# Structure Design, Kinematics Analysis, and Effect Evaluation of a Novel Ankle Rehabilitation Robot
## 2 Type Synthesis and Structure Design
### Type Synthesis of the Mechanism
The rotation matrix R of the moving coordinate system {_P-uvw_} relative to the fixed platform coordinate system {_O-xyz_} can be simplified as  
\[{}^{o}\textbf{R}_{p}=\textbf{Rot}_{(u)}(\alpha)\textbf{Rot}_{(v)}(\beta) \tag{3}\]  
The rotation matrix contains sine and cosine functions of the variable rotation angles _\(\alpha\)_, \(\beta\)_, and \(\gamma\). Referring to Figure 3, a vector closure equation for each limb \(i\) can be written as  
\[\textbf{a}_{i}+\textbf{d}_{i}+\textbf{r}_{i}+\textbf{q}_{i}=\textbf{p}+\textbf {b}_{i}\ (i=1,2) \tag{4}\]  
where \(\textbf{a}_{i}\), \(\textbf{p}\) and \(\textbf{b}_{i}\) are the position vectors of the fixed and moving platform expressed in the fixed reference frame; \(\textbf{q}_{i}\), \(r_{i}\) and \(\textbf{q}_{i}\) are the _i_-th limb vectors. The third limb satisfied the constraint  
\[\textbf{a}_{3}+\textbf{d}_{3}=\textbf{p}+\textbf{b}_{3} \tag{5}\]  
In the plane of the _i_-th (_i_ = 1, 2) limb, vector \(\textbf{m}_{i}\) is always perpendicular to vector \(\textbf{n}_{i}\). Therefore,  
\[\textbf{m}_{i}^{\textbf{T}}\cdot\textbf{n}_{i}=0 \tag{6}\]  
where \(\textbf{m}_{i}\) is axial vector of the _i_-th rotation pair; \(\textbf{n}_{i}\) is the direction vector of \(B_{i}\) to \(A_{i}\). Figure 3: 2-CRS+PU parallel mechanism. (**a**) Coordinate system. (**b**) Geometric relationship of the moving and fixed platform. The position parameters of the fixed platform point \(A_{i}\) in the coordinate system {\(O\)-\(xyz\)} can be expressed as  
\[{}^{o}A_{1}=(-\sqrt{3}a,-a,0),\ ^{o}A_{2}=(\sqrt{3}a,-a,0),\ ^{o}A_{3}=(0,2a,0)\]  
The position parameters of the moving platform point \(B_{i}\) in the coordinate system {\(P\)-\(uvw\)} can be expressed as  
\[{}^{p}B_{1}=(-\sqrt{3}b,\ -b,\ 0),\ ^{p}B_{2}=(\sqrt{3}b,\ -b,\ 0),\ ^{p}B_{3}=(0,\ 2b,\ 0)\]  
According to the coordinate transformation formula, there is  
Figure 1:0  
where \({}^{o}\textbf{{B}}_{i}\) is point \(B_{i}\)(\(i\) = 1, 2, 3) position in the coordinate system {\(O\)-\(xyz\)}, and \({}^{o}\textbf{{P}}_{p}\) is the position vector from \(P\) to \(O\). According to the closed-vector relation, there is  
\[\left\{ \begin{array}{l}\textbf{{a}}_{i}+\textbf{{d}}_{i}+\textbf{{q}}_{i}+ \textbf{{r}}_{i}+\textbf{{s}}_{i}=\textbf{{p}}+\textbf{{b}}_{i}\\ \textbf{{a}}_{3}+\textbf{{q}}_{3}=\textbf{{p}}+\textbf{{b}}_{3}\end{array} \right. \tag{8}\]  
From the constraints relationship,  
\[\textbf{{m}}_{3}^{\textbf{T}}\textbf{{q}}_{3}=0 \tag{9}\]  
The geometric position relationship of the moving platform and fixed platform moving satisfied the constraint relationship as follows:  
\[\left\{ \begin{array}{l}\textbf{{m}}_{i}=({}^{o}\textbf{{B}}_{i}-{}^{o} \textbf{{B}}_{i+1})\times\textbf{{A}}_{i}\\ \textbf{{n}}_{i}={}^{o}\textbf{{B}}_{i}-{}^{o}\textbf{{A}}_{i}\\ \textbf{{m}}_{3}=\left[\begin{array}{ccc}1&0&0\end{array}\right]^{\text{T}} \qquad(i=1,2)\\ \textbf{{q}}_{3}={}^{o}\textbf{{B}}_{3}-{}^{o}\textbf{{A}}_{3}\end{array} \right. \tag{10}\]  
Bring Equation (10) into Equations (8) and (9), and then there is  
\[\left\{\begin{array}{l}\textbf{{d}}_{1}=\left(\sqrt{3}a+2b\sin\beta\sin \alpha-\sqrt{3}b\cos\beta+b\sin\beta\sin\alpha,\ a+y+b\cos\alpha,\ z+\sqrt{3}b\sin\beta+b\cos\beta\sin \alpha\right)^{\text{T}}\\ \textbf{{d}}_{2}=(\ -\sqrt{3}a+2b\sin\beta\sin\alpha+\sqrt{3}b\cos\beta+b\sin\beta \sin\alpha,\ a+y+b\cos\alpha,\ z-\sqrt{3}b\sin\beta+b\cos\beta\sin\alpha)^{ \text{T}}\\ \textbf{{d}}_{3}=\left(-2a+y-2b\cos\alpha,\ z-2b\cos\beta\sin\alpha\right)^{ \text{T}}\\ \text{where}\\ \qquad\qquad\qquad\qquad\qquad y=\frac{\left(-\sqrt{3}b\cos\beta+3b\sin\beta \sin\alpha\right)\left(\sqrt{3}a-\sqrt{3}b\cos\beta+b\sin\beta\sin\alpha\right) }{3b\cos\alpha}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{2b\sin\beta\sin \alpha+\sqrt{3}b\sin\beta+b\cos\beta\sin\alpha}{3b\cos\alpha}\\ \left\{