## 4 Simulation Analysis
### Linear Relationship Analysis  
Based on Tables 3-5, 100 sets of nominal poses and actual poses of the industrial robot were generated. Here, the actual poses do not include noise and are obtained solely from contaminated geometric parameters and forward kinematic equations. Assuming the C-LM algorithm is iterated 20 times, the cross-identification results of the TM-MDH model are shown in Tables 6 and 7. It was found that 14 parameter identification results were anomalous. In this paper, the definition of parameter anomaly is as follows: if the relative error between the identified deviation of a parameter and its given theoretical value exceeds 2%, the identification result of that parameter is considered anomalous. Analyzing the anomalous data of \(d_{2}\) and \(d_{3}\) in Table 6 and observing Figure 4, it can be concluded that as long as they satisfy Equation (29), their anomalies will not affect the actual pose of the industrial robot's end effector. In other words, for the same pose, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Joints** & \(\Delta\theta\)**/rad** & \(\Delta d\)**/mm** & \(\Delta a\)**/mm** & \(\Delta a\)**/rad** & \(\Delta\beta\)**/rad** \\ \hline
1 & \(\mathbf{-0.0022}\) & \(\mathbf{0.0104}\) & \(-0.7\) & \(0\) & / \\
2 & \(0.002\) & \(\mathbf{-0.0104}\) & \(-0.4\) & \(0.0035\) & \(0.001\) \\
3 & \(0.001\) & \(\mathbf{-0.0096}\) & \(0.5\) & \(0.0014\) & / \\
4 & \(0.004\) & \(-0.04\) & \(0.5\) & \(-0.0014\) & / \\
5 & \(0.0035\) & \(0.01\) & \(-0.1\) & \(0.00087\) & / \\
6 & \(\mathbf{-0.0044}\) & \(\mathbf{-0.03}\) & \(\mathbf{-0.01}\) & \(\mathbf{-0.0011}\) & / \\ \hline \hline \end{tabular}
\end{table} 
Table 6: \(\Delta\theta\), \(\Delta d\), \(\Delta a\), \(\Delta a\), and \(\Delta\beta\) identification results without noise (the values in bold font indicate anomalies). there exist an infinite number of sets of inverse solutions for \(d_{2}\) and \(d_{3}\), indicating that the solution exhibits singularity. \[f(d_{2},d_{3})=\Delta d_{2}^{x}+\Delta d_{3}^{x}-(\Delta d_{2}^{x}+\Delta d_{3}^{ x})=0 \tag{29}\]  
Here, \(\Delta d_{2}^{x}\) and \(\Delta d_{3}^{x}\) are the identification results of the deviations for parameters \(d_{2}\) and \(d_{3}\). \(\Delta d_{2}^{x}\) and \(\Delta d_{3}^{x}\) are the preset theoretical values for the deviations of parameters \(d_{2}\) and \(d_{3}\). Based on the above analysis, the preliminary results showed that there is a linear relationship between \(d_{2}\) and \(d_{3}\), as their deviations sum up to a constant value equal to the sum of the preset deviations. Similarly, it can be inferred that there are linear relationships between \(d_{1}\) and \(z\); \(d_{6}\) and \(z^{\prime}\); \(d_{6}\) and \(x^{\prime}\); \(\theta_{1}\) and \(\alpha\); \(\theta_{6}\) and \(\alpha^{\prime}\); and \(\alpha_{6}\) and \(\gamma^{\prime}\). To further validate these judgments, while keeping the other geometric parameters constant, we continuously adjusted the preset deviations of \(d_{2}\) and \(d_{3}\), and observed the value of \(f(d_{2},\ d_{3})\), as shown in Figure 5a. Upon observation, it was found that, for different preset deviations \(\Delta d_{2}^{x}\) and \(\Delta d_{3}^{x}\), the value of \(f(d_{2},d_{3})\) consistently remained at zero. This demonstrates that there exists a linear relationship between the link offsets \(d_{2}\) and \(d_{3}\) for the BRTIRUS0707A robot. Similarly, Figure 5 can also be used to determine the linear relationships between \(d_{1}\) and \(z\); \(d_{6}\) and \(z^{\prime}\); \(\theta_{1}\) and \(\alpha\); \(\theta_{6}\) and \(\alpha^{\prime}\); and \(\alpha_{6}\) and \(\gamma^{\prime}\). During the parameter identification process of the industrial robot, it was observed that there exists a linear relationship between \(d_{1}\) and \(z\) (i.e., \(z\) can be expressed in terms of \(d_{1}\)). This indicates the presence of parameter redundancy, suggesting that the introduction of the transformation parameter \(z\) would affect the identification of the actual structural parameter \(d_{1}\).