# Real-Time Tracking and Position Control of an Elastic Cable-Driven Winch System
## 2 Mathematical Model of an Elastic Cable Winch System  
As illustrated in Figure 1, a seemingly simple system consisting of a motor and a mass is modeled more realistically by incorporating the cable's elasticity. For mathematical modeling of the winch system, the definition of the system is as follows: the radius of the drum and the direction pulleys are defined as \(R\) and \(R_{m}\), and the moments of inertia are defined as \(I\) and \(I_{m}\), respectively. The cable is determined to be homogeneous and elastic, and its mass is mc. The load consists of the combined mass of the carrier box and any additional weights. The system is structured so that the load moves vertically with the change in cable length. The angular velocity and acceleration of the drum are demonstrated by \(\dot{\theta}\) and \(\bar{\theta}\), respectively. A cable system where the segment between points A and B has a length denoted by \(L_{1}\) and stiffness \(k_{1}\), while the segment between points B and C has a  
Figure 1: A schematic elastic cable winch system. length with \(k_{2}\) stiffness. These stiffness parameters are defined as follows: \(k_{1}=EA/L_{1}\), \(k_{2}=EA/L_{2}\). E represents the Young's modulus and A is the cross-sectional area of the cable. Assuming no slippage occurs between the cable and the pulleys, one full turn of the cable is wrapped around the directional pulleys located at points B and C. Due to the elasticity of the cable segments, the rotation angles of the pulleys at points B and C will not necessarily be identical, even if the pulleys share the same diameter. This characteristic introduces additional degrees of freedom to the system. In this revised configuration, the system exhibits four degrees of freedom: the rotational motions of the pulleys at points A, B, and C, denoted by \(\theta_{A}\), \(\theta_{B}\), and \(\theta_{C}\), and the vertical translation of the attached mass, represented by \(y\). The linear velocities of the cable at these points can be expressed as follows: \(V_{A}=R\dot{\theta}_{A}\), \(V_{B}=R_{m}\dot{\theta}_{B}\), \(V_{\rm C}=R_{m}\dot{\theta}_{C}\) where \(R\) and \(R_{m}\) denote the radii of the respective pulleys, and the overdot indicates the time derivative. \[T_{A}=\frac{1}{2}I\dot{\theta}_{A}^{2},\ T_{B}=\frac{1}{2}I_{m}\dot{\theta}_{B }^{2},\ T_{\rm C}=\frac{1}{2}I_{m}\dot{\theta}_{C}^{2},\ T_{y}=\frac{1}{2}m \dot{y}^{2}\]  
To calculate the kinetic energy of the unwound portion of the cable, it is essential to determine the linear velocity of its center of mass. The length of the unwound segment is given by \(L_{c}=h-(y+h_{m})\), with mass \(m_{c,CD}\), and the distance of its center of mass from the sensor is \(y_{c}=y+h_{m}+0.5*l_{c}=0.5(y+h_{m}+h)\). The velocity of the center of mass is thus \(\dot{y}_{c}=0.5\dot{y}\). Accordingly, the kinetic energy of this segment is approximately \(T_{CD}\cong\frac{1}{2}m_{c,CD}\dot{y}_{k}^{2}\). The linear velocity of the center of mass of the AB cable can be considered the average of these \(\left(\frac{(V_{A}+V_{B})}{2}\right)\). Thus, the kinetic energy of the AB segment is approximately \(T_{AB}\cong\frac{1}{2}m_{L1}\frac{(V_{A}+V_{B})}{2^{2}}\). The total kinetic energy of the system is as follows:  
\[T=T_{A}+T_{B}+T_{\rm C}+T_{y}+T_{AB}+T_{BC}+T_{CD} \tag{1}\]  
\[T=\frac{1}{2}I\dot{\theta_{A}}^{2}+\frac{1}{2}I_{m}\dot{\theta_{B}}^{2}+\frac {1}{2}I_{m}\dot{\theta_{C}}^{2}+\frac{1}{2}m\dot{\theta_{C}}^{2}+\frac{1}{2} m\dot{y}^{2}+\frac{1}{8}m_{L2}R_{m}{}^{2}\left(\dot{\theta}_{C}+\dot{ \theta}_{B}\right)^{2}+\frac{1}{8}m_{c,CD}\left(R_{m}\dot{\theta}_{c}+\dot{y} \right)^{2} \tag{2}\]  
In this formula, \(m_{L1}=\lambda L_{1}\) and \(m_{L2}=\lambda L_{2}\) are constants and \(m_{c,CD}=\lambda[h-(y+h_{m})]\) is variable.