Analysis of quasi-steady component in acceleration measurement data obtained onboard Foton M-2
Preprint, Inst. Appl. Math., the Russian Academy of Science
( , -2)

T.Beuselinck, C.Van Bavinchove, V.V.Sazonov, S.Yu.Chebukov
(., . , .., ..)

Russian Academy of Science, Keldysh Institute of Applied Mathematics
. ..

Moscow, 2008

Abstract

The paper presents the results of the investigation of the measurement data obtained onboard the spacecraft Foton M-2 by the triaxial accelerometer TAS3. TAS3 had a sample rate equal to 1000 readings per second and produced the data in a wide spectral range. We extracted the low-frequency componen t from those data and compared it with its calculation analog that was obtained by reconstruction of the spacecraft attitude motion. The spectral analysis of functions presenting the both results was done. It confirmed the influence of the Earth magnetic field upon the measurement data. When we made a correction for this influence and refined the position of the accelerometer onboard the spacecraft the results obtained in these both ways, coincided with each other very exactly (the mean-root-square error doesn't exceed 10-6 m/s2).

, -2 TAS-3. 1000 . . , . , , . , 10-6 /2 .



1. Two ways of determining quasi-steady residual accelerations onboard a spacecraft. This paper contains the analysis of the measurement data obtained onboard the spacecraft Foton M-2 by the triaxial accelerometer TAS3. The spacecraft was a free flyer. It was in orbit during the period 31.05.2005 16.06.2005. The accelerometer was produced by the company RedShift Design and Engineering BVBA (Sint Niklaas, Belgium). It was placed on the furnace Polizon and operated continuously during almost the whole flight. Its measurements served for monitoring of microgravity environment during technological experiments.

The residual accelerations onboard a free flyer can be decomposed into two components, vibration (high-frequency) and quasi-steady (low-frequency) ones. Usually, the spectrum of a vibration component contains frequencies from above a few hundredths of Hz. A quasi-steady component has the spectrum in the range from zero to a few thousandths of Hz. We analyze below only a quasi-steady acceleration component. The following reasons cause it: a spacecraft attitude motion, a gradient of the Earth gravitational field, and an atmosphere drag.

That component can be found by two ways. The first way consists in a low-frequency filtration of measurement data of an onboard accelerometer. This way makes high demands for sensitivity and stability of the accelerometer in a low-frequency range. Besides, this way gives the quasi-steady acceleration component only at the point of the accelerometer location. The second way is based on a reconstruction of a satellite real attitude motion and a subsequent calculation of the acceleration along the reconstructed motion by the well-known formula. Let us remind that formula and some related definitions.

Let a spacecraft be a rigid body and a point be fixed with its frame. The difference between the gravitational field strength at the point and the absolute acceleration of that point is called a residual acceleration at the point . We denote the difference by . This quantity plays a part of in orbital experiments. We assume the atmosphere drag is a sole nongravitational influence upon the spacecraft. Then is defined by the formula [1]

. (1)

Here, , the point is the spacecraft mass center, is the absolute angular rate of the spacecraft, the dot above a letter denotes differentiation with respect to time , is the gravitational parameter of the Earth, is the geocentric radius vector of the point , is the velocity of the point with respect to the Earth surface, is the atmosphere density at that point, is the spacecraft ballistic coefficient.

The reconstruction of the spacecraft attitude motion can be made by processing measurement data of onboard sensors. We can do with indirect measurements if we reconstruct a spacecraft attitude motion using a full system of motion equations of a rigid body. In particular, we reconstructed the motion of Foton-12 and Foton M-2 based on measurements of triaxial magnetometers [2, 3]. The measurement data were accumulated continually during the most part of the flight but the procedure deals with data segments of a few hours length. The measurement data on each such segment are processed jointly using the least squares method and integration of the spacecraft attitude motion equations. The procedure results in the solution of those equations that approximates measurements. Then, we calculate the acceleration at a prescribed point of the spacecraft as a function of time along the found solution by formula (1). This formula was derived for a general situation without any frequency restrictions. But it gives just a quasi-steady acceleration component in Fotons case [4].

The second way is rather universal. It allows determining the quasi-steady acceleration component at any point fixed with the spacecraft body but it does not take into account possible local acceleration features. We can follow various reasons when choice the point for application of formula (1) but one reason has to be picked out especially. We must consider as the points, where accelerometers were placed. Then we can compare results obtained in both discussed ways. It allows us to check the accelerometers and the calculation model.

