Passive Magnetic Stabilization of the Rotational Motion of the Satellite in its Inclined Orbit

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1 Applied Mahemaical Sciences, Vol. 9, 215, no. 16, HIKARI Ld, hp://dx.doi.org/ /ams Passive Magneic Sabilizaion of he Roaional Moion of he Saellie in is Inclined Orbi Assel Ismailova and Karlyga Zhilisbayeva Deparmen of Mechanics al-farabi Kazakh Naional Universiy, Kazakhsan Copyrigh c 214 Assel Ismailova and Karlyga Zhilisbayeva. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac The problem of perurbed roaional moion of he saellie is one of he mos ineresing, imporan and, a he same ime, mahemaically complex problems of celesial mechanics and space fligh dynamics. Among exising sabilizaion sysems, a passive magneic sabilizaion sysems have a special place, since hey have an excepional reliabiliy and are easy o manufacure. In his paper he problem of passive magneic sabilizaion of he roaional moion of he saellie is sudied. I is assumed ha passive magneic sysem provides is orienaion along he vecor of he geomagneic field srengh H. The geomagneic field is simulaed by he direc dipole model, considering differen orbis of inclinaion. In he considered model an effec of he graviaional orque is aken ino accoun. Resuls of compuaional experimens are presened. Keywords: Passive magneic sabilizaion, roaional moion, direc dipole model, orbi of inclinaion, coupled sysem of differenial equaions, numerical soluion 1 Inroducion Passive Magneic Sabilizaion (PMS) is a mos used sabilizaion echnique o sabilize small saellies, occupied an inclined orbi around he Earh, by using a combinaion of srong permanen magnes and hyseresis maerial.

2 792 Assel Ismailova and Karlyga Zhilisbayeva PMS echnique allows he saellie o sabilize by keeping one axis of he spacecraf aligned wih he field lines of he Earh magneic field in orbi. I is expeced ha perurbaion orques ending o produce urning momens abou he cener of mass of an orbiing saellie will arise from many differen causes. Among he more imporan are he graviaional orque, i.e., orque due o he Earh s graviaional field, and magneic orque, i.e., he orque due o he elecric and magneic fields around he Earh. As i was shown in [5] he graviaional orque is he major disurbance orque and exceeds he nex larges disurbance, i.e., magneic orque, by a facor of 6. For his reason, in he considered mahemaical model PMS echnique is used by aking ino accoun he effec of he graviaional orque. This model has been announced in [12]. The effec of perurbaion orques on a saellie s aiude depends upon he reference frame chosen; he space-sabilized vehicle will be subjec o a urning momen due o he variaion of he Earh s graviaional field over he saellie configuraion, unless he saellie is a homogeneous sphere for example, when he perurbaion orque vanishes. On he oher hand an earh-poining saellie can uilize his graviaional orque as a sabilizing effec, so as o provide a naural posiion of equilibrium. For small angular displacemens from he equilibrium posiion of an earh-poining saellie employing reacion-flywheel damping, governing equaions of moion have been given in [6]. During he las decades he small saellies have played an imporan role in he echnological developmen. The aracive shor period of design and low cos of hem and he capaciy o solve problems ha are usually considered as problems o big and expensive spacecrafs lead us o sudy he conrol problem of hese saellies. Acive hree-axis magneic aiude sabilizaion of a low Earh orbi saellie is considered in [8]. The conrol is creaed by ineracion beween he magneic momen generaed by magneorquers mouned on he saellie body and he geomagneic field. This problem is quie complex and difficul o solve. Sabilizaion as a conrol problem for a saellie is considered in [4], where he conrol is carried ou by a magneic momen of curren coils (magneorquers) mouned on he saellie body. The sabilizer consruced in his work solves he problems of magneic and graviaional sabilizaion. Review of sabilizaion echniques, in paricular, graviy gradien sabilizaion echnique, as he simples of he hree sabilizaion echniques, is discussed in [11]. Analyical models and some exac soluions of he problem of passive magneic sabilizaion relaed o mahemaical models of roaional moion of he saellie has been derived in [2]. Using nonlinear models of hyseresis behavior and nonlinear model of spacecraf aiude dynamics, he problems relaed o design and analysis of passive sabilizaion have been considered in [7]. Here some resuls of nonlinear and quaernions-based mahemaical model for saellie moion involving permanen magnes and hyseresis effecs

