Study of the accuracy of active magnetic damping algorithm

Transcription

1 Keldysh Institute Publication search Keldysh Institute preprints Preprint No 6 ISSN (Print) ISSN 7-9 (Online) Ovchinnikov MYu Roldugin DS Tkachev SS Study of the accuracy of active magnetic damping algorithm Recommended form of bibliographic references: Ovchinnikov MYu Roldugin DS Tkachev SS Study of the accuracy of active magnetic damping algorithm // Keldysh Institute Preprints 6 No 6 p doi:948/prepr-6--e URL:

2 Ордена Ленина ИНСТИТУТ ПРИКЛАДНОЙ МАТЕМАТИКИ имени МВ Келдыша Р о с с и й с к о й а к а д е м и и н а у к MYu Ovchinnikov DS Roldugin SS Tkachev Study of the accuracy of active magnetic damping algorithm Москва 6

3 МЮ Овчинников ДС Ролдугин СС Ткачев Исследование точности алгоритма активного магнитного демпфирования Рассматривается спутник оснащенный магнитной системой ориентации реализующей алгоритм демпфирования угловой скорости «-dot» Исследуется поведение спутника в конце переходных процессов Показано что спутник медленно вращается вокруг оси максимального момента инерции найдена величина скорости вращения Исследуется точность ориентации спутника в дипольной модели геомагнитного поля Проводится численное моделирование Ключевые слова: магнитная система ориентации демпфирование dot Mikhail Ovchinnikov Dmitry Roldugin Stepan Tkachev Study of the accuracy of active magnetic damping algorithm ttitude motion of a satellite equipped with an active magnetic attitude control system is considered Control system implements «-dot» damping algorithm Satellite behavior is analyzed in a steady-state motion Slow spinning around the principal axis of maximum inertia is proven angular velocity is found ttitude accuracy in dipole geomagnetic field model is studied Numerical analysis is carried out Key words: magnetic attitude control system damping dot The work was supported by RFR grants --8 and Contents Introduction Equations of motion and geomagnetic field models Satellite rotation stability Stabilization accuracy 8 Satellite on near equatorial orbit 8 Near polar satellite 4 Numerical simulation 4 Conclusion ibliography ppendix Some relations 6

4 Introduction Magnetic attitude control system is implemented on almost every modern small satellite Even big and complex spacecraft utilize this system quite frequently Magnetic control system has two main tasks: angular velocity damping and reaction wheel angular momentum unload This work is devoted to the first problem ttitude acquisition phase was considered by authors in [] under fast rotating satellite assumption In this paper satellite angular velocity is comparable with the orbital one This attitude mode is relevant due to the most used magnetic damping algorithm «-dot» [4] Its name implies that the control dipole moment is opposite to the geomagnetic induction vector change Satellite acquires approximate rotation information with respect to geomagnetic field Control system should spin satellite in the opposite direction effectively cancelling the rotation However geomagnetic induction vector motion with respect to the satellite is due to its vector rotation in inertial space also So the satellite should end up spinning with approximately double orbital angular velocity Here this rotation is proven to be about the principal axis of the maximum moment of inertia Spinning velocity value is refined satellite attitude is assessed Equations of motion and geomagnetic field models The satellite is considered to be a rigid body ttitude is maintained using three orthogonal magnetorquers ctual attitude is available without any error Four reference frames are used: Inertial reference frame O a Y Y Y О a is Earth s center O a Y is directed along Earth s spin axis O a Y lies in the equatorial plane and is directed to the ascending node O a Y is directed so the system to be a right handed Inertial reference frame O a Z Z Z is got from frame O a Y Y Y turning by angle about O a Y This angle is a function of orbit inclination It is defined with averaged geomagnetic field model Inertial reference frame O a S S S is tied to satellite s orbit O a S is normal to the orbital plane O a S is directed to the ascending node O a S is directed so the system to be a right handed Frame O a S S S attitude with respect to O a Y Y Y is defined with rotation by angle i (inclination) about O a Y rotation from O a S S S to O a Z Z Z is defined with angle i about O a S Inertial reference frames are depicted in fig

