# Multicriteria Optimization of Lower Limb Exoskeleton Mechanism
## 5 Global Search and Multi-Criteria Optimization of Exoskeleton Link Dimensions  
Let us keep notation \(\zeta\), \(\eta\) for absolute coordinates of joints as it was in the previous sections. \(n=13\) variable parameters of the exoskeleton mechanism are varied within a given search area by using so-called random LP-\(\tau\) sequences, evenly distributed in \(n\)-dimensional parallelepiped [35; 38; 39]. A global search carried out specifying the following limits on variable parameters  
\[-1.50\leq\eta_{D}\leq\ 1.50\] \[-1.50\leq\xi_{G}\leq\ 1.50\] \[-1.50\leq\eta_{G}\leq\ 1.50\] \[0.05\leq r_{AB}\leq\ 0.30\] \[0^{\circ}\leq\phi_{0}\leq 360^{\circ}\] \[180^{\circ}\leq\Phi\leq 190^{\circ}\] \[0.40\leq L_{BC}\leq\ 1.00\] \[0.40\leq L_{CD}\leq\ 1.00\] \[0.50\leq x_{E}\leq\ 1.20\] \[-0.50\leq y_{E}\leq\ 1.20\] \[-1.50\leq\xi_{G}\leq\ 1.50\]0 \[-1.50\leq\xi_{G}\leq\ 1.50\]1  
The foot-point traces a straight line while the crank angle is in the range \(\varphi_{0}\leq\varphi\leq\varphi_{0}+\Delta\Phi\). For each set of \(n=13\) random variables analytical solutions for 5 design parameters \(x_{1},x_{2},\ldots,x_{5}\) are determined applying Formulas (26) and (35)-(38), and the trial table is obtained by calculating design criteria values. Analyzing the obtained trial table, 25 preliminary solutions are selected as specified in Table 1 and shown in Figure A2 (Appendix C). The criteria values \((c_{1},\ldots,c_{5})\) for the obtained solutions are found to vary within the following limits:  
\[-1.50\leq\xi_{G}\leq\ 1.50\]2 \[-1.50\leq\xi_{G}\leq\ 1.50\]3  
The parameter values for solution in Figure A2a are shown in Tables A1-A3(Appendix C). **Global Search within the Narrowed Search Area. ** Analyzing the results, we study the functionality of the mechanism. The most elusive was meeting criteria \(c_{5}\) (achieving internal swing with trajectory turned inward). Thus, a design criterion \(c_{6}\) was introduced to increase the step height \(h_{i}=\eta_{P_{i}}-\eta^{0}\) defined as  
\[-1.50\leq\xi_{G}\leq\ 1.50\]4  
that has to be maximized. (Note that we have not used \(h=\max h_{i}\) as a design criterion \(c_{6}\) since it does not reflect the foot trajectory that goes below the limit \(\eta=\eta^{0}\) as demonstrated in Figure A2. At the same time, when using the sum, these trajectories will have a negative sign, which decreases the sum). Arranging and cutting the obtained trial table and eliminating solutions with unacceptable criteria values, we are looking for compromise solutions that meet all design criteria, so we clarify new boundaries for design variables. Then, we carry out a new search of the design parameters within the new search area and obtain a new trial table. After several repetitions of this sequence of actions, we came to the following search area specified by the boundaries: \begin{table}
\begin{tabular}{c c c c c c} \hline
**LP\(\tau\)****\(\mathbf{\lambda}\)** & **Figure** & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & \(\mathbf{c_{3}}\) & \(\mathbf{c_{4}}\) & \(\mathbf{c_{5}}\) \\ \hline  
037 & & 0.03713 & \(-\)144,800 & 3,883,024 & 124,053 & 838,340 \\ \hline \end{tabular}
\end{table} 
Table 1: Trial table fragment: criteria values with the limits specified by (46). \[-0.77\leq\xi_{D}\leq-0.05\] \[-0.84\leq\eta_{D}\leq\ 0.05\] \[-0.95\leq\xi_{G}\leq-0.06\] \[-0.91\leq\eta_{G}\leq\ 0.04\] \[0.07\leq r_{AB}\leq\ 0.22\] \[-110^{\circ}\leq\varphi_{0}\leq-9^{\circ}\] \[180^{\circ}\leq\Phi\leq 190^{\circ}\] \[0.20\leq L_{BC}\leq\ 0.75\] \[0.19\leq L_{CD}\leq\ 0.60\] \[-0.77\leq\xi_{D}\leq-0.05\]0 \[-0.77\leq\xi_{D}\leq-0.05\]1 \[-0.77\leq\xi_{D}\leq-0.05\]2 \[-0.77\leq\xi_{D}\leq-0.05\]3  
As a result, we obtained a number of solutions presented in Table 2, having an internal transfer segment of the foot so that in the swing phase, the foot trajectory is turned inward. Part of these solutions are plotted in \[-0.77\leq\xi_{D}\leq-0.05\]4 The solutions in Table 2 are arranged by criterion \(c_{6}\) in descending order.