RU2651342C1 - Method of sequential determination of certain trajectories of movement of material objects in three-dimensional space - Google Patents

Abstract

FIELD: measurement; test.

SUBSTANCE: invention relates to a method for determining the trajectories of the movement of material objects in three-dimensional space in the selected directions. To implement the method, objects with a specific set of motion parameters are registered, on the basis of which a sample of trajectories is formed, aligning the trajectories as necessary, forming a two-dimensional orthogonal projection of the set of points by which the formed sample of trajectories is represented, order the points of the formed projection in a certain way, determine the averaged trajectory in the selected beam.

EFFECT: provides allocation of areas of space corresponding to the trajectories of the movement of objects to the selected goals of their movement.

3 cl, 5 dwg

Description

The present invention relates to the field of data analysis, in particular to the task of processing and analysis of multidimensional spatial trajectories of moving objects that have common goals, and which are close in final coordinates. The present invention can be used to detect hidden targets of moving objects to identify steady traffic flows in solving the problem of sectorization of the airspace of an extended zone of an airport in a three-dimensional case. Also, the present invention can be used to create automatic (unmanned) aircraft approach systems for landing (including when landing in a complex geographical landscape, poor visibility, on the deck of a ship) and in determining the optimal (normal) trajectories of objects in space based on radar data.

Currently, in connection with the ever-increasing volume of data, an important task is the creation of methods and algorithms that allow you to process large amounts of data quickly and automatically. To analyze large data sets, new methods are required that allow you to recognize hidden patterns of motion and identify moving objects with similar motion characteristics and / or the same end goal.

Of greatest interest are tasks related to the analysis of multidimensional spatial motion trajectories. In particular, these tasks are becoming increasingly important in aviation due to the constant increase in air traffic, the need to optimize the congestion of runways and runways and improve existing air traffic control systems (ATCM).

The trajectories of the aircraft are represented by a four-dimensional description of the state of the aircraft in time, where states can be understood as the position of the center of mass or other characteristics of movement such as speed, height or weight. In the general case, the spatial trajectory is a multidimensional description of the path of a moving object.

According to the Federal Air Transport Agency of the Russian Federation (Rosaviation) and the International Civil Aviation Organization (ICAO), a significant number of accidents are associated with the organization of air traffic in the vicinity of airports. In this regard, to ensure flight safety in the context of an ever-increasing workload of airports and maintain a high level of flight safety, it is necessary to modernize and optimize the operation of existing air traffic control systems.

From the patent US 6393358 B1, the TRACON ( Terminal Radar Approach Control ) automatic radar system is known, the use of which is aimed at simplifying the work of dispatching services, ensuring flight safety and more efficient airspace congestion in the extended area around the airport. Such systems make it possible to ensure traffic safety in airspace and improve airport throughput. Using the TRACON radar, metadata is recorded that contains information about the spatial coordinates of the position of the center of mass of the aircraft, based on which the trajectories of their movement are calculated.

A feature of the air traffic control problem is that due to separation constraints, aircraft of different weights cannot follow a single runway in a caravan. From the application US 20140019033 A1, a method of air traffic planning is known, including determining a network consisting of nodal points and main landing directions (an arrival network of nodes and legs), which is used to optimize the schedule of arriving aircraft.

To optimize traffic in the extended area of the airport, data mining methods can be used. In particular, using data analysis methods, the problem of sectorization of the airspace of an extended zone of an airport, i.e. division of space into sectors or areas of responsibility of dispatching services. Moreover, the airspace sectorization should be carried out taking into account the established separation of flights along routes by highlighting the characteristic traffic flows in the set of aircraft paths recorded by the radar.

It is important to note that the complexity of the problem being solved is related to the spatial geometry of the motion paths (their potential intersections, curvature and torsion) (see Fig. 1a) and possible spatial intersections of the bundles of multidimensional trajectories with each other.

Thus, when developing methods for analyzing trajectory data, it is desirable to use a measure of proximity that would allow us to separate trajectories of a similar shape, even if they intersect in space. The measure of proximity used should reflect the similarity (or difference) of the shape of the trajectories in three-dimensional space.

The solution to the sectorization problem is currently at the initial stage and is associated with the solution to the problem of distinguishing bundles of trajectories, that is, groups of motion trajectories that have similar characteristics and common goals. (In the case of landing trajectories, it is also necessary to additionally take into account the direction of approach and the given runway).

