Response characteristics of visual altitude control system in Bombus terrestris

Flying insects satisfy both their maneuverability and stability requirements by continuously varying their wing kinematics. Smaller objects have, in general, difficulty in controlling their flight with stability,because they have lower moments of inertia and are more sensitive to high frequency disturbances. The control system must therefore possess sufficient steady state and transient characteristics. Although basic flight mechanisms of insects have been summarized (e.g. Azuma, 1992) 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.

Bumblebees Bombus terrestris L. were chosen as the model insect because of their general availability, and noted flight capability. We obtained colonies of the bumblebees from a commercial supplier (Arysta LifeScience Corporation, Tokyo, Japan). The environment in which the colonies were placed and experiments were performed was maintained at around 20°C,when the bees are fully active. Only female bumblebees were selected because females are larger and more robust than males. We used a total of 12 bumblebees for the measurements, all 1-2 cm long and 1.5-2.5 mN in weight.

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 long×0.7 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.

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.

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(ω) 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).

Our measurement system is shown in Fig. 2. The vertical force generated by the bee was measured using a load cell (UL-10GR, Minebea Co., Tokyo, Japan). The iron wire glued to the bee was rigidly attached to the load cell with a clamp (8.5 mN weight). This load cell was hard-wired to an amplifier (CSA-507B, Minebea Co.), using a 100 V AC power system. The output signal was digitalized using an A/D converter (NR110,KEYENCE, Osaka, Japan) and stored on a PC. The resonance frequency of the force measurement system was approximately 60 Hz in our experimental conditions, and is much higher than the visual oscillation frequency (at most 7.4 Hz) and lower than the wingbeat frequency (typically 150-200 Hz). The resonance in this force measurement system, therefore, was not coupled with the measurement data. The sampling frequency of the A/D converter was 2 kHz. We recorded the signals for enough time to observe more than one cycle of the visual oscillation in each experiment. The output signal of the amplifier was imported as channel 1 (CH1). We used a light signal to identify the visual stripe motion, whose output was imported as channel 2 (CH2). This light signal circuit was connected to the controller of the flight arena. The phase and frequency of the visual oscillation were identified from the CH2 waveform,which enabled us to synchronize the data of the input (visual oscillation) and the output (force variation).

where A_input and A_output are the input and output signal amplitudes, respectively. The gain is observed to be attenuated as the frequency increases. Fig. 6B shows the phase differences between the input and output. The phase lag is observed to be enlarged with increasing frequency. We showed the gain attenuation (G_a), attenuation ratio in amplitude(A_output/A_input), and phase differences (θ_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.

The bumblebees responded to visual oscillations in the vertical direction,and varied their vertical forces. Typical results of the temporal transitions of the vertical force (F₀) and a visual stripe position(z_v) are shown in Fig. 7. The dynamic properties of the force measurement system were not compensated at this stage. The value of F₀ was calculated by dividing the measured force by the weight of the bee, and was equivalent to g in an airplane. The upward direction was defined as the positive direction for both F₀ and z_v. The noticeable fine oscillation in the red lines (F₀), the frequency of which is approximately 150 Hz in common throughout A-D, is most certainly due to the wingbeat of the bumblebee. We removed this unwanted oscillation by using Fourier analysis, and fitted sine curves to the respective F₀ data (broken lines in blue). Fig. 7 shows that the filtered line of F₀ has approximately the same frequency as the line of z_v (bold broken lines in green) at each. This indicates that the bumblebees showed the typical frequency response. We focused on their response characteristics in terms of amplitude and phase differences with respect to z_v. In Fig. 7, the amplitude of the force (A_F0) is observed to remain nearly constant, whereas the phase of F₀(θ_F0) gradually lagged behind that of z_v with increasing ω.

We plotted

where m is the mass of a bumblebee, z_b is the hypothetical vertical position of the bee,

Fig. 10B shows the phase differences (θ). The mean values of G and θ 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 θ 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.

where G₀ and θ₀ 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^-τ_es from those of B(s). We showed the resultant values of G₀ and θ₀ in Table 4.

