## 4 The Improved Self-Collision Detection Technology
### Comparison of Kinematic Modeling Results  
In this section, we evaluate the performance of our proposed kinematic modeling in a mobile manipulator with the UR-like arm in Figure 1. Namely, we mainly focus on the difference between the mathematical model and the actual physical model of the mobile manipulator robot. As a mature industrial robot, the UR manipulator has been extensively researched and developed. At present, the industrial application of the UR arm mainly includes UR3, UR5, and UR10. The basic shapes of these robots are similar, but there are great differences in structural parameters and working performance. Therefore, we selected the manipulator similar to the UR5 arm as the object of this experiment for comparing the kinematic modeling methods. The basic structure of the manipulator subsystem is shown in Figure 1, and the coordinate frames established based on different kinematic methods are the same as Figure 2.  
To verify the modeling performance in this study, we established the kinematic modeling of the UR robot based on the classical kinematic modeling method and the optimized kinematic modeling method mentioned in this study. The corresponding basic parameters of the UR5 arm with different kinematic modeling methods are shown in Table 1.  
According to these basic parameters of the manipulator subsystem, we can build different kinematic modeling of the mobile manipulator robot with the UR5-like arm, which are shown in Figure 6a,b. In the classical kinematic modeling method, the initial angle of each joint in the manipulator robot is setting as \((90^{\circ},0^{\circ},0^{\circ},0^{\circ},180^{\circ},0^{\circ})\), the location of each component in the manipulator is the same as the result in Figure 1. The position and posture of all coordinate frames in the based-fixed frame were solved by these two methods. Due to the difference in the establishment of these coordinate frames, we solve the position of some joint-fixed coordinate frames of the manipulator in the world frame, which can be expressed as: \begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & & \multicolumn{4}{c}{**(a) Classical Kinematic Modeling Method**} \\ \hline \(Joint\) & \(d_{i}^{c}\) **(mm)** & \(a_{i}^{c}\) **(mm)** & \(\alpha_{i}^{c}\) **(rad)** & \(Off\) **(rad)** \\ \(J_{1}\) & 89.2 & 0 & \(\pi/2\) & 0 \\ \(J_{2}\) & 0 & \(-425\) & 0 & \(-\pi/2\) \\ \(J_{3}\) & 0 & \(-392.25\) & 0 & 0 \\ \(J_{4}\) & 109.15 & 0 & \(\pi/2\) & \(-\pi/2\) \\ \(J_{5}\) & 94.65 & 0 & \(-\pi/2\) & 0 \\ \(J_{6}\) & 82.3 & 0 & 0 & 0 \\ \hline \multicolumn{5}{c}{**(b) Optimized Kinematic Modeling Method**} \\ \hline \(Start\) & \(End\) & \(d_{x}^{i}\) **(mm)** & \(d_{y}^{i}\) **(mm)** & \(d_{z}^{i}\) **(mm)** & \(q_{x}^{i}\) **(rad)** & \(q_{y}^{i}\) **(rad)** & \(q_{z}^{i}\) **(rad)** \\ \(O_{b}\) & \(O_{1}\) & 0 & 0 & 89.2 & 0 & 0 & \(q_{1}\) \\ \(O_{1}\) & \(O_{c12}\) & 0 & \(-120\) & 0 & 0 & \(q_{2}\) & 0 \\ \(O_{c12}\) & \(O_{2}\) & 0 & 0 & 425 & 0 & 0 & 0 \\ \(O_{2}\) & \(O_{c23}\) & 0 & 120 & 0 & 0 & \(q_{3}\) & 0 \\ \(O_{c23}\) & \(O_{3}\) & 0 & 0 & 392.25 & 0 & 0 & 0 \\ \(O_{3}\) & \(O_{4}\) & 0 & \(-89.15\) & 0 & 0 & \(q_{4}\) & 0 \\ \(O_{4}\) & \(O_{5}\) & 0 & 0 & 94.65 & 0 & 0 & \(q_{5}\) \\ \(O_{5}\) & \(O_{6}\) & 0 & \(-82.3\) & 0 & 0 & \(q_{6}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The key parameters obtained for different kinematic modeling methods. Figure 6: Kinematic modeling of the mobile manipulator robot with different methods (the \({}^{\star\star}\) is the midpoint of these driving wheels, and the \({}^{\star\star}\) is the gravity center of the mobile platform). **(a)** The classical kinematic modeling of the mobile manipulator robot; **(b)** The optimized kinematic modeling of the mobile manipulator robot. \[{}^{0}T_{1}=^{1}T_{Cla}=^{1}T_{Opt}=\left[ \begin{array}{cccc}540.0,&0.0,&569.2 \end{array} \right]^{T}; \tag{23}\] \[{}^{0}T_{2}=^{2}T_{Cla}=^{O_{c23}}T_{Opt}=\left[ \begin{array}{cccc} 540.0,&0.0,&994.2\end{array} \right]^{T};\] \[{}^{0}T_{3}=^{3}T_{Cla}=^{3}T_{Opt}=\left[