Magnetic Torquer Attitude Control via Asymptotic Periodic Linear Quadratic Regulation

A method of using magnetic torque rods to do 3axis spacecraft attitude control has been developed. The goal of this system is to achieve a nadir pointing accuracy on the order of 0.1 to 1.0 deg without the need for thrusters or wheels. The open-loop system is under-actuated because magnetic torque rods cannot torque about the local magnetic field direction. This direction moves in space as the spacecraft moves along an inclined orbit, and the resulting system is roughly periodic. Periodic controllers are designed using an asymptotic linear quadratic regulator technique. The control laws include integral action and saturation logic. This system's performance has been studied via analysis and simulation. The resulting closed-loop systems are robust with respect to parametric modeling uncertainty. They converge from initial attitude errors of 30 deg per axis, and they achieve steady-state pointing errors on the order of 0.5 to 1.0 deg in the presence of drag torques and unmodeled residual dipole moments. Introduction All spacecraft have an attitude stabilization system. They range from passive spin-stabilized 1 or gravitygradient stabilized 2 systems to fully active three-axis controlled systems . Pointing accuracies for such systems may range from 10 deg down to 10 deg or better, depending on the spacecraft design and on the types of sensors and actuators that it carries. The most accurate designs normally include momentum wheels or reaction wheels. This paper develops an active 3-axis attitude stabilization system for a nadir-pointing spacecraft. It uses only magnetic torque rods as actuators. Additional components of the system include appropriate attitude sensors and a magnetometer. The goal of this system is to achieve pointing accuracy that is better than a gravity gradient stabilization system, on the order of 0.1 to 1 deg. Such a system will weigh less than either a gravity-gradient system or a wheelbased system, and it will use less power than a wheel∗ Associate Professor, Sibley School of Mech. & Aero. Engr. Associate Fellow, AIAA. Copyright  2000 by Mark L. Psiaki. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. based system. Thus, it will be ideal for small satellite applications, where weight and power budgets are severely restricted. There are two classic uses of magnetic torque rods in attitude control. One is for momentum management of wheel-based systems . The other is for angularmomentum and nutation control of spinning , momentum-biased , and dual-spin spacecraft . The present study is one of a growing number that consider active 3-axis magnetic attitude stabilization of a nadir-pointing spacecraft . Reference 5 also should be classified with this group because it uses similar techniques. Reference 7, the earliest such study, presents a 3-axis proportional-derivative control law. It computes a desired torque and projects it perpendicular to the Earth's magnetic field in order to determine the actual torque. Projection is necessary because the magnetic torque, nm, takes the form nm = m × b (1) where m is the magnetic dipole moment vector of the torque rods and b is the Earth's magnetic field. Equation (1) highlights the principal problem of magnetic-torque-based 3-axis attitude control: the system is under-actuated. A rigid spacecraft has 3 rotational degrees of freedom, but the torque rods can only torque about the 2 axes that are perpendicular to the magnetic field vector. The system is controllable if the orbit is inclined because the Earth's magnetic field vector rotates in space as the spacecraft moves around its orbit. It is a time-varying system that is approximately periodic. This system's under-actuation and its periodicity combine to create a challenging feedback controller design problem. The present problem is different from the problem of attitude control when thrusters or reaction wheels provide torque only about 2 axes. References 15 and 16 and others have addressed this alternate problem, in which the un-actuated direction is defined in spacecraft coordinates. For magnetic torques, the un-actuated direction does not rotate with the spacecraft. Various control laws have been considered for magnetic attitude control systems. Some of the controllers are similar to the original controller of Martel et al. . Time-varying Linear Quadratic Regulator (LQR) formulations have been tried , as has fuzzy control 9 and sliding-mode control . References 9 and 13 patch together solutions of time-