Simulation Environment Set up and Overview

Recalling from Chapter 1, the main parameters, whose effects on the follower’s for- mation keeping performance are investigated, are the following specifications of the leader-follower relative position information: Error magnitude, sampling period, and delay. These entities are used as the parameters of the simulation-based analysis and are also addressed as the independent variables of the analysis. The simulation-based analysis relies on varying these independent variables among repeated runs of the simulation. By this means the effects of the independent variables on the formation flight performance are found. The performance of the formation flight is characterized by the maximum position error during a predefined duration of the formation flight. Figure 7.1 depicts the overview of the simulation environment, which was set up

Leader aircraft Flight dynamics + Autopilot +

Path following Relative calculation

Random error addition Sample & hold

Delay rL rF rLF Wake vortex disturbance

rLF_d Follower aircraft_{Flight dynamics +}

Autopilot + Formation control position Formation conditions flight Vd, γd, χd Enabled or disabled rLF(corrupted) model (true) e_r(t) k, d, T Leader aircraft Flight dynamics + Autopilot +

Path following Relative calculation

Sample & Delay

rL

rF

rLF

Wake vortex disturbance

r_dLF Follower aircraft_{Flight dynamics +}

Autopilot + Formation control position Formation conditions flight Vd, γd, χd Enabled or disabled rLF model (true) er(t) k, d, T Random number source Adding deficiencies k hold by T by d (perturbed)

Figure 7.1: Simulation environment, set up for analyzing the effects of the independent

variables on the performance of formation flight. Shown are only the signals, which are of direct relevance to the analysis.

and used in order to carry out the analysis outlined above. The simulation environ- ment includes closed loop leader and follower aircraft models, which are made up of their nonlinear flight dynamics, driven by the automatic pilots and corresponding guidance algorithms. The leader and the follower aircraft are commanded to fly a common formation speed, at a predefined constant altitude (γd = 0) and a common course. Additionally, the follower aircraft is commanded to maintain the formation at a desired relative position

rLF_d with respect to the leader aircraft. These conditions

of the formation flight are kept constant for both scales of aircraft in this work. Using the actual or true positions of the leader and follower aircraft, the true relative position of the follower with respect to the leader aircraft is calculated using eqs. (5.9) and (5.10). The relative position vector’s components are expressed in the leader’s kinematic reference frame. The subscript addressing the reference frame, on which the vector’s components are decomposed, is not shown for simplicity. The difference between the desired relative position and the actual relative position is called actual or true relative position error. This term is shown by er in order to distinguish it from the relative position error, which is seen by the follower’s formation flight controller. The true relative position and the true relative position error do not

contain the deficiencies of the relative position vector, that the follower aircraft’s formation controller processes. The term er, which is a vector with components ex,

ey and ez, is recorded as a time series at the end of each simulation run in order to assess the performance of the formation flight. The scalar term er is also used in the subsequent sections of the text, in order to refer to any component of the vector er.

As shown in figure 7.1 the simulation environment includes a module, which adds the predetermined amounts of deficiencies to the relative position information. The output of the module is the perturbed relative position information, which includes the added deficiencies. The perturbed relative position information is then processed by the follower aircraft’s formation controller. The module receives the actual values of the deficiencies as input. The values of the deficiencies are the parameters of the simulation and they are varied within predefined ranges among each individual run of the simulation sets. The parameters remain constant during each run of the simulation, and they are shown by the terms k, d, and T . The term d represents the delay of the relative position information, which is measured in seconds. The term T is the period of the sample and hold action, measured in seconds. The term k is a dimensionless, non-negative real number, controlling the magnitude of the error added to the relative position information. The module uses a random number generator as the error source, which generates random numbers of Gaussian probability distribution. The term k is multiplied by the output of the random number generator, before the addition of the error to the true relative position. Therefore, any value of the parameter k, which is less than 1 but greater than zero, attenuates the error magnitude added to the relative position information. For k = 0, no error is added to the relative position information. For the values of k, which are greater than 1, the random error values are amplified by the factor given by k. As the relative position information is a vector with components on longitudinal, lateral and vertical axes of the leader kinematic reference frame, the random number source generates the numbers for all three axes. The numbers generated for all axes are uncorrelated with each other, by means of initializing each random number generator with different seeds. However, the term k affects the magnitudes of all the components commonly and in the same fashion.

The simulation environment also includes the wake disturbance model, which gen- erates the wind velocities acting on the follower aircraft, based on the true relative position between the aircraft. The simulations were run with both wake vortex dis- turbance disabled and enabled. The initial sets of simulation runs were carried out with disabled wake vortex effects in order to solely obtain the effects of the inde- pendent variables on the formation flight, without the interfering effects of external disturbances. Simulations with the wake vortex effects were performed in order to find out the impact of this external disturbance on the results obtained without the wake vortex disturbance.

the aircraft does not play a role on the formation flight performance. That is, the same result will be obtained if the follower aircraft is commanded to maintain the formation at different relative positions with respect to the leader. However, in a simulation with the wake vortex effects, the relative position of the aircraft during the formation flight gain importance as an additional parameter affecting the performance of the formation flight. This is due to the nonuniform flow field at the leader’s wake, which produces different disturbance on the follower aircraft at different relative positions. A separate analysis for describing the relation between the formation keeping performance and the formation relative position is beyond the scope of this work. In order to limit the parameters of the simulation-based analysis only to those characterizing the relative position information, a single commanded relative position is used in this work. The commanded relative position of the formation was selected as the relative position, which is used during probe and drogue aerial refueling. Mao and Eke [138] state the typical separation between the tanker and receiver aircraft during the aerial refueling as approximately two wingspans of longitudinal and a quarter wingspan of vertical separation with the receiver below and aft of the tanker and the aircraft are laterally aligned.