Orbit Boosting of an Electrodynamic Tethered Satellite with Input-Shaped Current

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1 Orbit Boosting of an Electrodynamic Tethered Satellite with Inpt-Shaped Crrent Arn Banerjee Lockheed Martin Advanced Technology Center Palo Alto, CA and William Singhose and David Blackbrn Mechanical Engineering Dept Georgia Institte of Technology, Atlanta, GA Introdction Ever since the proposition of the tethered satellite concept by Colombo et al. [1], se of the tether as a crent carying condctor moving in the earth s magnetic field to generate a proplsive force has been investigated by several researchers. Penzo and Amman [2] gave a model of the electrodynamic forces on a tether. Netzer and Kane [3] sed this with a seqence of crrent implses for orbit boost with a hinged rigid body model of the tether. Lanoix, Misra, Modi, and Tyc [4] made a stdy of the parameters inflencing electrodynamic forces on tethers. Vas, Kelley, and Scarl [5] discssed electrodynamic reboost of the space station. Williams et al. [6] sed a flexible tether model of a elecrodynamic tether for deployment and retrieval control of a tethered satellite. The stability of electrodynamic tethered satellites in elliptic orbits has been investigated by Pelaez and Andres [7]. This paper re-examines the se of the electrodynamic force to boost the orbit of a tethered satellite, the objective being to start and end the orbit transfer with the tether aligned as close to the local vertical as possible. The process is exected by trning on the crrent in the tether for a time to generate the force-time implse needed for the orbit transfer. Unfortnately, the electromagnetic force indced in the tether is perpendiclar to it, and will case in-plane and ot-of-plane librations, as well as string vibrations of the tether. Redction of vibrations associated with any manever by shaping the inpt to the system has been demonstrated for robstness to model error in Ref. [8]. References [9],[10] are examples of extensions of the concept. In this paper we do a robst shaping of the crrent inpt into the tether moving in the earth s magnetic field, to achieve both orbit transfer and redced tether oscillations. A particlar point of interest in modeling is the consideration, sggested in Ref. [4], of a two-stage tether, part of it being crrent-condcting and the remainder not. This is examined in the nmerical simlations. Modeling The tethered satellite system is modeled as a system of n orbiting particles P 1, P 2,, P n connected by springs, basically as in Ref. [1]. We se an earth-centered inertial frame with basis vectors n i EP and define the inertial components of the position vector p 1 from earth E to the first particle P 1, as the first three generalized coordinates; for the remaining particles we se the inertial components of the relative position vectors of one particle from its inboard particle as its generalized coordinates: EP q p 1 n i 1,2,3 (1) i i

2 1 P j q 3 j 3 i p n i j 2,..., n ; i 1,2,3 (2) Interaction force de to springs applied by P j on P j-1 does not then reqire sbtracting what wold otherwise be two nearly eqal large nmbers, and is simply F P j / P j 1 s k ( p 1 P j L j ), j 2,..., n Here k is the longitdinal stiffness of the tether segment of length L and j is the nit vector in the direction from P j-1 to P j. External force on particle P j of mass m j de to gravity is E m jp Fg, j 1,...,n E E 1.5 ( p p ) Aerodynamic drag is compted from Fa 0.5C dav v, j 1,..., n (5) The electrodynamic force on a tether segment of length L moving in the earth s magnetic field vector B with a crrent I flowing along the segment direction j is given by F P j e j I L B (6) The magnetic field of the earth [2] can be evalated at the midpoint R j of the tether segment as de to a magnetic dipole and can be written as 3 e R e[cos n 2sin r ] B ER j ER j 1.5 ( p p ) sin n r m r r r m m m sin (cos e t n1 sin et n 2 ) cos n 3 Here r is the nit vector from E to R j and R e is the earth eqatorial radis, e is earth s magnetic constant, 3.5e-5 Wb/m 2 m is the nit vector along the earth s magnetic axis, e is the earth rotation rate and is the tilt angle between the earth s spin axis and the magnetic axis. Once the electromagnetic force vector is compted at the midpoint of a tether segment, it can be apportioned by half and lmped on the two ends of the segment. The eqations of motion becomes then (3) (4) (7) m a F s F g F a F e, j 1,...,n (8) where the terms within the anglar brackets <> are trned off if either aerodynamic drag is ignored or if no crrent is flowing in a given segment of the two-stage tether. In this paper electrodynamic force was considered to be acting only on the half of the tether nearest the main satellite, while aerodynamic force was assmed to act only on the lower half. The exact eqations of three-dimensional motion are generated with the symbol maniplation code, AUTOLEV [11]. Integration of these differential eqations allows the measrement of tether vibrations as well as the tether libration, which are given by the in-plane angle and ot-of-plane angle, which can be evalated for this model as follows.

