Magnetic attitude control for satellites

Transcription

1 Magnetic attitude control for satellites Jan Tommy Gravdahl, Department of Engineering Cyernetics Norwegian University of Science and Technology N-7491 TRONDHEIM, Norway Astract In this paper it is proven that three-axis stailization of satellites using magnetorqers is uniformly gloally asymptotically stale (UGAS). The application of Matrosovs theorem allows for removal of the assumption that the geomagnetic (internal and external) eld is periodic in order to estalish this staility result. The results are applied to a model of a Norwegian pico-satellite. I. INTRODUCTION Active control of the attitude of a satellite can e achieved with a numer of different actuators: reaction wheels, thrusters or magnetic coils. The development of a Norwegian pico-satellite required a control system that is oth inexpensive and lightweight. This motivated the use of magnetic coils. As shown in [1], magnetic threeaxis stailization of a spacecraft can e achieved y using feedack from magnetic eld measurements and angular velocity. Asymptotic staility can then e estalished y assuming that the internal geomagnetic eld (the component of the eld having its cause in the iron core of the earth), as seen from the spacecraft, is periodic and then applying the Krasovskii-LaSalle theorem. However, the magnetometer on oard the satellite will not only measure the internal geomagnetic eld, ut also the external magnetic eld and a disturance eld induced y the spacecraft electronics, thus prohiiting the assumption of a periodic eld. In this paper periodicity of the magnetic eld will not e assumed, and UGAS will e proven for this time-varying system y using Matrosov's theorem. II. THE NCUBE SATELLITE On mission from The Norwegian Space Center and Andøya Rocket Range, four Norwegian universities and educational institutions have since 2001 participated in a program to develop a pico satellite. The satellite, ncue, is eing designed and uilt y MSc-students, in accordance with the Cuesat concept: Mass is restricted to 1 kg and size is restricted to a cue measuring 0:1 m on all sides. A prototype of ncue is shown in Figure 1. The four partners are Narvik University College, Norwegian University of Science and Technology (NTNU), Agricultural University of Norway, and University of Oslo. The Attitude Determination and Control System (ADCS) is the responsiility of the Department of Engineering Cyernetics at NTNU. The main mission of the satellite is to demonstrate ship trafc surveillance from a LEO satellite using the maritime Automatic Identication System (AIS) recently introduced Fig. 1. Prototype of ncue illustrating the size and solar panels. y the International Maritime Organization (IMO). The AIS system is ased on VHF transponders located on oard ships. These transponders roadcast the position, speed, heading and other relevant information from the ships at regular time intervals. The main ojective of the satellite is to receive, store and retransmit at least one AIS-message from a ship. Another ojective of the satellite project is to demonstrate reindeer herd monitoring from space y equipping a reindeer with an AIS transponder during a limited experimental period. In addition, the satellite should maintain communications and digipeater operations using amateur frequencies. A third ojective is to demonstrate efcient attitude control using passive gravity gradient stailization and active magnetic torquers. The satellite will e placed in a low earth sun synchronous orit with a perigee of approximately 700 km, and as circular as possile. The inclination will e close to 98. The launch is scheduled to early For further information on the ncue project and detailed description of the satellite the reader is referred to [2] or [3] The attitude and angular velocity of the spacecraft will e determined y integrating measurements of the magnetic eld and measurements of the sun vector in an extended Kalman lter. The magnetic eld will e measured using a

