## 3 Model-Free Adaptive Control with Prescribed Performance Method
### Dynamic Description and Controller Design  
Since the one-DOF upper-limb exoskeleton covers one active joint on the elbow in order to assist human elbow extension and flexion movement, the dynamic system is simplified as a one-DOF rigid link model (as shown in Figure 3). It can be expressed as  
\[M\ddot{q}+C\dot{q}+G+\tau_{d}=\tau \tag{17}\]  
where \(\tau_{d}\) denotes the sum of the external disturbance and dynamic model uncertainty; \(M\ddot{q}\) is the inertia torque; \(C\dot{q}\) represents the joint friction torque; \(G\) represents the gravitation torque; \(\dot{q}\) and \(\ddot{q}\) denote the first derivative and second derivative of the joint position \(q\), respectively. Figure 3: Rigid link model. According to the reference [43], the state form of the dynamic system is discretized as  
\[\dot{q}(k)=\frac{\tau(k)-D(k)}{M} \tag{18}\]  
\[D(k){=}G(k)+C\dot{q}(k)+\tau_{d}(k) \tag{19}\]  
Subsequently, \(T\) is set as the sample period of the system. \(\tau(k)\) denotes the input signal \(u(k)\). \(\Delta q(k)=q(k)-q(k-1)\) represents the output signal \(u(k)\). Then, Equation (18) can be transformed into Equation (20):  
\[y(k+1)=y(k)+\frac{T^{2}u(k)}{M}{+}\Delta f(k) \tag{20}\]  
where \(\Delta f(k)=-T^{2}D(k)/M\). As previously mentioned, \(y(k+1)\) thereby demonstrates that the system is a generalized Lipschitz system, with respect to the output signal \(y(k)\) and the input signal \(u(k)\), and the controller can be designed to be obeyed in the MFAC-SMC-PP control framework. Hence, the dynamic of the one-DOF upper-limb exoskeleton can be translated to the system described in Equation (5). However, the upper-limb exoskeleton may restart according to the user's mind, and the tracking trajectory control should also be reset. In this situation, the time constant \(k\) can be easily reset to 0, but the joint position \(y(k)\) may not be easily reset to \(y(0)\). Then, the original MFAC-SMC-PP controller may not work since the initial value of \(e(0)\) may escape from the boundary. Hence, a novel prescribed performance method is introduced below. Inconsistently with the original MFAC-SMC-PP control framework, a tuning function is adopted to modify the tracking error \(e(k)\) into a novel variable \(b(e)\), thereby demonstrating that the prescribed performance controller is not correlated with the initial error. As highlighted by the related literature [24; 34], the tuning function is defined as  
\[ \begin{split} b(e)&=\varphi(t)e(t)\\ \varphi(t)&=\left\{\begin{array}{cc}\sin\Bigl{(} \frac{\pi t}{B_{in}}\Bigr{)}&t<k_{in}\\ 1&\quad\text{\emph{other}}\end{array}\right.\end{split}  \tag{21}\]  
where \(k_{in}\) denotes a constant which should consider the maximal output of actuators and the convergence time designed by the user [34]. Then, the visual state variable of \(z\) can be determined as  
\[b(e)=\rho(k)S(z) \tag{22}\]  
In the proposed method, the prescribed performance function satisfies:  
\[\rho(k)=\rho(k+1)=\rho \tag{23}\]  
where \(\rho\) is a constant which can replace \(\rho_{\infty}\) from Equation (6). The tracking error can be bounded as  
\[-\rho<e(k)<\rho \tag{24}\]  
Selecting the identical strictly increasing function of \(S(z)\), Equation (9) can be rewritten as  
\[z(k)=\frac{1}{2}\ln\Bigl{(}\frac{\rho+b(e(k))}{\rho-b(e(k))}\Bigr{)} \tag{25}\]  
By complying with the designing order of the MFAC-SMC-PP control framework, the identical sliding mode variable of \(s(k)\) and the equivalent reaching law are defined.