## 3 ABP Determination Models and BPE Computing Method
### Definition of the Thermal Distortion Errors of the Reflector  
The ABP determination model given in Section 3.2 was deduced on the assumption that all antenna elements remained at the ideal P&A values. However, in practice, the actual ABP will deviate from the ideal ABP due to the thermal distortion effects of the antenna reflectors exposed outside the spacecraft. Before developing the actual ABP determination model, the necessary first step is to define the TDEs of the reflector. In this study, only the rigid motion of the ideal reflector is considered. Thus, the reflector's TDEs are defined as a result of the rigid motion of the actual reflector with respect to the ideal reflector. Specifically, the TDEs of the reflector include the translation error of the origin of the actual reflector's BFCS relative to the origin of the ideal reflector's BFCS, and the misalignment errors of each axis of the actual reflector's BFCS are relative to the corresponding axis of the ideal reflector's BFCS. The transformation from the ideal reflector's BFCS to the actual reflector's BFCS can be reduced to a translation followed by a rotation. The origin of the ideal reflector's BFCS first translates in the direction of \((x_{i,ma},y_{i,ma},z_{i,ma})\), and then the translated result rotates in a 3-1-2 convention with three Euler angles \((\psi_{i,ma},\varphi_{i,ma},\theta_{i,ma})\) (as measured counterclockwise from the positive \(z\) -, \(x\) -, and \(y\)-axes) to arrive at the actual reflector's BFCS (note: subscript \(i=\{\text{SSR},\text{ FRS},\text{ MR}\}\) is used to distinguish the TDEs of the SSR, FSR, and MR). Finally, for ease of description, we define the \(x\)-, \(y\)-, and \(z\)-axes of the reflector's BFCS as the roll, pitch, and yaw axes, respectively. Therefore, \(\varphi_{i,ma},\theta_{i,ma}\), and \(\psi_{i,ma}\) are called the roll, pitch, and yaw misalignment angles, and \(x_{i,ma},y_{i,ma}\), and \(z_{i,ma}\) are called the roll, pitch, and yaw axial translations. Based on the above definitions, the 7-by-7 GTM \(\mathbf{M}_{i^{\prime}}^{\prime}\) which transforms the pointing vectors between representations from the actual BFCS \(i^{\prime}\) (whose origin is denoted as \(l^{\prime}\)) to the actual BFCS \(j^{\prime}\) (whose origin is denoted as \(l^{\prime}\)), can be written in the following form:  
\[\left\{ \begin{array}{l}\mathbf{M}_{i^{\prime}}^{\prime}=\left[\begin{array}{ ccc}\mathbf{Rot}_{i^{\prime}}^{\prime}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 1}\\ \mathbf{O}_{3\times 3}&\mathbf{Rot}_{i^{\prime}}^{\prime}&\mathbf{I}_{i^{\prime}}^{\prime}\\ \mathbf{O}_{1\times 3}&\mathbf{O}_{1\times 3}&1\end{array}\right]\\ \\ \mathbf{Rot}_{i^{\prime}}^{\prime}=\mathbf{Rot}_{j}^{\prime}\left(\varphi_{j,ma}, \theta_{j,ma},\psi_{j,ma}\right)\mathbf{Rot}_{i}^{\prime}\left(\mathbf{Rot}_{i}^{\prime }(\varphi_{i,ma},\theta_{i,ma},\psi_{i,ma})\right)^{\text{T}}\\ \\ \mathbf{I}_{i^{\prime}}^{\prime}=\mathbf{Rot}_{j}^{\prime}\left(\varphi_{j,ma},\theta_ {j,ma},\psi_{j,ma}\right)\left(-\mathbf{I}_{j}^{\prime}+\mathbf{I}_{j}+\mathbf{Rot}_{i}^{ \prime}\mathbf{I}_{i^{\prime}}\right)\end{array} \right.