If objects move towards common (specific, given) goals, their final coordinates are close. Thus, the trajectories of the objects form convergent beams of multidimensional spatial motion trajectories, as, for example, in the case of landing trajectories of aircraft landing on a given runway (see Fig. 1b). In the general case, the trajectories in such beams have multiple intersections. The beam convergence is determined by the threshold parameter. The threshold parameter for the beam of trajectories does not exceed the width of the runway.

An analysis of the civil aviation safety of the Russian Federation, conducted by the Federal Air Transport Agency (flight safety inspection department), shows that there are cases of failure to maintain a safe descent path at the final stage of approach, maneuvering at the stage of landing (increased speed, deviations from the glide path) and rolling out Runways pose the greatest threat to flight safety, since even a slight deviation from a safe route can lead to an airplane crash, especially If the airport is located in the area of complex geographical landscape.

The present invention proposes a method of sectorization of airspace in the three-dimensional case, which allows you to determine the reference trajectory of the aircraft landing (the so-called averaged trajectory or centroid) and emissions - the beam paths that are farthest from the central one, the movement of which is potentially at risk.

Trajectory analysis of aircraft movements is often used in control systems and safety of air traffic (airtraffic management). In particular, based on the analysis of data on successful landings of the same type of aircraft, further trajectories are displayed (see US 9020662) and trajectories that are most optimal in the given meteorological conditions (see US 8977484) are determined.

However, the above methods have a number of drawbacks, associated primarily with a significant reduction in the dimensionality of the analyzed data, as well as with the use of the Euclidean measure aircraft trajectories as a measure of proximity. The Euclidean measure of proximity does not take into account these features and does not allow the separation of intersecting, but different in geometry trajectories of motion. Clustering algorithms [1], which are currently used to solve the sectorization problem, cannot be applied in practice, because do not allow to obtain sustainable results.

The present invention proposes a new method [2, 3] for processing multidimensional trajectories of moving objects with common goals and close final coordinates in order to select objects with common goals. The present invention allows, when applying a computer to the data on the recorded characteristics of the movement of an object (for example, an aircraft), to select asymptotically converging bundles of trajectories of multidimensional spatial motion trajectories - landing the aircraft on a given runway. The method is based on determining the geometric asymptote tangent to the distinguished beam of trajectories in a space of lower dimension. A bunch of trajectories (after determining the tangent to it in a space of lower dimension) is highlighted as a result of the reverse transition to the space of the original dimension. Thus, there is no loss of data dimension. Such an approach when applied to real data (for example, radar data) allows you to get stable results and solve the problem of sectorization of the airspace of the airport zone in the three-dimensional case.

The fundamental difference between the present invention and the known analogues is the use as a measure of proximity of spatial trajectories of an experimental measure of similarity of motion trajectories - a measure of cosine both at the stage of determining the tangent to the beam of trajectories, and when highlighting the beam of trajectories, the central trajectory of the beam.

The use of the cosine measure allows one to take into account the spatial geometry of multidimensional trajectories and to separate intersecting, but different in geometry trajectories of motion and, thus, accurately distinguish bundles of trajectories (to solve the problem of sectorization of air space in the three-dimensional case) and to determine averaged trajectories in the selected bundles of trajectories (central trajectories) or centroids).

Averaged trajectory (central trajectory, centroid) should be understood as the possible trajectory of the aircraft, associated with the asymptotic behavior of the trajectories of a converging beam of multidimensional trajectories at a given threshold parameter, which is the spatial averaging of all trajectories in the selected beam.

Outliers should be understood as the trajectories of asymptotically converging beams of multidimensional trajectories that are farthest from the corresponding centroids as the cosine. In the context of solving security problems, we can say that the trajectory of an asymptotically converging bundle of multidimensional trajectories distinguished in this way is boundary (minimally admissible), i.e. potentially dangerous trajectory.

The problem to be solved within the framework of the present invention is associated with the further improvement of the methods for controlling and controlling the movement of objects in predetermined directions (for example, in the case of aircraft landing). The technical result achieved by solving such a problem is to determine the number of objects moving targets and identifying areas of space corresponding to the trajectories of objects moving towards such goals. The practical aspect of this result can be associated with increasing the safety of the movement of objects (in particular, flights of aircraft), for example, preventing conflict situations when setting the trajectories of movement of objects in a limited area of space and reducing the number of emergency (emergency) situations.

