# Industrial Robot Positioning Performance Measured on Inclined and Parallel Planes by Double Ballbar
## 4 Positional Analysis in the Creo Parametric System 5.0.4.0  
To verify the influence of the selected orientation of the circle on the motion of the robot's individual joints, a positional analysis was created in the Creo Parametric system 5.0.4.0. For the purpose thereof, a simulation model of the LR Mate 200 iC robot was created with the kinematic constraints applied. Zero positions of the individual robot joints and their range of motion were defined, corresponding to those of the real robot. The position of the TCP and of the individual coordinate systems was also defined, as was the case with the experiments on a real robot. The simulation conditions are given in Table 3.  
The center of the required circular path was located at the zero of the local coordinate system, while its position relative to the WCS robot is shown in Table 3. The circular path itself was created through an inverse kinematics task, by defining the so-called servomotors in the individual axes of motion of the Cartesian coordinate system. The cosine function was used for the direction of the X and Y axes, with its general shape in the Creo Parametric system being the following (1):  
\[\mathrm{q}=\mathrm{A}\cdot\cos\left(360\frac{\mathrm{x}}{\mathrm{T}}+\mathrm{ B}\right)+\mathrm{C} \tag{1}\]  
where the coefficients A = amplitude [mm], B = phase, C = offset [mm] and T [s] represent a period [36]. For the X, Y-axis direction, the required radius of the circle of 100 mm, and the clockwise movement, the given function has the following form (2), (3):  
\[\mathrm{q}=100\cdot\cos\left(360\frac{\mathrm{x}}{360}\right) \tag{2}\]  
\[\mathrm{q}=100\cdot\cos\left(360\frac{\mathrm{x}}{360}+90\right) \tag{3}\]  
Other axes are defined as constants. Movement in direction of the Z-axis = 0 mm, rotations around the Y and Z axes are also zero (P = \(0^{\circ}\); R = \(0^{\circ}\)). The rotation around the X-axis has a defined value W = \(-180^{\circ}\), which ensures that the Z-axis of the tool coordinate system has the correct orientation and is perpendicular to the measuring plane or the plane in which the required circle is located.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Type of Industrial Robot** & **Fanuc LR Mate 200 iC** \\ \hline The radius of circular paths & 100 mm \\ Centre of circle relative to WCS (X; Y; Z; W; P; R) & 200; 400; \(-200\); \(0\); \(-45\); 0 \\ Coordinates of the TCP (X; Y; Z) & 0; 0; 74.65 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation conditions for positional analysis.  
The results of the positional analysis of the course of motion of individual robot joints are shown in the form of a graph in Figure 8. From the individual trend lines, one can see that all joints of the robot, including the J4 joint, participated in the defined motion. The above graph applies to the clockwise movement of the TCP and to the \"NOFLIP\" robot arm configuration. For the second configuration (\"FLIP\"), the graph is similar, with all the joints of the robot participating in the movement. The absolute value of the range of motion of each of the robot's joints is shown in the graph in Figure 9. Because the TCP moved along the same path in both configurations, the displayed range value for both configurations is identical.  
The present positional analysis confirmed the hypothesis of the robot's individual joints' motion in connection with the location of the required circle on an inclined plane. The next step was to verify the possibilities of practical implementation of such measurement with the Renishaw Ballbar.