## 6 Discussion and Future Work  
In the benchmark environment shown in Figure 4, the lengths of \(l_{1}\), \(l_{link}\) and \(l_{max}\) in pixels are 200, 350, and 440, respectively. It results in an index value of 0.8 for 3D environment. The maneuvering ability resulting in an index value of 0.8 means that it wastes almost no maneuvering space as long as it is physically attainable to maneuver. As seen in Figure 21, if the middle path is followed or the manipulator links are mapped onto the middle path generated by the potential fields, the collision with obstacles are inevitable with the larger link lengths. In the proposed algorithm in this paper, this problem is solved by mapping the manipulator links onto a new path consisting of the end points of the beams in real-time. The method is able to drive manipulators in a wide range: from few number of links to a huge number of links. The method's complexities drive up almost linearly with the number of links. The method is extensible since adding new features to the method has little influence on existing features consisting of a number of \"if-else\" statements. The method does not deal with the manipulator position using a kind of a propagation procedure between the links. On the contrary, the method in this paper handles the control in a simpler manner. It does not need to determine each link's position by means of being tangent and it is capable of dealing with the tight maneuvering by reducing the control strategy to keeping the links' end points on the manipulator path. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Example** & \multicolumn{4}{c}{\(d_{p}=8\) (pixels)} & \multicolumn{4}{c}{\(d_{p}=16\) (pixels)} \\ \hline  & \(n_{\text{B}}\) & \(\begin{array}{c}t_{PFC}\\ \textbf{(ms)}\end{array}\) & \(t_{BC}\) (ms) & \(\begin{array}{c}t_{MPC}\\ \textbf{(ms)}\end{array}\) & \(t_{\text{N}}\) (ms) & \(n_{\text{B}}\) & \(\begin{array}{c}t_{PFC}\\ \textbf{(ms)}\end{array}\) & \(t_{BC}\) (ms) & \(\begin{array}{c}t_{MPC}\\ \textbf{(ms)}\end{array}\) & \(t_{N}\) (ms) \\ \hline First & 13,128 & 13,280 & 32,500 & 281 & 30,251 & 6564 & 13,022 & 15,738 & 117 & 14,623 \\ Second & 14,592 & 14,722 & 32,763 & 123 & 39,264 & 7296 & 13,822 & 15,124 & 56 & 29,578 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of the beams and the calculation times of the proposed algorithm. Figure 21: Collision with obstacles when manipulator is following the middle path. As mentioned before, the obstacles' shapes are rectangular in the first example without tiny or different shaped obstacles scattered in the workspace. The crucial point to present in the paper is to show high maneuvering ability of the manipulator and the arrangement of the workspace with rectangular-shaped obstacles is the best to show this ability. However, in the second example small-sized obstacles are used. One issue is shown in Figure 22a. Although the algorithm finds the middle path, the critical regions and the path for the manipulator perfectly well, there is no way that the manipulator passes through the obstacles. Since the piece of the middle path in that region is too short with respect to the link length. Figure 22b shows the subsequent configurations of the links trying to maneuver in vain, the collisions can be seen clearly. Recalling Figure 22a, there is actually large free space seen on the left-hand side of the middle path. If the middle path is modified to accommodate this space, collision-free navigation can be done easily. The real-time method presented here is a very efficient and robust with respect to maneuvering in a tight workspace cluttered with many obstacles. Since the proposed method has special cases, it is not a complete method which means that being successful all the time it it is physically possible. As a future work, there is a need for further exploration to be able to have a complete algorithm. We think that we are very close to having a complete algorithm.