RU2684733C2 - Method for determining position of object with cut from two measuring points in azimuth, elevation angle and distance - Google Patents

Abstract

FIELD: aviation.

SUBSTANCE: invention relates to the field of determining the coordinates of aircraft and can be used in military technology. Result is achieved by converting three measured values - azimuth α₁, the elevation angle ε₁ and distance D₂ or α₂, ε₂, D₁ in rectangular coordinates of the measurement object. First two of them (α₁; ε₁) also determine the line passing through the first measuring point in the direction of the object. Geometrical place of points for which the value of D₂ constant, there is a sphere of radius D₂ with the centre in the second measuring point. Position of the object in this case will be determined by the point of intersection of the line and the sphere. Accuracy of the proposed method for determining the coordinates of an object is determined by the accuracy of the binding of measuring points, mutual position thereof, the terrain relief, and the accuracy of the measured parameters and is calculated from certain dependencies.

EFFECT: achieved technical result is determination of the coordinates of aircraft in the production of external trajectory measurements of the distance measuring and direction-finding method from two measuring points in azimuth, elevation angle and distance, and an assessment of its accuracy.

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Description

The invention relates to the field of detection and determination of the coordinates of various objects (aircraft for various purposes, projectiles, rockets) and can be used in military equipment.

Currently, there are various methods for determining the coordinates of objects using the methods of transmitting and receiving radio signals from terrestrial radio beacons (patent of the Russian Federation No. 2436232) [1], method of triangulating targets (patent of the Russian Federation No. 2423720) [2]. The disadvantages of these methods are the complexity of information processing from the points of detection of objects, the use of only the active radar range of electromagnetic waves, a large number of measured parameters.

The method of automated determination of the coordinates of unmanned aerial vehicles (patent of the Russian Federation No. 2523446) [3] consists of using all-round cameras, a video monitor, a computer and a laser range finder to illuminate an aircraft. Circular view cameras are placed symmetrically and sent in different directions, so as to observe 360 ° along the horizon in the optical range of electromagnetic waves. The appearance of the aircraft is recorded automatically as an obstacle occurring on the frame of the video sequence relative to the previous one, and the data obtained are processed on a computer, where the angular coordinates of the aircraft relative to the center of the angle-measuring device are calculated, which by means of rotating mechanisms directs the laser range finder to the aircraft to measure its distance. The measured distance is supplied to the processing device and display information where a determination object rectangular coordinates (X _{r O,} h _{O T,} Y _{T G).}

The method of determining the coordinates of aircraft based on the use of two directional angles and one elevation angle (patent of the Russian Federation No. 260149) [4], consists in measuring the specified parameters from two points of conjugate observation, with subsequent recalculation of the values (α ₁ , α ₂ , ε ₂ ) to the coordinates of the object.

A common disadvantage of these methods for determining the coordinates of aircraft is the redundancy of information about the position of the object, incomplete use of measurement data and, as a consequence, non-maximum accuracy, as well as the lack of methods for estimating the accuracy of coordinates.

The task of the present invention is to develop a method for obtaining the coordinates of the object being measured with the minimum necessary information for obtaining the result and an assessment of the accuracy of the object positioning.

The problem is solved as follows.

At present, the distance-finding direction finding method for determining the coordinates of objects is widely used in the practice of optical and radar external trajectory measurements (Fig. 1). It is based on measuring the angular coordinates of the object in the horizontal (azimuth A or directional angle α) and vertical planes (elevation ε), as well as the distance to it D (Fig. 1). In the simplest case, such organization measurement only one measuring point with known coordinates (X _r; h _F; Y _F), to uniquely determine the space coordinates of the aircraft (X _{r O;} h _{F O;} Y _{F D)} on dependences

