## 3 Aircraft Elastic Deformation Estimation and Inertial Network Algorithm Design
### Distributed Data Fusion Algorithm
#### 3.2.1 Measurement Fusion Algorithm  
Taking node 1 as an example, the measurement equation is as follows:  
\[m_{1}=Hx_{1}=HC_{i}^{1}x_{i} \tag{15}\]  
where \(C_{i}^{1}(i=2\dots n)\) is the dynamic rotation transformation matrix, \(x_{1}\) is the 3-dimensional local vector of node 1 (the acceleration or angular velocity of node 1), \(m_{1}\) is the measured value of 6 sensors of node 1 and \(H\) is the installation matrix of nodes. The single node  
Figure 3: Distributed inertial network data fusion algorithm. of the network in this paper selects the dodecahedron installation of the six sensors, the installation matrix can be determined as follows:  
\[H= \begin{bmatrix}0.5257&0&0.8507\\ -0.5257&0&0.8507\\ 0.8507&0.5257&0\\ 0.8507&-0.5257&0\\ 0&0.8507&0.5257\\ 0&0.8507&-0.5257\\ \end{bmatrix}  \tag{16}\]  
The three-dimensional local vector \(x\) in the measurement equation is estimated by the least squares method. \[\hat{x}_{1}=\left(H^{T}H\right)^{-1}\!H^{T}\!m_{1} \tag{17}\]  
The covariance matrix of the estimated value is of the form  
\[P_{x}=\left(H^{T}H\right)^{-1}\!H^{T}RH\!\left(H^{T}H\right)^{-1} \tag{18}\]  
where \(R\) is the covariance matrix of the measurement error. Because the sensors in a single node are independent, the covariance matrix is the diagonal matrix of the variance of each sensor. All sensors of a single node are independent, and their measurement results satisfy the Gaussian probability distribution in the same frame. The errors of the local inertial state estimates are also a Gaussian distributed random vector. The Gaussian probability density function of the multi-dimensional (independent and identically distributed) normal distribution is as follows. \[p(x)=\frac{1}{\left(2\pi\right)^{D/2}\!|\Lambda|^{1/2}}\exp\!\left(-\frac{1}{2 }(x-\mu)^{T}\Lambda^{-1}(x-\mu)\right) \tag{19}\]  
where \(\Lambda\) is the covariance matrix of random variable \(x\), \(D\) is the dimension of random variable \(x\) and \(\mu\) represents mathematical expectation. The dimension of the local inertial state vector is 3, and \(P_{x,l}\) is the covariance matrix of the error of the local inertial state estimate. Taking node 1 as an example, the probability density distribution function of the local inertial state can be written as  
\[\left\{ \begin{array}{l}P_{1}(x)=\frac{1}{\sqrt{(2\pi)^{3}det_{P_{x,1}}}}\exp \!\left[-\frac{1}{2}(\hat{x}_{1}-x_{1})^{T}P_{x,1}^{-1}(\hat{x}_{1}-x_{1}) \right]\\ P_{2}(x)=\frac{1}{\sqrt{(2\pi)^{3}det_{P_{x,2}}}}\exp\!\left[-\frac{1}{2}(\hat {x}_{2}-C_{1}^{2}x_{1})^{T}P_{x,2}^{-1}(\hat{x}_{2}-C_{1}^{2}x_{1})\right]\\ \vdots\\ P_{n}(x)=\frac{1}{\sqrt{(2\pi)^{3}det_{P_{x,l}}}}\exp\!\left[-\frac{1}{2}(\hat {x}_{n}-C_{1}^{n}x_{1})^{T}P_{x_{1}n}^{-1}(\hat{x}_{n}-C_{1}^{n}x_{1})\right] \end{array} \right. \tag{20}\]  
The objective of the fusion of inertial measurements is to generate the optimal estimates of all the local states of inertia, defining the optimality criterion as the maximum of the conditional probabilities. Then, since the measurements of all nodes are independent, the conditional probability density function of the true local inertial state of each node can be written as  
\[L=P_{1}(x)P_{2}(x)\cdots P_{n}(x) \tag{21}\]  
The state corresponding to the maximum value of the established likelihood function is obtained.