# Dynamics of a 3-UPS-UPU-S Parallel Mechanism
## 3 The Velocity and Displacement Analysis of the 3 UPS-UPU-S Parallel Mechanism  
This section gives the kinematic analysis of the 3 UPS-UPU-S. As shown in Figure 2b, with the definition of the twist matrix, the kinematic screw equation of a multi-rigid-body system can be obtained through  
\[\mathbf{S}\mathbf{\omega}=\mathbf{V} \tag{15}\]  
with  
\[\mathbf{S}=\text{diag}\Big{[}\Big{[}\mathbf{S}^{1}\Big{]}_{6\times 5}\quad\Big{[}\mathbf{S} ^{2}\Big{]}_{6\times 6}\quad\cdots\quad\Big{[}\mathbf{S}^{5}\Big{]}_{6\times 3} \Big{]}_{30\times 26} \tag{16}\]  
and  
\[\mathbf{\omega}=\Big{[}\mathbf{\omega}^{1}\Big{]}_{5\times 1}^{T}\quad\big{[}\mathbf{ \omega}^{2}\big{]}_{6\times 1}^{T}\quad\cdots\quad\big{[}\mathbf{\omega}^{5}\big{]}_{3 \times 1}^{T}\Big{]}_{26\times 1}^{T} \tag{17}\]  
where \(\mathbf{\omega}^{N}\) is the relative angular velocity vector of each joint of each kinematic chain \(\mathit{loop}(N)\). \(\mathbf{S}^{N}\) in Equation (16) is the unit screw matrix of the single kinematic chain \(\mathit{loop}(N)\), and \(\text{diag}[]\) means a diagonal matrix with \(\mathbf{S}^{1},\mathbf{S}^{2},\cdots,\mathbf{S}^{5}\). As shown in Figure 2a, according to Equation (5), the kinematic unit screw matrix \(\mathbf{S}^{1}\) of the U\(\underline{\text{UPU}}\) chain can be expressed as  
\[\mathbf{S}^{1}= \begin{bmatrix}\mathbf{e}_{1}^{1}&\mathbf{e}_{2}^{1}&\mathbf{0}&\mathbf{e}_{4}^{1} &\mathbf{e}_{5}^{1}\\ \mathbf{r}_{A}^{1}\times\mathbf{e}_{1}^{1}&\mathbf{r}_{A}^{1}\times\mathbf{e}_{2}^{1}&\mathbf{e}_{ 3}^{1}&\mathbf{r}_{C}^{1}\times\mathbf{e}_{4}^{1}&\mathbf{r}_{C}^{1}\times\mathbf{e}_{5}^{1} \end{bmatrix}  \tag{18}\]  
where \( \begin{bmatrix}\mathbf{0}\\ \mathbf{e}_{3}^{1}\end{bmatrix} \) gives the unit velocity screw of a prismatic joint. In a similar way, as shown in Figure 2c, the kinematic screw matrix \(\mathbf{S}^{K}\) of the \(K\)th UPS chain is  
\[\mathbf{S}^{K}= \begin{bmatrix}\mathbf{e}_{1}^{K}&\mathbf{e}_{2}^{K}&\mathbf{0}&\mathbf{e}_{4}^{K} &\mathbf{e}_{5}^{K}&\mathbf{e}_{6}^{K}\\ \mathbf{r}_{A}^{K}\times\mathbf{e}_{1}^{K}&\mathbf{r}_{A}^{K}\times\mathbf{e}_{2}^{K}&\mathbf{e}_{ 3}^{K}&\mathbf{r}_{C}^{K}\times\mathbf{e}_{4}^{K}&\mathbf{r}_{C}^{K}\times\mathbf{e}_{5}^{K}&\bm {r}_{C}^{K}\times\mathbf{e}_{6}^{K}\end{bmatrix} ,(K=2,3,4) \tag{19}\]  
\[\mathbf{S}^{5}= \begin{bmatrix}\mathbf{e}_{1}^{5}&\mathbf{e}_{2}^{5}&\mathbf{e}_{3}^{5}\\ \mathbf{r}_{C}^{5}\times\mathbf{e}_{1}^{5}&\mathbf{r}_{C}^{5}\times\mathbf{e}_{2}^{5}&\mathbf{r}_{ C}^{5}\times\mathbf{e}_{3}^{5}\end{bmatrix}  \tag{20}\]  
Figure 2: Kinematics of the 3 UPS-UPU-S parallel mechanism: (**a**) kinematics of the UPS chain; (**b**) kinematics of the S chain; (**c**) kinematics of the U\(\underline{\text{UPU}}\) chain. The velocity matrix in Equation (15) can be expressed as  
\[\mathbf{V}=\left[ \begin{bmatrix}C\mathbf{V}^{1}\end{bmatrix} _{6\times 1}^{T}& \begin{bmatrix}C \mathbf{V}^{2}\end{bmatrix} _{6\times 1}^{T}&\cdots& \begin{bmatrix}C\mathbf{V}^{5} \end{bmatrix} _{6\times 1}^{T}\end{bmatrix}_{30\times 1}^{T} \tag{21}\]  
where \(\overset{C}{0}\mathbf{V}^{K}\) contains all the absolute velocity screw of the joints in \(K\)th kinematic chain. In addition, the absolute velocity in Equation (19) can be calculated through  
\[\overset{C}{0}\mathbf{V}^{K}=\overset{P}{0}\mathbf{V}-\overset{P}{C}\mathbf{V}^{K} \tag{22}\]  
where \(\overset{P}{0}\mathbf{V}\) is the velocity screw of the mass center, \(P\), on moving platform which is given in solving the inverse kinematics, and \(\overset{P}{C}\mathbf{V}^{K}\) is the relative velocity screw of joint \(P\) in regard to joint \(C\), which can be gained through  
\[\overset{P}{C}\mathbf{V}^{K}=\left[ \begin{matrix}\mathbf{0}\\ \overset{C}{0}\mathbf{\omega}^{K}\times\mathbf{r}_{P}^{K}\end{matrix} \right] \tag{23}\]  
where \(\overset{C}{0}\mathbf{\omega}^{K}\) is the output absolute angular velocity of the \(K\)th kinematic chain connected with the moving part, while \(\mathbf{r}_{P}^{K}\) presents the absolute position of point \(P\) in the absolute coordinate frame.