RU2529016C1 - Method of locating mobile object in navigation measurements - Google Patents

Abstract

FIELD: physics, navigation.

SUBSTANCE: invention relates to methods of determining and predicting the position of an object in space. Dynamic properties of an object are used to predict the region in space of possible location of the object during subsequent navigation measurements. The corrected position of the object in space during the next navigation measurements is the intersection of regions in space of the subsequent navigation measurements with predicted regions.

EFFECT: high accuracy of locating moving objects in space during navigation measurements based on use of dynamic properties of said objects.

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Description

The invention relates to methods for determining and predicting the location of an object in space and can be used to improve the accuracy of the location of moving objects in navigation measurements. The method can be used in navigation systems, in short-range navigation and landing systems, in control systems and automatic control of moving objects, in air traffic control systems and collision avoidance systems, in systems of automatic docking of moving objects, in radar systems, for example, in situations when the target is identified and its potential for moving in space is known, and the frequency of updating radar information is low.

Currently, various methods for determining the location of an object are known.

So, from the description of the patent of the Russian Federation No. 2202102 (published on 04/10/2003), a method is known for determining the coordinates of moving ground objects, light aircraft, boats, and yachts. The method includes measuring in the calibration cycle the control values of the horizontal projections of the total vector of the Earth’s magnetic field and the magnetic field of the object, measuring during the working cycle the average values of the projections of the acceleration of gravity and the projections of the total vector of the Earth’s field and the magnetic field of the object. Taking into account the correction coefficients, the values of the horizontal projections of the Earth's field strength vector are determined. Determine the angle of direction. The increments of coordinates during the working cycle are determined. Relative coordinates are determined by summing the increments of the coordinates. The coordinates of the object are determined by summing the relative coordinates and the coordinates of the starting point. Using the receiver of the satellite navigation system, the coordinates of the object are measured, taking into account which the relative coordinates and the coordinates of the starting point are corrected. Correcting coefficients are determined, taking into account which, in each working cycle, the angle of the direction of movement and the increment of the path are corrected. EFFECT: increased accuracy of measuring the direction of movement and coordinates of the object and simplification of device calibration.

Also, from the RF patent No. 2399065 (published on 09/10/2010), a method is known for determining the location of a moving object using a hybrid navigation system that combines a satellite navigation system receiver for receiving a navigation signal that allows you to determine the position of satellites, the speed of satellites, the pseudo-distance to the object of observation, and the estimation of Doppler shift the carrier frequency of the navigation signal, the measuring device of the inertial navigation system and the computing device located at a moving object from which the angular velocities along each orthogonal axis X, Y and Z, linear accelerations along the orthogonal axes, which consist in the fact that the initial location of the moving object and the initial value of its velocity vector are known, the received data from satellites, which located in the radio visibility zone, they are used to form two rotation matrices by means of a computing device, one of which is the rotation matrix for satellite coordinates, the other is the rotation matrix for satellite speeds, used Using the generated rotation matrices by means of a computing device, the coordinates of the satellites and their speeds are converted from a geocentric fixed coordinate system to a local coordinate system, receive data from the measuring device of the inertial navigation system of the moving object — angular velocities and linear accelerations of the moving object, using the received data on the angular velocities of the moving object, form the rotation matrix R from the coordinate system associated with the moving object, in cial coordinate system using a rotation matrix R formed converted data obtained on linear accelerations of the mobile object; form a transition matrix for the state vector, a correlation matrix of inertial measurement errors Q, characterizing the measurements performed by the measuring device of the inertial navigation system, a correlation matrix of measurement errors W, characterizing the data received from the receiver of the satellite navigation system using the obtained transformed data on linear accelerations of the moving object, as well as location values, speeds, formed rotation matrices, by means of a computing device and calculating a location of the mobile object and its speed; the converted satellite data, the transition matrix for the state vector, the correlation matrix of inertial measurement errors Q, the correlation matrix of measurement errors W, the calculated location of the moving object and its speed of movement are used to form the matrix H, which describes the linear relationship of all measurements with the components of the state vector, for calculating the predicted values of the state vector dx and for calculating the correlation matrix of errors in estimating the components of the state vector P using the calculated values i, the state vector and its correlation matrix of errors are calculated, according to the results of the calculated state vector, the current location of the object is determined, characterized in that the correlation matrix of inertial measurement errors Q and the correlation matrix of measurement errors W are multiplied by the corresponding weight coefficients using a computing device, when calculating the predicted values of the state vector dx and the correlation matrix of its errors is performed by the computing device matrix by the approximate formula:

(W + H · P · HT) -1≈ 1 / W (1-H · P · HT / W),

where T is the matrix transposition operation, while the weighting coefficients are chosen so that the condition H · P · H ^T / W <1 is satisfied, the elements of the correlation matrix of errors of the state vector are compared with a given threshold value if at least one of the elements exceeds a given value threshold, then to calculate the state vector of the current stage, the calculated values of the error matrix of the state vector of the previous stage are used, if the threshold is exceeded, at least one of the elements returns the dispersion matrix calculated on the previous present step.

The closest analogue to the invention is a method for determining the coordinates of moving objects, based on the reception of signals from spacecraft of global navigation satellite systems, measuring pseudorange, introducing corrections and calculating the coordinates of moving objects, characterized in that n measurements of pseudorange and coordinates of a moving object along a known route determine the points x _io , y _io , z _io on a known route corresponding to the shortest distance to the points with measured coordinates x _i *, y _i *, z _i *, and determine the corrections by solving a system of 3 n equations

x = a_xxi

x_i+a_xyi

y_i+a_xzi

z_i,

y = a_yxi

x_i+a_yyi

y_i+a_yzi

z_i,

z = a_zxi

x_i+a_zyi

y_i+a_zzi

z_i,
где i = 1, n;

x = a _xxi

x _i + a _xyi

y _i + a _xzi

z _i

y = a _yxi

x _i + a _yyi

y _i + a _yzi

z _i

z = a _zxi

x _i + a _zyi

y _i + a _zzi

z _i
where i = 1, n;

x_i = b_xxi(x_i - x_io) + b_xyi(y_i - y_io) + b_xzi(z_i - z_io),and _xxi , and _xyi , and _xzi , and _yxi , a _yyi , a _yzi , a _zxi , a _zyi , a _zzi are the conversion factors from a special topocentric coordinate system centered at the point X _io , Y _io , Z _io , X axes _ti and Y _ti lying in the horizontal plane and the Z _ti axis directed vertically upward, with the point x _i *, y _i *, z _i * lying in the X _ti , Z _{ti plane} , and the tangent to the known route lying in the Y plane _ti , Z _ti , in the geocentric coordinate system;

x _i = b _xxi (x _i - x _io ) + b _xyi (y _i - y _io ) + b _xzi (z _i - z _io ),

z_i = b_zxi(x_i - x_io) + b_zyi(y_i - y_io) + b_zzi(z_i - z_io),

z _i = b _zxi (x _i - x _io ) + b _zyi (y _i - y _io ) + b _zzi (z _i - z _io ),

y_i - n сопутствующих неизвестных;

x,

y,

z - определяемые поправки (патент РФ №2145423, опубликован 10.02.2000).b _xxi , b _xyi , b _xzi , b _zxi , b _zyi , b _zzi - conversion factors of coordinates from a geometric coordinate system to a special topocentric coordinate system with the specified characteristics;

y _i - n associated unknowns;

x

y

z - defined amendments (RF patent No. 2145423, published 02.10.2000).

