## 3 Standardization ISO 9283--Path Accuracy
### Mathematical Background of CRX-10 iA  
An important part of the robot analysis is a complete kinematic model of the mechanical system. It is a description of the position and orientation of the working end point in time and the corresponding course of position-taking by individual parts of the mechanism. The Fanuc CRX-10 iA robot's structure is a serial kinematic RRR. That is why the kinematics was described using the Denavit-Hartenberg convention, enabling the description of the robot's kinematics using DH parameters. The kinematic model, Figure 8, makes it possible to study the processes of the robot's movement in time-space [30]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **CRX-10 iA** \\ \hline Type & 6 axes (J1, J2, J3, J4, J5, J6) \\ Reach & 1249 mm \\ Axis & Motion range (maximum speed) \\ Axis J1 & 380\({}^{\circ}\) (120\({}^{\circ}\)/s) \\ Axis J2 & 360\({}^{\circ}\) (120\({}^{\circ}\)/s) \\ Axis J3 & 570\({}^{\circ}\) (180\({}^{\circ}\)/s) \\ Axis J4 & 380\({}^{\circ}\) (180\({}^{\circ}\)/s) \\ Axis J5 & 360\({}^{\circ}\) (180\({}^{\circ}\)/s) \\ Axis J6 & 450\({}^{\circ}\) (180\({}^{\circ}\)/s) \\ Max. speed in collaborative mode & 1000 mm/s \\ Repeatability & \(\pm\)0.04 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of robot Fanuc CRX-10 iA. Figure 7: Measuring with GEPARD (receiver–transmitter). Inverse transformation yielded the courses of position-taking by tilting drives that enabled measurement. The last part of the robot will move at a constant speed with linear approximation. The values of speed and acceleration of individual drives are obtained by its derivative. \[\dot{q}(t),\ddot{q}(t) \tag{8}\]  
The speed of each material point of the part, as well as that of its local coordinate system, can be calculated from the following equation:  
\[\omega_{i}(t)=\omega_{i-1}(t)+\omega_{i-1}^{i} \tag{9}\]  
where \(\omega i\) is the angular speed of the \(i\)th coordinate system, \(\omega i-1\) is the angular speed of the previous coordinate system, and \(\omega_{i-1}^{i}\) is the relative angular speed of two adjacent coordinate systems, which can be expressed as follows:  
\[\omega_{i-1}^{i} \begin{cases}k_{i-1}\dot{q}\ R\\ 0\ T\end{cases}  \tag{10}\]  
The angular speed of the naught coordinate system is zero. In this way, velocities of all six local coordinate systems can be expressed, and thus, those of the robot parts from its base to its endpoint. The derivation of translational speed necessitates the derivation of the positional vector between two coordinate systems. \[\frac{d\text{P}_{\text{i}}}{dt}=\frac{d\text{P}_{\text{i}-1}}{dt}+\frac{d\text {P}_{\text{i}-1}^{i}}{dt} \tag{11}\]  
A robotic system enables setting the trajectory by setting individual points and then setting the type of movement. In our case, a linear movement at a constant speed is selected between the extreme points of the measuring cube. Figure 8: Kinematic model of CRX-10 iA. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link** & \(\alpha_{\text{i}}\) & \(\alpha_{\text{i}}\) & \(\phi_{\text{i}}\) & \(\phi_{\text{i}}\) & \(\phi_{\text{i}}\) \\ \hline
1 & \(-\pi/2\) & 0.218 & q\({}_{\text{i}}\) & 0.159 \\
2 & 0 & 0.540 & q\({}_{\text{2}}\) & 0.000 \\
3 & \(-\pi/2\) & 0.000 & q\({}_{\text{3}}\) & 0.218 \\
4 & \(\pi/2\) & 0.000 & q\({}_{\text{4}}\) & 0.540 \\
5 & \(-\pi/2\) & 0.000 & q\({}_{\text{5}}\) & 0.150 \\
6 & 0 & 0.000 & q\({}_{\text{6}}\) & 0.160 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: D-H parameters Fanuc CRX-10 iA. Dynamic parameters were simulated in Creo. The courses of position-taking, speed, and acceleration describing the movement of individual drives during linearity measurement are shown in Figures 9-11. For the given collaborative robot model, a continuous linear path containing measuring points was simulated. The results of the position analysis of the course movement of individual joints are shown in Figure 9. The individual trend lines justify a belief that all robot joints participated in the movement defined.