5 4 Fig Inertial reference frames Ox x x is bound reference with axes directed along the principal axes of inertia Satellite attitude with respect to any inertial frame is defined using angles (rotation sequence --) and absolute angular velocity components is bound reference frame Quaternions are used for numerical simulation Direction cosines matrix is cos cos sin sin sin sin cos cos sin sin cos sin sin cos cos cos sin cos sin sin sin cos sin sin cos sin cos cos cos () Dynamical equations of the satellite motion e with inertia tensor J diag( C) are dω J dt ω Jω M () where M is control magnetic torque These equations are complemented with kinematics d sin cos dt cos d cos sin dt d tg sin cos dt ()

6 Geomagnetic field is represented [] with right dipole model and the averaged model (simplified dipole model) Right dipole implies Earth s field to be of the dipole k in O a Y Y Y frame Induction vector in frame O a S S S is sin usini e S sin usini sini (4) r cosi where u is argument of latitude; e m 4 m is Earth s dipole strength (currently e T km 7 ) 4 kg m А - s - is magnetic constant Simplified dipole model provides the most compact field representation in inertial space Induction vector has constant length and uniformly moves on the circular cone Induction vector in frame O a Z Z Z is sin sin u Z sin cos u () cos where sin i e r cone half-opening angle is sin i tg sin i sin i Satellite is subected to «-dot» control Dipole moment of magnetorquers is given by dx m k dt Satellite rotation stability Here simplified dipole model is used Satellite motion is considered in O a Z Z Z frame Control dipole moment of magnetorquers is dz m m m kω Z k () dt Satellite angular velocity is always considered fast when «-dot» algorithm is studied ω ω In this case second term in () may be omitted If only first part in () is implemented satellite angular velocity is asymptotically damped to zero If second term is preserved angular velocity is damped approximately to double orbital velocity In the end of damping process second term becomes of the same order as the first and cannot be omitted

7 6 Suppose the satellite is spun about its third axis and find conditions providing the existence and stability of the motion Control dipole moment is this case is u u cos cos m k sin sin u sin u This implies and u The satellite and geomagnetic induction vector rotation rates are exactly the same Suppose and obtain linearized equations in the vicinity of motion u Direction cosines matrix is cosu sin u sin u cosu sin u cosu cosu sin u cosu sin u sin u cosu geomagnetic induction vector in bound frame sin cos x sin cos sin cos control dipole moments components are cos sin sin cos m k sin cos cos m k sin sin Dimensionless linearized equations of motion are cos sin cos cos cos sin sin sin cos () Here argument of latitude is used instead of time angular velocity is referred to the orbital one k C C C C C Equation for is decomposed It is omitted in ongoing analysis Characteristic polynomial for () is P C ()

8 coefficients are C C C C C C 4 cos sin cos sin sin 4 sin 4 cos 4 sin 6 6 sin 7 Necessary stability conditions are C This is obviously true for C 4 and C C if so C is either maximum or minimum moment of inertia Remaining C are also positive in this case Further investigation of polynomial () is complicated by overburdened coefficients form x x x where x C Note that equations () have the Parameter may be considered small since magnetic control system frequently employs weak magnetorquers Write characteristic exponents in the same form Zero approximation ( ) exponents are easily found 4 i i Here i is imaginary unit Here necessary stability condition was used Characteristic polynomial coefficients are C cos sin 4 4 sin 4sin 4 cos 6 sin Terms of the order of O are omitted Characteristic polynomial is P