However, the proposed technical solutions require either additional navigational measurements, or reference points with previously known geodetic coordinates and the presence of a communication channel between a moving object and a reference point, or fully known and with a given accuracy predicted trajectories of the object.

The technical result of the invention is to increase the accuracy of determining the location of a moving object in space based on the use of its dynamic characteristics.

The proposed method for determining the location (coordinates) of moving objects during navigation measurements differs from the known ones in that, based on the dynamic properties of the object, a space region of the possible location of the object is predicted at the time of subsequent navigation measurements. The corrected location of the object in space during subsequent navigational measurements is the intersection of the space regions of the subsequent navigational measurements with the predicted areas.

центра масс подвижного объекта в момент времени t₁-t_n, по данным результатов навигационных измерений определяют область

пространства, в котором находится объект в момент времени t₁-t_n:The claimed technical result is achieved due to the implementation of the method for determining the location of a moving object in space, characterized in that at least three navigation measurements are carried out, as a result of which the coordinates are determined $$\overline{x},\overline{y},\overline{z}$$

the center of mass of a moving object at time t ₁ -t _n , according to the results of navigation measurements, determine the area

the space in which the object is located at time t ₁ -t _n :

$${\Lambda }_{\mathrm{one}}=\left\{\begin{array}{l}({\overline{x}}_{\mathrm{one}}-h_X)+m\Delta x;\\ ({\overline{y}}_{\mathrm{one}}-h_Y)+n\Delta y;\\ ({\overline{z}}_{\mathrm{one}}-h_Z)+k\Delta z\end{array}\right\}$$

...

$${\Lambda }_n=\left\{\begin{array}{l}({\overline{x}}_n-h_X)+m\Delta x;\\ ({\overline{y}}_n-h_Y)+n\Delta y;\\ ({\overline{z}}_n-h_Z)+k\Delta z\end{array}\right\}$$

while navigational measurements determine the following conditions:

- the error of each subsequent measurement and the measurement values themselves do not depend on the results of previous measurements and their errors, they are statistically independent;

- the error of each measurement does not depend on the location of the object in space, or this dependence can be neglected within several consecutive measurements of coordinates and assume that hx; hy; hz = const for several consecutive navigation measurements,

- then determine the predicted area of space that the object occupies through the time interval Δt, taking into account the parameters of the restriction of movement, which are determined by means of on-board meters located on the moving object, while the following are taken as the restriction parameters:

- heading speed V _{K MAX} - the maximum possible heading speed when moving the object (at each stage of the movement);

- maximum course acceleration and _{K MAX} - the maximum possible acceleration of the object at the heading (at each of the stages of movement);

- maximum vertical speed - V V _MAX ;

- a _{In MAX} - the maximum possible vertical acceleration (at each of the stages of movement);

- the maximum possible change in the yaw angle Ψ _MAX for the time Δt between subsequent navigation measurements;

- the maximum possible change in pitch angle Θ _MAX during the time Δt between successive navigation measurements and the general conditions for restricting the location of the object according to the data of navigation measurements and dynamic properties of the object are expressed as a system of inequalities:

Values V _{K MAX} ; a _{K MAX} ; V _{B MAX} ; a _{B MAX} ; Θ _MAX ; Ψ _MAX in the system of inequalities of generalized conditions for restricting the location of an object according to the data of navigation measurements, taking into account the influence of wind, can be written as the readings of airborne meters as follows:

Where

Λ _PR1 - the region of space predicted at time t ₁ that a moving body at time t ₂ through Δt can occupy lies within the cone formed by the tangents to the surfaces of the spheres Λ ₁ and Λ ₂ ;

further, the true, extreme, position of the course is increased by the angle of the maximum possible deviation of the object in time Δt from the pitch angles and the angle of the course, resulting in a predicted space Λ _{PR2 of the} possible location of the object at time t ₂ in the form of a convex elliptic cone with aperture angles θ _MAX and Ψ _MAX , moreover, the height of the specified cone is determined by the maximum possible heading speed of the object during Δt, and the true location of the object after the second navigation measurement through the interval Δt (pr winning space) is determined by the intersection of areas of space

the joint probability is p (0.95) × p (0.95) = 0.9025, and the payoff space Λ _PR2 ∩Λ ₂ ∩Λ _PR1 should be increased in all three coordinates x; y; z entering the space Λ ₂ as a percentage equal to the percentage of the random variable p (0.95) / s (0.9025), then the location of moving objects is corrected according to the results of three navigation measurements as follows:

according to the results of the first and second navigation measurements according to the above algorithm, Λ _{2 COR} is determined - the gain in determining the location after the second navigation measurement, while the correction is carried out according to speed and maximum deviation angles;

on the basis of the law of distribution of a random variable - errors of navigation measurements - determine the region of space Λ ' _2COR (p (0.95)) with a probability of at least 0.95, while the true position of the object Λ' _2COR (p (0.95)) is inside the volume of the space Λ ₂ ;

upon receipt of the data of the third navigation measurement - region Λ ₃ - determine the region Λ _{3 COR} - gain in determining the location after the third navigation measurement, using the readings of the second and third measurements as initial data, while the correction is carried out according to speed and maximum deviation angles, and as the initial for forecasting take the area of the adjusted space of the location of the object at the time of the second navigation measurement Λ ' _2KOR (p (0.95));

after which, on the basis of the law of distribution of a random variable — the error of navigation measurements — a region of space Λ ' _3COR (p (0.95)) is determined with a probability of at least 0.95, while Λ' _3COR (p (0.95)) is inside volume of space Λ ₃ ,

on the basis of all three navigational measurements, the Λ _{3KOR USK} region is again determined - the gain in determining the location after the third navigation measurement, while the correction takes place according to the acceleration values, and the Λ ' _{2KOR region} (p (0.95)) is taken as the initial for prediction the already adjusted volume of the space of the possible location of the object at the time of the second navigation measurement;

