## 2 Notion  
Continuum space robots consists of a satellite base and a continuum manipulator.  
The three-dimensional model of the continuum manipulator is depicted in Figure 1, comprising a three-segment, cable-driven continuum manipulator. Each segment has a fixed length, with a central axis consisting of an elastic flexible rod controlled by three driving cables, allowing it to bend in any direction, forming an arc. The first segment passes through 9 cables, the second through 6, and the third through 3. Equally spaced support discs are employed to maintain the curvature of the central rod, with the number of perforations on each disc matching the number of cables passing through each segment. Actuated by motors on the base, the cables are pulled to induce bending deformation in the manipulator, facilitating pose control of the robotic system.  
The three-dimensional model of the continuum space robot system is illustrated in Figure 2. The base is a free-floating satellite, and the manipulator is a continuum manipulator. The advantages of this configuration lie in the flexibility and adaptability of the cable-driven continuum manipulator, making it more suitable for exploration and operations in complex and space-constrained environments, as depicted in Figure 3.  
Define the symbols as follows: \(\ q_{0}=\begin{bmatrix}x_{0}&y_{0}&z_{0}&\varphi_{0}&\theta_{0}&\psi_{0} \end{bmatrix}^{\text{T}}\) is the virtual joint variable corresponding to the base, where \(\mathbf{r}_{0}=\begin{bmatrix}x_{0}&y_{0}&z_{0}\end{bmatrix}^{\text{T}}\) is the coordinates of the  
Figure 1: Three-dimensional model of the continuum manipulator.  
Figure 3: On-orbit service task scenarios.  
Figure 2: Three-dimensional model of the continuum space robot.  
center of mass of the base in the inertial frame and \(\left[\begin{matrix}\varphi_{0}&\theta_{0}&\psi_{0}\end{matrix}\right]^{\text{T}}\) is the Euler angles describing the orientation of the base relative to the inertial frame, using a 3-2-1 sequence. \(\mathbf{V}_{0}=\left[\begin{matrix}\mathbf{v}_{0}^{\text{T}}&\mathbf{\omega}_{0}^{\text{T} }\end{matrix}\right]^{\text{T}}\) is the velocity of the base. \(\mathbf{q}=\left[\begin{matrix}\varphi_{1}\theta_{1}\varphi_{2}\theta_{2}\varphi_ {3}\theta_{3}\end{matrix}\right]^{\text{T}}\) is the joint variable, where \(\varphi_{i}\) and \(\theta_{i}\) are the bending plane angle and bending angle of the \(i\)-th (\(i=1\), \(2\), \(3\)) segment of the continuum manipulator. \(\mathbf{q}\) is the joint velocity. \(\overline{\mathbf{q}}=[\mathbf{q}_{0}^{\text{T}}\ \mathbf{q}^{\text{T}}]^{\text{T}}\) is the generalized virtual joint variable. \(\dot{\overline{\mathbf{q}}}\) is the generalized virtual joint velocity. \(\mathbf{P}_{0}\) and \(\mathbf{L}_{0}\) are the linear momentum and angular momentum of the base, respectively. \(\mathbf{P}_{i,\mathbf{d}}\), \(\mathbf{L}_{i,\mathbf{d}}\), \(\mathbf{P}_{i,\mathbf{r}}\), and \(\mathbf{L}_{i,\mathbf{r}}\) are the linear momentum and angular momentum of the elastic rod and support disk of the \(i\)-th segment of the continuum robot, respectively.  
\(\mathbf{M}_{0\text{T}}\) and \(\mathbf{M}_{0\text{T}}\) are the inertia matrices for translation and rotation of the base, respectively. \(\mathbf{M}_{i,\mathbf{dt}}\), \(\mathbf{M}_{i,\mathbf{dr}}\), \(\mathbf{M}_{i,\mathbf{rt}}\), and \(\mathbf{M}_{i,\mathbf{rr}}\) are the inertia matrices for the translation and rotation of the elastic rod and support disk of the \(i\)-th segment of the continuum robot, respectively.