Since all inputs and outputs are received in the moving frame of the y, we considered the feedback mechanism from that perspective. Thus neither were the y's inputs directly observed nor its outputs directly measured, but inferred using both information about the y's velocity and tunnel-frame inputs.
As the y moves, it experiences visual motion on its eyes, which we will call vv for visual velocity. It is necessary as well to dene a direction convention, and so we dened positive to be the direction experienced by the y when it is moving forward, known in the literature as progressive visual motion. We likewise dened positive airspeed va to be in the same direction (Figure 3.1).
ground spd.
vg
total force f
+
va vwwind vel.
vpprojector vel.
visual vel.
vv
Figure 3.1: A diagram of how tunnel-frame inputs and outputs are transformed into y- frame inputs and outputs. The two inputs, visual velocity vv and airspeed va, and one control force output fc are dened as positive along the direction of the arrows (a). A block diagram (b) indicates how tunnel-frame disturbances (right) are transformed into the moving frame of the y, with example traces showing how a wind disturbance would be propagated. The goal is to nd an abstract model of the box labeled y controller. Color conventions introduced for dierent quantities in this gure are used throughout the paper.
In the xed geometry of the arena, we dened the visual velocityvv as a linear term, but the y probably measures angular rate across the retina [43], averaged over portions of the visual sphere. The mechanism for visual motion detection is likely a correlator [72] combined with spatial averaging by tangential cells [73, 23]. More detail is given in the introduction to Chapter 5. That the y is likely measuring angular visual rates has implications for feedback control in varying geometries, to which we will return in the Discussion.
Because the dynamics of the y can not be approximated as primarily viscous in nature at the timescale considered here, we performed modeling in the domain of forces. Previous studies on free-ight forward ight [48, 47] have modeled y visual ight speed response in velocity domain only. But in this work, because the response from the antennae is much
faster, it is necessary to model the dynamics in the force domain. Equating forces and accelerations along the x-axis, the force-balance equation for the y is
fc+fd=mv˙g, (3.1)
wherefc is the active control force generated by the y in response to sensory stimuli and fd is the passive drag force arising from baseline wing kinematics . The y changes its wing kinematics from baseline motions to alter the active force fc, but we do not address the specic changes that occur in this work. Results from [74] indicate that passive wing damping drag force fd is roughly proportional with airspeed va, as do our results and a simulation on an aerodynamic apping-wing fruit y model based on data taken from a dynamically-scaled model in a tow-tank [49] (and see Figure 3.2). A dynamic element with force proportional to velocity is known in control engineering as a dashpot or viscous damper. Call this coecient of proportionality b, so that fd = −bva. If inertial forces dominate (short timescales), the force balance equation (Equation 3.1) reduces tofc=mv˙g, and if viscous forces dominate (longer timescales), then it reduces to fc = bva. The time constant of exponential relaxation to asymptote velocity in response to a steady force in the full force balance equation is equal to mb ≈ .13 s. Considering delay from vision reported in [48] is 0.081 - 0.1 s, the slower velocity-domain viscous dynamics may be sucient for the purposes of purely visual modeling. But the faster antenna response we consider in this work necessitates inertial dynamics so the slower viscous domain dynamics approximation cannot hold.
Moving to the domain of active sensory feedback, we seek a model for both drag and active control forces. But rather than perform an exhaustive search among all possible nonlinear models, we considered a set of simple, plausible linear models. If a satisfactory one could be found, no further analysis would be necessary. And in general if they do not have strong nonlinearities, nonlinear systems can be approximated by linear ones in a large- enough neighborhood around equilibrium. Thus complicated nonlinear spiking ensembles of neurons performing computations in the nervous system of the y may be abstracted as linear. Because the drag was roughly linear, this was considered as further evidence that linear models could be sucient. The tting approach was to perform least-squares ts on each candidate model. For clarity, the results are segregated by functionality into delay and
gain boxes, but most likely both operations are being performed by a single sensory-motor transduction cascade.
To estimate velocities and accelerations from position information, we ltered using causal linear lters and a rst-order hold input model (linear interpolation between input data points). All ltering and tting computations were performed using the python-control package version 0.3 (http://python-control.sourceforge.net/) and SciPy (http://www.scipy.org).
Accelerations were calculated from position using a lter of the form (τ s+1)s2 2, whereas ve- locities were estimated using the lter (τ s+1)s 2. When the information was already in nal form, such as wind velocity measurements from the hotwire anemometer, it too was ltered, but with the lter (τ s+1)1 2 so that any phase lags induced by the ltering were replicated in all data. The time constantτ used in the lters was 3 ms.