## 3 Results and Discussions
### Mono-Objective Formulation  
Another method can be used for the resolution of a multiobjective optimization problem. This technique, called the scalarization method [28], consists of attributing a weight to each criterion that corresponds to the priority given to this objective in the optimization process. This method allows a single solution. Based on this technique, the modified optimization aims to find the optimal design vector \(\mathbf{I}^{*}\) which minimizes the new objective function \(\mathcal{F}\), defined as a weighted sum of the three subfunctions introduced above. The optimization problem is then formulated as follows:  
\[\min(\mathcal{F}(\mathbf{I})),\]  
Subject to  
\[0<T_{min}\leq T_{i}(j)\leq T_{max},\;\;i=1.4,\;\;j=1..n \tag{18}\] \[\alpha_{i}>\alpha_{lim},\qquad i=1..4\] \[\mathcal{P}_{i}\;\epsilon\;\text{fixed frame},\;i=1..4\]  
Figure 7: Bi-objective visualizations of the Pareto front. The problem constraints are handled using a penalty approach. \(\mathcal{F}\) is then expressed as follows:  
\[\mathcal{F}(\mathbf{I})=\beta_{1}\mathcal{C}_{1}(\mathbf{I})+\beta_{2}\mathcal{C }_{2}(\mathbf{I})+\beta_{3}\mathcal{C}_{3}(\mathbf{I})+\varphi_{1}+\varphi_{2} +\varphi_{3}, \tag{19}\]  
where \(\wp_{i,\ i=1,2,3}\) are the penalty functions given by the Equations (15)-(17) and \(\beta_{1}\), \(\beta_{2}\) and \(\beta_{3}\) define the weight of each criterion. These coefficients must verify Equation (20). \[\sum_{i=1}^{3}\beta_{i}=1, \tag{20}\]  
Three strategies have been used in the literature to compute the values of the coefficients \(\beta_{i}\)[29], namely, the equal weights method [30], the rank order centroid weights method [31] and the rank-sum weights method [31]. The equal weights method was used in our previous work [32]. In this paper, a deeper study of the appropriate values of these weighting coefficients is conducted in order to identify the impact of each coefficient on the quality of the final solution and the optimization process. A mapping of the global objective function and those corresponding to the three criteria is computed for all the possible coefficient values inside the interval (0,1) with respect to Equation (20), using the Particle Swarm Optimization algorithm (PSO) [33]. The computed variations are illustrated in Figure 8.  
Figure 8: Variation of (**a**) the cable tension criterion, (**b**) the elastic stiffness criterion, (**c**) the dexterity criterion and (**d**) the objective function along the prescribed movement vs \(\beta_{1}\) and \(\beta_{2}\) (\(\beta_{3}=1-(\beta_{1}+\beta_{2})\)). Referring to Figure 8d, the objective function decreases with \(\beta_{1}\). This means that the cable tension criterion \(\mathcal{C}_{1}\) has the greatest impact on the objective function \(\mathcal{F}\) compared to the other coefficients \(\beta_{2}\) and \(\beta_{3}\). Thus, having a structure with a minimum cable tension distribution is easier than generating good dexterity and elastic stiffness. As illustrated in Figure 8c, the dexterity criterion \(\mathcal{C}_{3}\) is nearly equal to \(0.3\) (\(\mathcal{C}_{3}\in\ [0.3,\ 0.33]\) for \(\beta_{3}=0\) and \(\mathcal{C}_{3}=0.28\) for \(\beta_{3}=1\)). Thus, according to Equation (13), the robot dexterity remains between \(0.67\) and \(0.72\). In other words, it takes good values even when \(\beta_{1}\) and \(\beta_{2}\) are large (\(\beta_{3}\) is close to zero). This is contrary to the elastic stiffness criterion \(\mathcal{C}_{2}\),whose minimum value is equal to \(0.58\) for \(\beta_{2}=1\), which corresponds to an elastic stiffness equal to \(0.42\) (according to Equation (12)). Based on this mapping study, since the dexterity has acceptable values for any chosen \(\beta_{3}\), the lowest coefficient is given to the criterion \(\mathcal{C}_{3}\). The remaining weight is divided equally between the two other criteria. Thus, \(\beta_{3}=0.1,\ \ \beta_{1}=\beta_{2}=0.45\). Several methods exist in the literature allowing the resolution of optimization problems [34; 35].