3 Passive magneic sabilizaion of he roaional moion of he saellie 793 are presened. In [9] an analyical model for he passive magneically conrolled aiude dynamics of he RAX nanosaellie is analyzed. Some wih numerical simulaions are derived o assess he properies of he aiude dynamics. In his paper we discuss he problem of passive magneic sabilizaion of he roaional moion of he saellie. The effec of he graviaional orque on his sabilizaion is aken ino accoun. In Secion 1 we derive he mahemaical model and formulaion of he considered problem. Derivaion of he elecric and he magneic fields are given in Secion 2. In Secion 3, a compuaional analysis of roaional moion of he saellie in is inclined orbi, based on he proposed mahemaical model, is explained. Some imporan conclusions are given in he final Secion 4. 2 The mahemaical model and problem formulaion The problem of passive magneic sabilizaion of he roaional moion of he saellie is sudied. Is aken ino accoun he effec of he graviaional orque on passive magneic sabilizaion. In he presen mahemaical model we will discuss he roaional moion during he period of is fligh along roaional moion of a saellie during he period (, T f ) of fligh along is inclined orbi around he Earh s cener, and shall base he discussion on Euler s equaions of moion for he roaion of a rigid body around is cener of mass. Differen from previously considered works, we assume ha he body of he saellie is supposed o be non-symmerical which means, in paricular, ha he componens J x, Jȳ and J z of principal momen of ineria are arbirary. To deermine he saellie aiude in space we use he following reference frames. O a XȲ Z is he geocenric reference frame, O a is he Earh cener, O a Ȳ axis is direced along he Earh spin axis, O a Z lies in he Earh equaorial plane and is direced o he poin of he Vernal Equinox, O a X axis is direced so he sysem o be a righ-handed. Oxyz is he orbial reference frame, O is he saellie cener, Oy lies in he orbial plane, is perpendicular o he radius vecor and direced as he orbial velociy does, Oz axis is direced along he radius-vecor of a saellie, Ox is direced such ha he reference frame is righ-handed. O xȳ z is he moving reference sysem, is axes are direced along he principal axes of ineria of he saellie. We will define he aiude of he moving reference sysem O xȳ z wih respec o he orbial reference frame Oxyz via he angles Ψ, Θ and Φ. The muual orienaions of hese reference frames are deermined by he

4 794 Assel Ismailova and Karlyga Zhilisbayeva following linear ransformaions wih direcion cosine marices: x α 1 α 2 α 3 x X a 1 a 2 a 3 y = β 1 β 2 β 3 ȳ = b 1 b 2 b 3 z γ 1 γ 2 γ 3 z Ȳ Z c 1 c 2 c 3 x y z (2.1) Le u = ω π + ν be he argumen laiude, ω π be he argumen of periapsis, ν be he rue anomaly, i be he orbi inclinaion and Ω be longiude of he ascending node of he orbi from he poin of he Vernal Equinox. We will use he following relaionships beween he direcion cosines, he angles Φ, Ψ, Θ, and he above defined parameers [1]: α 1 = cos Θ cos Φ, α 2 = cos Θ sin Φ, α 3 = sin Θ, β 1 = cos Ψ sin Φ + sin Ψ sin Θ cos Φ, β 2 = cos Ψ cos Φ sin Ψ sin Θ sin Φ, β 3 = sin Ψ cos Θ, γ 1 = sin Ψ sin Φ cos Ψ sin Θ cos Φ, γ 2 = sin Ψ cos Φ + cos Ψ sin Θ sin Φ, γ 1 = cos Φ cos Θ. a 1 = sin u sin Ω + cos u cos Ω cos i, a 2 = cos Ω sin i, a 3 = cos u sin Ω + sin u cos Ω cos i, b 1 = cos u sin i, b 2 = cos i, (2.2) b 3 = sin u sin i, c 1 = cos Ω sin u sin Ω cos u cos i, c 2 = sin Ω sin i, c 3 = cos Ω cos u sin Ω sin u cos i. The dynamic and kinemaic equaions of he angular moion of he saellie in an inclined orbi is described by J x ṗ + (J z Jȳ)qr = M m x + M g x, Jȳ q + (J x J z )pr = M mȳ + M gȳ, J z ṙ + (Jȳ J x )pq = M m z + M g z, (, T f ), p() = p ; q() = q ; r() = r ; (2.3) Φ = p (q cos Φ r sin Φ) an Θ sin Ψ ω cos Θ (1 e 2 ) (1 + e cos 3/2 ν)2, (q cos Φ r sin Φ) ω Ψ = + sin Ψ an Θ cos Θ (1 e 2 ) (1 + e cos 3/2 ν)2, ω Θ = q sin Φ + r cos Φ + cos Ψ (1 e 2 ) (1 + e cos 3/2 ν)2, (, T f ), Φ() = Φ ; Ψ() = Ψ ; Θ() = Θ. (2.4) Here p, q and r are he projecions of he saellie s angular velociy ono he axis of he moving reference sysem O xȳ z. Furher, M m := (M m x, M mȳ, M m z ) T and M g := (M g x, M gȳ, M g z ) T are he magneic and graviaional momens, respecively, ω is he angular velociy of he orbial moion and e is he eccenriciy. Noe ha in he considered mahemaical model no only each sysem of differenial equaions (2.3) and (2.4) are he coupled sysem, bu also he

