## 3 Exoskeleton Manipulator Structure for Wrist Rehabilitation
### Inverse Kinematics  
The 3 UPS/S parallel manipulator is composed of three expandable links, 1, 1, and 1, as shown in Figure 3. The coordinate axes are affixed to the platforms of the exoskeleton for analytical purposes. Joints B1, B2, and B3 are kept 120\({}^{\circ}\) apart, as well as A1, A2, and A3, therefore, these joints appear at the vertices of an equilateral triangle. The frames {b} and {m} are fixed to the base and moving platforms. Figure 3: Exoskeleton kinematics. To calculate the rotational motion of the 3 UPS/S robot, the Euler angle representations were used. The designed parallel robot has three rotational DOFs. This robot is expected to rotate about the centre of the top platform. Thus, the centre of the top platform is at a fixed distance from the base platform or the length of the central leg of the robot is fixed. Thus, the \(4\times 4\) homogeneous transformation matrix in Equation (1), which includes the orientation and position, can be used to show this transformation [3]. \[\mathrm{\prescript{m}{b}[T]=\left[ \begin{array}{ccc}c\phi c\phi c\phi c\phi s \Psi-s\phi c\Psi&c\phi s\phi c\Psi+s\phi s\Psi&0\\ s\phi c\phi&s\phi s\psi+c\phi c\Psi&s\phi s\phi c\Psi-c\phi s\Psi&0\\ -s\phi&c\phi s\Psi&c\phi c\Psi&1\\ 0&0&0&1\end{array} \right] \tag{1}\]  
During flexion and extension motions, the top platform of the robot is rotated only by the \(\Psi\) angle about the X-axis. Then, during ulnar and radial deviation motions, the top platform of the robot is rotated only by the \(\theta\) angle about the Y-axis. Since this robot is used only to perform said motions, there is no need of rotation about the \(Z\)-axis; hence, the angle \(\phi\) about the \(Z\)-axis is always zero. The fixed length of the central leg of the robot is denoted by \"1\" (l = 171.43 mm). The radius of base and top platforms is r1 = 82 mm and r2 = 45.5 mm, respectively. The coordinate frame {b} (Figure 3) is fixed to the base platform's centre, with the \(\mathrm{X_{b}}\) and \(\mathrm{Y_{b}}\) axes located on its plane. The frame {m} is attached to the moving platform, where the \(\mathrm{X_{m}}\) and \(\mathrm{Y_{m}}\) axes rest on the moving platform plane. \(Z\)-axes of both frames are normal to the platform planes. The mechanism has three DOFs. As a result, the moving platform has three rotational DOFs. The symbol \"l,\" indicates leg length. Equation (2) provides the coordinates of the universal joints with respect to the frame {b}. \[\overrightarrow{\mathrm{O_{b}A_{1}}} =(-\mathrm{r_{1}},0,0)^{\mathrm{T}}\] \[\overrightarrow{\mathrm{O_{b}A_{2}}} =(\mathrm{r_{1}}\cos 60,\mathrm{r_{1}}\sin 60,0)^{\mathrm{T}}\] \[\overrightarrow{\mathrm{O_{b}A_{3}}} =(\mathrm{r_{1}}\cos 60,-\mathrm{r_{1}}\sin 60,0)^{\mathrm{T}} \tag{2}\]  
The coordinates of the centres of spherical joints with respect to frame {m} are given in Equation (3):  
\[\overrightarrow{\mathrm{O_{m}B_{1}}} =(-\mathrm{r_{2}},0,0)^{\mathrm{T}}\] \[\overrightarrow{\mathrm{O_{m}B_{2}}} =(\mathrm{r_{2}}\cos 60,\mathrm{r_{2}}\sin 60,0)^{\mathrm{T}}\] \[\overrightarrow{\mathrm{O_{m}B_{3}}} =(\mathrm{r_{2}}\cos 60,-\mathrm{r_{2}}\sin 60,0)^{\mathrm{T}} \tag{3}\]  
The spherical joint's position vector with respect to base frame {b} was obtained using Equation (4):  
\[\left[\overrightarrow{\mathrm{O_{b}B_{i}}}{1}\right]=\mathrm{\prescript{m}{b}[ T]\left[\overrightarrow{\mathrm{O_{m}B_{i}}}{1}\right]} \tag{4}\]  
The leg lengths or the actuation needed to rotate the moving platform was found out using Equation (5):  
\[\mathrm{l_{i}}=\left\|\overrightarrow{\mathrm{O_{b}A_{i}}}-\overrightarrow{ \mathrm{O_{b}B_{i}}}{1}\right\|\ i=1,2,3 \tag{5}\]  
For a serial link robot, solving the forward kinematics is easy and the analytical form exists. However, for a parallel link robot, solving the forward kinematics is very complex and the analytical form may not exist. For a serial link robot, the inverse kinematics is typically complex and the analytical form may not exist; however, for a parallel robot, the inverse kinematics is easy to determine and the analytical form exists.