## 3 Aircraft Elastic Deformation Estimation and Inertial Network Algorithm Design  
### Aircraft Elastic Deformation Estimation Algorithm  
The elastic deformation of flight affects the performance of inertial navigation. If the error angle between the master node and the child node cannot be accurately estimated, and the deflection deformation at the child node cannot be compensated, it will lead to the child node (such as the weapon system) to misestimate the navigation parameters such as attitude, speed and position and reduce the accuracy of the inertial network fusion algorithm. Therefore, the Kalman filter is used to estimate the misalignment angle, the elastic deformation angle and the elastic deformation angle rate in real time by using the sampling data of the gyroscope and accelerometer of the child node and the master node. When the misalignment angle reaches a certain accuracy, the error of the child node is corrected once.  
The state equation and measurement equation can be written in the form  
\[\dot{X}=FX+w \tag{9}\] \[Z=HX+v\]  
The Kalman filter state vector is given by  
\[X=\left[\begin{array}{cccccccccccc}\Phi_{x}&\Phi_{y}&\Phi_{z}&\delta V_{x}& \delta V_{y}&\delta V_{z}&\epsilon_{x}&\epsilon_{y}&\epsilon_{z}&\
abla_{x}& \
abla_{y}&\
abla_{z}\\ \mu_{x}&\mu_{y}&\mu_{z}&\theta_{x}&\theta_{y}&\theta_{z}&w_{x}&w_{y}&w_{z}& \end{array}\right]^{\mathsf{T}} \tag{10}\]  
where \(\Phi_{x}\quad\Phi_{y}\quad\Phi_{z}\) is attitude angle error; \(\delta V_{x}\quad\delta V_{y}\quad\delta V_{z}\) is velocity error; \(\epsilon_{x}\quad\epsilon_{y}\quad\epsilon_{z}\) is gyroscope bias error; \(\
abla_{x}\quad\
abla_{y}\quad\
abla_{z}\) is accelerometer bias error; \(\mu_{x}\quad\mu_{y}\quad\mu_{z}\) is initial installation angle error; \(\theta_{x}\quad\theta_{y}\quad\theta_{z}\) is elastic deformation angle; \(w_{x}\quad w_{y}\quad w_{z}\) is elastic deformation angular velocity.  
The measurement vector is given by  
\[Z=\begin{bmatrix}\varphi_{x}&\varphi_{y}&\varphi_{z}&\delta V_{x}&\delta V_{y} &\delta V_{z}\end{bmatrix}^{\mathsf{T}} \tag{11}\]  
The state transition matrix \(F\) is obtained by equivalent transformation of inertial navigation error equation [33], the \(F\) matrix is as follows.  
\[F=\begin{bmatrix}-\begin{bmatrix}(\omega_{en}^{n}+\omega_{ie}^{n})\times\end{bmatrix} &0_{3\times 3}&-C_{bs}^{n}&0_{3\times 3}&0_{3\times 3}&0_{3\times 3}&0_{3 \times 3}\\ \begin{bmatrix}f^{n}\times\end{bmatrix}&-\begin{bmatrix}(\omega_{en}^{n}+2 \omega_{ie}^{n})\times\end{bmatrix}&0_{3\times 3}&C_{bs}^{n}&0_{3\times 3}&0_{3 \times 3}&0_{3\times 3}\\ \begin{matrix}0_{3\times 3}&0_{3\times 3}&0_{3\times 3}&0_{3\times 3}&The values of \(F_{1}\) and \(F_{2}\) in Formula (12) are given by  
\[F_{1}=\begin{bmatrix}-\beta_{x}^{2}&0&0\\ 0&-\beta_{y}^{2}&0\\ 0&0&-\beta_{z}^{2}\end{bmatrix},\;F_{2}=\begin{bmatrix}-2\beta_{x}&0&0\\ 0&-2\beta_{y}&0\\ 0&0&-2\beta_{z}\end{bmatrix} \tag{13}\]  
The measurement matrix \(H\) is given by the relationship between the selected state vector and the measurement vector.  
\[H=\begin{bmatrix}I_{3\times 3}&0_{3\times 3}&0_{3\times 6}&-C_{bs}{}^{n}&-C_{bs}{}^{n }0_{3\times 3}\\ 0_{3\times 3}&I_{3\times 3}&0_{3\times 6}&0_{3\times 3}&0_{3\times 3}&0_{3 \times 3}\end{bmatrix} \tag{14}\]  
The Kalman filter is used to estimate the misalignment angles, installation error angle, elastic deformation angle and elastic deformation angular velocity, and then the navigation state of the child node is compensated by using the estimated error state.