# Dynamics of a 3-UPS-UPU-S Parallel Mechanism
## 4 The Analysis of Acceleration and Its Mathematical Expression
\begin{split}{}^{W}_{U}A_{P}&=\begin{bmatrix} \begin{smallmatrix}W\beta\\ V\mathbf{a}_{P}-\begin{smallmatrix}W\mathbf{\omega}\end{smallmatrix}\times\begin{smallmatrix} W\\ V\mathbf{v}\end{smallmatrix}\end{bmatrix}\\ =\begin{bmatrix}\begin{smallmatrix}V\beta\\ V\mathbf{\beta}\end{smallmatrix}\\ \
u\mathbf{a}_{P}-\begin{smallmatrix}V\mathbf{\omega}\end{smallmatrix}\times\begin{smallmatrix} (V\mathbf{\omega}\times\begin{smallmatrix}W\\ V\mathbf{\omega}\times\mathbf{r}_{UP}\end{smallmatrix})\end{bmatrix}+\begin{bmatrix} \begin{smallmatrix}W\beta\\ V\mathbf{\beta}\end{smallmatrix}\\ \
u\mathbf{a}_{P}-\begin{smallmatrix}W\mathbf{\omega}\end{smallmatrix}\times\begin{smallmatrix} (V\mathbf{\omega}\times\mathbf{r}_{UP})\end{smallmatrix}\end{bmatrix}+\begin{bmatrix} \begin{smallmatrix}V\omega\times\begin{smallmatrix}W\\ V\mathbf{\omega}\times\mathbf{r}_{UP}\end{smallmatrix}\\ \
u\mathbf{\omega}\times\begin{smallmatrix}(V\mathbf{\omega}\times\mathbf{r}_{UP})-\begin{smallmatrix} W\mathbf{\omega}\times\begin{smallmatrix}V\mathbf{\omega}\times\mathbf{r}_{UP}\end{smallmatrix} \end{smallmatrix}\end{bmatrix}\end{bmatrix}\end{split} \end{split} \tag{32}\]  
Substituting Equation (26) into Equation (28) yields  
\[{}^{V}_{U}A_{P}= \begin{bmatrix}\begin{smallmatrix}V\beta\\ V\mathbf{\beta}\end{smallmatrix}\\ {}^{W}_{U}\beta\times\mathbf{r}_{UP}\end{bmatrix}  \tag{33}\]  
Additionally, substituting Equation (26) into Equation (31) gives  
\[{}^{W}_{V}A_{P}= \begin{bmatrix}\begin{smallmatrix}W\beta\\ V\beta\\ V\beta\end{smallmatrix}\times\mathbf{r}_{VP}\end{bmatrix}  \tag{34}\]  
Let  
\[{}^{V}_{U}L_{P}= \begin{bmatrix}\begin{smallmatrix}V\mathbf{\omega}\\ V\mathbf{\omega}\\ U\mathbf{\omega}_{P}\end{smallmatrix}\end{bmatrix} ,{}^{W}_{V}L_{P}= \begin{bmatrix} \begin{smallmatrix}W\mathbf{\omega}\\ V\mathbf{\omega}\\ V\mathbf{v}_{P}\end{smallmatrix}\end{bmatrix}  \tag{35}\]  
where \({}^{V}_{U}\mathbf{v}_{P}={}^{V}_{U}\mathbf{\omega}\times\mathbf{r}_{UP}\) and \({}^{W}_{V}\mathbf{v}_{P}={}^{W}_{V}\mathbf{\omega}\times\mathbf{r}_{VP}\). With the operation of _Lie bracket_ on \({}^{V}_{U}L_{P}\) and \({}^{W}_{V}L_{P}\) defined by  
\[ \begin{bmatrix}{}^{V}_{U}\mathbf{L}_{P},{}^{W}_{V}\mathbf{L}_{P}\end{bmatrix} = \begin{bmatrix}\begin{smallmatrix}V\mathbf{\omega}\times\begin{smallmatrix}W\\ V\mathbf{\omega}\\ V\mathbf{\omega}\times\begin{smallmatrix}W\\ V\mathbf{v}_{P}- \begin{smallmatrix}W\mathbf{\omega}\times\begin{smallmatrix}V\\ V\mathbf{v}_{P}\end{smallmatrix}\end{smallmatrix} \  
Since \({}_{W-2}^{W}\mathbf{L}_{p}={}_{W-2}^{W-1}\mathbf{L}_{p}+{}_{W-1}^{W}\mathbf{L}_{p}\) and \({}_{W-3}^{W}\mathbf{L}_{p}={}_{W-3}^{W-2}\mathbf{L}_{p}+{}_{W-2}^{W-1}\mathbf{L}_{p}+{}_{W-1}^ {W}\mathbf{L}_{p}\), Equation (35) can be expanded to  
\[ \begin{array}{l}{}_{W-4}^{W}\mathbf{A}_{P}={}_{W-4}^{W-3}\mathbf{A}_{P}+{}_{W-3}^{W-2 }\mathbf{A}_{P}+{}_{W-1}^{W-1}\mathbf{A}_{P}+{}_{W-1}^{W}\mathbf{A}_{P}+{}_{W-1}^{W}\mathbf{L}_{ p}+{}_{W-2}^{W-2}\mathbf{L}_{p}+{}_{W-1}^{W}\mathbf{L}_{p}\Big{]}\\ \quad\quad+{}_{W-3}^{W-3}\mathbf{L}_{P},{}_{W-3}^{W-2}\mathbf{L}_{p}+{}_{W-2}^{W-1}\mathbf{ L}_{p}+{}_{W-1}^{W}\mathbf{L}_{p}\Big{]}\end{array}  \tag{41}\]  
Through the same procedure, the acceleration of the point \(P\) from frame 0 to frame \(N\) in screw form can be derived as  
\[ \begin{array}{l}{}_{0}^{N}\mathbf{A}_{P}={}_{0}^{1}\mathbf{A}_{P}+{}_{1}^{2}\mathbf{A}_{ P}+\cdots+{}_{N-2}^{N-1}\mathbf{A}_{P}+{}_{N-1}^{N}\mathbf{A}_{P}\\ \quad\quad+{}_{W-3}^{W-3}\mathbf{L}_{P}+{}_{W-2}^{W-1}\mathbf{L}_{P}+{}_{W-1}^{W}\mathbf{ L}_{p}\Big{]}+{}_{W-2}^{W-1}\mathbf{L}_{P}+{}_{W-1}^{W}\mathbf{L}_{p}\Big{]} \end{array}  \tag{42}\]  
When \(P\) is superimposed with the origin of the absolute coordinate frame at this time instant, the acceleration in screw form in accordance with the definition of the unit screw could be gained:  
\[