# Independent Random Recurrent Neural Networks for Infrared Spatial Point Targets Classification
## 2 Infrared Radiation Sequence Model
### Attitude Motion Model  
To calculate the geometric projection area of the ballistic target in the LOS direction, it is common practice to split the target surface into a number of small pieces and then accumulate the projected area of each small piece. Assume that the target surface is divided into N small-area slices, the normal vector and area of each slice are respectively \(\overrightarrow{n}_{i}\) and \(\mathrm{a}_{i}\), and the vector of LOS is \(\overrightarrow{n}_{los}\). The key to calculating the projection area sequence is to transform \(\overrightarrow{n}_{i}\) and \(\overrightarrow{n}_{los}\) into the reference coordinate system and determine the rotational transformation relationship of \(\overrightarrow{n}_{i}\) over time in the reference coordinate system.  
Assume that the corresponding vector of \(\overrightarrow{n}_{i}\) in the reference coordinate system (\(X\), \(Y\), \(Z\)) is \(\overrightarrow{n}_{i}^{\,}\), the conversion relationship between the two can be described by the rotation matrix \(R_{init}\) determined by the Euler angle (\(\varphi\),\(\rho\),\(\varphi\)) [22]. The order of Euler angles is \(zxy\), \(R_{init}\) can be mathematically expressed as  
\[R_{init}=\left[\begin{array}{ccc}\cos\phi&-\sin\phi&0\\ \sin\phi&\cos\phi&0\\ 0&0&1\end{array}\right]\left[\begin{array}{ccc}1&0&0\\ 0&\cos\theta&-\sin\theta\\ 0&\sin\theta&\cos\theta\end{array}\right]\left[\begin{array}{ccc}\cos \varphi&0&\sin\varphi\\ 0&1&0\\ -\sin\varphi&0&\cos\varphi\end{array}\right]. \tag{3}\]  
At time \(t_{0}\), the azimuth and elevation angles of the target local axis \(z\) in the reference coordinate system are \(\alpha_{0}\) and \(\beta_{0}\) respectively, as shown in Figure 1, and \(R_{init}\) can be determined by the Euler angle (\(-\alpha_{0}\), \(0\), \(0.5\pi\)-\(\beta_{0}\)), so \(\overrightarrow{n}_{i}^{\,}=R_{init}\cdot\overrightarrow{n}_{i}\).  
The target nutation contains two rotational motions: spinning and coning, as shown in Figure 1. Let the azimuth and elevation angles of the nutation axis be \(\alpha_{n}\) and \(\beta_{n}\), and the angular velocities of the coning and spinning are \(\omega_{n}\) and \(\omega_{s}\) respectively. According to the Rodriguez formula [25], the nutation rotation matrix \(R(t)\) of the target at the time \(t\) is  
\[R(t)=R_{coni}(t)\cdot R_{spin}(t) \tag{4}\]  
where  
\[\begin{split} R_{cmi}(t)&=I+\hat{e}_{1}\sin\omega_{n}t+ \hat{e}_{1}^{2}(1-\cos\omega_{n}t),\\ R_{spin}(t)&=I+\hat{e}_{2}\sin\omega_{s}t+\hat{e}_{2}^{2 }(1-\cos\omega_{s}t).\end{split} \tag{5}\]  
In the Equation (5), \(\hat{e}_{1},\hat{e}_{2}\) are antisymmetric matrices, which are defined as  
\[\begin{split}\hat{e}_{1}=&\left[\begin{array}{ ccc}0&-\sin\beta_{n}&\sin\alpha_{n}\cos\beta_{n}\\ \sin\beta_{n}&0&-\cos\alpha_{n}\cos\beta_{n}\\ -\sin\alpha_{n}\cos\beta_{n}&\cos\alpha_{n}\cos\beta_{n}&0\\ 0&-\sin\beta_{0}&\sin\alpha_{0}\cos\beta_{0}\\ \sin\beta_{0}&0&-\cos\alpha_{0}\cos\beta_{0}\\ -\sin\alpha_{0}\cos\beta_{0}&\cos\alpha_{0}\cos\beta_{0}&0\\ \end{array}\right],\\ \end{split} \tag{6}\]  
Therefore, the vector \(\overrightarrow{n_{i}}^{r}\) will change to \(\overrightarrow{n_{i}}^{r}\), that is, \(\overrightarrow{n_{i}}^{r}=R(t)\cdot\overrightarrow{n_{i}}^{r}=R_{cmi}(t) \cdot R_{spin}(t)\cdot\overrightarrow{n_{i}}^{r}\). The geometric projection area of the target at this time is  
\[A_{proj}=\sum_{i=1}^{N}a_{i}\cdot\max\left(\cos\left(-\overrightarrow{n_{ los}},\overrightarrow{n_{i}}^{r}\right)0\right), \tag{7}\]  
where \(\cos\left(-\overrightarrow{n_{los}},\overrightarrow{n_{i}}^{r}\right)\) is the cosine of the angle between the vector \(\overrightarrow{n_{los}}\) and \(\overrightarrow{n_{i}}^{r}\).