2 loop
controls relatively slow responses. We applied this control model to the
measured bumblebee system. The K2 loop in
Fig. 12 is likely comparable
to the system that conveys an intention in the bees (for example, an intention
to reach the destination as soon as possible, or an intention to fly as far
away as possible). The reflexive response measured in our study is most likely
corresponding to the open-loop output of the K1 loop,
Y1 in Fig.
12.
The performance of this reflexive control system in the bumblebees can be
estimated on the basis of the frequency response results. When we analyze
control characteristics of an open-loop system, `crossover frequency' and
`stability margin' are important (Franklin
et al., 2002). The crossover frequency is, in general, divided
into the gain crossover frequency and the phase crossover frequency. The
stability margin is also divided into the gain margin and the phase margin.
The gain crossover frequency (gc) is defined as the
frequency that produces a gain of 0 dB. The phase crossover frequency
(pc) is defined as the frequency that produces a phase of
-180°. The gain margin (GM) is the difference between the gain curve and 0
dB at pc. The phase margin (PM) is the difference in phase
between the phase curve and -180° at gc. Each of
gc, pc, GM, and PM in our measurement data
are shown in Fig. 13,
revealing that the measured bumblebee system has sufficient stability margins
for both gain and phase. When PM is less than 90°, gc is
less than or equal to the bandwidth, bw
(Franklin et al., 2002). The
bandwidth is defined as the frequency at which the gain in the closed-loop
response is attenuated 3 dB from the steady state. Larger bw
means that the output of the closed-loop control system can follow the input
up to higher frequencies. In other words, larger gc yields
larger bw, and the quick response characteristics are
improved. In addition, PM is known as an indicator of damping characteristics;
larger PM yields a smaller peak gain in a closed-loop control system
(Franklin et al., 2002).
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Empirical studies guide designers of control systems in their choice for PM
and GM values. In the case of designing a servo control system, for example,
output should be controlled to track a target value, and PM of 40
60°
and GM of 1020 dB are desirable. In the case of designing a regulator
control system, on the other hand, output should be kept constant to minimize
the effect of disturbances, and PM larger than 20° and GM of 310 dB
are required. In Fig. 13, it
is observed that gc=9 rad s-1,
pc=25 rad s-1, GM=20 dB and PM=45°. These
results indicate that the measured control system in the bumblebees is
analogous to an ideal servo control system.
Comparison between human beings and bumblebees in terms of dynamic control characteristics
Dynamic flight control analysis has long been studied in the field of
aeronautical engineering. McRuer and Graham
(McRuer and Graham, 1964)
studied the dynamic control characteristics of a human pilot operating an
aerospace vehicle. They analyzed the frequency response of the human pilot,
and revealed that the open-loop control characteristics of the combined system
(human pilot-vehicle system) can be approximated by using a simple transfer
function. This transfer function model is called `the crossover model', and is
represented as the following expression, where Hp and
Hc mean the describing functions of the human pilot and
the controlled element (vehicle), respectively:
 | (16) |
The crossover frequency, gc, is equivalent to the loop
gain, and accounts for the adaptive compensation of the pilot for the
controlled element gain. An effective time delay, e, includes
the lags due to transport delays and high frequency neuromuscular dynamics. In
general, e is approximately 0.2 s in humans, ten times larger
than in the bumblebees. The simplest describing function of the pilot is
represented as follows:
 | (17) |
where Kp is pilot static gain, TL
is lead time constant, and TI is lag time constant. Eqn 17
indicates that the human pilot can be adjusted for variation in lead-lag
characteristics of the controlled element by modifying TL
and TI, so as to keep `the crossover model'. Although more
complicated and fitted models have been developed (e.g.
Davidson and Schmidt, 1992;
Kleinman et al., 1970), the
crossover model is still widely accepted among researchers of man-machine
interfaces.
McRuer and Graham proposed design guidance for vehicle systems, on the
basis of the crossover model. They suggested that an appropriate vehicle
system needs PM of approximately 40°, and that the gain slope near
gc on a Bode plot should be -20 dB/decade. Because PM is
directly correlated with the dynamic stability, the requirement for PM is
principal in designing the system. The requirement for the gain slope is
derived from the crossover model, because 1/s gives a gain slope of
-20 dB/decade. If the system is designed to satisfy these conditions, the
human pilot does not need to compensate for the dynamics of the vehicle
system, and the handling quality is improved. However, the both requirements
for PM and gain slope are simultaneously satisfied under limited condition,
because the gain and phase characteristics of a system are correlated with
each other (Franklin et al.,
2002). When the gain slope becomes steeper, the phase lag
inevitably enlarges, and the positive phase margin may be lost.
We compared the control characteristics of the measured bumblebee system
with the crossover model. The frequency response data of the two systems are
shown in Fig. 14. The data of
the human pilot-vehicle system were quoted from McRuer and Jex
(McRuer and Jex, 1967). Our
results reveal that the gain slope is approximately -40 dB/decade in the
bumblebee response, whereas it is approximately -20 dB/decade in the human
response. Considering that the measured bumblebee system is prominently
dominated by the effect of (gc/s)2
(because gc9), the control rule in the visual altitude
control system of the bumblebee could be called as `the square crossover
model'. Because the bumblebee system possesses a steeper gain slope than the
human pilot-vehicle system, the gain at <gc is
higher in the bumblebee system than in the human pilot-vehicle system. The
behavior at low frequencies generally determines the attenuation of
disturbances and the performance of tracking low frequency reference signals.
These are called steady-state characteristics, and high gain is required to
improve these characteristics. The measured control system in the bumblebee
is, therefore, revealed to possess superior steady-state characteristics in
comparison with the human pilot-vehicle system. The steeper gain slope also
produces lower gain at high frequencies. Such behavior is desirable because
the robust stability, which is more important than the attenuation of
disturbances at high frequencies, is improved.
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Evaluation of the control performance must also include the transient
characteristics as well as the steady-state characteristics. The transient
characteristics are divided into damping characteristics and quick response
characteristics. The adequate PM in the bumblebee system indicates excellent
damping characteristics, which are comparable to those of an ideal control
system for humans. The quick response characteristics depend on the gain
crossover frequency. In a human pilot-vehicle system, the gain crossover
frequency (gc,human) is roughly 2-5 rad s-1
(McRuer and Jex, 1967), which
is approximately half of the gain crossover frequency in the measured
bumblebee system (gc,bumblebee). The bumblebee system is,
therefore, revealed to possess superior quick response characteristics in
comparison with the human pilot-vehicle system.
In conclusion, we have measured and analyzed the frequency response characteristics of visual altitude control system in the bumblebees. The measured bumblebee system is revealed to have superiority in both the steady-state and transient characteristics, in comparison with the human pilot-vehicle system. Such excellence will be the evidence that the bumblebees can effectively control their flight with stability and maneuverability.
F0
0
e
bw
gc
n
pc
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Acknowledgments |
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References |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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