# Dynamics of a 3-UPS-UPU-S Parallel Mechanism
## 4 The Analysis of Acceleration and Its Mathematical Expression  
In this section, the analysis of acceleration and its mathematical expression is given. As depicted in Figure 3, suppose that \(U\), \(V\) and \(W\) are three rigid bodies in a serial kinematic chain connected by a number of joints successively, and there is an arbitrary point, \(P\), attached to the link \(L_{\text{W}}\). Therefore, the velocities of \(P\) attached to these two frames observed from frame \(U\) are  
\[\left\{ \begin{array}{l}\overset{V}{u}\mathbf{v}_{p}=\overset{V}{u}\mathbf{\omega} \times\mathbf{r}_{UP}\\ \overset{V}{v}\mathbf{v}_{p}=\overset{W}{v}\mathbf{\omega}\times\mathbf{r}_{VP}\end{array} \right. \tag{28}\]  
where the \(\overset{V}{u}\mathbf{\omega}\) and \(\overset{W}{v}\mathbf{\omega}\) are the relative angular velocity of joint \(J_{U}\) and \(J_{V}\), \(\mathbf{r}_{UP}\) and \(\mathbf{r}_{VP}\) is the position vector in the absolute coordinate frame. Differentiating both sides of Equation (26) with respect to time, the acceleration of point \(P\) rigidly attached to frame \(V\) and frame \(W\) observed in frame \(U\) could be derived as  
\[\left\{ \begin{array}{l}\overset{V}{u}\mathbf{a}_{P}=\overset{V}{u}\mathbf{\beta} \times\mathbf{r}_{UP}+\overset{V}{u}\mathbf{\omega}\times\left(\overset{V}{u}\mathbf{ \omega}\times\mathbf{r}_{UP}\right)\\ \overset{W}{u}\mathbf{a}_{P}=\overset{W}{V}\mathbf{\beta}\times\mathbf{r}_{VP}+\overset{W} {V}\mathbf{\omega}\times\left(\overset{W}{V}\mathbf{\omega}\times\mathbf{r}_{VP}\right) \end{array} \right. \tag{29}\]  
where the \(\overset{V}{U}\mathbf{\beta}\) and \(\overset{W}{V}\mathbf{\beta}\) are the relative angular acceleration of the joints with respect to their fore joints. It is well known that the acceleration could not be expressed in screw form due to its implicated motion. Therefore, to express the acceleration by one matrix, the implicated motion is removed to satisfy the definition of a screw. Suppose that \(\overset{V}{U}\mathbf{A}_{P}\) containing all acceleration information of the point \(P\) from coordinate \(U\) to coordinate \(V\) in screw form can be expressed as  
\[\overset{V}{U}\mathbf{A}_{P}= \begin{bmatrix}\overset{V}{u}\mathbf{\beta}\\ \overset{V}{u}\mathbf{a}_{P}-\overset{V}{U}\mathbf{\omega}\times\overset{V}{u}\mathbf{ \omega}\end{bmatrix} = \begin{bmatrix}\overset{V}{u}\mathbf{\beta}\\ \overset{V}{u}\mathbf{a}_{P}-\overset{V}{u}\mathbf{\omega}\times\left(\overset{V}{u} \mathbf{\omega}\times\mathbf{r}_{UP}\right)\end{bmatrix}  \tag{30}\]  
Figure 3: Acceleration analysis of multi-rigid body system. Suppose that \(V^{W}A_{P}\) containing all acceleration information of the point \(P\) from coordinate \(V\) to coordinate \(W\) in screw form can be expressed as  
\[{}^{W}_{V}A_{P}= \begin{bmatrix}\begin{smallmatrix}W\beta\\ V\beta\\ V\mathbf{a}_{P}-\begin{smallmatrix}W\mathbf{\omega}\end{smallmatrix}\times\begin{smallmatrix} W\\ V\mathbf{v}\end{smallmatrix}\end{bmatrix} = \begin{bmatrix}\begin{smallmatrix}W\beta\\ V\mathbf{a}_{P}-\begin{smallmatrix}W\\ V\mathbf{\omega}\end{smallmatrix}\times\begin{smallmatrix}W\\ V\mathbf{\omega}\end{smallmatrix}\end{bmatrix}  \tag{31}\]  
Therefore, \({}^{W}_{U}A_{P}\), which contains all acceleration information of the point \(P\) from coordinate \(U\) to coordinate \(W\) in screw form can be expressed as  
\[