Development of a methodological approach and software for information processing and experimental assessment of the accuracy of the “SazhenTA” measuring system
Abstract
Keywords
For citation:
Simonov S.M., Kisin Yu.K., Nekrasov A.V., Shamshin M.V. Development of a methodological approach and software for information processing and experimental assessment of the accuracy of the “SazhenTA” measuring system. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2020;(1):96106. https://doi.org/10.38013/254205422020196106
Introduction
With introduction of the SazhenTA trajectory measuring system into the measuring complex, a vital task is seen in the development of methodological approach and mathematical software support (MSS) of processing and experimental assessment of the measurement accuracy of this system.
The SazhenTA system, consisting of two sets, is deployed in the launch site area for taking measurements at the initial segment of aircraft (A/C) flight. The azimuth and elevation angle are measured in the visiblespectrum and infrared ranges. The tasks of determining A/C trajectory coordinates by the measured angular parameters and assessing the accuracy of the obtained trajectory are considered. The software is developed on the algorithmic programming languages PASCAL, C#, which allows to adapt its application to the operating system MSVS OS accepted and implemented in the Armed Forces of the Russian Federation.
The content of the paper is determined by the topicality of the task consisting in estimation of the SazhenTA system accuracy during measuring complex operation in the sea range conditions. Taking into account the nature of flight test objects, reference facilities are selected, based on which an experimental assessment of the SazhenTA system accuracy is made. At the first stage, this task of experimental assessment of the accuracy required development of a methodological approach and software for processing the SazhenTA system information on determining experimental trajectory, and on the second stage  creation of respective algorithms and programs for determining experimental accuracy of the SazhenTA measuring system.
Algorithms for determining aircraft trajectory by the end method and leastsquare method based on the SazhenTA system information
The task of determining trajectory coordinates is implemented as per two measuring facilities by the leastsquare method (LSM) [1] and the end method, using intersection of two beams determined by the azimuths and elevation angles measured by two SazhenТА systems in accordance with [1–3].
Consideration is given to the task (α_{1}, γ_{1}, α_{2}, γ_{2}) → (x, y, z) of determining aircraft (A/C) coordinates by azimuths and elevation angles measured by two SazhenТА systems. As pointed out in [2, 3], this task is equivalent to the task of determining A/C coordinates for two phase direction finders which measure direction cosines, as considered in the known monograph on the methods of statistical processing of trajectory measurements [1].
The association of direction cosines (cos θ_{x}, cos θ_{z}) with azimuth and elevation angle (α, γ) [2, 3] is determined by the formula:
cos θ_{χ} = sin γ · cos α, (1)
cos θ_{z} = sin γ · sin α,
In [1], there are formulas for determining object coordinates for two phase direction finders which measure direction cosines. A/C coordinates in the local coordinate system (LCS) of the 1st measuring facility (MF) can be determined by the formula:
Thus, it is necessary to determine parameter D1 – distance between the A/C and standing point of the 1st MF.
The method of solving the task is illustrated in Fig. 1 [1], showing angles δ, φ, ψ used in subsequent calculations.
Fig. 1. Determining A/C coordinates as per two measuring facilities
The following designations are used in Fig. 1: O_{1}X_{1}Y_{1}Z_{1} and O_{2}X_{2}Y_{2}Z_{2} of the LCS of two measuring facilities numbered 1 and 2. Points O_{1} and O_{2} – location points of the measuring facilities. Point S – A/C current location point. Parameter D_{1} is determined as follows:
where b – distance between the two MF, b_{1}^{0} – unit vector (shown in Fig. 1 and calculated in accordance with [1]). Triangle angles O_{1}O_{2}S are determined from relations:
where Φ_{1}, Φ_{2} – matrices of direction cosines for transition from the measuring facilities’ LCS to the Greenwich coordinate system (GCS) [1];
D^{0}_{1} = (cos θ_{χ1}, cos θ_{y1}, cos θ_{z1}), D2 = (cos θ_{χ2}, cos θ_{y2}, cos θ_{z2}).
The GCS and LCS coordinate systems are defined in accordance with [1, p. 194, 198].
Geocentric reference (Greenwich) coordinate system (GCS) is a 3D rectangular coordinate system OX_{г}Y_{г}Z_{г}, with its datum point lying in the centre of Earth’s ellipsoid, axis OX_{r} of which is located in the equatorial plane and directed towards zero (Greenwich) meridian, axis OZ_{r} is directed along the Earth’s rotation axis towards the North pole, and axis OY_{r} complements the coordinate system to a righthanded one.
