## 5 Cable Elasticity  
So far we have considered cables to be inextensible. With this model, it can be clearly seen from Figure 4b that, at the launch instant, tensions suddenly change due to the instantaneous variation of the EE mass \(m\). Though inextensible-cable models can be very useful for predicting the dynamic feasibility of a trajectory for a CDPR [3; 4; 5; 6; 7; 8; 9; 10], tension discontinuity is unrealistic. Thus, we consider now a more realistic model where cables behave as linear elastic springs: by this second approach, tension discontinuities disappear, but the sudden variation of the gravity load applied on the EE causes oscillations. We introduce \(\mathbf{p}_{I}(t)=[x_{I}(t),y_{I}(t),z_{I}(t)]^{T}\) and \(\mathbf{p}_{A}(t)=[x_{A}(t),y_{A}(t),z_{A}(t)]^{T}\) as the positions of the EE as it moves along the ideal (\(\Gamma_{I}\)) and the actual (\(\Gamma_{A}\)) trajectory, respectively: that is, \(\Gamma_{I}\) is the  
Figure 4: (**a**) Optimal trajectory: 3D view of the path \(\Gamma_{opt}\). (**b**) The cable tensions along \(\Gamma_{opt}\). Figure 3: Pareto front for the two objectives \(E\) and \(\Delta t_{LT}\). trajectory defined in the planning phase considering ideal inextensible cables, while \(\Gamma_{A}\) is the trajectory obtained when elastic effects are introduced. Accordingly, cable lengths become  
\[\rho_{i,A}=\|\mathbf{p}_{A}-\mathbf{a}_{i}\|,\quad\rho_{i,I}=\|\mathbf{p}_{I}- \mathbf{a}_{i}\|,\quad i=1,2,3 \tag{40}\]  
We also define \(\Delta\rho_{i}=\rho_{i,A}-\rho_{i,I}\). We use the following linear elastic model:  
\[\tau_{i,A}= \begin{cases}\frac{K_{0}}{\rho_{i,I}}\Delta\rho_{i},&\Delta\rho_{i }\geq 0\\ 0,&\Delta\rho_{i}<0\end{cases} ,\quad i=1,2,3 \tag{41}\]  
where \(K_{0}\) is the stiffness of a unit length of cable (assumed equal for all cables). The cable-vector directions are  
\[\mathbf{e}_{i,A}=\frac{\mathbf{p}_{A}-\mathbf{a}_{i}}{\rho_{i,A}},i=1,2,3 \tag{42}\]  
and the actual structure matrix is  
\[\mathbf{M}_{A}= \begin{bmatrix}\mathbf{e}_{1,A}&\mathbf{e}_{2,A}&\mathbf{e}_{3,A}\end{bmatrix}  \tag{43}\]  
Since any real system is affected by energy dissipation, damping needs to be taken into account to obtain a realistic model. A complete model of all dissipative forces in a real system is beyond the scope of the present work; therefore, for simplicity we group all damping effects in a linear term, as  
\[\mathbf{F}_{d}=K_{d}\dot{\mathbf{p}}_{A} \tag{44}\]  
where \(K_{d}\) is the damping coefficient and \(\dot{\mathbf{p}}_{A}\) is the velocity of the EE as it moves along the actual trajectory. This dissipative force could be due, for instance, to air drag. So, the general equilibrium equation for the elastic model is  
\[\mathbf{M}_{A}\boldsymbol{\tau}_{A}=m\left(\mathbf{g}-\ddot{\mathbf{p}}_{A} \right)-\mathbf{F}_{d} \tag{45}\]  
where \(\boldsymbol{\tau}_{A}=[\tau_{1,A},\tau_{2,A},\tau_{3,A}]^{T}\). We can then combine Equations (41), (44) and (45) to obtain a set of ordinary differential equations. For given initial values of \(\mathbf{p}_{A}(t)\) and \(\dot{\mathbf{p}}_{A}(t)\), this set can be numerically solved to yield the actual trajectory and actual cable tensions. To provide an example, we use the optimal trajectory parameters found in Section 4.2, as expressed in Equations (37) and (39); here, the robot starts moving from rest at \(C\) with a transition trajectory, reaches the target trajectory \(\Gamma\) and finally performs the launch. The other parameters are set as \(K_{0}=10{,}000\) N and \(K_{d}=0.01\) Ns/m; the value of \(K_{0}\) is obtained considering polymer cables having diameter \(\approx\)1 mm (using the data presented in [45] for reference), while the value of \(K_{d}\) is only a reasonable parameter for simulation. The results are shown in Figures 5-7. Figure 5: Ideal trajectory (dashed line) and actual trajectory (continuous). Figure 6: Plots of the cable tensions over time, with the inextensible model (dashed line) and elastic model (continuous).