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Forward Dynamics of Fixed-Wing Aircraft with Attitude Reconstruction via Novel Quaternion- Integration Procedure

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. 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 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- normalisation 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. This can be done due to the isomorphism of the Lie algebras of the SO(3) group and the unit quaternion Sp(1) group. Besides avoiding DAE formulation by reducing integration process to standard three ODEs problem (by using incremental rotation vector as integration variable at Lie algebra level), the proposed algorithm may also exhibit numerical advantages. Depending on the numerical case at hand - as it is in the presented flight example with 'slower' dynamics and steady rotation component - the method allows for utilization of longer integration steps and overall better numerical accuracy. This is because numerical integration of three non-linear ODEs in terms of incremental rotation vector within the proposed novel algorithm follows a non-linear kinematical rotation update more accurately than integration of a four (linear!) ODEs (as it appears in the standard algorithm), resulting in more accurate simulation results. The effect of better numerical accuracy will be more accented if overall flight vehicle dynamics allows for longer integration steps and aircraft motion pattern involves steady rotational component within its 3D motion.

Time integration schemes ; Rotational quaternions ; Lie groups ; Symplectic group Sp(1) ; Aerospace Applications

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