on the basis of the law of distribution of a random variable - the error of navigation measurements - determine the region of space Λ ' _{3 USK} (p (0.95)) with a probability of at least 0.95, while Λ' _{3 USK} (p (0.95)) is inside volume of space Λ ₃ ;

determine the intersection of the space volumes of the found adjusted volumes of the object location space Λ ' _3COR (p (0.95)) with the Λ' _{3COR USC} (p (0.95)), the intersection space of the indicated volumes Λ _{3COR FINISH.} will be the final corrected position of the object at time t ₃ ;

further, on the basis of the law of distribution of a random variable - the error of navigation measurements - determine the region of space Λ ' _{3COR OKON} (p (0.95)) with a probability of at least 0.95, while Λ' _{3KOR OKON} (p (0.95)) is inside the volume space Λ ₃ and for the space region of the third dimension, the values of the range of coordinate indices x are obtained; y; z (i ... j; r ... p; q ... f) entering the space of the object’s finally corrected position at time t ₃ Λ ' _{3 CORRECT OKON} (p (0.95)):

m = i ... j is the range of indices of the coordinate x included in the space Λ ' _{3 CORRECT OKON} ;

n = r ... p is the range of indices of the y coordinate that are included in the space Λ ' _{3 CORRECT OKON} ;

k = q ... f is the range of indices of the coordinate z included in the space Λ ' _{3 COR CORRECT} ;

as the final result of the location of the object in space with a probability of at least p (0.95), consider the geometric midpoint of the corrected region of the space Λ ' _{3 COR OKON} with the corrected coordinates x _{(ji) / 2} ; y _{(pr) / 2} ; z _{(fq) / 2} .

The essence of the above method is as follows.

- результаты навигационного измерения его местоположения в какой-то момент t. Пусть для навигационной системы погрешность измеряемой величины определена с вероятностью р(0,95), т.е. с вероятностью 95% можно говорить о том, что погрешность измерения навигационного параметра не превысит своих максимальных значений h_X; h_Y; h_Z, известных из тактико-технических характеристик навигационной системы.Let x, y, z, be the true coordinates of the object in the base coordinate system associated with the center of mass of the object, and $$\overline{x};\overline{y};\overline{z}$$

- the results of a navigation measurement of its location at some point t. Let for the navigation system the error of the measured quantity is determined with probability p (0.95), i.e. with a probability of 95% we can say that the measurement error of the navigation parameter does not exceed its maximum values h _X ; h _Y ; h _Z , known from the tactical and technical characteristics of the navigation system.

For example, for global satellite navigation systems GPS or GLONASS with a probability of at least p (0.95), it can be argued that the true value will differ from the measured value by no more than 8-10 meters in the horizontal plane and 1.5 ÷ 2 times higher error in the vertical plane.

The results of any navigation measurements can be presented as a spatial region, where with a probability p (0.95) the measurement error does not exceed the values of h _X ; h _Y ; h _Z. For global satellite navigation systems, such a region of space can be interpreted as a vertical ellipsoid. Suppose that during navigation measurements the following conditions are satisfied:

the error of each subsequent and the measurement values themselves do not depend on the results of previous measurements and their errors, they are statistically independent;

the error of each measurement does not depend on the location of the object in space, or this dependence can be neglected in the framework of several successive measurements of coordinates and assume that h _X ; h _Y ; h _Z = const for several successive navigation measurements.

, где m=0…M=2h_X/Δx; n=0…N=2h_Y/Δy; k=0…K=2h_Z/Δz; m; n; k -целые числа (индексы разбиения по координатам). Значения h_X ; h_Y; h_Z - максимальная погрешность измерения координат движущегося тела с вероятностью не менее p(0,95), т.е. $$\left|\overline{x}-x_1\right|<h_x;\left|\overline{y}-y\right|<h_y;\left|\overline{z}-z_1\right|<h_z$$

с вероятностью p(0,95); Δx;Δy;Δz-интервалы разбиения (можно трактовать как интервалы разбиения для обеспечения необходимой точности дискретизации пространства). Suppose that according to the data of the results of navigation measurements at time t, a region of space Λ is given. It can be represented in discrete form as $$\Lambda =\left\{(\overline{x}-h_X)+m\Delta x;(\overline{y}-h_Y)+n\Delta y;(\overline{z}-h_Z)+k\Delta z\right\}$$

where m = 0 ... M = 2h _X / Δx; n = 0 ... N = 2h _Y / Δy; k = 0 ... K = 2h _Z / Δz; m; n; k are integers (partition indexes by coordinates). Values h _X ; h _Y ; h _Z is the maximum measurement error of the coordinates of a moving body with a probability of at least p (0.95), i.e. $$|\; |\; |\overline{x}-x_{\mathrm{one}}|\; |\; |<h_x;|\; |\; |\overline{y}-y|\; |\; |<h_y;|\; |\; |\overline{z}-z_{\mathrm{one}}|\; |\; |<h_z$$

with probability p (0.95); Δx; Δy; Δz-partition intervals (can be interpreted as partition intervals to provide the necessary accuracy of space discretization).

Consider successive navigation measurements at time instants t ₁ -t _n defining regions of the space Λ ₁ -Λ _{n of} the object’s location at time instants t ₁ -t _n :

$${\Lambda }_{\mathrm{one}}=\left\{\begin{array}{l}({\overline{x}}_{\mathrm{one}}-h_X)+m\Delta x;\\ ({\overline{y}}_{\mathrm{one}}-h_Y)+n\Delta y;\\ ({\overline{z}}_{\mathrm{one}}-h_Z)+k\Delta z\end{array}\right\}$$

...

$${\Lambda }_n=\left\{\begin{array}{l}({\overline{x}}_n-h_X)+m\Delta x;\\ ({\overline{y}}_n-h_Y)+n\Delta y;\\ ({\overline{z}}_n-h_Z)+k\Delta z\end{array}\right\}$$

- данные навигационных измерений в момент времени t₁-t_n. $${\overline{x}}_{\mathrm{one}}\mathrm{...}{\overline{x}}_n,{\overline{y}}_{\mathrm{one}}\mathrm{...}{\overline{y}}_n,{\overline{z}}_{\mathrm{one}}\mathrm{...}{\overline{z}}_n$$

- data of navigation measurements at time t ₁ -t _n .

To predict the area of each subsequent navigational measurement Λ _OL , parameters are selected to limit the location of the object due to its dynamic properties, which can be: maximum speed and acceleration (depending on the tasks, there can be vertical and horizontal, directional and lowering, other linear and angular speed and acceleration); maximum change in azimuth (or yaw) angle; elevation angle (or roll) or other parameters.

