RU2578671C1 - Method of determining angular orientation in global radio navigation systems - Google Patents

Abstract

FIELD: space.

SUBSTANCE: invention relates to navigation spacecraft signals of global navigation satellite systems and can be used for determining angular orientation in space. Disclosed method enables determination of angular orientation by results of measurements of phase shifts of signals from spacecraft number not less than three. Method is implemented by selecting integer ambiguities of measured phase shift, for which there is a direction-finding of spacecraft in two bases. Unknown value of angular orientation of object is determined based on the criterion of maximum likelihood function, which takes into account the position of spacecraft, vectors, which connect the antenna, bearings spacecraft signals and their mutual connection.

EFFECT: technical result is fewer passes possible values of parameters of phase ambiguity.

1 cl, 6 dwg

Description

The present invention relates to the field of navigation by signals from spacecraft of global radio navigation satellite systems and can be used to determine the angular orientation of the aircraft in space. Known methods for determining the orientation of objects from the signals of the spacecraft global radio navigation systems, based on the reception of signals from spacecraft with antennas of at least three located on the object so that they do not lie on one straight line, while the antennas receive signals from spacecraft, based which determine the phase differences of the carrier frequency of the signals received at the separated antennas, in which (phase differences) contains information about the angles between the directions to the space apparatuses and vectors formed by antennas, on the basis of which the orientation problem is solved using information about the location of the antennas relative to the object, while the antennas receive signals from three spacecraft, and information about the coordinates of the object and spacecraft is known [1, 2, 3].

The disadvantage of this method is the need for simultaneous measurements of phase shifts between spaced antennas of the object, which increases the time required to determine the angular position.

A known method for determining the angular orientation of an object from the signals of global satellite radio navigation satellite systems [4], based on the reception of signals from spacecraft global satellite radio navigation satellite systems on spaced antennas, measuring the phase shift between the received signals from each spacecraft, selecting the values of integer ambiguities of phase measurement measurements shifts, solving a system of equations for determining the angular orientation, while the desired angular orientation of The object is determined from the condition of the maximum likelihood function.

The disadvantage of this method is to view a large number of combinations of integer ambiguities of phase shift measurements, which does not take into account information about the position of spacecraft, vectors connecting antennas.

The objective of the invention is to reduce the number of combinations when using the exhaustive method of resolving phase ambiguity in determining the angles between the directions to the spacecraft and the vectors formed by the lines connecting the antennas, and using the likelihood function, which takes into account the position of the spacecraft and the vectors formed by the lines connecting the antennas.

The problem is solved in that in the method for determining the angular orientation of the signals from spacecraft global satellite navigation systems, based on the reception of signals from spacecraft to antennas, at least three in number, located on the object so that they do not lie on one straight line, measuring phase the shift between the received signals from each spacecraft, the selection of integer ambiguities, which allows to determine the possible values of the angular orientation, determining the value of the claim According to the invention, the possible values of the angular orientation are determined using the measured phase shifts and expressions that take into account the position of the spacecraft, the vectors formed by the lines connecting the antennas, and their mutual connection.

The proposed method is illustrated by figures 1 ÷ 6.

In FIG. 1 shows a rectangular Earth coordinate system and shows the relationship of rectangular and geodetic coordinates.

FIG. 2 illustrates the orientation of the normal Earth coordinate system relative to the Earth system through geodetic coordinates.

In FIG. 3 shows the orientation of the antennas with respect to the direction finder coordinate system.

In FIG. 4 shows a direction finding diagram of two radiation sources.

In FIG. 5 shows the orientation of the direction finder coordinate system with respect to the associated aircraft coordinate system.

FIG. 6 illustrates the procedure for determining heading, pitch, and roll angles in a normal earth coordinate system.

The angular orientation of the object through the angles of roll, course and pitch is determined in the normal (earth) coordinate system in accordance with [5]. The normal earth coordinate system is connected with the earth coordinate system, in which the coordinates of objects in the GNSS environment are determined.

