## 5 Simulation Analysis  
The fully flexible base-link-joint space robot system shown in Figure 1 is used as an example for simulation and analysis. Fully flexible space robot base size \(l_{0}=1.5\) m, base mass \(m_{0}=40\) kg, base rotational inertia \(J_{0}=30\) kg \(\cdot\) m\({}^{2}\), base elasticity coefficient \(k_{b}=500\) N/m. Link density \(\rho_{1}=3.5\) kg/m, \(\rho_{2}=1.1\) kg/m, link length \(l_{k}=1.5\) m, link bending stiffness \(El_{k}=100\) N/m\({}^{2}\). Motor rotor moment of inertia \(J_{\rm mk}=0.1\) kg \(\cdot\) m\({}^{2}\), joint elasticity coefficient \(k_{\rm mk}=50\) Nm/rad, \(k=1,2\). The simulation validation is implemented in MATLAB software, mainly using the 4 th order Runge-Kutta method, taking a step size of 0.01 s. Let 0 s be the initial simulation time. Before the controller is turned on, the base and joints of the fully flexible space robot do not move, and the base, links and joints do not vibrate. Its initial configuration \(q_{\rm b}=0\) m, \(\mathbf{q}=[0.5,0.3,0.9]^{\rm T}(\rm rad)\), \(\delta=[0,0.0,0]^{\rm T}(\rm m)\), \(\mathbf{q}_{\rm m}=[0.3,0.9]^{\rm T}(\rm rad)\). \(\dot{\mathbf{q}}_{\rm b}=0\) m/s, \(\dot{\mathbf{q}}=[0,0,0]^{\rm T}(\rm rad/s)\), \(\dot{\mathbf{\delta}}=[0,0,0,0]^{\rm T}(\rm m/s)\), \(\dot{\mathbf{q}}_{\rm m}=[0,0]^{\rm T}(\rm rad/s)\). Assume that a space mission requires a fully flexible space robot base attitude and joints to track the following desired trajectory \(\mathbf{q}_{\text{d}}=[q_{\text{d0}},q_{\text{d1}},q_{\text{d2}}]^{\text{T}}\):  
\[\left\{ \begin{array}{l}q_{\text{d0}}=\cos(\frac{\pi}{5}t)-1\\ q_{\text{d1}}=\sin(\frac{\pi}{5}t)\\ q_{\text{d2}}=\cos(\frac{\pi}{5}t)\end{array} \right..\]  
In order to control the fully flexible space robot base and joints to track the preset desired trajectory to accomplish the space mission, the following simulation is conducted. Turn on the IFSM proposed in this paper at 0 s, and observe the movement of the system base and joint, as well as the vibration of the base, link and joint. The control parameters are selected as \(\gamma_{1}=40\), \(\beta_{1}=3\), \(m_{1}=7\), \(n_{1}=5\), \(p_{1}=7\), \(q_{1}=9\), \(\gamma_{2}=3\), \(\beta_{2}=5\), \(m_{2}=7\), \(n_{2}=5\), \(p_{2}=7\), \(q_{2}=9\), because the design purpose of IFSM controller is to suppress multiple vibration of fully flexible space robot and realize motion control. The common rigid controller has low reliability in controlling the fully flexible space robot, and it is easy to cause control failure. In order to verify this problem, and verify the effectiveness of the flexible vibration suppressor in the IFSM controller, the flexible vibration suppressor is closed in IFSM, and the IFSM without vibration suppression (IFSM-NV) is used to re-simulate. IFSM-NV is composed of Equations (28) and (29) and \(\tau_{\text{m}}=(I+K_{\text{c}})\tau_{\text{ns}}-K_{\text{c}}\tau\). The control parameters during simulation are the same as IFSM. In addition, compared with the traditional motion and vibration integrated sliding mode controller (ISM), IFSM has the functions of fixed time convergence and resisting external interference. In order to verify the advantages of IFSM control, ISM is used for comparative simulation. ISM is composed of traditional sliding mode controller and flexible vibration suppressor proposed in this paper, where switching function \(\mathbf{s}_{\text{r}}=\dot{\mathbf{e}}_{\text{r}}+\lambda\mathbf{e}_{\text{r}}\), sliding mode controller based on hybrid trajectory \(\left(\mathbf{\tau}_{\text{qj}}\right)_{\text{r}}=M(\dot{\mathbf{q}}_{\text{h}}- \lambda\dot{\mathbf{e}}_{\text{r}}-\varepsilon\text{sgn}\mathbf{s}_{\text{r}})+C\dot{ \mathbf{q}}\). The control parameters are selected as \(\varepsilon=2,\lambda=1\). At the same time, the flexible vibration suppressor is closed in ISM (ISM-NV) for simulation, and the control effect of ISM-NV is observed on the fully flexible system.