Derivation of All Attitude Error Governing Equations for Attitude Filtering and Control

This article presents the full analytical derivations of the attitude error kinematics equations. This is done for several attitude error representations, obtaining compact closed-forms expressions. Attitude error is defined as the rotation between true and estimated orientations. Two distinct approaches to attitude error kinematics are developed. In the first, the estimated angular velocity is defined in the true attitude axes frame, while in the second, it is defined in the estimated attitude axes frame. The first approach is of interest in simulations where the true attitude is known, while the second approach is for real estimation/control applications. Two nonlinear kinematic models are derived that are valid for arbitrarily large rotations and rotation rates. The results presented are expected to be broadly useful to nonlinear attitude estimation/control filtering formulations. A discussion of the benefits of the derived error kinematic models is included.