# Real-Time Tracking and Position Control of an Elastic Cable-Driven Winch System
## 2 Mathematical Model of an Elastic Cable Winch System
The difference in the translations at the ends of the AB cable is \(\Delta_{1}=R\theta_{A}-R_{m}\theta_{B}\) and the potential energy of this part is \(U_{AB}=\frac{1}{2}k_{1}\Delta_{1}^{2}\), for the BC cable \(\Delta_{2}=R_{m}\theta_{B}-R\theta_{C}\) and \(U_{BC}=\frac{1}{2}k_{2}\Delta_{2}^{2}\), and for the CD cable \(\Delta_{3}=R\theta_{C}-y\) and \(U_{CD}=\frac{1}{2}k_{3}\Delta_{3}^{2}\) where \(k_{3}=\frac{EA}{L_{C}}=\frac{EA}{h-(y+h_{m})}\). The potential energy of the CD part of the cable is \(U_{mc}=m_{c,CD}\frac{(y+h_{m}+h)}{2}\). The total potential energy is  
\[U=U_{AB}+U_{BC}+U_{CD}+U_{m}+U_{mc} \tag{3}\]  
\[U=\frac{1}{2}k_{1}\Delta_{1}^{2}+\frac{1}{2}k_{2}\Delta_{2}^{2}+\frac{1}{2}k_{ 3}\Delta_{3}^{2}+mg(y+0.5h_{m})+m_{c,\ \ CD}g\frac{(y+h_{m}+h)}{2} \tag{4}\]  
The motion equations of the crane system are derived using Lagrangian mechanics. The Lagrangian function of the system is defined as the difference between the kinetic energy \(T\) and the potential energy \(U\):  
\[L=T-U\]The equations of motion of the system are derived using the Lagrange equation. \[\frac{d}{dt}\left[\frac{\partial L}{\partial q_{i}}\right]-\frac{\partial L}{ \partial q_{i}}=Q_{i} \tag{6}\]  
The generalized coordinates are \(q_{1}=\theta_{A}\), \(q_{2}=\theta_{B}\), \(q_{3}=\theta_{C}\) and \(q_{4}=y\), resulting in a total of four equations of motion. \[ \begin{array}{c}\ddot{\theta}_{A}\left(I+\frac{1}{4}m_{L1}R^{2}\right)+ \ddot{\theta}_{B}\left(\frac{1}{4}m_{L1}RR_{m}\right)+\theta_{A}\left(k_{1}R^{2 }\right)+\theta_{B}(-k_{1}RR_{m})=\tau\\ \ddot{\theta}_{A}\left(\frac{1}{4}m_{L1}RR_{m}\right)+\ddot{\theta}_{B}\left( I_{m}+\frac{1}{4}m_{L1}{R_{m}}^{2}+\frac{1}{4}m_{L2}{R_{m}}^{2}\right)+\ddot{ \theta}_{c}\left(\frac{1}{4}m_{L2}{R_{m}}^{2}\right)+\\ \theta_{A}\left(-k_{1}RR_{m}\right)+\theta_{B}\left(k_{1}+k_{2}\right){R_{m}} ^{2}+\theta_{C}\left(-k_{2}{R_{m}}^{2}\right)=0\\ \ddot{\theta}_{B}\left(\frac{1}{4}m_{L2}{R_{m}}^{2}\right)+\ddot{\theta}_{c} \left(I_{m}+\frac{1}{4}m_{L2}{R_{m}}^{2}+\frac{1}{4}m_{C,CD}{R_{m}}^{2}\right) +\ddot{y}\left(\frac{1}{4}m_{C,CD}{R_{m}}\right)\\ +\theta_{B}\left(-k_{2}{R_{m}}^{2}\right)+\theta_{C}\left(k_{2}+k_{C,CD}\right) {R_{m}}^{2}-y\left(k_{C,CD}{R_{m}}\right)=0\\ \ddot{y}\left(m+\frac{1}{4}m_{C,CD}\right)+\ddot{\theta}_{c}\left(\frac{1}{4} m_{C,CD}{R_{m}}\right)+\frac{1}{8}\lambda\left({R_{m}\dot{\theta}_{c}+\dot{y}} \right)^{2}+g\left(m+\frac{m_{C,CD}}{2}\right)\\ +\frac{k_{C,CD}}{2_{C,CD}}\left({R_{m}\theta_{c}-y}\right)^{2}-k_{C,CD}\left( {R_{m}\theta_{c}-y}\right)=0\end{array}  \tag{7}\]  
The system was initially modeled as a four-degree-of-freedom (4-DOF) structure to reflect its physical characteristics accurately. However, the experimental setup includes only a single motor integrated with an encoder and a laser distance sensor for measuring the vertical position of the load. Since no sensors are available to measure the angular positions of the guide pulleys, the motions of these components cannot be directly observed. As a result, the 4-DOF model contains unobservable states, limiting its practical applicability for control design. Therefore, a reduced two-degree-of-freedom (2-DOF) model, incorporating only the measurable variables (the drum angle and the vertical position of the load) has been adopted. The elongation occurring in the cable segments between the winch drum at point A and the guide pulley at point B, as well as between the guide pulleys at points B and C, is negligible compared to the elongation observed between the pulley at point C and the load. Therefore, the stretching behavior of the cable is primarily considered over the segment between the pulley and the load. This assumption enables the derivation of a simplified mathematical model, which forms the theoretical basis for subsequent controller design and analysis. The stretching of the cable takes place mainly between the directional pulley and the mass. For this reason, the elongation of the cable is expressed as \(\Delta=\theta R-y\).