5 Passive magneic sabilizaion of he roaional moion of he saellie 795 sysem (2.3)-(2.4) is also a coupled one. The sysem of dynamic equaions (2.3) has previously been sudied by several auhors assuming various symmery condiions. In he presened mahemaical model, we invesigae he coupled sysem of dynamic and kinemaic equaions (2.3)-(2.4) no only in is general form, bu also assume ha he funcion ν = ν() in (2.3)-(2.4) is no a given daa, bu is a soluion of he Cauchy problem for he following differenial equaion: { ω ν = (1 e 2 ) (1 + e cos 3/2 ν)2, (, T f ) (2.5) ν() = ν. Therefore one of goals of he sudy is o solve he coupled sysem of differenial equaions (2.3)-(2.4) wih respec o he unknown angles Φ, Ψ, Θ, and he projecions p, q and r of he saellie s angular velociy, aking ino accoun ha he rue anomaly funcion of ime ν = ν() is he soluion of he Cauchy problem (2.5). The magneic and graviaional momens M m := (M m x, M mȳ, M m z ) T and M g := (M g x, M gȳ, M g z ) T in (2.3) will be calculaed analyically. 3 Derivaion of he elecric and he magneic fields According o he mahemaical heory of magneic field of Earh, he poenial of he inernal geomagneic field U m which varies inversely wih he disance from he cener of he Earh o a poin in space, is represened by he following series spherical harmonics (see [1]): U m = n=1 ( ) R n+2 e P R n+1 n m (cos θ R ) (gn m cos(mλ R ) + h m n sin(mλ R )). (3.1) m= Here R e > is he mean equaorial radius of he Earh, R, λ R, θ R are he spherical coordinaes of he saellie in he space where he poenial is calculaed (R is he radius-vecor of he saellie, λ R is he geographical longiude and θ is he angle beween he radius-vecor and he spin axis of he Earh), P m n (cos θ R ) is he associae Legendre funcion of he firs kind of degree n and order m, g m n and h m n are he Gaussian coefficiens. Then he vecor of geomagneic field srengh wih poenial (3.1) is defined as follows H = U m. As numerous measuremens he Earh s magneic field on he surface of he globe, as well as near-earh space show, in he firs approximaion, he

6 796 Assel Ismailova and Karlyga Zhilisbayeva Earh s magneic field close o he field of a uniformly magneized sphere or field of a dipole placed in he cener of he Earh, wih he Norh Magneic Pole corresponding o he Souh Geographic Pole and vice versa. In view of his approximae model, cerain erms of he series (3.1) have precise physical inerpreaions. Le us consider hose erms used in he invesigaion of saellie dynamics under he effec of he geomagneic field. If we ake ino consideraion he coefficien g1 only (n = 1, m = ), he field described by he series (3.1) is he field of a dipole siuaed a he cener of he Earh and oriened aniparallel o is spin axis, i.e. oriened in he direcion Norh o Souh. This is he, so called, direc dipole model. This model is able o ake ino accoun he wo principle feaures of he behavior of he local vecor H of magneic field srengh during he moion of a saellie along is orbi, namely, non-uniform roaion of H wih respec o he inerial reference sysem and variaions in is magniude. According o he dipole field he vecor H can be represened as follows [1]: H = µe { k E 3( k E e R ) e R }/R 3, where µ e is he Earh s magneic momen, ke is he uni vecor of he dipole axis, e R is he uni vecor in he direcion of he radius-vecor. By means of projecions of his vecor on he geocenric reference sysem O a XȲ Z his yields : H X = 3µ e R sin i cos i 3 sin2 u, H Ȳ = 3µ e R (1 3 3 sin2 i sin 2 u), H Z = µ e sin i sin u cos u.(3.2) R3 Le us consider he simples pracical mehod of realizing passive saellie sabilizaion. This involves providing orienaion along he vecor H of he local geomagneic field srengh. In his case, a resoring magneic orque, arise from he ineracion beween a permanen magne insalled on-board he saellie and he local geomagneic field. The magneic momen of he magne is chosen o be sufficienly srong o govern he moion of he saellie wih respec o he vecor H. Wihin he framework of he direc dipole, he magneic orque is defined as follows: Mm = I H. Here he vecor I = (I x, I ȳ, I z ) T is he magneic momen of he saellie. Noe ha his momen arises due o he permanen magne on board. Then using (2.1) and (2.2) in (3.2), afer simple calculaions, we find he coordinaes H x, H z, H z, of

