## 4 Analysis and Discussion of Results
### Test No. 3: Accuracy and Repeatability  
Test no. 3 was used to evaluate the accuracy and repeatability according to ISO 9283 [63]. The test results for thirty repetitive movements going through a sequence of five positions are shown in Figure 16. This test was conducted with the maximum payload, an environmental temperature of 20 \({}^{\circ}\)C, and at 10% and 35% of the nominal speed. Upon reaching each position, the angles of each joint, position and orientation of the effector are recorded. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Position Still with Active Drive**} & \multicolumn{4}{c}{**Position after Disabled Drive**} \\ \hline  & **P1** & **P2** & **P3** & **P4** & **P5** & **P1** & **P2** & **P3** & **P4** & **P5** \\ \hline Joints [-] & \(3.51\times 10^{-2}\) & \(1.41\times 10^{-2}\) & \(1.29\times 10^{-2}\) & \(8.16\times 10^{-3}\) & \(1.41\times 10^{-2}\) & \(4.08\times 10^{-2}\) & \(7.79\times 10^{-2}\) & \(5.35\times 10^{-2}\) & \(1.83\times 10^{-2}\) & \(4.93\times 10^{-2}\) \\ Position [mm] & \(1.42\times 10^{-1}\) & \(9.99\times 10^{-2}\) & \(8.80\times 10^{-2}\) & \(1.78\times 10^{-2}\) & \(7.25\times 10^{-2}\) & \(1.65\times 10^{-1}\) & \(6.27\times 10^{-1}\) & \(1.71\times 10^{-1}\) & \(8.44\times 10^{-2}\) & \(1.06\) \\ Orientation [-] & \(3.51\times 10^{-2}\) & \(1.73\times 10^{-2}\) & \(1.73\times 10^{-2}\) & 0.00 & \(1.83\times 10^{-2}\) & \(4.04\times 10^{-2}\) & \(1.10\times 10^{-1}\) & \(8.16\times 10^{-3}\) & \(2.31\times 10^{-2}\) & \(2.54\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Root mean square error (RMSE) associated with test no. 2. Figure 16: (a) Position plan for test no. 3; (b) Sequence of movement. Table 5 represents the deviation of joint angle, position, and orientation from the desired position during the test. With the data recorded during the test, we calculated the following [53]:  
\[\text{P}\text{P}=\sqrt{\left(\overline{\text{x}}-\text{x}_{\text{c}}\right)^{2}+ \left(\overline{\text{y}}-\text{y}_{\text{c}}\right)^{2}+\left(\overline{\text {z}}-\text{z}_{\text{c}}\right)^{2}}, \tag{35}\]  
where \(\overline{\text{x}}\), \(\overline{\text{y}}\) e \(\overline{\text{z}}\) are the averages of the positions obtained and x\({}_{\text{c}}\), y\({}_{\text{c}}\) e \(\text{z}_{\text{c}}\) are the positions commanded. The positioning repeatability (RPt) is calculated as:  
\[\text{RP}_{\text{t}}=\bar{\text{I}}+3\cdot\text{S}_{\text{t}}, \tag{36}\]  
\[\bar{\text{I}}=\frac{1}{\text{n}}\cdot\sum_{\text{j=1}}^{\text{n}}\text{l}_{ \text{j}}, \tag{37}\]  
\[\text{l}_{\text{j}}=\sqrt{\left(\overline{\text{x}}-\text{x}_{\text{j}}\right) ^{2}+\left(\overline{\text{y}}-\text{y}_{\text{j}}\right)^{2}+\left(\overline {\text{z}}-\text{z}_{\text{j}}\right)^{2}}, \tag{38}\]  
\[\text{S}_{\text{i}}=\sqrt{\frac{\sum_{\text{j=1}}^{\text{n}}\left(\text{l}_{ \text{j}}-\bar{\text{I}}\right)}{\text{n}-1}} \tag{39}\]  
where x\({}_{\text{j}}\), y\({}_{\text{j}}\) e \(\text{z}_{\text{j}}\) are the angles of the jth attained pose. Similarly, the accuracy and repeatability of orientation are given by:  
\[\text{AP}_{\text{O}}=\sqrt{\left(\overline{\text{O}}-\text{O}_{\text{c}_{ \text{i}}}\right)^{2}}, \tag{40}\]  
\[\text{RP}_{\text{O}}\ =\pm 3\cdot\text{S}_{\text{O}}=\pm 3\cdot\sqrt{\frac{\sum_{ \text{j=1}}^{\text{n}}\left(\text{O}_{\text{j}}-\overline{\text{O}}\right)}{ \text{n}-1}}, \tag{41}\]  
The results of the calculated precision and repeatability are compiled in Table 6.  
The results of test no. 3 for accuracy and repeatability show a maximum deviation of 0.021\({}^{\circ}\) in the angles of each joint. A maximum variation of up to 0.07\({}^{\circ}\) is shown in the angles of each joint during the cycles; a maximum deviation of 0.085 mm is shown from the average of the end positions of the effector; and a variation of 0.179 mm is shown during the cycles of the movement. These results are most affected in cycles where the speed percentage is 35%.