# Real-Time Tracking and Position Control of an Elastic Cable-Driven Winch System
## 3 Controller Design
### Sliding Mode Control  
A sliding mode control (SMC) law is derived to control the desired set point of a cable-driven winch system having nonlinear properties. Next, a sliding manifold is selected to define the relationship between state variables and prescribed dynamics. Then, the control law is designed to drive the state trajectories to the manifold and restrict them  
Figure 2: The real\(-\)time block diagram of position control and PID velocity control. there. The controller is designed using the dynamic equation of the cable-driven system in state-space form. The state variables are chosen as  
\[X=\left[x_{1}\ x_{2}\ x_{3}\ x_{4}\right]^{T}=\left[\theta\ \dot{\theta}\ y\ \dot{y}\right]^{T} \tag{15}\]  
With the system input as U (output of the controller) and the output as \(y\), the system is written in the state-space form as follows:  
\[ \begin{array}{c}\dot{x}=A(x)+B(x)U\\ y=C(x)\end{array}  \tag{16}\]  
where \(x_{1}\), \(x_{2}\) are the position and the velocity of the winch, respectively. \(x_{3}\) denotes the altitude of the mass, and \(x_{4}\) stands for the load velocity. The elements of the matrices \(A\) and \(B\) are as follows  
\[A(x)= \begin{bmatrix}x_{2}\\ -\frac{2 bx_{3}}{K}\frac{R}{R_{m}}-\frac{EA}{K(h-x_{3})}(x_{1}R-x_{3})\\ x_{4}\\ -\frac{EA}{2L(h-x_{3})}(x_{1}R-x_{3})^{2}+\frac{EA}{L(h-x_{3})}(x_{1}R-x_{3}) \end{bmatrix}  \tag{17}\]  
\[B(x)= \begin{bmatrix}0&\frac{\eta}{K}&0&0\end{bmatrix} ^{T} \tag{18}\]  
where \(K=\left(I+2I_{m}\frac{R^{2}}{R_{m}^{2}}\right)\), \(L=(m+m_{c})\), and \(\eta\) is the efficiency factor. The torque of the winch system is equal to the multiplication of \(\eta\) and the control signal in voltage, i.e., \(T_{s}=\eta U\). \[C(x)=y \tag{19}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameters** & **Value** & **Unit** \\ \hline Motor & Armature resistance (Ra) & 3.65 & [\(\Omega\)] \\ Maxon & Armature inductance (La) & 24.3 \(\times\) 10\({}^{-3}\) & [H] \\ ECmax- & Torque constant (Kt) & 0.31 \(\times\) 10\({}^{-3}\) & [Nm/A] \\
303487 & EMF constant (Kb) & 0.02429 & [Vs/rad] \\ \hline Winch & Drum radius (R) & 0.01 & [m] \\ Inertia moment (I) & 2409.25 \(\times\) 10\({}^{-9}\) & [kgm\({}^{2}\)] \\ \hline Direction & Radius (Rm) & 0.022 & [m] \\ pulley & Inertia moment (Im) & 1665.88 \(\times\) 10\({}^{-9}\) & [kgm\({}^{2}\)] \\ \hline \multirow{4}{*}{Elastic Cable} & Diameter (d) & 1 & [mm] \\  & Length at time zero (Lo) & 1.70 & [m] \\  & Cable length between drum-pulley (L\({}_{1}\)) & 0.3 & [m] \\  & Cable length between pulleys (L\({}_{2}\)) & 0.4 & [m] \\  & Mass per unit (qc) & 0.0013 & [kg/m] \\  & Max. working height (h) & 1.00 & [m] \\  & Modulus of elasticity (E) & 3.3354 \(\times\) 10\({}^{7}\) & [N/m\({}^{2}\)] \\ \hline \multirow{3}{*}{Load} & Carrier box weight (ma) & 0.077 & [kg] \\  & Min-max payload (mb) & 0–0.200 & [kg] \\  & The maximum total mass (M) & 0.277 & [kg] \\ \hline Distance & Sensing range & 50–5000 & mm \\ sensor & Voltage output & 0–10 & V \\  & LR-TB5000C & Sensitivity & \(\pm\)1 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the components used in the winch system. The sliding mode manifold can be chosen as below. \[\sigma=Ce+\dot{e}(C>0) \tag{20}\]  
where  
\[e={x_{3}}^{r}-x_{3},\ \dot{e}={x_{4}}^{r}-x_{4}\]  
To ensure the stability of the error dynamics, a Lyapunov function is selected as below. \[ \begin{array}{l}V=\frac{1}{2}\sigma^{T}\sigma>0\\ \dot{V}=\sigma^{T}\dot{\sigma}\leq 0\end{array}  \tag{21}\]  
Choosing \(\dot{\sigma}=-D\sigma\) (D > 0 and positive definite constant),  
\[\dot{V}=\sigma^{T}\dot{\sigma}=-\sigma^{T}D\sigma<0 \tag{22}\]  
Canceling the term \(\sigma^{T}\) on both sides of Equation (22),  
\[\dot{\sigma}+D\sigma=0 \tag{23}\]  
Taking the time derivative of (22) and substituting the system equations, the following is derived:  
\[\dot{\sigma}=C\dot{e}+\ddot{e}=C\dot{e}+\dot{x}_{2}^{r}-f-bu \tag{24}\]  
Next, the equivalent control is calculated, which satisfies the following relationship.