UNCONTROLLED ATTITUDE MOTION OF THE ORBITAL STATION MIR IN THE LAST YEARS OF ITS FLIGHT

Transcription

1 UNCONTROLLED ATTITUDE MOTION OF THE ORBITAL STATION MIR IN THE LAST YEARS OF ITS FLIGHT () () () () Belaev MYu, Stazhkov VM, Babkn EV, Sazonov VV () Rocket and Space Corporaton Energa, 4a Lenn st, Korolev, Moscow regon, 4070 Russa, tel +7(095)55868 () Keldysh Insttute of Appled Mathematcs RAS, 4 Musskaya sq, Moscow, 5047 Russa, sazonov@keldyshru, tel +7(095)5078, fax +7(095)97077 ABSTRACT We analyze the real atttude moton of the orbtal staton Mr durng long perods of ts uncontrolled flght wthout a crew n The moton was reconstructed by processng on board measurements of the Earth magnetc feld strength Every day, there was a run of measurements The run duraton was about an orbtal revoluton The measurement data, obtaned durng a run, were processed jontly by means of the least squares method and ntegraton of the staton atttude moton equatons The processng resulted n the estmaton of the ntal condtons of the moton as well as parameters of the equatons Daly processng allowed to carry out expermental nvestgaton of several types of regular uncontrolled atttude moton MONITORING OF UNCONTROLLED ATTITUDE MOTION OF THE STATION Startng wth 999, the orbtal staton Mr had long perods of the flght wthout a crew The problem arose for ts operaton durng them wth mnmum fuel consumpton and wth acceptable output of the solar arrays As a rule, the onboard computer was swtched off durng those perods Ths fact restrcted possbltes of controllng the staton atttude moton The problem was solved by the use of two modes of uncontrolled moton One was a rotaton of the staton around ts longtudnal axs wth the angular rate of 0-0 deg/s, the axs oscllatng near the normal to the orbtal plane The second mode began as the baxal rotaton of the staton around ts longtudnal and transverse axes wth angular rates of 05 deg/s, the ntal orentaton of the staton beng arbtrary The baxal rotaton was also carred out n the cases when the absolute value of the staton angular rate exceeded a preset lmt (04 deg/s) or when the energy output was too small The frst mode of the moton was used rarely Two condtons should be met f one uses t Frst, the Sun should not be near the orbtal plane Otherwse ths mode dd not guarantee an acceptable energy output Second, the realzaton of the mode used the onboard computer Thus, the frst mode could be used only at the begnnng of the uncontrolled flght The requred ntal condtons of the atttude moton were realzed before swtchng off the computer Usually the frst mode exsted at the begnnng of the next prolonged perod of the uncontrolled flght, when the Sun was far from the orbtal plane The second mode of moton was the man one, snce baxal rotaton of the staton was realzed wthout the onboard computer The atttude moton of the staton durng uncontrolled flght was checked by the measurements of the Earth's magnetc feld strength These measurements were carred out by the onboard traxal magnetometers Usually a run of measurements was carred out every day Its duraton was about 90 mn The actual atttude moton of the staton on the tme nterval contanng the obtaned data was reconstructed by standard methods (see for example [-]) The measurements of the Earth's magnetc feld and ther processng formed the bass of the montorng of uncontrolled moton of the staton The montorng allowed us to nvestgate expermentally the evoluton of the staton atttude moton on longed tme ntervals The goal was to study the stablty of the frst above mode and to reveal the stable fnal modes, whch fnshed motons startng wth baxal rotaton Bellow some results of ths nvestgaton are presented More detals can be found n [, 4] DETERMINATION OF ATTITUDE MOTION OF THE STATION BY MEASUREMENT DATA The staton s assumed to be a rgd body whose center of mass performs a Kepleran ellptc geocentrc moton The elements of ths moton arose from the rado trackng of the staton Three rght-hand Cartesan coordnate systems were used n order to descrbe the atttude moton of the staton and measurement data The structural coordnate system Oy y y s rgdly connected wth the staton Pont O s the staton's center of mass Axs Oy s parallel to the longtudnal axs of the base unt and s drected to the Kvant unt Axs Oy s drected along the asymmetrc solar battery of the base unt Prncpal central axes of nerta of the staton form the coordnate system Ox x Axes Ox make small angles wth axes Oy (,, ) Axes Ox and Ox

