# Reconfiguration for the Maximum Dynamic Wrench Capability of a Parallel Robot
## Appendix  
Inertia matrix:  
\[\begin{array}{l}\mathbf{M_{1}}=\left[ \begin{array}{cc}m_{\mathbf{B}}\mathbf{I_{3 \times 3}}&\mathbf{0}_{\mathbf{3\times 3}}\\ \mathbf{0}_{\mathbf{3\times 3}}&\mathbf{I_{B}}\end{array} \right],\\ \\ \mathbf{M_{2}}=\left[ \begin{array}{cc}\mathbf{diag}\left[m_{12},\ldots,m_{62} \right]&\mathbf{diag}\left[\begin{array}{cc}m_{12}\mathbf{s}_{1}^{T} \widetilde{d}_{12}^{T}&\ldots,&m_{62}\mathbf{s}_{6}^{T}\widetilde{d}_{62}^{T} \end{array}\right]\\ \mathbf{diag}\left[\begin{array}{cc}m_{12}\widetilde{d}_{12}\mathbf{s}_{1}& \ldots,&m_{62}\widetilde{d}_{62}\mathbf{s}_{6}\end{array}\right]&\mathbf{diag} \left[\begin{array}{cc}\mathbf{I_{1}}&\ldots,&\mathbf{I_{6}}\end{array}\right] \end{array} \right],\\ \\ \text{and }\mathbf{I_{i}}=\mathbf{I_{1i}}+\mathbf{I_{2i}}-m_{i2}l_{i1}^{2} \widetilde{s}_{i}^{2}+m_{i2}l_{i1}\left(\widetilde{s}_{i}^{T}\widetilde{d}_{i2 }+\widetilde{d}_{i2}^{T}\widetilde{s}_{i}\right),\,i=1,\ldots,6\]  
Coriolis and centrifugal terms:  
\[ \begin{array}{l}\mathbf{D_{1}}=\left[\begin{array}{cc}\mathbf{0}_{\mathbf{3 \times 3}}&\mathbf{0}_{\mathbf{3\times 3}}\\ \mathbf{0}_{\mathbf{3\times 3}}&\left(\mathbf{I_{B}}\otimes\mathbf{\omega_{B}}\right)^{T} \end{array}\right],\\ \\ \end{array} \]  
\[\begin{array}{l}\mathbf{D_{2}}=\left[\begin{array}{cc}\mathbf{a}_{\mathbf{k} _{\times}}&\mathbf{a}_{\times}^{T}\mathbf{a}_{\times}\mathbf{a}_{\times}^{T} \mathbf{a}_{\times}\mathbf{a}_{\times}&\ldots,&m_{62}\widetilde{s}_{62}^{T} \widetilde{s}_{62}^{T}\widetilde{s}_{62}^{T}\widetilde{s}_{62}^{T}\widetilde{ s}_{62}^{T}\widetilde{s}_{62}^{T}\widetilde{s}_{62}^{T}\widetilde{s}_{62}^{T} \widetilde{gravity vector:  
\[\mathbf{G_{1}}=\left[ \begin{array}{c}-m_{\rm B}\mathbf{g}\\ \mathbf{0_{3\times 1}}\end{array} \right],\] \[\mathbf{G_{2}}=\left[ \begin{array}{cc}\left[\begin{array}{cc}-m_{ \rm 12}\mathbf{g}^{T}\,{}_{\mathbf{1}}^{N}\mathbf{Rs_{1}}&\ldots&-m_{\rm 62}\mathbf{g}^{T}\,{}_{\mathbf{6}}^{N}\mathbf{Rs_{6}} \end{array}\right]\\ \left[\begin{array}{cc}-\mathbf{g}^{T}\,{}_{\mathbf{1}}^{N}\mathbf{R}(m_{\rm 11}\tilde{\mathbf{d}}_{ \mathbf{1}1}+m_{\rm 12}l_{\rm 11}\tilde{\mathbf{s}}_{\rm i}+m_{\rm 12}\tilde{\mathbf{d}}_{ \mathbf{12}})^{T}&\ldots&-\mathbf{g}^{T}\,{}_{\mathbf{6}}^{N}\mathbf{R}(m_{\rm 61}\tilde{\mathbf{d}}_{ \mathbf{61}}+m_{\rm 62}l_{\rm 61}\tilde{\mathbf{s}}_{\rm i}+m_{\rm 62}\tilde{\mathbf{d}}_{ \mathbf{62}})^{T}\end{array}\right]^{T}\end{array} \right]\] The actuating torques: \(\tau=\left[ \begin{array}{cc}\tau_{1}&\ldots&\tau_{6}\end{array} \right]^{T}\). \[\mathbf{J_{1}}=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{cc} \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc} \begin{array}[  
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