## 2 Disturbance Torque Model of the Sun-Tracking Solar Array
### Dynamics Model of the Solar Array  
As shown in Figure 1, the origin of the inertial frame is assigned to the center of the spacecraft-rocket separation interface, and the origin of the body-fixed frame of the sun-tracking solar array is assigned to the center of the installation interface. The fundamental frequency of the spacecraft body is much greater than that of the sun-tracking solar array, and the mass and moment of inertia of the spacecraft body are also generally one or more orders of magnitude greater than that of the sun-tracking solar array. Therefore, when the sun-tracking rotates at a low speed, the spacecraft body can be assumed as a rigid body with constrained translational and rotational displacements. The solar array is discretized by the finite element method, and the position vector and the absolute velocity vector of the \(j\)th node can be expressed as follows:  
\[\left\{ \begin{array}{c}R_{j}=\rho_{P}+\rho_{j}+\delta_{j}\\ v_{j}=\omega_{l}\times\rho_{j}+\omega_{l}\times\delta_{j}+\dot{\delta}_{j}\end{array} \right. \tag{8}\]  
where \(\rho_{P}\) is the vector from the origin of the body-fixed frame to the origin of the inertial frame, \(\rho_{j}\) and \(\dot{\delta}_{j}\) are the vector of the \(j\)th node in the body-fixed frame relative to the origin before deformation and the vector diameter generated by the flexible deformation of the solar array, \(\omega_{l}\) is the rotating velocity matrix of the solar array. In terms of the modal coordinates, \(\delta_{j}\) can be written as follows:  
\[\delta_{j}=\Phi_{j}\eta \tag{9}\]  
Figure 1: Schematic diagram of the solar array drive system. where \(\eta=\left[ \begin{array}{ccccc}\eta_{1}&\eta_{1}&\cdots&\eta_{s}\end{array}  \right]^{T}\) is the modal generalized coordinates matrix of the solar array, \(\Phi_{j}=\left[ \begin{array}{ccccc}\varphi_{1}&\varphi_{2}&\cdots&\varphi_{s} \end{array} \right]\) is the \(j\)th modal shape matrix of the solar array, \(s\) is the modal truncation order. It is known that  
\[\omega_{l}\times\rho_{j}=\widetilde{\omega}_{l}\rho_{j}=-\widetilde{\rho}_{j }\omega_{l} \tag{10}\]  
where \(\widetilde{\omega}_{l}\) and \(\widetilde{\rho}_{j}\) are the skew symmetric matrices of \(\omega_{l}\) and \(\rho_{j}\), respectively. Therefore, the absolute velocity vector of the \(j\)th node can also be expressed as:  
\[v_{j}=\widetilde{\omega}_{l}\rho_{j}+\widetilde{\omega}_{l}\delta_{j}+ \dot{\delta}_{j} \tag{11}\]  
The kinetic energy of the solar array can be expressed as:  
\[T=\frac{1}{2}\sum\limits_{j=1}^{N}v_{j}^{T}v_{j}m_{j} \tag{12}\]  
where \(N\) is the number of the nodes in the solar array, \(m_{j}\) is the mass of the \(j\)th node. Substituting Equation (11) into Equation (12), the kinetic energy equation of the solar array can be obtained as follows:  
\[T=\frac{1}{2}\left(\omega_{l}^{T}J\omega_{l}-\eta^{T}F_{\omega\omega}\eta+ \dot{\eta}^{T}\dot{\eta}\right)+\frac{1}{2}\left(\eta^{T}H_{\omega}^{T}\omega _{l}+\omega_{l}^{T}H_{\omega}\eta+\dot{\eta}H^{T}\omega_{l}+\omega_{l}^{T}H_{ \omega}\dot{\eta}+\dot{\eta}^{T}F_{\omega}\eta+\eta^{T}F_{\omega}\dot{\eta}\right) \tag{13}\]  
where \(J=\sum\limits_{j=1}^{N}\widetilde{\rho}_{j}^{T}\widetilde{\rho}_{j}m_{j}\) is the moment of inertia of the solar array before the deformation, \(H=\sum\limits_{j=1}^{N}\widetilde{\rho}_{j}\[PERSON]_{j}m_{j}\) is the modal angular momentum of the solar array, \(H_{\omega}=\sum\limits_{j=1}^{N}\widetilde{\rho}_{j}\widetilde{\omega}_{l} \[PERSON]_{j}m_{j}\), \(F_{\omega}=\sum\limits_{j=1}^{N}\[PERSON]_{j}^{T}\widetilde{\omega}_{l}\[PERSON]_{j}m_{j}\), and \(F_{\omega\omega}=\sum\limits_{j=1}^{N}\[PERSON]_{j}^{T}\widetilde{\omega}_{l} \widetilde{\omega}_{l}\widetilde{\omega}_{l}\[PERSON]_{j}m_{j}\). The elastic potential energy and the energy loss of the solar array can be expressed as follows:  
\[\left\{ \begin{array}{l}V=\frac{1}{2}\eta^{T}\Lambda\eta\\ C=\frac{1}{2}\eta^{T}D\dot{\eta}\end{array} \right.