Periodic Approximation of the Rotational Motion of the Photon-12 Satellite

The results of repeated processing of magnetic measurements carried out on the Photon-12 satellite (which was in orbit from September 9 to 24, 1999) are described. The processing was performed in order to reconstruct the uncontrolled rotational motion of this satellite. In re-processing, the simplified mathematical model of rotational motion was used. The actual orbit of a satellite (the apogee height is 380 km, the perigee height is 220 km) is replaced by a circular orbit; the expression for the aerodynamic moment acting on a satellite is simplified. The system of differential equations underlying the new model is autonomous and proved to be sufficiently accurate to reconstruct the satellite motion based on magnetic measurements in the case in which the satellite angular velocity was not very low and grew gradually. In the last third of the flight, when the satellite motion became virtually stable and had a sufficiently high angular velocity, this system could be reduced to a generalized-conservative system. Such a reduction makes more definite the set of its solutions suitable for approximate description of the actual motion of a satellite. In some segments of motion, the combining of which covers about 3 days, it was possible to use, for this purpose, the periodic solutions continued from the Lyapunov periodic solutions.


INTRODUCTION
This paper is a continuation of works [1,2]. In paper [1], the magnetic measurements carried out in 1999 on the Photon-12 satellite were reprocessed. The original processing was done soon after the flight [3] using the detailed model of rotational motion of this satellite. The model adopted in [1] is simpler than the model of [3], but is comparable with that model in accuracy when reconstructing the motion with a not very low angular velocity. The model of [1] allowed reconstructing the evolution of the actual motion of Photon-12 with growing angular velocity. An even simpler model is used below in which the satellite's actual orbit (the apogee height is 380 km, the perigee height is 220 km) is replaced by a circular orbit. This replacement made it possible to describe the rotational motion of a satellite by autonomous differential equations, which, in the case of steady-state motion with a sufficiently high angular velocity (the result of the evolution indicated above), could be reduced to a generalized-conservative system. Such a reduction made it possible to approximately, but fairly accurately, describe the actual motion of a satellite over time spans up to 3.5 h by periodic solutions extrapolated from Lyapunov's periodic solutions. Periodic solutions suitable for this purpose were studied in [2]. Their similarity to the virtually rotational motion of Photon-12 was also noted there. However, this similarity has not been investigated in detail. Such an investigation is performed below.

EQUATIONS OF A SATELLITE ROTATIONAL MOTION
We consider the satellite to be an axisymmetric solid body, the center of masses of which moves over an unchanged circular orbit. To write down the equations of satellite motion, we introduce four right-hand Cartesian coordinate systems.
The system is formed by the major central axes of inertia of a satellite. Point is satellite 's center of masses, and the axis is the axis of material symmetry of a satellite. This axis is close to the longitudinal axis of a satellite and is directed from the lander to the instrumental module. The satellite's moment of inertia relative to the axis is denoted by the equal moments of inertia relative to the and axes are denoted by The auxiliary coordinate system serves for writing the equations of a satellite rotational motion. The axis coincides with the axis; the and axes are obtained from the and axes by 1 2 3 Ox

Oy
3 Oy turning the system at angle around the axis The kinematic bound between the and systems is specified by the condition that the projection of the absolute angular velocity of the second of them on the axis is zero. The projections of this angular velocity on the and axes are denoted by and Let absolute angular velocity of a satellite in the system have components Then, and Here and below, the dot indicates differentiation with respect to time t.
The measurement data of onboard magnetometers are interpreted in the instrumental coordinate system . This system can be transferred into the system by two successive turns. The first turn is performed at angle around the axis while the second turn, at angle , is performed around the axis which took place after the first turn. In the general case, in order to specify the position of one coordinate system relative to another, three angles are needed. In this case, it could be possible to introduce another angle of turning the instrumental system around its axis obtained after the first two turns. However, since the direction of one of axes and can be chosen arbitrarily, it is convenient to assume the third angle is equal to zero, thereby fixing the system position relative to the system The matrix of transition between these coordinate systems is denoted as where is the cosine of the angle between the axes and The elements of this matrix are expressed in terms of angles and by the formulas The rotational motion of a satellite is studied with respect to the orbital coordinate system Its axes and are directed along the geocentric velocity and radius vector of point . The matrix of transition from the system to the orbital system is denoted by where is the cosine of the angle between the axes and The elements of this matrix will be expressed as a function of angles , and which will be introduced such that the system is transferred into the system by three successive turns: (1) at angle around the axis (2) at angle around the new axis and (3) at angle around the new axis coinciding 1 2 3 Oy y y χ 1 . Oy 1 2 3 Ox x x 1 2 3 Oy y y cos sin , sin cos .
w w w w 1 2 3 Oz z z Oy y y OX X X 1 2 3 Oy y y ψ with the axis So, is the angle between the axis and the plane while is the angle between the projection of axis on the plane and the axis These two angles specify the direction of the axis in the orbital coordinate system. There exist the relationships The system of equations of rotational motion of a satellite is formed by the dynamic Euler equations for angular velocities and and by the kinematic Poisson equations for the first and third rows of matrices In the Euler equations, the gravitational and restoring aerodynamic moments are taken into account, as well as the constant moment along the axis When calculating the aerodynamic moment, the atmosphere is to be considered motionless in absolute space; its density along the orbit is constant; the outer shell of a satellite is assumed to be a sphere with the center on the axis Under the assumptions that have been made, the aerodynamic moment is characterized by a single scalar parameter. The system of equations of rotational motion has the form [4,5] (2) Here, is the orbital frequency, is the aerodynamic parameter, and and are constant quantities. In Eq. (2), the explicit form of the solution of one of the Euler equations is used with the initial condition . The choice of will be indicated below. In the numerical integration of Eqs. (2), the time measurement unit is 10 3 s and the units of measurement of other quantities are and Variables and are interdependent and are bound by the conditions of orthogonality of matrix For this reason, initial conditions and are expressed in terms of angles , and The elements of the second row of matrix 1 .    are calculated as the vector product of its third and first rows. Formulas (1) and the relation allow us to find functions and and the motion of the system by solving Eqs. (2).