2 Fig. 2. Photo of one of the torque coils yaw which is the rotation around the x-, y-, and z-axis respectively. The orit reference frame is denoted o. Body Reference Frame The ody frame shares it's origin with the orit frame and is denoted. The rotation etween the orit frame and the ody frame is used to represent the spacecraft's attitude. It's axes are locally dened in the spacecraft, with the origin in the center of gravity or the center of the volume. The nadir side of the spacecraft, intended to point towards the earth, is in the z-axis direction. The rotation matrix R from frame a to frame is denoted R a. Rotation matrices are memers of the special orthogonal group of order three: R 2SO(3)= R R2R 33 ; R T R=I; det R = 1 (1) Fig. 3. Photo of the nadir surface of Ncue. The gravity oom is coiled up and restrained inside a ox. digital magnetometer and the sun vector will e measured y monitoring the current output from the solar panels which cover all the sides of the cue. For actuation of the satellite, magnetic torque coils will e used. A prototype coil is shown in Figure 2. A gravity oom is constructed from measuring tape which will act as the oom rod. A tip-mass made of lead is attached to the oom. The coiled up oom is shown in the lower middle in Figure 3. For details on sensors and actuators and the construction of the satellite in general, see [4]. A. Satellite model III. MODELING 1) Coordinate frames and kinematics: Earth-Centered Inertial (ECI) Reference Frame The origin of this frame is located in the center of the earth. This reference frame will e denoted i, and the earth rotates around its z-axis. The x-axis points towards the vernal equinox. Orit Reference Frame The orit frame origin coincides with the spacecraft center. The origin rotate at an angular velocity! o relative to the ECI frame and has its z-axis pointed towards the center of the earth. The x-axis points in the spacecraft's direction of motion tangentially to the orit. The satellite attitude is descried y roll, pitch and where I is the 3 3 identity matrix. A transformation of a vector r from frame a to frame is written r = R ar a. From the orthogonal property in (1), it can e shown [5] that the time derivative of R a can e written as _R a = S (! a a) R a = R as! a (2) where! a is the angular velocity of frame relative to frame a represented in frame, and S (!) =! is the cross product operator. Generally, the matrix R o can e written as R o = c 1 c 2 c 3 ; (3) where c 1 = c T ix c iy iz c are column vectors. The c 3 vector is the projection of the z o -axis in the ody frame. If c 3 = (0 0 1) T, the z -axis is aligned with the z o -axis. In this paper c iz will e frequently used as a measurement of deviation etween the z -axis and the z o -axis. From (2) and (3), it follows that _c i = S(c i)! o: (4) 2) Dynamics: Using Euler's moment equation, the attitude dynamics of the satellite can e derived as J _! i +! i J! i = ; (5) where J is the inertia matrix of the satellite,! i is the angular velocity of the -frame with respect to the i-frame, decomposed in the -frame, and = m + g + a + s + m is the torque acting on the satellite. The torque m generated y the magnetorquers can e modelled as m = m B ; (6) where m is the magnetic dipole moment generated y the coils and B = (B x B y B z ) T is the local geomagnetic eld vector. The magnetic dipole moment is given y m = m x + m y + m z N xi x A x N y i y A y N z i z A z A m x m y m z A ;

3 Disturance Magnitude [Nm] Gravity gradient 1: Aerodynamic drag 3: Solar radiation 1: Internal electronics eld TABLE I DISTURBANCE TORQUES FOR NCUBE where N k is the numer of windings in the magnetic coil on the axis in the k-direction, i k is the current in the coil and A k is the coil area. The disturance torques g ; a ; s and m will e dened later. The angular velocity of the ody frame with respect to the orit frame can e found y! o =! i R o! o io =! i! o c 1: B. Disturance torques A satellite is suject to small ut persistent disturance torques. Unless actively resisted the disturances will reorient the satellite. The main disturances are riey discussed elow. The discussion is ased on [6] and [7]. a) Gravity gradient torque: The gravity gradient torque g, written in the -frame as g = 3! 2 0c 3 Jc 3 ; (7) where! 