Earth coordinate system OXYZ is a right-angled Cartesian coordinate system whose origin and axes are fixed with respect to the Earth. The origin of the coordinate system is located in the center of mass of the Earth, the OZ axis is directed to the International Conditional Origin (the conventional Earth pole, as defined by the recommendations of the International Earth Rotation Service), the OX axis lies in the plane of the initial astronomical meridian established by the International Bureau of Time (at the intersection of the equator and initial meridian), the OY axis complements the system to the right (PZ-90.11). In this coordinate system, the position of a point in space is determined by the values of the X, Y, Z coordinates (see Fig. 1). The geodetic coordinates B, L, and H refer to a common ellipsoid. In this case, the center of the Earth-wide ellipsoid coincides with the center of mass of the Earth, its minor axis (rotation axis) is aligned with the OZ axis, and the OX and OY axes are located in the equatorial plane so that the OX axis passes through the initial meridian. The Earth-wide coordinate system rotates with the Earth. Geodetic latitude B is defined as the angle between the normal to the ellipsoid passing through the given point and the equator plane; geodetic longitude L - dihedral angle between the plane of the initial meridian and the plane of the meridian passing through the point (positive direction of counting longitudes from the initial meridian to the east, from 0 ° to 360 °); the geodesic height H is the segment of the normal to the common terrestrial ellipsoid from its surface to the point. Positive heights correspond to points located above the ellipsoid.

The normal (terrestrial) coordinate system O _g X _g Y _g Z _g is the earth coordinate system whose axis O _g Y _{g is} directed upward along the local vertical and the axis O _g Z _{g is} directed east, the axis O _g X _g coincides with the direction to north. The local vertical is understood as a straight line coinciding with the direction of gravity at the point in question (see Fig. 2).

(антенн может быть три и более, но для простоты понимания рассматриваются три антенны). Угол между вектором r_AB, образованными антеннами А, В, и вектором r_АС, образованным антеннами А, С, равен Ω, при этом расстояния между антеннами А, В и А, С (базы) равны соответственно |r_AB|=b₁, |r_AC|=b₂.The essence of the proposed method can be explained as follows. A direction finder is installed at the facility, which has, for example, three antennas

(there can be three or more antennas, but three antennas are considered for ease of understanding). The angle between the vector r _AB formed by the antennas A, B and the vector r _AC formed by the antennas A, C is equal to Ω, while the distances between the antennas A, B and A, C (bases) are equal to | r _AB | = b _1, respectively , | r _AC | = b ₂ .

Antennas A, B, C are located in the plane OX _p Z _p of the coordinate system OX _p Y _p Z _p , in which the direction of the axis OX _p coincides with the direction of the vector r _AB , the axis OY _{p is} perpendicular to the plane of the antenna, and the axis OZ _p complements the coordinate system to the right (see Fig. 3).

объекта

в земной системе координат и измеряются разности фаз сигналов

и

здесьAntennas receive signals from three or more spacecraft, for example, from k spacecraft R ₁ , R ₂ , R _3, ..., R _k , which determine the coordinates of spacecraft

facility

in the earth's coordinate system and the phase differences of the signals are measured

and

here

i - spacecraft number

- разность фаз сигналов, принимаемых антеннами А, В от космического аппарата S_i, излучающего сигнал с длиной волны λ_i;

- the difference in phases of the signals received by antennas A, B from the spacecraft S _i, emitting a signal with the length λ _i of the wave;

- разность фаз сигналов, принимаемых антеннами А, С от космического аппарата S_i, излучающего сигнал с длиной волны λ_i (верхний индекс относится к указанию, на какой базе выполнено измерение, а второй - номер космического аппарата).

- the phase difference of the signals received by antennas A, C from the spacecraft S _i emitting a signal with a wavelength of λ _i (the upper index refers to the indication on which basis the measurement was made, and the second is the number of the spacecraft).

A direction finding diagram of two radiation sources is shown in FIG. four.

объекта

и разности фаз сигналов

принимаемых антеннами А, В и А, С, запишем выражения, характеризующие углы между векторами баз, образованными антеннами r_АВ=r_B-r_А, r_АС=r_C-r_А, где r_А, r_B, r_C - координаты антенн в земной системе координат, и векторами наблюдения источников излучения r_0i=r_i-r₀, r_0j=r_j-r₀.Knowing the coordinates of spacecraft with numbers in the earth coordinate system

facility

and phase differences of signals

received by antennas A, B and A, C, we write the expressions characterizing the angles between the base vectors formed by the antennas r _AB = r _B -r _A , r _AC = r _C -r _A , where r _A , r _B , r _C - coordinates antennas in the earth's coordinate system, and the observation vectors of radiation sources r _0i = r _i -r ₀ , r _0j = r _j -r ₀ .

