# Reconfiguration for the Maximum Dynamic Wrench Capability of a Parallel Robot
## 2 Dynamics of the Stewart-Platform Robot with Sliding Lockable Base Joints
vector \(I_{i1}\) of the lower segment to the constant length vector \(I_{i2}\) of the upper segment, the length vector of link \(i\) can be written as  
\[I_{i}=l_{i}\boldsymbol{s_{i}}=I_{i1}+I_{i2}=\left(l_{i1}+l_{i2}\right) \boldsymbol{s_{i}} \tag{1}\]  
where \(\boldsymbol{s_{i}}\) is an unit vector along the \(i\)th link axis and is expressed in the i\({}_{1}\)-frame; \(l_{i}\), \(l_{i1}\) and \(l_{i2}\) are the respective length magnitudes of link \(i\), as well as the lower segment and the upper segment of link \(i\). Because in the parallel robot structure there exist multiple closed-loop chain mechanisms, the geometric constraint equation based on a vector loop relationship can be formulated as  
\[\boldsymbol{p}+\overset{N}{{}_{B}}\boldsymbol{R}\boldsymbol{b_{i}}= \boldsymbol{a_{i}}+\overset{N}{{}_{i}}\boldsymbol{R}I_{i} \tag{2}\]  
In Equation (2), the rotation matrix \(\overset{N}{{}_{L}}\boldsymbol{R}\) with subscripts \(L\in\{B,i(i=1,...,6)\}\) represents the transformation from a local \(L\)-frame to the inertial \(N\)-frame. The time derivative of this rotation matrix can be expressed as  
\[\overset{N}{{}_{L}}\boldsymbol{R}=\overset{N}{{}_{L}}\boldsymbol{R}\widetilde {\boldsymbol{\omega}}_{L} \tag{3}\]  
in which \(\widetilde{\boldsymbol{\omega}}_{L}\) is a skew symmetric matrix formed from the angular velocity \(\boldsymbol{\omega}_{L}=\left[ \begin{array}{ccc}\omega_{x}&\omega_{y}& \omega_{z}\end{array} \right]_{L}^{T}\) with respect to a local body-fixed \(L\)-frame, _i.e._,  
\[\widetilde{\boldsymbol{\omega}}_{L}=\left[ \begin{array}{ccc}0&-\omega_{z}& \omega_{y}\\ \omega_{z}&0&-\omega_{x}\\ -\omega_{y}&\omega_{x}&0\end{array} \right]_{L} \tag{4}\]  
Moreover, the angular velocity \(\boldsymbol{\omega}_{L}\) can be related to the time derivative of the Euler angles using a corresponding velocity transformation matrix \(\boldsymbol{C}_{L}\) such that  
\[\boldsymbol{\omega}_{L}=\boldsymbol{C}_{L}\dot{\boldsymbol{\gamma}}_{L} \tag{5}\]  
Taking a derivative on the constraint Equation (2), and using the fact that no spin is allowed about its link axis due to the connection of each link to the base by an universal joint, _i.e._,  
\[\boldsymbol{\omega}_{i}^{T}\boldsymbol{s}_{i}=0 \tag{6}\]  
Figure 2: Coordinate systems of link and moving platform. From Equation (14), \(\mathbf{v_{2}}\) can be parameterized by \(\mathbf{v_{1}}\) as  
\[\mathbf{v_{2}}=\mathbf{J_{1}}\mathbf{v_{1}} \tag{16}\]  
and then by differentiating Equation (16), the parameterized acceleration is  
\[\dot{\mathbf{v}_{2}}=\dot{\mathbf{J_{1}}}\mathbf{v_{1}}+\mathbf{J_{1}}\dot{\mathbf{v_{1}}} \tag{17}\]  
On the basis of the developed structured Boltzmann-Hamel-d'Alembert formulism for a parallel mechanism [23; 24; 25], the dynamics of the PR can be given as  
\[(\mathbf{M_{1}}\dot{\mathbf{v}_{1}}+\mathbf{D_{1}}\mathbf{v_{1}}+\mathbf{G_{1}})+\dot{\mathbf{J_{1}}}( \mathbf{M_{2}}\dot{\mathbf{v_{2}}}+\mathbf{D_{2}}\mathbf{v_{2}}+\mathbf{G_{2}})=\mathbf{Jacob}^{T}L^{-1 }\mathbf{\tau}-\left[ \begin{array}{c}\mathbf{N_{2}}\mathbf{R}\mathbf{f}\\ \mathbf{n}\end{array} \right] \tag{18}\]  
in which \(\mathbf{v_{1}}=\left[ \begin{array}{cc}\dot{\mathbf{p}}^{T}&\mathbf{\omega_{B}^{T}}\\ \end{array} \right]^{T}\) are the velocities of the moving platform, \(\mathbf{v_{2}}=\left[ \begin{array}{cc}\dot{l_{11}}&\ldots&\dot{l_{61}}&\mathbf{ \omega_{1}^{T}}&\ldots&\mathbf{\omega_{6}^{T}}\end{array} \right]^{T}\) are the sliding velocities and the angular velocities of the six links. Moreover, the matrices and vectors in Equation (18) are fully derived and explained in the Appendix. Also, it is noted that the applied dynamic wrench force \(f\) and wrench moment \(n\) are expressed in terms of the moving platform-fixed \(B\)-frame.