Such a comparison is made below for the accelerometer TAS3 located onboard Foton M-2. The results, obtained in these both ways, coincided with each other very exactly after we refined the accelerometer position and corrected the filtered data for the influence of the Earth magnetic field. This influence was revealed by spectral analysis of the filtered and calculated data as well as the Earth magnetic field strength in the spacecraft fixed coordinate system.

2. Calculation of quasi-steady accelerations by reconstruction of spacecraft attitude motion. The method of the reconstruction consists in following [3]. We assign a time interval and, using the measurement data, construct on it the functions approximating the components of the strength of the local magnetic field in the spacecraft structural coordinate system . The axis is the longitudinal axis of the spacecraft and is directed from the landing capsule to the device unit. We suppose that the local magnetic field coincide with the Earth one at the point and calculate its components in the Greenwich coordinate system along the spacecraft orbit basing on the analytical model IGRF2005. Certain relations should link two sets of functions obtained. The condition of the closest fit of these relations on the interval defines the solution to the spacecraft attitude motion equations that approximates the real motion.

The gravitational and some other torques are taken into account in those equations. The equations are written in the coordinate system formed by the principal central axes of inertia of the spacecraft. The angles between the axes and did not exceed several degrees. Denote by the matrix of transition from the system to the system , where was the cosine of the angle between axes and . The phase vector of the attitude motion equations consists of the quantities , , and the components of the spacecraft angular rate in the system . The quantities are calculated by formulas , etc. The matrix of transition from the system to the structural coordinate system is denoted by . Here, is the cosine of the angle between axes and . We consider the solution to the motion equations minimizing the functional

, (2)

,

as an approximation of the real attitude motion of the spacecraft on the interval . Here, are constant shifts in the measurement data. Functional (2) is minimized on the initial conditions of the solution at the point and parameters of the mathematical model. The latter include the parameters of the motion equations, the shifts , and three angles specifying the transition matrix . Usually, we take min and min.

The example of reconstructing the attitude motion of the spacecraft is presented in Fig. 1. This figure consists of two parts. Fig. 1a illustrates the agreement of the functions and by the found spacecraft motion. Here, the solid lines present the plots of the functions defined in (2); the marks indicate the points , . The quality of the agreement is characterized by the standard deviation , where is the minimum value of functional (2). We have in this example.

Fig. 1b presents the plots of the angular rate components . One can see from the plots that the spacecraft motion was similar to Eulers regular precession of an axisymmetric rigid body with the symmetry axis . Foton M-2 was not exactly axisymmetric but it had close inertia moments regarding to the axes and . One can also treat that motion as the motion near the stationary rotation of a triaxial rigid body around its principal central axis of the minimal inertia moment. In this motion

 

, , , (3)

, .

 

Here, , , and are arbitrary constants, , are the moments of inertia of the spacecraft with respect to the axes , i.e. its principal central moments of inertia. Foton M-2 had , ; the constants , and for each processed interval are evaluated as

, .

 

The accuracy of formulas (3) is characterized by the quantities

 

, .

 

The motion in Fig. 1 is characterized by the values deg./s, deg./s, deg./s, and deg./s.

Fig. 1 illustrates the satellite motion in the last hours of the magnetic field measurements. The satellite motion was reconstructed in the same manner for preceding days too [3]. Table 1 presents some results obtained in 13 time intervals. Each interval has the length of 270 min. The table contains their initial points (the date and time) and the respective values of , , , , and . Fig. 1 corresponds to interval 13.

 

Table 1. Basic results of processing the Mirage measurements

 

Inter-val

Date

05/06.2005

UTC

,

,

deg./s

,

deg./s

,

deg./s

,

deg./s

1

31

23:25:30

2947

0.200

0.017

0.107

0.045

2

1

11:11:08

1318

0.312

0.014

0.082

0.045

3

2

00:11:50

1428

0.441

0.013

0.099

0.038

4

2

11:12:25

1566

0.521

0.012

0.066

0.029

5

3

00:13:07

1038

0.645

0.016

0.070

0.024

6

3

11:13:43

1231

0.745

0.0070

0.056

0.016

7

4

00:14:24

1381

0.789

0.0059

0.094

0.029

8

4

13:15:06

1111

0.849

0.0067

0.145

0.013

9

5

10:36:15

1340

0.931

0.0059

0.147

0.011

10

6

11:17:34

1094

1.008

0.0072

0.146

0.011

11

7

09:18:45

1136

1.066

0.0039

0.131

0.0099

12

8

09:20:02

1210

1.111

0.0058

0.114

0.010

13

9

09:21:20

1147

1.149

0.0021

0.112

0.010

 

The table shows that the angular rate of the satellite increased and formulas (3) became more precise coupled with this increase (note the behavior of and ). The final mode of the attitude motion was formed a few days before the flight termination. There were deg./s and deg./s [5].

