# 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
### Position Motion of Each Rigid Body in the System
The vector radius of the base point of the conjoined base with respect to the base point of the overall base of the system is  
\[\mathbf{r}_{i}=\sum_{k:B_{k}\in(B_{i})}\mathbf{d}_{rk}+\mathbf{r}_{0} \tag{12}\]  
where, \(k\) is the subscript of all rigid bodies \(B_{k}\) on the path from \(B_{i}\) to \(B_{0}\). The coordinate matrix of the above formula in the overall basis \(\underline{\epsilon}\) is  
\[\underline{r}_{i}=\sum_{k:B_{k}\in(B_{i})}\underline{d}_{rk}+\underline{r}_{0 }=\sum_{k:B_{k}\in(B_{i})}\underline{A}^{k}\underline{d}^{\prime}{}_{rk}+ \underline{r}_{0} \tag{13}\]  
The equation only contains the direction cosine matrix \(\underline{A}^{k}\) of all rigid bodies on the path from the rigid body \(B_{i}\) to \(B_{0}\) and constant coordinate matrix of the path matrix in the conjoined basis. Thus, the centroid coordinate matrix of \(B_{i}\) is a function of the generalized coordinates of all rigid internal joints on the path from \(B_{i}\) to \(B_{0}\). Thus, the above equation can be written as  
\[\underline{r}=\underline{d}^{T}\underline{1}_{N}+\mathbf{r}_{0}\underline{1}_{N} \tag{14}\]  
The velocity of the centroid can be calculated through differentiation, i.e.,  
\[\dot{\mathbf{r}}_{i}=\sum_{k:B_{k}\in(B_{i})}\dot{\mathbf{d}}_{rk}+\dot{\mathbf{r}}_{0} \quad(i=1,\ldots,N) \tag{15}\]  
Since the vector is fixed to the rigid body \(B_{k}\), substituting \(\dot{\mathbf{d}}_{ki}=\mathbf{\omega}_{k}\times\mathbf{d}_{ki}\) into the above equation yields  
\[\dot{\underline{r}}=-\underline{d}^{T}\times\mathbf{\omega}+\dot{\mathbf{r}}_{0} \underline{1}_{N} \tag{16}\]The relationship between the velocity of the base point of a rigid body \(B_{i}\) and the derivative of the generalized coordinates can be recorded as  
\[\dot{\underline{r}}=\underline{\alpha}\underline{\dot{q}}+\underline{v} \tag{17}\]  
The angular velocity of the above equation is changed to:  
\[\Delta\dot{\underline{r}}=\underline{\alpha}\Delta\underline{\dot{q}} \tag{18}\]  
where  
\[\underline{\alpha}=-(\underline{H}^{\Omega}T\times\underline{d})^{T} \tag{19}\]  
\[\underline{v}=\dot{r}_{0}\underline{1}_{N}-\underline{d}^{T}\times\omega_{0} \underline{1}_{N} \tag{20}\]  
\[\underline{\alpha}= \begin{pmatrix}\underline{H}_{1}^{\Omega T}\times d_{11}&0&0 &0&0&0&0&\\ \underline{H}_{1}^{\Omega T}\times(d_{12}+d_{22})&\underline{H}_{2}^{\Omega T }\times d_{22}&0&0&0&0&\\ \underline{H}_{1}^{\Omega T}\times(d_{13}+d_{33})&0&\underline{H}_{3}^{ \Omega T}\times d_{33}&0&0&0&\\ \underline{H}_{1}^{\Omega T}\times(d_{14}+d_{34}+d_{44})&0&\underline{H}_{3} ^{\Omega T}\times(d_{34}+d_{44})&\underline{H}_{4}^{\Omega T}\times d_{44}&0&0 &\\ \underline{H}_{1}^{\Omega T}\times(d_{15}+d_{55})&0&0&0&\underline{H}_{5}^{ \Omega T}\times d_{55}&0&\\ \underline{H}_{1}^{\Omega T}\times(d_{16}+d_{66})&0&0&0&0&\underline{H}_{6}^{ \Omega T}\times d_{66}\end{pmatrix} \]  
The acceleration of the center of mass can be found by taking two derivatives with respect to time:  
\[\ddot{r}_{i}=\sum_{k:B_{k}\in(B_{i})}\ddot{\underline{d}}_{rk}+\tilde{r}_{0} \quad(i=1,\dots,N) \tag{21}\]  
Since the vector is fixed to the rigid body \(B_{k}\), substituting \(\ddot{\underline{d}}_{ki}=\dot{\mathbf{\omega}}_{k}\times\mathbf{d}_{ki}+\mathbf{\omega}_ {k}(\mathbf{\omega}_{k}\times\mathbf{d}_{ki})\) into the above equation yields  
\[\ddot{\tilde{r}}_{i}=-\sum_{k:B_{k}\in(B_{i})}\mathbf{d}_{ki}\times\dot{\mathbf{ \omega}}_{k}+\sum_{k:B_{k}\in(B_{i})}\mathbf{\omega}_{k}(\mathbf{\omega}_{k}\times\bm {d}_{ki})+\tilde{r}_{0} \tag{22}\]  
i.e.,  
\[\ddot{\underline{r}}=\underline{\alpha}\ddot{\underline{q}}+\underline{\mathbf{w}} \tag{23}\]  
where  
\[w_{i}=\ddot{r}_{0}+\sum_{k:B_{k}\in(B_{i})}\left[\mathbf{\omega}_{k}\times\left( \mathbf{\omega}_{k}\times\mathbf{d}_{ki}\right)+\mathbf{\sigma}_{k}\times\mathbf{d}_{ki}\right] \tag{24}\]