## 4 Simulation for Robot Model  
To improve errors based on the direct approach, the simulation was performed based on PUMA robot parameters before and after compensation using a neural network. For the comparison, two robot models were created: the nominal model and the error model. All DH parameters of the error model added to the nominal model with uniformly distributed noise. Let \(\varepsilon_{length}\) and \(\varepsilon_{angular}\) be the length error magnitude (mm) and angular error offset (deg). The noises added to the nominal model parameters with length error \(\Delta\varepsilon_{length}\)(mm) and angular error \(\Delta\varepsilon_{angular}\)(deg) are given as:  
\[\Delta\varepsilon_{length}=G(x_{l},\sigma_{l})*\varepsilon_{length}\, \tag{18}\]  
\[\Delta\varepsilon_{angular}=G(x_{a},\sigma_{a})*\varepsilon_{angular}\, \tag{19}\]  
where \(G(x\bullet,\sigma\bullet)=\ \frac{1}{2\pi\alpha\bullet^{2}}e^{-x \bullet^{2}/2\phi\bullet^{2}}\), \(x\bullet,\sigma\bullet\) are the mean and standard deviation. The geometric errors in the robot-link could be written as \(\Delta a_{i},\ \Delta a_{i},\ \Delta d_{i},\ \Delta\theta_{i}\) while the original parameters of the robot-link \(i^{th}\) were denoted as \(a_{i},\ \alpha_{i},\ d_{i},\ \theta_{i}\). The parameters of error model are as following: \(a_{i}{}^{r}=\ a_{i}+\Delta a_{i}\ (mm)\), \(\alpha_{i}{}^{r}=\ \alpha_{i}+\Delta\alpha_{i}\ (mm)\), \(d_{i}{}^{r}=\ d_{i}+\Delta d_{i}\ (mm)\).  
Where \(\Delta\bullet\) is the noise determined in Equations (18) and (19). The actual values for two kinematic models are shown in Tables 1 and 2, respectively. In this simulation, \(\varepsilon_{length}=10\) mm, and \(\varepsilon_{angular}=2 deg\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link No.** & \(\boldsymbol{\theta_{i}/rad}\) & \(\boldsymbol{d_{i}/mm}\) & \(\boldsymbol{a_{i}/mm}\) & \(\boldsymbol{\alpha_{i}/rad}\) \\ \hline
1 & \(\theta_{1}\) & 0 & 0 & 1.5708 \\
2 & \(\theta_{2}\) & 0 & 432 & 0 \\
3 & \(\theta_{3}\) & 150 & 20 & \(-\)1.5708 \\
4 & \(\theta_{4}\) & 432 & 0 Firstly, the simulation was based on the neural network combined with the laser tracker system. The simulation describes the use of an artificial neural to estimate the robot's end-effector where the actual positions of end-effector in the workspace are known. The robot is moved to all 3D grid points, all position errors in the 3D grid are measured and recorded by the laser tracker system, and these position errors are stored in the memory to train the neural network. The simulation results are shown in Figure 7. The errors have been improved after the compensation. However, a direct approach using a laser tracker is not well suited for vision-based robot applications because of the cost-effectiveness of the external measuring device and the elaborate setup for the calibration procedure.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link No.** & \(\theta_{y}\)**/rad** & \(d_{j}\)**/mm** & \(a_{j}\)**/mm** & \(\alpha_{i}\)**/rad** \\ \hline
1 & \(\theta_{1}\) & 2.42463 & 3.433 & 1.65256 \\
2 & \(\theta_{2}\) & 5.92042 & 433.966 & 0.031139 \\
3 & \(\theta_{3}\) & 150.023 & 20.0033 & \(-\)1.441 \\
4 & \(\theta_{4}\) & 436.297 & 0.317539 & 1.5975 \\
5 & \(\theta_{5}\) & 5.57796 & 1.30921 & \(-\)1.4774 \\
6 & \(\theta_{6}\) & 4.17643 & 6.27515 & 0.1111 \\ \hline \hline \end{tabular}
\end{table}
Table 2: DH parameters of the error model.  
Figure 7: The error compensation results. (**a**) position error (mm) (**b**) compensated position error (mm) (**c**) angular error (deg) (**d**) compensated angular error (deg).  
Secondly, to illustrate the validity of the proposed method, this section performs error compensation in the simulation environment of the PUMA 560 robot model. For comparison, two robot models were created the same as above. The data generated in this simulation for solving the hand-eye vision problem and training the neural network is also described in this section.