Fig. 2a illustrates the residual acceleration calculated by formula (1) for the motion in Fig. 1. Calculations were made for the point with , where the sensors of the accelerometer TAS3 should be located. The plots in the figure represent time the components of the vector as functions of time. Here and below, components of vectors are referred to the structural coordinate system. Calculating the last term in formula (1), we used the ballistic coefficient obtained by processing trajectory measurements [3]. The atmosphere density in (1) was calculated according to GOST R (state standard) 25645.166-2004 Model of the upper atmosphere for ballistic calculations. The matrices of different intervals somewhat differed from each other. The acceleration was calculated in each interval using the matrix obtained just for this interval.

3. Filtration of low-frequency component from TAS3 data. The accelerometer TAS3 measured an apparent acceleration . Its sensitive axes were parallel to the axes of structural coordinate system but axes, corresponding to and , had opposite directions. TAS3 had a sample rate equal to 1000 readings per second and produced the data in a wide spectral range. The low-frequency filtration of the data was made using finite Fourier series independently for each vector component.

Let and be natural numbers, be a segment of the scalar measurement data. We refer the measurement to the instant , , and seek the low-frequency component, contained in these data, in the form

. (4)

Here, are coefficients. They are found by the least squares method. The simple explicit formulas are available to calculate them [1]. Some oscillations with relatively high frequencies are often revealed in function (4) that was obtained in this way. In order to remove them, some terms in (4) are modified using the correctional multipliers

.

Here, is the integer part of the number . As a rule, we don't use expressions (4) directly but deal with their values

, , . (5)

We refer to these values as the filtered data. We denote the vector components of the filtered acceleration data by .

In all examples below, expressions (4) were constructed using data segments with a length of 270 min. They were certain of the segments listed in Table 1. The above procedure was applied at s, , and . The spectrum of functions, obtained in this way, locates within the limits from 0 to 0.017 Hz. TAS3 measurements have erroneous constant biases in each vector component. We changed on that reason the coefficient in (4) to obtain zero mean value of data (5). Fig. 3a presents the example of the filtered data from TAS3 measurements. It illustrates the same time interval as Fig. 2a. Each coordinate system in Fig. 3a contains a couple of plots. The plot of expression (4) has greater oscillations.

TAS3 measurements contain not only erroneous constant biases but an erroneous infra low-frequency component too. Such a component has frequencies less than 0.00005 Hz. It is lacking in calculated accelerations. One should guess it by comparing the plots in Fig. 3a with the respective plots in Figs. 2a. This effect takes place for the other intervals of Table 1. To obtain the likeness between the filtered low-frequency component in TAS3 data and its calculated analog, we eliminated the infra low-frequency component from data (5). First, we smoothed these data by the expression

,

where the coefficients were found by the least squares method. We took in the case of . The function represented the sought ultra low-frequency component. Then we replaced the quantities in (5) by the quantities . Just new data (5) are referred bellow as filtered ones. These new data are again the values of certain new expression (4).

Fig. 3b presents the plots of the functions related to interval 13. Fluent curves in Fig. 3a present the plots of the functions . When and , the described method of filtering does not change the amplitudes of harmonic components in the measurement data with frequencies from to Hz; the filtered data dont contain harmonics with frequencies higher than Hz and lower than Hz.

Fig. 2b gives a comparison of low-frequency component in TAS3 data on interval 13 with its calculated analog. The plots, drawn by fine lines, were drawn using the filtered data; the plots, drawn by thick lines, repeat corresponding plots in Fig. 2a. The thick lines were obtained from the respective lines in Fig. 3b by the following way. First, we changed the sign of the function (thereby, we made the transform ). Then, we added the constant biases to the functions to obtain the equalities . The operator of mean value was defined above.

Fig. 2b shows the functions and are close. This fact is valid for intervals 7 13 in Table 1. The oscillations of and in them have large amplitudes and frequencies increasing coupled with . It is difficult to see proximity in the case of functions , or , . This is valid for all intervals in Table 1. C. Van Bavinchove, one of TAS3 creators, supposed the discrepancy was caused of the Earth magnetic field influence. The next sections contain the analysis confirming this hypothesis.

4. Spectral analysis of low-frequency acceleration component. Judging from the plots in Figs. 2 and 3, the low-frequency component of the acceleration onboard Foton M-2 can be represented as a linear combination of a few harmonics (cyclic trends) with frequencies that are incommensurable in the general case. The representation promises to be especially exact in intervals 7 13 in Table 1. Searching for such harmonics is a typical problem of the time series analysis [8, 9]. In our case this problem was solved as follows.