9 8 Taking into account that satisfy () first order approximation becomes 4 k k k k k k k Taking into account coefficients and zero order approximation this yields cos sin 4 cos Due to inertia properties and hence 4 However if and only if So only rotation around the maximum axis of inertia is asymptotically stable Stabilization accuracy Simplified dipole model provides general qualitative result stable rotation with double angular velocity Direct dipole model is more accurate and has one important peculiarity Geomagnetic induction vector rotation is non-uniform We use this property to refine both direction of spinning satellite axis in inertial space and this rotation rate Poincare method for periodical solutions of differential equation is used for this purpose [6] Satellite on near equatorial orbit Satellite motion is represented with respect to O a S S S frame Small parameter is orbit inclination i Geomagnetic induction vector is decomposed retaining the second order of small parameter sin u e e e S sin u i i r r r Equation of motion have the form i i where x x f x g x g x () i i Periodical solution of () is also of the form x x x x ()

10 9 Note that x is () solution for i (generating periodical solution) symptotical stability of solution x u may be proven ust like in previous section so this is the only periodical solution ngular velocity is arbitrary Substituting () into () we obtain for different orders of magnitude of i x f x i f i x x g x x x f f g T i x x x x x g x x x x x x x First order approximation equations are decomposed into two systems sin u sin u cos u cosu z Fz () where z e k r C is a new parameter F First order does not allow the exact value to be determined Second order approximation equations are used z Fz G u z cosu 9 where G u is periodical matrix Equation for provides condition of periodical solution existence 8 Satellite spins a little slower than in the simplified dipole magnetic field Equilibria perturbation is characterized by equations () Unperturbed system equilibrium position is asymptotically stable General solution of () asymptotically tends to zero Partial solution has governs system behavior It may be written as z p a cosu b sinu acos u b sin u (4)

11 Constant vectors a and b are obtained by substituting (4) into () a E44 F q b F E p 44 a E4 4 F q b F E44 p where q p q q p p Calculations lead to b a q and p calculation involves block matrix inverse E44 F E E F E F F () F E4 4 F E F F E F F Since () becomes E44 F F F F E F F 44 Therefore z F q F p cosu F q F p sinu sin u cosu Since is small solutions and sin u cos u are mainly governed by a and b Rotation axis deviates by the angle from the orbit normal i u i u acos cos sin8 cos cos8 asin i u i u cos sin 9 cos cos9 asin 4i i (6) Therefore angle between the maximum moment of inertia axis and normal to the orbit is approximately equal to the half of the inclination

12 Near polar satellite Near polar orbits are of more interest for magnetically actuated satellites Consider this case and new small parameter i Geomagnetic induction vector is decomposed as sin u sin u e e e S sin u sin u r r r Generating solution equations are divided into two systems Out of plane motion allows trivial solution again Rotation around maximum moment of inertia persist Rotation rate becomes varying This rate is governed by the equation f u f u (7) where f u sin u f u sin u Solution of (7) is (assume u ) u Fu e f xe dx where Fx (8) u sin 4 (9) F u f x dx u u Integral in (8) poses a problem Rewrite it as u u u x x x sin x 4 4 sin Fx sin g u f x e dx e dx x e dx sin u u u x sin x 4 4 e e e dx Hence u Fu Fu x sin x 4 e e e e dx Note that sinx ngular velocity is bounded by following relations u sinu sin sin sin 4 4 u u u u 4 4 e e e e e e ()

13 Influence of the second term in (9) vanishes as argument of latitude rises This term may be omitted given appropriate time interval and u u Fx x g u f xe dx sin xe dx u e sin u 8 u 6 sin u 4cosu e 6 8 Substituting into (9) and (8) brings angular velocity of rotation around the maximum moment of inertia u sinu u 4 e sin u 8 u e 6 sin u 4cos u e 6 8 () ngle in generating solution is found integrating () First order equations are x F u x g u where F F u u u u i 4 i are trigonometric functions (sine and cosine) They appear in relations for F components linearly (exact expression of F can be found in appendix) These equations cannot be solved pproximate solution of the first order equations is found using Poincare method again Small parameter is (control torque value) Solution of the equations of motion is represented as x x x x y y x u u matrix F is u u F F F Generating solution y and first order approximation y are found from equations z F z () z F z F u z g u () and cosu where z Independent equations yield are constant Expressions in ()-() contain u We use () to find its approximate value This expression is decomposed assuming small parameter and sufficient time interval

