The conflict of airships is understood as their convergence in space and time, during which there is a violation of the specified minimum separation distance (echeloning). An increase in the intensity of flights inevitably leads to an increase in conflicts between airships. The minimum separation distance between aircraft is determined by their protective spatial zone with regulated geometry. Thanks to the proposed mathematical and software, it became possible to reduce the protective space zone and thereby reduce the minimum permissible interval between aircraft and increase flight safety, as well as to optimize the structure of airspace to improve throughput and thereby increase flight safety.
The relevance of maintaining traffic safety in the aeronautical environment is consistently high. It is caused by several processes accompanying the development of aviation technologies [1–4].
The intensity of air traffic is growing
exponentially (5–6% per year). An increase in
traffic leads to an increase in the frequency of
delays. Air traffic safety is based on methods and
algorithms for detecting and preventing conflict
situations [5, 6]. For this purpose, there is a need
to improve the existing air traffic control systems
on the route and improve the algorithms of their
functioning. Modern concepts are aimed at
increasing the safety of flights, providing aircraft
with the ability to fly within clearly defined
airspace along arbitrary routes [7]. However, they
do not fully satisfy the modern requirements of air
traffic safety, as they do not ensure full autonomy
of aircraft movement, and reliable resolution of
conflict situations in the airspace is not performed
[
The analysis of available sources of
information shows that the existing concepts
related to flight autonomy, such as Free Flight,
A3, TCAS, S&A, ADS-B, and ASAS are
imperfect. This is because systems of this level are
multi-faceted complex systems with a hierarchical
organization scheme, which contains technical,
organizational, informational, management,
socio-technical and energetic components. These
concepts can solve only partial problems of air
traffic safety. The results of recent research
revealed certain existing shortcomings and
limitations of these concepts in the
implementation of autonomous flight [
Eurocontrol has formulated a strategy for the
development of the air traffic organization for the
coming decades. An important role is given to the
development of new principles of air traffic
management and airspace organization [
The technology of self-organizing systems is
currently considered the only technology capable
of offering adequate methods, architecture, and
instrumental support for the software
implementation of the most complex modern
systems. This technology has great prospects, first
of all, about systems that are characterized by
openness, large dimensions, the autonomy of its
component subsystems and their network
organization, as well as mobility [
It can be said that the process of
selforganization of the system confirms that the
effectiveness of rational and purposeful actions is
weakening, the organization itself “creates” itself,
sometimes opposing conscious leadership.
Research into the processes of self-organization
provides an opportunity not only to identify the
mechanism of autonomous cyclical self-support
but also to find an opportunity to start this
mechanism [
Let’s consider a special case when two planes
in the airspace have to pass each other at a safe
distance. For this case, an analogy can be drawn
with connected pendulums that can move freely at
the same height. We impose a condition or “bind”
two bodies not rigidly, but so that they cannot
touch each other. At the same time, it is also not
necessary that the bodies do not diverge over a
long distance (that is, the plane remains on its
route) [
The goal is to improve the mathematical and
software of the automated aircraft traffic control
system in terms of conflict prevention in
autonomous flight conditions [
The task is to find mathematically two harmonics for the case when the lengths of the pendulums are the same. As an example of linear oscillators with nonlinear coupling, consider the equation where the quadratic terms on the right-hand side describe the relationship.
Let the deviation of one pendulum be x, and the other—y, as shown in Fig. 1. In the absence of a spring, the gravitational force acting on the first pendulum is proportional to its deflection. If there were no spring here, then a certain natural frequency ω0 would appear for one pendulum, and the equation of motion, in this case, would take the form
The second pendulum, in the absence of spring, would swing exactly like the first. However, in the presence of a spring, in addition to the restoring force arising as a result of gravity, there is an additional force from the spring that tends to “pull” the pendulums. This force depends on the excess of deviation x over deviation y and is proportional to their difference, that is, it is equal to some constant, dependent only on the geometry, multiplied by (x-y). The same force, but in the opposite direction, acts on the second pendulum. Therefore, the equations of motion that we must solve will be as follows: (3) (4) (1) (2)
To find the motion at which both pendulums oscillate at the same frequency, we must determine how much each of them deviates. In other words, pendulum A and pendulum B will oscillate with the same frequency and with certain amplitudes of A and B, the ratio of which is fixed. Let’s check how suitable this solution is: x=Aeiωt y=Beiωt.