0 =R 3 0 1=2, = GM is the Earth's gravitational coefcient and R 0 is the distance to the Earth's center, will affect a non symmetric ody in the Earth's gravity eld. This effect can e exploited, and will e in the case of ncue, with a gravity oom for passive stailization. ) Magnetic elds: The electronics in the satellite may create an unwanted residual magnetic dipole. This eld will interact with the Earth's geomagnetic eld. The resulting torque can e written as m = DB; where D is the residual dipole of the satellite and B is the Earth's magnetic eld. Also, disturances may have their source in the external geomagnetic eld. This eld varies on a faster time-scale than the internal geomagnetic eld. Moreover, the external eld may very in an unpredictale manner. c) Other: Other disturances that are taken into account for the ncue satellite, ut not discussed in detail here, include aerodynamic torque, a, solar radiation torque, s ; and oom distortion. In Tale I, the worst case numerical values for ncue are presented. A. Energy considerations IV. CONTROLLER DESIGN An important tool in control theory is the use of energyased controllers ased on Lyapunov designs and passivity [5]. In this section expressions for the satellite's energy is presented, and a suitale Lyapunov function candidate and its derivative is found. The energy of the satellite can e divided into kinetic and potential energy. The kinetic energy is mainly due to rotation in the inertial and orit frame, while the most important source to potential energy is the gravity gradient and gyro effects due to revolution aout the Earth. The expressions for kinetic and potential energy is ased on [8], [9] and [10]. From a control perspective the rotation of the ody frame with respect to the orit frame is most interesting. Assuming a near circular orit, and therefore a constant orital rate! o, the kinetic energy can e written E kin = 1 2! o T J! o : (8) The potential energy due to the gravity gradient is E gg = 3! 2 o c T 3 Jc 2 3 I z ; (9) and the potential energy due to the revolution of the satellite aout the Earth is given y E gyro = 1 2!2 o I x c T 1 Jc 1 : (10) Dening x =! o c 21 c 31 c 13 c 23 T 2 R 7 ; and using (8), (9) and (10), it can e seen that the energy function V dened y V (x) = E kin + E gg + E gyro = 1 2! o T J! o + 3 2!2 o (I x I z ) c (I y I z ) c 2 23 (11) + 1 2!2 o (I x I y ) c (I x I z ) c 2 31 satises V (0) = 0: The simplications in the last two terms of (11) follows from the fact that R o is orthogonal. In order to ensure that V is positive denite, that is V > 0 8x 6= 0; we require that I x > I y > I z : For use in the staility analysis of the controller we need an expression for the time derivative of (11) _V =! T o J _! o + 3! 2 o c T 3 J_c 3! 2 o c T 1 J_c 1 : (12) It follows from (5) and (7) that the satellite dynamics, considering the gravity gradient and magnetic coil torques only, can e written as J _! i +! i J! i = 3! 2 0c 3 Jc 3 + m: (13) Using (4), (13) and the relations! i =! o +! 0 c 1 and! o T S(! o ) = 0; (12) is written _V =! T o 3! 2 0S(c 3)Jc 3 + m! 0 JS(c 1)! o! 0 S(c 1)J! o! 2 0S(c 1)Jc 1 (14) +3! 2 o c 3 T JS(c 3 )! o! 2 o c 1 T JS(c 1 )! o:

4 Since S T (x) = S(x), (14) is reduced to _V =! o T m : (15) Remark 1: The equilirium given y x =! o c 21 c 31 c 13 c 23 T = 0; corresponds to four equiliria! o c 3 c 1 T = (0 c o 3 c o 1) T for the satellite, [8]. B. Detumling When the satellite is released from the launcher it will have an initial angular velocity. Before the oom can e deployed, angular velocity must e reduced and the ody frame must e aligned with the orit frame. During detumling the kinetic energy of the satellite is dumped and the angular velocity of the ody frame with respect to the inertial frame is to e reduced to a value elow! o < T : The only sensor availale in this mode will e the magnetometer. After the rate detumling phase the satellite may have an aritrary attitude. Before the oom can e deployed we must ensure that the ody z -axis is aligned with the orit z o -axis. If the oom is deployed in the opposite direction it may e difcult to turn the satellite. For oom deployment we require that the deviation etween the z and z o axes is less than 30. The ojective of the rate detumling controller is to dissipate the kinetic energy of the satellite. A controller which uses only rate measurements from the magnetometer is suggested elow. The controller is proposed in [10], [11] and [9]. Proposition 1: The control law m = k _B m c ; (16) where m c = (0 0 m c ) T will dissipate the kinetic energy of the satellite and align it with the local geomagnetic eld. Proof: To prove that the energy is dissipated, Lyapunov theory will e used. The proof is ased on [10] and [11]. Comining (6) and (16), the control torque m gives m = m B = k _B m c B : (17) We note that the magnetic eld vector B can e written as B = R i Bi, and consequently _B = _R ib i + R i _ B i = B! i + R i _ B i : (18) Near the North and South Poles, B is approximately constant. Equation (18) can therefore e approximated as _B B! i : This assumption is valid only in the polar regions. When the oom is stowed, the gravity gradient will e very small and can e neglected, however the constant term in (16) will contriute to the potential energy. Thus, the sum of kinetic T and potential energy U is V = T + U = 1 2! i T J! i + jm c j B + m T c B : (19) Assuming a constant magnitude of the geomagnetic eld in the polar regions, the time derivative of (19) is _V =! i T m + m T c _ B ; and using (17) it follows that _V =! T i k _B = k B _ T B _ m c B + m T c _ B which is negative semidenite. We conclude that energy is dissipated and the angular velocities are reduced. Remark 2: The z -axis will tend to point along B i. This is not shown in the analysis aove, ut a proof can e found in [9] or [11]. Near the poles B i points vertically upwards, meaning that in the polar regions the deviation of the z - axis from nadir will e relatively small. This can e utilized for oom deployment. C. Stailization We will here use the same control law as in [10] and [1] with the correction suggested y [12].. Asymptotic staility of Wisniewski's controller was proven y assuming Earth's magnetic eld to e periodic and then using the Krasovskii-LaSalle theorem [13]. As discussed in Section III-B.0., the magnetic eld may vary with time in an unpredictale manner. Our contriution to previous work is to use Matrosov's theorem [14] as stated in [15] in order to prove uniform gloal asymptotic staility (UGAS) of the equilirium without assuming periodicity of the geomagnetic eld B (t): Proposition 2: The control law m = H! o B ; (20) makes the origin, x = 0 of the system (13), (4) GUAS. Proof: Dene I z V 1 = V = 1 T 2! o J! o !2 o! 2 o c T 3 Jc 3 I x c T 1 Jc 1 ; (21) where the LFK in equation (11) has een used. Using (20) and calculating the time derivative of (21) along the trajectories of (13), (4) results in _V 1 =! o T H! o B B (22) =! o T S T B HS B! o = Y 1 0; as shown in equations (12) to (15). The Lyapunov function candidate (21) is positive denite and its time derivative (22) is negative semidenite. It follows that the origin is UGS, and Assumption 1 of Theorem 1 in [15] is satised. Moreover, Assumption 2 is satised for i = 1. Dene the auxiliary function V 2. V 2 = c 3 J T S T (c 3 )J! o; where the superscript on c 1;2 has een dropped for notational convenience. Now, _V 2 = _c 3 J T S T (c 3 )J! o c 3 J T S T (_c 3 )J! o c 3 J T S T (c 3 )J _! o:

5 As the system is stale, it follows that the states are ounded. Using the same notation as in [15], we let the numer denote a generic ound on continuous functions, and _ V 2 can e upper ounded as _V 2 c 3 J T S T (c 3 )J _! o + 1! o ; and using (13) further ounded as _V 2 c T 3 J T S T (c 3 ) 3! 2 0S(c 3 )Jc 3! 2 0S(c 1 )Jc 1 +2! o The second term in _ V 2 satises! 2 0c T 3 J T S T (c 3 )S(c 1 )Jc 1 =! 2 0c T 3 J T S(c 3 )S(c 1 )Jc 1 so that =! 2 0c T 3 J T c 1 c T 3 c T 3 c 1 I Jc 1 =! 2 0c T 3 J T c 1 c T 3 Jc 1 =! 2 0 c T 3 J T c 1 2 < 0 _V 2 3! 2 0c T 3 J T S T (c 3 )S(c 3 )Jc 3 + 2! o : Now, we see that Y 1 0 implies _V 2 2! 2 0c 3 J T S T (c 3 )S(c 3 )Jc 3 = Y 2 0; and Assumption 2 and 3 are satised for i = 2. Dening the auxiliary function V 3 as V 3 = c 1 J T S T (c 1 )J! o; and calculating its time derivative in the same manner as for _ V 2 results in _V 3 = _c 1 J T S T (c 1 )J! o + c 1 J T S T (_c 1 )J! o +c 1 J T S T (c 1 )J _! o c 1 J T S T (c 1 )J _! o + 3! o c 1 J T S T (c 1 ) 3! 2 0S(c 3 )Jc 3! 2 0S(c 1 )Jc 1 + 4! o! 2 0c 1 J T S T (c 1 )S(c 1 )Jc 1 + 4! o + 5 jc 3 j ; where the same technique as for _ V2 has een used. Now Y 1 Y 2 0 implies _V 3! 