## 6 Point Correction Verification  
The point's information was taken through the AT901 Leica Tracker, which was used to develop the NRTEC system. The graph in Figure 9 shows that the system was able to match the planned path between two points. Figure 8: NRTEC repeatability test. Figure 7: NRTEC physical cell. Points p1 and p2 are the same points as reported by the robot, measured by the tracker, and corrected by NRTEC. \"Planned\" in red indicates the planned path that forms the robot trajectory, \"Uncorrected\" are the points where the robot reaches and \"Corrected\" are the same points which were corrected by NRTEC after receiving the information from the Leica Tracker (AT901) and sent back to the robot. The point selected is on the first instance where the system is able to be in the close proximity to the planned path. The system is further able to correct itself and achieve the correction within 0.02 mm. For experimental purposes, 18 different positions were chosen to form the path, as shown in Table 1. There were positions in +ve and -ve of each axis (X, Y, Z) and +ve and -ve in rotation around each axis (RX, RY, RZ). The following table shows one of the samples of the positions in the robot reference frame that forms the path for the experiment. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**X (mm)** & **Y (mm)** & **Z (mm)** & **RX (deg)** & **RY (deg)  
The graph in Figure 10 shows the improved correction over the period, which was achieved due to the prediction and learning capability in neural network-based MPC (model predictive control). The system was able to achieve an accuracy of 0.02 mm. Figure 10 shows the output of the NRTEC system at the end of the calibration process. The residual error was 2.2792 before correction, which was reduced to 0.0262; the graph in middle shows that information. The graph at the top of Figure 10 shows the positional distance between the corrected and uncorrected points. It shows the distance between all 18 points before and after correction. The corrected version of points has been taken after the completion of the 15 th cycle. Each cycle contained the path that was formed by the 18 points. The graph in the middle left shows the error correction improvement from the first cycle to the 15 th cycle. The achievement of correction after completing the first cycle of the path was 0.04 and the error was reduced to 0.0262 after the 15 th cycle. The graph in the middle shows the comparison between the errors before starting the process and after finishing each cycle. The error before starting the process was 2.2795, which reduced to 0.0262 after finishing the process. The graph in the middle right shows the path that was planned for the process. The blue line shows the planned path, the green line shows the path that was taken before correction and the red line shows the path that was taken by the robot after the correction and compensation. The graphs on the bottom show the histogram of the errors in each axis (X, Y, Z) and the rotation (RX, RY, RZ). The histograms are presented for all the 18 points at the end of the process, which means they show the data from the last cycle. It means that the calibration model developed was able to achieve the accuracy of individual components X, Y, Z and RX, RY, RZ within the range of \(\pm\)0.15 mm and \(\pm\)0.02\({}^{\circ}\), respectively. The experiment was performed on different data sets that were collected in different robot positions. Each data set has several parameters; for example, it should have variable positions in +ve and -ve direction of each axis and should have variable reachable rotations RX, RY, RZ. Figure 11 illustrates the performance of the system on eight different robot paths. The 15 cycles of error correction were performed for each robot path to achieve the correction.