## 3 Controller Design
### Output Constraints Tracking Control
#### 3.3.1 Model-Based (MB) Impedance Control  
Firstly, the control design with known system parameters is considered. We define the tracking error as \(z_{1}=\zeta(t)-\zeta_{d}(t)\), and a new assistant error variable as \(z_{2}=\zeta(t)-\chi(t)\), where \(\chi(t)=[\chi_{1}(t),\chi_{1}(t),\cdot\cdot\cdot,\chi_{n}(t)]^{T}\) is an assistant vector to be defined later. From Equation (1), \(z_{2}\) is given as  
\[\dot{z}_{2}=M^{-1}(\zeta)[\tau+f-\tau^{dis}-C(\zeta,\dot{\zeta})\dot{\zeta}-G( \zeta)]-\hat{\chi} \tag{27}\]  
We consider the following BLF:  
\[V_{1}(t)=\sum_{i=1}^{n}\left(\frac{\vartheta_{i}}{2}\ln\frac{\kappa_{b,i}^{2} (t)}{\kappa_{b,i}^{2}(t)-z_{1,i}^{2}(t)}+\frac{1-\vartheta_{i}}{2}\ln\frac{ \kappa_{a,i}^{2}(t)}{\kappa_{a,i}^{2}(t)-z_{1,i}^{2}(t)}\right) \tag{28}\]  
where \(\vartheta_{i}\) is given by  
\[\vartheta_{i}=\left\{\begin{array}{cc}0,&z_{1,i}\leq 0\\ 1,&z_{1,i}>0\end{array}\right. \tag{29}\]  
Using the coordinate transformation of the error  
\[\xi_{a,i}=\frac{z_{1,i}}{\kappa_{a,i}},\quad\xi_{b,i}=\frac{z_{1,i}}{\kappa_{ b,i}} \tag{30}\]  
there is  
\[\xi_{i}=\vartheta_{i}\xi_{b,i}+(1-\vartheta_{i})\xi_{a,i} \tag{31}\]  
Then, (28) is rewritten as  
\[V_{1}(t)=\sum_{i=1}^{n}\frac{1}{2}\ln\frac{1}{1-\xi_{i}^{2}} \tag{32}\]  
Figure 2: Exoskeleton output constraint diagram.  
Differentiating (32), we have  
\[\dot{V}_{1}(t) =\sum_{i=1}^{n}\left(\frac{\xi_{b,j}\vartheta_{i}}{(1-\xi_{b,i}^{2}) \kappa_{b,j}}(z_{2,j}+\chi_{i}-\xi_{d,i}-z_{1,i}\frac{\dot{\kappa}_{b,j}}{ \kappa_{b,j}})\right.\] \[+\left.\frac{\xi_{b,i}\vartheta_{i}}{(1-\xi_{b,i}^{2})\kappa_{b,j }}(z_{2,i}+\chi_{i}-\xi_{d,i}-z_{1,i}\frac{\dot{\kappa}_{b,j}}{\kappa_{b,j}})\right) \tag{33}\]  
We choose the virtual controller \(\chi_{i}\) as  
\[\chi_{i}=\xi_{d,i}-\varsigma_{i}z_{1,i}-\xi_{i}z_{1,i} \tag{34}\]  
where  
\[\xi_{i}=\sqrt{\left(\frac{\dot{\kappa}_{b,i}(t)}{\kappa_{b,i}(t)}\right)^{2}+ \left(\frac{\dot{\kappa}_{a,i}(t)}{\kappa_{a,i}(t)}\right)^{2}+\beta} \tag{35}\]  
\(\beta\) is a small positive constant.  
Using (33) and (34), we can obtain  
\[V_{1}(t)\leq\sum_{i=1}^{n}-\varsigma_{i}\frac{\xi_{i}^{2}}{1-\xi_{i}^{2}}+\sum _{i=1}^{n}\varphi_{i}z_{1,i}z_{2,i} \tag{36}\]  
where the diagonal matrix \(\varphi\in\Re^{n\times n}\) satifies  
\[\varphi_{i}=\frac{\varsigma_{i}}{\kappa_{b,i}^{2}-z_{1,i}^{2}}+\frac{1- \varsigma_{i}}{\kappa_{a,i}^{2}-z_{1,i}^{2}} \tag{37}\]  
We construct the BLF candidate \(V_{2}\) as  
\[V_{2}=V_{1}+\frac{1}{2}z_{2}^{T}M(\xi)z_{2} \tag{38}\]  
Then, the derivative of \(V_{2}\) is  
\[\dot{V}_{2} =\dot{V}_{1}+\frac{1}{2}z_{2}^{T}\dot{M}(\xi)z_{2}+z_{2}M(\zeta) \dot{z}_{2}\] \[\leq\sum_{i=1}^{n}-\varsigma_{i}\frac{\xi_{i}^{2}}{1-\xi_{i}^{2} }+\sum_{i=1}^{n}\varphi_{i}z_{1,i}z_{2,i}+z_{2}^{T}(\tau+f-\tau^{dis}-G(\xi)- M(\xi)\dot{\chi}-C(\xi,\dot{\xi})\chi) \tag{39}\]  
We choose the model-based control law as  
\[\tau_{mb}=\tau^{dis}-f+M(\zeta)\dot{\chi}+C(\zeta,\dot{\xi})\chi+G(\zeta)- \varphi z_{1}-\mathrm{K}z_{2} \tag{40}\]  
where \(\mathrm{K}\) denotes the gain matrix, and \(\mathrm{K}=\mathrm{K}^{T}>0\).  
From (39) and (40), there is  
\[\dot{V}_{2}\leq\sum_{i=1}^{n}-\varsigma_{i}\frac{\xi_{i}^{2}}{1-\xi_{i}^{2}}-z _{2}^{T}\mathrm{K}z_{2} \tag{41}\]  
From the Barbalat lemma, the asymptotic stability of the system can be obtained.