3 ρ p EPi / p EPi ν ρ V d p P1 Pn sin 1 Pi τ ν ρ / ρ V ν (d / p P1 Pn Pi ) (9) 1 P P P P tan [( p 1 n d ν) τ /( p 1 n d ν) ( ρ)] The bowing deflection of the tether at the midpoint can be measred, for example, for a forelement tether connecting five particles by P1 P3 P1 P5 P1 P5 p p / p (10) Initial and Transfer Orbit A parametric stdy of the electrodynamic forces on tethered satellites have been given in Ref. [4]. Based on this work, orbit plane inclination of 50 deg was chosen for a 20 km tether with a large mass connecting an end mass. The initial condition for the system mass center was selected for a circlar orbit at 500 Km altitde. To raise the orbit of the system mass center, the reqired change in orbit anglar momentm was eqated to the time-integral of the moment of the component of the electrodynamic force in the direction of the in-plane velocity vector. Approximating that the force remains constant, time dration of the plse to raise the orbit from a radis R 1 to R 2 can be compted for a total system mass M from R 2 R 1 t (11) F(R1 R 1) 2M where F F e τ. Simlations sggested that a good approximation to this time is that given by the total implse time for a Hohmann transfer, namely, by the set of eqations ( / R )(R R ) /(R R ) (12) v v 2 ( / R 2 )(R 2 R 1) /(R1 R 2 ) (13) t M( v 1 v2 ) /( Fe τ ) (14) Inpt Shaping of Crrent One way of orbit transfer and vibration sppression sing on-off thrsters has been described in Ref. [10]. Once the crrent level and the reqired dration of the rectanglar force plse is compted, the simlation shows the orbit plsations and the in-plane and ot-of-plane librations as well as the violin string vibrations of the tether. The tether crrent shaped profile is then simply a convoltion of the rectanglar plse with the three implse seqence [8] for each of these for vibration modes.

4 Simlation of Orbit Transfer Nmerical simlation of orbit transfer of a tethered satellite with a two-stage electrodynamic tether sing inpt-shaped crrent will be reported in the paper. The initial position and velocity of the particles, needed for integration of the differential eqations of motion, can be derived sing AUTOLEV for an assmed set of orbital elements that define the orbit of the system mass center, sing well-known procedres of orbital mechanics [12]. The system parameters sed for the simlation are: Mass of satellite 300 kg Tether mass 20 Kg End Mass 5 Kg Initial circlar orbit altitde 500 Km Tether total length 20 Km Length of tether segment condcting crrent, pper 10 Km Length of tether with aerodynamic drag, lower 10 Km The simlation can be sed to stdy sensitivities of the dynamics to system parameters sch as orbit radis, mass of the main satellite, length and axial stiffness of the tether. Then an optimal crrent inpt shaper can be designed to sppress the libration and vibration of the tether to certain levels, no matter what combination of parameters the system actally has. The periods of vibration for the nshaped crrent are: Orbit altitde plsation period: 5817 sec In-Plane period: 3398 sec Ot-of_Plane period: 2828 sec Tether String period: 652 sec Figre 2 shows a raising of the orbit altitde by 3.5 Km with crrent shaping, and orbit altitde plsations withot shaping. In effect, this shows that the shaped electrodynamic force circlarizes an otherwise elliptic orbit. Figre 3 and 4 plot the in-plane and ot-of-plane angles with and withot shaping. Figre 5 shows the bowing of the tether with a violin string mode, redced by crrent shaping from the nshaped sitation. The shaped crrent profile that prodced these reslts is shown in Fig. 6. It is seen that simple zero vibration and derivative shaping of the crrent is qite effective in orbit raising as well as redcing amplitdes of tether librations and string vibrations excited. References 1. G. Colombo, E. M. Gaposchkin, M. D. Grossi, and G. C. Weiffenbach, "Shttle-Borne Skyhook, A New Tool for Low-Orbital Altitde Research," Smithsonian Astrophysical Observatory, Cambridge, Massachsetts September Penzo, P.A., and Ammann, P.W., Tethers in Space Handbook, NASA, Washington, DC, VVC