The set result in general is achieved by the claimed method of sequentially determining the averaged trajectories of the movement of material objects in three-dimensional space in selected directions, including a sequence of the following steps:

(1) registration of objects with a set of motion parameters, including at least spatial coordinates and speed at successive times, the selection from the registered set of objects whose motion parameters are characterized by a decrease in the speed of movement, indicating the end of the movement, the formation of a sample of motion paths selected objects based on a set of parameters of their movement, determining the need to align the sampling paths in time and, if necessary, vnivanie;

(2) representation of the trajectories of the formed sample in the form of a set of points making up each trajectory, the formation of a two-dimensional orthogonal projection of the set of points;

(3) at least a single ordering of the points of the formed projection in the direction of increasing or decreasing the value of one of the coordinates, analysis for the ordered points of the formed projection of the most likely model of orthogonal linear regression with the determination of the geometric asymptote for a given threshold parameter and highlighting in the generated sample asymptotically converging with the threshold parameter of the beam of trajectories of motion of material objects in three-dimensional space based on the measure of proximity of the trajectories of the selected to the geometric asymptote defined above, and the proximity measure is determined by the cosine at a given threshold parameter and is calculated in the range of values from 0 to +1, and the selected beam contains at least two trajectories of the motion of material objects in three-dimensional space;

(4) the definition in the selected beam of the average trajectory of the material object in three-dimensional space from the condition that the sum of the squares of the distances from such a trajectory to all the trajectories of the beam calculated according to the cosine is minimal;

(5) removing the selected beam from the generated sample of trajectories and repeating steps (3) ÷ (5).

Implementations of the method suggest that the material object is an aircraft (LA), and the highlighted direction is the landing path of such an apparatus, and the threshold parameter is determined by the width of the runway.

The invention is illustrated by a schematic flowchart depicting the steps for implementing the method (FIG. 1), and FIG. 2-5, illustrating the feasibility of the claimed method. In particular, in FIG. 2a shows a general view of beams of multidimensional spatial trajectories of aircraft landing on the aerodrome strip, FIG. 2b - bundles of landing trajectories corresponding to different runways. A three-dimensional representation of the landing trajectories recorded by the radar is shown in FIG. 3a, in FIG. 3b — the partition of these trajectories into five clusters, performed by the method of polynomial regressions [4, 5]. Further, the trajectories of each cluster are considered a sample of trajectories in which the selection of beams is possible. In FIG. 4, projections onto the coordinate axes of the sample paths corresponding to the pink cluster in FIG. 3b. FIG. 5 illustrates the successive steps of extracting beams of multidimensional spatial trajectories in the sample of FIG. four.

Revealing the possibility of realizing the claimed purpose and the practical feasibility of the claimed solution, the present description establishes that the trajectories of the movement of material objects in three-dimensional space in the selected directions or multidimensional trajectories of the movement of objects (for example, aircraft (LA), submarines, etc.), represented by multidimensional vectors are calculated on the basis of metadata recorded at the corresponding time instants, containing, in particular, spatial coordinates (e.g. measures position coordinates of the center of mass, speed, altitude, etc.). The sequence of recorded metadata determines the multidimensional spatial trajectories of the object (which are hereinafter referred to briefly as trajectories of motion). Multidimensionality is determined by the long trajectory and the number of metadata parameters used.

It should be noted that the present invention is used in the analysis of data on the movement of objects to specific (identical, set) goals. An example of such a movement is an aircraft landing on a given runway. In the general case, if objects move toward a common goal, then their motion paths form a bundle of multidimensional spatial motion paths (hereinafter referred to briefly as a path bundle), and in this case, the motion paths may intersect. For the purposes of disclosing the feasibility of the present invention, it is established that a beam of trajectories is considered convergent (and is briefly called a convergent beam of trajectories) if the trajectories of the beam have a common goal and are close in final coordinates (for example, when analyzing aircraft landing trajectories - on the landing plane), when In this, the convergence of the beam of trajectories is determined by the threshold parameter. When analyzing the flight paths of the aircraft in the airport area (see Fig. 2a), it is possible to distinguish converging bundles of trajectories - landings on given runways (see Fig. 2b), while in the general case the threshold parameter does not exceed the width of the runway.

The achievement of the set result is related to the above-mentioned features of the analyzed data and is due to the sequential determination of the geometric asymptotes of the corresponding bundles of trajectories by determining the most plausible orthogonal linear regressions, which, in turn, is performed when we proceed to consider a two-dimensional projection of the points of the motion paths. As a result of this transition, there is a significant reduction in the dimension of the source data. It should be noted that such a reduction in the dimension of the data does not lead to loss of information, since after determining the geometric asymptote, the transition back to the original space occurs, where, when the threshold parameter is limited and in accordance with the cosine measure between the sample paths and the geometric asymptote defined in the space of smaller dimension, the corresponding her bunch of trajectories.

