Model for Position control of hovering Eristalis nemorum males
The excellent following behaviour of Eristalis nemorum males is suggesting an innate fast and accurate servo control system. The complete system is very complex, including eyes, neurons in different layers and the motor control of the wings. However a largely simplified model may be of use to gain more insight and to be able to relate the experiments. To begin with, the geometry of the hovering male will be defined (Fig 1). It will be assumed that the male is looking at some fixation point of the female. The angle θ_{e} is the angle between the direction from the eye of the male to the fixation point and the stripe of the male face. While hovering, the stripe at the face of the male is nearly vertical, the angle beeteen the stripe and the vertical is called μ. The angle of the wings with a horizontal plane is called φ. The distance between male and female is called r. The horizontal position of female and male are x_{F} and x_{M}

Figure 1 : Geometry of hovering
The model given here is meant to be of use for the following behaviour in the horizontal xdirection. From Fig. 1 for small angles θ_{e} the horizontal position error is x_{F}x_{M}=r*θ_{e}+r*μ_{ }, with r the distance between male and female. The angle μ is assumed to be small and constant and may therefore be neglected in the servo model. For simplicity it will be assume that the distance r is constant. The proposed model is given in Fig 2. Here K_{r} is the gain of the control system.
Figure 2 : The model
The velocity is determined by the control signal via two first order systems in series, with time constants τ_{1} and τ_{2}. In these first order systems all neural and mechanical delays are summarized. The velocity is integrated to obtain the male position. The performance of a control system may be characterized by the response to harmonic input signals of a range of frequencies. The frequency response H(ω) of the model (closed loop) is the response with the female position as input and the male position as output. H(ω) may have the characteristics of a lowpass or a bandpass filter determined by the parameters K_{r }, τ_{1} and τ_{2}. From H(ω) gain and delay may be calculated as a function of frequency ω. For small values of the frequency ω the delay of the model is approximately 1/K_{r} For Track 090831_1656_A10 of Fig. 4 the delay is 0.047 (s) and therefore K_{r} is approximately 21. τ_{1} and τ_{2}.are of the order of the measured delay, but for reliable estimates more results are needed.
Measurement of Gain and Delay
An attempt has been made to estimate the parameters of the model. In Fig 3 the Bode diagram has been given for two choices of parameters. Here a plot of delay versus frequncy has been given and not the standard phase versus frequency plot. The results of the measurements (see documentation), have also been displayed in Fig 3. The parameters K_{r}=44, τ_{1}= 0.032 (s) and τ_{2}=0.01 (s) deliver a fit of the measurements. Many other sets of parameters will also deliver a fit of the same quality. However, the measurements do show a large variation in the resulting delay. The uncertainty of the measured delay is of the order of a few ms, so perhaps in some cases the male is predicting the position of the female. In these cases feedforward should be added to the model. The results for 7 Hz are from one film with two males and one female. Here it has been assumed that Male1, nearest to the female, is following Male2. When Male2 is following Male1 the results for gain and delay are changed, but the order of magnitude for the parameters does not change.
Figure 3 : The gain H and delay as a function of frequency for the model with τ_{2} = 0.01 s . The experimental values are indicated by X