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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

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Response characteristics of visual altitude control system in Bombus terrestris

Kensaku Tanaka^* and Keiji Kawachi

Department of Aeronautics and Astronautics, the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

View larger version (40K): [in a new window]   Fig. 1. (A) The flight arena. During the experiments, the bumblebee was fixed inside this arena, and was given a visual stimulus in the vertical direction. (B) Magnified view of one wall of the arena. The inside wall contains 16 rows of LEDs in the horizontal direction, and 32 in the vertical direction. The diameter of each LED is 5.5 mm. We displayed horizontal stripe patterns 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. The oscillation amplitude was kept constant at 33 mm (6 dots of the LEDs), whereas the oscillation frequency was varied at 0.9, 1.8, 3.6 and 7.4 Hz (i.e. 5.6, 11, 22 and 46 rad s-1).

View larger version (12K): [in a new window]   Fig. 2. The measurement system. We measured the vertical force of the bumblebee using a load cell. This load cell was hard-wired to an amplifier using a 100 V AC power system. The output signal was digitalized by an A/D converter, and then stored on a PC. 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 flight arena controller. The phase and frequency of the visual oscillation were identified using the CH2 waveform, which enabled us to synchronize the data of the input (visual oscillation) and the output (force variation).

View larger version (12K): [in a new window]   Fig. 3. The dynamic properties of the force measurement system were verified by measuring the step response. (A) Before the step input. A 3 mN weight [load (I)] was loaded on the load cell, for the purpose of adjusting the weight condition to the experiments. In addition, a 5 mN weight [load (II)] and an electromagnet were prepared. Load (II) was iron, and pushed up the load cell by the magnetic force of the electromagnet. The magnetic force worked on load (II) was approximately 7 mN. Note that the magnitude of the additional force does not have an influence on the step response result. (B) After the step input. When the electromagnet was switched off, load (II) was immediately detached from the load cell, which signified the step input. Variations in the voltage of the electromagnet circuit and the output of the force measurement system were recorded at a sampling frequency of 10 kHz.

View larger version (5K): [in a new window]   Fig. 4. The resultant voltage variations in the step response measurement. The voltage of the electromagnet circuit varies almost instantly at t=0 s, which determines the time of the step input. The voltage of the load cell-amplifier output varies with a time lag, and includes a resonance.

View larger version (7K): [in a new window]   Fig. 5. The step response characteristics of the experimental data and M(s) (Eqn 2). The agreement is observed to be highly reasonable.

View larger version (12K): [in a new window]   Fig. 6. The frequency response characteristics of our force measurement system, M(j). This style of figure is called a `Bode plot', and shows the magnitude and phase differences plotted 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. (A) The gain in decibels; (B) the phase differences. Note that the gain is attenuated and the phase lag enlarges with increasing frequency, which should be compensated for in the analysis.

View larger version (40K): [in a new window]   Fig. 7. Typical measurement data of the force response of the bees. The red lines represent the raw measurement data of the vertical force (F0), without any compensations. The source of fine oscillations in F0 is certainly the wingbeat of the bee; the oscillation frequency is approximately 150 Hz in common throughout A-D. The broken blue lines represent the filtered data of F0. The bold broken green lines represent the visual stripe position. The upward direction is defined as the positive direction for both F0 and zv. (A) Visual oscillation frequency is 0.9 Hz (5.6 rad s-1). The phase of F0 obviously precedes that of zv. (B) Visual oscillation frequency is 1.8 Hz (11 rad s-1). The phase of F0 still precedes that of zv. (C) Visual oscillation frequency is 3.6 Hz (22 rad s-1). The phase difference between F0 and zv is small. (D) Visual oscillation frequency is 7.4 Hz (46 rad s-1). The phase of F0 obviously lags behind that of zv. Although the variation in the phase through A-D is clear, the amplitude of F0 is relatively constant.