Local coordinate system (LCS) is a 3D rectangular coordinate system OX_{м}Y_{м}Z_{м} associated with the Earth’s surface, with its datum point having geodetic coordinates: В_{м} (geodetic latitude), L_{M} (geodetic longitude), h_{M} (elevation above the surface of Earth’s ellipsoid); axis OY_{м} is directed along the outward normal towards the Earth’s ellipsoid surface; axis OX_{м} lies in the plane tangential to the Earth’s ellipsoid in point O and makes with the datum point meridian plane angle А_{м} (geodetic azimuth), which is reckoned clockwise from a direction to the North; axis OZ_{м} complements the coordinate system to a righthanded one.
For LCS of the SazhenTA system, A_{м} = 0.
The formulas of direction cosines association with azimuth and elevation angle (α_{1}, γ_{1}, α_{2}, γ_{2}) → (cos θ_{x1}, cos θ_{z1}, cos θ_{x2}, cos θ_{z2}) are assigned by relation (1).
In this way, the task (α_{1}, γ_{1}, α_{2}, γ_{2}) → (x, y, z) of determining object coordinates by azimuths and elevation angles measured by two measuring facilities is solved as well. The output results are the trajectory parameters in the GCS.
Determination of trajectory parameters by azimuths and elevation angles measured by two measuring facilities is also performed by the LSM in the GCS.
The LSM task is solved iteratively, with a system of normal equations being solved at each step [1, 4, 5].
For each instant of the measurement period, trajectory coordinates to be assessed are determined iteratively by the formula:
where Xˆ_{(k+1)}, Xˆ_{(k)} – assessments of trajectory coordinates in the GCS at iterations k +1 and k;
ΔΧ_{(k+1)} – corrections to trajectory parameters at iteration k + 1;
A_{(k)} – matrix of partial derivatives from the measured parameters of two MF as per assessed trajectory parameters in the GCS from the kth iteration;
H – measurement vector (measured parameters (α_{1,} γ_{1}, α_{2}, γ_{2});
Y(Xˆ_{(k)}) – vector of measured parameters calculated as per assessment of coordinates at the kth iteration;
W – inverse matrix to the covariance matrix of measurement errors K_{y};
K_{y} – diagonal square matrix of the 4th order with dispersion of rated errors of the SazhenTA system on the diagonal.
The initial approximation Xˆ_{(0)} is assigned as per the data of a priori estimated trajectory.
The matrix of partial derivatives from the measured parameters of two MF as per assessed trajectory parameters in the GCS has a block view:
where A_{1}, A_{2} – respective matrices of partial derivatives for the first and second MF.
At that, each one of the said matrices has the following view:
where B – matrix of partial derivatives from the measured parameters as per trajectory parameters in the LCS (with MF index number omitted);
Ф – matrix of direction cosines of MF’s LCS association with the Greenwich coordinate system [1];
x, y, z – A/C coordinates in the LCS.
The elements of B matrix are calculated by the formulas:
When calculating estimated values of the vector elements of measured parameters Y(Xˆ(k)) as per coordinates at the kth iteration, azimuth and elevation angle for each MF are determined by the formulas, with the iteration index omitted:
where x, y, z – A/C coordinates in the LCS (for a given MF).
When calculating arctangent, respective procedure of the programming language is applied, determining angle within an interval of (0,2).
During calculations, one should take into account possible transition for the estimated azimuth value from the 1st to the 4th quarter, and vice versa.
Transition in the measured parameters from azimuth and elevation angle to parameters in the form of direction cosines by formula (1) makes it possible to operate with continuous functions and ease the requirements to initial approximation in the iteration process. The programs implement processing of both types of measured parameters.
The accuracy of A/C trajectory parameters is determined by the covariance matrix:
where k corresponds to the number of the last iteration in formula (6).
The considered methods of determining trajectory parameters by the angular measurements are applicable both for the active and the passive segments of A/C flight.
It is also possible to determine A/C trajectory parameters by the results of measurements taken by one SazhenTA system – as the coordinates of intersection point of a beam, determined by azimuth and elevation angle, with the A/C flight plane. This method is applicable at a segment of measurements where trajectory parameters lie ‘almost’ in the same plane. To find trajectory segments lying in ‘almost’ the same plane, a special procedure is applied. In the said procedure, plane is determined by three selected trajectory points, and distances from other trajectory points to the selected plane are calculated. If for all the points of selected trajectory segment the calculated distances do not exceed small number ε, then the trajectory points lie ‘almost’ in the same plane. Number ε is selected at the level of errors of trajectory parameters determination by the LSM as per two measuring facilities in accordance with matrix K_{x}.
