## 3 Study Cases
### Case 2: Cam-Linkage Mechanism  
The second optimization problem is the synthesis of a cam-linkage mechanism. The design problem was presented by [PERSON] et al. [22]. The conceptual model for the cam-linkage mechanism is shown in Figure 5, where \(r_{1},\ r_{2},\...,\ r_{8}\) are the link lengths; \(\beta,\ \gamma,\ \eta\) are related to angular displacements; \([x_{0},y_{0}]\) is the Cartesian position of the mechanism with respect to the \(x-y\) coordinate system; \(e\) and the angle \(\alpha\) define the displacement position of the slider. The dimensional synthesis problem for this mechanism consists of finding the link lengths and the cam profile that approximate the coordinates \([x_{p}^{i},y_{p}^{i}]\) of the coupler link point \(P\) to the coordinates \([x_{p}^{i},y_{p}^{i}]\) of the rehabilitation trajectory. In this work, two rehabilitation trajectories are considered for the synthesis process representing a change in the human anatomy. The Pearson correlation between both trajectories is 97%. The trajectory points are shown in Figure 6, where Figure (a)a,b are considered as tra  
Figure 4: four–bar linkage precision points for the synthesis process. Figure 5: cam–linkage mechanism. 2, respectively, for the synthesis process. The Cartesian coordinates of the trajectory points are shown in Tables A2 and A3. In this synthesis problem, the design variable vector is given by (21):  
\[x=[r_{1},r_{2},r_{3},r_{4},r_{5},r_{6},r_{7},r_{8},\alpha,\beta,\gamma,\eta,x_{ 0},y_{0},e] \tag{21}\]  
The dimensional synthesis optimization problem is presented in (22)-(34). \[\underset{x}{\text{Min}}\;w_{1}\frac{\sum_{i=1}^{n_{f}}(\theta_{1}^{i}-\bar{ \theta}_{1}^{i})^{2}}{\Theta_{ref}}+w_{2}\frac{R_{0}}{R_{ref}} \tag{22}\]  
subject to:  
\[g_{1}: r_{1}+r_{2}-r_{3}-\sqrt{s_{min}^{2}+e^{2}}\leq 0 \tag{23}\] \[g_{2}: r_{1}+r_{3}-r_{2}-\sqrt{s_{min}^{2}+e^{2}}\leq 0\] (24) \[g_{3}: r_{1}+\sqrt{s_{max}^{2}+e^{2}}-r_{2}-r_{3}\leq 0\] (25) \[g_{4}: r_{2}^{2}+r_{3}^{2}-(d_{AC})_{min}^{2}-2r_{2}r_{3}\cos\pi/6\leq 0\] (26) \[g_{5}: -r_{2}^{2}-r_{3}^{2}+(d_{AC})_{max}^{2}+2r_{2}r_{3}\cos(\pi-\pi/ 6)\leq 0\] (27) \[g_{6}: r_{5}^{2}+r_{6}^{2}-(d_{DF})_{min}^{2}-2r_{5}r_{6}\cos\pi/6\leq 0\] (28) Figure 5:0 (29) Figure 5:1 (30) Figure 5:2 (31) Figure 5:3 (32) Figure 5:4 (33) Figure 5:5  
The first term of the weighted objective function (22) minimizes the error between the crank angles \(\theta_{1}^{i}\) and the desired angles \(\bar{\theta}_{1}^{i}\)\(\forall\)\(i=\{1,2,\ldots,n_{f}\}\), where \(n_{f}\) is the number of precision points, and \(\bar{\theta}_{1}^{i}\) is defined by (35):  
Figure 5:6  
The second term minimizes the base radius \(R_{0}\) of the cam (see Figure 5). Both design objectives are weighted by \(w_{1}=0.3\) and \(w_{2}=0.7\), and normalized by \(\Theta_{ref}=0.4246\) [rad] and \(R_{ref}=0.3782\) [m] according to [22]. Figure 5:7 cam–linkage precision points for the synthesis process. On the other hand, the first three inequality constraints (23)-(25) guarantee a complete revolution of the crank angle \(\theta_{2}\) (Grashof criterion), where \(s_{min}\) and \(s_{max}\) are the minimum and maximum values of \(s(\theta_{1})\) in the precision points. The inequality constraints (26)-(29) provide high efficiency of the force transmission from the input link (crank link) to the output link, where \(d_{AC}\) is the distance between points A and C, and \(d_{DF}\) is the distance between points D and F. The inequality constraints (30)-(32) guarantee the path feasibility in the output link, where \(d_{OD}\) is the distance between points D and F, and \(d_{Be}\) is the distance between points B and e. The inequality constraint (33) defines the minimum curvature radius of the crank profile. The design variable vector \(x\) is bounded by \(x_{min}\) and \(x_{max}\). For more details about the constraints and boundary limits, see [29].