# Multicriteria Optimization of Lower Limb Exoskeleton Mechanism
## 2 Lower Limb Exoskeleton Mechanism Structure and Kinematic Synthesis Problem Formulation  
Human walking can be analyzed in three planes: the sagittal plane, coronal plane, and transverse plane [6; 12]. Among these, the sagittal plane motion is predominant. The designed leg exoskeleton aids in hip and knee flexion/extension movements while the wearer  
Figure 1: Prof. [PERSON]’s Foot-Path Diagram. stays in place. This practice focuses on motions within the sagittal plane. Consequently, a planar linkage with angular outputs can effectively facilitate these movements. The kinematic scheme of the six-bar Stephenson III type lower-limb exoskeleton mechanism is illustrated in Figure 2a with the input link \(AB\) and the foot \(P\) mounted on coupler \(EF\). Rotation of the crank \(AB\) is described by \(\varphi_{i}\),  
\[\varphi_{i}=\varphi_{0}+\Delta\Phi\frac{i-1}{N-1},\quad i=\overline{1,N}. \tag{1}\]  
\[\varphi_{i}=\varphi_{0}+\Delta\Phi\frac{i-1}{N-1},\quad i=\overline{1,N}. \tag{2}\]  
Here, \(\varphi_{0}\) is an initial angular position of the crank \(AB\) with respect to the horizontal axis. \(\Delta\Phi\) is the maximum rotation angle, \(\Delta\Phi>\pi\), in order to supply the overlap of support phases of alternating two legs. **Kinematic analysis** of the mechanism is provided in Appendix A. The rotation of the crank \(GF\) is represented by the angle \(\theta_{i}\), \(i=\overline{1,N}\). \(\vec{r}_{P}[x_{P},y_{P}]\) is a local radius vector, rigidly associated with the moving coordinate system \(Exy\); and \(\vec{R}_{P_{i}}[\xi_{P_{i}},\eta_{P_{i}}]\) and \(\vec{R}_{E_{i}}[\xi_{E_{i}},\eta_{E_{i}}]\) signify the absolute positions of the foot center \(P_{i}\) and the point \(E\) respectively. Then the relationship for \(\vec{R}_{P_{i}}\) can be expressed as:  
\[\vec{R}_{P_{i}}=\vec{R}_{E_{i}}+\Gamma(\theta_{i})\vec{r}_{P}= \begin{bmatrix} \xi_{E_{i}}\\ \eta_{E_{i}}\end{bmatrix} + \begin{bmatrix}x_{P}\cos\theta_{i}-y_{P}\sin\theta_ {i}\\ x_{P}\sin\theta_{i}+y_{P}\cos\theta_{i}\end{bmatrix} , \tag{3}\]  
Given the desired absolute coordinates of the foot \(P_{i}^{*}\) (Figure 2b):  
\[\xi_{P_{i}}^{*}=\xi^{0}+L\frac{i-1}{N-1},i=\overline{1,N};\]  
\[\eta_{P_{i}}^{*}=\eta^{0},\]  
with stride \(L\) and desired start position \(P_{1}^{*}(\xi^{0},\eta^{0})\), the synthesis condition is stated as follows:  
\[\vec{\Delta}_{i}\equiv\vec{R}_{P_{i}}-\vec{R}_{P_{i}}^{*}=\vec{0},\quad i= \overline{1,N}. \tag{4}\]  
Figure 2: Kinematic scheme of a lower-limb exoskeleton mechanism (**a**) and illustrative aid for the synthesis task (**b**). Considering the constraint Equation (3) one can write in scalar form as follows:  
\[\delta_{i}^{\xi} =x_{p}\cos\theta_{i}-y_{p}\sin\theta_{i}-\xi^{0}+\xi_{E_{i}}-L\frac{ i-1}{N-1}=0 \tag{5}\] \[\delta_{i}^{\eta} =x_{p}\sin\theta_{i}+y_{p}\cos\theta_{i}-\eta^{0}+\eta_{E_{i}}=0\]  
Given the mechanism link lengths except \((x_{p},y_{p})\), the synthesis task consists in determining the parameters \(x_{p},y_{p},\xi^{0},\eta^{0},L\) that satisfy these \(2N\) constraints approximately (since in general case we cannot meet these \(2N\) Equation (5) with 5 unknowns exactly).