Let data (5) be the filtered data of an acceleration vector component. Expression (4) that generated them contains harmonics with a fixed set of frequencies. This set has a formal sense and does not reflect itself spectral properties of the data. In order to reveal these properties let us try to fit data (5) by the function

 

, (6)

 

where , , , and are parameters. We will seek the values of these parameters by the least squares method. We make up the following expression

(7)

and minimize it over , , , and . The function has a lot of local minima and only part of them corresponds to real harmonics. To find such minima, we solve a number of identical linear least squares problems and calculate the function


at points of a sufficiently fine uniform grid on the interval . Then the plot of this function is drawn and the approximate values of minimum points are found. The abscissas of significant (in the value of ) minima are the frequencies of desired harmonics. Let the frequencies be found in this way. We seek the trend corresponding to them in the form

, (8)

where , , , and are parameters. The values of these parameters are found by minimization of the function specified by relations (7) and (8) using Gauss-Newton's method. This least squares problem is nonlinear. The initial approximation to its solution is formed by the frequencies and the solution of the linear least squares problem (7), (8) over , , with these frequencies.

In order to verify the found solution by simple means, we considered so-called Schuster's periodogram [6, 7]

 

,

along with the function . Let data (5) under study be generated by function (8), where . Then , the periodogram has local maxima at points , while . Thus, studying the periodogram maxima one can evaluate the frequencies and amplitudes of harmonic components in data (5).

We present below the plots of the functions

,

instead of functions and . The minima of the function expresses the root mean square error of approximation of data (5) by sole cyclic trend (6), while the maxima of function estimate the amplitude .

Consider as an example the results of spectral analysis of the acceleration in Fig. 3b. The plots of the functions and for the acceleration components and are shown in Figs. 4a, 5a. The component has essentially the same frequency properties as and so it is not considered in detail. The minimum points of the functions differ from the maximum points of the respective functions no more than Hz.

Each function or contains several harmonics. Constructing appropriate expressions (8), we take into account all clear-cut harmonics (corresponding to well pronounced extrema of and ) and some of slightly definite ones. To analyze these expressions, we introduce the following designations. We denote by expression (8) approximated the function . Plots of the functions serve to check the approximation. We refer to the quantity as the amplitude of a harmonic with the frequency in (8). The frequencies and amplitudes of harmonics of are denoted as and . We also use analogous designations in the case of functions and defined in Section 2. We take Hz and m/s as the units for frequencies and acceleration amplitudes respectively.

The plots of functions , , and () are given in Figs. 4b, 5b. We see the approximation is sufficiently exact. This fact confirms the accuracy of finding the frequencies and amplitudes that are listed in Table 2. Here, the frequencies with identical subscripts are approximately equal and empty cells mean that corresponding harmonics are absent in a respective function.

Following the least squares method, we estimate the accuracy of determination of the quantities and by corresponding standard deviations. These standard deviations seem to be not adequate from the probabilistic point of view in this situation but they give useful information. The frequency has the least standard deviation equal to 0.00021; standard deviations of the frequencies and dont exceed 0.001; standard deviations of the other frequencies are within the limits . Standard deviations of the amplitudes and dont exceed 0.3 and 0.15 correspondingly.

The standard deviations of the frequencies look too small. We point out for comparison that frequency estimations as minima of or maxima of have errors with the upper bound . We have in our case. This value looks too much great as the accuracy estimate of the frequencies .

Certain of the found frequencies admit the obvious interpretation. The frequencies are caused by spacecraft orbital motion. The orbital frequency (the reciprocal quantity to the orbital period) equals 0.185 so

. (9)

Return to formulas (3). The motion, which they describe, is called the nutational motion and its circular frequency is called the nutation frequency. This circular frequency corresponds to the cyclic frequency and we have for interval 13. Hence,

, . (10)

Just the harmonic with the greatest amplitude has the frequency . The spacecraft nutational motion causes it. This result agrees with formula (1), where the first two terms predominate.

To interpret some other frequencies, let us assume that the spacecraft performs exact Eulers regular precession of an axisymmetric rigid body. Then we have to put in (3). Eulers precession is described usually by the nutation angle , the precession angle and the angle of a proper rotation, the quantities , , and being constants in the exact precession. Foton M-2 had [3]

, , .

 

A vector that is a constant in the absolute space has time-dependent components in the system . These components are sums of constant terms and four harmonics with the frequencies

, , , .

