# Design and Simulation of On-Orbit Assembly System Based on Insect-Inspired Transportation
## 5 Research on Robot Assembly Process
 \tag{28}\]  
\[ \begin{array}{l}\frac{dV_{i}}{dt}=\dot{\theta}^{T}J_{i}^{T}\left[\begin{array} []{cc}I_{3}&ad^{T}(r_{ci})\\ 0&I_{3}\end{array}\right]\left(\begin{array}{c}0\\ m_{i}g\end{array}\right)\\ =\dot{\theta}^{T}J_{i}^{T}\left(\begin{array}{cc}P_{i}\times r_{ci}\\ P_{i}\end{array}\right)\\ \frac{dV_{i}}{dt}=\dot{\theta}^{T}\frac{\partial V_{i}}{\partial\theta}\end{array}  \tag{29}\]  
\(r_{ci}\) represents the position vector of the center of mass relative to the end effector \(ad(r_{ci})\) denotes an operator that performs an affine transformation on vector \(r_{ci}\). \(\overline{\omega}_{i}\) stands for angular velocity, \(I_{3}\) stands for a \(3\times 3\) identity matrix. Therefore, the Lagrangian function expression of the robot is:  
\[ \begin{array}{l}L=T-V=\frac{1}{2}\dot{\theta}^{T}N(\theta)\dot{\theta}-V( \theta)\\ =\frac{1}{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}V_{i}(\theta)N_{ij}\dot{ \theta}_{i}\dot{\theta}_{j}-\sum\limits_{i=1}^{n}P_{i}{}^{T}r_{ci}(\theta)\end{array}  \tag{30}\]  
\[ \begin{array}{l}\frac{\partial L}{\partial\dot{\theta}_{i}}=\sum\limits_{j=1 }N_{ij}\dot{\theta}_{j}\\ \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{\theta}_{i}}\right)=\sum \limits_{j=1}^{N}N_{ij}\ddot{\theta}_{i}+\sum\limits_{j=1}^{dN_{ij}}\dot{\theta }_{j}\\ =\sum\limits_{j=1}^{\infty}N_{ij}\ddot{\theta}_{j}+\sum\limits_{j=1}^{\infty} \sum\limits_{k=1}^{\infty}\frac{\partial N_{ij}}{\partial\dot{\theta}_{k}} \dot{\theta}_{j}\end{array}  \tag{32}\]  
\[ \begin{array}{l}\frac{\partial L}{\partial\dot{\theta}_{i}}=\frac{1}{2} \frac{\partial}{\partial\dot{\theta}_{i}}\left(\sum\limits_{j=1}^{n}\sum \limits_{k=1}^{n}N_{jk}\dot{\theta}_{j}\dot{\theta}_{k}\right)\\ +\sum\limits_{j=1}^{n}I_{j}^{T}\left[\begin{array}{cc}P_{j}\times r_{ci}\\ P_{j}\end{array}\right]=\\ \frac{1}{2}\dot{\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{\infty}\frac{\partial N _{jk}}{\partial\dot{\theta}_{i}}\dot{\theta}_{j}+\sum\limits_{j=1}^{n}I_{ij}^{T }\left[\begin{array}{cc}P_{j}\times r_{cj}\\ P_{j}\end{array}\right]\end{array}  \tag{33}\]From the above formulas, it can be concluded that:  
\[ \begin{array}{l}\underset{j=1}{\overset{\bar{\Sigma}}{\sum}}\,N_{ij}\ddot{\theta }_{j}+\underset{j=1}{\overset{\bar{\Sigma}}{\sum}}\,\underset{k=1}{\overset{ \bar{\alpha}N_{ij}}{\partial\theta_{k}}}\dot{\theta}_{k}\dot{\theta}_{j}\\ -\frac{1}{2}\underset{j=1}{\overset{\bar{\Sigma}}{\sum}}\,\underset{k=1}{ \overset{\bar{\alpha}N_{jk}}{\partial\theta_{i}}}\dot{\theta}_{j}\dot{\theta}_{ k}-\underset{j=1}{\overset{\bar{\Sigma}}{\sum}}\,I_{ij}^{\bar{\gamma}}\left[\begin{array}{c}P_{j} \times\tau_{cj}\\ P_{j}\end{array}\right]=Q\end{array}  \tag{34}\]  
After simplification, it can be concluded that  
\[ \begin{array}{l}\underset{j=1}{\overset{n}{\sum}}\,N_{ij}\ddot{\theta}_{j}+C _{ij}(\theta,\dot{\theta})\dot{\theta}_{j}+P_{i}(\theta,\dot{\theta})=Q_{i}\\ i=1,2,3,\ldots,n\end{array}  \tag{35}\]  
Equation (35) is the Lagrangian equation for the robot's dynamics. From left to right, the three terms on the left side of the equation represent inertial force, centrifugal force, Coriolis force, and the constraint force caused by the flexible body. The right side of the equation represents the Driven force. We have completed the simulation of the third phase of the on-orbit robotic assembly process using the previously described method. In the second phase, the robot grasps one unit and guides it in a cruising motion toward another unit. When it reaches the connection point of the two assembly units, one of the moving legs damps down on the unit and adjusts its attitude to maintain a 120\({}^{\circ}\) angle with the load-bearing assembly unit. Meanwhile, the other moving leg functions as a robotic arm and grips the third assembly unit, thereby completing the on-orbit assembly process of the robotic system. Figure 5 illustrates a schematic diagram of the robotic system completing the assembly process. The assembly process of the robotic assembly unit is demonstrated in Figure 5a.