Parameter is known:
Nevertheless, this parameter and parameters and are determined from the processing of measurement data, along with unknown initial conditions of satellite motion; i.e., they serve as matching parameters.
Equations (2) were derived from similar equations of rotational motion [1] by transferring to the circular orbit and using the orbital coordinate system as a system, with respect to which the satellite motion is considered. To compensate all performed simplifications, the satellite motions were reconstructed, in which the angular velocity component was sufficiently great. The processing of measurements with using equations (2) was carried out according to the scheme [1] and gave close results.

RECONSTRUCTION OF THE ROTATIONAL MOTION OF A SATELLITE BASED ON THE MAGNETIC MEASUREMENTS
Mirage instruments with several three-component magnetometers were installed onboard the Photon-12.
Since the motion of a satellite was uncontrollable, the measurement data of these instruments and Eqs. (2) could be used to determine the actual rotational motion of a satellite using the conventional statistical techniques. The technique used below is as follows [1,4]. Using measurements taken over a certain time span we have constructed functions which specified, on this span, the components of the vector of the local magnetic-field strength in the coordinate system The root-mean-square approximation errors did not exceed 200 ( Oe). Then, the pseudomeasurements were calculated as where n = 0, 1, 2, …, N. Usually, min, min. Pseudomeasurements served as the initial information for finding solutions of Eqs. (2) describing the actual motion of a satellite.
In accordance with the least-squares method, the reconstruction of the actual motion of a satellite was considered to be the solution of Eqs. (2), which delivered a minimum to the functional  (3), it is necessary to specify the bound of the system with the Greenwich coordinate system, in which the actual satellite orbit is specified, and the EMF strength is calculated. For this purpose, the actual satellite orbit was approximated by a circular orbit on the span . The approximation was constructed using the least-squares method on the basis of fairly accurate values of an actual phase vector of satellite's center of masses specified on a uniform grid with a step of 3 min. The circular orbit was specified by five elements: radius, average motion (orbital frequency ), initial value of the latitude argument, and ascending node longitude and inclination. Along the circular orbit, the EMF strength components were calculated in the coordinate system for moments These components were used in the repeated calculation of functional (3) in the process of its minimization. Using the solution of Eqs. (2), these components were recalculated into the system and then into the system The initial point of the processed data segment was always used as in Eqs. (2).
The acceptability of transition to a circular orbit, as well as the expected accuracy of approximating pseudomeasurements by the extremal of functional (3), were estimated by minimizing the function with respect to and Here, is the scaling factor and are estimates of constant biases in pseudomeasurements. The first radical in the formula for represents the corrected magnitude of the measured magnetic field strength on a satellite at moment the second radical represents the calculated EMF strength magnitude at point at the same moment. Function characterizes the proximity of magnitudes of measured and calculated magnetic-field strength on the span To calculate , it is necessary to know the orbital motion of a satellite only. Minimum value of this function provides a preliminary estimate of the accuracy of magnetic measurements.
At min, for the majority of segments of pseudomeasurements, standard deviation = lies within the limits of 1600-2000γ. In calculating with the use of the actual orbit, is lower and lies within the limits of 1400-1600γ.
Oy y y Two examples of minimizing function are presented in Fig. 1. In the upper part of the figure, the EMF strength vector magnitudes calculated from the corrected pseudomeasurements and based on the IGRF model are compared. The calculation from the pseudomeasurements is presented by markers; the calculation based on the IGRF model is presented by solid lines. The plots in the lower part of the figure characterize the deviations of pseudomeasurements from the model. These plots represent the broken lines, the vertices of which correspond to approximation errors. The figure caption indicates the values of next to them, in parentheses, the values of obtained using the actual orbital motion are indicated. Pseudomeasurements, the scale of which was corrected by multiplier , were substituted into the functional (3). The values of this multiplier were updated for each processed data segment.