2 correspond to the mnmum and maxmum moments of nerta, respectvely The moments of nerta wth respect to axes Ox and Ox are close to each other In the orbtal coordnate system OX X X axs OX s drected along the geocentrc radus vector of pont O Axs OX s the normal to the plane of the orbt We specfy the poston of the system Ox x wth respect to the orbtal one by two sets of angles The frst set s formed by the angles γ, δ and β [-], ntroduced n the followng manner The system OX X can be transformed nto the system X x Ox by three successve rotatons, () through angle δ + π / about the OX axs, () through angle β about new OX axs, and () through angle γ about the new OX axs concdng wth the Ox axs The angles γ and β specfy the drecton of the OX axs n the system Ox x, and the angle δ specfes the rotaton of the staton about ths axs wth respect to the orbtal system The second set s formed by the angles ψ, θ and ϕ determned n the followng manner [] The system OX X X can be transformed nto the system Ox x by three consequent rotatons, () through angle ψ about the OX axs; () through angle θ about the new OX axs; () through angle ϕ about the new OX axs concdng wth the Ox axs The angles ψ and θ specfy the drecton of the Ox axs n the orbtal coordnate system and the angle ϕ specfes the rotaton of the staton about ths axs We specfy the poston of the system Ox x wth respect to the structural system by the angles α c, β c and γ c The structural system should be successvely rotated through these angles about the axes Oy, Oy and Oy n order to be transformed nto the system Ox x We desgnate the matrx of transton from the Ox x system to the orbtal system as, j a j Here a j s the cosne of the angle between the axes OX and Ox j The elements of the matrx are expressed as functons of the angles ψ, θ, ϕ or the angles γ, δ β We also desgnate the matrx of transton from the Ox x, j system to the structural system as b j The matrx b j was close to the unt one [] Its elements are expressed as functons of the angles α c, β c, γ c The gravtatonal and restorng aerodynamc torques were taken nto account n the equatons of the staton atttude moton The equatons had the form []: ω& ω& ω& µ ( ω ω ηaa ) + κ ( v p v p ) λ λκ ( ωω ηaa ) + ( v p v + λµ + λµ p ( λ + λµ )( ωω ηaa ) + λκ ( v p v ) p ω ω ω a a 0a ω ω ω a a 0a () aω aω ω 0a ω ω + ω a a 0a ω ω + ω a a 0a aω aω + ω 0a I I I λ, µ, η I I r + v v κ E ρ v + µ e Here, dots over letters denote dfferentaton on tme t, ω and v (,, ) are the components of the absolute angular rate of the staton and of ts velocty relatve to the Earth's surface n the Ox x system Parameters p characterze the aerodynamc torque actng upon the staton, ω 0 s the module of the absolute angular rate of the orbtal coordnate system, I are the moments of nerta of the staton wth respect to the Ox axes, µ e s the gravtatonal parameter of the Earth, r s the geocentrc dstance of the pont O, ρ s the atmosphere densty at ths pont, and E s a scale factor When Eqs () are numercally ntegrated, tme s measured n terms of 000 s, length n 000 km, velocty s expressed n km/s, and the unt of measurng the angular rate s 000 s - The atmosphere densty s calculated n kg/m accordng to model [5], and 0 E 0 Elements a are expressed through a and a by formulas a aa aa, etc The varables a and a are not ndependent, they are constraned by the relatons of orthogonalty of the matrx a j For ths reason, we expressed the ntal condtons for a and a through the angles γ, δ and β Thus, the ntal condtons of a soluton of Eqs () were expressed by 6 quanttes The parameters λ and µ n Eqs (), as well as the angles α c, β c and γ c were known: λ 0 686, )

