JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 18, No. 6, November-December 1995 Magnetic Precession and Product-of-Inertia Nutation Damping of Bias Momentum Satellites Hari B. Hablani* Rockwell International Corporation, Downey, California 90241-7009 This paper is concerned with large-angle precession of Earth-pointing bias momentum satellites in circular orbits and small-angle precession in eccentric orbits. In the first part, ideal torque requirements are determined for precessing a momentum vector by a large yaw angle. It is then shown that a quasi-quarter-orbit open-loop magnetic precession is feasible only when the spacecraft rotation for Earth pointing is halted. For this circumstance, general relationships are developed to size a pitch dipole using a tilted dipole model of the geomagnetic field and quasi-quarter-orbit apart true anomalies whereat a pitch dipole reverses its polarity. The second part of the paper considers elliptic orbits. A pitch dipole sizing equation is developed for small-angle open-loop roll/yaw magnetic precession. This equation is then used for designing a closed-loop bang-bang precession scheme, working in concert with nutation damping via a roll/pitch product-of-inertia. A literal relationship is devised between the desired nuta- tion damping coefficient (or time constant), roll/pitch product-of-inertia, and rate gain of a linear pitch controller. I. Introduction F OR efficient operation of Earth-pointing spacecraft, thermal radiators are placed so as to face the cold side of the orbit, and solar arrays are placed so as to face the warm side. Every few months, however, due to Earth's motion around the sun and due to Earth's oblateness, orbit planes of some spacecraft regress by nearly 180 deg, causing a transfer of the sun from one side of the orbit to the other side. This in turn exposes the radiators to the sun, and the array may begin to be occulted by the spacecraft bus. Moreover, if the solar array is fixed to the spacecraft at a constant angle with the orbit plane, it could become excessively off-normal to sun rays. To restore thermal equilibrium and adequate power generation, there- fore, some Earth-pointing spacecraft are rotated about the yaw axis by 180 deg. Now consider a different scenario. For relatively loose pointing requirements, say, 0.5 deg in roll and pitch, for oceanogra- phy, Earth observation, geodesy, surveillance and reconnaissance, aeronomy, or scientific purposes, bias momentum satellites with magnetic attitude control systems have been used in the past and proposed for some future missions. Clearly, then, the 180 deg yaw rotations mentioned earlier of such spacecraft will necessitate a pre- cession of bias momentum from one side of the orbit normal to the other. Although this may be accomplished using jets, doing so may not be in harmony with other mission requirements; the jets may perturb the spacecraft ephemeris, contaminate its surroundings and sensitive instruments, or limit its life span, propellant being irretriev- able after expulsion. For this reason and for minimum cost, weight, and simplicity of the attitude control hardware/software, the yaw precession may instead be performed with electromagnets. Sizing a pitch dipole for this purpose is one objective of the paper. The 180 deg yaw precession is, of course, not required for all Earth- pointing bias momentum satellites; the satellites in equatorial orbit, for example, are unikely to require one. At times, bias momentum spacecraft are designed to traverse el- liptic orbits in order to focus on a certain zone of Earth, the atmo- sphere, and the magnetic or gravity field or to perform any other intended operation for a long duration over the slow portion of the orbit. A spacecraft in such orbits may experience strong, impulse- like atmospheric torque near perigee, inducing small (within 10 deg) roll/yaw errors in each pass. Hence the second objective of the paper Received Nov. 4, 1993; revision received March 13, 1995; accepted for publication May 7, 1995. Copyright © 1995 by Hari B. Hablani. Pub- lished by the American Institute of Aeronautics and Astronautics, Inc., with permission. * Senior Engineering Specialist, Advanced Programs, Avionics and Soft- ware Group, Space Systems Division, 12214 Lakewood Boulevard. Asso- ciate Fellow AIAA. is to size a pitch dipole for a small-angle closed-loop magnetic pre- cession so that the spacecraft attains its nominal Earth-pointing at- titude before diving again into the atmospheric pocket near perigee. The problem of small-angle precession arose, for instance, in the case of a multinational spacecraft AMPTE/IRM (active magneto- spheric particle tracer explorers/ion release module) in a highly el- liptic orbit, for which the electromagnet changed its polarity through time-tagged (that is, open-loop) commands. 1 The closed-loop mag- netic precession studied here is executed in conjunction with active nutation damping effected through a roll/pitch product-of-inertia and momentum wheel speed modulation. The development of a suitable relationship between various parameters involved in this process is the last objective of the paper. Magnetic precession is a classical concept; to our knowledge, however, the available literature does not furnish a relationship to de- termine the pitch dipole strength for a general spacecraft orbit, gen- eral orientation of geomagnetic field, desired precession angle, and the number of orbits allocated for the precession; see the compan- ion paper (Ref. 2) for a review of the past, significant contributions. Space limitations forbid including this review here, but certainly, formal techniques such as optimal control as well as others have been used successfully to design the magnetic controllers; however, because of their elaborate and rigorous mathematical framework, these techniques are unsuitable for preliminary conceptual verifica- tion and parametric studies. Simple sizing equations are therefore desired and hence developed in this paper for just this purpose. The ideal torque required for a 180 deg yaw precession of an Earth- pointing bias momentum satellite in a circular orbit is determined in Sec. II. This ideal torque profile about the roll and yaw axes is compared with the one producible by the electromagnets interact- ing with the geomagnetic field. An unexpected conclusion emerges that, due to the Earth-pointing requirement during precession, it ap- pears infeasible to generate the ideal torque profile magnetically in an open-loop fashion. Section III then formulates a magnetic yaw precession after halting the once-per-orbit rotation rate a) Q . This in- ertial yaw precession resembles the precession of spin-stabilized spacecraft, but the analysis presented here is general and fills a gap in the literature. Focusing on the elliptic orbits next, Refs. 1 and 3-5 do demon- strate precession control with a given pitch dipole, but none of these studies provides a way of sizing the dipole for precessing a mo- mentum vector by a specified linear angle. A quasi-quarter-orbit, open-loop magnetic controller for eccentric orbits is therefore for- mulated in Sec. IV, and a literal relationship is developed to size the pitch dipole. For an autonomous implementation of this scheme, the onboard magnetic controller of Rajaram and Goel 6 could be used. But inasmuch as they do not address damping of high-frequency 1321