In order to improve the reliability of obtaining trajectory information, the number of measuring points is increased to two or three. Moreover, on each of them similar measurements are made of three spherical coordinates of the object (α _i , ε _i , D _i ). In this case, there is some redundancy of information, determined by the difference between the number of registered parameters and the number of degrees of freedom of the object equal to three. In the case of two measuring points, six parameters are synchronously recorded (α ₁ , ε ₁ , D ₁ ) (α ₂ , ε ₂ , D ₂ ), where the index shows the number of the measuring point. Most often, the second triple of numbers is used, by analogy with the first, to calculate the coordinates of an object from dependencies (1) and serves only to control calculations. At the same time, almost any triple of numbers from six measured parameters allows determining the coordinates of an object. With this method of processing, most of the information received is lost, and the capabilities of measuring equipment are not fully utilized. In the case of obtaining incomplete information from the measuring means, it is not possible to use the canonical dependencies of the ranging-direction-finding method (1). Therefore, there is a need to develop a method for determining the position of an object with incomplete measurements, one of which is the case of registration of three quantities - α ₁ , ε ₁ , D ₂ or α ₂ , ε ₂ , D ₁ . All further calculations for definiteness are made for the first case, the second option is obtained from the first by replacing the indices 1 by 2 and 2 by 1.

It is known from theory that a pair of numbers (α ₁ ; ε ₁ ) geometrically uniquely defines a straight line in space passing through the first measuring point in the direction of the object. The locus of points for which the value of D ₂ is a constant is a sphere centered at the second measuring point. The position of the object in this case is determined by the point of intersection of the line and the sphere (Fig. 2).

A straight line and a sphere in space can be mutually arranged in one of the following ways:

1. A straight line passes through a sphere. In this case, these points will have two common points.

2. The straight does not pass through the sphere. In this case, these objects have no common points.

3. Direct concerns sphere. In this case, these objects have one common point - the touch point.

Most often, the first case with two common points is implemented in practice. Here it is necessary to introduce additional conditions for choosing one of them. As such a condition can be used the criterion of proximity to the true solution, determined at least approximately. In the case under consideration, as a true value, it is advisable to use the solution obtained from dependencies (1). The second and third cases in practice are less common and occur with large measurement errors of the recorded parameters.

The equation of a straight line passing through the first measuring point in the direction of the object has the form

(xX _r1) / [cos ε ₁ cos α _1] = (hh _r1) / sinε ₁ = (yY _r1) / [cos ε ₁ sin α _1] = k.

where k is the proportionality coefficient;

(X _{Г 1} , h _{Г 1} , Y _{Г 1} ) - rectangular coordinates of the first measuring point.

By alternately expressing the variables x, h, and y from this equation, we get

x = X _G1 + k cos ε ₁ cos α ₁ ; h = h _Г1 + k sin ε ₁ ; y = Y _Г1 + k cos ε ₁ sin α ₁ .

Substituting the resulting expressions into the sphere equation, we will have

(X _G1 + k cos ε ₁ cos α ₁ -X _G2 ) ² + (h _G1 + k sin ε ₁ -h _G2 ) ² + (Y _G1 + k cos ε ₁ sin α ₁ -Y _G2 ) ² = D ² ₂ ,

where (X _{D 2} ; h _{D 2} ; Y _{D 2} ) are the rectangular coordinates of the second measuring point.

Squaring and opening the brackets, we get

(X _G1 -X _G2 ) ² +2 (X _G1 -X _G2 ) k cos ε ₁ cos α ₁ + k ² cos ² ε ₁ cos ² α ₁ +

+ (h _G1 -h _G2 ) ² +2 (h _G1 -h _G2 ) k sin ε ₁ + k ² sin ² ε ₁ +

+ (Y _G1 -Y _G2 ) ² +2 (Y _G1 -Y _G2 ) k cos ε ₁ sin α ₁ + k ² cos ² ε ₁ sin ² α ₁ = D ² ₂ .

This is a quadratic equation for the proportionality coefficient k. We give these terms

k ² (cos ² ε ₁ cos ² α ₁ + sin ² ε ₁ + cos ² ε ₁ sin ² α ₁ ) +

+ 2k [(X _G1 -X _G2 ) cos ε ₁ cos α ₁ + (h _G1 -h _G2 ) sin ε ₁ + (Y _G1 -Y _G2 ) cos ε ₁ sin α ₁ ]

= r ² ₂ - (X _G1 -X _G2 ) ² - (h _G1 -h _G2 ) ² - (Y _G1 -Y _G2 ) ² .

