# Analysis of Distributed Measurement Method for Array Antenna Position
## 2 Scheme and Principle
### Error Analysis of Distributed POS Technology
#### 2.3.3 Attitude Error Equation  
In the navigation system, the attitude of the carrier coordinate system relative to the navigation coordinate system can be expressed by the direction cosine matrix \(C_{b}^{n}\). The relationship between the calculated value \(\widetilde{C}_{b}^{n}\) and the real value \(C_{b}^{n}\) of the matrix can be expressed as follows:  
\[\widetilde{C}_{b}^{n}=C_{b}^{n}+\delta C_{b}^{n}=(I-[\Phi\times])C_{b}^{n} \tag{19}\]  
where,\(\Phi\times=\left[ \begin{array}{cc}0&-\delta_{\gamma}&\delta_{\alpha}\\ \delta_{\gamma}&0&-\delta_{\beta}\\ -\delta_{\alpha}&\delta_{\beta}&0\end{array} \right]\) is the skew symmetric matrix of attitude error angle in real coordinate system, \(\delta_{\alpha}\) is the attitude error around the x-axis,\(\delta_{\beta}\) is the attitude error around the y-axis, \(\delta_{\gamma}\) is the attitude error around the z-axis, \(I\) is the unit matrix of \(3\times 3\). In the navigation coordinate system \(n\), the relationship between the calculated value and the real value of the rotation angular velocity \(\omega_{in}^{n}\) of the navigation coordinate system \(n\) relative to the geocentric inertial coordinate system \(i\) satisfies the following formula:  
\[\widetilde{\omega}_{in}^{n}=\omega_{in}^{n}+\delta\omega_{in}^{n} \tag{20}\]  
Assuming that only the drift error of the gyroscope is considered, the relationship between the measured carrier angular velocity and the real value of the gyroscope is as follows. \[\widetilde{\omega}_{ib}^{b}=\omega_{ib}^{b}+\epsilon^{b} \tag{21}\]  
In carrier coordinate system \(b\), the angular velocity \(\omega_{nb}^{b}\) of carrier coordinate system \(b\) relative to navigation coordinate system \(n\) can be expressed as follows:  
\[\omega_{nb}^{b}=\omega_{ib}^{b}-C_{n}^{b}\omega_{in}^{n} \tag{22}\]From the Poisson equation, the transfer equation of direction cosine matrix can be expressed as the following formula:  
\[\dot{C}_{b}^{n}=C_{b}^{n}[\omega_{nb}^{b}\times] \tag{23}\]  
where, \(\omega_{nb}^{b}\times=\left[ \begin{array}{ccc}0&-\omega_{nbz}^{b}&\omega_{nbx} ^{b}\\ \omega_{nhc}^{b}&0&\omega_{nby}^{b}\\ -\omega_{nbx}^{b}&\omega_{nby}^{b}&0\end{array} \right]\). By substituting Equation (22) into Equation (23), we can obtain:  
\[\dot{C}_{b}^{n}=C_{b}^{n}[(\omega_{nb}^{b}-C_{n}^{b}\omega_{in}^{n})\times] \tag{24}\]  
Similarly, the calculated value of direction cosine matrix obeys Equation (25):  
\[\dot{\overline{C}}_{b}^{n}=\overline{C}_{b}^{n}[(\widetilde{\omega}_{nb}^{b}- \overline{C}_{n}^{b}\widetilde{\omega}_{in}^{n})\times] \tag{25}\]  
Combined with the above formula, we can obtain:  
\[\dot{C}_{b}^{n}+\delta\dot{C}_{b}^{n} =\overline{C}_{b}^{n}[(\widetilde{\omega}_{nb}^{b}-\overline{C} _{n}^{b}\widetilde{\omega}_{in}^{n})\times] \tag{26}\] \[=(I-[\Phi\times])C_{b}^{n}\Big{[}((\omega_{nb}^{b}+\varepsilon^{b })-C_{b}^{n}(I+[\Phi\times])(\omega_{in}^{n}+\delta\omega_{in}^{n})]\times \Big{\}}\]  
\[\delta C_{b}^{n}=-[\Phi\times]C_{b}^{n} \tag{27}\]  
By deriving the two ends of Equation (27) and combining Equation (23), we can obtain:  
\[\delta\dot{C}_{b}^{n}=-[\dot{\Phi}\times]C_{b}^{n}-[\Phi\times]C_{b}^{n}[ \omega_{nb}^{b}\times] \tag{28}\]  
The following formula can be obtained by neglecting the second order small quantity and substituting Equation (23) and Equation (28) into Equation (26):  
\[ \begin{array}{l}(I-[\Phi\times])C_{b}^{n}[\omega_{nb}^{b}\times]-[\dot{\Phi }\times]C_{b}^{n}\\ =(I-[\Phi\times])C_{b}^{n}\Big{[}(\omega_{nb}^{b}+\varepsilon^{b}-\delta \omega_{in}^{b}-C_{n}^{b}[\Phi\times]\omega_{in}^{n}]\times\Big{\}}\end{array}