For example, the following parameters can be used to limit movement: the heading speed V _{K MAX} is the maximum possible heading speed when moving an object (at each stage of the movement); maximum directional acceleration and _{K MAX} - the maximum possible acceleration of the object at the heading (at each of the stages of movement); maximum vertical speed - V V _MAX ; and _{in MAX, the} maximum possible vertical acceleration (at each of the stages of movement); the maximum possible change in the yaw angle ψ _MAX during the time Δt between subsequent navigation measurements; the maximum possible change in pitch angle θ _MAX during the time Δt between subsequent navigation measurements; if necessary, it is possible to use the maximum roll angle γ _MAX for the time Δt between subsequent navigation measurements.

Generalized conditions for restricting the location of the object according to the data of navigation measurements and dynamic properties of the object can be written as follows:

(1)

(one)

wherein :

(2)\tab\tab\tab\tab

(2) \ tab \ tab \ tab \ tab

t ₁ -t ₃ - time instants of three consecutive navigation measurements.

These inequalities should be interpreted as follows.

The first inequality is the speed the object travels for the time Δt between the first and second dimensions cannot exceed the maximum possible heading speed.

The second inequality is similar, but between the second and third dimension.

The third inequality is the course acceleration of the object should not exceed the maximum possible.

The fourth to sixth inequality is similar to 1-3 inequalities for vertical speed and acceleration.

The seventh-eighth inequality determines the maximum possible angle of inclination of the object from the vertical (which actually determines the pitch angle) for the second and third dimensions.

The ninth to tenth inequality determines the maximum possible deviation angle for the second and third dimensions.

The maximum values of possible speeds, accelerations and deviation angles in expressions (1) - (2) should take into account the strength of the wind and its effect on a moving object. For specific technical tasks and objects, the conditions for restricting the location of an object due to its dynamic properties may have a slightly different look.

As the maximum possible speeds, accelerations and deviation angles, you can take both physically impossible for a moving object (for example, for a car), or not recommended at this stage of movement (for example, flight stage) for reasons of traffic safety (for example, critical angles of attack, glide paths, critical rate of descent, etc.). As a rule, for example, on board an aircraft there are warning systems about critical modes of movement (exceeding the maximum values of speeds, accelerations and deviation angles). Due to this, with a probability of at least p (0.95), it can be argued that the object does not exceed the specified critical values.

To increase the accuracy of the location of the object in space, it is possible to use not the limiting values of speeds, accelerations and changes in the angular positions of the object, but the current real data of the on-board meters (BI) of the object's motion parameters (speeds, angles, accelerations). Maximum parameters V _{K MAX} ; a _{K MAX} ; V _{B MAX} ; a _{B MAX} ; Θ _MAX ; Ψ _{MAX is} determined by the current readings of the on-board meters, their errors, the readings of the wind sensors and the direction of the wind, as well as the increment of the measured parameters during the time between the navigation measurement and the closest information received from the on-board meter.

In this case (and taking into account the influence of the wind), the values of V _{K MAX} ; a _{K MAX} ; V _{B MAX} ; a _{B MAX} ; Θ _MAX ; Ψ _MAX in the system of inequalities (1) of the general conditions for restricting the location of the object according to the data of navigation measurements can be written as the readings of on-board meters as follows:

(3)

(3)

Where

In these expressions, the values of the errors in the readings of the on-board meters and the frequency of updating the information on-board meters are determined by the tactical and technical characteristics (TTD) of specific devices. Often, the value of measurement errors in the TTD is expressed as a percentage of the values of the measured parameters.

Thus, using restrictions on the ability to move an object (1) - (3), it is possible to predict the area of the space of the possible location of each subsequent after the initial navigation measurement.

и t₂: $${\overline{x}}_2;{\overline{y}}_2;{\overline{z}}_2$$

в базовой системе координат и соответственно области пространства возможных точных измерений координат Δ₁ и Δ₂ (см. фиг.1). Λ_ПР1 - прогнозируемая в момент t₁ область пространства, которую может занимать движущееся тело в момент времени t₂ через Δt (интервал обновления навигационной информации) ввиду своих динамических свойств по возможности перемещения. Λ₁ ∈Λ_ПР1, т.е. Λ_ПР1 вбирает в себя Λ₁. Так как после первого измерения курс объекта не очевиден, то пространство Λ_ПР1 определяется максимально возможной скоростью передвижения объекта за время Δt.Let two navigation measurements be known at time t_one: $${\overline{x}}_{\mathrm{one}};{\overline{y}}_{\mathrm{one}};{\overline{z}}_{\mathrm{one}}$$

 and t₂: $${\overline{x}}_2;{\overline{y}}_2;{\overline{z}}_2$$

 in the base coordinate system and, accordingly, the space region of possible accurate measurements of coordinates Δ_oneand Δ₂(see figure 1). Λ_PR1 - predicted at time t_one the region of space that a moving body can occupy at time t₂ through Δt (interval for updating navigation information) due to its dynamic properties as far as possible. Λ_one ∈Λ_PR1, i.e. Λ_PR1 incorporates Λ_one. Since the course of the object is not obvious after the first measurement, the space Λ_PR1 is determined by the maximum possible speed of the object in time Δt.

и $${\overline{x}}_2;{\overline{y}}_2;{\overline{z}}_2$$

, является псевдокурсом. Истинное максимально возможное (с точностью р(0,95)) положение курса лежит в пределах конуса, образованного касательными к поверхностям сфер Λ₁ и Λ₂, как это представлено на фиг.1. Для простоты графического представления, будем считать, что h_X=h_Y=h_Z. и Λ₁; Λ₂; Λ₃ являются сферами (хотя для глобальных спутниковых навигационных систем правильно было бы считать, что h_X=h_Y, а h_Z≈1,5÷2 h_X). The direction defined by the line connecting the points of navigational measurements $${\overline{x}}_{\mathrm{one}};{\overline{y}}_{\mathrm{one}};{\overline{z}}_{\mathrm{one}}$$

 and $${\overline{x}}_2;{\overline{y}}_2;{\overline{z}}_2$$

 is a pseudo-course. The true maximum possible (with accuracy p (0.95)) position of the course lies within the cone formed by the tangents to the surfaces of the spheres Λ_oneand Λ₂, as shown in figure 1. For simplicity of the graphical representation, we assume that h_X= h_Y= h_Z. and Λ_one; Λ₂; Λ₃ are spheres (although for global satellite navigation systems it would be correct to assume that h_X= h_Y, and h_Z≈1.5 ÷ 2 h_X)

The true, extreme position of the course must be increased by the angle of the maximum possible (for the time Δt) deviation of the object at pitch angles and course angle. Consider the section of the spheres Λ ₁ and Λ ₂ relative to the axis of the pseudo-course in the horizontal and vertical plane. As a result, we obtain the predicted space Λ _{PR2 of the} possible location of the object at time t ₂ in the form of a convex elliptic cone (the opening angles are determined by θ _MAX and ψ _MAX ) _. The height of this cone is determined by the maximum possible heading speed of the object during the time Δt. The true location of the object after the second navigation measurement through the interval Δt will be determined not by the region Λ ₂ , but by the intersection of the regions of the spaces Λ _PR2 ∩Λ ₂ ∩Λ _PR1 .

