## 3 Nonlinear Tension Model of the DYNAMICA Cable  
### Nonlinear Cable Tension Formulation  
This section aims to establish the nonlinear tension formulation, and the dynamic modeling presented in the previous section will be based on this model to analyze the feasible workspace of the CDPR system. The linear tension formulations are described simply as a linear spring, and they are commonly used in CDPR systems. These formulations cannot satisfy the nonlinear characteristics of the elastic cable and the effect of dynamic phenomena with high cable length variations. Therefore, a nonlinear tension model is considered in this study. To provide a total nonlinear tension formulation that is suitable for highly dynamic applications, nonlinear tension is proposed as follows. According to [25], applying the [PERSON]'s law definition, the stress-strain relationship can be expressed as:  
\[\sigma(t)=E\varepsilon(t) \tag{15}\]  
where \(\varepsilon(t)=\frac{\Delta I(t)}{I(t)}\), \(I\) is the cable length, \(\sigma\) represents the stress, and \(E\) and \(\varepsilon\) are the modulus of elasticity and strain, respectively. Assuming that cross-sectional area \(A\) is a constant, then the cable tension can be written as follows:  
\[T(t)=A\sigma(t) \tag{16}\]  
where \(\sigma(t)=\frac{T(t)}{A}\), \(T\) is the cable tension. Equation (16) can be rewritten as the equation shown below by the time derivative of the cable tension:  
\[\ddot{T}(t)=EA\dot{\varepsilon}(t) \tag{17}\]  
Finally, the nonlinear cable tension equation is obtained by integrating Equation (18):  
\[T(t)=EA\ln\left(\frac{l(t)}{l_{0}}\right)+T_{0} \tag{18}\]  
where \(l(t)\) denotes the cable length, \(l_{0}\) is the initial length of the cable, and \(T_{0}\) is the initial cable tension. This equation is valid for highly dynamic systems, such as CDPR systems. For the CDPR system using cables as polymer materials, the nonlinear elastic properties of the cable or viscosity are generated by the dynamic phenomenon with high cable length variation. The linear tension model can be used to determine the cable tension. However, it is not suitable for determining the feasible workspace of the CDPR system because it does not satisfy the nonlinear elastic properties of the polymer cable. Moreover,the operation of the robot is also affected by dynamic phenomena associated with high cable length variations. Therefore, kinematics and dynamics Equation (12) can be extended by considering Equation (18). Hence, the modified equation that includes the non-linear tension is given as:  
\[[J]^{T}\bigg{(}T+EA\ln\bigg{(}\frac{L(t)}{L_{0}}\bigg{)}+T_{0}\bigg{)}+\left\{ \begin{array}{c}m\overset{\rightarrow}{g}\\ 0\end{array} \right\}=W_{R} \tag{19}\]  
where \([J]^{T}\)represents the transposed matrix of \(J\). To find solutions for the cable tensions, by inverting Equation (19) then multiplying by the pseudoinverse matrix of \(J\) and adding the term of the homogeneous solution [11], the equation of tension can be determined according to the equation below. \[T+ES\ln\bigg{(}\frac{L(t)}{L_{0}}\bigg{)}+T_{0}=[J]^{P}\bigg{(}W_{R}-\left\{ \begin{array}{c}mg\\ 0\end{array} \right\}\bigg{)}+\left([I_{8}]-[J]^{P}[J]\right)\big{\{}z\big{\}} \tag{20}\]  
where \([J]^{P}\) represents the Moore-Penrose pseudoinverse Jacobian matrix and \([I_{8}]\) is the [8 \(\times\) 8] identity matrix. Equation (20) combines two parts, with the term on the right-hand side representing a particular solution and the second term representing the homogeneous solution. Unlike rigid arm mechanisms, the CDPR robot uses flexible cables to hold the end-effector. Therefore, the vector \(\{z\}\) is an arbitrary vector and was chosen to ensure that all cable tensions were positive. To maintain the positive cable tension and resolve the problem of actuation redundancy in the CDPR workspace, many methods have been proposed, such as the closed-form force distribution method [26] and an improved force distribution algorithm [27].