The amplitudes of the harmonics have the order , , , and respectively when . There are , , , in our example. The harmonic with the frequency proved to be appreciable. We see in Table 2 that


Table 2. Frequencies and amplitudes of harmonic components in the calculated and measured accelerations. Interval 13

 

Frequency

interpretation

1

 

 

 

 

 

 

 

0.158

0.983

 

 

 

 

2

0.371

2.011

 

 

0.367

0.531

 

 

 

 

 

 

3

 

0.509

1.681

 

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

0.698

0.699

 

 

 

 

5

 

0.862

1.357

 

 

 

 

 

 

 

 

 

 

6

 

 

 

2.044

0.478

2.035

0.600

 

 

 

 

 

 

7

 

 

 

 

 

 

 

 

 

2.215

0.214

2.215

0.173

8

2.376

20.05

 

 

 

 

2.375

20.22

2.374

0.674

2.375

0.785

9

 

 

 

2.535

0.440

2.530

0.655

 

 

2.536

0.206

2.536

0.158

10

 

2.683

1.255

2.725

0.588

2.720

0.926

 

 

2.705

0.204

2.705

0.168

11

2.867

2.280

2.887

2.621

2.887

4.475

2.924

2.387

2.891

0.819

2.892

0.664

12

 

 

3.074

1.884

3.075

1.860

 

 

3.075

2.070

3.075

1.694

13

 

 

3.251

2.018

3.249

2.948

3.223

1.691

3.261

0.764

3.262

0.614

14

 

 

 

 

 

 

 

 

 

3.371

0.212

3.370

0.174

15

 

 

 

 

 

 

 

3.769

0.790

 

 

 

 

16

 

 

4.746

0.632

4.750

0.606

 

 

4.751

0.523

4.751

0.643

17

 

 

 

 

 

 

 

 

 

5.300

0.210

5.300

0.259

18

 

 

 

6.143

0.474

6.147

0.357

 

 

6.145

0.364

6.145

0.447

 

 


. (11)

We see also that

 

, . (12)

 

The harmonics with the frequencies , , and can be explained by the last two terms in formula (1). In particular, the components of the last term that describes the atmosphere drag are presented in the geocentric absolute coordinate system by periodical functions with the orbital period. The second column in Table 2 summarizes our interpretation of some found frequencies.

We performed in the same way the spectral analysis of the functions plotted in Fig. 2a. Its results are presented in Tables 2 and Figs. 6, 7. We omitted plots relating to the function because it has the same frequency properties as . Accuracy characteristics of the found harmonics are following. The frequency has the least standard deviation equal to 0.00011; standard deviations of the frequencies and dont exceed 0.001; standard deviations of the other frequencies are within the limits . Standard deviations of the amplitudes and dont exceed 0.14 and 0.04 respectively.

One can see from Table 2 that the functions contain harmonics with about the same frequencies as the functions . Therefore we used the same principle of the frequency numbering. The close frequencies are in the same line in Table 2. It is not surprising that the frequencies of functions satisfy the relations (9) (12). However amplitudes of some corresponding harmonics in and differ markedly. The greatest discrepancy of amplitudes takes place for harmonics with the frequencies and . There is only one good coincidence of amplitudes. It takes place for harmonics with the frequency . We see some coincidence in the case of frequencies . Some discrepancy in the case of frequencies and can be explained by our pared-down using the TAS3 geometrical characteristics. The single-axis sensors for different directions had slightly different coordinates in this device whereas we use the same coordinates for each sensor.

It is worth to note that the discrepancy between corresponding frequencies of functions and are distinctly smaller than errors in their interpretation in terms of , and . Possibly, the inaccuracy of the interpretation is caused by some fine details of the motion.

Now, we turn to the spectral analysis of the components of the magnetic field strength. We investigated the functions calculated by formulas (2) and plotted in Fig. 1a. The investigation of the functions gave the same results. The analysis was made according to the scheme above. Its results are presented in Table 3 and Figs. 8, 9. The table and figures are arranged in the same manner as Table 2 and Figs. 4 7. The functions and have the same frequency properties, so we cited the plots for only. The quantities and in Table 3 have the standard deviations equal to 0.0062 and 7000 respectively. The frequencies and have the least standard deviations equal to 0.00019; standard deviations of the frequencies and dont exceed 0.0004; standard deviations of the other frequencies are within the limits . Standard deviations of the amplitudes , except , dont exceed 300.

The functions contain some harmonics with about the same frequencies as the functions and . The first column of Table 3 gives in brackets the number of a close frequency from Table 2. Therefore it was not surprising that some frequencies, found in the functions , admit the obvious interpretation. Namely, we have the relations

, , ,

for frequencies of harmonics with large amplitudes and we have the relations

,

for frequencies of harmonics with small amplitudes.

