## 4 Performance Analysis  
### Performance Index  
The main movements of the ankle joint are plantar flexion/dorsiflexion andeversion/inversion movements. The range of motions of the ankle joint is shown in Table 2[8]. The initial dimension parameters of the mechanism are as follows: \(a=180\) mm, \(b=120\) mm, \(c=80\) mm, \(d=100\) mm, \(l_{\text{a}}=180\) mm and \(l_{\text{b}}=180\) mm. In the 1T2R motion mode, the mechanism carries out the training of ankle joint activity. According to Table 2, the search space of the mechanism is \(\alpha\in(-40^{\circ},\,24^{\circ})\), \(\beta\in(-18^{\circ},\,18^{\circ})\), and \(Z\in(250\) mm, \(350\) mm). In the 2T1R motion mode, the mechanism carries out ankle joint proprioception training. According to Table 2, the search space of the mechanism is \(\alpha\in(-18^{\circ},\,18^{\circ})\), \(X\in(-75\) mm, \(75\) mm), and \(Z\in(194\) mm, \(320\) mm).  
It is assumed that the number of points in the search space is \(W\). While solving the workspace of the mechanism, the number of points in the reachable workspace is \(W_{i}\). Then, the performance index \(W_{V}\) of the reachable workspace is as follows:  
\[W_{V}=\frac{W_{i}}{W} \tag{28}\]  
Dexterity is an important index to evaluate the kinematic performance of the mechanism. The reciprocal of the condition number of the Jacobian matrix is used to analyze the dexterity index of the mechanism. The range of the condition number of the Jacobian matrix is \(1\) to \(\infty\); thus, the range of the dexterity index is \(0\) to \(1\)[31]. The greater the dexterity index, the better the motion transmission performance of the mechanism.  
To evaluate the dexterity of the mechanism in the 1T2R motion mode, the mechanism Jacobian matrix \(\mathbf{J}\) is divided into the translational Jacobian matrix \(\mathbf{J}_{y}\) and the rotational  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Type of Motion** & **Range of Motions** \\ \hline Dorsiflexion & \(24.68\pm 3.25^{\circ}\) \\ Plantar flexion & \(40.92\pm 4.32^{\circ}\) \\ Inversion & \(16.29\pm 3.88^{\circ}\) \\ Eversion & \(15.87\pm 4.45^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Allowable range of motions of ankle joint.  
Figure 4: Singularity configuration of the mechanism in the 2T1R motion mode.  
Jacobian matrix \(I_{\omega}\), and then the translational dexterity and the rotational dexterity of the mechanism are analyzed, respectively, as follows:  
\[L_{Clv}=\frac{1}{\text{cond}(I_{v})}=\frac{\sigma_{\text{min}}(I_{v})}{\sigma_{ \text{max}}(I_{v})} \tag{29}\]  
\[L_{Cl\omega}=\frac{1}{\text{cond}(I_{\omega})}=\frac{\sigma_{\text{min}}(I_{ \omega})}{\sigma_{\text{max}}(I_{\omega})} \tag{30}\]  
The level of the load capacity is related to whether the mechanism can achieve the expected load goal. The relationship between the force or moment vector \(F\) acting on the moving platform and the driving force \(f\) of the driving pair can be expressed as  
\[F=Gf \tag{31}\]  
where the matrix \(G\) is the force Jacobian matrix of the mechanism, with \(G=J^{\text{T}}\).  
\(\left\|F_{\text{max}}\right\|\) represents the maximum load capacity at the limit of the mechanism when the modulus of the driving force vector \(f\) is unit 1. In the 2T1R motion mode, \(\left\|F_{\text{max}}\right\|\) is used to evaluate the load capacity performance of the mechanism. The larger the value, the better the load capacity performance of the mechanism. Its expression is as follows:  
\[\left\|F_{\text{max}}\right\|=\sqrt{\lambda_{\text{max}}\Big{(}G^{\text{T}}G \Big{)}} \tag{32}\]