## 2 System Kinematics Analysis  
The structure of the flexible-base flexible-link and flexible-joint space robot and the satellite are shown in Figure 1. The space robot comprises a base \(B_{0}\), a flexible link \(B_{1}\) near the base, and a flexible link \(B_{2}\) away from the base. The center of mass of the base is \(O_{0}\), the center of the hinge connecting the base and link \(B_{1}\) is \(O_{1}\), the center of the hinge connecting link \(B_{1}\) and \(B_{2}\) is \(O_{2}\), and the center of mass of the captured satellite is \(O_{3}\). \(XOY\) is the inertial coordinate system, and \(x_{i}O_{i}y_{i}(i=0,1,2,3)\) is the local coordinate system of \(B_{i}\). \(\theta_{0}\) is the attitude angle of the base, and \(\theta_{j}(j=1,2)\) is the rotation angle of link \(B_{j}\). \(\theta_{m1}\) and \(\theta_{m2}\) are the rotor angles of the motor, and \(\theta_{3}\) is the attitude angle of the satellite. \(P\) and \(P^{\prime}\) are the capture points of the space robot end effector and satellite, respectively. \(r_{0}\), \(r_{P}\), \(r_{P^{\prime}}\) and \(r_{3}\) represent the position vectors of the mass center of the base, the capture point of the space robot end effector, the captured point of the satellite and the mass center of the satellite in the inertial coordinate system, respectively. \(r_{j}\) is the position vector of any point on link \(B_{j}\) in the inertial coordinate system. Let \(x_{j}\) be any distance in the symmetry axis direction of the link \(B_{j}\), and let \(v_{j}(j=1,2)\) be the elastic deformation of link \(B_{j}\) at time \(t\). According to the hypothesis of [PERSON] [26], the flexible base and joint are assumed to be a massless linear telescopic spring and linear torsion spring, respectively. The flexible links are equivalent to the Euler Bernoulli simply supported beam. The elastic coefficients of the base, joints and bending rigidity of the links are fixed, which are expressed as \(k_{\text{b}}\), \(k_{\text{mj}}\) and \(EI_{j}(j=1,2)\), respectively. The deformation of link \(B_{j}\) is \(v_{j}\big{(}x_{j},\text{T}\big{)}=\sum\limits_{k=1}^{n}\phi_{jk}\big{(}x_{j} \big{)}\delta_{jk}(t)\), where \(\phi_{jk}\) and \(\delta_{jk}\) represent the \(k\)th-order modal function and coordinates of link \(B_{j}\), respectively, and \(n\) is the number of reserved modes (\(n=2\) in the paper). The captured satellite is assumed to be a single rigid body system. According to the geometric position relationship of the system, the position vectors \(\mathbf{r}_{0}\), \(\mathbf{r}_{1}\), \(\mathbf{r}_{2}\), \(\mathbf{r}_{\text{p}}\), \(\mathbf{r}_{3}\) and \(\mathbf{r}_{\text{p}^{\prime}}\) are:  
\[\left\{ \begin{array}{l}\mathbf{r}_{0}=(x_{0},y_{0})^{\text{T}}\\ \mathbf{r}_{1}=\mathbf{r}_{0}+(l_{0}+x_{\text{b}})\mathbf{e}_{0}+x_{1}\mathbf{e}_{1}+v_{1}\mathbf{e }_{1}^{\prime},0\leq x_{1}\leq l_{1}\\ \mathbf{r}_{2}=\mathbf{r}_{0}+(l_{0}+x_{\text{b}})\mathbf{e}_{0}+l_{1}\mathbf{e}_{1}+x_{2}\mathbf{ e}_{2}+v_{2}\mathbf{e}_{2}^{\prime},0\leq x_{2}\leq l_{2}\\ \mathbf{r}_{\text{p}}=\mathbf{r}_{0}+(l_{0}+x_{\text{b}})\mathbf{e}_{0}+l_{1}\mathbf{e}_{1}+l_{ 2}\mathbf{e}_{2}\\ \mathbf{r}_{3}=(x_{3},y_{3})^{\text{T}}\\ \mathbf{r}_{\text{p}^{\prime}}=\mathbf{r}_{3}+l_{3}\mathbf{e}_{3}\end{array} \right. \tag{1}\]  
where \((x_{0},y_{0})\) are the base centroid coordinate, \(l_{0}\) is the distance between the rotation centers \(O_{0}\) and \(O_{1}\), \(x_{\text{b}}\) is the base flexible deformation, \((x_{3},y_{3})\) are the satellite centroid coordinate, \(l_{3}\) is the distance between the rotation centers \(O_{3}\) and \(P^{\prime}\), \(\mathbf{e}_{0}\), \(\mathbf{e}_{j}\), \(\mathbf{e}_{j}^{\prime}\)\((j=1,2)\) and \(\mathbf{e}_{3}\) are base vectors. Taking the differentiation of Equation (1) with respected to time \(t\) leads to:  
\[\left\{