# Camera Calibration in High-Speed Robotic Assembly Operations
## 4 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & **P9** & & & **P8** & & & **P7** & \\
**Data Set** & & **P6** & & & **P5** & & & **P4** & \\  & & **P3** & & & **P2** & & & **P1** & \\ \hline \multirow{3}{*}{H2 40 px} & \(-123\) & \(-124\) & 561 & 0 & \(-125\) & 564 & 122 & \(-123\) & 557 \\  & \(-129\) & \(-1\) & 585 & 0 & \(-1\) & 587 & 128 & \(-1\) & 580 \\  & \(-131\) & 131 & 600 & 0 & 133 & 607 & 132 & 131 & 600 \\ \hline \multirow{3}{*}{H2 50 px} & \(-151\) & \(-152\) & 554 & 0 & \(-153\) & 555 & 150 & \(-151\) & 549 \\  & \(-160\) & \(-1\) & 584 & 0 & \(-1\) & 586 & 159 & \(-1\) & 577 \\  & \(-164\) & 164 & 602 & 0 & 166 & 607 & 165 & 164 & 601 \\ \hline \multirow{3}{*}{H3 20 px} & \(-66\) & \(-67\) & 597 & 0 & \(-62\) & 596 & 66 & \(-66\) & 593 \\  & \(-67\) & \(-1\) & 609 & 0 & \(-1\) & 610 & 67 & \(-1\) & 606 \\  & \(-68\) & 68 & 621 & 0 & 61 & 620 & 68 & 67 & 616 \\ \hline \multirow{3}{*}{H3 30 px} & \(-98\) & \(-99\) & 593 & 0 & \(-98\) & 590 & 97 & \(-98\) & 589 \\  & \(-100\) & \(-1\) & 607 & 0 & \(-1\) & 609 & 100 & \(-1\) & 602 \\  & \(-102\) & 102 & 621 & 0 & 102 & 623 & 102 & 101 & 619 \\ \hline \multirow{3}{*}{H3 40 px} & \(-128\) & \(-129\) & 584 & 0 & \(-130\) & 586 & 127 & \(-128\) & 579 \\  & \(-134\) & \(-1\) & 607 & 0 & \(-1\) & 609 & 132 & \(-1\) & 601 \\  & \(-137\) & 136 & 623 & 0 & 138 & 630 & 137 & 136 & 622 \\ \hline \multirow{3}{*}{H3 50 px} & \(-158\) & \(-159\) & 577 & 0 & \(-160\) & 579 & 157 & \(-157\) & 572 \\  & \(-167\) & \(-1\) & 607 & 0 & \(-1\) & 609 & 165 & \(-1\) & 600 \\  & \(-170\) & 170 & 624 & 0 & 172 & 628 & 171 & 170 & 623 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Height** & **Px** & **Angle [\({}^{\circ}\)]** & **Error** \\ \hline  & 20 & 81.75414348 & 2.19\% \\
1 & 30 & 76.05467223 & 4.93\% \\
1 & 40 & 75.04858306 & 6.19\% \\  & 50 & 72.38921295 & 9.51\% \\  & 20 & 74.61077611 & 6.74\% \\
2 & 30 & 73.879802451 & 7.65\% \\  & 40 & 73.32914035 & 8.34\% \\  & 50 & 70.86462849 & 11.42\% \\  & 20 & 74.0954911 & 7.38\% \\  & 30 & 74.25758952 & 7.18\% \\
3 & 40 & 73.72102328 & 7.85\% \\  & 50 & 71.68477916 & 10.39\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: First approach results—law of cosines. The error in column 4 was calculated concerning the imposed angle\({}_{\text{real}}\) of 80\({}^{\circ}\) as in the previous case using Formula (9). \[Error=\left|\frac{Angle_{experimental}-Angle_{real}}{Angle_{real}}\right|\cdot 1000\ [ \%] \tag{9}\]  
Regarding the measurement results and the methods implemented for the calculation of the camera rotation angles, the following aspects should be noted:  
* The method using the law of cosines is easier to implement and requires far more computational resources because it uses less data. In the case of the 20 pixels measurement perimeter, for example, the law of cosines approach uses four points, while the planar interpolation approach uses nine measurement points. This observation applies to all measurement perimeters and all measurement heights. Also, the calculation formula for the lay of cosines approach is simpler. * However, since it uses fewer measurement points, the law of cosines approach is less accurate. The error levels calculated for the first approach, shown in Table 3, are ranging from 2.19% to 11.42%, with an average of 7.48%. * It is worth noting that the law of cosines itself has some limitations, generating high level of errors for low camera rotation angles. For the most part, the higher level of errors is caused by rounding off the numbers used in calculations and by the uncertainties in the working environment. * For this study, it was taken into account the fact that the robotic vision feature is used for acquiring data from working environments that are not deterministic. This means that these type of cameras will be used in a setup that has a high number of undetermined variables and deviations.