7 Passive magneic sabilizaion of he roaional moion of he saellie 797 he geomagneic field srengh vecor H in he moving reference sysem O xȳ z: H x = µ e R {(3 cos i cos Ω 3 sin3 u 3 cos 2 u cos Ω sin u + sin Ω 3 sin 2 i sin 2 u sin Ω +6 cos 2 (i/2) cos i cos u sin 2 u sin Ω) sin Θ + [(3(1 + cos i) cos u sin i sin u 1 +3 sin 2 i sin 2 u) sin Φ sin u [(1 + 3 cos i cos u sin 2 u + 3 cos 2 i cos u sin 2 u 3 sin 2 i sin 2 u) cos Ω 3(cos 2 u + cos i sin 2 u) sin u sin Ω] cos Φ] cos Θ} sin i, Hȳ = µ e R {(cos Ψ sin Φ + cos Φ sin Θ sin 3 Ψ)[3(cos2 u + cos i sin 2 u) sin u sin Ω (1 +3 cos i cos u sin 2 u + 3 cos 2 i cos u sin 2 u 3 sin 2 i sin 2 u) cos Ω] (cos Φ cos Ψ + sin Φ sin Θ sin Ψ)(3(1 + cos i) cos u sin i sin u sin 2 i sin 2 u) sin u +[3 cos 2 u cos Ω sin u 3 cos i cos Ω sin 3 u 6 cos 2 (i/2) cos i cos u sin 2 u sin Ω (3.3) +(3 sin 2 i sin 2 u 1) sin Ω] cos Θ sin Ψ} sin i, H z = µ e R {3[cos Θ cos Ψ cos Ω + (sin Φ sin Ψ cos Φ cos Ψ sin Θ) sin Ω] cos i 3 sin3 u 3[cos Θ cos Ψ cos Ω + (cos Φ cos Ψ sin Θ sin Φ sin Ψ) sin Ω] cos 2 u sin u (3 sin 2 i sin 2 u 1)[(cos Ψ cos Ω sin Θ + sin u sin Ψ) cos Φ cos Ω sin Φ sin Ψ +(sin u sin Θ sin Φ + cos Θ sin Ω) cos Ψ] + 3(1 + cos i)[(cos Φ cos Ψ cos Ω sin Θ cos Ω sin Φ sin Ψ + cos Θ cos Ψ sin Ω) cos i (cos Ψ sin Θ sin Φ + cos Φ sin Ψ) sin i] cos u sin 2 u} sin i. We assume ha he vecor I is locaed along he O z axis, i.e. I := (,, I ) T. Then for he projecions M m x, M mȳ, M m z, of he magneic orque M m = I H of he Earh we have: M m x = I Hȳ, M mȳ = I H x, M m z =. Using (3.3) we obain: M m x = I µ e R {(cos Ψ sin Φ + cos Φ sin Θ sin 3 Ψ)[3(cos2 u + cos i sin 2 u) sin u sin Ω (1 +3 cos i cos u sin 2 u + 3 cos 2 i cos u sin 2 u 3 sin 2 i sin 2 u) cos Ω] (cos Φ cos Ψ + sin Φ sin Θ sin Ψ)(3(1 + cos i) cos u sin i sin u sin 2 i sin 2 u) sin u +[3 cos 2 u cos Ω sin u 3 cos i cos Ω sin 3 u 6 cos 2 (i/2) cos i cos u sin 2 u sin Ω +(3 sin 2 i sin 2 u 1) sin Ω] cos Θ sin Ψ} sin i, M mȳ = I µ e R 3 {(3 cos i cos Ω sin3 u 3 cos 2 u cos Ω sin u + sin Ω 3 sin 2 i sin 2 u sin Ω +6 cos 2 (i/2) cos i cos u sin 2 u sin Ω) sin Θ + [(3(1 + cos i) cos u sin i sin u 1 +3 sin 2 i sin 2 u) sin Φ sin u [(1 + 3 cos i cos u sin 2 u + 3 cos 2 i cos u sin 2 u 3 sin 2 i sin 2 u) cos Ω 3(cos 2 u + cos i sin 2 u) sin u sin Ω] cos Φ] cos Θ} sin i, M m z =. (3.4) Formulas (3.4) will be used below in our compuaional experimens. The disurbing orque M g = (M g x, M gȳ, M g z ) T produced by he gravia-

