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Research Papers

Lie Group Forward Dynamics of Fixed-Wing Aircraft With Singularity-Free Attitude Reconstruction on SO(3)

[+] Author and Article Information
Zdravko Terze

Department of Aeronautical Engineering,
Faculty of Mechanical Engineering
and Naval Architecture,
University of Zagreb,
Ivana Lučića 5,
Zagreb 10002, Croatia
e-mail: zdravko.terze@fsb.hr

Dario Zlatar

Department of Aeronautical Engineering,
Faculty of Mechanical Engineering
and Naval Architecture,
University of Zagreb,
Ivana Lučića 5,
Zagreb 10002, Croatia
e-mail: dario.zlatar@fsb.hr

Milan Vrdoljak

Department of Aeronautical Engineering,
Faculty of Mechanical Engineering
and Naval Architecture,
University of Zagreb,
Ivana Lučića 5,
Zagreb 10002, Croatia
e-mail: milan.vrdoljak@fsb.hr

Viktor Pandža

Department of Aeronautical Engineering,
Faculty of Mechanical Engineering
and Naval Architecture,
University of Zagreb,
Ivana Lučića 5,
Zagreb 10002, Croatia
e-mail: viktor.pandza@fsb.hr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 26, 2016; final manuscript received July 20, 2016; published online December 2, 2016. Assoc. Editor: Andreas Mueller.

J. Comput. Nonlinear Dynam 12(2), 021009 (Dec 02, 2016) (11 pages) Paper No: CND-16-1101; doi: 10.1115/1.4034398 History: Received February 26, 2016; Revised July 20, 2016

This paper proposes an approach to formulation and integration of the governing equations for aircraft flight simulation that is based on a Lie group setting, and leads to a nonsingular coordinate-free numerical integration. Dynamical model of an aircraft is formulated in Lie group state space form and integrated by ordinary-differential-equation (ODE)-on-Lie groups Munthe-Kaas (MK) type of integrator. By following such an approach, it is assured that kinematic singularities, which are unavoidable if a three-angles-based rotation parameterization is applied for the whole 3D rotation domain, do not occur in the proposed noncoordinate formulation form. Moreover, in contrast to the quaternion rotation parameterization that imposes additional algebraic constraint and leads to integration of differential-algebraic equations (DAEs) (with necessary algebraic-equation-violation stabilization step), the proposed formulation leads to a nonredundant ODE integration in minimal form. To this end, this approach combines benefits of both traditional approaches to aircraft simulation (i.e., three angles parameterization and quaternions), while at the same time it avoids related drawbacks of the classical models. Besides solving kinematic singularity problem without introducing DAEs, the proposed formulation also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation and when aircraft motion pattern comprises steady rotational component of its 3D motion. This is due to the fact that a Lie group setting and applied MK integrator determine vehicle orientation on the basis of integration of local (tangent, nonlinear) kinematical differential equations (KDEs) that model process of 3D rotations (i.e., vehicle attitude reconstruction on nonlinear manifold SO(3)) more accurately than “global” KDEs of the classical formulations (that are linear in differential equations part in the case of standard quaternion models).

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Figures

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Fig. 1

Trajectory r of aircraft's mass center

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Fig. 2

Time evolution of aircraft's mass center negative altitude r3(t) (above) and the ground projection of aircraft's mass center trajectory (r1 and r2) in the global frame (below)

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Fig. 3

Components of aircraft velocity vector in the global frame v

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Fig. 4

Attitude of the aircraft via angles Ψ : bank, elevation, and azimuth angle (angles are introduced as −π≤( φ, ψ)<π and −π/2≤θ<π/2)

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Fig. 5

Attitude of the aircraft via Tait–Bryan angles Θ (angles are introduced as −π≤( Θ1,Θ3)<π and −π/2≤Θ2<π/2)

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Fig. 6

Elements of the aircraft rotation matrix R

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Fig. 7

Properties of the rotation matrix R∈SO(3). Numerical errors of diagonal elements of product RRT=I and determinant detR=+1.

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Fig. 8

Convergence in the norm of an error in aircraft's attitude quaternions

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Fig. 9

Convergence in the norm of an error in the position of aircraft's mass center

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Fig. 10

Convergence in the norm of an error in the angular velocity

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