Estimation of the asymptotes of the trajectory bundle

,

(K ₀ - эмпирический параметр) считается асимптотически сходящимся с параметром порога, если для векторов

, представляющих пучок траекторий

,

, выполняется условие асимптотического схождения пучкаFormally, a trajectory bundle

,

( K ₀ is an empirical parameter) is considered asymptotically converging with the threshold parameter, if for vectors

representing a bundle of trajectories

,

, the condition of asymptotic convergence of the beam

,

- координаты точек траекторий, которые почти совпадают, т.е. параметры L _i ,

подлежат определению,

- евклидова мера расстояния в трехмерном пространстве

, ε - порог (который при рассмотрении сходящихся пучков, образуемых траекториями посадки ЛА, не превосходит ширины ВПП).Where

,

- coordinates of the points of the trajectories, which almost coincide, i.e. parameters L _i ,

to be determined

- Euclidean measure of distance in three-dimensional space

, ε is the threshold (which, when considering convergent beams formed by the aircraft landing paths, does not exceed the width of the runway).

(трехмерное пространство координат), удовлетворяющая условию (1). Траектории в асимптотически сходящихся пучках имеют касательную в окрестности конечных точек

, x[L _i ;i] всех траекторий пучка с порогом ε (1)) [6], так что асимптотически сходящиеся пучки траекторий могут быть идентифицированы посредством определения касательных им геометрических асимптот в точках их фокусов.Considering the claimed approach to determining the number of convergent sheaves of multidimensional trajectories, it should be taken into account initially that the trajectories in asymptotically converging sheaves have a typical shape (profile) and a characteristic geometric asymptote in the region of convergence of the trajectories (1) [6]. The geometric asymptote of the converging beam of multidimensional landing trajectories of aircraft is the line in

(three-dimensional coordinate space) satisfying condition (1). Trajectories in asymptotically convergent sheaves have a tangent in a neighborhood of finite points

, x [ L _i ; i ] of all trajectories of the pencil with threshold ε (1)) [6], so that asymptotically convergent bundles of trajectories can be identified by determining tangent geometric asymptotes to them at the points of their foci.

рассеивается во множество точек этих траекторийSince the discrete points of the beam trajectories lie densely in the vicinity of the asymptotes, the basis of the proposed method for determining the number of beams of convergent multidimensional spatial trajectories of objects is that a set of sample vectors of multidimensional trajectories

scattered to many points of these trajectories

(2), например, с помощью алгоритма RANSAC (Random Sample and Consensus - случайная выборка и консенсус) [7, 8], анализируются модели ортогональной линейной регрессииThe set of points (2) should be ordered by the values of one of the coordinates (in the direction of increase - ascend or decrease - descend ). In this case, ordering by the remaining coordinates of all points representing a converging bundle of trajectories along a certain profile occurs. After that, for scattered 3D data

(2), for example, using the RANSAC algorithm ( Random Sample and Consensus ) [7, 8], models of orthogonal linear regression are analyzed

, вычисляемого по ортогональной проекции точки z=(x,y,z) из (2) на линию

. Таким образом, модель (3) симметрична относительно координат x,y,z. Для выдвижения гипотезы относительно модели ортогональной линейной регрессии (3) достаточно любой пары точек из (2). Окончательная модель (3) подтверждается наибольшим относительным количеством (процентом) рассеянных данных

(2). Для данных целей может быть использован алгоритм MLESAC (Maximum Likelihood Estimation Sample Consensus - консенсус выборок с оценкой по максимуму правдоподобия) [9, 10] - вероятностная версия алгоритма RANSAC. Этот алгоритм оценивает правдоподобие модели (3), представляя распределение расстояния рассеянных данных

от модели

(3) как смесь распределения данных, подтверждающих модель (3) (inliers), и распределения данных, отклоняющих эту модель (outliers). Считая, что рассеянные данные

(2) независимые, получаем выражение для логарифма правдоподобия в видеwhere ∧ is the conjunction, θ = { а ₁ , b ₁ , с ₁ , d ₁ , and ₂ , b ₂ , с ₂ , d ₂ } is the vector of parameters of these models for a given threshold of Euclidean distance

calculated from the orthogonal projection of the point z = ( x , y , z ) from (2) onto the line

