# Screw Dynamics of a Multibody System by a
## 3 Screw Kinematics and Dynamics of a Limb of a Schoenflies Manipulator
### Screw Dynamics Analysis of the Moving Platform  
The constraint force screws from the four joints to the end actuator are demonstrated in Figure 6.  
It can be seen from Equation (14) that the momentum screw of the moving platform is  
\[\mathbf{\Gamma}=\begin{bmatrix}m\mathbf{v}_{O_{M}}\\ \mathbf{r}_{O_{M}}\times\left(m\mathbf{v}_{O_{M}}\right)+\mathbf{J}_{M}\mathbf{\omega}\end{bmatrix} \tag{35}\]  
The sum of the external force screws acting on the moving platform is  
\[\mathbf{\mathfrak{s}}=\sum_{i=1}^{4}\mathbf{\mathfrak{s}}_{x_{i}}+\mathbf{\mathfrak{s}}_{ Mg} \tag{36}\]  
Substituting Equations (38) and (39) into Equation (14) presents  
Figure 5: Constraint torque of the UPU kinematic chain.  
\[\frac{d}{dt}\begin{bmatrix}m\
u_{O_{M}}\\ \mathbf{r}_{O_{M}}\times\left(m\
u_{O_{M}}\right)+\mathbf{J}_{M}\mathbf{\omega}\end{bmatrix} =\begin{bmatrix}-\mathbf{F}_{C_{1}}-\mathbf{F}_{C_{2}}-\mathbf{F}_{C_{3}}-\mathbf{F}_{C_{4}}+M \mathbf{g}\\ \mathbf{r}_{C_{1}}\times\left(-\mathbf{F}_{C_{1}}\right)+x_{5}\mathbf{e}_{p_{1}}+\mathbf{r}_{ C_{2}}\times\left(-\mathbf{F}_{C_{2}}\right)+\\ \mathbf{r}_{C_{3}}\times\left(-\mathbf{F}_{C_{3}}\right)+\mathbf{r}_{C_{4}}\times\left(- \mathbf{F}_{C_{4}}\right)+x_{6}\mathbf{e}_{p_{4}}+\mathbf{r}_{O_{M}}\times M\mathbf{g}\end{bmatrix} \tag{37}\]  
The constraint force \(\mathbf{F}_{C_{i}}(i=1,2,3,4)\) can be transformed by Equations (32) and (34). Therefore, Equation (37) contains six scalar equations in the unknowns denoted by a vector \(\mathbf{x}=\begin{bmatrix}\ x_{1}&\ x_{2}&\ x_{3}&\ x_{4}&\ x_{5}&\ x_{6}\end{bmatrix}^ {T}\). From Equations (32), (34), and (37), we obtain that  
\[\frac{d}{dt}\begin{bmatrix}m\
u_{O_{M}}\\ \mathbf{r}_{O_{M}}\times\left(m\
u_{O_{M}}\right)+\mathbf{J}_{M}\mathbf{\omega}\end{bmatrix} =-\begin{bmatrix}\sum\limits_{i=1}^{4}\mathbf{K}_{i}x_{i}+\sum\limits_{i=1}^{4}\bm {H}_{i}+M\mathbf{g}\\ \sum\limits_{i=1}^{4}\left(\mathbf{r}_{C_{i}}\times\mathbf{K}_{i}\right)x_{i}+\sum \limits_{i=1}^{4}\left(\begin{array}{c}\mathbf{r}_{C_{i}}\times\mathbf{H}_{i}-x_{5} \mathbf{e}_{p_{1}}\\ -x_{6}\mathbf{e}_{p_{4}}-\mathbf{r}_{O_{M}}\times M\mathbf{g}\end{array}\right)\end{bmatrix} \tag{38}\]  
Rearranging Equation (38) yields  
\[\left[\begin{array}{c}\sum\limits_{i=1}^{4}\mathbf{K}_{i}x_{i}\\ \sum\limits_{i=1}^{4}\left(\mathbf{r}_{C_{i}}\times\mathbf{K}_{i}\right)x_{i}\\ -x_{5}\mathbf{e}_{p_{1}}-x_{6}\mathbf{e}_{p_{4}}\end{array}\right]=-\frac{d}{dt} \begin{bmatrix}m\
u_{O_{M}}\\ \mathbf{r}_{O_{M}}\times\left(m\
u_{O_{M}}\right)+\mathbf{J}_{M}\mathbf{\omega}\end{bmatrix} -\begin{bmatrix}\sum\limits_{i=1}^{4}\mathbf{H}_{i}+M\mathbf{g}\\ \sum\limits_{i=1}^{4}\mathbf{r}_{C_{i}}\times\mathbf{H}_{i}-\mathbf{r}_{O_{M}}\times M\mathbf{ g}\end{bmatrix} \tag{39}\]  
Thus, Equation (39) can be expressed in matrix multiplication form,  
\[\mathbf{A}\mathbf{x}=\mathbf{b} \tag{40}\]  
where \(\mathbf{A}\) is the coefficient matrix, and \(\mathbf{b}\) is a column vector: \(\mathbf{A}=\begin{bmatrix}\mathbf{A}_{1}\\ \mathbf{A}_{2}\end{bmatrix}=\begin{bmatrix}\mathbf{K}_{1}&\mathbf{K}_{2}&\mathbf{K}_{3}&\mathbf{K} _{4}&0&0\\ \mathbf{r}_{C_{1}}\times\mathbf{K}_{1}&\mathbf{r}_{C_{2}}\times\mathbf{K}_{2}&\mathbf{r}_{C_{3}} \times\mathbf{K}_{3}&\mathbf{r}_{C_{4}}\times\mathbf{K}_{4}&-\mathbf{e}_{p_{1}}&-\mathbf{e}_{p_{4}} \end{bmatrix}\) and \(\mathbf{b}=-\frac{d}{dt}\begin{bmatrix}m\
u_{O_{M}}\\ \mathbf{r}_{O_{M}}\times\left(m\
u_{O_{M}}\right)+\mathbf{J}_{M}\mathbf{\omega}\end{bmatrix} -\begin{bmatrix}\sum\limits_{i=1}^{4}\mathbf{H}_{i}+M\mathbf{g}\\ \sum\limits_{i=1}^{4}\mathbf{r}_{C_{1}}\times\mathbf{H}_{i}-\left(\mathbf{r}_{O_{M}}\times M \mathbf{g}\right)\end{bmatrix}\).  
From Figure 5, we know that \(\mathbf{e}_{p_{1}}\) and \(\mathbf{e}_{p_{4}}\) are located in the \(xoy\) plane. Therefore, only \(x_{i}(i=1,2,3,4)\) need to be calculated because they are required in the calculation of driving forces.  
Figure 6: The constraints force screws applied to the moving platform.