In Fig. 11, the frequency responses of B₀(s) and B(s)[i.e. B₀(jω) and B(jω), 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 θ.

The frequency response characteristics of visual altitude control system were measured for bumblebees. The tethered bumblebees responded to visual oscillations in the vertical direction, and varied their vertical forces(F₀) according to the oscillation frequency (ω). 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(⁠

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 ω=46 rad s^-1 (Table 1). This is much smaller than the gain variation through ω (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 G_a. 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 ωis 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.

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 A_zv (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).

In this study, the visual stripes were oscillated in the open-loop condition. The tethered bumblebees could not feed back the variation in the vertical force to the stripe position. The results therefore do not represent the will of the bees to optimize their entire flight, but most certainly represent the conditional reflexes at each instant. For the purpose of characterizing this reflexive response of the bees, we utilized a model of an airplane control system. Fig. 12 shows a block diagram of the most advanced control system in airplanes (Mclean, 1990; 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 K<sub>1</sub> loop reflexively controls quick responses. The outside loop including K<sub>2</sub> 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 (ω_gc) is defined as the frequency that produces a gain of 0 dB. The phase crossover frequency(ω_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 ω_pc. The phase margin (PM) is the difference in phase between the phase curve and -180° at ω_gc. Each ofω _gc, ω_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°, ω_gc is less than or equal to the bandwidth, ω_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 ω_bwmeans that the output of the closed-loop control system can follow the input up to higher frequencies. In other words, larger ω_gc yields larger ω_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).

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 10∼20 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 3∼10 dB are required. In Fig. 13, it is observed that ω_gc=9 rad s^-1,ω _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.

where K_p is pilot static gain, T_Lis lead time constant, and T_I 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 T_Land T_I, 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 nearω _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 (ω_gc/s)²(because ω_gc∼9), 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 ω<ω_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.

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 (ω_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 (ω_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.

List of abbreviations and symbols
 
• A

  amplitude

 
• A_F2

  raw amplitude of force response

 
• \({\bar{A}}_{\mathrm{F}_{2}}\)

  corrected amplitude of force response

 
• A_zb

  amplitude of position response

 
• A_zv

  amplitude of visual oscillation

 
• B(s)

  open-loop transfer function of bumblebee

 
• B₀

  linear part of B(s)

 
• F₀

  non-dimensional raw vertical force

 
• \({\bar{F}}_{0}\)

  non-dimensional corrected vertical force

 
• g

  acceleration of gravity

 
• G

  gain

 
• G_a

  gain attenuation

 
• G₀

  gain of B₀(s)

 
• GM

  gain margin

 
• H

  open-loop transfer function of human pilot-vehicle system

 
• H_c

  describing function of controlled element in human pilot-vehicle system

 
• H_p

  describing function of human pilot

 
• j

  imaginary unit

 
• K_p

  pilot static gain

 
• m

  mass of bumblebee

 
• M(s)

  transfer function of force measurement system

 
• PM

  phase margin

 
• s

  parameter in Laplace transform

 
• t

  time

 
• T

  time constant

 
• T_I

  lag-time constant

 
• T_L

  lead-time constant

 
• z_b

  hypothetical vertical position of bumblebee

 
• z_v

  vertical position of visual stripes

 
• ζ

  attenuation coefficient

 
• θ

  phase

 
• θF0

  raw phase of force response

 
• \({\bar{{\theta}}}_{\mathrm{F}_{0}}\)

  corrected phase of force response

 
• θ0

  phase of B₀(s)

 
• τe

  effective time delay

 
• ω

  input frequency (stimulus frequency)

 
• ωbw

  bandwidth

 
• ωgc

  gain crossover frequency

 
• ωn

  resonance frequency

 
• ωpc

  phase crossover frequency

The present work was supported in part through the 21st Century COE Program, `Mechanical Systems Innovation', and Grant-in-Aid for Scientific Research (S), 18100002, 2006, by the Ministry of Education, Culture, Sports,Science and Technology, Japan.