Functional (3) was minimized with respect to 11 quantities: the initial conditions for solving system (2) and and parameters and The final stage of minimization was performed by the Gauss-Newton method. The accu- We denote these standard deviations as and The motion of the Photon-12 satellite was reconstructed over 25 time intervals [5]. Some reconstruction results related to intervals 10-25 are given in Table 1; the reconstructions of motion on intervals 14 and 17 are presented in Figs. 2-4. Universal Coordinated Time (UCT) was used in captions to the figures and in the table. Table 1 presents the initial points of intervals and their actual length T. Nominal length min was chosen in such a way that the obtained values of were no more than double the typical values of indicated above, or, in other words, in order that the accepted model of motion be sufficiently adequate. Because the accepted model is simplified, it was used to reconstruct the motions of a satellite with a noticeable angular velocity, which arose several (~2) days after the uncontrolled motion beginning. Table 1 contains the data for the intervals, on which the angular velocity was high enough. Figure 2 illustrates the typical quality of the approximation of pseudomeasurements. Here, solid lines depict the plots of functions at with markers indicating points Quantitatively, the approximation of pseudomeasurements is characterized by standard deviations the values of which are given in Table 1 and in the figure caption. The values of in Table 1 are slightly higher than those in [1]. Nevertheless, the achieved accuracy is sufficient for the purposes of this work. Figure 3 shows the plots of the angular velocity components and The component of angular velocity in the motion shown in Fig. 3a monotonically decreases, while it grows monotonically in the motion shown in Fig. 3b. The limits of variation of this variable are indicated in the figure caption.
As the component of increased, the motion of a satellite became more and more similar to the regular Euler precession of an axisymmetric solid body. The formation of a regular precession with slowly increasing angular velocity and slightly changing value of was completed after several days of flight.
The accurate regular Euler's precession can take place only in the case in which a satellite is axisymmetric, and the major torque of external forces, applied to it, is zero. Then, quantities and remain unchanged during the motion. For the used model, the first condition was met, while the second one was not. For this reason, we can only say about the motions close to the regular precession. It is convenient to characterize such motions by the following quantities: as well as by quantities and , determined by similar formulas. For the latter ones, the following formulas apply: Standard deviations and characterize the proximity of the satellite's motion to a regular precession with parameters and The values of quantities , and are indicated in Table 1. Figure 4 shows the plots of the time dependence of angles and specifying the axis position with respect to coordinates These angles were calculated using the formulas  The satellite motion over the angles became stable by interval 7 only, with the axis ceasing to intersect the orbital plane and remaining located in the halfspace Z 2 > 0 during the entire subsequent flight. The motion in Figs. 3 and 4 is already clearly similar to the regular Euler precession.
Additional information about the satellite motion is provided in Fig. 5. Here, for the two reconstructions discussed above, the plots are presented for angle between the axis and the vector of the angular momentum of a satellite in its motion relative to the center of masses, as well as for angle between this vector and the axis In the regular Euler precession, In the examples in Fig. 5, this angle varies within fairly narrow limits, near the values which are indicated in the figure caption. The angular momentum vector oscillates near the axis (see the plots of angle ), the angular velocity of which is less than 5°/day. Motions of such a type took place on September 15-19-on intervals 10 and 14-18, in particular. The estimates of parameters and and their standard deviations for these intervals are presented in Table 2. Here, the angles are expressed in radians, while quantities and are presented in units of 10 -6 s -2 . The data in this table are typical for all intervals 1-25 [5]. The standard deviations of initial conditions of reconstructions on θ = + θ = − ψ = ψ = + + p ε intervals 1-25, expressed in radians and 0.001 s -1 , lie in the ranges [5] and These examples shows an approximation of pseudomeasurements that is less accurate than in [1,3]. The values of in Table 1 are slightly higher than in [1,3]. Nevertheless, the reconstructions of this work reproduce all features of a satellite's rotational motion.
Intervals 10 and 14-18 are of particular interest for the further analysis, because the motions on them allow an acceptable approximation of a generalizedconservative mechanical system by periodic motions.