At the modelling stage, the calculations for selection of trajectory segment, the points of which lie ‘almost’ in the same plane, are made as per estimated trajectory. The estimated trajectory is provided by the A/C designer. During modelling, the parameters of this trajectory are taken as the true values.
Let us suppose that there is a trajectory segment with points lying in ‘almost’ the same plane, and there are measurements for this segment taken by one SazhenTA system. The end formulas for solving the task of determining trajectory by one SazhenTA system will be obtained by finding the coordinates of intersection point of beam (l_{1}), determined by angles (α_{1}, γ_{1}), and the plane.
The plane can be defined by three points in the GCS: M_{1} = (X_{1}, y_{1}, Z_{1}), M_{2} = (X_{2}, y_{2}, Z_{2}), M_{3} = (X_{3}, y_{3}, Z_{3}).
In a general plane equation A · x + B · y + C · z + D = 0, plane coefficients are determined in accordance with equation (8) [6, 7]:
In the LCS of the first measuring facility, beam (l_{1}) lies on a straight line determined by equation in the parametric form:
In the GCS, straightline equation of beam (l_{1}) has the view:
Intersection of beam (l_{1}), determined by equation (10), with plane (П) corresponds to parameter t_{1}:
A/C coordinates in the GCS are determined by the formula:
The method of solving the task of determining A/C trajectory coordinates at a flight segment, the points of which lie ‘almost’ in the same plane, as per the data from one (first or second) SazhenTA system set, is illustrated in Fig. 2.
Fig. 2. Determining A/C trajectory coordinates at ‘almost’ flat segment of flight as per data from one (first or second) SazhenTA system set
The following abbreviations and designations are used in Fig. 2: OMF1 – optical measuring facility No. 1 with LCS coordinates O_{1}X_{1}Y_{1}Z_{1}; EFLSC – Earthfixed launch site coordinates; O_{c}X_{c}Y_{c}Z_{c} – EFLSC datum point and axes; M_{1}, M_{2}, M_{3}, M_{4} – points which angular measurements of azimuth and elevation angle were taken in: (α_{1}, γ_{1}, α_{2}, γ_{2}, α_{3}, γ_{3}, α_{4}, γ_{4}).
Figs. 3–5 show deviations of A/C coordinates determined by three methods from the reference. The numbers of measurements (as well as in all other figures) are given on the horizontal axis, and the vertical axis shows deviations of A/C coordinates in km.
The plots of Figs. 3–5 demonstrate experiential coincidence of the results of trajectory determination by the LSM and beams intersection as per two SazhenTA systems, as well as acceptable accuracy of trajectory determination by one SazhenTA system at ‘almost’ flat segment of flight.
Fig. 6 shows the distances in metres to the plane from the current trajectory points when determining A/C trajectory coordinates at an ‘almost’ flat segment of flight.
Fig. 7 shows the angle between beam and plane and the angle of beams intersection in degrees. If the angle between beam and plane is close to zero, in this case, an error of trajectory determination by one SazhenTA system will be great. In this case, the task of trajectory determination by the measurements taken by one SazhenTA system is poorly visible [1, 4, 5]. It is recommended in the operation manual to ensure that the intersection angle of beams (lines of sight) from two SazhenТА systems exceeds 10 degrees.
Fig. 6. Distances from plane to current trajectory points
Development of a methodological approach and software for experimental assessment of the accuracy of the SazhenTA measuring system
The efficiency of developed methodological approach and software is assessed on the basis of mathematical modelling. Experimental assessment of the accuracy is done at the level of measured parameters (azimuth and elevation angle) of the SazhenTA system and experimental trajectory parameters determined as per information of the SazhenTA system in accordance with the algorithms of the first section.
During mathematical modelling, estimated trajectory is taken as the true one. Based on this trajectory, angular measurements by two SazhenТА systems are modelled, with random errors of ‘measurements’ input by a pseudorandom number sensor with normal law of distribution and RMSD corresponding to the rated values, i. e. 5 arc seconds for the optical channel and 10 arc seconds for the infrared channel (all modelling calculations were performed with this RMSD level).
The methodology for assessing measurement accuracy with the use of a reference measuring facility is implemented in accordance with [1, Chapter 6]. Modelling of a reference trajectory is done by entering random errors into the estimated trajectory by means of pseudorandom number sensor with normal law of distribution and RMSD corresponding to the covariance matrix data obtained from the results of satellite navigation equipment information processing. Based on the reference trajectory data, the measured angular parameters of the SazhenТА systems are calculated.
Random process vector is calculated:
where – measured and reference vector, respectively, t_{i} – time instant of taking the measurements, i – measurement number, j – measurement session number.
