## 2 Preliminaries
### Mathematical Modeling  
As the airflow in the wind tunnel test section is exactly horizontal, the experimental setup demonstrates \"1g\" level flights, i.e., the flight path angle (\(\gamma\)) is always fixed at zero. The flight dynamics of the tailless aircraft model on the rig are modeled as a three degree-of-freedom (3-DOF) angular motion, which has been developed in ref. [32; 36; 37]. In modeling of this motion, a set of assumptions are made including: the rigid-body aircraft model, the symmetrical distribution of aircraft mass about the \(xz\) plane, the negligible aerodynamic interference induced by the rig and the limited dynamic influence of rig friction and support misalignment. The equations that describe the angular dynamics are derived from [PERSON]'s laws in the body-fixed frame as commonly defined for the aircraft [38]. \[\left[ \begin{array}{c}\dot{p}\\ \dot{q}\\ \dot{r}\end{array} \right]=J^{-1}M_{B}-J^{-1}\left(\left[ \begin{array}{c}p\\ q\\ r\end{array} \right]\times J\left[ \begin{array}{c}p\\ q\\ r\end{array} \right]\right) \tag{1}\]  
where \(J=\left[ \begin{array}{ccc}J_{x}&0&J_{xz}\\ 0&J_{y}&0\\ J_{xz}&0&J_{z}\end{array} \right]\). The CG of the tailless aircraft model coincides well with the aerodynamic reference point. The total external moment acting on the aircraft model \(M_{B}\) is the sum of the aerodynamic moments \(M_{A}\), the rig friction torque \(M_{F}\), and the support misalignment induced torque \(M_{G}\). \[M_{B}=M_{A}+M_{F}+M_{G} \tag{2}\]  
The kinematic equations of the angular motion are expressed with the aerodynamic angles \(\alpha\), \(\beta\), and the bank angle about the velocity vector (\(\mu\)) that defines the rotation between the body and wind frame. \[\left[ \begin{array}{c}\dot{\alpha}\\ \dot{\beta}\\ \dot{\mu}\end{array} \right]=\left[ \begin{array}{ccc}-\tan\beta\cos\alpha&1& -\tan\beta\sin\alpha\\ \sin\alpha&0&-\cos\alpha\\ \sec\beta\cos\alpha&0&\sec\beta\sin\alpha\end{array} \right]\left[ \begin{array} []{c}p\\ q\\ r\end{array} \right]+\left[ \begin{array}{c}d_{21}\\ d_{22}\\ d_{23}\end{array} \right] \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline
**Name** & **Symbol** & **Value** \\ \hline Scale ratio & \(K_{L}\) & 10\% \\ Span & b & 2.0 m \\ Mean chord & \(\overline{z}\) & 0.6571 m \\ Wing area & S & 0.8742 m\({}^{2}\) \\ Mass & m & 16.85 kg \\ Moment of inertia & \(J_{x},J_{y},J_{z}\) & 1.1017, 0.7380,1.6920 kg\(\cdot\)m\({}^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Physical properties of the dynamically scaled aircraft model. Figure 2: Tailless aircraft model flying on a three degree-of-freedom (3-DOF) dynamic rig in the wind tunnel. where  
\[\left[ \begin{array}{c}d_{21}\\ d_{22}\\ d_{23}\end{array} \right]=\frac{1}{M\Omega}\left(\left[ \begin{array}{c}W\cos \gamma\cos\mu\sec\beta\\ W\cos\gamma\sin\mu\\ -W\cos\gamma\cos\mu\tan\beta\end{array} \right]\right.\]  
\[\left.+\left[ \begin{array}{c}-L\sec\beta\\ Y\cos\beta\\ L(\tan\beta+\tan\gamma\sin\mu)+Y\tan\gamma\cos\mu\cos\beta\end{array} \right]\right)\]  
It is noted that the last terms in Equation (3), i.e., \(\left[ \begin{array}{cc}d_{21}&d_{22}&d_{23}\end{array} \right]^{T}\) are the translational terms (aerodynamic lift \(L\), aerodynamic side-force \(Y\), weight \(W\), aircraft mass \(M\), and airspeed \(V\)) that influence the rotational motion [39]. As widely accepted [40; 41; 42], the influence is negligible when designing the attitude controller. The aerodynamic forces and moments are expressed as:  
\[F_{A}=\left[ \begin{array}{c}L\\ D\\ Y\end{array} \right]=\frac{1}{2}\rho V^{2}S\left[ \begin{array}{c}C_{L}\\ C_{D}\\ C_{Y}\end{array} \right] \tag{4}\]  
\[M_{A}=\left[ \begin{array}{c}l\\ m\\ n\end{array} \right]=\frac{1}{2}\rho V^{2}S\left[ \begin{array}{c}b\cdot C_{l} \\ \overline{c}\cdot C_{m}\\ b\cdot C_{n}\end{array} \right] \tag{5}\]  
where \(D\) is the aerodynamic drag; \(l,m,n\) represent the aerodynamic moment of roll, pitch, and yaw; \(\rho\) is the air density; \(S,b,\overline{c}\) denote the reference wing area, wingspan and mean aerodynamic chord respectively.