# Dynamics of a 3-UPS-UPU-S Parallel Mechanism
## 5 The Analysis of Dynamics of the 3 UPS-UPU-S Parallel Mechanism  
For the 3 UPS-UPU-S parallel mechanism, there are four actuators at the prismatic joints to drive the moving legs, fixed legs and moving platform. As aforementioned in Section 2, the 3 UPS-UPU-S has two rotational degrees of freedom. When it rotates, the inertia force would be generated, and in order to give accurate control, this section will yield the dynamic analysis based on the Newton-Euler equation. The Newton-Euler equation can group all the forces and torques into a single equation. Hence, the three constraint forces and three constraint torques can be calculated in a straightforward way. When the external loads, constraint forces and self-weights of all rigid bodies are considered in the calculation of wrenches, the equation of the balance conditions for a single-rigid-body system is  
\[ \begin{bmatrix}\mathbf{F}\\ \mathbf{T}\end{bmatrix} = \begin{bmatrix}m\mathbf{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{J}\end{bmatrix}\begin{bmatrix}\mathbf{a}\\ \mathbf{\beta}\end{bmatrix} + \begin{bmatrix}m\mathbf{g}\\ \mathbf{\omega}\times\mathbf{J}\mathbf{\omega}\end{bmatrix}  \tag{51}\]  
where \(\mathbf{T}\) is the resultant torque vector, \(\mathbf{F}\) is the resultant force vector of a single-rigid-body, \(\mathbf{I}\) is the 3 rd-order identity matrix, and \(m\) is the mass of a single-rigid-body at its mass center. \(\{\mathbf{a},\mathbf{\beta}\}\) in Equation (51) is the absolute linear acceleration and absolute angular acceleration vector at the mass center according to the absolute coordinate frame, and \(\mathbf{\omega}\) is the absolute angular velocity vector at the mass center in the absolute coordinate frame. The acceleration \(\{\mathbf{a},\mathbf{\beta}\}\) can be calculated by Equation (50), and the angular velocity \(\mathbf{\omega}\) can be obtained using Equation (24). The moments of inertia \(\mathbf{J}\) of a single-rigid-body at its mass center in the absolute coordinate frame (0-coordinate frame) can be obtained through  
\[\mathbf{J}=\mathbf{R}_{0n}\mathbf{J}_{n}\mathbf{R}_{0n}^{T} \tag{52}\]  
where \(\mathbf{J}_{n}\) is the moments of inertia of a single-rigid-body at its mass center in the relative coordinate (n-coordinate frame), and \(\mathbf{R}_{0n}\) is the rotation matrix from the n-coordinate frame to the 0-coordinate frame, which could be calculated by Equation (22). The fixed leg, \(L_{AB}\), has one universal joint and one prismatic joint, which is shown in Figure 4a, and the Newton-Euler equations for the fixed leg, \(L_{AB}\), can be derived as  
\[ \begin{bmatrix}\mathbf{F}_{A}-\mathbf{F}_{B}\\ \mathbf{a}\mathbf{r}_{\text{D}}\times\mathbf{F}_{A}-{}_{\text{D}}\mathbf{r}_{\text{B}}\times \mathbf{F}_{B}+\mathbf{T}_{A}-\mathbf{T}_{\text{B}}\end{bmatrix} = \begin{bmatrix}m_{\text{D }}\mathbf{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{J}_{\text{D}}\end{bmatrix}\begin{bmatrix}\mathbf{a}_{\text{D}}\\ \mathbf{\beta}_{\text{D}}\end{bmatrix} + \begin{bmatrix}m_{\text{D}}\mathbf{g}\\ \mathbf{\omega}_{\text{D}}\times\mathbf{J}_{\text{D}}\mathbf{\omega}_{\text{D}}\end{bmatrix}  \tag{53}\]  
The moving leg, \(L_{BC}\), consists of universal joint (or spherical joint) and one prismatic joint, which is shown in Figure 4b, and the Newton-Euler equations for the fixed leg, \(L_{BC}\), can be deduced as  
\[ \begin{bmatrix}\mathbf{F}_{B}-\mathbf{F}_{C}\\ {}_{B}\mathbf{r}_{E}\times\mathbf{F}_{B}-{}_{E}\mathbf{r}_{\text{C}}\times\mathbf{F}_{C}+\mathbf{T} _{B}-\mathbf{T}_{\text{C}}\end{bmatrix} = \begin{bmatrix}m_{E}\mathbf{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{J}_{E}\end{bmatrix}\begin{bmatrix}\mathbf{a}_{E}\\ \mathbf{\beta}_{E}\end{bmatrix} + \begin{bmatrix}m_{E}\mathbf{g}\\ \mathbf{\omega}_{E}\times\mathbf{J}_{E}\mathbf{\omega}_{E}\end{bmatrix}  \tag{54}\]  
When the moving leg, \(L_{BC}\), is PU, the \(\mathbf{T}_{\text{C}}\) is zero. The moving platform, \(L_{P}\), consists of one universal joint and four spherical joints, which is shown in Figure 4d. The Newton-Euler equations for the moving platform, \(L_{P}\), can be expressed as  
\[\left[