## 2 Autonomous Orbit Determination Model
### Rotation Correction Model
#### 2.2.1 Theoretical Analysis of the Influence of the Overall Constellation Rotation  
At a certain time, the position and velocity of a certain satellite in the constellation are  
\[X=\begin{pmatrix}x&y&z&\dot{x}&\dot{y}&\dot{z}\end{pmatrix}^{T} \tag{6}\]If there is an overall rotation error with a rotation quantity of \(\theta\big{(}\theta_{\alpha},\theta_{\beta},\theta_{\gamma}\big{)}\) in the constellation for autonomous orbit determination, then according to the criteria for coordinate transformation, the estimated value \(X^{\prime}\) of the satellite position state is  
\[X^{\prime}=\begin{pmatrix}x^{\prime}\\ y^{\prime}\\ z^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}=\begin{bmatrix}R(-\theta)&0\\ 0&R(-\theta)\\ \end{bmatrix}\begin{pmatrix}x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ z\end{bmatrix}=R^{*}X \tag{7}\]  
where \(R\) represents the rotation matrix of the constellation; then,  
\[R(-\theta)=R_{\gamma}(-\theta_{\gamma})R_{\beta}\big{(}-\theta_{\beta}\big{)} R_{\alpha}(-\theta_{\alpha})\]  
\[R_{\alpha}(-\theta_{\alpha})=\begin{bmatrix}1&0&0\\ 0&\cos\theta_{\alpha}&-sin\theta_{\alpha}\\ 0&sin\theta_{\alpha}&\cos\theta_{\alpha}\end{bmatrix}\]  
\[R_{\beta}\big{(}-\theta_{\beta}\big{)}=\begin{bmatrix}\cos\theta_{\beta}&0& sin\theta_{\beta}\\ 0&1&0\\ -sin\theta_{\beta}&0&\cos\theta_{\beta}\end{bmatrix}\]  
\[R_{\gamma}(-\theta_{\gamma})=\begin{bmatrix}\cos\theta_{\gamma}&-sin\theta_{ \gamma}&0\\ sin\theta_{\gamma}&\cos\theta_{\gamma}&0\\ 0&0&1\end{bmatrix}\]  
where \(R_{\alpha}\), \(R_{\beta}\), and \(R_{\gamma}\) represent the rotation matrices around the \(X\)-axis, \(Y\)-axis, and \(Z\)-axis, respectively (same below). When \(\theta\) is a small quantity, we have  
\[R(-\theta)\approx\begin{bmatrix}1&-\theta_{\gamma}&\theta_{\beta}\\ \theta_{\gamma}&1&-\theta_{\alpha}\\ -\theta_{\beta}&\theta_{\alpha}&1\end{bmatrix} \tag{8}\]  
Therefore,  
\[R^{*}\approx\begin{bmatrix}1&-\theta_{\gamma}&\theta_{\beta}&0&0&0\\ \theta_{\gamma}&1&-\theta_{\alpha}&0&0&0\\ -\theta_{\beta}&\theta_{\alpha}&1&0&0&0\\ 0&0&0&1&-\theta_{\gamma}&\theta_{\beta}\\ 0&0&0&\theta_{\gamma}&1&-\theta_{\alpha}\\ 0&0&0&-\theta_{\beta}&\theta_{\alpha}&1\end{bmatrix} \tag{9}\]  
Noting that  
\[X^{\prime}=R^{*}X\approx\begin{pmatrix}x+z\theta_{\beta}-y\theta_{\gamma}\\ y-z\theta_{\alpha}+x\theta_{\gamma}\\ z+y\theta_{\alpha}-x\theta_{\beta}\\ \dot{x}+\dot{z}\theta_{\beta}-y\theta_{\gamma}\\ \dot{y}-\dot{z}\theta_{\alpha}+\dot{x}\theta_{\gamma}\\ \dot{z}+\dot{y}\theta_{\alpha}-\dot{x}\theta_{\beta}\end{pmatrix}=\begin{pmatrix} x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ \dot{z}\end{pmatrix}+\begin{bmatrix}0&z&-y\\ -z&0&x\\ y&-x&0\\ 0&\dot{z}&-\dot{y}\\ -\dot{z}&0&\dot{x}\\ \dot{y}&-\dot{x}&0\end{bmatrix}\begin{pmatrix}\theta_{\alpha}\\ \theta_{\beta}\\ \theta_{\gamma}\end{pmatrix} \tag{10}\]  
\[H=\begin{bmatrix}0&z&-y\\ -z&0&x\\ y&-x&0\\ 0&\dot{z}&-y\\ -\dot{z}&0&\dot{x}\\ \dot{y}&-\dot{x}&0\end{bmatrix} \tag{11}\]\[X^{\prime}\approx X+H\theta \tag{12}\]  
That is, the satellite position and velocity state errors caused by the constellation rotation error are  
\[\delta X=X^{\prime}-X\approx H\theta=\begin{pmatrix}z\theta_{\beta}-y\theta_{ \gamma}\\ x\theta_{\gamma}-z\theta_{\alpha}\\ y\theta_{\alpha}-x\theta_{\beta}\\ z\theta_{\beta}-\dot{y}\theta_{\gamma}\\ \dot{x}\theta_{\gamma}-\dot{z}\theta_{\delta}\\ \dot{y}\theta_{\alpha}-\dot{x}\theta_{\beta}\\ \end{pmatrix} \tag{13}\]