# Structure Design, Kinematics Analysis, and Effect Evaluation of a Novel Ankle Rehabilitation Robot
## 2 Type Synthesis and Structure Design
### Type Synthesis of the Mechanism  
Ankle rehabilitation mechanism should be universal, and meet the size requirements of different adults' lower limbs and meanwhile satisfy three-dimensional rotation. Therefore, the mechanism can be considered as three rotations around the corresponding axes and one translation degree of freedom along the \(z\) axis. Its movement mode is the rotation about three mutually perpendicular intersecting axes. Therefore, the ankle rehabilitation mechanism should be capable of three-dimensional rotation. In addition, the rotation center should be coincident with the ankle joint center so as to avoid second injury in rehabilitation process. Supposing the movement mode is \(\mathbb{R}^{x}\mathbb{R}^{y}\mathbb{R}^{z}\mathbb{T}^{z}\), according to the movement and constraint relationship of the screw theory [18], the expected number of degrees of freedom is \(M(w,p)\), where \(w\) and \(p\) denote the number and the property of the degree of freedom required. Let the screw system of the mechanism be \(\left\{\mathcal{S}_{M}|\left(\mathcal{S}_{1}^{m},\mathcal{S}_{2}^{m},\cdots, \mathcal{S}_{w}^{m}\right)\right.\). This means that if \(\mathcal{S}_{M}\) is known, \(M(w,p)\) can be determined. By solving the reciprocal screw of \(\mathcal{S}_{M}\), the reciprocal screws of the screw system of the mechanism can be determined. The reciprocal screws of the screw system of the limbs \(\left\{\mathcal{S}_{P}^{jl}|\left(\mathcal{S}_{j1}^{lr},\mathcal{S}_{j2}^{lr},\cdots,\mathcal{S}_{ji}^{lr}\right)\right.\) (where \(i\leq 6-w\) is the number of the reciprocal screws of the j-th jth limb) are a subset of those of the mechanism, and the relation can be described as follows. \[ \begin{split}& M(w,p)\Leftrightarrow\left\{\mathcal{S}_{M} \middle|\mathcal{S}_{1}^{m},\mathcal{S}_{2}^{m},\cdots,\mathcal{S}_{w}^{m} \right\}\Leftrightarrow\\ &\left\{\mathcal{S}_{r}\middle|\mathcal{S}_{1}^{r},\mathcal{S}_{2 }^{r},\cdots,\mathcal{S}_{6-w}^{r}\right\}\Leftrightarrow\\ &\left\{\mathcal{S}_{1}^{r}\middle|\mathcal{S}_{1}^{r},\mathcal{S} _{2}^{r},\cdots,\mathcal{S}_{ji}^{r}\right\}\Leftrightarrow\\ &\left\{\mathcal{S}_{lj}^{m}\middle|\mathcal{S}_{1}^{m},\mathcal{ S}_{2}^{m},\cdots,\mathcal{S}_{j(6-i)}^{m}\right\}\Leftrightarrow\left\{\text{PM}s \right\}\end{split}  \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline
**Rotation Direction** & **Movement Model** & **Movement Range** \\ \hline \multirow{2}{*}{Around \(X\)-axis} & Plantar flexion & 40–45\({}^{\circ}\) \\  & Dorsiflexion & 25–30\({}^{\circ}\) \\ \multirow{2}{*}{Around \(Y\)-axis} & Abduction & 15\({}^{\circ}\) \\  & Adduction & 35–40\({}^{\circ}\) \\ \multirow{2}{*}{Around \(Z\)-axis} & Eversion & 25–30\({}^{\circ}\) \\  & Inversion & 25–30\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The movement ranges of human ankle joint. Figure 1: The movements of the ankle joint. The fixed coordinate system {_O-xyz_} is fixedly connected to the fixed platform, and its coordinate origin is located at the geometric center \(O\) of the fixed platform. The moving coordinate system {_P-uvw_} is fixedly connected to the moving platform, and its coordinate origin is located at the geometric center \(P\) of the moving platform. \(y\) axis and \(v\) axis are parallel to \(A\)1\(A\)2 and \(B\)1\(B\)2, respectively; the \(x\) axis and \(u\) axis coincide with _OA_3 and _PB_3, respectively; the direction of \(w\) axis and \(z\) axis is according to right hand rule, as shown in Figure 3.  
The orientation of the moving platform {_P-uvw_} relative to fixed coordinate system {_O-xyz_} can be expressed as rotation matrix \(R\). We define three rotation angles _\(\alpha\)_, \(\beta\)_, \(\gamma\) as roll, pitch, and yaw from the moving platform to the fixed platform. Since the 2-CRS-PU parallel mechanism does not rotate around the \(w\) axis, \(\gamma=0^{\circ}\).