# Dynamics of a 3-UPS-UPU-S Parallel Mechanism
## 4 The Analysis of Acceleration and Its Mathematical Expression
\begin{array}{l}{}_{0}^{N}\mathbf{A}={}_{0}^{1}\beta\mathbf{\xi}_{1}+{}_{2}^{2}\beta \mathbf{\xi}_{2}+\cdots+{}_{N-2}^{N-1}\beta\mathbf{\xi}_{N-1}+{}_{N-1}^{N}\beta\mathbf{ \xi}_{N}\\ \quad\quad+{}_{0}^{W}\mathbf{\xi}_{1},{}_{1}^{2}\omega\mathbf{\xi}_{2}+{}_{3}^{2}\omega \mathbf{\xi}_{3}+\cdots+{}_{N-2}^{N-1}\omega\mathbf{\xi}_{N-1}+{}_{N-1}^{N}\omega\mathbf{ \xi}_{N}\Big{]}\\ \quad\quad+{}_{2}^{W}\mathbf{\xi}_{2},{}_{2}^{2}\omega\mathbf{\xi}_{3}+\cdots+{}_{N-2 }^{W-2}\omega\mathbf{\xi}_{N-1}+{}_{N-1}^{N}\omega\mathbf{\xi}_{N}\Big{]}+\cdots\\ \quad\quad+{}_{N-2}^{W-2}\omega\mathbf{\xi}_{N-2},{}_{N-2}^{N-1}\omega\mathbf{\xi}_{N-1 }+{}_{N-1}^{N}\omega\mathbf{\xi}_{N}\Big{]}+{}_{N-2}^{N-1}\omega\mathbf{\xi}_{N-1},{}_ {N-1}^{N}\omega\mathbf{\xi}_{N}\Big{]}\end{array}  \tag{43}\]  
where \({}_{N}^{N+1}\beta=\frac{d_{N}^{N+1}\omega}{dt}\) and \({}_{N}^{N+1}\mathbf{\xi}_{i+1}={}_{ \begin{bmatrix}{}_{N}^{N+1}\mathbf{e}\\ {}_{0}^{N}\times{}_{N}^{N+1}\mathbf{e}\end{bmatrix} }\). To simplify the expression of Equation (43), \({}_{N}^{N+1}\omega\) and \({}_{N}^{N+1}\beta\) can be denoted by \(\omega_{N+1}\) and \(\beta_{N+1}\) to express the relative angular velocity and acceleration of the \((N+1)\)th frame with respect to the \(N\)th frame without any ambiguity. Equation (43) can be written in a matrix multiplication form as  
\[{}_{0}^{N}\mathbf{A}=\mathbf{\mathsf{S}}\mathbf{\beta}+\mathbf{\omega}^{T}\mathbf{S}_{L}\mathbf{\omega} \tag{44}\]  
where  
\[\mathbf{\mathsf{S}}= \begin{bmatrix}\mathbf{\mathsf{S}}_{1}&\mathbf{\mathsf{S}}_{2}&\cdots& \mathbf{\mathsf{S}}_{N}\end{bmatrix}  \tag{45}\]  
which is the unit screw matrix of the kinematic chain, and  
\[\mathbf{\omega}= \begin{bmatrix}\omega_{1}&\omega_{2}&\cdots&\omega_{N}\end{bmatrix} ^ {T} \tag{46}\]  
which consists of all relative angular velocities, and  
\[\mathbf{\beta}= \begin{bmatrix}\beta_{1}&\beta_{2}&\cdots&\beta_{N}\end{bmatrix} ^ {T}= \begin{bmatrix}\frac{\mathrm{d}\omega_{1}}{\mathrm{d}t}&\frac{\mathrm{d} \omega_{2}}{\mathrm{d}t}&\ldots&\frac{\mathrm{d}\omega_{N}}{\mathrm{d}t}\end{bmatrix}  ^{T} \tag{47}\]  
which is a vector consisting of all relative angular accelerations, and  
\[\mathbf{S}_{L}=\begin{bmatrix}\mathbf{0}&[\mathbf{\mathsf{S}}_{1},\mathbf{\mathsf{S}}_{2}]&[ \mathbf{\mathsf{S}}_{1},\mathbf{\mathsf{S}}_{3}]&[\mathbf{\mathsf{S}}_{1},\mathbf{\mathsf{S}}_{3} ]&\cdots&[\mathbf{\mathsf{S}}_{1},\mathbf{\mathsf{S}}_{N-1}]&[\mathbf{\mathsf{S}}_{1},\mathbf{ \mathsf{S}}_{N}]\\ \mathbf{0}&\mathbf{0}&[\mathbf{\mathsf{S}}_{2},\mathbf{\mathsf{S}}_{3}]&[\mathbf{\mathsf{S}}_{2}, \mathbf{\mathsf{S}}_{3}]&\cdots&[\mathbf{\mathsf{S}}_{2},\mathbf{\mathsf{S}}_{N-1}]&[\mathbf{ \mathsf{S}}_{2},\mathbf{\mathsf{S}}_{N}]\\ \mathbf{0 The linear acceleration of the mass/geometry center of the end effector is  
\[{}_{0}^{N}\mathbf{a}_{\text{C}}={}_{0}^{N}\mathbf{a}+{}_{0}^{N}\mathbf{\beta}\times\mathbf{r}_{ \text{C}}+{}_{0}^{N}\mathbf{\omega}\times{}_{0}^{N}\mathbf{v}_{\text{C}} \tag{49}\]  
where \({}_{0}^{N}\mathbf{\omega}=\mathbf{\omega}_{1}+\mathbf{\omega}_{2}+\cdots+\mathbf{\omega}_{N}{}_ {0}^{N}\mathbf{\beta}=\mathbf{\beta}_{1}+\mathbf{\beta}_{2}+\cdots+\mathbf{\beta}_{N}\), \(\mathbf{r}_{\text{C}}=\mathbf{r}_{N}+\mathbf{r}_{\text{NC}}\), and \({}_{0}^{N}\mathbf{v}_{\text{C}}={}_{0}^{N}\mathbf{\omega}\times(\mathbf{r}_{N}+\mathbf{r}_{ \text{NC}})\). Accordingly, the absolute acceleration in screw form of the center point of the end effector is expressed as  
\[{}_{0}^{N}\mathbf{A}_{\text{C}}= \begin{bmatrix}{}_{0}^{N}\mathbf{\beta}\\ {}_{0}^{N}\mathbf{a}_{\text{C}}\end{bmatrix} = \begin{bmatrix}{}_{0}^{N}\mathbf{\beta}\\ {}_{0}^{N}\mathbf{a}+{}_{0}^{N}\mathbf{\beta}\times\mathbf{r}_{\text{C}}+{}_{0}^{N}\mathbf{ \omega}\times{}_{0}^{N}\mathbf{v}_{\text{C}}\end{bmatrix}  \tag{50}\]