Aircraft attitude reconstruction via novel quaternion-integration procedure

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Unit quaternion representation is widely used in flight simulation to overcome the limitations of the standard numerical ordinary-differential-equations (ODEs) based on three-parameters rotation variables (such as Euler angels), as they may impose kinematic singularities during aircraft's attitude reconstruction. However, these benefits do not come without a price, since the classical way of integrating rotational quaternions includes solving of differential-algebraic equations (DAEs) that requires post-integration numerical stabilization of the additional algebraic constraint enforcing the quaternion unitary norm. This can pose a problem in the case of longer flight simulations since improper numerical treatment of the quaternion-normalization constraint may induce numerical drift into the simulation results. As a remedy, the proposed novel algorithm circumvents DAE problem of quaternion integration by shifting update-integration-process from configuration manifold to the local tangential level of the incremental rotations (reducing thus integration to standard three ODEs problem at tangential Lie algebra level). This can be done due to the isomorphism of the Lie algebras of the rotational SO(3) group and the configuration manifold unit quaternion Sp(1) group. Besides avoiding DAE formulation by reducing integration process to standard three ODEs problem, the proposed algorithm also exhibits numerical advantages as it is discussed in the presented example.