The following cases of predicting the position of the object are possible, they are schematically shown in figures 1-3.

In the first case (figure 1), the region Λ ₂ completely belongs to the region Λ _PR2 , and there is no gain in determining the location of a moving object.

In the second case (figure 2) Λ _PR2 ∩Λ ₂ ∩Λ _PR1 <Λ ₂ , and there is a gain in determining the location of a moving object.

In the third case (figure 3), the intersection of Λ _PR2 and Λ ₂ is an empty set. In this case, the result of the second navigation measurement is not consistent with the data predicted by the dynamic characteristics of the location of the object. This means that either the error in this navigation measurement is large (falls out of the confidence interval in p (0.95)), or the dynamic properties of a moving object are incorrectly (or with a large error). The probability that both navigation measurements at times t ₁ and t ₂ lie in the interval p (0.95) is somewhat lower. Since in the initial conditions it is assumed that navigation measurements at different times are statistically independent, the joint probability is p (0.95) x p (0.95) = 0.9025. If you know the distribution law of the random value of the error in navigation measurements, then the gain space Λ _PR2 ∩Λ ₂ ∩Λ _PR1 should be increased (in all three coordinates x; y; z, but so that they fit into the space Λ ₂ ) in a percentage ratio the percentage of a random variable p (0.95) / p (0.9025). This somewhat reduces the gain in determining the location of the object, but the adjusted volume of the space of the possible location of the moving object will be determined with a probability of at least p (0.95). An example of the resulting volume of space is shown in figure 4.

We denote the gain in positioning as Λ _2COR = Λ _PR2 ∩Λ ₂ ∩Λ _PR1 and Λ ' _2COR (p (0.95)) - the gain in location with a probability of at least 0.95.

For practical implementation, it is proposed to use the following algorithm for correcting the location of moving objects based on the results of three navigation measurements:

1. Based on the results of the first and second navigation measurements using the above algorithm, we determine Λ _2KOR - the gain in determining the location after the second navigation measurement. Moreover, to predict the location we use the readings of airborne meters based on inequalities (1) - (3). Correction occurs in speed and maximum deflection angles.

2. Based on the distribution law of the random variable, the error of navigation measurements, we determine the region of space Λ ' _{2 COR} (p (0.95)) with a probability of at least 0.95. Moreover, Λ ' _{2 COR} (p (0.95)) is inside the volume of the space Λ ₂ .

3. Upon receipt of the data of the third navigation measurement (region Λ ₃ ), similarly to point 1, we determine the region Λ _3COR — a gain in determining the location after the third navigation measurement, using the readings of the second and third measurements as initial data. Correction occurs in speed and maximum deflection angles. The difference from point 1 is that the region not Λ ₂ , but Λ ' _2COR (p (0.95)) is taken as the initial one for forecasting - the already adjusted volume of the space of the possible location of the object at the time of the second navigation measurement.

4. Based on the law of distribution of a random variable - the error of navigation measurements, we define the space region Λ ' _3COR (p (0.95)) with a probability of at least 0.95. Moreover, Λ ' _{3 COR} (p (0.95)) is inside the volume of the space Λ ₃ .

5. Based on all three navigation measurements, we again define the region Λ _{3KOR USK} - the gain in positioning after the third navigation measurement. In this case, the correction occurs according to the values of accelerations. The area not Λ ₂ but Λ ' _2COR (p (0.95)) is taken as the initial one for forecasting — the already adjusted volume of the space of the possible location of the object at the time of the second navigation measurement.

6. Based on the law of distribution of a random variable - the error of navigation measurements, we define the region of space Λ ' _{3KOR USK} (p (0.95)) with a probability of at least 0.95. Moreover, the Λ ' _{3 COR of USC} (p (0.95)) is located inside the volume of the space Λ ₃ .

7. We define the intersection of the volumes of space found in paragraphs 4 and 6, that is, Λ ' _3COR (p (0.95)) ∩ Λ' _{3COR USC} (p (0.95)) = Λ _{3COR FINISH.} -final (according to the proposed algorithm) adjusted position of the object at time t ₃ .

8. On the basis of the law of distribution of a random variable - the error of navigation measurements, we define the region of space Λ ' _{3 COR OKON} (p (0.95)) with a probability of at least 0.95. Thus Λ _{'3KOR WINDOWS} (p (0.95)) is within the volume space Λ _3.

The proposed algorithm is one of the options for correcting the location of the object based on data from navigation measurements and imposing restrictions associated with the dynamic properties of a moving object. This method of determining the location of an object in space, refining navigation data by imposing restrictions associated with the dynamic properties of a moving object, can be called the dynamic recurrent correction method.

To increase the accuracy of determining the location of a moving object, you can use not only the first three measurements, but also the results of the subsequent fourth and fifth measurements. In this case, the first correction will be delayed by 5 steps (5Δt), and each subsequent measurement result will go with a delay of 2 steps (2Δt), where Δt is the update frequency of the navigation information.

Using the method of determining the location of an object in space, updating the navigation data due to the imposition of restrictions associated with the dynamic properties of a moving object, depends on the accuracy characteristics of the navigation system, the accuracy of the readings of on-board meters of motion parameters and wind speed and force.

The conditions for the ability to use dynamic recurrence correction can be considered an excess of the accuracy of determining the motion parameters compared to the accuracy of determining the location of the navigation system (figure 5):

Thus, gaining from the use of dynamic recurrent correction is possible for different values of the speeds and lateral displacements of the object, if it is possible to provide accurate forecasting (accurate measurements of its motion parameters).

There may be situations depicted in figure 3, when the data of subsequent navigational measurements are not consistent with the data predicted by the dynamic characteristics of the location of the object. In this case, with probability P (95%) it can be argued that such a measurement value should not be. However, such cases are possible due to failures of navigational aids, as well as during the passage of aircraft through the "affected areas", where due to re-reflections or other destabilizing factors, the measurement error increases sharply. Some similar cases can be considered misses, but when they are repeated in the sample of accumulated values of navigation measurements of a more critical value, the navigation system is considered to be failed. For these purposes, you can use any statistical evaluation criterion, with the known statistical law of the probability distribution of the error of navigation measurements.

