4DoF Dynamic Lateral Aircraft Model

This 4DoF model contains the roll rate, yaw rate, roll angle, and the Dutch roll, spiral, and roll flight modes. The states are sideslip angle \(\beta\), roll rate \(p\), yaw rate \(r\), roll angle \(\phi\). The lateral and directional modes must be coupled because sideslip generates yawing and rolling moments.

$$\begin{bmatrix}\dot{\Delta\beta} \\\dot{\Delta p} \\\dot{\Delta r} \\\dot{\Delta\phi} \\\end{bmatrix}=\begin{bmatrix}\frac{y_\beta}{u0} & \frac{y_p}{u0} & -(1-\frac{y_r}{u0}) & \frac{gcos\theta_0}{u_0}\\L_\beta & L_p & L_r & 0\\N_\beta & N_p & N_r & 0\\0 & 1 & 0 & 0 \\\end{bmatrix}\begin{bmatrix}\Delta\beta \\\Delta p \\\Delta r \\\Delta\phi \\\end{bmatrix}+\begin{bmatrix}0&\frac{y_{\delta_r}}{u_0}\\L_{\delta_a}&L_{\delta_r}\\N_{\delta_a}&N_{\delta_r}\\0&0\end{bmatrix}\begin{bmatrix}\delta_a\\\delta_r\end{bmatrix}$$

$$
\begin{bmatrix}
p\\r\\A_y
\end{bmatrix}=
\begin{bmatrix}
0&1&0&0\\
0&0&1&0\\
Y_\beta&Y_p&Y_r&0
\end{bmatrix}
\begin{bmatrix}
\beta\\p\\r\\phi
\end{bmatrix}+
\begin{bmatrix}
0&0\\
0&0\\
0&Y_{\delta_R}
\end{bmatrix}
\begin{bmatrix}
\delta_a\\
\delta_r
\end{bmatrix}$$

The model parameters are defined in the following table.

Parameter
\(Y_\beta=\frac{QSC_{y\beta}}{m}\)
\(Y_p=\frac{QSbC_{yp}}{2mu_0}\)
\(Y_r=\frac{QsbC_{yr}}{2mu_0}\)
\(Y_{\delta_r}=\frac{QSC_{y\delta_r}}{m}\)
$zvL_\beta=\frac{QSbC_{l\beta}}{I_{xx}}$
\(L_p=\frac{Qsb^2C_{lp}}{2I_{xx}u_0}\)
\(L_r=\frac{Qsb^2C_{lr}}{2I_{xx}u_0}\)
\(L_{\delta_a}=\frac{QSbC_{l\delta_a}}{I_{xx}}\)
\(L_{\delta_r}=\frac{QSbC_{l\delta_r}}{I_{xx}}\)
\(N_\beta=\frac{QSbC_{n\beta}}{I_zz}\)
\(N_p=\frac{QSb^2C_{np}}{2I_{zz}u_0}\)
\(N_r=\frac{QSb^2C_{mr}}{2I_{xx}U_0}\)
\(N_{\delta_a}=\frac{QSbC_{n\delta_a}}{I_{zz}}\)
\(N_{\delta_r}=\frac{QSbC_{n\delta_r}}{I_{zz}}\)

Where \(b\) is the span,

[[Deriving 4DoF Lateral Equations of Motion]]
[[X-29 Lateral Directional Model]] – 4DOF X-29 dynamics
[[Dynamic Inversion Lateral Directional Example]] – example using 3DOF
[[Wing Rock in Fighter Aircraft]] – potentially hazardous aero-inertial condition. only uses 2 of the aircraft states.
[[F-15 Active Lateral Directional Model]] – 4-DOF model
[[Aircraft Motion Variables]] – explains L, N, p, r, and $\phi$
[[Spiral Mode]]
[[Piper Cherokee]] – includes lateral model in this form

• AircraftDutchRoll
• parkModelingVortexLattice2013
• WingRockPrediction

Backlinks:

[[Aerodynamic Models]]
[[Deriving 4DoF Lateral Equations of Motion]]
[[Dutch Roll]]
[[Lateral Directional Unsteady Modeling]]
State-Space Model