## 6 Conclusions  
The FLHBR system possesses highly nonlinear dynamics and many degrees of freedom that are not easy to manipulate. The FLHBR system is also unavoidably subjected to all kinds of external disturbances such as contact with the ground and different ground situations. As a result, accurately modeling the dynamics and walking stability of FLHBR systems are greatly difficult. The study first presents the complete derivations of a mathematical model for highly nonlinear FLHBR systems, and proposes the robust control by the feedback linearization technique to greatly improve the shortcoming of the traditional singular perturbation approach that requires to meet the difficult complete condition for the discriminant function, and the restriction of the traditional H-infinity technique that needs to solve the Hamilton-Jacobi equation.  
This study first proposes the very valuable formulas of nonlinear exponential convergence rate and convergent radius for the highly nonlinear FLHBR system. Finally, through the demonstration of the Matlab simulation, the responses are shown to have good tracking performance as well as better robustness performance as compared with the traditional singular perturbation method. In the final section, we compare some simulations of the proposed feedback linearization approach with the traditional PID approach. The simulation results show that the transient dynamics of the proposed approach including the peak time, the rise time, the settling time and the maximum overshoot specifications is superior to the traditional PID approach.  
In future works, we hope that a real FLHBR system using the proposed main theorem can be implemented via hardware devices. Based on the important contribution that this article has first proposed on the convergence rate formula of the general nonlinear system, we may use the particle swarm optimization and linear quadratic regulator algorithms to achieve the more optimal performances for nonlinear FLHBR system under the guarantee of globally exponential stability in the near future.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & & **Peak Time** & **Settling Time** & **Rise Time** & **Maximum Overshoot** \\ \hline \multirow{4}{*}{PID approach} & x1 & 0.4 & 3.5 & 0.2 & 3.8 \\ \cline{2-6}  & x2 & 0.05 & 4.5 & 0.02 & 2.1 \\ \cline{2-6}  & x3 & 0.1 & 5.8 & 0.05 & 3.7 \\ \cline{2-6}  & x4 & 0 & 3.4 & 0 & 0 \\ \cline{2-6}  & x5 & 0.4 & 3.8 & 0.3 & 3.8 \\ \hline \multirow{4}{*}{Proposed approach} & x1 & 0 & 0.75 & 0 & 0 \\ \cline{2-6}  & x2 & 0 & 0.75 & 0 & 0 \\ \cline{2-6}  & x3 & 0 & 0.8 & 0 & 0 \\ \cline{1-1} \cline{2-6}  & x4 & 0 & 0.8 & 0 & 0 \\ \cline{1-1} \cline{2-6}  & x5 & 0 & 0.9 & 0 & 0 \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of transient dynamics for proposed approach and PID approach.  
**Author Contributions:** Conceptualization, K.-H.C.; methodology, C.-C.C.; software, Y.-F.H.; validation, Y.-F.H.; formal analysis, Y.-F.H.; investigation, Y.-F.H.; resources, K.-H.C.; data curation, K.-H.C.; writing--original draft preparation, C.-C.C.; writing--review and editing, K.-H.C. and Y.-F.H.; visualization, Y.-F.H.; project administration, Y.-F.H. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.