# Smooth Interpolation Design with Consideration of Corner Tolerance Constraints for Robotics
## 5 Results of Simulations and Experiments
### Simulation Results for the CTC Method  
To confirm that the VPO algorithm not only satisfies the kinematic constraints but can also be used to specify the tolerance at the corner of the corner, another experiment, Case 3, was performed. The NC code for Case 3 is shown in Table 3. The velocity and acceleration  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameters** & **X-Axis (mm)** & **Y-Axis (mm)** & **Z-Axis (mm)** & **Orientation (deg)** \\ \hline Velocity (mm/s, deg/s) & 2000 & 2000 & 2000 & 500 \\ \hline Acceleration (mm/s\({}^{2}\), deg/s\({}^{2}\)) & 3500 & 3500 & 3500 & 2000 \\ \hline Jerk (mm/s\({}^{3}\), deg/s\({}^{3}\)) & 50,000 & 50,000 & 50,000 & 30,000 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Kinematic constraints of robot manipulator.  
Figure 11: Corner smoothing results with various _OVLP_ parameters for the trajectory. (**a**) Entire trajectory. (**b**) Corner A. (**c**) Corner B. (**d**) Corner C. (**e**) Corner D. (**f**) Corner E.  
Figure 12: Velocity and acceleration profiles of the TCP for a simple polygon.  
of the TCP for Case 3 are shown in Figure 13a. The feed rate reached 150 mm/s with a maximum acceleration of \(-1430\) mm/s\({}^{2}\) and a cycle time of 5.45 s. Figure 13b presents the profiles of axial velocity, axial acceleration, and axial jerk. The maximum axial acceleration and jerk occurred on the \(y\)-axis and were \(-1684\) mm/s\({}^{2}\) and 20,000 mm/s\({}^{3}\), respectively. For orientation planning, the rotational angles were calculated from Equations (1), (6), (14) and (15), obtaining \(17.8^{\circ}\), \(33.2^{\circ}\), \(49.7^{\circ}\), \(25.1^{\circ}\), \(22.2^{\circ}\), and \(38.6^{\circ}\). The results for the \(x\), \(y\), and \(z\) rotations are presented in Figure 14. The maximum velocity was 48.48 deg/s, the maximum acceleration was 509.7 deg/s\({}^{2}\), and the maximum jerk was 5309 deg/s\({}^{3}\). Hence, the position and orientation of the end-effectors of the robot manipulator satisfy the kinematic constraints. Finally, the rotation matrix and position were substituted back into Equation (19), and the joint commands were obtained through inverse kinematics. Figure 15 presents the joint commands, angular velocity, angular acceleration, and angular jerk. The maximum values of velocity, acceleration, and jerk are \(-69.7\) deg/s, \(-932\) deg/s\({}^{2}\), and \(-9701\) deg/s\({}^{3}\), respectively.  
To validate the effectiveness of the CTC equation in the VPO algorithm, the results for Case 3 (Table 3) were analyzed. The corner tolerance is the shortest distance from the sharp corner to the ultimate trajectory. By inputting the overlapping time calculated with the CTC equation to the VPO algorithm, the corner tolerance at each corner can be correctly constrained within the specified tolerance. When the corner tolerances of A, B, C, D, and E on the trajectory in Figure 16a were set to 4.2, 3.4, 2.6, 0, and 4.0 mm, respectively, the resulting trajectories at each corner in Figure 16b-f satisfied these constraints with overlapping times of 0.237, 0.226, 0.181, 0, and 0.22 s, respectively. Hence, the simulatio  
Figure 14: Simulation results for the orientation profile. (**a**) Position. (**b**) Velocity, acceleration, and jerk.  
Figure 13: Kinematic profile of the TCP and axial direction. (**a**) Velocity and acceleration. (**b**) Velocity, acceleration, and jerk.  
the proposed VPO algorithm, unlike the conventional ADAI approach, enables separately setting corner tolerances at each corner to satisfy the specified constraints.