The frequencies and appear both in the functions and in the functions . But their presence in is much more greater the corresponding harmonics have much more greater amplitudes. It is worth to compare this fact with the following one. The frequency is present in functions and too; the amplitudes of corresponding harmonics are approximately equal in all these functions and are twice greater than amplitudes of harmonics with frequencies , in . Thus transition doesn't change the amplitudes for the frequency , which is absent in the functions , and essentially increases the amplitudes for the frequencies , , which are present in the functions . This situation is illustrated by comparison of Figs. 5a, 7a, and 9a. The comparison shows that the function inherits the frequencies from the functions and . The same inheritance takes place in the case of functions , and (compare corresponding columns in Tables 2, 3). The analogous inheritance in the case of functions , , and is not so pronounced (see Figs. 4a, 6a, and 8a) against a background of the large amplitudes of the harmonics with the frequency in and . But if we calculate amplitude ratios for harmonics with frequencies closed to in and , we find the influence of the magnetic field has here the same order as in the case of the functions and . Quantitative characteristics of the influence will be described below.

 

Table 3. Frequencies and amplitudes of harmonic components

in the magnetic field strength.

 

Frequency

interpretation

,

,

,

1

 

0.026

13960

 

 

 

 

2(1)

0.193

3297

 

 

 

 

3(2)

0.339

20240

 

 

 

 

4(3)

 

0.510

13164

 

 

 

 

5(4)

 

0.700

1775

 

 

 

 

6(5)

 

0.868

6032

 

 

 

 

7(6)

 

 

 

2.039

3697

2.039

3707

8(8)

 

 

2.365

3362

2.364

3394

9(9)

 

 

 

2.526

3892

2.526

3851

10(10)

 

 

 

2.717

5831

2.717

5849

11(11)

 

 

2.887

28042

2.887

28067

12(13)

 

 

3.245

14010

3.245

14011

13

 

 

 

3.433

1648

3.433

1671

14

 

 

 

3.566

2790

3.566

2770

 

The analogous analysis was made for interval 9 from Table 1 to investigate the influence of variations of on the results obtained. New results proved to be in a good agreement with the previous ones. We have and based on for interval 9. The transition increases the amplitudes for frequencies and which are present in the functions . The transition increases the amplitude for frequency , which is present in the function .

5. Correction of filtered TAS3 measurement data. As long as the main frequencies of the functions are obtained by joining up the main frequencies of the functions and , we can assume that the Earth magnetic field influenced upon TAS3 measurements linearly. This assumption gives hope to us that TAS3 filtered data can be corrected by the formulas

,

where are constants. We suppose here and below in this Section that the sign of the component has been changed.

If we make a correction for the magnetic field, it is naturally to make simultaneously some other corrections, namely, the correction for infra low-frequency errors, the correction for the shift of TAS3 time scale, the correction for the error in the spacecraft ballistic coefficient and the correction for misalignment of sensitive TAS3 axes with respect to the axes . We specify the last correction by the vector of infinitesimal rotation of TAS3 sensitive axes with respect to the system . The components of can be regarded both to the system and to the system formed by sensitive axes of TAS3. The correction of the ballistic coefficient is specified by means of multiplication of it by the factor : . This correction compensates short time variations of and within a long interval in which was defined. Taking into account all these corrections and assuming they allow removing all possible errors, we can write

,

(13)

,

,

.

Here, the functions compensate infra low-frequency errors in filtered data, is the shift of TAS3 time scale with respect to the time scale used for description of spacecraft attitude motion, the functions and are defined by relations (see (1), are unit vectors along the axes )

, ,

 

the quantities set the origin of TAS3 coordinate system with respect to the spacecraft mass center, are the coordinates of the TAS3 sensor for

the axis in the TAS3 own coordinate system,

mm, mm, mm,

mm, mm, mm,

mm, mm, mm,

 

We considered relations (13) as equations for determining the unknown quantities , , , , , and . We look for these quantities in the following way. Let be given. We consider relations (13) at the points defined by formulas (5). The quantities are calculated at filtration and we dont exclude the infra low-frequency component from them because this corrections are provided by functions . The quantities and are calculated by interpolation using finite Fourier series. Those series were constructed beforehand basing on the proper solution of spacecraft motion equations. We obtained as a result the overdetermined linear system with the unknown quantities , , , , and . We treat the problem of finding its solution as a standard linear regression problem. We solve it by the least squares method for each at points of the uniform grid with the step 1 s and calculate the standard deviation of discrepancies in (13). The value is considered to be the required estimate of . The solution of the regression problem at gives us the required estimates of the quantities listed above. The standard deviations of those quantities, calculated at in the framework of a linear regression problem previously mentioned, are adopted as accuracy characteristics of the found estimates. We emphasize the standard deviations are calculated at fixed , which is supposed to be known, and are so-called conditional standard deviations. The unconditional standard deviation of the estimate is calculated by the formula

 

.

