## 6 Conclusions  
The following are the final main remarks derived from this study:  
* This study proposes a new adaptive controller for a class of simplified biped robots with state restrictions that are handled with the application of state-dependent gains in a class of state feedback controllers. * The proposed controller was inspired by a traditional state feedback formulation that contains gains that are adjusted using the online evaluation of the tracking error and its derivative. Intending to simplify the implementation of the proposed controller, the application of a robust and exact differentiator based on the STA contributed to the state feedback, resulting in a class of output-based controllers with gains that also depend on the estimated derivative of the tracking error. * The stability analysis resulted in the time-dependent gain adjusting law, which may guarantee the tracking of the reference trajectories. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Characteristics**} & \multicolumn{2}{c}{**Controller**} \\ \cline{2-3}  & **Adaptive** & **PID** \\ \hline \(J_{e,h}\) & \(2.5892\times 10^{4}\) & \(9.0636\times 10^{3}\) \\ \(J_{c,h}\) & \(7.4752\times 10^{5}\) & \(9.6471\times 10^{6}\) \\ \(J_{e,k}\) & \(1.8016\times 10^{4}\) & \(1.1642\times 10^{3}\) \\ \(J_{c,k}\) & \(8.1580\times 10^{5}\) & \(1.0912\times 10^{6}\) \\ \(J_{e,a}\) & \(2.8016\times 10^{4}\) & \(1.2569\times 10^{3}\) \\ \(J_{c,a}\) & \(9.0570\times 10^{5}\) & \(1.1258\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Norm \(L_{2}\) of the error and control signals of both joints. Figure 11: Norm of the voltage evolutions with the application of the regular PID and the adaptive controller proposed in this study. * The application of the proposed controller justified the exertion of a regular gait cycle for a simulated biped robot that was developed in a virtualized version of the BRD that satisfies the dimensions of an averaged human being. Consider the Lyapunov candidate function given by  
\[ \begin{gathered} V\big{(}\xi,E,\bar{k}_{1},\bar{k}_{2}\big{)}=\sum_{ i=1}^{n}V_{i}\big{(}\bar{\xi}_{i},E_{i},\bar{k}_{1,i},\bar{k}_{2,i}\big{)}\\ \\ V_{i}\big{(}\bar{\xi}_{i},E_{i},k_{1,i},k_{2,i}\big{)}:=V_{1,i}(\bar{ \xi}_{i})+V_{2,i}(E_{i})+V_{3,i}\big{(}\bar{k}_{1,i},\bar{k}_{2,i}\big{)}\end{gathered}  \tag{75}\]  
with \(V_{1,i}(\bar{\xi}_{i})=\xi_{i}^{\top}P_{1,i}\bar{\xi}_{i}\), \(V_{2,i}(E_{i})=E_{i}^{\top}P_{2,i}E_{i}\) and \(V_{3,i}\big{(}\bar{k}_{1,i},\bar{k}_{2,i}\big{)}=\pi_{1,i}\bar{k}_{1,i}^{2}+ \pi_{2,i}\bar{k}_{2,i}^{2}\) where \(\bar{k}_{1,i}=K_{P,i}(t)-K_{P,i}^{*}\) and \(\bar{k}_{2,i}=K_{D,i}(t)-K_{D,i}^{*}\). The term referred to as \(\bar{\xi}_{i}\) is described by \(\bar{\xi}_{i}=\Big{[}|\delta_{1,i}|^{1/2}\text{sign}(\delta_{1,i})\ \ \delta_{2,i}\Big{]}^{\top}\). The time derivative of each function \(V_{i}\big{(}\bar{\xi}_{i},e_{i},\bar{k}_{1,i},\bar{k}_{2,i}\big{)}\) is:  
\[ \begin{gathered}\frac{d}{dt}V_{i}(t)=2\bar{\xi}_{i}^{\top}(t)P_{1,i}\frac{d}{dt}\bar{\xi}_{i}(t)+2E_{i}^{\top}P_{2,i}\frac{d}{dt}E_{i}+\\ 2\pi_{1,i}\bar{k}_{1,i}\frac{d}{dt}\bar{k}_{1,i}+2\pi_{2,i}\bar{ k}_{2,i}\frac{d}{dt}\bar{k}_{2,i}\end{gathered}  \tag{76}\]  
Notice that \(V_{1,i}(\bar{\xi}_{i})\) is continuous; however, it is not differentiable if \(\delta_{1,i}=0\). The ideas proposed by [20] are reconsidered here for handling the class of no regular Lyapunov functions. Let us continue the analysis of the second stage \(V_{2,i}(E_{i})\). From the calculus of the full-time derivative for the second function, one obtains \(\frac{d}{dt}V_{2,i}(t)=2E_{i}^{\top}(t)P_{2,i}\frac{d}{dt}E_{i}(t)\) using [PERSON]'s derivative. The direct substitution of \(\frac{d}{dt}\xi_{i}\) and \(\frac{d}{dt}E_{i}\) in the expression corresponding to the full-time derivative of \(V_{i}(t)\) yields:  
\[