## 3 ABP Determination Models and BPE Computing Method
### Definition of Basic Coordinate Systems  
This part defines the ideal body-fixed coordinate system (BFCS) for each antenna element. It should be pointed out that the QOF system is a black box system for users other than the instrument manufacturer, and the beam transmission process in it is confidential. Therefore, we equate the rotation of the lightweight moving part in the QOF system to the circular motion of an equivalent coaxial feed (ECF). The fixed rotation axis around which the ECF is rotated is called the feed rotation axis (FRA). According to Figure 4, the definitions for five basic BFCSs are given:  
1. ECF coordinate system \(A\)-\(x_{a}y_{a}z_{a}\)  
The origin \(A\) of the coordinates is placed at the geometric centroid of the ECF, and the \(z_{a}\)-axis coincides with the outward-pointing normal at the entrance of the ECF. Ideally, the antenna beam is assumed to enter the ECF along the negative \(z_{a}\)-axis. Particularly, when the ECF is located at the \"zero\" position, the corresponding ECF coordinate system is denoted as \(A0\)-\(x_{a0}y_{a0}z_{a0}\). The ICS coordinates of the origin \(A0\) are \((x_{A0},y_{A0},z_{A0})\), and the virtual mounting matrix of the \"zero-position\" ECF is \(\mathbf{R}_{\mathrm{ICS}}^{\mathrm{ECF0}}=\mathbf{R}_{z}(\alpha_{a0})\mathbf{R }_{y}(\beta_{a0})\mathbf{R}_{z}(\gamma_{a0})\).
2. FRA coordinate system \(B\)-\(x_{b}y_{b}z_{b}\)  
The FRA coordinate system is an auxiliary coordinate system established to describe the circular rotation of the ECF, where the \(z_{b}\)-axis is specified as the rotation axis of the ECF. The ICS coordinates of the origin \(B\) are \((x_{B},y_{B},z_{B})\), and the virtual mounting matrix of the FRA is \(\mathbf{R}_{\mathrm{ICS}}^{\mathrm{FRA}}=\mathbf{R}_{z}(\alpha_{b})\mathbf{R }_{y}(\beta_{b})\mathbf{R}_{z}(\gamma_{b})\).
3. SSR coordinate system \(C\)-\(x_{c}y_{c}z_{c}\)  
The origin \(C\) of the coordinates is placed at the geometric centroid of the SSR, and the \(z_{c}\)-axis is specified as the outward-pointing normal of the SSR. The ICS coordinates of the origin \(C\) are \((x_{C},y_{C},z_{C})\), and the mounting matrix of the SSR is \(R_{\mathrm{ICS}}^{\mathrm{SSR}}=\mathbf{R}_{z}(\alpha_{c})\mathbf{R}_{y}( \beta_{c})\mathbf{R}_{z}(\gamma_{c})\)  
Figure 4: Definition of basic BFCSs that are used in deducing the ideal ABP determination model.  
(4)  
FSR coordinate system \(D\)-\(x_{d}y_{d}z_{d}\)  
The origin \(D\) of the coordinates is placed at the virtual focus of the FSR, and the \(z_{d}\)-axis is specified as the focal axis of the FSR. The ICS coordinates of the origin \(D\) are \((x_{D},y_{D},z_{D})\), and the mounting matrix of the FSR is \(\mathbf{R}_{\rm ICS}^{\rm FSR}=\mathbf{R}_{z}(\alpha_{d})\mathbf{R}_{y}(\beta_{d})\mathbf{R}_{z} (\gamma_{d})\).  
(5)  
MR coordinate system \(E\)-\(x_{e}y_{e}z_{e}\)  
The origin \(E\) of the coordinates is placed at the vertex of the parent paraboloid of the MR, and the \(z_{e}\)-axis is specified as the focal axis of the MR. The MR is mounted on the Earth-facing panel of the GEO satellite, and its focal axis points to the Earth. It should be noted that the MR coordinate system is also taken as the ICS.