# Asymmetric Constrained Control of a Cervical Orthotic Device Based on Barrier Sliding Modes
## 4 Mathematical Description of the Cervical Orthotic Device  
The mathematical description consider to be fulfilled by the mechanical orthosis device is obtained through the Euler-Lagrange approach. Therefore, the following mathematical model can describe the complete dynamics of the orthosis [23]  
\[\ddot{q}(t)=F(\dot{q}(t),t)+G(q(t),t)+\Psi(q(t),t), \tag{2}\]  
with \(q(0)=q_{0}\) and \(\dot{q}(0)=q_{a0}\) being the initial conditions, \(q_{0},q_{a0}\in\mathbb{R}^{n}\). \(q\in\mathbb{R}^{2n}\) is the vector of generalized coordinates and \(\dot{q}\in\mathbb{R}^{2n}\) its corresponding derivative. Function \(F:\mathbb{R}^{2n}\times\mathbb{R}^{+}\to\mathbb{R}^{n}\) is the drift term and it satisfies the Lipchitz condition and the term \(G:\mathbb{R}^{n}\times\mathbb{R}^{+}\to\mathbb{R}^{n\times n}\) is a known matrix associated with the input signal \(u\in\mathbb{R}^{n}\). The nonlinear function \(\Psi:\mathbb{R}^{2n}\times\mathbb{R}^{+}\to\mathbb{R}^{n}\) represents uncertainties and non-modeling dynamics that satisfies the next condition  
\[\|\Psi(q,t)\|\leq\Psi_{0}+\Psi_{1}\|q\|^{2},\quad\Psi_{0}.\Psi_{1}\in\mathbb{ R}^{+}. \tag{3}\]  
defining \(x_{1}=q\) and \(x_{2}=\dot{q}\), by the state variable approach, system in (2) becomes (the time dependence is omitted to facilitate the reading of the manuscript)  
\[\dot{x}_{1}=x_{2} \tag{4}\] \[\dot{x}_{2}=f(x,t)+g(x_{1})u+\Psi(x,t),\]  
where \(x_{1}= \begin{bmatrix}x_{1,1}&x_{1,2}&\ldots&x_{1,n}\end{bmatrix} ^{\top}\) and \(x_{2}= \begin{bmatrix}x_{2,1}&x_{2,2}&\ldots&x_{2,n}\end{bmatrix} ^{\top}\). As it was mentioned before, the orthotic device should be restricted in movement and velocity. Therefore, the following assumption is taking into account in this manuscript. **Assumption 1**. : _The states are bounded by_  
\[x_{1}^{-}\leq\|x_{1}(t)\|^{2}\leq x_{1}^{+},\quad x_{2}^{-}\leq\|x_{2}(t)\|^{2 }\leq x_{2}^{+},\quad\forall t\geq 0,\]  
_with \(x_{1,+}\), \(x_{2,+}\) being positive known constants and \(x_{2,-}\) and \(x_{1,-}\) being negative constants. For \(n\)-dimensional systems, it is possible to assume that each state is bounded individually by_  
\[k_{\mathrm{c}_{1,i}}^{-}\leq x_{1,i}\leq k_{\mathrm{c}_{1,i}}^{+}, \qquad k_{\mathrm{c}_{2,i}}^{-}\leq x_{2,i}\leq k_{\mathrm{c}_{2,i}}^{+}\] \[k_{\mathrm{c}_{1,i}}^{+},k_{\mathrm{c}_{2,i}}^{+}\in\mathbb{R}^{ +},\qquad k_{\mathrm{c}_{1,i}}^{-},k_{\mathrm{c}_{2,i}}^{-}\in\mathbb{R}^{-},\]  
_previous equation implies an asymmetric box-type constraints for the second order system in (2)._  
**Remark 1**. : _The definition of the constraints in Assumption 1, without loss of generality, implies asymmetric-type box constraints centered at zero. For positive asymmetric constraints, a transformation can be implemented to apply the control design proposed in this manuscript._  
**Assumption 2**. : _The nonlinear function \(f(x,t)\) accepts the next upperbound_  
\[\|f(t,x)\|^{2}\leq f_{0}+f_{1}\|x\|^{2},\quad f_{0},f_{1}\in\mathbb{R}^{+},\]based on Assumption 1, previous inequality is equivalent to_  
\[f^{-}\leq\|f(t,x)\|^{2}\leq f^{+},\]  
_where \(f^{-}=f_{0}+f_{1}x^{-}\) and \(f^{+}=f_{0}+f_{1}x^{+}\)_  
As a consequence of the restrictions imposed over the physical system, the tracking trajectories should be restricted according the next assumption. **Assumption 3**. : _The vector of target signals \(y_{r}\) and its first time derivative \(\dot{y}_{r}\) are continuous and bounded. Moreover, the \(i\)th components of \(y_{r}\) and \(\dot{y}_{r}\) called \(y_{r,i}\) and \(\dot{y}_{r,i}\), respectively, accepts the following bounds_  
\[ \begin{array}{l}k_{c_{1},i}^{-}\leq y_{r,i}^{-}\leq y_{r,i}\leq y_{r,i}^{+} \leq k_{c_{1},i}^{+}\\ k_{c_{2},i}^{-}\leq dy_{r,i}^{-}\leq\dot{y}_{r,i}\leq dy_{r,i}^{+}\leq k_{c_{2 },i}^{+}\end{array} \]  
Based on Assumptions 1 and 3, the tracking error defined as \(z=x-\bar{y}_{r}\) should be bounded inside an asymmetric box-type constraints. Here \(\bar{y}_{r}:=\left[y_{r}^{\top}\dot{y}_{r}^{\top}\right]^{\top}\) and \(x:=\left[x_{1}^{\top}x_{2}^{\top}\right]^{\top}\).