# Smooth Interpolation Design with Consideration of Corner Tolerance Constraints for Robotics
## 6 Conclusions  
This study proposed a VPO algorithm for designing robotic interpolators that uses the velocity overlap method on each axis to remove the discontinuities caused by projecting the velocity profile of the TCP onto each axis. A superior balance between smoothness and cycle time can be achieved by adjusting the overlapping time. This paper described the derivation of the CTC equation, which enables using the VPO method to individually control the tolerance for each corner. Other interpolation design methods, such as the conventional ADAI approach, do not incorporate these features. The experimental results indicate that the VPO algorithm outperformed the ADAI method by reducing the cycle time by 6.08-15.22% if the parameters of the methods were adjusted to obtain approximately equal contour error. Experiments were also conducted to confirm that the contour errors could be reduced by 9.19-12.30% if the cycle times of both algorithms were adjusted to be approximately equal. The results confirm the effectiveness of the VPO method.  
Conceptualization, H.-M.L., M.-S.T. and T.-H.Z.; methodology, H.-M.L. and T.-H.Z.; software, H.-M.L.; validation, H.-M.L. and M.-S.T.; formal analysis, H.-M.L. and T.-H.Z.; investigation, H.-M.L. and T.-H.Z.; resources, M.-S.T.; data curation, H.-M.L.; writing--original draft preparation, H.-M.L.; writing--review and editing, M.-S.T. and C.-C.C.; visualization, H.-M.L. and M.-S.T.; supervision, M.-S.T. and C.-C.C.; project administration, H.-M.L. and M.-S.T.; funding acquisition, M.-S.T. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Science and Technology Council (NSTC), R.O.C., under the contracts 111-2221-E-002-159-MY3 and 111-2218-E-002-032-.  
Not applicable.  
The data presented in this study are available upon request from the corresponding author.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\begin{tabular}{c} **Feed Rate** \\ **(mm/s)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **VPO** \\ **Acceleration Time (s)/** \\ **OVLP (\%)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **ADAI** \\ **Acceleration (mm/s)/** \\ **Time Constant (ms)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Point A Contour Error** \\ **ADAI (mm)/** \\ **VPO (mm)/** \\ **Reduced (\%)** \\ \end{tabular} } \\ \hline Case D & 100 & 0.35/58 & 1200/500 & 0.1841/0.1566/15.22 \\ Case E & 100 & 0.44/81 & 1600/700 & 0.1826/0.1693/7.28 \\ Case F & 150 & 0.33/84 & 1600/600 & 0.2005/0.1883/6.08 \\ \hline \end{tabular}
\end{table}
Table 6: Contour error of the VPO and ADAI trajectories with approximately equal cycle times.  
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\begin{tabular}{c} **Feed Rate** \\ **(mm/s)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **VPO** \\ **Acceleration Time (s)/** \\ **OVLP (\%)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Maximum** \\ **Acceleration (mm/s)/** \\ **Time Constant (ms)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **ADAI (s)/** \\ **VPO (s)/** \\ **Reduced (\%)** \\ \end{tabular} } \\ \hline Case A & 100 & 0.25/74 & 1200/500 & 10.05/8.98/10.60 \\ Case B & 100 & 0.35/81 & 1600/700 & 10.11/9.18/9.19 \\ Case C & 150 & 0.28/85 & 1600/600 & 7.32/6.42/12.30 \\ \hline \end{tabular}
\end{table}
Table 5: Cycle times of the VPO and ADAI trajectories with approximately equal maximum contour and tracking errors.