# Dynamics of a 3-UPS-UPU-S Parallel Mechanism
## 2 A Novel 2-Rotational-DoF Parallel Mechanism (3 UPS-UPU-S) and Its DoF
### Reciprocal Screws of Moving Platform and Its DOF  
To analyze the DoFs of a parallel mechanism with various kinematic chains, firstly, the mechanism should be divided into chains, and then the constraints of each chain acting on the moving platform should be explained [30; 31]. Foremost, the Plucker coordinates of each joint of each chain are carried out based on the absolute coordinate frame as shown in  
Figure 1: Geometry of the 3 UPS-UPU-S parallel mechanism: (a) front view; (b) top view of the parallel mechanism.  
Figure 1a. Therefore, the Plucker coordinates of joints in the UPUchain can be written as follows:  
\[\begin{array}{l}\mathbf{\xi}_{1 IPU}^{1}=\left[\begin{array}{cccccc}1&0&0&0&0&0\\ \end{array}\right]^{T}\\ \mathbf{\xi}_{1 IPU}^{2}=\left[\begin{array}{cccccc}0&1&0&0&0&R\\ \end{array}\right]^{T}\\ \mathbf{\xi}_{1 IPU}^{3}=\left[\begin{array}{cccccc}0&0&0&-\sin\theta&0&\cos \theta\\ \end{array}\right]^{T}\\ \mathbf{\xi}_{1 IPU}^{4}=\left[\begin{array}{cccccc}\cos\theta&0&\sin\theta&0&-R \sin\theta+L&0\\ \end{array}\right]^{T}\\ \mathbf{\xi}_{1 IPU}^{5}=\left[\begin{array}{cccccc}0&1&0&-L\cos\theta&0&R-L \sin\theta\\ \end{array}\right]^{T}\end{array} \tag{6}\]  
where \(L\) is the length of the kinematic chain, and \(\theta\) is angle between the UPU chain and the \(z\)-axis. Therefore, the screws of the chain are now expressed by  
\[\begin{array}{l}\text{S}=\left[\mathbf{\xi}_{UPU}^{1}&\mathbf{\xi}_{IPU}^{2}&\mathbf{ \xi}_{UPU}^{3}&\mathbf{\xi}_{UPU}^{4}&\mathbf{\xi}_{UPU}^{5}\right]\end{array} \tag{7}\]  
According to the reciprocal screw theory, the inverse screw \(\mathbf{\xi}^{r}\) can be obtained through  
\[\begin{array}{l}\mathbf{S}^{T}\text{N}\mathbf{\xi}^{r}=0\end{array} \tag{8}\]  
where \(\text{N}=\begin{bmatrix}0&\mathbf{I}_{3\times 3}\\ \mathbf{I}_{3\times 3}&0\end{bmatrix}\) and \(\mathbf{I}_{3\times 3}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\).  
Therefore, the inverse screw of the UPU chain is  
\[\mathbf{\xi}_{UPU}^{r}=\left[\begin{array}{cccccc}0&1&0&0&0&R-\frac{L}{\sin \theta}\end{array}\right]^{T} \tag{9}\]  
Based on screw theory, \(\mathbf{\xi}_{UPU}^{r}\) can be expressed in a wrench form as \(\mathbf{\xi}_{UPU}^{r}=\begin{bmatrix}\mathbf{F}\\ \mathbf{r}\times\mathbf{F}\end{bmatrix}\) and as shown in Equation (9), \(\mathbf{\xi}_{UPU}^{r}\) is expressed as a force vector parallel to the \(y\)-axis passing through point \((R-\frac{L}{\sin\theta}\quad b\quad 0)\), where \(b\) can be any value.  
Using the same procedure, the inverse screws of the UPS and S chains can be obtained  
\[\mathbf{\xi}_{UPS}^{r}=\begin{bmatrix}0&0&0&0&0&0\end{bmatrix}^{T} \tag{10}\]  
\[\mathbf{\xi}_{S}^{r}=\begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\end{bmatrix}^{T} \tag{11}\]  
Therefore, all the inverse screws provide the terminal constraints to the moving platform:  
\[\begin{array}{l}\mathbf{S}_{P}^{C}=\begin{bmatrix}0&1&0&0\\ 1&0&1&0\\ 0&0&0&1\\ 0&0&0&0\\ R-\frac{L}{\sin\theta}&0&0&0\end{bmatrix}\end{array} \tag{12}\]  
According to the reciprocal screw Equation (8), the free velocity screws of the moving platform \(\mathbf{\xi}_{P}\) demonstrate that  
\[\begin{array}{l}\left(\mathbf{S}_{P}^{C}\right)^{T}\text{N}\mathbf{\xi}_{P}=0\end{array} \tag{13}\]  
Therefore,  
\[\mathbf{\xi}_{P}=\begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0\end{bmatrix}^{T} \tag{14}\]