. Thus, model (3) is symmetric with respect to the x , y , z coordinates. To put forward a hypothesis regarding the model of orthogonal linear regression (3), any pair of points from (2) is sufficient. The final model (3) is confirmed by the largest relative amount (percentage) of scattered data

(2). For these purposes, the MLESAC algorithm (Maximum Likelihood Estimation Sample Consensus - consensus of samples with maximum likelihood estimation) can be used [9, 10] - the probabilistic version of the RANSAC algorithm. This algorithm estimates the likelihood of a model (3) by representing the distribution of the distance of the scattered data

from model

(3) as a mixture of the distribution of data confirming the model (3) (inliers) and the distribution of data rejecting this model (outliers). Assuming scattered data

(2) independent, we obtain the expression for the likelihood logarithm in the form

where γ is the mixing parameter. The distribution of distances to the data confirming model (3) is represented by a Gaussian distribution

where σ is the standard deviation. The distribution of distances to the data rejecting the model (3) is described by a uniform distribution

where ρ _max is the largest distance to the data (determined by context). Minimizing the likelihood logarithm (4) allows us to estimate the parameter vector θ and the mixing parameter γ, which is ensured by iterations of the likelihood-maximization likelihood algorithm.

(3) одного из пучков при условии (1) помимо рассматриваемых в настоящей заявке алгоритмов могут быть использованы и другие методы.For a specialist it is obvious that when determining the geometric asymptote

(3) of one of the beams under condition (1), in addition to the algorithms considered in this application, other methods can be used.

(3) одного из его пучков при условии (1). Полученная таким образом геометрическая асимптота удовлетворяет условиюThe most plausible linear regression of scattered data from a sample of trajectories determines the geometric asymptote

(3) one of its bundles under condition (1). The geometric asymptote obtained in this way satisfies the condition

.

.

Trajectory beam selection

The tangent bundle tangent to the distinguished geometric asymptote is determined by finding the minimum of the objective function

,

,

,

- набор бинарных индикаторных переменных (т.е. если вектор x[i] назначен пучку k, то r[i;k]=1 и r[i;k]=0 в противном случае). Расстояние между геометрической асимптотой и траекториями выборки вычисляется по мере косинусаWhere

,

- a set of binary indicator variables (that is, if the vector x [ i ] is assigned to the pencil k , then r [ i ; k ] = 1 and r [ i ; k ] = 0 otherwise). The distance between the geometric asymptote and the sample paths is calculated as the cosine

.

.

After removing from the scattered data (2) those points that represent the trajectories of the selected beam, the procedure for determining the geometric asymptote is repeated and the next bunch of trajectories is selected. Since the definition of model (3) must be symmetric with respect to the x , y , z coordinates, when generating scattered data of the remaining trajectories in (2), sorting by the next spatial coordinate is performed as compared to that used in (2) in determining the previous asymptote (3). A possible dependence of the result of (3) on the direction of coordinates is eliminated by changing the direction of sorting in (2) from increasing to decreasing or vice versa. An analysis of the trajectories of the cluster ends with the determination of all beams in the cluster.

In the general case, the approach described above to determining the number of asymptotically converging bundles of multidimensional spatial motion trajectories consists of two stages. A significant reduction in the dimension of data at the first stage simplifies the detection of characteristic features in the data. When considering two-dimensional projections of scattered points of three-dimensional trajectories, the most plausible orthogonal linear regression of scattered data is determined, which corresponds to the geometric asymptote of one of the beams in the analyzed set of trajectories. At the second stage, the transition back to the space of the original data dimension is performed. In this case, the beam of trajectories is extracted from the analyzed sample of trajectories in accordance with its proximity to the selected asymptote as the cosine. Thus, with this approach, there is no loss of information about the source data.

Determination of the averaged trajectory (centroid) for a beam of trajectories

для пучка траекторий

определяется на основе однократного применения формулыCentroid

for beam paths

determined based on a single application of the formula

provided

using squared measure cosine

и

в исходном пространстве состояний.Such an estimate is effective in representing vectors

and

in the original state space.

It is the use of the cosine measure as a measure of proximity between multidimensional spatial trajectories (for example, aircraft landing trajectories) that allows you to separate intersecting, but different in geometry, multidimensional motion trajectories and provides accurate determination of centroids for asymptotically converging (potentially intersecting) bundles of multidimensional trajectories in solving the conditional optimization problem at which the standard deviation of the centroid from all trajectories in the beam is minimal as the cosine.

