# Design and Simulation of On-Orbit Assembly System Based on Insect-Inspired Transportation
## 3 Kinematics Model of Robot Moving Process  
When studying the gait control of an on-orbit assembly robot, it is necessary to master the corresponding relationship between the angle of each joint and the trajectory of each  
Figure 2: Structure of on-orbit assembly robot. (**a**) Schematic diagram of ants carrying heavy objects. (**b**) Configuration diagram. (**c**) Distribution of degrees of freedom. joint, so it is necessary to establish the kinematics model of the robot. The robot kinematics model is the prerequisite for gait planning and control research. The patrol phase of the on-orbit robot can be simplified as a five-link model. Each leg includes the thigh, calf, and sole, and joints include the hip joint and knee joint. As shown in Figure 3, the links L1 and L2 constitute the left leg, the links L4 and L5 constitute the right leg, and the link L3 is the main body part of the robot. The length of each connecting rod is set as \(L_{i}\) (\(i\) = 1, 2,..., 5), the included angle between each connecting rod and the vertical direction is set as \(I\) (\(i\) = 1, 2,..., 5), wherein the clockwise included angle is set as positive, the center of the contact point between the left leg sole and the ground is taken as the origin to establish a coordinate system, the mass of each connecting rod is \(m_{i}\) (\(i\) = 1, 2,..., 5), and the coordinate (\(x_{i}\), \(z_{i}\)) (\(i\) = 1, 2,..., 5) is the center of gravity of the \(i\)th connecting rod. Then, the coordinates at ankle joint A are:  
\[\left\{ \begin{array}{l}x_{A}=0\\ z_{A}=l_{1}\end{array} \right. \tag{1}\]  
Through the trigonometric function relationship, the coordinates of joints B, C, and D can be obtained. \[\left\{ \begin{array}{l}x_{B}=x_{A}+l_{2}\sin\theta_{2}\\ z_{B}=z_{A}+l_{2}\cos\theta_{2}\end{array} \right. \tag{2}\]  
\[\left\{ \begin{array}{l}x_{C}=x_{A}+l_{2}\sin\theta_{2}+l_{3}\sin\theta_{3}\\ z_{C}=z_{A}+l_{2}\cos\theta_{2}+l_{3}\cos\theta_{3}\end{array} \right. \tag{3}\]  
\[\left\{ \begin{array}{l}x_{D}=x_{A}+l_{2}\sin\theta_{2}+l_{3}\sin\theta_{3}- l_{4}\sin\theta_{4}\\ z_{D}=z_{A}+l_{2}\cos\theta_{2}+l_{3}\cos\theta_{3}-l_{4}\cos\theta_{4}\end{array} \right. \tag{4}\]  
The coordinates of each joint and the coordinates of the center of gravity of each connecting rod are derived with respect to time to obtain the speed of each point, and, finally, the coordinates of the center of gravity of the robot are obtained:  
\[\left\{ \begin{array}{l}x_{g}=\frac{m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}+m_{4}x_ {4}+m_{5}x_{5}}{m_{1}+m_{2}+m_{3}x_{3}+m_{4}+m_{5}}\\ z_{g}=\frac{m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}+m_{4}+m_{5}}{m_{1}+m_{2}+m_{3}+m_ {4}+m_{5}}\end{array} \right. \tag{5}\]  
Inverse kinematics solution is a process of solving the rotation angle of each joint required to achieve this attitude by knowing the attitude coordinates of each point of the robot. According to the link length and joint position information, the solution formulas of the robot are deduced as follows. \[\theta_{2}=\arccos\frac{X_{B}^{2}+Z_{B}^{2}-L_{1}^{2}-L_{2}^{2}}{2L_{1}L_{2}} \tag{6}\]  
Figure 3: Five-link model of on-orbit assembly robot. \[\theta_{3}=\arccos\frac{X_{B}^{2}+Z_{B}^{2}-L_{1}^{2}-L_{2}^{2}}{2L_{1}L_{2}}+ \tag{7}\] \[\arccos\frac{\left(Z_{C}-Z_{A}\right)^{2}+\left(X_{C}-X_{A}\right)^ {2}-L_{2}^{2}-L_{3}^{2}}{2L_{2}L_{3}}\] \[\theta_{4}=\arccos\frac{X_{B}^{2}+Z_{B}^{2}-L_{1}^{2}-L_{2}^{2}}{2 L_{1}L_{2}}+\arccos\frac{\left(Z_{C}-Z_{A}\right)^{2}+\left(X_{C}-X_{A} \right)^{2}-L_{2}^{2}-L_{3}^{2}}{2L_{2}L_{3}}\] \[+\arccos\frac{\left(Z_{D}-Z_{B}\right)^{2}+\left(X_{D}-X_{B} \right)^{2}-L_{1}^{2}-L_{2}^{2}}{2L_{3}L_{4}} \tag{8}\]  
\[\theta_{5}=\arccos\frac{L_{4}^{2}+L_{5}^{2}-\left(Z_{E}-Z_{C}\right)^{2}-\left( X_{E}-X_{C}\right)^{2}}{2L_{4}L_{5}} \tag{9}\]  
Therefore, through the above calculation, the forward and inverse kinematics equations of the five-link biped robot moving forward are obtained.