If we substitute it into equation (1) and add similar terms, we get (5)
When deriving these equations, we reduced the common factor eiωt and divided everything by m. Now we see that we have two equations for what would seem to be two unknowns. However, in reality, there are no two unknowns here, because the general scales of motion cannot be found in these equations. They can only give us the ratio of A to B, and both equations must give the same value. The requirement that the equations be consistent with each other requires the frequency: it must be something very special.
But finding the frequency in this particular case is quite easy. If we multiply both equations, we get
On both sides, the product AB can be truncated, except when either A or B is zero, which means no motion at all. But if there is movement, then other coefficients must be equal to each other, which leads to a quadratic equation. As a result, two possible frequencies are obtained: і
Moreover, if we substitute these values of frequencies again in equation (5), then for the first frequency we will get A = B, that is, the spring will not stretch at all and both pendulums oscillate with the frequency ω0, as if there was no spring at all. In another solution, when A = -B, the spring increases the restoring force, and the frequency increases.
This problem was solved in stochastics by
referring to H. Haken (he is the founder of the
synergistic approach in the deterministic
formulation of the problem) [
As a result of modeling the movement of pendulums in the ideal case, in the absence of noise, interference, and other distortions, we have the following curves (Fig. 2a). But in real life, a system of connected moving objects is exposed to various types of disturbances. In addition, there are always errors in the measurements of the movement parameters of these objects. This is reflected in the curves (Fig. 2b). a b Figure 2: An example of stochastic oscillations of pendulums with different SLEs of noise: a) deterministic foundations; b) a real signal, which includes a deterministic basis and noises.
We will perform wavelet filtering of the amplitudes of stochastic pendulums taking into account the analytical relationships between their determined bases to increase the accuracy of data evaluation.
To filter the parameters of mathematical pendulums by classical and proposed (taking into account analytical connections between their deterministic bases) methods, we use a multiscale analysis with a biorthogonal spline wavelet of the eighth order as a basis function.
Based on B-splines, several biorthogonal bases of bursts are constructed. For example, the scaling function and the Malla-Zong burst are defined in terms of their Fourier transforms as follows:
The smoothing function (t) and the wavelet function ψ (t) are shown in Fig. 3.
Note that in the case of biorthogonal bursts, different filters must be used for forward and reverse transformation.
The function for calculating the coefficients of wavelet filters for stochastic pendulums, taking into account the linear relationship between them, has the form: where is the coefficient of the low-frequency wavelet filter (Low); λ—is a weighting factor characterizing the rigidity of the analytical connection; y—input readings of the signal; —oscillation frequency of the first pendulum; α—is the weight coefficient of the equation of motion of the first pendulum (4).
We will conduct statistical modeling and build a comparative table of filtering by the classic and proposed methods (Table 1). We can see that the accuracy advantage of the new method is many times higher than that of the classical method. The accuracy indicators of flight data measurement do not deteriorate, but on the contrary, improve.
To find the optimal coefficients, we will compile the following system of equations:
We will conduct statistical modeling and build a comparative table of filtering by the classic and proposed methods (Table 1). We can see that the accuracy advantage of the new method is many times higher than that of the classical method. The accuracy indicators of flight data measurement do not deteriorate, but on the contrary, improve.
As evidenced by the data presented in Table 1, the advantage in filtering the accuracy of the first pendulum by the created method is 26–28 percent compared to the classical filtering method. At the same time, the accuracy of filtering the second pendulum by the developed method is 50–59 percent higher than the accuracy of the classical filtering method.
Let’s graphically display the results of modeling the dependence of the Root mean square deviation deviation (RMS) of the filtered signal relative to its deterministic base on the mean square deviation deviation of the input noise (in the range from 0.1 to 0.25) for both pendulums (Fig. 7) m.s. d.of the input signal where: 1—the MSD of the filtered signal from the deterministic base of the pendulum oscillation, obtained by the classical filtering method; 2— MSD of the filtered signal from the deterministic basis of the pendulum oscillation, obtained by the proposed filtering method.
