## 6 An Illustrative Example
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(\mathbf{\Phi}_{i}\) & \(\mathbf{i=1\dots 3}\) & \(\mathbf{x\in\mathbb{R}^{2}}\) & \(\mathbf{P\in\mathbb{R}^{20}}\) & \(\mathbf{G\in\mathbb{R}^{10}}\) & **Parameter Bounds** \\ \hline \multirow{4}{*}{\(\Phi_{1}=\int\limits_{0}^{t_{f}}e^{2}dt\) with: \(e=\theta_{2_{d}}-\theta_{2}\)} & \(\frac{x_{1}=\theta_{2}}{x_{2}=\theta_{2}}\) & \(p_{1,2,3,4}=L_{1,2,3,4}\) & \(g_{1}=p_{2}\leq p_{4}\) & \(0.1\leq p_{1,2,3,4}\leq 1\) [m] \\ \cline{2-10}  & & \(x_{2}=\theta_{2}\) & \(p_{5,6,7}=l_{2,3,4}\) & \(g_{2}=p_{4}\leq p_{1}\) & \(0\leq p_{5,6,7}\leq 1\) [m] \\ \cline{2-10}  & & \(p_{8,9,10,11}=\theta_{1},\varphi_{2,3,4}\) & \(g_{3}=p_{1}\leq p_{3}\) & \(-\pi\leq p_{8,9,10,11}\leq\pi\) [rad] \\ \cline{2-10}  & & \(p_{12,13,14}=m_{2,3,4}\) & \(g_{4}=p_{3}+p_{2}\leq p_{4}+p_{1}\) & \(0.5\leq p_{12,13,14}\leq 2\) [kg] \\ \cline{2-10}  & & \(p_{15,16,17}=l_{2,3,4}\) & \(g_{5}=t_{7}-0.1\leq 0\) & \(0.01\leq p_{15,16,17}\leq 1\) [kgm\({}^{2}\)] \\ \cline{2-10}  & & \(p_{18,19,20}=k_{p},k_{i},k_{d}\) & \(g_{6}=u_{\max}-300\leq 0\) & \(0\leq p_{18,19,20}\leq 15\) [–] \\ \cline{2-10}  & & \(g\pi=p_{5}\leq p_{2}\) & & & & \\ \cline{2-10}  & & & \(g\pi=p_{6}\leq p_{3}\) & & **Control algorithm** \\ \hline \multirow{4}{*}{\(\Phi_{3}=\hat{\theta}_{2}(t_{p})-\hat{\theta}_{2_{d}}\)} & \multirow{4}{*}{} & \multirow{4}{*}{} & \multirow{4}{*}{} & \multirow{4}{*}{} & \multirow{4}{*}{} & \multirow{4}{*}{} & \multirow{4}{*}{} \\  & & & & & & & & & \\ \cline{1-10}  & & & & & & & & & \\ \cline{1-10}  & & & & & & & & & \\ \cline{1-10}  & & & & & & & & & \\ \cline{1-10}  & & & & & & & & & \\ \cline{1-10}  & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of data for the optimization problem formulation for the case of study in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\mathbf{p}_{i}\) & \(p_{1}\) & \(p_{2}\) & \(p_{3}\) & \(p_{4}\) & \(p_{8}\) & \(p_{5}\) & \(p_{6}\) & \(p_{7}\) & \(p_{12}\) & \(p_{13}\) \\ \hline Design variable & \(L_{1}\) & \(L_{2}\) & \(L_{3}\) & \(L_and (9) for a four-bar mechanism. The motion equations are computed for an evaluation period of two seconds considering values in Table 2.  
\[\mathbf{A}(\theta_{2})\ddot{\theta}_{2}+\mathbf{B}(\theta_{2},\dot{\theta}_{2})\dot{ \theta}_{2}+\mathbf{C}(\theta_{2})\mathbf{g}=u(\mathbf{P}) \tag{10}\]  
The solving process of the constrained multi-objective optimization process is carried out as illustrated in Figure 4 using the algorithm described in Figure 6 and Algorithm 1. The parameters of the implemented heuristic algorithm are a crossover range of \(DE_{CR}\in[0.7,0.9]\), a mutation range of \(DE_{F}\in[0.1,0.8]\), a crowding factor of \(DE_{CRW}=0.6\), a population of \(DE_{NI}=1000\) individuals, and a maximum number of iterations of \(DE_{IT_{max}}=1000\). The algorithm was executed 10 independent times. The results are illustrated in Figure 8, with the Pareto front after filtering the 10 executions. This figure illustrates the distribution of the obtained solutions and one solution is selected as the illustrative example to be analyzed in this section. In the initial stage of the algorithm execution, a random population is generated, as described in Algorithm 1 within the parameter space illustrated in Figure 5. At this point, the heuristic algorithm focuses on moving all the individuals in the population to the feasible space illustrated in Figure 5. This process is mainly related to finding feasible solutions that fulfill strict design requirements by using the constraint manager of the implemented heuristic algorithm. Figure 9 illustrates how the Grashof condition and the center of mass position (\(g_{4}\) and \(g_{7}\)) are not fulfilled for the first iterations, and these values change from one unfeasible value to another every iteration. Once the heuristic algorithm finds feasible solutions that fulfill the constraints (\(g_{i}\leq 0\)), the frequency change of these values decreases dramatically, and they do not go back to non-feasible values.