## 4 Adaptive Robust Model Free Control
### Design of Control Based on ANFSMC  
Let \(x_{1}=\theta_{k}\) and \(x_{2}=\dot{\theta}_{k}\). Equation (7) is rearranged into a state space form:  
\[ \begin{array}{l}\dot{x}_{1}=x_{2};\\ \dot{x}_{2}=\bar{\mathrm{M}}^{-1}(u-\mathrm{F}(t)+\mathfrak{o}_{GK});\\ y=x_{1};\end{array}  \tag{10}\]  
where \(u=\mathfrak{o}_{u}\). To realize the desired gait trajectory \(\theta_{d}\), we define \(x_{d}=\theta_{d}\); then, the tracking error is obtained as  
\[e=x_{1}-x_{d} \tag{11}\]  
The NFTSMC design process contains two steps. Firstly, a suitable non-singular fast terminal sliding surface is designed [29]:  
\[s=e+\gamma_{1}e^{[a]}+\gamma_{2}e^{[\frac{m}{n}]} \tag{12}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) are positive constants, \(m\) and \(n\) must satisfy the condition \(1<m/n<2\) and are generally positive odd numbers, and the value of \(a\) satisfies \(a>m/n\). \(e^{[c]}=|e|^{c}sign(e)(c>0)\), where \(sign(\cdot)\) means symbolic function. Secondly, to propel the system state toward the sliding surface within limited time, it is necessary to preserve the ideal sliding motion, which means \(\dot{s}=0\)[29], and thus  
\[\dot{s}=\dot{e}+\gamma_{1}a|e|^{a-1}\dot{e}+\gamma_{2}\frac{m}{n}|\dot{e}|^{ \frac{m}{n}-1}(\ddot{x}_{1}-\ddot{x}_{d}) \tag{13}\]  
Substituting Equation (7) into Equation (13), we can have  
\[\dot{s}=\dot{e}+\gamma_{1}a|e|^{a-1}\dot{e}+\gamma_{2}\frac{m}{n}|\dot{e}|^{ \frac{m}{n}-1}(\bar{\mathrm{M}}^{-1}u-\bar{\mathrm{M}}^{-1}\mathrm{F}(t)+\bar {\mathrm{M}}^{-1}\mathfrak{o}_{GK}-\ddot{x}_{d}) \tag{14}\]  
where \(u\) can be designed as [40]:  
\[u=u_{eq}+u_{cor} \tag{15}\]  
where \(u_{eq}\) denotes equivalent control law and \(u_{cor}\) represents correction control law. \(u_{eq}\) is applied to control object component and can be obtained from the ideal sliding motion \(\dot{s}=0\). Substitute Equation ( 15 ) into Equation (14), where \(u=u_{eq}\), and solve for \(u_{eq}\) as follows:  
\[u_{eq}=-\bar{\mathrm{M}}\frac{1}{\gamma_{2}}\frac{n}{m}((e^{[2-\frac{m}{n}]})+ \gamma_{1}a|e|^{a-1}e^{[2-\frac{m}{n}]})+\hat{\mathrm{F}}(t)-\mathfrak{o}_{GK }+\bar{\mathrm{M}}\ddot{x}_{d} \tag{16}\]To guarantee reaching the sliding surface, conventional \(u_{cor}\) is designed [29] as  
\[u_{cor}=-\bar{\mathbf{M}}(\eta+Z)sign(s) \tag{17}\]  
where \(\eta>0\) denotes the converging factor and \(Z>0\) is the upper bound of disturbances and uncertainties. To ensure the convergence of the system, the sliding gain \(\eta+Z\) in Equation (17) must be greater than the superior limit of the perturbation, which is impossible to be acquired in advance in practical applications. However, too large control gain will lead to significant chattering on the switching manifold. To relieve the fundamental chattering, we design an adaptive reaching law to approximate the upper limit \(\widehat{Z}\):  
\[\widehat{\dot{Z}}=s\gamma_{2}\frac{m}{n}|e|^{\frac{m}{n}-1}sign(s) \tag{18}\]  
**Theorem 1**. : _A nonlinear system Equation (10) with disturbances and uncertainties is considered. If the defined ANFTSMC-TDE control law in Equations (15)-(18) is employed, the convergence and stability of the system are ensured during the whole process._  
Proof: Define the Lyapunov function as  
\[V_{2}=V_{1}+\frac{1}{2\kappa}\widetilde{w}^{2} \tag{19}\]  
where \(\widetilde{w}=w^{*}-\hat{w}\). Differentiate the Lyapunov function \(V_{2}\), which leads to:  
\[ \begin{split}\dot{V}_{2}&=\dot{V}_{1}+\frac{1}{ \kappa}\widetilde{w}\hat{w}\\ &=s(\dot{e}+\gamma_{1}a|e|^{a-1}\dot{e}+\gamma_{2}\frac{m}{n}|e|^ {\frac{m}{n}-1}(\bar{\mathbf{M}}^{-1}u-\bar{\mathbf{M}}^{-1}\mathrm{F}(t)\\ &+\bar{\mathbf{M}}^{-1}\mathfrak{o}_{\mathcal{C}k}-\bar{\mathbf{ \lambda}}_{d}))+(\widehat{Z}-Z)\widehat{Z}+\frac{1}{\kappa}\widetilde{w}\hat{w }\end{split}  \tag{20}\]  
where  
\[