5 3. Netzer, E. and Kane, T.R., Electrodynamic Forces in Tethered Satelite Systems Part I: System Control, IEEE Transactions of Aerospace and Electronic Systems, vol. 30, no. 4, October 1994, pp Lanoix, E., Misra, A., Modi, V., Tyc, G., Efect of Electromagnetic Forces on the Orbital Dynamics of Tethered Satelites, Proceedings of the AAS/AIAA Space Flight Mechanics Conference, vol. 105, ED., C. Klever, B. Neta, C. Hall, and J. Hansen, Univelt, SanDiego, 2000, pp Vas, I.E., Keley, T.J., and Scarl, E.A., Space Station Reboost with Electrodynamic Tethers, Jornal of Spacecraft and Rockets, March-April, 2000, pp Wiliams, P., Watanabe, T., Blanksby, C., Trivailo, P., Fji, H., Libration Control of Flexible Tethers Using Electromagnetic Forces and Movable Atachment, Jornal of Gidance, Control, and Dynamics, vol. 27, no.5, 2004, pp Pelaez, J. and Andres, Y.N., Dynamic Stability of Electrodynamic Tethers in Inclined Orbits,Jornal of Gidance, Control, and Dynamics, vol. 28, no.4, 2005, pp Singer, N. C. And Seering, W. P., Preshaping Command Inpts to Redce System Vibration, Jornal of Dynamic Systems, Measrement, and Control, vol. 112, March 1990, pp Singhose, W., Banerjee, A.K, and Seering, W. E., Slewing Flexible Spacecraft with Deflection Limiting Inpt Shaping, Jornal of Gidance, Control, and Dynamics, 1997, vol. 20, no. 2, pp Banerjee, A. K., and Diedrich, B., Spacecraft Vibration Redction Folowing Thrster Firing for Orbit Adjstment, Jornal of Gidance, Control, and Dynamics, Jly-Agst, 2003, pp Kane, T. and Levinson, D., AUTOLEV User s Manal, 2001, Online Dynamics, Inc., Snnyvale, CA. 12. Prssing, J.E. and Conway, B.A., Orbital Mechanics, Oxford University Press.

6 X Electro Dynamic Force Crrent Y Magnetic Field Orbit Path Earth Figre 1. Schematic of the Tethered Satellite Model Orbital Altitde (km) With Shaping Withot Shaping 498 Figre 2. Orbit Altitde vs. Time with and withot crrent shaping

7 In-Plane Angle (deg) With Shaping Withot Shaping -15 Figre 3. Tether in-plane angle with and withot crrent shaping Ot of Plane Angle (deg) With Shaping Withot Shaping -10 Figre 4. Tether ot-of-plane angle with and withot crrent shaping

8 Midpoint Deflection (m) With Shaping Withot Shaping 0 Figre 5. Tether violin string vibrations with and withot crrent shaping Crrent (A) Shaped Inpt Unshaped Inpt -1.6 Figre 6 Shaped and nshaped tether crrent profiles

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