# 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 same setup as illustrated in Figure 1 is used to calibrate the theoretical robotic vision system. Let the camera mounted on the flange of the robotic arm be navigated from position \(i\) to \(j\) and at each position an image of the calibration object is captured. Then a classical camera calibration method is used to estimate matrix \(Q\) for each position. The navigation of the camera mounted robot flange from position \(i\) to \(j\) (refer Figure 1) can be expressed by the following system equation. \[P_{i}XQ_{i}=P_{j}XQ_{j}. \tag{22}\]  
In Equation (22), the values of \(Q_{i}\) and \(Q_{j}\) are known with the help of the camera calibration approach suggested by [PERSON] [7]. Assuming the theoretical value of matrices \(X\) (denoted as \(X^{\prime}\)), matrix \(P_{j}\) can be estimated from \(P_{i}\) by using the system equation defined in Equation (23). \[P_{j}=(P_{i}X^{\prime}Q_{i})(X^{\prime}Q_{j})^{-1}. \tag{23}\]  
After obtaining all the required matrices, the matrices \(A\) and \(B\) of the theoretical robotic vision system can be evaluated from Equations (2) and (4). The main objective for theoretical estimation of \(X\) is to analyse the performance of hand-eye calibration algorithms under systematic errors. The advantage of opting theoretical setup is that the matrix \(X=X^{\prime}\) is accurately known and can be used as the reference matrix while performing uncertainty analyses by adding systematic errors on matrices \(A\) and \(B\). One of the most practical ways to introduce errors in matrix \(A\) is to systematically perturb the angles of robotic manipulator. After each additive uncertainty, the erroneous transformation matrix \(\hat{P}\) is evaluated with the help of the kinematics model. Similar approach is to be applied for matrix \(B\), that is, the systematic error is introduced on perfectly detected calibration pattern in the images. After introducing a systematic error each time, the camera calibration algorithm is used to estimate erroneous matrix \(\hat{Q}\). Now the expression defined in Equation (22) is modified to:  
\[\hat{P}_{i}X\hat{Q}_{i}=\hat{P}_{j}X\hat{Q}_{j} \tag{24}\]  
Solving this system equation gives  
\[\hat{P}_{j}^{-1}\hat{P}_{i}X =X\hat{Q}_{j}\hat{Q}_{i}^{-1} \tag{25}\] \[\hat{A}X =X\hat{B}, \tag{26}\]  
where erroneous matrix \(\hat{A}\) defines a cumulative form of kinematic and non-kinematic errors discussed in Section 2.1, while erroneous matrix \(\hat{B}\) defines the cumulative accuracy error discussed in Section 2.2. Transformation matrix \(X\) is unknown and need to be estimated with the help of hand-eye calibration algorithms for different robot positions (\(\eta\)). Note that to estimate \(X\), the value of \(\eta\) should not be less than three, hence we have opted \(\eta=4\) and \(\eta=5\) in this study. After introducing systematic errors in transformation matrices \(P\) and \(Q\), the system equation is solved using three popular hand-eye calibration methods. The first two methods belong to the first computation category based on the individual estimation of rotation matrix and translation vector [11; 12], and the other belonging to the second computation category based on combined estimation [14]. Note that uncertainties in the angles of the robot manipulator are obtained by introducing \(\delta_{A}\) error in all the six joint angles (\(\theta\)'s) of the robot in the form of normally distributed random numbers (\(N(\mu_{A},\sigma_{A})\)) simulating noise, where \(\mu_{A}\) (in degrees) ranges between \([-1,1]\) with step-size of \(10^{-1}\) and \(\sigma_{A}=0.05\mu_{A}\). Similarly, the uncertainty in the calibration pattern is obtained by introducing \(\delta_{B}\) error in all the detected image corner points (\(\alpha_{i},\beta_{i}\)) in the form of normally distributed random numbers lying within the window size of \([0,5]\) having unity step-size.