# Design and Dynamic Simulation Verification of an On-Orbit Operation-Based Modular Space Robot
## 3 Dynamic Modeling of Spatial Modular Robot Based on R-W Method
### Establishment of Kinetic Equations  
In the tree system composed of \(N\) rigid bodies, the mass of rigid body \(B_{i}\) is \(m_{i}\) and the moment of inertia is \(\mathbf{J}_{i}\). The principal vector of the external force on the rigid body and the principal moment relative to the center of mass are \(\mathbf{F}_{i}^{\circ}\) and \(\mathbf{M}_{i}^{\circ}\), respectively. According to the velocity variation principle, the variational form of the dynamic equation of the system is  
\[\sum_{i=1}^{N}\left[\Delta\dot{\mathbf{r}}(-m_{i}\ddot{r}_{i}+\mathbf{F}_{i}^{\circ})+ \Delta\mathbf{\omega}_{i}\cdot\left(-\mathbf{J}_{i}\cdot\dot{\mathbf{\omega}}_{i}-\mathbf{ \varepsilon}_{i}+\mathbf{M}_{i}^{\circ}\right)\right]+\Delta P=0 \tag{25}\]  
where \(\Delta P\) represents the virtual power exerted by the internal forces between objects in the system.  
The topology of the robot system in the initial state is described in Figure 17.  
According to the described connection relationship between modules, one first determines the rotation axis vector of \(M_{1}\) and then obtains the orientation vectors of all modules.  
Transformed into a multi-body system problem description, the system topology is shown in Figure 18, where \(B_{1}\) is the object composed of module one and the platform, \(B_{2}\)-\(B_{9}\) are new objects formed by connecting adjacent modules, and \(B_{10}\) is the passive sub-module of module 9. \(H_{1}\) is the virtual hinge from \(B_{1}\) to the inertial coordinate system, \(H_{2}\)-\(H_{10}\) are all rotation hinges of the module, and its constraint vector is the projection of the rotation axis vector in the inertial system. Thus, according to the topological configuration, the system topology description matrix is obtained.  
The correlation matrix and pathway matrix are:  
The array of internal objects is \(\underline{L}=\big{(}0\quad 1\quad 2\quad 3\quad 4\quad 5\quad 6\quad 7\quad 8\quad 9\big{)}\).  
The mass of the platform base is set to be 800 kg. According to the mission design, the robot system reconfiguration process is as shown in the following three stages.  
Reconfiguration stage 1: movement of modules No. 3, No. 4, No. 6, and No. 7. The initial configuration of the modules is shown in Figure 19.  
Figure 16: Mission initial state.  
Figure 17: Module topology description.  
Figure 18: System topology description.  
Figure 19: Initial position.  
Figure 20: Reconfiguration process A.  
Figure 21: Reconfiguration process B.  
Figure 22: New topology.  
Figure 19: Initial position.  
The array of internal objects is \(\underline{L}=\big{(}0\quad 1\quad 2\quad 3\quad 4\quad 5\quad 1\quad 7\quad 8\quad 9 \big{)}\).  
The masses of \(B_{1}\) and \(B_{6}\) have changed and need to be reacquired during the kinetic analysis.  
Reconfiguration stage 3: Module 9 is connected to the capture module in the functional cabin, and the capture module is defined as module 10, thereby forming a new robot system with capture function. At this time, the topology of the robot system also changes, adding a module and a rotating hinge, which will not be elaborated here. After the docking process, the motions of modules 6, 10, 3, and 4 are controlled. The rotation angles are -120\({}^{\circ}\), 120\({}^{\circ}\), 120\({}^{\circ}\), \(-\)240\({}^{\circ}\), \(-\)120\({}^{\circ}\), respectively, and one can get the final robot configuration shown in Figure 23.  
The program flow chart is shown in Figure 24.  
Figure 24: Simulation process.  
Figure 23: Reconfiguration of Process C.  
Figure 27: Time response of control torque of all module joints.