In addition, by using the cosine measure as a measure of proximity between multidimensional trajectories in the beam, it is possible to determine the so-called outliers, i.e. beam paths farthest from the centroid as the cosine.

The possibility of practical implementation of the claimed method is considered by the example of data analysis of 116 aircraft flight paths recorded by the TRACON radar on January 1, 2006 over the San Francisco Bay (they are freely available https://c3.nasa.gov/dashlink/resources/132/) . The origin coincides with the position of the radar, the time interval between the registration points is about 5 s. Only 160 of the last points of each trajectory are taken into account in the work, which excludes accidental aircraft maneuvers before approach.

In FIG. 3a presents an initial three-dimensional representation of the analyzed data. Further, these data are divided into clusters (see Fig. 3b) according to the method of polynomial regressions [4, 5] in accordance with the similarity of shape and speed modes. The distribution of trajectories in clusters is as follows: 16 trajectories in the pink cluster, 13 in the green, blue, black and red clusters - 3, 37 and 38, respectively. The trajectories of each cluster correspond to a certain sample of trajectories, in which it is possible to select bundles of trajectories corresponding to certain landing profiles.

As an example, we define bundles of sample paths defined by a pink cluster (see Fig. 4). It should be noted that the bundles of trajectories of the analyzed sample intersect significantly. Their other feature is the presence of almost linear sections in their tails away from the focus. Since all the trajectories in the sample have the same time direction, the analysis of these trajectories uses scattered data of reduced trajectories with some quotum ≈ 0.4 points, counting from the foci of the beams.

In FIG. 5a shows time-aligned scattered data of two-dimensional projections of reduced sampling paths. In FIG. 5b is the result of their linear regression using the MLESAC algorithm. In FIG. 5c, the asymptote of the first beam (blue line) is determined. It should be noted that in this case, the results of the orthogonal linear regression of the scattered data of shortened and full trajectories coincide. The trajectories of the first (blue) beam (see Fig. 5d) are removed from the sample based on the proximity of the trajectories to the blue asymptote (Fig. 5c) as the cosine. After removing from the sample the trajectories of the first selected beam in the remaining part of the sample, the following asymptote (see Fig. 5e-h) and the corresponding bundle of trajectories are similarly determined. Further, the trajectories of the second (green) beam are removed from the sample under consideration on the basis of the proximity of the motion paths to the green asymptote as the cosine. In FIG. 5h, three beams of motion paths (blue, green, and blue) selected in the sample under consideration as a result of successive iterations are presented. The thick red line in the center of each beam corresponds to the central path in the beam.

Literature

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Claims (8)

1. A method for sequentially determining the averaged trajectories of the movement of material objects in three-dimensional space in selected directions, comprising a sequence of the following steps: (1) registration of objects with a set of motion parameters, including at least spatial coordinates and speed at successive times, the selection from the registered set of objects whose motion parameters are characterized by a decrease in the speed of movement, indicating the end of the movement, the formation of a sample of motion paths selected objects based on a set of parameters of their movement, determining the need to align the sampling paths in time and, if necessary, vnivanie; (2) representation of the trajectories of the formed sample in the form of a set of points making up each trajectory, the formation of a two-dimensional orthogonal projection of the set of points; (3) at least a single ordering of the points of the formed projection in the direction of increasing or decreasing the value of one of the coordinates, analysis for the ordered points of the formed projection of the most likely model of orthogonal linear regression with the determination of the geometric asymptote for a given threshold parameter and highlighting in the generated sample asymptotically converging with the threshold parameter of the beam of trajectories of motion of material objects in three-dimensional space based on the measure of proximity of the trajectories of the selected to the geometric asymptote defined above, and the proximity measure is determined by the cosine at a given threshold parameter and is calculated in the range of values from 0 to +1, and the selected beam contains at least two trajectories of the motion of material objects in three-dimensional space; (4) the definition in the selected beam of the average trajectory of the material object in three-dimensional space from the condition that the sum of the squares of the distances from such a trajectory to all the trajectories of the beam calculated according to the cosine is minimal; (5) removing the selected beam from the generated sample of trajectories and repeating steps (3) ÷ (5). 2. The method according to p. 1, in which the material object is an aircraft, and the highlighted direction is the landing trajectory of such an apparatus. 3. The method according to p. 2, in which the threshold parameter is determined by the width of the runway.

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