Formation Flight Performance Representation

As stated earlier, the performance of the follower aircraft’s formation keeping is rep- resented by the maximum position error during a predefined period of automated formation keeping. The maximum position error is determined separately for each axis: Longitudinal, lateral and vertical. Therefore, the time history of the true rela- tive position error, er, is taken from the simulation component-wise as output. The output is recorded for a period of time, before which the transients due to the initial- ization of the simulation are already died out. An example time history of er, which was obtained from a 10-minute formation flight simulation with higher-scale aircraft is shown in figure 7.2. The simulation was performed with maximum sampling period and minimum delay values shown in table 7.1. Another example time history, which is taken from a simulation with maximum delay and minimum sampling period values, is given in Appendix B, on figure B.3.

In order to obtain the maximum position error during the recorded period of formation flight, the method of simply taking the maximum value in the ex, ey and

ez time histories is not applied. Because this method ignores the information in the entire data set, except the single points on each er component, at which the relative position error reaches to a maximum. In order to facilitate the information carried by the entire length of the data, the maximum error is approximated as three times the standard deviation of the time series for each axis, applying the three-sigma rule (Section 3.1). Therefore, the maximum, true relative position error during the formation flight, er,max, is represented component-wise as given in eqs. (7.1) to (7.3). The terms σex(t), σey(t), and σez(t) represent the standard deviations of the respective

0 50 100 150 200 250 300 350 400 450 500 550 600 −0.2 0 0.2 Time (s) ex (m ) 0 50 100 150 200 250 300 350 400 450 500 550 600 −0.1 0 0.1 Time (s) ey (m ) 0 50 100 150 200 250 300 350 400 450 500 550 600 −0.2 0 0.2 Time (s) ez (m ) Time history 3 sigma

Figure 7.2: Example time history of er taken from the simulation environment.1

ex,max = 3 σex(t) (7.1)

ey,max = 3 σey(t) (7.2)

ez,max = 3 σez(t) (7.3)

Figures 7.2 and B.3 also depict the three-standard-deviation-based maximum ap- proximations to both sides of the time histories. It can be seen that the approximated maximums closely match the actual maximums.

1_{The time history was obtained from a single simulation run, in which the following parameter}

set was used: k = 1, d = 0 s, T = 1 s. Higher-scale aircraft models were used and wake vortex effects were disabled. The lower frequency and magnitude of ey is due to the slower closed-loop dynamics

on lateral control channel than that on longitudinal and vertical channels, as depicted by figure 5.2 on page 86. The development of the error, as well as the reaction against the already-developed error is slower due to the slower lateral closed-loop dynamics.

7.1.3.1 Determination of the Simulation Duration

Another aspect of determining the maximum relative position error during the forma- tion flight simulation is the duration of the simulation, based on which the maximum error is calculated. It is desired to obtain a time history from the simulation, which is sufficiently long, in order to allow the deficiencies of the relative position information to manifest their effect on the formation keeping to a more complete extent, as well as, in order to give sufficient time to the follower aircraft to react. On the other hand, it is desired to have a simulation duration, which is sufficiently short, in order to reduce the size of data needs to be stored, as well as, in order to shorten the overall time required for the simulation. Taking the above considerations into account, the time span, during which the simulation output er is recorded, was iteratively deter- mined as 600 seconds for the higher-scale aircraft. The time span, during which er is recorded is simply referred to by simulation duration throughout the text.

0 0.2 0.4 0.6 0.8 1 0 5 · 10−2 0.1 0.15 T (s) ey,max (m ) 600 s_{1800 s} 3600 s 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 ·10−2 d(s) ey,max (m )

Figure 7.3: Change of determined ey,max with respect to different simulation durations.

The plots are based on simulations with higher-scale aircraft and k = 1.

Figure 7.3 shows the effect of a subset of simulation parameters on the lateral error component, which was determined according to eq. (7.2) with varying durations of simulation. Only the lateral error component is shown here, because the closed loop dynamics is the slowest on the lateral axis. Therefore it requires a longer observation duration for a more complete representation of the formation keeping performance. Equivalent plots for the remaining error components are given in Appendix B on figure B.4.

On figure 7.3, error representations based on 10-minute, 30-minute and 1-hour sim- ulations are shown with different markers. Simulations were performed with higher- scale aircraft, using reduced number of delay and sampling period values at constant

k. The plot on the left depicts the change of ey,max with respect to the sampling

period, T . The figure on the right depicts the change of ey,max with respect to the delay d. As shown, at any specific T or d value, the magnitude of ey,max varies with different durations of the simulation. This change is present, due to the random nature of the error time history, added to the relative position information that the

follower aircraft’s formation controller processes. Due to this randomness and the fact that the er,max is determined using the entire er(t) recorded, depending on the error time history, the magnitudes of the er,max components can be estimated higher or lower with simulations of different duration. For instance, the determined ey,max on a 600-second simulation could be higher than that on a 1800-second simulation, should the error time series contain a section, which causes less position error at the part of the simulation, where 600 s < t ≤ 1800 s. This random change of the error magnitude depending on simulation duration can also be exemplified by comparing the right-hand side plots of the y-component on figure 7.3 and the x-component on figure B.4 on page 166. Recalling that the random error sources are initialized with different seeds for each axis, it can be seen on the y-component, that the magnitude of the error is the greatest at the 600-second simulation. However on the x-component, the error magnitude is the lowest at the 600-second simulation.

Unlike the magnitude of the components of er,max, the dependency of er,max on the parameters d and T remain nearly unchanged at different durations of the simulation. Therefore, the change of formation keeping performance can be described nearly as well with an 10-minute simulation, as with simulations of 30-minute and 1-hour durations.

Having the duration of the simulation for the higher-scale aircraft determined as 600 seconds, the simulation duration for the lower-scale aircraft was derived from that of the higher-scale aircraft. The closed-loop dynamics of the lower-scale aircraft is approximately 2.5 times faster than that of the higher-scale aircraft on all three axes2_{. This means, that the lower-scale follower aircraft will react to the gathered}

relative position information in a shorter period of time. Therefore, a shorter simu- lation duration can be used for obtaining a representation of the formation keeping performance. Hence, the simulation duration of the lower-scale aircraft was selected as 1/2.5 times that of the higher-scale aircraft. This yields a simulation duration of 240 seconds, i.e., 4 minutes.

7.1.3.2 Simulation Data Recording

The final aspect considered as part of the formation keeping performance represen- tation is the frequency, at which the simulation outputs are recorded. The recording rate of the simulation output er was selected as 5 Hz for the higher-scale aircraft. This recording rate enables the capture of the time variations on all components of er without causing notable loss of accuracy. A sample simulation with a parameter set 2_{Figure 8.2 on page 127 shows the time responses of the closed-loop formation keeping system}

to a unit increase of commanded relative position. The figure also depicts the unit step responses of second-order transfer functions, which approximate the system responses. Table 8.1 on page 127 presents the natural frequencies of the fitted transfer functions for each scale of aircraft on each axis. The natural frequencies of the lower-scale aircraft is 2.5 times higher on each axis than those of the higher-scale aircraft.

of k = 1, d = 1 s and T = 0.01 s showed that, selecting the recording rate 40 times the value mentioned above only brings an accuracy gain of under 0.075 % on all three components of er,max. For the simulations with the lower-scale aircraft, the recording rate was selected 2.5 times faster than that of the higher-scale aircraft, according to the logic outlined above. This corresponds to a recording rate of 12.5 Hz.