## 4 Experimental Validations
### Compliant Path Generator
#### 4.1.2 Validity of Decoupling Constraint Functions  
In this section, we further investigate the effect of coupling constraints on achieving full decoupling. Figure 7 plots the horizontal and vertical displacement distributions of the design result shown in Figure 5b under different load cases \(F_{I1}\) and \(F_{I2}\). Figure 8 draws the same type of displacement distributions related to the design result optimized using the optimization mode (13) with coupling constraints. Note that the two permissible coupling indices [\(\delta\)] and [\(\gamma\)] involved in Equation (13) are both set to 0.01 here. Figure 9 shows the evolution of the structural configuration optimized using Equation (13), and Figure 10 plots the convergence histories.  
From Figures 7 and 8, one could visually determine whether the coupling of inputs or outputs exists or not. As depicted in Figure 7b, the undesired vertical displacement at the output point is produced by the load \(F_{I1}=1\), and this output coupling issue is successfully solved in Figure 8b. In addition, Figure 7c shows unexpected horizontal displacements at the output point and the input point \(I1\), and the latter is exceptionally large, both of which are well suppressed in Figure 8c. Desired horizontal and vertical displacements at the output point \(O1(O2)\) are exhibited in Figures 7a,b and 8a,b, from which we could see that the design result with totally decoupled kinematics has almost the same expected output displacements as the one with the movement coupling issue.  
Figure 5: Design results obtained by using optimization models (**a**) Equation (17) and (**b**) Equation (19).  
Figure 6: Plots of the resulting strain energy densities related to optimization modes (**a**) Equation (17) and (**b**) Equation (19).  
Figure 8: Obtained displacement distributions using Equation (13): (**a**) horizontal and (**b**) vertical displacement results generated by \(F_{I1}\), (**c**) horizontal and (**d**) vertical displacement results generated by \(F_{I2}\).  
Figure 7: Obtained displacement distributions using Equation (19): (**a**) horizontal and (**b**) vertical displacement results generated by \(F_{I1}\), (**c**) horizontal and (**b**) vertical displacement results generated by \(F_{I2}\).  
Tables 1 and 2 provide the exact values of desired and undesired displacements as well as output and input coupling indices, and we can evaluate the output performances of the two design results shown in Figures 7 and 8 quantitatively. The absolute values of all the undesired displacements \(\widetilde{u}_{\mathcal{O}2,1}\), \(\widetilde{u}_{\mathcal{I}2,1}\), \(\widetilde{u}_{\mathcal{O}1,2}\) and \(\widetilde{u}_{\mathcal{I}1,2}\) related to the path generator drawn in Figure 8 decrease significantly, and accordingly the values of the coupling indices \(\delta_{2,1}\), \(\delta_{12}\), \(\gamma_{2,1}\) and \(\gamma_{1,2}\) are all quite small and not bigger than \([\delta]=[\gamma]=0.01\). Such results manifest that both input and output couplings are under control.  
Figure 9: Evolution of the structural configuration optimized using Equation (13).