# Multicriteria Optimization of Lower Limb Exoskeleton Mechanism
## 3 The Method of Kinematic Synthesis
### Analytical Solution on 5 Synthesis Parameters
Thereby, due to finding 5 design variables \(\vec{x}=A^{-1}(\vec{P})\vec{b}(\vec{P})\), (that found from the kinematic requirement--generating the desired foot path, \(\vec{P}\) is the vector of 13 design parameters), the dimensionality of the optimization problem is reduced to 13 design parameters instead of 18 unknowns. However, solving Equation (29) does not provide an exact generation of the desired foot path, but supplies an approximate solution; thus, we need to improve the accuracy by using the rest 13 \"non-linear\" parameters. So, we have to solve the following optimization task:  
\[S^{0}(\vec{P})=\min_{\vec{x}}S=S(\vec{P},\vec{x}(\vec{P}))=S(\vec{P},A^{-1}( \vec{P},\vec{b}(\vec{P})))\to min. \tag{39}\]  
However, achieving the best accuracy conflicts with some additional design requirements, such as force transmission, matching leg anatomy, etc. As the preliminary design had shown, improving the accuracy leads to worsened force transmission and decreased step height (small swing phase height limits overcoming large obstacles). Moreover, the swing phase trajectory tends to go beneath the ground level (the trajectory becomes external relative to the mechanism frame), etc. For these reasons, multiple design criteria must be considered, and a multi-criteria optimization task has to be solved. For these purposes Sobol and Statnikov method is applied [35], based on  
* Generating the trial table with a great number of random LP-\(\tau\) sequences of design parameters evenly distributed within the specified search area and
* Iterative approaching the compromise solutions, truncating the trial table by strengthening (\"narrowing\") restrictions on criteria. The trajectory of the foot center \(P\) called a step cycle, consists of two phases: the \"support phase\" and the \"transfer phase\" (swing). During the support phase, the foot center \(P\) should trace a horizontal straight line. The _main criterion_ of synthesis is the accuracy of straight-line generation during the support phase, which has to be minimized:  
\[c_{1}=\max_{i=\overline{1,N}}|\eta_{P}^{0}-\eta_{P_{i}}|. \tag{40}\]
* The chassis height \(H=-x_{4}\) has to be minimized or (which is the same condition) \(\eta^{0}=-H\) to be maximized \[c_{2}=x_{4}.\] (41)
* In order to achieve the best force transmission, the worst transference angle has to be maximized \[c_{3}=\min_{0\leq\eta\leq 2\Pi}(\mu_{\text{BCD}},\mu_{\text{FFG}}).\] (42) where \[x_{2}=\frac{b_{6}-mx_{3}kx_{4}-k_{\beta}x_{5}}{N} \tag{36}\]0 (43) \[x_{2}=\frac{b_{6}-mx_{3}kx_{4}-k_{\beta}x_{5}}{N} \tag{36}\]1  
\(L_{BC},L_{CD},L_{FG},L_{EF}\) are lengths of corresponding links, \(|BD|,|EG|\) are distances between the centers of corresponding joints. * In order to match human anatomy, the hip-to-shin relation \(L_{FP}/L_{FG}\) (the ratio of the thigh to the lower leg) has to be around one (\(c_{4}=1\)) \[x_{2}=\frac{b_{6}-mx_{3}kx_{4}-k_{\beta}x_{5}}{N} \tag{36}\]2 (44)For external transfer of the foot center \(P\), the following penalty function \(c_{5}\) has to be minimized  
\[x_{2}=\frac{b_{6}-mx_{3}kx_{4}-k_{\beta}x_{5}}{N} \tag{36}\]3  
The meaning of this penalty function is as follows. If \(\eta_{P_{i}}<\eta^{0}\), then the penalty function \(c_{5}\) will be increased to positive value \(\max((\eta^{0}-\eta_{P_{i}},0)\). Otherwise, if \(\eta_{P_{i}}\leq\eta^{0}\), then \(\max((\eta^{0}-\eta_{P_{i}}),0)=0\), thus nothing will be added to the function \(c_{5}\).