8 798 Assel Ismailova and Karlyga Zhilisbayeva.5.5 i= i=15 i=3.5 p i=45 i=6 i=75 p Figure 1: Behavior of he projecion p() of he saellie s angular velociy for orbi inclinaions i = ; 15 ; 3 ; 45 ; 6 ; 75. ional field of he Earh, has he form [2] M g x = 3µ R 3 (J z Jȳ)γ 2 γ 3, M gȳ = 3µ R 3 (J x J z )γ 3 γ 1, M g z = 3µ R (J 3 ȳ J x )γ 1 γ 2. (3.5) Here µ = GM e is he graviaional parameer of he Earh (G is he graviaional consan and M e is he mass of he Earh), ( i, j, k) are he uni vecors along he xȳ z axes, respecively, R = P/(1 + e cos ν) is he radius vecor of he saellie and P is he focal parameer of orbi. 4 Compuaional analysis of roaional moion of he saellie in is inclined orbi This secion deals wih he numerical soluion of he coupled sysem of differenial equaions (2.3)-(2.5). We will use here he fourh-order explici Runge- Kua mehod, which is popular as each sage can be calculaed wih one funcion evaluaion (see, for example, [3]). For his aim his mehod firs is applied

9 Passive magneic sabilizaion of he roaional moion of he saellie i=.4 i=45.4 i= q i=.12 i=45.12 i= r Figure 2: Behavior of he projecions q() and r() of he saellie s angular velociy for orbi inclinaions i = ; 45 ; 75. o he Cauchy problem (2.5) for funding numerical values of he rue anomaly funcion ν(). Then hese values are used in he coupled sysem (2.3)-(2.4). To show he algorihm of he Runge-Kua mehod, we rewrie he coupled sysem (2.3)-(2.4) in he vecor form, inroducing he vecor V := (p, q, r, Φ, Ψ, Θ) T, V = V(): { V = Fi (V, ), i = 1, 6, V() = V, V = (p, q, r, Φ, Ψ, Θ ) T. (4.1) We apply he second order accurae midpoin Runge-Kua mehod, using he derivaive a he iniial ime =, o approximae he derivaive a he ime midpoin j+1/2 := j + τ/2 of he uniform ime mesh ω τ := { j [, T f ] : j = jτ, τ = T f /J}: { K1 = τf i (V j, j ), K 2 = τf i (V j + K 1 /2, j + h/2), V j+1 = V j + K 2, j =, J. (4.2) The resuls of compuaional experimens are shown in Fig Specifically, Figures 1 illusrae he behavior of he projecion p() of he saellie s angular velociy ono he axis of he moving reference sysem O x. The firs figure, corresponding o he inclinaion i =, show ha sabilizaion of he