3 µ 068, α c 0 47, β c 0 05, γ c 0 0 (the angles are expressed n radans) These values were used at the ntal stage of processng the measurement data Only the parameters p and ntal condtons of the staton moton were determned at ths stage At the second stage, we refned also the quanttes λ, µ, α, β c and γ c Below, we descrbe the results of the second stage The magnetometer measurements were processed n the followng way On the bass of the data obtaned durng a run, functons h (t) (,, ) were constructed, specfyng the components of the local strength H (t) of the Earth's magnetc feld n the structural system and n the proper tme nterval t0 t t Usng the Kepleran approxmaton of the staton orbtal moton and the analytc model of the Earth's magnetc feld, components H (t) (,, ) of H (t) n the orbtal systems were calculated The sets of functons h (t) and H (t) should be lnked by certan relatons Ths condton allows to formulate a crteron for fndng the soluton of Eqs (), whch approxmates the actual staton moton on the nterval t0 t t Namely, the soluton, whch mnmze the functonal τ + [ h ˆ ( n ) h ( τ n )] ( N + ) n 0 ( p + p + p ) + ε [( λ λ ) + ( µ µ + Φ ε N ) ( c α c ) + ( β c β c ) + ( γ c γ c ) α ] () j, k N + j jk N n 0 h ˆ ( t) H ( t) a ( t) b, τ [ h ( τ ) hˆ ( τ )] k n n n ( N n) t0 + nt N s a sought approxmaton Here, are the constant shfts n the measurement data, ε 0 00, ε 0 0 The parameters ε, ε are expressed n measurement unts of physcal quanttes used n ntegraton of Eqs () These parameters were used for takng nto account a pror values of the staton parametrs As a rule, there was N 60 Functonal () was consdered as a functon of 4 varables: 6 ntal condtons at the nstant t 0 and 8 model parametrs Mnmzaton of Φ was carred out by Levenberg- Marquardt method The maxmal error n determnng the staton atttude n descrbed way dd not exceed 4 The check of accuracy was carred out by usng star sensors measurements Sometmes, neghborng measurement runs were separated by a few orbt revolutons Measurement data of c those runs could be smoothed by the same soluton of Eqs () Dfferent magnetometers were used usually n such runs, therefore the method of processng were modfed The functons h (t) were constracted and the shfts were determned for each nterval separately ROTATION ABOUT THE AXIS OF THE MINIMAL MOMENT OF INERTIA Let the staton orbt be crcular, axs Ox, be the axs of materal symmetry of the staton, and the atmosphere be absent Then the staton atttude moton can be descrbed by Eqs (), n whch ω 0 const, η ω 0, µ p p p 0 These equatons admt a famly of partal solutons whch, n terms of the angles ψ, θ, and ϕ, can be wrtten n the form π ψ ±, θ 0, ϕ ( c m ω0 ) t + c () ω c, ω ω 0 Here, c and c are arbtrary constants Famly () s called a cylndrcal precesson It descrbes the staton statonary rotatatons around the axs Ox drected along the axs OX, g the normal to the orbt plane The shortcomng of a cylndrcal precesson conssts n ts nstablty for saffcently small c The smaller λ, the greater the module of border (wth respect to stablty) values of c Ths fact does famly () useless for prolonged satelltes wth λ << For the staton Mr the border values were acceptable For λ 0 69, ψ π /, solutons () are nstable exponentally at 0 deg/ s < c < 08 deg/ s [6] The mode, based on famly (), was used four tmes; September 7 4, 999, February 0 9 and June 7, 000, January 8 February, 00The examples of reconstructon of the staton moton n ths mode are presented n Fgs and Here and below t was assumed that t 0 0 ; the actual values of t 0 are shown n the fgure captons Fgs and are naturally separated n three parts, the left, mddle, and rght ones The rght parts llustrate the concordance of the functons h (t) and H (t) by the extremal of functonal () The sold lnes are the plots of functons ˆ ( t), the markers correspond to the ponts ( τ n, h ( τ n ) ) ( n 0,, K, N ) Quanttatvely the ft of these functons s characterzed by the standard devaton σ Φ mn /(N 4), where Φ mn s the value of functonal () at the extremal The values of σ are h

