## 3 Federated Kalman Filter Design
### Error Model of Strapdown Inertial Navigation System
\begin{array}{rl}\delta\dot{V}_{\rm E}=&f_{\rm N}\varphi_{\rm U}-\rm f _{\rm U}\varphi_{\rm N}+(\frac{V_{\rm N}\rm tan\,L-V_{\rm U}}{R_{\rm N}+\rm H} )\delta V_{\rm E}+(2\omega_{\rm ie}\rm sin\,L+\frac{V_{\rm Fe}\rm tan\,L}{R_{ \rm N}+\rm H})\delta V_{\rm N}+(2\omega_{\rm ie}V_{\rm N}\rm cos\,L+\\ \frac{V_{\rm E}V_{\rm N}\rm sec^{2}\,L}{R_{\rm N}+\rm H}+&2\omega_{\rm ie}\rm V _{\rm U}\sin\,L)\delta L-(2\omega_{\rm ie}\rm cos\,L+\frac{V_{\rm E}}{R_{\rm N }+\rm H})\delta V_{\rm U}+\frac{V_{\rm E}V_{\rm U}-V_{\rm E}V_{\rm N}\rm tan\,L }{(R_{\rm N}+\rm H)^{2}}\delta\rm H+\
abla_{\rm E}\\ &\delta\dot{V}_{\rm N}=&f_{\rm U}\varphi_{\rm E}-\rm f_{\rm E}\varphi_{\rm U}-2 (\omega_{\rm ie}\rm sin\,L+\frac{V_{\rm Fe}\rm tan\,L}{R_{\rm N}+\rm H})\delta V _{\rm E}-\frac{V_{\rm E}V_{\rm N}\rm V_{\rm N}}{R_{\rm M}+\rm H}-\frac{V_{\rm N }\rm V_{\rm U}}{R_{\rm M}+\rm H}-\\ &(2\omega_{\rm ie}\rm cos\,L+\frac{V_{\rm E}\rm sec^{2}\,L}{R_{\rm N}+\rm H})V_{ \rm E}\delta L+\frac{V_{\rm N}V_{\rm U}+V_{\rm E}\rm tan\,L}{(R_{\rm N}+\rm H )^{2}}\delta\rm H+\
abla_{\rm N}\\ &\delta\dot{V}_{\rm U}=&f_{\rm N}\varphi_{\rm E}+\rm f_{\rm E}\varphi_{\rm N} +2(\omega_{\rm ie}\rm cos\,L+\frac{V_{\rm E}}{R_{\rm N}+\rm H})\delta V_{\rm E}+2 \frac{V_{\rm N}\rm N\delta V_{\rm N}}{R_{\rm M}+\rm H}-2\omega_{\rm ie}V_{\rm E }\rm sin\,L\delta L-\\ &\frac{V_{\rm E}V_{\rm E}+\rm V_{\rm N}V_{\rm N}}{(R_{\rm N}+\rm H)^{2}}\delta H +\
abla_{\rm U}\end{array} \right. \tag{2}\]  
where \(\delta\rm H\) is the height position errors in the navigation coordinate system; \(\rm f_{E}\), \(\rm f_{N}\), and \(\rm f_{U}\) are the specific force of accelerometer in the east, north, and upward directions in the n system, respectively; and \(\
abla_{\rm E}\), \(\
abla_{\rm N}\), and \(\
abla_{\rm U}\) are the specific force error of accelerometer in the east, north, and upward directions in the n system, respectively. The position error equation of the strapdown inertial navigation system is shown in Equation (3):  
\[\left\{ \begin{array}{rl}\delta\dot{\rm L}=\frac{\delta\
abla_{\rm N}}{R_{\rm M }+\rm H}-\frac{V_{\rm N}}{(R_{\rm M}+\rm H)^{2}}\delta\rm H\\ \delta\dot{\rm\lambda}=\frac{\rm sec}{R_{\rm N}+\rm H}\delta V_{\rm E}+\frac{V_{ \rm E}\rm sec\,L\rm tan}{R_{\rm N}+\rm H}\delta L-\frac{V_{\rm E}\rm sec\,L}{(R_{ \rm N}+\rm H)^{2}}\delta\rm H\\ \delta\dot{\rm H}=&\delta\dot{\rm V}_{\rm U}\end{array} \right. \tag{3}\]  
In this study, the errors of the inertial measurement unit (IMU) were calibrated before use, and it was considered that the errors of IMU after calibration and compensation were random zero bias and white noise. The random zero bias constant error equations of the gyroscope and accelerometer are shown in Equations (4) and (5), respectively:  
\[\dot{\varepsilon}_{\rm x} =0,\ \dot{\varepsilon}_{\rm y} =0,\ \dot{\varepsilon}_{\rm z} =0 \tag{4}\] \[\dot{\
abla}_{\rm x} =0,\ \dot{\
abla}_{\rm y} =0,\ \dot{\
abla}_{\rm z} =0 \tag{5}\]  
where \(\varepsilon_{\rm x}\), \(\varepsilon_{\rm y}\), \(\varepsilon_{\rm z}\), and \(\
abla_{\rm x}\), \(\
abla_{\rm y}\), and \(\
abla_{\rm z}\) are the constant zero bias of the gyroscope and the constant zero deviation of the accelerometer on the three measurement axes of the inertial measurement unit, respectively.