## 2 Lateral-Directional Dynamics  
Figure 1 shows the configuration of the research aircraft, featuring an LSR fuselage and a high-sweep wing specifically designed for supersonic flight. The aircraft employs aileron and rudder for roll and yaw control, respectively, in accordance with standard control procedures. Table 1 provides an overview of the general properties of the aircraft. On the basis of computational fluid dynamics (CFD) simulations, a high-dimensional linear interpolation table is constructed to create an accurate aerodynamic model. The nonlinear lateral-directional dynamic equations [20] can then be formulated using this model:  
\[ \begin{array}{l}\dot{\beta}=\frac{\gamma}{m^{\mathrm{V}}}+\left(p\sin\alpha-r \cos\alpha\right)+\frac{\phi}{\phi}\cos\phi\cos\theta\\ \dot{p}=\frac{\left(l_{q}-l_{z}\right)l_{z}-l_{zz}}{l_{x}l_{z}-l_{xz}}rq+\frac {\left(l_{x}-l_{y}+l_{z}\right)l_{xz}}{l_{x}l_{z}-l_{xz}}pq+\frac{l_{z}}{l_{x} l_{z}-l_{xz}}L+\frac{l_{xz}}{l_{x}l_{z}-l_{xz}}N\\ \dot{r}=\frac{l_{x}\left(l_{x}-l_{y}\right)+l_{xz}}{l_{x}l_{z}-l_{xz}}pq-\frac {\left(l_{x}-l_{y}+l_{z}\right)l_{xz}}{l_{x}l_{z}-l_{xz}}rq+\frac{l_{xz}}{l_{x} l_{z}-l_{xz}}L+\frac{l_{x}}{l_{x}l_{z}-l_{xz}}N\\ \dot{\phi}=p+\tan\theta(q\sin\phi+r\cos\phi)\end{array}  \tag{1}\]  
Equation (1) is notable for its strong coupling between roll and yaw dynamics, which encompasses various forms of coupling, including aerodynamic coupling, control coupling, motion coupling, and inertial coupling. In the case of an LSR fuselage aircraft, the moment of inertia coupling can become particularly pronounced during rapid roll maneuver flight. Define  
\[\Gamma=I_{xz}/\left(l_{x}I_{z}-l_{xz}^{2}\right) \tag{2}\]  
From a stability perspective, the influence of the inertial coupling term is dependent on the angular velocity. During a rapid roll maneuver, if an aircraft has upward pitching angular velocity, the term \(\left(l_{x}-l_{y}+l_{z}\right)l_{xz}q\) will destabilize the aircraft. Conversely, if the aircraft has downward pitching angular velocity, the term \(\left(l_{x}-l_{y}+l_{z}\right)l_{xz}q\) will stabilize it. Therefore, the inertial coupling moment affects both the aircraft's stability and its dynamic response. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Explanation** & **Value** \\ \hline \(S\) & Reference area & 1.2 m\({}^{2}\) \\ \(c\) & Mean aerodynamic chord & 0.5 m \\ \(b\) & Wingspan & 2.1 m \\ \(m\) & Mass & 280 kg \\ \(l_{xz}\) & Product of inertia \(xz\) & 6 kg\(\cdot\)m\({}^{2}\) \\ \(l_{x}\) & Moment of inertia around the \(x\)-axis & 9 kg\(\cdot\)m\({}^{2}\) \\ \(l_{y}\) & Moment of inertia around the \(y\)-axis & 300 kg\(\cdot\)m\({}^{2}\) \\ \(l_{z}\) & Moment of inertia around the \(z\)-axis & 300 kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General properties of the research aircraft [21]. Figure 1: Plan view of the research LSR aircraft. For a wing-level flight with zero sideslip angle and roll angle, by applying the small disturbance theory, the nonlinear model in Equation (1) can be linearized as follows:  
\[ \begin{bmatrix}\dot{\beta}\\ \dot{p}\\ \dot{r}\\ \dot{\phi}\end{bmatrix}  = \begin{bmatrix}\overline{Y}_{\beta}&\sin\alpha+\overline{Y}_{p}& \overline{Y}_{r}-\cos\alpha&g\cos\theta/V\\ \overline{L}_{\beta}&\overline{L}_{p}&\overline{L}_{r}&0\\ \overline{N}_{\beta}&\overline{N}_{p}&\overline{N}_{r}&0\\ 0&1&\tan\theta&0\end{bmatrix}\begin{bmatrix}\beta\\ p\\ r\\ \phi\end{bmatrix}  \tag{3}\] \[+ \begin{bmatrix}\overline{Y}_{\delta_{a}}&\overline{Y}_{\delta_{r }}\\ \overline{L}_{\delta_{a}}&\overline{L}_{\delta_{r}}\\ \overline{N}_{\delta_{a}}&\overline{N}_{\delta_{r}}\\ 0&0\end{bmatrix}\begin{bmatrix}\delta_{a}\\ \dot{\delta}_{r}\end{bmatrix} \]  
The parameters of the moments and forces induced by the sideslip angle, roll angular rate, and yaw angular rate can be defined as  
\[