14 9 sinu (4) This leads to 9 sin u 9 sin8u 9 cosu9 cos8u F g 9 u 9 Matrix F u G u G u G u F is written as u cos6u sin6u sin6u cos6u sin6u sin6u cos6u sin6 u components G G u u u u u i i 4 6 (exact expression for F can be found in appendix) y Generating solution for () is cosu sin u sin u cosu D cos8u E sin8u cosu 4sin u Dsin8u E cos8u cosu sin u where 8 D E are constant They i i are found from the existence conditions of periodical solution of equations () F u z g u does not contain frequencies This implies that heterogeneous part F u and u and 8 (Eigen frequency of F ) g contain only rational numbers so Constants D E are found from absence of frequency 8 in G uz g u (matrix u G may be omitted) so E D 4 9 pproximate solution of equations of motions is 4 9 i cos8 u 4 9 i sin8 u is governed by one of the expressions () () (4) can be found integrating xis of the maximum moment of inertia deviates from the orbit normal by the angle

15 i u i u Decomposing this analogous to (6) provides 4 acos cos 4 9 cos8 cos 4 9 sin8 slightly less for near-polar orbits 4 Numerical simulation 4 9 i Deviation is Numerical simulation was carried out with following parameters: J 46 kg m Inertia tensor Orbit inclination 7 altitude 7 km Initial conditions: attitude angles angular velocity components Control gain N m s/т kg m (approximately ) Fourth order Runge-Kutta with constant time step s was used Fig brings simulation results for the right dipole model Fig presents analogous result for inclined dipole model Fig Near polar orbit right dipole model Fig Near polar orbit inclined dipole model Fig verifies analytical results xis of maximum moment of inertia is stabilized with -7 accuracy while analytical expression provides 6 Inclined dipole model observes almost three times worse accuracy Horizontal line on figures corresponds to angular velocity 8 Near equatorial orbit yields same resultsю Figures 4 and provide results for orbit inclination 6 Right dipole model is used in both cases Gravitational torque is taken into account for Fig

16 Fig 4 Inclination 6 right dipole model Fig Inclination 6 gravitational torque rbitrary orbits that far from near polar or near equatorial yield spinning axis pointing accuracy of about - Gravitational torque acts as a restoring one Moreover its magnitude is almost comparable with the control magnetic torque s a result stabilization accuracy is better Conclusion Magnetically actuated satellite with «-dot» damping algorithm is considered Slow motion of the satellite is studied Rotation around the maximum moment of inertia is proved to be asymptotically stable in simplified dipole geomagnetic field Rotation velocity is found in right dipole field Maximum moment of inertia axis attitude in inertial space is found ibliography Ovchinnikov MY et al Investigation of the effectiveness of an algorithm of active magnetic damping // Cosm Res V pp 7 76 Ovchinnikov MY Roldugin DS Dual-spin satellite motion in magnetic and gravitational fields // Keldysh Institute Preprints p Stickler C Magnetic Control System for ttitude cquisition // Ithaco Inc Rep N Stickler C lfriend KT Elementary Magnetic ttitude Control System // J Spacecr Rockets 976 V pp 8 87 Pichuzhkina V Roldugin DS Geomagnetic field models for satellite angular motion // Keldysh Institute Preprints 6 87 p 6 Malkin IG Some problems in the theory of nonlinear oscillations Oak Ridge: US tomic Energy Commission Technical Information Service p

17 ppendix Some relations f f f f 4 f f f f 4 Fu cos u 9 f cos u cos cos u cos u f sin u sin u sin f cosu cosu cos u cos u cos 8 8 F u 9 g sin u g g g 4 9 sin u g g g g 4 sin u sin u 9 g cos4u cos6u cos6u cos u g sin6u sin 4u sin 6 u g cos u cos6u cos4u cos6 u 4 4 4