# Multicriteria Optimization of Lower Limb Exoskeleton Mechanism
## 3 The Method of Kinematic Synthesis
### Analytical Solution on 5 Synthesis Parameters  
Let us explore a scenario where \(L=x_{5}\) is an additional variable subject to optimization. Referring to (16a)-(16d), from the condition \(\frac{\partial\xi}{\partial x}=\vec{0}\) we obtain  
\[Nx_{1}-kx_{3}-mx_{4}-\sum_{i=1}^{N}\biggl{(}\frac{i-1}{N-1}\cos\theta_{i} \biggr{)}x_{5}=b_{5}, \tag{22}\]  
where  
\[b_{5}=\sum_{i=1}^{N}\bigl{(}-\xi_{P_{i}}\cdot\cos\theta_{i}-\eta_{P_{i}}\cdot \sin\theta_{i}\bigr{)};\]  
\[Nx_{2}+mx_{3}-kx_{4}+\sum_{i=1}^{N}\biggl{(}\frac{i-1}{N-1}\sin\theta_{i} \biggr{)}x_{5}=b_{6}, \tag{23}\]  
where  
\[b_{6}=\sum_{i=1}^{N}\bigl{(}\xi_{P_{i}}\cdot\sin\theta_{i}-\eta_{P_{i}}\cdot \cos\theta_{i}\bigr{)};\]  
\[kx_{1}-mx_{2}-Nx_{3}-\sum_{i=1}^{N}\biggl{(}\frac{i-1}{N-1}\biggr{)}x_{5}=b_{7}, \tag{24}\]  
where  
\[b_{7}=-\sum_{i=1}^{N}\xi_{P_{i}};\]  
\[mx_{1}+kx_{2}-Nx_{4}=b_{8}, \tag{25}\]  
where  
\[b_{8}=-\sum_{i=1}^{N}\eta_{P_{i}}.=b_{4}.\]  
The last equation is derived from (10), \(j=5\):  
\[ \begin{split}&\sum\biggl{(}\frac{i-1}{N-1}\cos\theta_{i}\biggr{)}x_{ 1}-\sum\biggl{(}\frac{i-1}{N-1}\sin\theta_{i}\biggr{)}x_{2}\\ &-\sum\biggl{(}\frac{i-1}{N-1}\biggr{)}x_{3}+\sum\biggl{(}\frac {i-1}{N-1}\xi_{P_{i}}\biggr{)}x_{4}+\sum\biggl{(}\frac{i-1}{N-1}\biggr{)}^{2} x_{5}=0.\end{split}  \tag{26}\]  
Finally, taking into account  
\[\sum\biggl{(}\frac{i-1}{N-1}\biggr{)}=N/2 \tag{27}\]  
and  
\[\sum\biggl{(}\frac{i-1}{N-1}\biggr{)}^{2}=\frac{N(2N-1)}{6(N-1)}, \tag{28}\]  
we obtain the following system of linear equations  
\[A\vec{x}=\vec{b}, \tag{29}\]where  
\[A= \begin{bmatrix}N&0&-k&-m&-k_{\alpha}\\ 0&N&m&-k&k_{\beta}\\ k&-m&-N&0&-N/2\\ m&k&0&-N&0\\ k_{\alpha}&-k_{\beta}&-N/2&0&-\frac{N(2N-1)}{6(N-1)}\end{bmatrix}  \tag{30}\]  
\[\vec{b}= \begin{bmatrix}b_{5}&b_{6}&b_{7}&b_{8}&-k_{\mu}\end{bmatrix} ^{T} \tag{31}\]  
and  
\[k_{\alpha}:=\sum\biggl{(}\frac{i-1}{N-1}\cos\theta_{i}\biggr{)} \tag{32}\]  
\[k_{\beta}:=\sum\biggl{(}\frac{i-1}{N-1}\sin\theta_{i}\biggr{)} \tag{33}\]  
\[k_{\mu}:=\sum\biggl{(}\frac{i-1}{N-1}\xi_{P_{i}}\biggr{)}. \tag{34}\]  
The Hessian \(H_{S}=\frac{d^{2}S}{dx^{2}}\), and it is non-negative definite (for the proof, refer to Appendix B). \[H_{S}= \begin{bmatrix}N&0&-k&-m&-k_{\alpha}\\ 0&N&m&-k&k_{\beta}\\ -k&m&N&0&N/2\\ -m&-k&0&N&0\\ -k_{\alpha}&k_{\beta}&N/2&0&\frac{N(2N-1)}{6(N-1)}\end{bmatrix} .\]  
Therefore, the last condition supplies the minimum of the function \(S\), if \(\det H_{S}\
eq 0\). For analytical solution in this case we refer to Equations (16a)-(16d):  
\[x_{1}=\frac{b_{5}+kx_{3}+mx_{4}+k_{\alpha}x_{5}}{N} \tag{35}\]  
\[x_{2}=\frac{b_{6}-mx_{3}kx_{4}-k_{\beta}x_{5}}{N} \tag{36}\]  
\[x_{3}=\frac{Nb_{7}+\frac{N^{2}}{2}x_{5}-kb_{5}-kk_{\alpha}x_{5}+mb_{6}-mk_{ \beta}x_{5}}{k^{2}+m^{2}-N^{2}} \tag{37}\]  
\[x_{4}=\frac{Nb_{8}-mb_{5}-mk_{\alpha}x_{5}-kb_{6}+kk_{\beta}x_{5}}{k^{2}+m^{2} -N^{2}}, \tag{38}\]  
assuming \(k^{2}+m^{2}\
eq N^{2}\); and \(x_{5}\) can be found directly by substituting (35)-(38) into (26). **4. Additional Synthesis Conditions and Multiple Optimization Criteria**  
Note that when determining the solution \(x\) according to (29), the matrix \(A\) and the column vector \(b\) depend on 13 design variables \(p_{1},\ldots,p_{13}\) (mechanism link dimensions). \[p_{1}:=\xi_{D},p_{2}:=\eta_{D},p_{3}:=\xi_{G},p_{4}:=\eta_{G},p_{5}:=r_{AB},p_ {6}:=\varphi_{0},p_{7}:=\Phi,\]  
\[p_{8}:=L_{BC},p_{9}:=L_{CD},p_{10}:=x_{E},p_{11}:=y_{E},p_{12}:=L_{EF},p_{13}: =L_{FG},\]  
where  
* \(\xi_{D},\eta_{D},\xi_{G},\eta_{G}\) are the absolute coordinates of frame joints \(D\) and \(G\);
* \(r_{AB}\) is the length of crank \(AB\);
* \(x_{E},y_{E}\) are the local coordinates of joint \(E\) relative to the \(Bx_{2}y_{2}\) moving coordinate system, with the \(Bx_{2}\) axis along line \(BC\). This is because \(\theta_{i},\xi_{E_{i}},\eta_{E_{i}}\) in the source Equation (5) depend on \(p_{1},\ldots,p_{13}\).