Analysis of graphs of functions in Fig. 7 shows that the proposed mathematical and software allows significant gain in reducing the SLE of the filtered signal. The first graphic window displays the results of the accuracy assessment of data filtering of the first pendulum, and the second graphic window shows the results of the second pendulum.
One of the design variations of conflict detection algorithms is the shape and size of the “protected zone”. The protected zone is actually defined by the threshold values used in the conflict detection logic. That is, mathematical models of three-dimensional volumes, as a rule, are not supported in real-time implementations.
Rather, airspace volumes serve as conceptually useful concepts for understanding the logic of conflict detection.
A “flattened spheroid” is a mathematically
convenient shape for a protected zone. This
volume is usually obtained by considering some
vertical division as equivalent to some horizontal
division [
For example, in cruise flight, a vertical deviation of 1,000 feet may be considered the equivalent of five miles of horizontal separation.
Vertical units are simply scaled to the equivalent horizontal division.
This is mathematically convenient, since a single value can be calculated to characterize the separation between two aircraft. Fig. 9 illustrates a “flattened spheroid”. criteria in both axes are checked separately. Fig. 10 illustrates the conceptual scope that results from this scheme. Five miles of zonal separation and 1000 feet of vertical separation are still appropriate examples of separation parameters.
Although the braided spheroid is mathematically simpler, the cylindrical shielded zone has the advantage of being consistent with existing separation criteria used today. In addition, it is most likely a division of the airspace that corresponds more to the perception of the pilots.
The RTCA CD&R working group aims to simultaneously use two protected zones for each aircraft—the Protected Airspace Zone (PAZ) and the Near Mid-Air Collision (NMAC) zone. The PAZ would define the desired airspace separation standards. For example, a criterion of five miles per 1,000 feet would define a PAZ. The NMAC zone is intended for tighter closure of the aircraft.
A smaller protected zone can be used to generate high-level notifications.
A “cylindrical” protected zone is another common shape used in conflict detection algorithms. For this protected zone, the separation
Thanks to the developed mathematical and software, it became possible to reduce the protective space zone (Fig. 11) and thereby reduce the minimum permissible interval between aircraft and increase flight safety.
The expected increase in air traffic density, a change in the dynamics of the relative movement of aircraft and a decrease in echeloning norms increases the probability of dangerous convergence of aircraft. Under these conditions, the role of systems for detecting and preventing potentially conflict situations is growing significantly. 1—for stochastic formulation of the problem; 2— after classic filtering; 3—after the suggested filtering.
An important role is given to the development of new principles of air traffic management and airspace organization, which are designed to ensure high throughput of the route network and the ability to fly on the most efficient trajectories with a guaranteed level of safety.
The aircraft monitors the current flight path of the aircraft from the point of view of possible conflicts. (In practice, the automatic ASAS function will monitor all relevant traffic and alert the flight crew of any conflicts. The bottom line is that the flight crew is responsible for separation from all aircraft, so the aircraft, not the ground systems or the controller must detect conflicts.) The aircraft entity adjusts its flight path as necessary to resolve any conflict, avoiding loss of separation with any other aircraft as a result of a change in trajectory.
An increase in the intensity of flights inevitably leads to an increase in conflicts between airships. A conflict of airships means such a convergence of them in space and time, during which there is a violation of the specified minimums of separation distance (echeloning). The minimum separation distance between aircraft is determined by their protective spatial zone with regulated geometry.
The possibility of using already existing methods of processing results, as well as the characteristic features of the behavior of wavelet transformation in the time-frequency domain allow to significantly expand and supplement the capabilities of such systems. It can be concluded that the wavelet transformation taking into account analytical connections provides a more accurate and informative picture of the results of simulation and experiment. Allows you to better clean the input data of the movement of the aircraft from noise and random distortions.
Thanks to the proposed mathematical and software, it became possible to reduce the protective space zone and thereby reduce the minimum permissible interval between aircraft and increase flight safety, as well as to optimize the structure of airspace in order to improve throughput and thereby increase flight safety. 6. References