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Comparative study of attitude control methods based on Euler angles, quaternions, angle-axis pairs and orientation matrices

This paper presents a comparative study about the attitude control methods based on four commonly used error indicators, namely the triad of 3-2-1 deviational Euler angles, the error quaternion, the deviational angle-axis pair and the orientation error matrix. These error indicators are used here with the same backstepping control law to have a common basis of comparison. This control law makes the controller track a restoring angular velocity generated here specifically for each error indicator. This comparative study shows that all these error indicators can be used satisfactorily even in the critical orientations associated with them by taking special measures. For the deviational Euler angle triad, a critical orientation is a singularity, at which the angles become indefinite. Unless the vehicle is stationary, this indefiniteness is resolved here by applying L'Hopital's Rule on the angular velocity information. The Euler angle triad has also a multiplicity problem. It is solved here by using the novel criterion of minimal deviation angles. For the other error indicators, a critical orientation is an antipodal orientation, which is opposite to the desired one. In an antipodal orientation, the error quaternion and the deviational angle-axis pair cannot be determined through the customary formulas. They are determined here by using the novel specially introduced formulas. Besides, they may suffer from the unwinding phenomenon in an ordinary orientation. This phenomenon is prevented here by keeping the scalar part of the error quaternion non-negative and the deviation angle between 0 degrees and 180 degrees. For the orientation error matrix, a stationary and undisturbed antipodal orientation is an unstable equilibrium, in which the ordinary backstepping control law becomes ineffective for driving the system into action. This problem is solved here by adding an extra term to the ordinary backstepping control law.