## 2 Distributed Inertial Network System and Aircraft Elastic Deformation
### Inertial Network Dynamic Measurement Model  
If the aircraft structure is rigid, the dynamic relationship between different node frames can be described by fixed rotation transformations, which can be precisely determined from the geometry of node locations at the time of installation. In high-speed flight and high dynamic maneuvers, the airframe should be considered as a flexible structure. If the flexible structure of an aircraft is ignored, continuing to use the static rotation matrix will lead to a large deviation of local state estimation. Accordingly, it is necessary to develop the dynamic relationships between the network nodes and to estimate these dynamic transformation matrices during flight [31].  
This paper adopts a calculation method of a dynamic rotation transformation matrix. This method is based on the development of analytical dynamic models of the transformation matrices. The body frame of node 1 is used as a reference frame to represent the relative rotation motion of the other frames and the measured angular velocities [31].  
The initial transformation matrix is measured when the aircraft is stationary on the ground. Taking the dynamic conversion between 1 and 2 nodes as an example, \(l\) is the inertial coordinate system, 1 represents the local body coordinate system of node 1 and 2 represents the local body coordinate system of node 2.  
\[\left\{\begin{array}{l}w(x)^{1}_{l/2}=w(x)^{1}_{l/1}+w(x)^{1}_{l/2}\\ w(y)^{1}_{l/2}=w(y)^{1}_{l/1}+w(y)^{1}_{l/2}\\ w(z)^{1}_{l/2}=w(z)^{1}_{l/1}+w(z)^{1}_{l/2}\end{array}\right. \tag{1}\]  
Figure 2: Installation and communication relationship of inertial network nodes.  
The angular rate is represented by a skew-symmetric matrix, and the superscript \(1\) denotes that the angular rate vectors are expressed in terms of the 1 body coordinates.  
\[\Omega^{1}_{1/2}=\Omega^{1}_{1/1}+\Omega^{1}_{1/2} \tag{2}\]  
Considering the differential equation of the rotation matrix \(\Omega^{1}_{1/2}=-\dot{C}^{2}_{1}C^{1}_{2}\), (2) can be written as  
\[\dot{C}^{2}_{1}=\left(\Omega^{1}_{1/1}-\Omega^{1}_{1/2}\right)C^{2}_{1} \tag{3}\]  
The measured values of the gyroscope and accelerometer are the angular velocity and acceleration of the node local body system relative to the inertial coordinate system in the node local body coordinate system. In the above Formula (3), \(\Omega^{1}_{1/1}\) can be estimated from the measured value of the gyroscope at node 1, and \(\Omega^{1}_{1/2}\) cannot be obtained directly from the measured value of the gyroscope at node 2.  
The skew-symmetric matrix \(\Omega^{1}_{1/2}\) can be represented by rotation transformation matrix as follows:  
\[\Omega^{1}_{1/2}=C^{1}_{2}\Omega^{2}_{1/2}C^{2}_{1} \tag{4}\]  
Clearly, the rotation transformation matrix is an orthogonal matrix; Equations (3) and (4) form a nonlinear matrix differential equation group, where the initial matrices can be measured when the aircraft is stationary, and these differential equations are solved at each measurement time to obtain a dynamic rotation transformation matrix. In the above formula, node \(1\) is usually the main node installed at the center of gravity of the aircraft.  
Similarly, the rotation transformation matrices \(C^{i}_{1}(i=3,4\ldots n)\) can be obtained by diagonal multiplication, such as.  
\[C^{3}_{2}=C^{3}_{1}C^{1}_{2}=C^{3}_{1}\left(C^{2}_{1}\right)^{-1} \tag{5}\]