# Significance of Camera Pixel Error in the Calibration Process of a Robotic Vision System
###### Abstract  
Although robotic vision systems offer a promising technology solution for rapid and reconfigurable in-process 3D inspection of complex and large parts in contemporary manufacturing, measurement accuracy poses a challenge for its wide deployment. One of the key issues in adopting a robotic vision system is to understand the extent of its measurement errors which are directly correlated with the calibration process. In this paper, a possible source of practical and inherent measurement uncertainties involved in the calibration process of a robotic vision system are discussed. The system considered in this work consists of an image sensor mounted on an industrial robot manipulator with six degrees of freedom. Based on a series of experimental tests and computer simulations, the paper gives a comprehensive performance comparison of different calibration approaches and shows the impact of measurement uncertainties on the calibration process. It has been found from the error sensitivity analysis that minor uncertainties in the calibration process can significantly affect the accuracy of the robotic vision system. Further investigations suggest that inducing errors in image calibration patterns can have an adverse effect on the hand-eye calibration process compared to the angular errors in the robot joints. robotic vision system; inspection and measurement; calibration uncertainty; camera calibration; hand-eye calibration +
Footnote †: journal: Applied  
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Footnote 1: email: [EMAIL_ADDRESS]  
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hand, estimating matrix \(B\) with the help of camera calibration is a common practice and well documented in the computer vision community [5; 6; 7; 8; 9]. In the calibration process, the camera mounted on the robot flange captures images of a calibration board (checkerboard) from a number of different positions and orientations. A simple yet effective algorithm suggested by [PERSON] [7] is adopted in this work, which is briefly discussed in Section 2.2. To investigate the major sources of measurement uncertainties and its overall impact on a common robotic vision system, a practical setup constituting of a camera, a six-axis robotic arm and a calibration object is used. The system calibration requires accurate estimation of the relative position and orientation of the camera with respect to the flange, with the help of existing hand-eye calibration algorithms [10; 11; 12; 13; 14; 15]. The practical setup used in this paper to calibrate the robotic vision system is illustrated in Figure 1. The developed robotic vision system follows the classical approach to estimating all the three transformation matrices required in this experiment. That is, the camera mounted on the flange of the robotic arm is navigated from position \(i\) to \(j\) to capture images of the calibration object. Note that \(A\) and \(B\) are transformation matrices that can be easily known by recording the pose of the flange relative to its base for \(\eta\) number of positions and capturing images of the calibration object from the same \(\eta\) positions, respectively. To calculate unknown matrix \(X\) by solving system equation \(AX=XB\), the minimum value of \(\eta\) should not be less than three [10]. In Figure 1, let \(P\) be a transformation matrix that defines the transformation of coordinates from the hand coordinates (flange of the robot) to the robot world coordinates (robot base). The matrix \(P\) of dimension \(4\times 4\) can be defined as:  
\[P= \begin{bmatrix}R_{P}&T_{P}\\ 0&1_{r}\end{bmatrix}  \tag{1}\]  
where \(R_{P}\) is a \(3\times 3\) rotation matrix, and \(T_{P}\) is a \(3\times 1\) translation vector. Let the matrix \(P\) be known for positions \(i\) and \(j\), then matrix \(A\) can be estimated using the following expression:  
\[A=P_{j}^{-1}P_{i}. \tag{2}\]  
Figure 1: Classical framework for evaluating unknown matrix \(X\) for a robotic vision system.