The angle between the observation vectors of spacecraft with numbers i, j is equal to:

The angle between the vectors r _AB , r _AC formed by antennas A, B and A, C is equal to:

For the distance between the antennas | r _AB |, | r _AC | and values of the observation vectors KA | r _0i |, | r _0j | conditions are met

выражения для пеленгов источников излучения космических аппаратов i, j

имеют вид:The fulfillment of these conditions allows us to consider the front of the received electromagnetic wave to be flat, and, accordingly, for the measured phase differences of the received signals

expressions for bearings of radiation sources of spacecraft i, j

have the form:

- целочисленные значения из множеств:here

- integer values from the sets:

where M _i is an integer number of wavelengths λ _i that fit on a length b ₁ ;

N _i is the integer number of wavelengths λ _i that fit on the length b ₂ ;

M _j is the integer number of wavelengths λ _j that fit on the length b ₁ ;

N _j is the integer number of wavelengths λ _j that fit on the length b ₂ .

The expression for the angular distance between radiation sources through bearings of the sources relative to the vectors of the bases and the angle Ω of their mutual orientation, has the form:

hereinafter i, j are the numbers of radiation sources;

- пеленг i-го источника, определенный на первой базе и на второй базе;

- bearing of the i-th source, defined on the first base and on the second base;

- пеленг j-го источника, определенный на первой базе и на второй базе.

- bearing of the j-th source, defined on the first base and on the second base.

The expression for the angular distance between the vectors of the direction finder bases as a function of bearing values and the angle Δθ _{i, j} between radiation sources has the form:

For bearings as functions of the values of the phase uncertainty parameters, the following expressions exist:

Option 1 for the condition:

- bearing of the j-th source on the second base through bearings of the i-th source and bearing of the j-th source on the first base is equal to:

- bearing of the i-th source in the first base through bearings of the j-th source and the bearing of the i-th source in the second base is equal to:

- bearing of the i-th source in the second base:

- bearing of the j-th source on the first base:

Option 2 for the condition:

- bearing of the j-th source on the second base through bearings of the i-th source and bearing of the j-th source on the first base is equal to:

- bearing of the i-th source in the first base through bearings of the j-th source and the bearing of the i-th source in the second base is equal to:

- bearing of the i-th source in the second base:

- bearing of the j-th source on the first base:

становится вероятностной процедурой.The calculation of the angular distances Ω, Δθ _{i, j is} carried out with errors, the bearing measurements are disturbed by the errors of measuring the phase differences and, as a result, the decision is made on the choice of the appropriate parameters for eliminating the phase ambiguity

becomes a probabilistic procedure.

при измеренных значениях пеленгов

и заданных угловых расстояниях между векторами баз (Ω) и источниками излучения (Δθ_i,j). В допущении нормального закона распределения погрешностей измерений косинусов углов пеленгации, угла между векторами баз и угла между векторами наблюдения источников излучения функция правдоподобия имеет вид:We define the likelihood function as the conditional probability density of the ambiguity parameters

at measured values of bearings

and given angular distances between the base vectors (Ω) and radiation sources (Δθ _{i, j} ). Under the assumption of the normal distribution law of measurement errors of cosines of direction-finding angles, the angle between the base vectors and the angle between the observation vectors of radiation sources, the likelihood function has the form:

- среднеквадратическая погрешность оценки косинусов углов Ω, Δθ_i,j по измерениям на объекте установки и расчетах по данным эфемерид космических аппаратов;where σ _Ω ,

- the standard error of the estimate of the cosines of the angles Ω, Δθ _{i, j} from measurements at the facility and calculations according to the data of the ephemeris of spacecraft;

- среднеквадратическая погрешность измерения косинусов пеленгов источников k=i, j на базах b=b1, b2;

- the standard error of the measurement of the cosines of the bearings of the sources k = i, j on the bases b = b1, b2;

Δ - residuals of the corresponding variables equal to:

The maximum of the likelihood function corresponds to the minimum value of the exponent, that is, it corresponds to the minimum of the function defined as the sum of the exponents:

с параметрами неоднозначности

выбираются параметры

при которых достигается минимум

что соответствует максимуму функции правдоподобия. При параметрах фазовой неопределенности, равных

значения

с максимальной вероятностью соответствуют истинным значениям пеленгов источников излучения.From the resulting set of possible values

with ambiguity parameters

parameters are selected

at which a minimum is achieved

which corresponds to the maximum likelihood function. With phase uncertainty parameters equal to

values

with maximum probability correspond to the true values of bearings of radiation sources.

The criterion for finding a solution corresponding to the maximum likelihood function can be used if there is a cross-correlation of measurement errors and the laws of error distribution that differ from the normal law. In this case, the form of the likelihood function changes, but the search for the optimal solution is still based on the search for the maximum likelihood function, taking into account the position of spacecraft, vectors connecting antennas, bearings of spacecraft and their mutual connection.

