# Analysis of Distributed Measurement Method for Array Antenna Position
## 2 Scheme and Principle
### Distributed Position and Orientation System (POS) Technology  
In order to obtain the phase center of the array antenna of each node with high precision, the paper first uses the distributed POS technology introduced by [PERSON] et al. [6]. The working principle of distributed POS technology is shown in the figure below, which is composed of a main POS and sub-POS. The main POS uses the high-precision inertial measurement unit (IMU), which can accurately measure the motion parameters of the carrier. The sub POS uses low-precision IMU. In the distributed POS technology, the system's accuracy is generally guaranteed by fast-transfer alignment technology, which can achieve the alignment accuracy of 3 angles in 5 s. As shown in Figure 4, the main POS and the sub-POS are respectively installed near different loads on both sides of the airframe or wing to measure the motion parameters of the load center. However, the accuracy of the sub-POS is not high, and the measurement error accumulates with time, so the main POS needs to transfer and align it continuously to achieve the high-precision measurement of the sub nodes. Assuming that the aircraft wing is regarded as a flexible beam, the high-precision measurement of the deformation of the flexible beam is conducive to improving the measurement accuracy of distributed POS, so the deformation model of the flexible beam needs to be established. Assuming that the flexible beam is composed of several small rectangular blocks, each rectangular block has four nodes, which are marked (i = 1,2,3,4) in turn anticlockwise; then the deflection \(\omega\) of each rectangular block, the rotation \(\theta_{x}\) around the X axis and the rotation \(\theta_{y}\) around the Y axis are found; then the displacement vector of the four nodes is:  
\[\alpha^{e}=\left[a_{1}\,a_{2}\,a_{3}\,a_{4}\right]^{T} \tag{7}\]  
Figure 4: Schematic diagram of distributed position and orientation system (POS) technology scheme. where, \(\alpha_{i}=\left\{ \begin{array}{l}\omega_{i}\\ \theta_{xi}\\ \theta_{yi}\end{array} \right\}=\left\{ \begin{array}{l}\omega_{i}\\ \left(\frac{\partial\omega}{\partial y}\right)_{i}\\ -\left(\frac{\partial\omega}{\partial x}\right)_{i}\end{array} \right\}(i=1, \ 2,\ 3,\ 4)\)  
The displacement function \(\omega\) formula of the flexible beam is as follows:  
\[ \begin{array}{ll}\omega=&a_{1}+a_{2}x+a_{3}y+a_{4}x^{2}+a_{5}xy+a_{6}y^{2} +a_{7}x^{3}\\ &+a_{8}x^{2}y+a_{9}xy^{2}+a_{10}y^{3}+a_{11}x^{3}y+a_{12}xy^{3}\end{array}  \tag{8}\]  
The formula can also be expressed as:  
\[\omega=Pa \tag{9}\]  
where, \(P=\left[1\ x\ y\ x^{  
\(\delta\mathbf{a}^{{}^{\prime}}=\left[\delta\psi\ \delta\ \delta\ \delta\gamma\ \right]\) is the attitude measurement, which is the difference between the attitude angle of the main POS and the attitude angle of the sub IMU after being compensated by the deflection angle measured by the grating and the formula is shown as follows:  
\[\delta\mathbf{a}^{{}^{\prime}}=\mathbf{a}_{s}-\mathbf{a}^{{}^{\prime}}_{m} \tag{13}\]  
where, \(\mathbf{a}_{s}\) is the attitude of the subsystem in navigation coordinates, \(\mathbf{a}^{{}^{\prime}}_{m}\) is the attitude angle of the main system after the conversion of the deflection angle at the load point measured by the FBG, which can be obtained by the inverse solution of the attitude transfer matrix \(\mathbf{C}_{b_{m}}^{l_{m}\pi}=\mathbf{C}_{\theta}\mathbf{C}_{b_{m}}^{l_{m}}\). \(\mathbf{C}_{\theta}\) can be obtained from the deflection angle \(\partial_{r}^{b_{m}}\) at the load point measured by FBG.