View larger version (8K): [in a new window]   Fig. 8. Summary of all the results of the force response data. The influence of the load cell dynamics was compensated for the respective measurement data, according to Eqn 4 and Eqn 5. We represented the characteristics of the corrected force response () in terms of amplitude () and phase (). (A) is mainly distributed between 0.1 and 0.5 at each , and the mean values are approximately 0.3 for all . The bumblebees did not change throughout . (B) In contrast, clearly decreases with increasing . When is lower than 3.6 Hz (22 rad s-1), F0 in all the data is positive, i.e., the phase of is earlier than that of zv. When is around 3.6 Hz (22 rad s-1), the mean value of is approximately 0°, meaning that the phases of and zv are almost synchronized. When the visual stripes oscillate at 7.4 Hz (46 rad s-1), in all the data is negative, i.e. the phase of lagged behind that of zv.

View larger version (22K): [in a new window]   Fig. 9. Typical results for the position response of the bees. We obtained the hypothetical variation in position of the bees (zb, red lines) according to Eqn 7. The bold broken green lines represent the visual stripe position (zv). We focused on the amplitudes of zb (Azb) and zv (Azv), and the phase differences between zb and zv (). (A) When =0.9 Hz (5.6 rad s-1), Azb is larger than Azv, and is larger than -180°. (B) When =1.8 Hz (11 rad s-1), Azb is a little smaller than Azv, and is larger than -180°. (C) When =3.6 Hz (22 rad s-1), Azb becomes much smaller than Azv, and is approximately -180°. (D) When =7.4 Hz (46 rad s-1), the oscillation in zb is hardly perceptible. The value of is, in fact, much smaller than -180°.

View larger version (8K): [in a new window]   Fig. 10. All the results of the position response of the bees were summarized on a Bode plot. (A) Gain of the response (G), calculated from Eqn 8. (B) Phase differences between zb and zv (). The gain is attenuated at approximately -40 dB/decade. The phase lag is observed to enlarge with increasing frequency.

View larger version (12K): [in a new window]   Fig. 11. The measured response data fitted with a transfer function, derived from a hypothesis that the expression was a product of a linear part and a non-linear exponential part. The exponential part affects the phase lags only (see solid and broken lines). We obtained the simplest transfer function: B(s)=[9/(s+3)]2e-0.02s. It is observed that B(s) fits well for both the gain and phase data.

View larger version (5K): [in a new window]   Fig. 12. Block diagram of a typical flight control system of airplanes. The inside loop including K1 is called a `stabilization control loop'. This K1 loop controls quick responses reflexively. The outside loop including K2 is called a `guidance and navigation control loop'. The K2 loop controls relatively slow responses such as selection of a flight course. This K2 loop is likely comparable to the system that conveys an intention in the bees. The reflexive responses measured in our study are most likely corresponding to the output of the K1 loop, Y1.

View larger version (10K): [in a new window]   Fig. 13. Gain crossover frequency (gc), phase crossover frequency (pc), gain margin (GM), and phase margin (PM) of the measured bumblebee system. gc is defined as the frequency that produces a gain of 0 dB, and pc as the frequency that produces a phase of -180°. GM is the difference between the gain curve and 0 dB at pc, and PM is the difference in phase between the phase curve and -180° at gc. The dynamic stability of a control system can be quantified on the basis of GM and PM. When both GM and PM are positive, the control system is dynamically stable. Our results (GM20 and PM45) indicate that the measured bumblebee system has substantial dynamic stability.

View larger version (18K): [in a new window]   Fig. 14. The measured control characteristics of the bumblebee system are compared with the characteristics of a human pilot-vehicle system (McRuer and Jex, 1967). For the human system, the control characteristics are fitted with a transfer function, H(s)= gc,human/s·e-e,humanS (the crossover model). On the other hand, the control characteristics in the bumblebee system can be approximated as B(s)=[gc,bumblebee/(s+3)]2.e-e,bumblebeeS, which could be called `the square crossover model'. The bumblebee system is observed to possess higher gain at <gc than the human system, indicating higher performance in terms of the steady-state characteristics. The gain crossover frequency in the bumblebee system (gc,bumblebee) is approximately twice as large as that in the human pilot-vehicle system (gc,human). Because larger gc causes larger bandwidth in the system, the bumblebee system is revealed to possess superior quick response characteristics. We already verified that the bumblebee system possesses substantial phase margin (PM; Fig. 13), indicating that the system possesses excellent damping characteristics. The bumblebee system was, therefore, revealed to have superiority in terms of the steady-state and transient (i.e. quick response and damping) characteristics, in comparison with the human pilot-vehicle system.