An exhaustive phase ambiguity elimination algorithm that implements autonomous consideration for each base requires viewing (2M _i +1) of the recovery values of the total phase delays for one base and (2N _i +1) values for the second base, so (2M _i +1) (2N _i +1) combinations of the position of the radiation source in space, from which one is selected that satisfies the selection criterion. The number of combinations equal to (2M + 1) (2N + 1) is proportional to the area of the rectangle with sides (2b ₁ / λ + 1, 2b ₂ / λ + 1).

где

и

Для этого случая количество комбинаций параметров (m, n) пропорционально площади эллипса с полуосями

Коэффициент уменьшения просматриваемых комбинаций, определяемый как отношение возможных комбинаций параметров уточненного алгоритма устранения фазовой неоднозначности к количеству комбинаций возможных неоднозначностей по базам, оценочно равен обратной величине отношения площади прямоугольника к площади, вписанного в него эллипса, то есть равен π/4.The condition for direction finding of a source along two bases (for example, when the bases are orthogonal, i.e., Ω = 90 °), namely, the intersection of the cones of possible positions of the source in space along the first and second bases with the ambiguity parameters (m, n) has the form:

Where

and

For this case, the number of combinations of parameters (m, n) is proportional to the area of the ellipse with half shafts

The reduction coefficient of the viewed combinations, defined as the ratio of possible combinations of parameters of a refined algorithm for eliminating phase ambiguity to the number of combinations of possible ambiguities in the bases, is estimated to be the reciprocal of the ratio of the area of the rectangle to the area of the ellipse inscribed in it, i.e., π / 4.

The expression for the angular distance between two radiation sources is determined under two conditions:

и

при пеленгации их одновременно двумя базами. В этом случае коэффициент уменьшения комбинаций параметров неоднозначности равен произведению коэффициентов уменьшения комбинаций параметров неоднозначности по каждой базе, то есть он оценочно равен

чем достигается поставленная цель - уменьшение количества просмотра комбинаций неоднозначности при переборном методе устранения фазовой неоднозначности при определении углов между направлениями на космические аппараты и векторами, образованным антеннами.which corresponds to the condition for the existence of signal sources on bearings equal to

and

when direction finding them simultaneously with two bases. In this case, the reduction coefficient of combinations of ambiguity parameters is equal to the product of the reduction coefficients of combinations of ambiguity parameters for each base, that is, it is estimated to be

what the goal is achieved is to reduce the number of viewing ambiguity combinations in the exhaustive method of eliminating phase ambiguity in determining the angles between the directions to the spacecraft and the vectors formed by antennas.

The directing cosines of the observation vectors of signal sources relative to the aircraft in the earth's coordinate system are:

Let the unknown direction cosines of the base vectors in the earth coordinate system be equal to:

for the first base -

for the second -

then the system of equations of direction finding of radiation sources at the first base has the form:

the system of equations of direction finding of radiation sources at the second base has the form:

We introduce the following notation:

The system of equations for the unknown direction cosines of the base vector will take the form:

The system of equations by the substitution method reduces to solving a quadratic equation with respect to the variable z of the form:

Az ² + 2Bz + C = 0

The roots of the equation:

from here: x _1,2 = u _x z _1,2 + ν _x

y _1,2 = u _y z _1,2 + ν _y

here

B = u _x v _x + u _y ν _y

D = d ₁₁ d ₂₂ -d ₂₁ d ₁₂

The first solution for guide vector vectors:

second solution:

выполняется аналогично и так же получается два решения относительно направляющих косинусов вектора второй базы соответственно:

и

Solution for guide vectors of the second base

It is performed similarly and two solutions are obtained with respect to the direction cosines of the second base vector, respectively:

and

The choice of the right solution is carried out by attracting the measurement results from a third radiation source. For this, a second pair of sources is determined in which at least one was not used to find the direction cosines of the base vectors, and the procedure for choosing the ambiguity parameters and the solution of the problem of determining the direction cosines of the bases are repeated for it. From the obtained values of the guide cosines of the base vectors, the values that coincide in the solutions for the first pair and the second pair of radiation sources are taken as the true value.