2 0c 1 J T S T (c 1 )S(c 1 )Jc 1 = Y 3 0; and Assumption 2 and 3 are satised for i = 3: Finally Y i = 0; i = f1; 2; 3g =) x = 0;and Assumption 4 of [15] is satised and the result follows. Remark 3: In the two equiliria! o c 3 c 1 = (0 c o 3 c o 1) the oom is pointing in the wrong direction. V. SIMULATIONS Now the controllers for detumling and stailization will e simulated. The parameters of the model used in the simulations are: Body size cm, oom length: 1:5 m, moments of inertia, oom stowed: I x = 0:0621 kgm 2 ; I y = 0:0606 kgm 2 ; I z = 0:0031 kgm 2 ; moments of inertia, oom deployed: I x = 0:3210 kgm 2 ; I y = c 3z [rad/s] w o Power consumption [W] Orits Fig. 4. Detumle mode simulation 0:1806 kgm 2 ; I z = 0:0031 kgm 2 ; maximum magnetic moment from the coils: 0:1 Am 2 : Feedack from the magnetic eld is done using the The International Geomagnetic Reference Field, IGRF. It is an approximation, near and aove the Earth's surface, to that part of the Earth's magnetic eld which has its origin in the earths core. The IGRF species the numerical coefcients of a truncated spherical harmonic series and together with an orit estimator, an estimate of the magnetic eld is made. A. Detumling mode The detumling mode controller (16) was simulated with initial values:! i = (0:1 0:1 0:09)T rad=s and controller parameters: k = 10 4 and m c = 0:01 Am 2 : In Figure 4 it is shown that the angular velocities are quickly reduced and the z -axis aligns itself with the geomagnetic eld vector. B. Stailization mode In Figure 5, the Euler angles of the satellite using the stailizing controller (20) is shown. A disturance (white noise) with amplitude 10 5 T has een added to the magnetometer measurements, and as can ee seen the satellite is stailized. The magnetic moments from the coils are shown in Figure 6. The total energy V of the satellite as well as the angular velocity is shown in Figure 7. VI. FURTHER WORK Ongoing work in the ADCS-part of this project includes nal design of the determination system and implementation of the complete ADCS on microcontrollers. The nal satellite will undergo assemly and testing during autumn 2004 and launch is planned for spring VII. CONCLUDING REMARKS It has een shown that the equilirium of a time-varying system consisting of spacecraft attitude dynamics and a nonlinear control law using feedack from magnetic eld measurements and angular velocity is UGAS. Magnetic w x w y w z

6 coils are used as actuators and the staility result is estalished y using Matrosovs theorem. The theoretical results have een con rmed y simulations on a model of the picosatellite ncue. VIII. ACKNOWLEDGEMENTS The former Dept. of Engineering Cyernetics students B. Busterud, K.M. Fauske, K. Svartveit, F. Indergaard and E.J. Øvery have all contriuted to the design of the ADCS for ncue. R EFERENCES The stailized attitude of ncue Magnetic moment from the coils Fig. 5. Time The magnetic moment from the three coils Angular velocity Energy Fig. 6. Time Fig. 7. Energy and angular velocity [1] R. Wisniewski and M. Blanke, Fully magnetic attitude control for spacecraft suject to gravity gradient, Automatica, vol. 35, pp , [2] we site of the Ncue project, March 1st, [3] Å.-R. Riise, B. Samuelsen, N. Sokolova, H. Cederlad, J. Fasseland, C. Nordin, J. Otterstad, K. Fauske, O. Eriksen, F. Indergaard, K. Svartveit, P. Fureotten, E. Sæther, and E. Eide, Ncue: The rst norwegian student satellite, in In Proceedings of The17th AIAA/USU Conference on Small Satellites, [4] J. Gravdahl, E. Eide, A.Skavhaug, K. Svartveit, K. M. Fauske, and F. M. Indergaard, Three axis attitude determination and control system for a picosatellite: Design and implementation in Proceedings of the 54th International Astronautical Congress. Bremen, Germany: IAF, [5] O. Egeland and J. Gravdahl, Modeling and Simulation for Automatic Control. Trondheim, Norway: Marine Cyernetics, [6] J. Wertz, Space Mission Analysis and Design, 3rd ed. London: Kluwer academic pulishers, [7] R. Kristiansen, Attitude control of mini satellite, Master's thesis, NTNU, [8] P. Hughes, Spacecraft Attitude Dynamics. New York: Wiley, [9] P. Soglo, 3-aksestyring av gravitasjonsstailisert satelitt ved ruk av magnetspoler, Master's thesis, NTNU, [10] R. Wisniewski, Satellite attitude control using only elecromagnetic actuation, Ph.D. dissertation, Dept. of Control Engineering, Aalorg University, Denmark, [11] O. Egeland, Norwegian ionospheric small satellite experiment attitude control and determination system, Sintef, Tech. Rep W, [12] C. Damaren, Comments on "fully magnetic attitude control for spacecraft suject to gravity gradient", Automatica, vol. 38, p. 2189, [13] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, [14] V. Matrosov, On the staility of motion, J. Appl. Math. Mech, vol. 26, pp , [15] A. Loria, E. Panteley, D. Popovic, and A. Teel, An extension of matrosov's theorem with application to stailization of nonholonomic control systems, in Proccedings of the 41st IEE Conference on Decision and Control, Las Vegas, NV, 2002, pp