 

The results of solution of the regression problem are presented in Table 4 and Figs. 10 12. These results were obtained for some intervals from Table 1. They were obtained at but they almost coincide with the results for and . Table 4 contains the estimates of the quantities , , , , and as well as their standard deviations. The unit of and is radian, the unit of and is m/(sOe).

Figs. 10a, 11a, and 12a contain the plots of the functions and defined by the left-hand sides and right-hand sides of formulas (13). Thick lines depict the plots of the functions ; fine lines depict the plots of the


Table 4. Estimations of TAS3 adjusting parameters. The unit of and is m/(sOe)

 

Interval

*,

m/s

*,

s

,

s

,

mm

*,

mm

,

mm

,

mm

,

mm

,

mm

1

0.764

48

3.0

22.3

19

121.0

5.5

219.8

4.5

0.929

0.023

2

0.675

37

2.4

13.7

13

96.9

2.9

229.1

2.8

1.089

0.016

4

0.801

22

2.5

16.7

13

109.8

2.5

231.0

2.4

1.179

0.020

6

0.748

32

1.7

1.6

12

86.0

1.9

227.2

1.9

1.043

0.020

8

0.781

25

1.7

7.0

5.0

94.3

0.74

241.2

0.73

1.095

0.015

9

0.999

32

1.8

8.1

5.8

63.6

0.83

238.8

0.82

0.939

0.016

10

0.742

23

1.2

8.4

4.1

96.1

0.57

235.8

0.57

0.900

0.012

11

0.745

19

1.4

10.9

4.3

96.8

0.60

226.1

0.60

1.078

0.013

12

0.952

23

1.9

9.8

6.0

69.4

0.85

236.4

0.84

0.895

0.017

13

0.734

15

1.2

7.8

4.6

104.2

0.66

229.8

0.64

1.040

0.014

 

Interval

1

0.002

0.020

0.039

0.017

0.0007

0.013

189.2

2.9

5.1

2.1

87.2

3.5

2

0.040

0.013

0.014

0.011

0.020

0.0085

197.9

1.9

16.6

1.9

101.8

2.3

4

0.099

0.016

0.010

0.0099

0.006

0.0076

184.6

1.7

15.8

1.9

97.2

2.4

6

0.060

0.015

0.017

0.0089

0.040

0.0064

191.0

1.6

5.3

1.7

99.5

2.2

8

0.008

0.012

0.034

0.0037

0.026

0.0025

186.9

2.8

16.7

1.4

98.2

1.5

9

0.132

0.014

0.010

0.0043

0.018

0.0028

188.7

2.9

1.8

1.9

105.2

2.0

10

0.026

0.011

0.033

0.0030

0.024

0.0019

189.0

2.5

14.9

1.4

100.8

1.5

11

0.022

0.011

0.026

0.0032

0.012

0.0021

178.8

3.0

13.5

1.4

96.4

1.5

12

0.161

0.015

0.021

0.0044

0.022

0.0029

185.9

3.1

6.6

1.9

101.8

2.0

13

0.043

0.013

0.040

0.0034

0.013

0.0022

184.1

2.6

18.1

1.4

99.6

1.5



Table 4 (continuation). Estimations of TAS3 adjusting parameters. The unit of and is m/(sOe)

 

Interval

1

6.6

3.2

104.4

1.6

26.7

3.9

21.9

3.6

1.9

2.3

169.4

2.8

2

3.1

2.4

105.9

1.7

29.2

2.5

14.5

2.6

14.5

2.0

170.9

1.8

4

11.6

2.1

111.4

1.7

35.2

3.2

15.2

2.4

10.9

2.4

172.0

1.8

6

7.8

1.9

108.0

1.6

26.5

3.0

18.3

2.2

17.2

2.2

176.0

1.9

8

2.8

2.8

111.7

1.5

23.9

2.3

11.6

2.8

21.7

1.8

171.8

1.4

9

2.2

2.9

98.4

1.9

6.8

2.9

10.7

3.0

9.8

2.3

184.8

1.9

10

10.3

2.5

106.3

1.5

26.0

2.2

25.4

2.6

11.3

1.8

174.7

1.4

11

10.6

3.0

106.9

1.5

22.6

2.2

20.3

3.0

28.8

1.8

180.5

1.4

12

6.7

3.1

96.4

1.9

4.2

3.1

15.3

3.2

18.6

2.4

188.1

1.8

13

2.3

2.6

115.4

1.5

28.2

2.5

19.7

2.6

20.1

1.9

170.2

1.4



Table 5. Estimations of TAS3 adjusting parameters. The unit of and is m/(sOe)

