Six-DOF Motion Equations

Author: CislunarSpace

Site: https://cislunarspace.cn

Definition

Six-degree-of-freedom (6-DOF) motion equations describe complete motion state of an aircraft in 3D space, including three translational degrees (position) and three rotational degrees (attitude). The 6-DOF model is fundamental for controller design and simulation of stratospheric airships.

Coordinate Systems

FrameOriginAxesPurpose
Earth Frame (OE)(O_E)Ground pointEN,EE,EDE_N, E_E, E_DTrajectory description
Body Frame (OB)(O_B)Airship CGBx,By,BzB_x, B_y, B_zForce/moment description
Velocity Frame (OA)(O_A)Airship CGAx,Ay,AzA_x, A_y, A_zAerodynamic calculation

Kinematics Equations

Position Rate

[x˙y˙z˙]=REB[uvw]\begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{bmatrix} = \mathbf{R}_{EB} \begin{bmatrix} u \\ v \\ w \end{bmatrix}

Attitude Rate

[ϕ˙θ˙ψ˙]=[1sin⁡ϕtan⁡θcos⁡ϕtan⁡θ0cos⁡ϕ−sin⁡ϕ0sin⁡ϕsec⁡θcos⁡ϕsec⁡θ][pqr]\begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix} = \begin{bmatrix} 1 & \sin\phi\tan\theta & \cos\phi\tan\theta \\ 0 & \cos\phi & -\sin\phi \\ 0 & \sin\phi\sec\theta & \cos\phi\sec\theta \end{bmatrix} \begin{bmatrix} p \\ q \\ r \end{bmatrix}

Dynamics Equations

Newton-Euler Equations

Force balance along body axes:

m(u˙+qw−rv−gsin⁡θ)=Xm(\dot{u} + qw - rv - g\sin\theta) = X

m(v˙+ru−pw+gcos⁡θsin⁡ϕ)=Ym(\dot{v} + ru - pw + g\cos\theta\sin\phi) = Y

m(w˙+pv−qu+gcos⁡θcos⁡ϕ)=Zm(\dot{w} + pv - qu + g\cos\theta\cos\phi) = Z

References

  • Etkin B, Reid L D. Dynamics of Flight: Stability and Control[M]. Wiley, 2024.
  • Wang H. Stratospheric Airship Modeling and Control[M]. National Defense Industry Press, 2025.