Nonlinear Optimal Approach to Spacecraft Attitude Control Using Magnetic and Impulsive Actuations
M AGNETIC torques originating from the interaction between the Earth's magnetic field and the onboard electromagnetic dipole moments can be used as an actuation mechanism for attitude control purposes [1 3] In this regard, spacecraft operating in low Earth orbits (LEOs) can be equipped with three mutually perpendicular magnetic torquers. The most basic source of a magnetic torquer is a current loop; a planar loop of areaAwithN turns of wire carrying a current of I Aproduces amagnetic dipole moment of magnitude NIA in the direction normal to the plane of the loop, satisfying a right hand rule [3] Although this relation has the advantage of being a linear function of the current, sufficiently large quantities for A andN are necessary to produce the onboard dipole moments required by many spacecraft. For this reason, torque rods (which are coils of wire wrapped around ferromagnetic cores) are employed in practice to obtain greater dipole moments for a given amount of current [3] Because themagnitude of the Earth'smagnetic field decreases as the inverse cube of the distance from the center of the Earth, magnetic control torques are used almost exclusively in near Earth orbits wherein the magnitude of the Earth's magnetic field is roughly in the range of 20 50 mT [3] As a consequence, the control torques generated by magnetic actuation are typically on the order of 10 5 to 10 4 N m for LEOs, depending on factors like orbit inclination [3] Magnetic control torques are typically used for the detumbling [4] and momentum dumping [5,6] of spacecraft. In the former, the magnetic attitude control system detumbles the spacecraft immediately after launch by damping all angular velocity components of the spacecraft to zero; whereas in the latter, the primary task of the attitude control system is to dump excess wheel angular momentum induced by external disturbances [2,3] Other applications of magnetic torquers include initial acquisition, precession control, nutation damping, and momentum control [3] Attitude control using magnetic torques was first proposed in Ref. [7] in the 1960s, and then in Ref. [8] by developing the so called B dot control law for spacecraft detumbling and initial acquisition. A great deal of research has been devoted to the use ofmagnetic torques for both attitude control and momentum dumping thereafter. The early research was focused on approximating the time varying behavior of magnetic actuators with its time invariant equivalence assuming sufficiently slow closed loop dynamics [9,10] Periodic change in the properties of the geomagnetic field relative to spacecraft (as they orbit the Earth) has also inspired many researchers to employ a linear optimal control policy (time varying, periodic, and constant gains) to stabilize the rotational dynamics of spacecraft [11 13] For instance, Ref. [11] proposed an infinite time horizon periodic attitude controller, wherein the geomagnetic field was averaged over a certain time interval to consider the periodic nature of the problem, and a finite time horizon periodic controller, which incorporated the time history of the real geomagnetic field into the controller architecture. Lyapunov based nonlinear techniques were also under investigation for magnetic attitude control [14,15] in this regard, the periodicity of the Earth's magnetic fieldwas used to apply the Krasovski LaSalle theorem for the stability analysis. A survey of magnetic spacecraft attitude control using both linear and nonlinear design methods can be found in Ref. [16] Magnetic attitude control systems possess several advantages for near Earth missions, including 1) an essentially unlimited mission life due to use of a renewable source of actuation, 2) the absence of catastrophic failure modes, 3) the smoothness of application, 4) the possibility of smoothly modulating the control torquewithout inducing undesired coupling with flexible modes, and 5) significant savings in the overall weight and complexity of the system as compared to any other class of actuators owing to the absence ofmoving parts or plumbing [3,4,8] On the other hand, as the main drawback associated with magnetic attitude control systems,magnetic torques are constrained to lie in the plane orthogonal to the magnetic field (due to the crossproduct between the onboard magnetic dipole moments and the magnetic field from which they originate) consequently, only two out of three axes can be controlled at a given time instant [3] Moreover, controllability in such systems depends on orbit characteristics and the location of the spacecraft within the orbit [2] However, full threeaxis control for magnetic attitude control systems can be achieved over a complete orbit, provided that the spacecraft's orbital plane excludes the geomagnetic equatorial plane and the magnetic poles [17] Another disadvantage related tomagnetic attitude control is that moderately fast attitude maneuvers are almost limited to low orbit missions [2] To resolve the drawbacks inherently associated with magnetic actuation (namely, instantaneous underactuation and gain limitations) magnetic attitude control systems have been employed in conjunction Received 25October 2019; revision received30 January 2020; accepted for publication 30 January 2020; published online 27 February 2020. Copyright (c) 2020 by Esmaeil Sharifi and Christopher J. Damaren. Published by the American Institute ofAeronautics andAstronautics, Inc. with permission.All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533 3884 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. *Ph.D. Candidate, Spacecraft Dynamics and Control Laboratory, Institute for Aerospace Studies. Student Member AIAA. Professor and Director, Institute for Aerospace Studies. Associate Fellow AIAA.