## 5 Conclusions  
In this research, the post triangular processing of the midpoint interpolation method minimized the planning time and shortened the path length of the sampling-based algorithm.  
The proposed post triangular processing of the midpoint interpolation method was closer to the optimal path and somewhat solved the sharp path problem through the interpolation process. Furthermore, all path planning algorithms that plan a locally piecewise linear path could apply the proposed algorithm. This strength has significance in that it can be applied not only to the algorithm presented in this paper but also to various path planning algorithms. We validated the performance of the proposed method after it was applied to the RRT algorithm and its path planning results through simulation. It was verified that the path length was shortened by 18-38% (average 26%) depending on the threshold \(\varepsilon\) when applied to the RRT algorithm in the four different environment maps. As a result, the RRT algorithm applying the proposed post triangular processing of the midpoint interpolation method showed a more optimal path.  
The proposed post triangular processing of the midpoint interpolation method is based on a general global planning case in which a robot first discovers a global path from its start point to its destination point before beginning to navigate. In dynamic environments, not only local planners but also global planners must deal with kinodynamic problems in real-time. Furthermore, the method proposed in this paper is more efficient in terms of the optimality of robot path planning, but optimality is not guaranteed. Future work should determine how to make a path that is closer to the optimal path.  
Idea and conceptualization: [PERSON] and [PERSON]; Methodology: [PERSON] and [PERSON]; Software: [PERSON] and [PERSON]; Experiment: [PERSON] and [PERSON]; Validation: [PERSON], [PERSON] and [PERSON]; Investigation: [PERSON] and [PERSON]; Resources: [PERSON] and [PERSON]; Writing: [PERSON], [PERSON] and [PERSON]; Visualization: [PERSON] and [PERSON]; Project administration: [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1074974) and by the Ministry of Trade, Industry, and Energy (MOTIE) and the Korea Institute for Advancement of Technology (KIAT) through the International Cooperative R&D program (Project No. P0016096). It was also supported by the Technology development Program(S3041234) funded by the Ministry of SMEs and Startups (MSS, Korea) and by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) support program (IITP-2021-2020-0-01789) supervised by the IITP (Institute for Information & Communications Technology Planning & Evaluation).  
Not applicable.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Performance**} & \multirow{2}{*}{**RRT**} & **Proposed Method** & **Proposed Method** & **Proposed Method** \\  & & **(\(\varepsilon\): 50 px)** & **(\(\varepsilon\): 30 px)** & **(\(\varepsilon\): 10 px)** \\ \hline Path length (px) & 1536 (100\%) & 1284 (83\%) & 1261 (82\%) & 1190 (77\%) \\ Planning time (ms) & 1752 (100\%) & 1753 (100\%) & 1753 (100\%) & 1753 (100\%) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Experimental results of Map 4 (numbers in parentheses to (averages of repeating 100 times) are relative ratios to RRT results (values less than 1 are counted as 1)).