# Screw Dynamics of a Multibody System by a
## 3 Screw Kinematics and Dynamics of a Limb of a Schoenflies Manipulator
### Screw Dynamics Analysis of a Limb  
The momentum screw of the lower limb is  
\[\mathbf{\Gamma}_{l}= \begin{bmatrix}\mathbf{p}\\ \mathbf{r}_{C}\times\mathbf{p}+\mathbf{J}_{C}\mathbf{\omega}\end{bmatrix} = \begin{bmatrix}m_{1 }(\mathbf{\omega}_{i}\times\mathbf{r}_{1})\\ \mathbf{J}_{1}\mathbf{\omega}_{i}\end{bmatrix}  \tag{25}\]  
Figure 4: Angular velocity analysis of a single kinematic chain. where \(\tau_{1}\) is the position vector of the center of mass \(C_{\rm I}\) of the lower limb in the absolute coordinate frame. \(J_{\rm I}\) is the mass moment of inertia matrix of the lower limb relative to \(A_{i}(i=1,2,3,4)\) in the absolute coordinate frame. It is worth noting that the given \(J_{\rm I}^{\prime}\) is the mass moment of inertia matrix of the lower limb relative to its center, \(C_{\rm I}\), in the local coordinate frame, and they meet that \(J_{\rm I}=R^{\prime\,T}J_{\rm I}^{\prime}R^{\prime}+m_{\rm I}(\tau_{1}^{2}I- \tau_{\rm I}\tau_{1}^{T})\). For a kinematic chain, \(R^{\prime}\) is a transformation matrix from its local coordinate frame to the absolute coordinate frame. The momentum screw of the upper limb is  
\[\Gamma_{u}= \begin{bmatrix}\mathbf{p}\\ \mathbf{r}_{\rm C}\times\mathbf{p}+\mathbf{J}_{\rm C}\mathbf{\omega}\end{bmatrix} = \begin{bmatrix} m_{\rm II}\mathbf{v}_{C_{1}}+m_{\rm II}(\mathbf{\omega}_{i}\times\mathbf{n}_{\rm II})\\ J_{\rm II}\mathbf{\omega}_{i}\end{bmatrix}  \tag{26}\]  
where \(\mathbf{r}_{\rm II}\) is the position vector of the center of mass \(C_{\rm II}\) of the upper limb in the absolute coordinate frame. \(J_{\rm II}\) is the mass moment of inertia matrix of the upper limb relative to \(A_{i}(i=1,2,3,4)\) in the absolute coordinate frame. Similarly, \(J_{\rm II}^{\prime}\) meets \(J_{\rm II}=R^{\prime\,T}J_{\rm II}^{\prime}R^{\prime}+m_{\rm II}(\tau_{\rm II }^{2}I-\tau_{\rm II}\tau_{\rm II}^{T})\). The force screw acting on the single limb is  
\[\mathbf{\xi}_{i}= \begin{bmatrix}\mathbf{F}_{A_{i}}+\mathbf{F}_{C_{i}}\\ \mathbf{r}_{A_{i}C_{i}}\times\mathbf{F}_{C_{i}}+\big{(}T_{A_{i}}+T_{C_{i}}\big{)}\mathbf{e} _{p_{i}}\end{bmatrix}  \tag{27}\]  
Substituting Equations (25)-(27) into Equation (14) yields  
\[\frac{d}{dt}(\mathbf{\Gamma}_{I}+\mathbf{\Gamma}_{u})=\mathbf{\xi} \tag{28}\]  
or  
\[\frac{d}{dt} \begin{bmatrix}m_{\rm I}(\mathbf{\omega}_{i}\times\mathbf{\tau}_{1})+m_{ \rm II}\mathbf{v}_{C_{1}}+m_{\rm II}(\mathbf{\omega}_{i}\times\mathbf{\tau}_{\rm II})\\ J_{\rm I}\mathbf{\omega}_{i}+J_{\rm II}\mathbf{\omega}_{i}\end{bmatrix} =\mathbf{\xi}_{i}+ \mathbf{\xi}_{mg} \tag{29}\]  
where \(\mathbf{\xi}_{i}\) is the sum of the external force screw, including the constraints of joints \(A_{i}\) and \(B_{i}\). \(\mathbf{\xi}_{mg}\) is the force screw of gravity. Here, the viscous friction between the prismatic joint belongs to the internal force and therefore does not appear on the right side of the momentum screw Equation (14). Suppose \(D=\big{(}J_{\rm I}+J_{\rm II}\big{)}\dot{\mathbf{\omega}}_{1}-\mathbf{\tau}_{1}\times m _{1}\mathbf{g}-\mathbf{r}_{\rm II}\times m_{\rm II}\mathbf{g}\), we obtain that  
\[D=\mathbf{r}_{A_{i}C_{i}}\times\mathbf{F}_{C_{i}}+\big{(}T_{A_{i}}+T_{C_{i}}\big{)} \mathbf{e}_{p_{i}} \tag{30}\]  
Next, the dynamics equations of the UPU and UPS kinematic chains will be discussed separately. 1. _UPU kinematic chains_ According to the physical meaning of a reciprocal screw, \(\mathbf{\xi}_{A_{i}B_{i}C_{i}}^{\prime}(i=1,4)\) stands for the constraint torque around \(\mathbf{e}_{p_{i}}\) as shown in Figure 5 for \(i=1\). The terminal constraint of the UPU limbs is a torque parallel to the vector \(\mathbf{e}_{p_{i}}(i=1,4)\). Therefore, the constraint torques of the UPU limbs can be expressed by \(\big{(}T_{A_{i}}+T_{C_{i}}\big{)}\mathbf{e}_{p_{i}}(i=1,4)\).