# Significance of Camera Pixel Error in the Calibration Process of a Robotic Vision System
## 3 Results and Discussion
### Error Sensitivity Analysis of Theoretical Setup
The concept of error introduction in one of the detected corner point is illustrated in Figure 3, which can cause due to usage of low resolution camera. The norms of errors associated with estimation of matrix \(X\) using \(\hat{A}\) and \(\hat{B}\) are given by:  
\[U_{R}=\frac{\sum_{i=1}^{n}\sum_{k=1}^{o}||R_{X^{\prime}}-\hat{R}_{X _{i,k}}||_{F}}{n\times o} \tag{27}\] \[U_{T}=\frac{\sum_{i=1}^{n}\sum_{k=1}^{o}||T_{X^{\prime}}-\hat{T}_ {X_{i,k}}||_{E}}{n\times o} \tag{28}\]  
where \(n\) (=56) represents different combinations of flange positions, \(o\) (=1000) is simulation runs for normally distributed noise, \(||\cdot||_{F}\) is Frobenius norm and \(||\cdot||_{E}\) is Euclidean norm (in mm) obtained by solving system equation \(\hat{A}X=X\hat{B}\). \(R_{X^{\prime}}\) and \(T_{X^{\prime}}\) are the rotation matrix and translation vector of theoretical \(X^{\prime}\), while \(\hat{R}_{X}\) and \(\hat{T}_{X}\) are the rotation matrix and translation vector of transformation matrix \(X\) obtained after introducing uncertainty in the system. Ideally the value of \(U_{R}\) and \(U_{T}\) are zero, when both errors \(\delta_{A}\) and \(\delta_{B}\) are zero. Figure 4 illustrates the impact of measurement uncertainties on estimating matrix \(X\) from the system equation using three different hand-eye calibration algorithms, where change in the colour from navy-blue to red represents increasing errors in estimated transformation matrix \(X\) accurately. Figure 4a-f show the sensitivity of three hand-eye calibration methods to uncertainty in both matrices \(A\) and \(B\). The uncertainties in matrix \(A\) is introduced by adding normally distributed noise to all the six axes present in the robot kinematics model, while uncertainty in matrix \(B\) is introduced by adding noise to all the perfectly detected calibration points in the image. Based on three hand-eye calibration algorithms, the observations suggest that inaccuracies in matrices \(A\) and \(B\) have adverse impact on correct estimation of \(X\), especially for the case of smaller \(\eta\) as shown in Figure 4. It is seen that when \(\delta_{A}\
eq 0\) and \(\delta_{B}=0\), Horaud-Dornaika and Tsai-Lenz methods are able to estimate \(R_{X}\) accurately, while Andreff et al. method is not able to find the true value of \(R_{X}\) and the magnitude of error function \(U_{R}\) increases as \(\delta_{A}\) increases. This observation suggests that both methods belonging to the first computation category of hand-eye calibration algorithm are expected to estimate more accurate value of \(R_{X}\). On the other hand, when \(\delta_{A}=0\) and \(\delta_{B}\
eq 0\), then none of the methods are able to estimate \(T_{X}\) accurately. However, out of the three hand-eye calibration methods, the performance of the Horaud-Dornaika and Tsai-Lenz methods are better than the Andreff et al. method. One of the reasons behind such difference in the performance of two computation categories of the methods is an additive effect of propagated errors from rotation matrix to translation vector in the Andreff et al. method. Further, the Andreff et al. method evaluates matrix \(X\) by linearising the system equation. Although linearising the system equation can reduce the complexity of the system and can estimate matrix \(X\) in lesser time, such process is expected to reduce the useful constraints of the equation which can result in a wrong estimation of the solution. Figure 3: Introduction of error \(\delta_{B}\) in one of the perfectly detected image point \((\alpha_{i},\beta_{i})\). Adding a noise of \(|\gamma|\) pixel on image point \((\alpha_{i},\beta_{i})\) will result into region of \(\delta_{B}=\gamma\). The error in \(B\) can be estimated by evaluating area (in \(\upmu\upmu^{2}\)) of the uncertain region using pixel size value from Table 1.  
A close observation of Figure 4a-f reveals that introducing a minimal error of \(\delta_{B}=1\) has significantly impacted the accurate estimation of \(R_{X}\) and \(T_{X}\).