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Aircraft Attitude Dynamics in Terms of Quaternions Using a Non-Redundant ODE Lie-Group Formulation

Dynamic simulation procedures of flight vehicle (fixed-wing, rotorcraft, UAV, satellite) 3D manoeuvres need robust and efficient integration methods in order to allow for reliable, and possibly real-time, simulation missions. Since flight vehicle 3D manoeuvres necessary include complete 3D rotation domain, such procedures also require an efficiency way of dealing with large 3D rotation. Usually, in this context, the simulation procedures built around the standard numerical ordinarydifferential-equations (ODE) based on three-parameters rotation variables (such as Euler angels) have their limitations, as they impose discontinuities or even singularities in the flight vehicle attitude integral curves. Most commonly, the quaternion representation is widely used in flight simulation to overcome the mentioned deficiency [1]. However, if quaternions are used for a parameterisation of the rotation manifold, the standard model leads to integration of differential-algebraic equations (DAE) that requires additional stabilisation of the algebraic constraint due to quaternions normalisation equation. Recently, a method of integration of rotational quaternions based on non-vectorial geometric Lie-group integration that leads to a minimal-form ODE integration (avoiding thus DAE integration) has been introduced in [2]. By adopting such an approach, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form an ODE on Lie-algebra so(3) of the rotation group SO(3), after which the integration incremental update on the configuration quaternion group Sp(1) is determined by the exponential map. Consequently, only a system of three independent ODEs is integrated and hence no stabilization of the unit-length constraint is necessary.

Flight vehicle; Rotation Tensor; Quaternion-group Sp(1); Special Orthogonal Group SO(3); Integration of the Rotational Dynamics; Lie-algebra

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