## 7 Conclusions  
In this paper, a simple novel robust real-time algorithm is presented for path planning of redundant/hyper redundant manipulators within narrow terrains cluttered with obstacles in 3D environments. Using potential fields, a global path consisting of points in the middle of the free space from the start point to the goal point through obstacles is obtained. Having the path, two beam couples start from its path point and end at the obstacle border is generated. Critical regions consisting of critical corners that are formed from the path points whose beam couples satisfy certain criteria are obtained. Then, considering maneuvering direction of the manipulator for each critical region, path switching points on the same beam couple are determined as the middle point of the end point of the first critical region and the start point of the second critical region if the maneuvering direction changes between these consecutive critical regions. After determining critical regions and switching points for the selected beam couple, the beam switching points are determined considering the critical regions presence on both beam couples. Finally the path for the manipulator is obtained and the manipulator becomes ready to move. While advancing the end-effector of the manipulator, each link's end points are kept on the path until the end-effector reaches the goal. Compared to navigation methods in the literature utilizing redundancy of hyper-redundant manipulators, it is seen that the method presented in our paper significantly increases maneuvering ability of redundant/hyper-redundant manipulators utilizing narrow spaces through obstacles by using fewer number of DOFs. The method is simple and robust. While it almost uses the minimum number of links physically required for the task, there is no restriction on the number of links used. The exemplary computer simulations with detailed figures embody the noticeable advantage of the proposed method while verifying the effectiveness of it.  
The following are available online at [[https://www.mdpi.com/article/10.3390/app12178882/s1](https://www.mdpi.com/article/10.3390/app12178882/s1)]([https://www.mdpi.com/article/10.3390/app12178882/s1](https://www.mdpi.com/article/10.3390/app12178882/s1)). Video S1: A simulation of the first example without auxiliary elements. Video S2: A simulation of the first example with auxiliary elements. Video S3: A simulation of the second example without auxiliary elements. Video S4: A simulation of the second example with auxiliary elements.  
Conceptualization, E.S.C.; methodology, O.M. and E.S.C.; software, O.M. and E.S.C.; validation, O.M.; formal analysis, O.M.; investigation, O.M.; resources, O.M.; data curation, E.S.C.; writing--original draft preparation, O.M. and E.S.C.; writing--review and editing, O.M. and E.S.C.; visualization, O.M.; supervision, E.S.C.; project administration, E.S.C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
2D Two-Dimensional  
3D Three-Dimensional  
CPU Central Process Unit  
DOF Degrees of Freedom  
GA Genetic Algorithm  
RAM Random Access Memory  
RRT Rapidly-Exploring Random Tree  
USA United States of America  
WPF Windows Presentation Foundation