10 r 8 Assel Ismailova and Karlyga Zhilisbayeva p p 1 p 2 p 3 p 4 q q 1 q 2 q 3 q r 1 r 2 r 3 r i i i Figure 3: The values p( i ), q( i ) and r( i ) of he projecions p(), q() and r(), depending on he orbi inclinaion i, a he imes 1 = , 2 = , 3 = , 4 = angular velociy occurs when > The reason is ha for equaorial orbi he srengh vecor H remains consan a all poins of orbi and is orhogonal o he orbial plane. For oher orbis of inclinaions over 15, Passive Magneic Sabilizaion is no effecive o sabilize small saellies, as nex figures show. Similar effecs are observed from Figure 2. The values p( i ), q( i ) and r( i ) of he projecions of he angular velociy, depending on he orbi inclinaion i, a he iniial ( 1 = ), middle ( 2 = , 3 = ) and close o he final ( 4 = ) imes are ploed in Figure 3. 5 Conclusions During he roaional moion of he saellie along is orbi, he magniude and direcion of he vecor of he geomagneic field srengh H is varying. This makes impossible o make a saellie s orienaion as desired. The sysemaic perurbaions can cause oscillaions of a saellie near he resonance frequency. In his paper direc dipole model of he Earh s magneic field governed by he coupled sysem of differenial equaions (2.3)-(2.5) is considered. The problem of passive magneic sabilizaion of he roaional moion of he saellie is sudied by aking ino accoun he effec of he graviaional orque on his sabilizaion. An analysis of obained numerical resuls show ha as he saellie orbi ends o be polar (wih an inclinaion of exacly 9 ), an influence of he geomagneic field increases, since he srengh vecor H varies a each poin of he saellie s orbi. I is shown ha for near equaorial orbis of inclinaions under 1, Passive Magneic Sabilizaion is mos effecive o sabilize small saellies. However, his echnique is no enough effecive for saellie s orbis

11 Passive magneic sabilizaion of he roaional moion of he saellie 81 wih an inclinaion of more 1. In order o achieve desired sabilizaion, one needs o ake ino accoun damping momens. This issue will be subjec of nex sudy. Acknowledgemens. The research has been suppored by Izmir Universiy, and also personally, by Professor Alemdar Hasanov (Hasanoglu). We would like o express our deepes graiude for his suppor and guidance hroughou he research. References [1] V.V. Belesky, Arificial saellie moion abou cener of mass, Nauka, Moscow, 1965, 416. [2] Belesky, V.V., Khenov, A.A., Tumbling Moion of a Magneized Saellie, Nauka, Moscow, 1985, 288. [3] R.L. Burden, J. D. Faires, Numerical Analysis, PWS-Ken, Boson, 4h ediion, [4] V.A. Bushenkov, M.Yu. Ovchinnikov, G.V. Smirnov, Aiude sabilizaion of saellie by magneic coils, Aca Asronauica, 5(12), 22, hp://dx.doi.org/1.116/s (2)11-5 [5] M. Harris, R. Lyle Eds.,Spacecraf graviaional orques, NASA Repor, No. SP-824, UKA, [6] N. E. Ires, Principles of Aiude Conrol of Arificial Saellies, Aeronauical Research Council Repors and Memoranda, R.& M. No. 3276, UK Minisry of Aviaion, London, [7] S. Jayaram, D. Pais, Model-based Simulaion of Passive Aiude Conrol of SLUCUBE-2 Using Nonlinear Hyseresis and Geomagneic Models, Inernaional Journal of Aerospace Sciences, 1(4), 212, [8] F. Miranda, Guidance Sabilizaion of Saellies Using he Geomagneic Field, Inernaional Journal of Aerospace Engineering, Volume 212, Aricle ID , 9 pages, hp://dx.doi.org/1.1155/212/ [9] G. Park, S. Seagraves, N. H. McClamroch, A Dynamic Model of a Passive Magneic Aiude Conrol Sysem for he RAX Nanosaellie, AIAA Guidance, Navigaion, and Conrol Conference, Torono, Onario Canada, 21. hp://dx.doi.org/1.2514/ [1] B.V. Rauschenbakh, M.Yu. Ovchinnikov, S. McKenna-Lawlor, Essenial Spacefligh Dynamics and Magneospherics, Springer, 23. hp://dx.doi.org/1.17/

12 82 Assel Ismailova and Karlyga Zhilisbayeva [11] S. Rawashdeh, CubeSa Aerodynamic Sabiliy a ISS Aliude and Inclinaion, 26h Annual AIAA/USU Conference on Small Saellies, SSC12-VIII- 16, 212. [12] K. Zhilisbayeva, A. Ismailova, D. Tulekenova, On Influence of he Graviaional Momen on he Magneic Sabilizaion of he CubeSa in he Geomagneic Field, European CubeSa Symposium, Book of Absracs, Brussels, Belgium, 213, page 54. Received: December 24, 214; Published: January 25, 215