# Significance of Camera Pixel Error in the Calibration Process of a Robotic Vision System
## 2 Experimental Setup and Measurement Uncertainties
### Robot Base to Flange Calibration  
Robot calibration has been studied for more than three decades which brings compensation to the inaccuracy of robots. Although the technique of robot making has improved moderately during these years, the theory of robot calibration remains the same. It requires a mathematical model to be tailored to the individual robot which is to be calibrated. \begin{table}
\begin{tabular}{c c} \hline \hline
**Specification** & **Quantitative Value** \\ \hline Resolution & \(1392\times 1040\) \\ Pixel size & \(4.65\)\(\,\mathrm{\SIUnitSymbolMicro m}\) \\ Frame rate & \(20.8\)\(\,\mathrm{fps}\) \\ Focal length & \(12\)\(\,\mathrm{mm}\) \\ Sensor size & \(1/2^{\prime\prime}\) \\ Minimum working distance & \(150\)\(\,\mathrm{mm}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of 3 CCD Jai camera. This is a software model of the robot and is then applied to robot paths which replaces its nominal physical model. To calibrate and test a robot, it should be performed using high-end measurement equipment such as a laser tracker. Two types of parameters are commonly used for assessing the performance of an industrial robot: repeatability and accuracy. Modern robots can achieve a reasonably good repeatability, for example, a small robot of 5 kg payload typically has the repeatability of 0.02-0.03 mm. To improve the absolute accuracy of a robot the calibration is carried out in four steps:  
1. Determination of a mathematical model of the robot geometry and motion (kinematics and dynamics);
2. Measurement of the position and orientation of the robot end-effector (flange);
3. Identification of a relationship of the measurements and robot joint positions;
4. Modification of software control commands to compensate for the determined model. Among the aforementioned four steps, the determination of the robot model (step 1) and the identification process (step 3) are the most challenging. A complete calibration model should contain corrections of the geometric parameters (also referred as kinematic parameters), and necessary dynamics parameters, e.g., stiffness. A correction for the unknown error could be an offset or a linear approximation. In this section, applying the well known Denavit and Hartenberg (D-H) coordinate system to KUKA robot manipulator [17], the offset values associated with the parameters of KUKA robot are described. The D-H coordinate system is helpful in estimating the kinematics of the robot describing the relative position of the flange with respect to its base. The robot kinematics description is a geometric transformation, where the parameters of kinematics describe the motion of the robot manipulator without the consideration of forces and torques. The generic form of the convention for link \(l\) of the robot manipulator illustrated in Figure 2 can be described as follows:  
\[T_{l-1}^{l}=R_{z,\theta_{l}}t_{z,d_{l}}t_{x,a_{l}}R_{x,a_{l}}=\]  
\[\left[ \begin{array}{cccc}cos(\theta_{l})&-sin(\theta_{l})cos(\alpha_{l})&sin (\theta_{l})sin(\alpha_{l})&a_{l}cos(\theta_{l})\\ sin(\theta_{l})&cos(\theta_{l})cos(\alpha_{l})&-cos(\theta_{l})sin(\alpha_{l}) &a_{l}sin(\theta_{l})\\ 0&sin(\alpha_{l})&cos(\alpha_{l})&d_{l}\\ 0&0&0&1\end{array} \right]. \tag{5}\]  
Note that the link transformation \(T_{l-1}^{l}\) contains two basic rotations and two translations, specifically \(R_{z,\theta_{l}}\) is a rotation around z-axis by \(\theta_{l}\) degrees; \(t_{z,d_{l}}\) is a translation along z-axis by \(d_{l}\) distance; \(t_{x,a_{l}}\) is a translation along x-axis by \(a_{l}\) distance; and \(R_{x,a_{l}}\) is a rotation around x-axis by \(\alpha_{l}\) degrees. The KUKA robot manipulator used in this study has six links (joints), therefore its overall kinematics transformation is described using six transformation matrices. Ideally the relationship between each successive joints \(l-1\) and \(l\) or contiguous link in the form of transformation matrix \(T_{l-1}^{l}\) for KUKA robot is given below. Figure 2: D–H coordinate system of link \(l\) of robot. \[T_{0}^{1}=\left[