By comparing and statistically analyzing the space region of consecutive navigational measurements of the object’s location and the predicted space region of the possible object’s location (based on its dynamic properties of moving in space) at the time of the next measurement, we can draw conclusions about the reliability of the navigation aids.

Similarly, it is possible to identify errors in the readings of airborne meters of the parameters of the object’s movement (for example, DISS, bar altimeter, radio altimeter, etc.). For practical implementation, we can propose the following procedure for finding the space region of the possible location of a moving object using dynamic constraint conditions (see Fig. 6).

We define three-dimensional matrices A = {a _m ; a _n ; a _k }; B = {b _m ; b _n ; b _k }; C = {c _m ; c _n ; c _k } corresponding to regions Λ ₁ ; Λ ₂ ; Λ ₃ ; whose elements take values 0 or 1. Each element of the matrix A; B; C means a point in space when dividing the domains Λ ₁ ; Λ ₂ ; Λ in increments of Δx; Δy; Δz. Values of matrices A; AT; With "0" or "1" (1 - the object can be at a given point; 0 - can not). At the initial step of the calculations (obtaining the initial data), all matrix elements are considered equal to 1 (unadjusted values of the navigation measurement).

. In the basic procedure, all the possible values of the matrices A, B, C (points of the spaces Λ ₁ ; Λ ₂ ; Λ ₃ ) that satisfy the conditions of dynamic constraints (1) - (3) are simply enumerated. The calculation result is matrices B or C (depending on step 1-8 of the correction algorithm above), such that its elements B = {b _m ; b _n ; b _k } or C = {c _m ; c _n ; c _k } determine the ability to be in space $$\Lambda =\left\{(\overline{x}-h_x)+m\Delta x;(\overline{y}-h_y)+n\Delta y;(\overline{z}-h_z)+k\Delta z\right\}$$

.

In the example of the algorithm implementation (see Fig. 7) of the correction, the basic computational procedure for the second and subsequent calculation cycles should be modified. The meaning of the modification is that it does not check all the conditions of the equations of dynamic correction or not for the entire area of the space of the previous navigation dimension, but for the space of the previous dimension, already limited by the previous cycle of calculations. This reduces the number of calculations.

The gain in determining the location of the object can be estimated as the ratio of the number of matrix elements that are not equal to zero to the total number of matrix elements.

The proposed algorithm and basic computational procedure are only one of the possible options for implementing dynamic recurrent correction. The real algorithm for a particular object can take into account only part of the proposed correction conditions (1) - (3) or take into account its own conditions.

The proposed algorithm involves modifications of the basic computational procedure (enumeration of points in a region of space) in the primary and subsequent measurement cycles. In subsequent cycles of calculations, the region of the space of a possible location, already corrected by previous measurements, is taken as the initial region for forecasting. The algorithm for implementing dynamic recurrent correction allows you to process data from BI, wind sensors and a navigation system in real time as they arrive.

In practical implementation, it is necessary to provide for the possibility of stopping the supply of data from the navigation system. When the receipt of navigation data is stopped for a period longer than the update of the navigation information, the calculations must be started anew, from the initial calculation cycle.

When implementing the basic algorithm in real time, it is necessary to perform the entire volume of calculations in a time shorter than the minimum time for updating the information of the initial calculation data. To reduce the computational complexity of the basic algorithm, reduce the number of calculations and implement it in real time, you can optimize it as follows:

sequentially reduce the space partition interval of the possible location of the object Δx; Δy; Δz. Initially, you can make a calculation with a large split, then at the border of the space region of the specified location of the object, make a calculation with a smaller split;

use (or not use) the conditions for limiting the location of the object associated with acceleration. If a priori it is known that a moving object has positive accelerations (linear or angular), then the equations related to acceleration can not be used. Then in the basic computational procedure of the basic algorithm there will be not nine, but six cycles of calculation.

The incoming data from the on-board meters, wind sensors and the navigation system can be discrete with a certain update frequency, then it is necessary to carry out calculations at intervals of the minimum frequency of update information on-board meters. If the data streams from the airborne meters are analog, then changes to such data and recalculations can be carried out at a given threshold for changing such data (when converting the data format, set either threshold or differential restrictions). If the data streams are discrete and random (variable time of receipt of information), then changes to such data and recalculations must be carried out immediately upon their arrival (in real time).

As an example, one of the variants of the block diagram of the device is proposed, with which it is possible to implement a method of dynamic recurrent correction (Fig. 8).

Data from airborne meters, wind sensors, and the navigation system are fed through separate channels to the corresponding data format conversion blocks 1, 2, and 3, respectively. Blocks 1-3 converts data from on-board meters, wind sensors and a navigation system to be able to process numerical data in the on-board computer unit 6. Then, through separate channels, the converted data is sent to the unit of the time interval of data 5, in which time information is generated the intervals of data arrival, as well as all data are synchronized in time using the block synchronization of the calculation of time 4. From the block of the meter 5, the data are received in block b Orth calculator 6 (a solver according to the dynamic correction algorithm), which implements a dynamic correction algorithm for the location of a moving object. From block 6, the processed data is output, and also recorded in the memory buffer, which already stores data from previous measurements.

To convert the data format (by level and presentation format) depending on the type of on-board meter, ADCs can be used (for analog output signals from meters) or, if necessary, interface converters (for example, RS485 / RS232 or similar). As meters of time intervals, it is possible to use counters with clock generators and clamps for the temporary position of the pulse. As an on-board computer, an on-board computer or a separate mini-computer is used.

In the process of calculation (according to the above algorithm of moving objects locations correcting the results of three navigation measurements), we obtain a region of space Λ _{'3KOR WINDOWS} (p (0.95)), which is located within the volume space Λ _3, wherein for the third measuring space region obtained values:

m = i ... j - index range coordinates x, incoming into the space Λ _{'3KOR WINDOWS;}

n = r ... p - index range coordinates y, outside the space Λ _{'3KOR WINDOWS;}

k = q ... f - range of indices coordinates z, within the space of Λ _{'3KOR windows.}

окончательными выходными скорректированными данными третьего измерения (на выходе бортового вычислителя 6 по схеме на фиг.8) будут координаты $$x_{(j-i)/2};y_{(p-r)/2};z_{(f-q)/2}$$

.In such a case, the final result of processing improve accuracy navigation measurement with probability at least p (0.95) can be considered a geometrical center of the corrected area space Λ _{'3KOR WINDOWS,} i.e. instead of the values of the third dimension $${\overline{x}}_3;{\overline{y}}_3;{\overline{z}}_3$$

the final output adjusted data of the third dimension (at the output of the on-board computer 6 according to the scheme in Fig. 8) will be the coordinates $$x_{(j-i)/2};y_{(p-r)/2};z_{(f-q)/2}$$

.

