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Insect hovering position control

Many insect taxa may hover in an approximately constant position. Examples are :

• Eristalis interruptus males hovering above females .
• Bombylius major females hovering near a nesting hole of Andrena ground nesting bees.
• Anthrax anthrax females hovering near a possible nesting hole of a mason bee.
• Villa modesta Males hovering in courtship.
• Dasypoda hirtipes Male hovering courtship.
• Macroglossum stellatarum hovering near a flower taking nectar.
• Manduca sexta hovering before a flower taking nectar.
• Calopteryx splendens male courting a female.

In these examples the insect is performing position control. To gain further insight reference is made to control theory. The global structure of the control system is given in the picture below.

Hovering Control Global

The insect is visually detecting its position relative to the target (female, hole or flower). In the diagram the block Sensor designates the visual system calculating the distance to the target. The block Controller is indicating the neural system calculating the activation of the flying muscles from the distance error as input. The block System indicates the mechanical part of the insect. Due to neural delay and mechanical inertia the system may oscillate. The standard tests of control theory may be used to characterize the different blocks.

Testing the System

The system may be investigated in Open Loop or in Closed Loop conditions. An example of Closed Loop testing is performed with Macroglossum stellatarum hovering before a flower. In this case instead of the real flower the moth has been motivated to hover before a blue disk. The disk has been moved to and fro in the direction of the head sinusoidally and the position of the head of the moth was measured. The results of this experiment with different frequencies delivers a Bode diagram of the closed loop system. A test of the system in Open Loop may be performed by glueing the insect to a so called tether and measuring the Force due to a moving Target. In this case the characteristics of the System block will be estimated to obtain the Open Loop transfer function. The gain in the tethered experiments is dependent on the experimental circumstances..

Control model

From Control theory the block diagram is drawn as given here.

Feedback Loop with Target position y_T

The reference r is a variable internal to the insect. R will be assumed to be constant. If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. With consatnt reference signal r the laplace transform R(s)=0. This gives the following relations:

$Y(s)=P(s)U(s)\,\!$

$U(s)=C(s)E(s)\,\!$

$E(s)=-F(s)(Y(s)-YT(s))\,\!$

Solving for Y(s) in terms of YT(s) gives

$Y(s)=\left({\frac {F(s)P(s)C(s)}{1+F(s)P(s)C(s)}}\right)Y(s)=HT(s)YT(s).$

The expression $H(s)=F(s)P(s)C(s)$ is the so called loop-gain. HT(s) is closed-loop transfer function of the system.

$HT(s)={\frac {H(s)}{1+H(s)}}$

For Bombus terrestris Tanaka and Kawachi give

$H(s)={\frac {9*exp(-0.02*s)}{(1+s/3)(1+s/3)}}$.

For the discussion to follow the general expression to be used is

$H(s)={\frac {K*exp(-Td*s)}{(1+T1*s)(1+T2*s)}}$.

The closed loop will oscillate with frequency ω when H(jω)=-1. This implies

1. $arctan(\omega \cdot T1)+arctan(\omega \cdot T2)+\omega \cdot Td=\pi$
2. $K={\sqrt {1+(\omega \cdot T1)^{2}}}*{\sqrt {1+(\omega \cdot T2)^{2}}}$

When the oscillation frequency ω is given T1, T2 and Td may be chosen to fulfill relation 1. The gain K may be derived afterwards from relation 2.

Application

• Bombylius major females are shooting eggs in dark holes when hovering. Before shooting an egg the female is oscillating to and fro with a period of approximately 0.3 s. An animation (5 times retarded)of the shooting act is given here.

  

  Bombylius major egg ejection

  The model with Td=0.02 s is explaining this result with T1=T2=0.225 s and K=23.
• Villa modesta males are hovering near females. A male is hovering near a female oscillating to and fro for more than 8 seconds with a frequency of 5.3 Hz. In the animation given here one period of the movement is repeated (10 times retarded).

  

  Villa modesta male hovering near female (lower corner to the right)

  The model with Td=0.02 s is explaining this result with T1=T2=0.088 s and K=9.4. An other possibility from the infinite multitude of solutions is T1=T2=Td=0.039 s with K=2.7.
• Anthrax anthrax females are shooting eggs in dark holes while hovering. Bee hotels will be investigated by the female fly for possible solitary bee nest entrances. Before shooting an egg the female is oscillating back and forth with a period of approximately 0.20 s. An animation (6 times retarded) of the shooting act is given here.

  

  Anthrax anthrax female hovering before a nest entrance of a mason bee

• Eristalis interruptus males are hovering above females as given in this animation (3 times retarded).

  

  Eristalis interruptus male

  The position control has been tested here by moving the flower. The male is following the position of the female with a delay of 37 ms with frequency 1.90 Hz. The model will explain this result with K=5.7, Td=0.02 s, T1=T2=0.1 s.
• Dasypoda hirtipes male courtship oscillation near female. The frequency of oscillation is 4.5 Hz. An animation is given here.

  

  Dasypoda hirtipes male oscillating near a female.

  This oscillation of a male from the order Hymenoptera is resembling the oscillating Villa modesta order Diptera.
• Manduca sexta has been tested with moving dummy flowers. The flowers have been moved horizontally and vertically with frequencies 1,2 and 3 Hz. For 3 Hz the gain of the closed loop is 0.5 , the delay is 0.282 periods. The model will explain this result choosing T1=0.32, T2=0.09 and Td=0.01, however the modelled delay with these parameters is 0.378 instead of 0.282.