Using the values of the guiding cosines of the base vectors, the parameters of the phase ambiguities of the sources of signals from the spacecraft, which were not used to determine the orientation, are calculated. For source l, the ambiguity parameters are equal to:

- направляющие косинусы вектора наблюдения источника сигнала l относительно летательного аппарата в геоцентрической системе координат;here

- directing cosines of the observation vector of the signal source l relative to the aircraft in the geocentric coordinate system;

int [...] - the operation of determining the integer closest to the integer part of the operand;

sign (...) - the function takes a zero value with a positive value of the operand and is equal to one otherwise.

After determining the ambiguity parameters for all sources, a system of equations is solved to determine the refined angular orientation of the base vectors, taking into account the entire set of phase shifts of the received signals.

The system of equations that describes the relationship of the measured phase shifts with the calculated values that are determined through the guides of the cosines of the observation vectors of the radiation sources and direction finder bases has the form:

- номера космического аппарата;here

- spacecraft numbers;

- оценки косинусов пеленгов через измеренные полные фазовые задержки на базах, равные соответственно

- estimates of the cosines of the bearings through the measured total phase delays at the bases, equal respectively

e _0i × e _b1 , e _0i × e _b2 , are scalar products of the directing cosines of the observation vectors of the signal sources of the i-th spacecraft and unknown base vectors, equal by definition to the cosines of the angles between the corresponding vectors, equal to:

the last three equations are communication equations that determine the orientation of the base vectors relative to each other and the normalization condition for the direction cosines of the vectors.

As a result of solving this system of equations, the specified directional cosines of the base vectors in the earth coordinate system are found from the results of measurements of the phase differences of the signals from all spacecraft.

Второй орт

определяющий положение оси OY_П системы координат, находится как векторное произведение векторов е_b1 и е_b2 By definition, the direction cosines of the unit vector of the axis OX _P of the coordinate system OX _p Y _p Z _p coincide with the direction cosines of the first base, i.e.

Second Ort

determining the position of the axis OY _P of the coordinate system, is found as the vector product of the vectors e _b1 and e _b2

here ⊗ is the operation of the vector product.

The third unit e _Zп , which determines the position of the axis OZ _P and complements the orientation of the axes of the coordinate system OX _p Y _p Z _p to the right, is determined by the vector product of the vectors e _Xп and e _Yп :

совпадают с осями связанной с пеленгатором системы координат и определяют ориентацию ее относительно земной системы координат.Horta

coincide with the axes of the coordinate system associated with the direction finder and determine its orientation relative to the earth coordinate system.

The transformation of coordinates from the earth's coordinate system to the normal (earth) coordinate system and vice versa is carried out using the relations [9]:

- элементы сдвига (координаты центра нормальной системы координат в земной системе координат), равные:here

- shift elements (coordinates of the center of the normal coordinate system in the Earth coordinate system), equal to:

a - semimajor axis of the ellipsoid, m;

α is the compression of the ellipsoid;

B - longitude;

L is the latitude.

The relationship of the rectangular spatial coordinates X, Y, Z and geodesics L, B, H is described by the formulas:

where X, Y, Z are the rectangular coordinates of the point;

L, B, H - geodetic coordinates of the point (respectively latitude and longitude in rad, and height in m);

N is the radius of curvature of the first vertical, m;

e is the eccentricity of the ellipsoid.

The values of the radius of curvature of the first vertical and the square of the eccentricity of the ellipsoid are calculated respectively by the formulas:

e ² = 2α-α ²

where a is the semimajor axis of the ellipsoid, m;

α is the compression of the ellipsoid.

To convert spatial rectangular coordinates to geodesics, an iterative procedure for calculating the geodetic latitude and geodetic height is performed:

1) calculate the auxiliary value of D by the formula:

2) if D = 0, then

3) if D ≠ 0, then

wherein

4) if Z = 0, then

B = 0, H = D-a

5) if Z ≠ 0, find auxiliary quantities r, c, p by formulas and implement an iterative process:

start of the iterative process:

s ₁ = 0

M:

b = c + s ₁

if d is less than the set value of δ, then

B = b

if d is equal to or greater than the set value of δ, then

s ₁ = s ₂ and the calculations continue, starting from label M.

Notes

1. During coordinate transformations, the value of 0.0001 "(4.848136811 · 10 ^-6 ) is taken as the tolerance δ for termination of the iterative process. In this case, the error in calculating the geodetic height does not exceed 0.003 m.

2. The semimajor axis of the ellipsoid a = 6378136 m

Compression ratio α = 1 / 298.25784.

