# Microwave Staring Correlated Imaging Based on Unsteady Aerostat Platform
## 2 MSCI Method Based on Unsteady Aerostat Platform
### Real-Time Position Vector of Antenna  
To eliminate the influence of antenna motion, the real-time position vector \(\mathbf{r}^{n}\left(t_{l}\right)\) and \(\mathbf{r}^{s}\left(t_{l}\right)\) are introduced to the MSCI model based on unsteady aerostat platform, where \(\mathbf{r}^{n}\left(t_{l}\right)\) and \(\mathbf{r}^{s}\left(t_{l}\right)\) denote the real-time position vector of the \(n\)-th transmitter and the receiver at \(t_{l}\) in the \(l\)-th pulse in \(O_{t}X_{t}Y_{t}Z_{t}\) respectively. The complicated motion of the antenna array is decomposed into three-dimensional translations and three rotational components. The three-dimensional translations are along \(X_{t}\), \(Y_{t}\) and \(Z_{t}\) respectively. The rotational components are rotation of yaw angle, pitch angle, and roll angle, respectively. As the pulse repetition interval \(T_{P}\) is short, the translational speed and the rotational angular velocity will not change drastically during such a short period, so the assumption on the array motion is made that the antenna array motion is uniform translation and uniform rotation during each pulse repetition interval \(T_{P}\). Hence the translational speed of the antenna array during the \(l\)-th pulse is denoted as \(\mathbf{v}_{l}=\left[v_{l,x},v_{l,y},v_{l,z}\right]\), where \(v_{l,x},v_{l,y},v_{l,z}\) are the speeds of the three-dimensional translations along \(X_{t}\) axis, \(Y_{t}\) axis and \(Z_{t}\) axis respectively. The rotational angular velocity of the antenna array during the \(l\)-th pulse is denoted as \(\boldsymbol{\omega}_{l}=\left[\omega_{l,\theta},\omega_{l,\varphi},\omega_{l, \phi}\right]\), where \(\omega_{l,\theta},\omega_{l,\varphi},\omega_{l,\phi}\) are rotational angular velocity of the yaw angle, the pitch angle, and the roll angle respectively. If the motion of antenna array during each pulse is known, the real-time position vector of the antenna can be determined. \(T_{pos}\) denotes the repetition period of POS recording data. As Figure 3 shows, the pulse repetition interval \(T_{P}\) is far shorter than \(T_{pos}\) of POS, so there are many transmitting pulses between adjacent POS data. Assuming that the recorded time \(t_{i,pos}\) of the \(i\)-th POS data is in the \(l^{\prime}\)-th pulse and the recorded time \(t_{i+1,pos}\) of the next POS data is in the \(l^{\prime\prime}\)-th pulse, the real-time position vector of \(n\)-th transmitter \(\mathbf{r}^{n}\left(t_{l}\right)\) at \(t_{l}\) in the \(l\)-th (\(l^{\prime}\leq l\leq l^{\prime\prime}\)) pulse can be expressed as  
Figure 2: Graphical diagram for the altitude angles. \[\mathbf{r}^{n}\left(t_{l}\right)=\mathbf{r}^{n}\left(t_{i,pos}\right)+\Delta \mathbf{r}_{v}\left(t_{l}-t_{i,pos}\right)+\Delta\mathbf{r}_{\omega}^{n}\left(t _{l}-t_{i,pos}\right), \tag{2}\]  
where \(\Delta\mathbf{r}_{v}\left(t_{l}-t_{i,pos}\right)\) and \(\Delta\mathbf{r}_{\omega}^{n}\left(t_{l}-t_{i,pos}\right)\) are the displacement vectors of the \(n\)-th transmitter caused by translation and rotation during \(t_{l}-t_{i,pos}\), respectively. \(\mathbf{r}^{n}\left(t_{i,pos}\right)\) is the position vector of the \(n\)-th transmitter at \(t_{i,pos}\) and can be calculated by the following formula  
\[\mathbf{r}^{n}\left(t_{i,pos}\right)=\mathbf{r}^{s}\left(t_{i,pos}\right)+ \mathbf{C}\left(\theta_{t_{i,pos}},\varphi_{t_{i,pos}},\varphi_{t_{i,pos}} \right)\mathbf{r}_{b}^{n}, \tag{3}\]  
where \(\mathbf{r}_{b}^{n}\) is the position vector of the \(n\)-th transmitter in \(O_{b}X_{b}Y_{b}Z_{b}\). \(\mathbf{r}^{s}\left(t_{i,pos}\right)\) is the position vector of the receiver measured by the POS at \(t_{i,pos}\) in \(O_{t}X_{t}Y_{t}Z_{t}\). \(\mathbf{C}\left(\theta_{t_{i,pos}},\varphi_{t_{i,pos}},\varphi_{t_{i,pos}}\right)\) is the Direction Cosine Matrix (DCM) that transforms the coordinate from \(O_{b}X_{b}Y_{b}Z_{b}\) to \(O_{t}X_{t}Y_{t}Z_{t}\). The DCM can be expressed as  
\[\mathbf{C}\left(\theta_{t_{i,pos}},\varphi_{t_{i,pos}},\varphi_{t_{i,pos}} \right) =