## 8 Conclusions  
This study presented an in-depth analysis of the Directed Angular Restitution (DAR) method and its application to the locomotion of hexopod robots. The DAR method was developed to address the challenges of inverse kinematics and ensure smooth and continuous movement, allowing hexopod robots to achieve precise control over their trajectories.  
The HexV3-Mythbots model served as an exemplary platform to validate the DAR method. Detailed simulations demonstrated the ability of the robot to perform omni-directional locomotion, with the capability to adjust its orientation and speed through the parameters \(q_{5}\) and \(\omega\). This flexibility allows the robot to maneuver effectively in any direction, highlighting the robustness of the method in controlling multi-legged robots.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Phase** & **Leg 1** & **Leg 2** & **Leg 3** & **Leg 4** & **Leg 5** & **Leg 6** \\ \hline
1 & **Swing** & Support & **Swing** & Support & **Swing** & Support \\
2 & Support & **Swing** & Support & **Swing** & Support & **Swing** \\
3 & **Swing** & Support & **Swing** & Support & **Swing** & Support \\
4 & Support & **Swing** & Support & **Swing** & Support & **Swing** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Sequence diagram of tripod gait during a walking cycle.  
Experimental results with the Hex-Kurumy robot further validated the practical application of the DAR method. The remote control via a joystick, allowing for real-time adjustments of \(q_{5}\) and \(\omega\), showcased the ease with which the movement of the robot can be fine-tuned to meet various operational needs. The ability to switch between different motion patterns seamlessly underscored the adaptability and effectiveness of the method in real-world scenarios.  
In conclusion, the directed angular restitution method provides a significant advancement in the control of hexapod robots, enabling precise, smooth, and omnidirectional movement. The contributions of this work include the development and validation of the DAR method for inverse kinematics, the demonstration of enhanced omnidirectional locomotion capabilities, and practical implementation in physical robots, confirming its applicability and robustness. This work paves the way for further research into optimizing the DAR method for more complex terrains and integrating additional sensory inputs to enhance environmental interaction and adaptability.  
A.Y.A. and J.R.-L.: supervision, project administration, review, editing, funding acquisition. G.M.-G.: writing, review, and editing. A.Y.A. and J.R.-L.: methodology and implementation of the simulation results. A.Y.A. and J.R.-L.: supervision, review, and editing. J.A.L. and J.A.G.: studied the state-of-the-art, methodology, implementation of the simulation results, and writing. All authors have read and agreed to the published version of the manuscript.  
The authors are thankful for the support of CONACYT Mexico, through Project PCC-2022-319619 (\"Project supported by Fondo Sectorial de Investigacion para la Educacion\").  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors thank the support of CINVESTAV, Universidad de Guadalajara, Centro de Educacion Tecnologica Industrial and Instituto Tecnologico Nacional.  
The authors declare no conflicts of interest.