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 Foton’s 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 Euler’s 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

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 don’t 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  don’t exceed 0.001; standard deviations of the other frequencies are within the limits . Standard deviations of the amplitudes  and  don’t 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 Euler’s regular precession of an axisymmetric rigid body. Then we have to put  in (3). Euler’s 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