The structural diagram (Fig. 8) can be considered as a scheme for integrating on-board meters of the object’s motion parameters with the object’s navigation system.

Thus, the use of the method for determining the location of an object in the space according to the invention, clarifying the navigation data by imposing restrictions associated with the dynamic properties of a moving object, allows you to:

- increase the accuracy of the location of the object in space;

- does not require additional navigational measurements, or reference points with previously known geodetic coordinates, or an a priori known trajectory of the object;

- can be used for various values of speeds and angular displacements of the object;

- is algorithmic and does not require complex computational procedures;

- allows you to integrate equipment on-board measuring instruments for motion parameters and navigation equipment. It allows to increase the reliability of failure detection on-board measuring instruments for motion parameters and navigation equipment.

Claims (1)

A method for determining the location of a moving object in space, characterized in that at least three navigation measurements are carried out, as a result of which the coordinates are determined $$\overline{x}$$

, $$\overline{y}$$

, $$\overline{z}$$

the center of mass of the moving object at time t ₁ -t _n , according to the results of navigation measurements, determine the region Λ ₁ -Λ _{n of the} space in which the object is located at time t ₁ -t _n :

where m = 0 ... M = 2h _X / Δx; n = 0 ... N = 2h _Y / Δ _y ; k = 0 ... K = 2h _Z / Δz; m; n; k are integers (partition indexes by coordinates),
h _X ; h _Y ; h _Z is the maximum measurement error of the coordinates of a moving body with a probability of at least p (0.95), i.e. $$|\; |\; |\overline{x}-x_{\mathrm{one}}|\; |\; |<h_x$$

; $$|\; |\; |\overline{y}-y|\; |\; |<h_y$$

; $$|\; |\; |\overline{z}-z_{\mathrm{one}}|\; |\; |<h_z$$

with probability p (0.95);
Δx; Δy; Δz - partition intervals to ensure the necessary accuracy of the discretization of space,
$${\overline{x}}_{\mathrm{one}}...{\overline{x}}_n$$

, $${\overline{y}}_{\mathrm{one}}...{\overline{y}}_n$$

, $${\overline{z}}_{\mathrm{one}}...{\overline{z}}_n$$

- data of navigation measurements at time t ₁ -t _n ,
x ₁ ... x _n , y ₁ ... y _n , z ₁ ... z _n - data of the true position of the object at time t ₁ -t _n ,
while navigational measurements determine the following conditions:
- the error of each subsequent measurement and the measurement values themselves do not depend on the results of previous measurements and their errors, they are statistically independent;
- the error of each measurement does not depend on the location of the object in space, or this dependence can be neglected within several consecutive measurements of coordinates and assume that h _X ; h _Y ; h _Z = const for several consecutive navigation measurements,
- then determine the predicted area of space that the object occupies through the time interval Δt, taking into account the parameters of the restriction of movement, which are determined by means of on-board meters located on the moving object, while the following are taken as the restriction parameters:
- heading speed V _{K MAX} - the maximum possible heading speed when moving the object (at each stage of the movement);
- maximum course acceleration and _{K MAX} - the maximum possible acceleration of the object at the heading (at each of the stages of movement);
- maximum vertical speed - V _{B MAX} ;
- a _{B MAX} - the maximum possible vertical acceleration (at each stage of the movement);
- the maximum possible change in the yaw angle ψ _MAX during the time Δt between subsequent navigation measurements;
- the maximum possible change in pitch angle θ _MAX during the time Δt between subsequent navigation measurements,
and generalized conditions for restricting the location of the object according to the data of navigation measurements and dynamic properties of the object are expressed as a system of inequalities:

wherein:

V _{to MAX} values; a _{To MAX} ; V _{B MAX} ; a _{B MAX} ; Θ _MAX ; ψ _MAX in the system of inequalities of generalized conditions for restricting the location of an object according to the data of navigation measurements, taking into account the influence of wind, can be written as the readings of on-board meters as follows:

Where
$$V_{\mathrm{TO}}{}_{B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_K{}_{B\mathrm{AND}}\ge t_2}$$

; $$V_{\mathrm{TO}}{}_{B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_K{}_{B\mathrm{AND}}\ge t_3}$$

- the value of the heading speed according to the readings of the on-board meter at the time points of receipt of information from the on-board meter immediately after navigation measurements at moments t ₂ and t ₃ ;
$$V_{\mathrm{TO}}{}_{\mathrm{AT}ET}|\; |\; |{{}_{t=t}}_{{}_K{}_{\mathrm{AT}ET}\ge t_2}$$

$$V_{\mathrm{TO}}{}_{\mathrm{AT}ET}|\; |\; |{{}_{t=t}}_{{}_K{}_{\mathrm{AT}ET}\ge t_3}$$

- component of the heading speed due to the influence of wind (according to wind sensors) immediately after navigation measurements at moments t ₂ and t ₃ ;
ΔV _{K BI} - the error of the readings of the on-board heading speed meter;
$$V_{\mathrm{AT}}{}_{B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_{\mathrm{TO}}{}_{B\mathrm{AND}}\ge t_2}$$

; $$V_{\mathrm{AT}}{}_{B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_{\mathrm{TO}}{}_{B\mathrm{AND}}\ge t_3}$$

; $$V_{\mathrm{AT}}{}_{\mathrm{AT}ET}|\; |\; |{{}_{t=t}}_{{}_{\mathrm{TO}}{}_{\mathrm{AT}ET}\ge t_2}$$

; $$V_{\mathrm{AT}}{}_{\mathrm{AT}ET}|\; |\; |{{}_{t=t}}_{{}_{\mathrm{TO}}{}_{\mathrm{AT}ET}\ge t_3}$$

; ΔV _{IN BI} - values of the vertical speed according to the readings of the on-board meter, data from wind sensors and the error of the readings of the on-board meter of vertical speed, while all the values are similar to the values of the above values;
$$\Delta t_{\mathrm{TO}B\mathrm{AND}}=\left(t_{\mathrm{TO}B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_{\mathrm{TO}}{}_{B\mathrm{AND}}\ge t_3}-t_{\mathrm{TO}B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_{\mathrm{TO}}{}_{B\mathrm{AND}}}{}_{\ge t_2}\right)$$