 

Interval

*,

m/s

*,

s

,

s

,

mm

*,

mm

,

mm

,

mm

,

mm

,

mm

8

0.820

28

0.82

25.4

3.5

83.3

0.76

242.1

0.77

1.105

0.016

9

1.024

26

0.79

16.2

3.9

84.7

0.84

236.8

0.84

0.937

0.016

10

0.793

26

0.65

21.7

2.8

83.5

0.61

237.0

0.61

0.901

0.013

11

0.766

19

0.57

16.3

2.9

96.0

0.61

226.1

0.62

1.083

0.013

12

0.988

18

0.74

26.0

4.1

89.9

0.87

233.9

0.87

0.882

0.017

13

0.763

19

0.64

25.0

3.1

85.8

0.66

232.4

0.67

1.035

0.015

 

Interval

8

190.6

3.0

16.0

1.5

104.4

1.5

9

188.2

3.0

13.0

1.9

106.2

1.9

10

190.5

2.7

13.1

1.5

106.4

1.5

11

177.3

3.1

15.8

1.4

100.5

1.4

12

184.5

3.2

18.1

1.9

104.2

1.9

13

182.4

2.7

13.7

1.5

107.2

1.5

 

Interval

8

6.8

2.9

108.4

1.5

28.0

1.4

5.5

2.9

16.1

1.4

171.2

1.5

9

4.4

2.9

99.9

1.9

20.0

1.9

9.2

2.9

11.5

1.9

180.7

1.9

10

15.1

2.7

104.8

1.5

27.5

1.5

19.6

2.7

6.7

1.5

173.7

1.5

11

11.9

3.0

106.9

1.4

26.2

1.4

16.3

3.0

26.6

1.4

179.0

1.4

12

8.3

3.2

99.8

1.9

22.4

1.9

12.7

3.2

17.6

1.9

182.6

1.9

13

4.6

2.6

111.6

1.4

29.0

1.5

12.9

2.6

12.6

1.5

169.8

1.5

 

Table 6. Estimations of TAS3 adjusting parameters. The unit of quantities and is m/(sOe)

 

Interval

*,

m/s

*,

s

,

,

mm

*,

mm

,

mm

,

mm

,

mm

,

mm

8

0.781

25

0.79

7.0

5.0

94.3

0.74

241.2

0.73

1.095

0.015

9

1.005

25

0.79

6.8

5.8

88.8

0.84

236.4

0.83

0.930

0.016

10

0.742

24

0.61

8.9

4.1

92.2

0.58

236.3

0.57

0.900

0.012

11

0.745

18

0.56

10.3

4.3

100.7

0.60

225.4

0.60

1.077

0.013

12

0.958

16

0.72

6.9

6.0

99.3

0.86

232.7

0.85

0.874

0.017

13

0.734

17

0.62

8.9

4.6

95.5

0.65

231.3

0.64

1.036

0.014

 

Interval

8

0.035

0.0037

0.026

0.0024

186.9

2.8

16.7

1.4

98.2

1.5

9

0.023

0.0043

0.018

0.0028

188.2

2.9

11.9

1.9

102.4

2.0

10

0.032

0.0030

0.024

0.0019

189.0

2.5

13.3

1.4

101.2

1.5

11

0.027

0.0032

0.011

0.0021

178.7

3.0

14.9

1.4

95.9

1.5

12

0.034

0.0044

0.022

0.0029

185.5

3.1

17.6

1.9

98.3

2.0

13

0.037

0.0034

0.013

0.0022

184.2

2.6

14.6

1.4

100.7

1.5

 

Interval

8

2.8

2.8

111.4

1.4

22.7

1.4

11.5

2.8

22.5

1.4

171.9

1.4

9

1.3

2.9

101.6

1.9

19.0

1.9

13.3

3.0

14.4

1.9

181.6

1.9

10

10.8

2.5

105.4

1.4

23.5

1.4

24.8

2.6

11.2

1.4

175.1

1.4

11

10.2

3.0

107.9

1.4

24.5

1.4

20.8

3.0

29.4

1.4

179.9

1.4

12

4.6

3.1

100.9

1.9

18.8

1.9

18.6

3.2

23.6

1.9

183.4

1.9

13

2.8

2.6

113.4

1.4

25.1

1.4

19.0

2.6

18.5

1.4

171.3

1.4