## 2 Problem Description  
### ADS Model  
When establishing a model for the relevant parameters of an ADS, it is desired to unify the formula derivation in a single frame, namely the body frame, for example. In order to clarify the derivation process, Figure 1 shows the relationship between the body frame and other frames, as well as the representation of correlated parameters. The meanings of these parameters will be presented below, and the specific definitions are not elaborated upon for the sake of simplicity. The air data system obtains measurements, such as angle of attack \(\alpha\), sideslip angle \(\beta\), and airspeed \(V\), from its various sensors, and their interdependencies can be established by deriving the equations that describe the relationships between these measurements. Airspeed \(V\) and its components in the body frame \(\left[u,v,w\right]^{\mathrm{T}}\) can be represented as follows:  
\[\left\{ \begin{array}{l}u=V\cos\alpha\cos\beta\\ v=V\sin\beta\\ w=V\sin\alpha\cos\beta\end{array} \right. \tag{1}\]Taking the derivative of Equation (1) yields:  
\[\left\{ \begin{array}{l}\overset{\bullet}{u}=\overset{\bullet}{V}\cos\alpha\cos \beta-\overset{\bullet}{\alpha}V\sin\alpha\cos\beta-\overset{\bullet}{\beta}V\cos \alpha\sin\beta\\ \overset{\bullet}{v}=\overset{\bullet}{V}\sin\beta+\overset{\bullet}{\beta}V \cos\beta\\ \overset{\bullet}{w}=\overset{\bullet}{V}\sin\alpha\cos\beta+\overset{\bullet}{ \alpha}V\cos\alpha\cos\beta-\overset{\bullet}{\beta}V\sin\alpha\sin\beta\end{array} \right. \tag{2}\]  
After a matrix transformation, Equation (2) can be transformed into the following form:  
\[\left\{ \begin{array}{l}\overset{\bullet}{V}=\overset{\bullet}{u}\cos\alpha \cos\beta+\overset{\bullet}{v}\sin\beta+\overset{\bullet}{w}\sin\alpha\cos\beta \\ \overset{\bullet}{\alpha}=-\frac{1}{\
u\cos\beta}\left(\overset{\bullet}{u}\sin \alpha-\overset{\bullet}{w}\cos\alpha\right)\\ \overset{\bullet}{\beta}=-\frac{1}{\
u}\left(\overset{\bullet}{u}\cos\alpha \sin\beta-\overset{\bullet}{v}\cos\beta+\overset{\bullet}{w}\sin\alpha\sin\beta \right)\end{array} \right. \tag{3}\]  
Furthermore, there exists a relationship between the ground velocity vector \(\mathbf{V}_{\text{e}}\), air velocity vector \(\mathbf{V}_{\text{r}}\), and wind velocity vector \(\mathbf{V}_{\text{w}}\):  
\[\mathbf{V}_{\text{e}}=\mathbf{V}_{\text{r}}+\mathbf{V}_{\text{w}} \tag{4}\]  
The kinematics equation of the missile in the body frame can be written in the following form:  
\[\overset{\bullet}{\mathbf{V}_{\text{e}}}=\mathbf{A}+\text{T}_{\text{be}}\mathbf{g}-\mathbf{ \omega}\times\mathbf{V}_{\text{e}} \tag{5}\]  
where \(\mathbf{A}\) is the acceleration vector, \(T_{be}\) is the transformation matrix from inertial frame to body from, \(\mathbf{g}=[0,0,9.81]^{\text{T}}\), and \(\mathbf{\omega}\) is the angular velocity vector. Substituting Equation (4) into Equation (5) yields:  
\[\overset{\bullet}{\mathbf{V}_{\text{r}}}=\mathbf{A}+\text{T}_{\text{be}}\mathbf{g}-\mathbf{ \omega}\times\mathbf{V}_{\text{r}}-\overset{\bullet}{\mathbf{V}_{\text{w}}}-\mathbf{ \omega}\times\mathbf{V}_{\text{w}} \tag{6}\]  
As the wind velocity is relatively small during the missile's flight, we can denote the wind-related disturbance as \(\mathbf{d}\), \(\mathbf{d}=-\overset{\bullet}{\mathbf{V}_{\text{w}}}-\mathbf{\omega}\times\mathbf{V}_{\text{w}}\), then define \(\mathbf{V}_{\text{r}}=[u,v,w]^{\text{T}}\), \(\mathbf{A}=\left[A_{x},A_{y},A_{z}\right]^{\text{T}}\), \(\mathbf{\omega}=[p,q,r]^{\text{T}}\), and \(\mathbf{d}=[d_{u},d_{v},d_{w}]^{\text{T}}\). By substituting Equations (1) and (6) into Equation (3), it can be shown that:  
Figure 1: Relationship between different frames and correlated parameters. (**a**) Body frame and velocity frame. (**b**) Body frame and ground frame. \[\left\{