Simplifying the resulting expression and denoting by B the distance (base) between the measuring points, we get

k ² + 2k [(X _G1 -X _G2 ) cos ε ₁ cos α ₁ + (h _G1 -h _G2 ) sin ε ₁ + (Y _G1 -Y _G2 ) cos ε ₁ sin α ₁ ] + B ² -D ² ₂ = 0

where B ² = (X _G1 -X _G2 ) ² + (h _G1 -h _G2 ) ² + (Y _G1 -Y _G2 ) ² .

Solve this quadratic equation. Considering that the coefficient of the first degree of the parameter k is even

то прямая пересекает сферу в двух точках, если

то прямая не пересекает сферу, если

то прямая касается сферы в одной общей точке.If a

then the line intersects the sphere at two points if

then the line does not intersect the sphere if

then the line touches the sphere at one common point.

Finally

Thus, the coordinates of the object can be found on the dependencies

X _{Г О} = ƒ _X = X _Г1 + k _1,2 cos ε ₁ cos α ₁ ; h _{Г О} = ƒ _h + h _Г1 + k _1,2 sin ε ₁ ; Y _{Г О} = ƒ _Y = Y _Г1 + k _1,2 cos ε ₁ sin α ₁ .

Since the final dependence of the calculation results on the arguments is rather complicated, it is advisable to use the linearization method to estimate the coordinate accuracy. In the conditions of this method, the median coordinate errors can be found by the following relationship:

where E [X _G ] - the median error in determining the coordinates of X _g object;

∂ƒ _X / ∂α ₁ is the partial derivative of the coordinate X _g over the direction angle.

It shows how much the calculated value of X _g changes when α ₁ changes by one;

E [α ₁ ], E [ε ₁ ], E [D ₂ ] are the median measurement errors of the directional angle, elevation angle and range, respectively.

Analytical dependencies for calculating the median errors of coordinates h _g and Y _g have the same form

For the same reason, the partial derivatives included in expressions (2) - (3) should be determined using the three-point numerical differentiation formulas

or two points

Other partial derivatives must be determined by similar dependencies.

Thus, the proposed method of determining the position of an object by notching from two measuring points in azimuth, elevation and range allows determining the position of the aircraft in the range-finding direction finding method of organizing external directional measurements with two measuring points in the optical and radar ranges of electromagnetic waves with the minimum required information and evaluate its accuracy.

Information sources

1. Panov V.P., Prikhodko V.V. The method of transmitting and receiving radio signals of terrestrial beacons. - M .: FIPS. The patent for the invention №2436232, 10.12.2011

2. Bezyaev, B.C. A way to triangulate goals. - M .: FIPS. The patent for the invention №2423720, 07/10/2011

3. Shishkov S.V. The method of automated determination of the coordinates of unmanned aerial vehicles. - M .: FIPS. The patent for the invention №2523446, 05.26.2014

4. Shishkov S.V. The method of determining the coordinates of aircraft based on the use of two directional angles and one elevation angle. - M .: FIPS. Patent for invention №2601494, 04/20/2015

Claims (4)

The method of determining the position of the aircraft in the Cartesian coordinate system on the basis of serifs from two measuring points with geographical coordinates (X _G1 ; h _G1 ; Y _G1 ), (X _G2 ; h _G2 ; Y _G2 ) located at a distance B from each other, directional the angle α _1, the elevation angle ε ₁ and distance D ₂ followed by treatment with external trajectory information by geometric solutions sphere intersection problem with the center of the second measuring point with the line passing through the first measuring point in the direction to the object algebraically svodyasche Xia to search for the roots of a quadratic equation with respect to the proportionality factor k of the form k ² + 2k [(X _G1 -X _G2 ) cos ε ₁ cos α ₁ + (h _G1 -h _G2 ) sin ε ₁ + (Y _G1 -Y _G2 ) cos ε ₁ sinα ₁ ] + B ² -D ² ₂ = 0, and the subsequent determination of the desired coordinates of the aircraft (X _{G O} ; h _{G O} , Y _{G O} ) according to dependencies: X _{G O} = X _G1 + kcos ε ₁ cos α ₁ ; h _r = h _{O r1} + ksin ε _1; Y = Y _{T O r1} + kcos ε ₁ sin α _1.

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