4 shown n the fgure captons The mddle parts of the fgures present the plots of tme behavor of the components ω of the staton's absolute angular rate, and the left parts of the fgures present the plots of tme behavor of the angles θ, ψ and of the angle Λ arccos cosθ snψ between the Ox and OX axes Both fgures were obtaned as a result of the jont processng of two neghborng measurement runs Fg presents the segment of the frst use of the mode consdered, Fg the segment n the fnal part of the last use Large maxmum values of Λ n Fg can be explaned by a larger atmospherc drag, because at that tme the orbt of the staton was consderably lower than n 999 Maxmum values of Λ n the ntal part of the last use dd not exceed 8 [4] Aerodynamc torque had apprecable nfluence on the evoluton of the quanttes t ω dt t t ω t, Λ m max Λ( t) ( t0 t t) 0 0 Ths evaluton durng the frst three applcatons of the mode was descrbed n [] The frst and thrd applcatons were the most nterestng The frst applcaton was the most prolonged The evoluton of ω durng t was very fluent, parabolc At frst, ths quantty ncreased from the ntal value 0 44 deg/s ( ) to the value 0 88 deg/s (99909), then t decreased to the value 0 46 deg/s (999094) The quantty Λ m decreased (not monotoncally) from the nnal value 0 ( ) to the value (999097) and became stablze near ths value Durng the thrd applcaton, the quantty ω ncreased monotoncally from the value 0 4 deg/s (000067) to the value 0 07 deg/s (00006) The quantty Λ m decreased from the value (000067) to the value 9 (000060, n that measurement nterval ω 0 7 deg/s) Than t ncreased to 5 (00006) To all appearances, the border (wth respect to stablty) value of ω was passed (000060) 4 ROTATION ABOUT THE AXIS OF THE MAXIMAL MOMENT OF INERTIA As t was mentoned above, when the output from the solar arrays of the staton n ts uncontrolled flght became small enough, the baxal rotaton about ts longtudnal and transverse axes was appled Both angular rates was equal 05 deg/s, the orentaton of the staton was not taken nto account In partcular, all cases of the use the mode descrbed n the prevous Secton ended n ths way Startng wth the baxal rotaton, the atttude moton of the staton changed spontaneously durng several weeks, but sometmes regularty was revealed n ths varaton An example of a regular evoluton of the atttude moton of the staton s presented n [] The fnal moton was the rotaton of the staton around ts axs of the maxmal moment of nerta, namely, the Ox axs, whch oscllated near the normal to the orbtal plane That segment of uncontrolled moton was lasted from to The moton at the end of t s shown n Fg The fgure s smlar to Fgs,, but ts left plots present tme behavor of the angles γ, β, and of the angle Ψ arccos cosγ cos β between the axes Ox and OX The angular rate ω dd not reveal any evoluton durng the last two weeks of that segment The dscrbed fnal moton was not observed always (t s possble, the tme nterval was not suffcently long for ts realzaton), but t exsted before the staton's deorbtng Fg 4 shows the staton's moton n the last days of ts uncontrolled flght The staton angular rate was very large at that tme certan lmtatons were canceled n vew of the end of the flght Fve days before that tme, 0006, the moton was smlar: extremal values of ω and ω were practcally the same, extremal values of γ and β were a few degrees greater, but the quantty ω was less It oscllated wthn the lmts deg/s Twstng of the staton was caused by the aerodynamc torque REFERENCES Sarychev VA, Belyaev MYu, Kuz'mn SP, et al, Measurement of Center-of-mass Moton of Salyut-6 and Salyut-7 durng Slow Spnnng, Cosmc Research, 988, v 6, No, pp6-49 Sarychev VA Sazonov VV, Belyaev MYu and Efmov NI, Icreasng the Accuracy of Determnng the Rotatonal Moton of Orbtal Statons Salyut-6 and Salyut-7 by Measurement Data, Cosmc Research, 99, v 9, No, pp- Babkn EV, Belyaev MYu, Efmov NI, et al, Uncontrollable Rotatonal Moton of the Mr Orbtal Staton, Cosmc Research, 00, v 9, No, pp-7 4 Babkn EV, Belyaev MYu, Efmov NI, et al, Uncontrolled Atttude Moton of the Orbtal Staton Mr n the Last Months of Its Flght, Cosmc Research, 00, v 4, No, pp GOST (State Standard) 7-77: The Upper Atmosphere Model for Ballstc Calculatons, 977, Izd Standartov, Beletsk VV, Artfcal Satellte Moton Relatve to Its Center of Mass (n Russan), Moscow, Nauka, 965

5 Λ,θ, ψ (deg) ω, ω, ω (deg/s) h, h, h Fg The nstant t 0 on the plots corresponds to 5:8:7 Moscow Legal Tme 09999, σ 856γ Λ,θ,ψ (deg) ω, ω, ω (deg/s) h, h, h Fg The nstant t 0 on the plots corresponds to 0:: Moscow Legal Tme 0000, σ 806γ

6 Ψ, δ, β (deg) ω, ω, ω (deg/s) h, h, h Fg The nstant t 0 on the plots corresponds to 07:54:0 Moscow Legal Tme 07999, σ 8γ Ψ,γ, β (deg) ω, ω, ω (deg/s) h, h, h Fg 4 The nstant t 0 on the plots corresponds to :0: Moscow Legal Tme 000, σ 94 γ