Time-Optimal Magnetic Attitude Detumbling

The problem of minimum-time spacecraft attitude detumbling using magnetic rods is revisited in this paper within the context of optimal control theory. Two formulations are presented; the first one assumes the availability of the angular velocity measurements for feedback. The second formulation assumes the availability of only the ambient magnetic field measurements in the feedback. In both formulations, the constraint in this optimal control problem is a limit on the maximum magnetic dipole moment of the magnetic rods. It is shown that the time optimality will be achieved if the triple orthogonality condition between the torque, the dipole moment, and the magnetic field vector is fulfilled. This triple orthogonality is considered as a condition of optimality, which is often neglected in the existing B-dot law and its variants in the literature. The Pontryagin minimum principle is used to derive analytically the control logic, for each formulation, in this nonautonomous system, under the assumption of high angular velocity. The second formulation is shown to yield a new variant of the B-dot law. The Monte Carlo simulation results presented in this paper confirm that the two controls found in both formulations outperform most existing algorithms. In addition, the results show lower power consumption by the magnetic rods when using the proposed variant of the B-dot law as compared to existing B-dot laws.