# Autonomous Pointing Control of a Large Satellite Antenna Subject to Parametric Uncertainty
## 2 LSA Pointing Dynamics  
The LSA is fixed on the satellite body, as shown in Figure 1, and the attitude dynamics include two components: the attitude dynamic model of the flexible satellite and the antenna pointing in the satellite body coordinate system. The attitude dynamics of flexible satellite is governed by the following differential equations [23; 24; 25]:  
\[\left\{ \begin{array}{c}\boldsymbol{I}\ddot{\boldsymbol{\theta}}+\sum\limits _{i=1}^{m}\boldsymbol{B}_{roti}\ddot{\boldsymbol{\eta}}_{i}=\boldsymbol{T}+ \boldsymbol{T}_{d}\\ \ddot{\boldsymbol{\eta}}_{i}+2\xi_{i}\boldsymbol{\Lambda}_{i}\dot{\boldsymbol{ \eta}}_{i}+\boldsymbol{\Lambda}_{i}^{2}\boldsymbol{\eta}_{i}+\sum\limits_{i=1 }^{m}\boldsymbol{B}_{roti}\tilde{\boldsymbol{\theta}}=0\\ i=1,2,...,m\end{array} \right. \tag{1}\]  
where \(\boldsymbol{I}\) is the inertia matrix, \(\boldsymbol{\theta}\) is the vector of the satellite attitude angles, \(\boldsymbol{B}_{roti}\) is the rotational coupling coefficient of the \(i\)th flexible appendage, \(\boldsymbol{\eta}_{i}\) is the modal coordinate, \(\xi_{i}\) is the modal damping ratio, \(\boldsymbol{\Lambda}_{i}\) is the modal frequency, and \(\boldsymbol{T}\) and \(\boldsymbol{T}_{d}\) are the control torque and disturbance torque respectively. According to Craig-Bampton mode synthesis, the internal degree of freedom (DOF) of LSA is given by:  
\[\boldsymbol{X}_{Ia}=\boldsymbol{\phi}_{IIa}\boldsymbol{X}_{Ia}+\boldsymbol{ \phi}_{Ia}\boldsymbol{\eta}_{a} \tag{2}\]  
where \(\boldsymbol{\phi}_{IIa}\) is the constrained modal matrix, \(\boldsymbol{\phi}_{Ia}\) is the main modal matrix, \(\boldsymbol{X}_{Ia}\) represents interface DOF and \(\boldsymbol{\eta}_{a}\) is the LSA modal coordinate. Let \(i,j,k\) represent the LSA pointing DOF and \(\boldsymbol{L}=\left[\boldsymbol{L}_{i},\ \boldsymbol{L}_{j},\ \boldsymbol{L}_{k}\right]\) denotes the pointing DOF matrix. Then, the pointing DOF in the satellite body coordinate system is [9; 10]:  
\[\boldsymbol{\theta}_{a}=\boldsymbol{L}\boldsymbol{X}_{Ia}=\boldsymbol{L} \boldsymbol{\phi}_{IIa}\boldsymbol{X}_{Ia}+\boldsymbol{L}\boldsymbol{\phi}_{Ia }\boldsymbol{\eta}_{a}=\boldsymbol{\theta}+\boldsymbol{L}\boldsymbol{\phi}_{Ia }\boldsymbol{\eta}_{a} \tag{3}\]  
For each LSA pointing DOF, there is \(\boldsymbol{L}_{\
u}=\left[l_{\
u 1},\ l_{\
u 2},\ l_{\
u 3},\ldots l_{\
u n}\right]\)\(\
u=i,j,k\), where \(l_{\
u 1}\ldots l_{\
u n}\) represents the weight of each DOF in antenna pointing. Considering the thermal deformation error \(\boldsymbol{\theta}_{the}\), the installation error \(\boldsymbol{\theta}_{ins}\) and the deployment error \(\boldsymbol{\theta}_{rep}\), Equation (3) can be rewritten as:  
\[\boldsymbol{\theta}_{a}=\boldsymbol{\theta}+\boldsymbol{L}\boldsymbol{\phi}_{Ia }\boldsymbol{\eta}_{a}+\boldsymbol{\theta}_{the}+\boldsymbol{\theta}_{ins}+ \boldsymbol{\theta}_{rep} \tag{4}\]  
where \(\boldsymbol{\theta}_{a}=\left[\theta_{ax}\ \theta_{ay}\ \theta_{az}\right]^{T}\). Combining Equations (1) and (4) yields the following attitude dynamics with antenna pointing:  
\[\left\{ \begin{array}{c}\boldsymbol{I}\ddot{\boldsymbol{\theta}}+\sum\limits _{i=1}^{m}\boldsymbol{B}_{roti}\ddot{\boldsymbol{\eta}}_{i}=\boldsymbol{T}+ \boldsymbol{T}_{d}\\ \ddot{\boldsymbol{\eta}}_{i}+2\xi_{i}\boldsymbol{\Lambda}_{i}\dot{\boldsymbol{ \eta}}_{i}+\boldsymbol{\Lambda}_{i}^{2}\boldsymbol{\eta}_{i}+\sum\limits_{i=1 }^{m}\boldsymbol{B}_{roti}\tilde{\boldsymbol{\theta}}=0\\ i=1,2,...,m\\ \boldsymbol{\theta}_{a}=\boldsymbol{\theta}+\boldsymbol{L}\boldsymbol{\phi}_{Ia }\boldsymbol{\eta}_{a}+\boldsymbol{\theta}_{the}+\boldsymbol{\theta}_{ins}+ \boldsymbol{\theta}_{rep}\end{array} \right. \tag{5}\]  
Then, the LSA pointing dynamics are given by:  
\[\left\{