- the frequency of updating the readings of the on-board heading speed meter;
$$\Delta t_{\mathrm{AT}B\mathrm{AND}}=\left(t_{\mathrm{AT}B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_{\mathrm{AT}}{}_{B\mathrm{AND}}\ge t_3}-t_{\mathrm{AT}B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_{\mathrm{AT}}{}_{B\mathrm{AND}}}{}_{\ge t_2}\right)$$

- the frequency of updating the readings of the onboard vertical speed meter;
$${\Theta }_{B\mathrm{AND}}{}_{|\; |\; |t=t_{\Theta }{}_{B\mathrm{AND}}\ge t_2}$$

; $${\Theta }_{B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_{\Theta }{}_{B\mathrm{AND}}}{{}_{\ge t}}_{{}_3}$$

- the value of the angle of inclination of the object from the vertical (which actually determines the pitch angle) according to the readings of the on-board meter immediately after navigation measurements at moments t ₂ and t ₃ ;
$${\Theta }_{{}_{\mathrm{AT}ET}}{}_{|\; |\; |t=t_{\Theta }{}_{B\mathrm{AND}}\ge t_2}$$

; $${\Theta }_{\mathrm{AT}ET}|\; |\; |{{}_{t=t}}_{{}_{\Theta }{}_{B\mathrm{AND}}}{}_{\ge t_3}$$

- component of the angle of inclination of the object from the vertical due to the influence of wind (according to wind sensors) immediately after navigation measurements at moments t ₂ and t ₃ ;
ΔΘ _BI - the error of the on-board meter measuring the angle of inclination of the object from the vertical;
$${\Theta }_{B\mathrm{AND}}{}_{|\; |\; |t=t_{\Theta }{}_{B\mathrm{AND}}\ge t_2}$$

; $${\Theta }_{B\mathrm{AND}}|\; |\; |{{}_{t=t}}_{{}_{\Theta }{}_{B\mathrm{AND}}}{}_{\ge t_3}$$

; $${\Psi }_{{}_{\mathrm{AT}ET}}{}_{|\; |\; |t=t_{\psi }{}_{B\mathrm{AND}}\ge t_2}$$

; $${\Psi }_{\mathrm{AT}ET}|\; |\; |{{}_{t=t}}_{{}_{\psi }{}_{B\mathrm{AND}}}{}_{\ge t_3}$$

; Δψ _BI - the value of the angle of deviation according to the readings of the airborne meter, data from wind sensors and the error of the readings of the airborne meter of the angle of deviation;
Λ _PR1 - the region of space predicted at time t ₁ that a moving body at time t ₂ through Δt can occupy lies within the cone formed by the tangents to the surfaces of the spheres Λ ₁ and Λ ₂ ;
further, the true, extreme, position of the course is increased by the angle of the maximum possible deviation of the object in time Δt from the pitch angles and the angle of the course, resulting in a predicted space Λ _{PR2 of the} possible location of the object at time t ₂ in the form of a convex elliptic cone with aperture angles θ _MAX and ψ _MAX , and the height of the specified cone is determined by the maximum possible heading speed of the object during Δt, and the true location of the object after the second navigation measurement through the interval Δt (pr winning space) is determined by the intersection of areas of space
Λ _{2 COR} = Λ _PR2 ∩Λ ₂ ∩Λ _PR1 ,
the joint probability is p (0.95) × p (0.95) = 0.9025, and the gain space Λ _PR2 ∩Λ ₂ ∩Λ _PR1 should be increased in all three coordinates x; y; z entering the space Λ ₂ as a percentage equal to the percentage of the random variable p (0.95) / p (0.9025), then the location of moving objects is corrected according to the results of three navigation measurements as follows:
according to the results of the first and second navigation measurements according to the above algorithm, Λ _{2 COR} is determined - the gain in determining the location after the second navigation measurement, while the correction is carried out according to speed and maximum deviation angles;
on the basis of the law of distribution of a random variable - the error of navigation measurements - determine the region of space Λ ′ _{2 COR} (p (0.95)) with a probability of at least 0.95, while the true position of the object Λ ′ _{2 COR} (p (0.95)) is inside the volume of the space Λ ₂ ; upon receipt of the data of the third navigation measurement - region Λ ₃ - determine the region Λ _{3 COR} - gain in determining the location after the third navigation measurement, using the data of the second and third measurements as initial data, while the correction is carried out according to speed and maximum deviation angles, and as the initial one for forecasting is the area of the corrected location space of the object at the time of the second navigation measurement Λ ′ _2COR (p (0.95));
after which, on the basis of the distribution law of a random variable — the error of navigation measurements — a space region Λ ′ _3COR (p (0.95)) is determined with a probability of at least 0.95, while Λ ′ _3COR (p (0.95)) is inside volume of space Λ ₃ ,
on the basis of all three navigation measurements, the Λ _{3KOR USK} region is again determined - the gain in determining the location after the third navigation measurement, while the correction takes place according to the acceleration values, and the Λ ′ _{2KOR region} (p (0.95)) is taken as the initial one for forecasting - the already adjusted volume of the space of the possible location of the object at the time of the second navigation measurement;
on the basis of the law of distribution of a random variable - errors of navigation measurements - determine the region of space Λ ′ _{3KOR USK} (p (0.95)) with a probability of at least 0.95, while Λ ′ _{3KOR USK} (p (0.95)) is inside volume of space Λ ₃ ;
determine the intersection of the space volumes of the found adjusted volumes of the object location space Λ ′ _3COR (p (0.95)) with Λ ′ _{3COR USC} (p (0.95)), the intersection space of the indicated volumes Λ _{3COR OKON} will be the final adjusted position of the object at the moment time t ₃ ;
further, on the basis of the law of distribution of a random variable - the error of navigation measurements - determine the space region Λ ′ _{3COR OKON} (p (0.95)) with a probability of at least 0.95, while Λ ′ _{3KOR OKON} (p (0.95)) is inside the volume space Λ ₃ , and for the region of the space of the third dimension, the values of the range of coordinate indices x are obtained; y; z (i ... j; r ... p; q ... f) that enter the space of the object’s finally corrected position at time t ₃ Λ ′ _{3 CORRECT WINDOW} (p (0.95)):

m = i ... j is the range of indices of the coordinate x included in the space Λ ′ _{3 COR OKON} ;
n = r ... p is the range of indices of the coordinate y included in the space Λ ′ _{3 CORRECT OKON} ;
k = q ... f is the range of indices of the coordinate z included in the space Λ ′ _{3 COR OKON} ,
as the final result of the location of the object in space with a probability of at least p (0.95), consider the geometric midpoint of the corrected region of the space Λ ′ _{3 COR OKON} with the adjusted coordinates x _{(ji) / 2} ; y _{(pr) / 2} ; z _{(fq) / 2} .

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