# Task-Based Motion Planning Using Optimal Redundancy for a Minimally Actuated Robotic Arm
## 6 Conclusions  
In this paper, we presented a novel approach to motion planning for a minimally actuated serial robot (MASR) that minimizes not only the trajectory cost, but also the number of joints that need to be actuated. Using the Monte Carlo method, we integrate the beta distribution with the non-linear kinematics of the manipulator to achieve new numerical workspace analysis method, which makes it possible to find the minimal subset of joints that need to be activated to reach a given target, before finding the path itself. The preliminary search for the minimal workspace is also proven to be beneficial in creating a heuristic search graph, establishing start and goal nodes, and forming a consistent cost function. This method emerged as highly effective when planning a path for hyper redundant robots such as the MASR. The ability to quickly and accurately describe a hyper redundant manipulator's workspace boundary (up to 44.3% faster than previous methods) makes it possible to run a preliminary search to find the minimal subset of joints to be actuated (defined as \"minimal redundancy\") to reach a given target. The use of minimal redundancy guarantees that the dimensionality of the problem is kept at a minimum. Together with taking our numerical workspace as a search graph, path planning with an optimal heuristic search such as A\({}^{\star}\) becomes practical and efficient (reduces expansions number by a factor of 10). We showed that, in cases when the arm's end-effector is not fixed to the last link, and can move along the arm, the workspace volume grows substantially for the same set of joints, which translates into even more efficient planning (by reducing expansions up to a factor of 12.2). We believe that our minimal redundancy approach will enable heuristic searches for robotic manipulation to become more popular, and less impractical.  
Our future work will focus on expanding the workspace analysis to 3D space, while taking additional types of prismatic or spherical joints into consideration. We also plan to incorporate machine learning tools into our motion planning algorithm to improve the effectiveness of the search.  
Conceptualization, Y.G. and D.Z.; methodology, Y.G. and D.Z.; software, Y.G.; validation, Y.G.; investigation, Y.G.; writing--original draft preparation, Y.G.; writing--review and editing, Y.G. and D.Z.; supervision, D.Z. All authors have read and agreed to the published version of the manuscript.  
This study was supported by the Ministry of Science and Technology (MOST), grant number is 315654, the Helmsley Charitable Trust through the Agricultural, Biological and Cognitive Robotics Initiative, and by the Marcus Endowment Fund at Ben-Gurion University of the Negev.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Target No.** & **Last Link Expansions** & **Any Link Expansions** & **LL/AL Ratio** \\ \hline
4 & 152,332 & 27,738 & 5.5 \\ \hline
7 & 10,448 & 11,978 & 0.9 \\ \hline
9 & 34,548 & 11,044 & 3.1 \\ \hline
10 & 74,128 & 10,530 & 7.0 \\ \hline
12 & 4495 & 447 & 10.1 \\
13 & 1211 & 99 & 12.2 \\ \hline
16 & 4 & 9 & 0.4 \\ \hline
17 & 1851 & 276 & 6.7 \\ \hline
18 & 13,021 & 12,712 & 1.0 \\ \hline
20 & 32,299 & 9693 & 3.3 \\ \hline \end{tabular}
\end{table}
Table 3: Expansion within the A\({}^{\star}\) search for last link and any link. The target list corresponds to the numbering in Figure 8.