Vector components in (13) correspond to angular measurements (α_{1}, γ_{1}, α_{2}, γ_{2}) or experimental trajectory coordinates (x, y, z).
Fig. 8 illustrates implementation of the random process of deviations of experimental trajectory parameters by the LSM as per information of two measuring facilities from the reference data, in kilometres.
Fig. 9 illustrates implementation of the random process of deviations of two systems’ measurements from the reference data, in arc seconds.
The plots of Fig. 9 show that ‘random errors’ of the modelled angular measurements in a given implementation for two measuring systems do not exceed three RMSD of the rated values for the infrared channel, i. e. 10 arc seconds, even taking into account the reference trajectory errors.
Investigation into random processes is carried out, same as in [1, Chapter 6], under assumption that they are stationary and ergodic, and assessment of statistical characteristics is performed within one implementation, therefore, further on index j (measurement session number) is omitted.
Assessment of statistical characteristics is performed at the level of mathematical expectation and RMSD.
Random processes are represented in the form of three components of slowly varying (systematic) errors (SVE) and rapidly varying (random) errors (RVE), as well as the errors of reference data:
It is further assumed that, by the norm, condition is fulfilled for error vectors and the reference data error can be disregarded.
Assessment is done by approximation of experimental data from the LSM using algebraic polynomials or orthogonal Chebyshev polynomials [1, 8].
Smoothing out is performed independently by each vector coordinate .
Modelling of the introduction of slowly varying errors is performed as per angular measurements using the following formulas:
where v – SVE linear variation coefficient assigned in the modelling program.
The influence of systematic errors of angular measurements on determination of trajectory parameters is effected after processing as per algorithms of the fi rst section through comparison with the trajectory parameters without systematic errors in the angular measurements.
Singling out of slowly varying errors in implementation of a random process is done through approximation with the 3rd grade orthogonal Chebyshev polynomials with the use of WinLTX software package developed by REC ETU, St. Petersburg.
The modelling results are given in Fig. 10, in kilometres.
The plots of Fig. 10 show considerable influence of systematic measurement errors on the experimental trajectory and efficient singling out of the systematic errors through approximation of deviations between the measured and the reference data by means of orthogonal Chebyshev polynomials.
Fig. 11 illustrates in kilometres rootmeansquare deviations (RMSD) of experimental trajectory parameters by the LSM as per variance of covariance matrix K_{x} in accordance with (7).
Plots with designations RMSD X 2MF, RMSD Y 2MF, RMSD Z 2MF correspond to the RMSD obtained during processing of two measuring facilities.
Comparison of the plots in Figs. 11 and 8 shows that ‘random errors’ of the experimental trajectory in this implementation do not exceed three RMSD of experimental trajectory parameters by the LSM, even taking into account the reference trajectory errors.
Table 1 contains assessment of mathematical expectation and RMSD, in metres, of rapidly varying (random) errors of determining experimental trajectory as per the reference data.
Table 1
Assessment of mathematical expectation and RMSD of RVE as per reference data

X 
Y 
Z 
Math expectation 
0.79 
0.51 
1.18 
RMSD 
7 
118 
59 
The data in Table 1 correlate with the results of LSM measurements processing, taking into account the ‘reference’ trajectory errors.
When processing real information, experimental accuracy of the SazhenTA system is assessed applying satellite navigation equipment as data reference.
Conclusions
 Algorithms and programs for determining experimental parameters of the aircraft trajectory by azimuth and elevation angle, as measured by the SazhenTA system, are developed.
 Methodological approach and software for experimental assessment of the SazhenTA system accuracy with application of the reference data of satellite navigation equipment are developed.
 Performance capability of the proposed algorithms is confirmed through mathematical modelling.
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About the Authors
S. M. SimonovSergey Mikhailovich Simonov – Cand. Sci. (Engineering), Deputy Head for R&D of the First Scientific Centre
Research interests: methods for polygon testing of aircrafts.
Yu. K. Kisin
Yuri Konstantinovich Kisin – Cand. Sci. (Engineering), Senior Researcher, R&D Department
Research interests: methods for polygon testing of aircrafts.
A. V. Nekrasov
Alexander Vladimirovich Nekrasov – Head of R&D Department
Research interests: methods for polygon testing of aircrafts.
M. V. Shamshin
Mikhail Vladimirovich Shamshin – Junior Researcher, R&D Department
Research interests: methods for polygon testing of aircrafts.
Review
For citation:
Simonov S.M., Kisin Yu.K., Nekrasov A.V., Shamshin M.V. Development of a methodological approach and software for information processing and experimental assessment of the accuracy of the “SazhenTA” measuring system. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2020;(1):96106. https://doi.org/10.38013/254205422020196106