# Design and Simulation of On-Orbit Assembly System Based on Insect-Inspired Transportation
## 4 Dynamic Analysis and Simulation of Patrol Process  
In contrast to traditional biped robots, the on-orbit robot moves laterally on the solar panel and a complete walking cycle can be divided into two stages: the two-legged static support stage and the one-legged support stage. During the one-legged support period, one leg supports the other leg to move forward from the ground. The thigh rises first, followed by the lower leg falling. Then, the thigh falls forward and the lower leg swings backward until it touches the ground again. Using the Lagrange method, the dynamic equation of the one-leg support period is established. \[D(\theta)\ddot{\theta}+H(\theta,\dot{\theta})\dot{\theta}=T_{\tau}+T_{ext} \tag{16}\]  
In the process of in-orbit assembly in space, the objects are analyzed in a weightless state, without any effect from gravity and with the disregard of potential energy. The symmetrical positive definite moment of inertia matrix is represented by \(D(\theta)\in R^{5*5}\), while \(H(\theta,\dot{\theta})\in R^{5*5}\) stands for the matrix elements relating to centrifugal and Coriolis forces. The variables \(\theta,\dot{\theta},\ddot{\theta}\in R^{5*5}\) are generalized coordinates, velocities, and accelerations, whereas \(T_{\tau}\)= \([\tau_{1},\tau_{2},\tau_{3},\tau_{4},\tau_{5}]\) stands for the joint torque and \(T_{ext}\) represents the externally applied torque. The expression of the matrix elements is as follows:  
\[ \begin{array}{l}D_{ij}=p_{ij}\cos(\theta_{i}-\theta_{j})\\ H_{ij}=p_{ij}\sin(\theta_{i}-\theta_{j})\dot{\theta}_{j},i,j=1,2,...,5\end{array}  \tag{17}\]  
Let \(p_{ij}\) be defined as follows:  
\[p_{ij}= \begin{cases}I_{i}+m_{i}d_{i}^{2}+a_{i}\left(\sum\limits_{j=i+1}^{5}m_ {j}\right)l_{i}^{2}&j=i\\ a_{i}m_{j}d_{j}+a_{i}a_{j}\left(\sum\limits_{k=i+1}^{5}m_{k}\right)l_{i}l_{j}&j >i\\ p_{ji}&j<i\end{cases}  \tag{18}\]  
In the above formula, the mass, length, moment of inertia about the center of mass, and the distance from the center of mass to its rotating joint I of the \(i\)th link are, respectively, denoted by, and represented as \(m_{i}\), \(l_{i}\), \(l_{i}\), and \(d_{i}\). In addition, \(a_{j}\) stands for  
\[a_{j}=\left\{ \begin{array}{ll}0&j=3\\ 1&j=1,2,4,5\end{array} \right. \tag{19}\]  
In the two-leg support period, the robot's dynamic equation comprised a set of equations that constitute a Lagrange equation with a Lagrange multiplier and a closed chain constraint equation. This is due to the fact that the robot's legs are positioned on the same horizontal plane. \[\left\{ \begin{array}{l}D(\theta)\ddot{\theta}+H(\theta,\dot{\theta})\dot{ \theta}=J^{T}(\theta)\lambda+T_{\tau}\\ \Phi(\theta)=\left[\begin{array}{c}f_{1}\\ f_{2}\end{array}\right]=\left[\begin{array}{c}x_{e}-x_{b}-L\\ z_{e}-z_{b}\end{array}\right]=0\end{array} \right. \tag{20}\]  
The equation above can be described as follows: \(J\) is the Jacobian matrix, \(\lambda\) denotes the Lagrange multiplier, and \(L\) signifies the distance between the two feet during the two-leg support period, which is commonly known as the step size. The stage between the single-leg support period and the double-leg support period is the impact stage. In this stage, the swinging leg touches the ground in an instant, and there will be an impact force between the sole of the foot and the ground, which will lead to a sudden change in the speed of the joint. Assuming that there is a plastic collision between the swing leg and the ground (there is no rebound when touching the ground), there is no relative sliding between the support leg and the ground. Then, the collision occurs in a very short moment, and the speed of the heel in the basic coordinates immediately becomes zero after the collision, while the position remains unchanged. Using the Lagrange shock model, there are:  
\[D(\theta)\ddot{\theta}+H(\theta,\dot{\theta})\dot{\theta}=T_{\tau}+\delta F_{ cut} \tag{21}\]  
The equation defines \(\delta F_{cut}\) as the generalized external force exerted on the leg during the swing phase when it makes contact with the ground.