To determine the angular orientation of the aircraft, we transform the unit coordinates of the coordinate system OX _p Y _p Z _p from the earth coordinate system to the normal (earth) coordinate system. The conversion has the form:

where the coordinate transformation matrix M _{N is} given above.

(см. фиг. 5) из нормальной системы в связанную систему летательного аппарата по формуле:For the case when the coordinate system of the direction finder is rotated relative to the associated coordinate system of the aircraft in the direction of the angle Δψ, in pitch by the angle Δθ, in roll by the angle Δγ, the unit vectors are converted

(see Fig. 5) from the normal system to the associated system of the aircraft according to the formula:

here M _Δ is a matrix defined as follows:

where a ₁₁ = cosΔθ

a ₂₁ = sinΔγ sin sin ψ cos cos γ sin sin θ cos cos

a ₃₁ = sinΔγ sinins θ cosΔψ + cosΔγ sinΔψ

a ₁₂ = sinΔθ

a ₂₂ = cosΔγ

a ₃₂ = -sinΔγ cosΔθ

a ₁₃ = -cosΔθ

and ₂₃ = cosΔγ sin sin θ sin sin ψ + sinΔγ cos cos

a ₃₃ = cosΔγ cosΔψ-sinΔγ sin sin θ sin sin

положительное направление оси OY_п совпадает с направлением

а положительное направление оси OZ_п совпадает с направлением

) относительно нормальной (земной) системы координат OX_gY_gZ_g, имеет вид:The matrix of guide cosines that determines the rotation of the coordinate system associated with the direction finder OX _p Y _p Z _p (the positive direction of the axis OX _p coincides with the direction

the positive direction of the axis OY _n coincides with the direction

and the positive direction of the axis OZ _n coincides with the direction

) relative to the normal (terrestrial) coordinate system OX _g Y _g Z _g , has the form:

Where

The angles of the course, pitch and roll of the aircraft (see Fig. 6) in a normal earth coordinate system are determined by the expressions:

ψ = -arctg2 (c ₁₁ , c _3l )

θ = arcsin (c ₂₁ )

γ = -arctg2 (c ₂₂ , c ₂₃ )

The mathematical modeling confirmed the effectiveness of the proposed method for determining the angular orientation in the environment of global radio navigation systems.

Information sources

1. UDC 621.396.96: 629.783 Network satellite radio navigation systems / B.C. Shibshaevich, P.P. Dmitriev, N.V. Ivantsevich, and others; Ed. B.C. Shibshaevich, - 2nd ed., Rev. and add. - M .; Radio and communications, 1993 .-- 408 pp., Ill. - ISBN 5-526-00174-4, p. 205).

2. Patent No. 2185637, Russian Federation. The method of angular orientation of the object according to the signals of satellite radio navigation systems (options) / Aleshechkin A.M., Kokorin V.I., Fateev Yu.L. // Publ. 2002, bull. No. 20.

3. Patent No. 2379700, Russian Federation. The method of angular orientation of the object according to the signals of satellite radio navigation systems / Aleshechkin A.M., Kokorin V.I., Fateev Yu.L. // Publ. 2010, bull. No. 2.

4. Patent No. 2446410, Russian Federation. The method of angular orientation of the object according to the signals of satellite radio navigation systems / Aleshechkin A.M. // Publ. 03/27/2012.

5. GOST 20058-80 Dynamics of aircraft in the atmosphere. Terms, definitions and designations. Standards Publishing House, 1981.

Claims (1)

A method for determining the angular orientation of an aircraft, in which signals from spacecraft of global radio navigation systems are received by diversity antennas of at least three, phase shifts between received signals from each spacecraft are measured, integer ambiguities of phase shift measurements are selected, the angular orientation is determined from the maximum condition likelihood functions
characterized in that the antennas are located on the aircraft not on one straight line, and the phase ambiguity parameter values are selected only for cases of feasible direction finding of spacecraft, while the angular orientation values are determined taking into account the position of the spacecraft, vectors connecting the antennas, bearings of the spacecraft and their mutual relationship:

where i, j are the numbers of spacecraft (SC), with i ≠ j;
Δθ _{i, j} is the angle between the vectors formed by the lines connecting the object and the spacecraft with numbers i, j;
Ω is the angle between the vectors formed by the lines connecting the antennas and forming the first and second bases of the direction finder;

- bearing of the spacecraft number i at the first base;

- bearing of the spacecraft number i at the second base;

- bearing of the spacecraft number j at the first base;

- bearing of the spacecraft number j at the second base;
for condition

communication expressions for bearings:

for condition

communication expressions for bearings:

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