GENERALIZED-CONSERVATIVE MODEL OF A SATELLITE ROTATIONAL MOTION
As can be seen from Table 1, the values of have stabilized over time. Apparently, the moment of resistance proportional to [1], which was disregarded in Eqs. (2), has grown. On intervals from that part of Table 1 in which the values of are low, the solutions of Eqs. (2) can be used to reconstruct the satellite motion for The plots illustrating such reconstruction on intervals 10 and 14-18 are given in [5]. They are very similar to the plots obtained with using the original system (2), but have the increased values of These values, on indicated intervals, are presented in Table 3 together with the estimates of quantities and their standard deviations. In this table, and The comparison of the data in Tables 2 and 3   Below, in this Section, we let, in (2), and and assume to be a parameter. We supplement Eqs.
, a a  a a  a a a  a a  a a a  a  a  a  a  a a  a a a  a a  a a a  a a a a  where is an arbitrary constant; variables and are found by formulas (5). The ort of the Ox 1 axis in the OX 1 X 2 X 3 system has components and, as a result, solutions (4), (5) describe the state of rest of the Ox 1 axis in this system. For , these solutions coincide with known solutions, called conical, cylindrical and hyperboloidal precessions [6]. Their complicated shape and the unexpected period are associated with the method of introducing the coordinate system Oy 1 y 2 y 3 . This system, which is suitable in the task of processing magnetic measurements, is poorly suited for describing the satellite motion relative to the orbital coordinate system. It turned out that, in intervals 10 and 14-18, the periodic = 0 p  To describe the satellite motions in which the Ox 1 axis declines from the axis by less than 90°, the position of the Ox 1 x 2 x 3 system with respect to the OX 1 X 2 X 3 system can be conveniently specified by angles and introduced above, as well as by the 2 OX ψ θ third angle the turn at which around the axis completes the transformation of the OX 1 X 2 X 3 system into the Ox 1 x 2 x 3 system. The transformation OX 1 X 2 X 3 → Ox 1 x 2 x 3 can be presented as a superposition of transformations OX 1 X 2 X 3 → Oy 1 y 2 y 3 and Oy 1 y 2 y 3 → Ox 1 x 2 x 3 . The first transformation is specified by angles , and and the second transformation is specified by angle The turns at angles and are performed ϕ, 1 Ox ψ, θ δ, χ. δ χ The solutions of system (2a), expressed in terms of and on intervals 10 and 14-18, are similar to periodic solutions of system (6), studied in [2]. Solutions on these intervals differ from solutions on the other intervals in the fact, that the angular momentum of satellite's motion relative to the center = 1 1 Ox Oy ϕ = δ + χ.  (a) θ, ψ; Ω 2 , Ω 3 , 10 -3 s In [2], the symmetric periodic solutions of system (6) are constructed, which are continued from the Lyapunov solutions existing in the vicinity of its stationary solution For , this stationary solution transfers into the cylindrical precession. There exist two families of such periodic solutions, called "short-periodic" and "long-periodic." Short-periodic solutions were used for approximation. Parameters and of system (6) for each approximated interval were determined as a result of processing magnetic measurements ( Table 3). The value of period was found as a result of spectral analysis of the solutions of system (2a), which approximate magnetic measurements on interval under study. The variables of system (2a) in this solution were recalculated by the method described above into functions θ(t), and and the trial period was determined for each of these functions. Now, we describe the determination of a trial period for the example of function θ(t). This function (as well as the rest of the ones from the given set) was calculated on the uniform time grid s, Then, the approximation of this function was constructed by the expression . (a) θ, ψ; Ω 2 , Ω 3 , 10 -3 s approximating solution of boundary-value problem (6), (7). This problem was solved by the targeting method (see [2]). The amplitudes of corresponding expressions (8) served as an initial approximation of unknown initial conditions and for One can also use the values of variables and in the approximate solution at point determined by the relation Examples of constructed periodic solutions are given in Figs. 7 and 8 and in Table 4. The table indicates the parameters of solutions of problem (6) and (7) , and , as well as orbital frequency on the given interval and coefficient in the characteristic equation which

M h T Mh
( 1) ( 2 1) 0, a determines the multipliers of a periodic solution. Here, the unit of measurement of time is 1000 s, the unit of measurement of the angular velocity is 0.001 s -1 . As we can see, all found periodic solutions are orbitally stable in the first approximation. The figures present the plots of variables θ(t), , and in the periodic solutions on the span and the projections of the ort of the axis and of the ort of the satellite's angular momentum on the plane The projection of the ort of the angular momentum lies inside the projection of the ort of the axis The comparison of the motions, found by processing magnetic measurements, and the periodic solutions was carried out according to the following  , and in the periodic solution were approximated by the discrete Fourier series [7]. The even functions were expanded in a series in cosines, while the odd functions were expanded in a series in sines. Two hundred harmonics were used in each approximating expression. We denote the constructed series as , and We then composed the expression ψ( ), (a) θ, ψ; Ω 2 , Ω 3 , 10 -3 s eralized-conservative mechanical system. This fact serves as another justification to the use of simplified mathematical models in the problems of space-flight mechanics.