# Reconfiguration for the Maximum Dynamic Wrench Capability of a Parallel Robot
## 2 Dynamics of the Stewart-Platform Robot with Sliding Lockable Base Joints
Equation (18) can be further simplified using the kinematic relationships Equations (16) and (17) to eliminate \(\mathbf{v_{2}}\) and \(\dot{\mathbf{v_{2}}}\) so that the following equations of motion are yielded:  
\[\mathbf{M_{p}}\dot{\mathbf{v_{1}}}+\mathbf{D_{p}}\mathbf{v_{1}}+\mathbf{G_{p}}=\mathbf{f_{p}} \tag{19}\]  
where \(\mathbf{M_{p}}=\mathbf{M_{1}}+\mathbf{J_{1}^{T}}\mathbf{M_{2}}\mathbf{I_{1}}\), \(\mathbf{D_{p}}=\mathbf{D_{1}}+\mathbf{J_{1}^{T}}\mathbf{D_{2}}\mathbf{I_{1}}+\mathbf{J_{1}^{T}}\mathbf{M_ {2}}\mathbf{I_{1}}\), \(\mathbf{G_{p}}=\mathbf{G_{1}}+\mathbf{J_{1}^{T}}\mathbf{G_{2}}\), \(\mathbf{f_{p}}=\mathbf{Jacob}^{T}\mathbf{L}^{-1}\mathbf{\tau}-\left[ \begin{array}{c}\mathbf{N_{ 2}}\mathbf{R}\mathbf{f}\\ \mathbf{n}\end{array} \right]\). Because the expressions for the rotations of the moving platform are in terms of the quasi-velocities and the corresponding derivatives, a velocity transformation matrix relating quasi-velocities to the time derivative of angular orientation coordinates is required for further describing the orientations of the moving platform in space. The velocity transformation is defined as  
\[\mathbf{v_{1}}=\mathbf{C_{1}}\dot{\mathbf{x}} \tag{20}\]  
in which \(\dot{\mathbf{x}}=\left[ \begin{array}{cc}\mathbf{P}^{T}&\dot{\mathbf{\gamma}}_{B}^{T}\\ \end{array} \right]^{T}\), and \(\mathbf{\gamma_{B}}\), generally defined by a set of Euler angles, specifies the orientations of the moving platform. Also, the velocity transformation matrix \(\mathbf{C_{1}}\) is defined as  
\[\mathbf{C_{1}}=\left[ \begin{array}{cc}\mathbf{I_{3\times 3}}&\mathbf{0}\\ \mathbf{0}&\mathbf{C_{B}}\\ \end{array} \right] \tag{21}\]  
where \(\mathbf{C_{B}}\) is for the angular velocity transformation of the moving platform. Moreover, the applied wrenches can be expressed as  
\[\left[ \begin{array}{c}\mathbf{f}\\ \mathbf{n}\end{array} \right]=\left[ \begin{array}{cccc}(s\alpha_{f})(c\beta_{f}) &(s\alpha_{f})(s\beta_{f})&c\alpha_{f}&0&0&0&0\\ 0&0&0&(s\alpha_{n})(c\beta_{n})&(s\alpha_{n})(s\beta_{n})&c\alpha_{n}\\ \end{array} \right]^{T}\left[ \begin{array}{c}f_{w}\\ \mathbf{n}_{w}\end{array} \right]=\mathbf{T_{w}}\mathbf{Q_{w}} \tag{22}\]  
where \(f_{w}\) and \(n_{w}\) are the magnitudes of the dynamic wrench force and wrench moment; \(\alpha\) and \(\beta\) are the angles of the applied wrenches with respect to the \(B\)-frame, and the subscripts \(f\) and \(n\) in Equation (22) respectively refer to the dynamic wrench force and wrench moment. For the problem of the dynamic wrench capability, we wish to find the allowable maximum dynamic wrench of the reconfigurable parallel robot along a specified trajectory without the violation of constraints. Accordingly, the objective function is dynamic wrench capability that comprises the dynamic wrench force and wrench moment. Substituting Equation (20) together with its time derivative and Equation (22) into the equations of motion, Equation (19), gives rise to an inverse dynamics equation in true coordinates \((\mathbf{p},\tau_{B})\):  
\[\left[ \begin{array}{cc}\mathbf{Jacob}^{T}\mathbf{L}^{-1}&-\left[\begin{array}{cc} \mathbf{\frac{N}{B}}\mathbf{R}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}_{3\times 3}\end{array}\right]\mathbf{T}_{w}\end{array} \right] \left[ \begin{array}{c}\mathbf{\tau}\\ \mathbf{Q}_{w}\end{array} \right]=\overline{\mathbf{M}}_{p}\ddot{\mathbf{x}}+\overline{\mathbf{ D}}_{p}\dot{\mathbf{x}}+\overline{\mathbf{G}}_{p} \tag{23}\]  
where \(\overline{\mathbf{M}}_{p}=M_{p}\mathbf{C_{1}},\overline{\mathbf{D}}_{p}=\overline{\mathbf{M}}_ {p}\dot{\mathbf{C_{1}}}+\mathbf{D}_{p}\mathbf{C_{1}},\overline{\mathbf{G}}_{p}=\mathbf{G}_{p}\), or more compactly,  
\[\mathbf{A}_{eq}\mathbf{\zeta}=\mathbf{b}_{eq} \tag{24}\]  
in which \(\mathbf{\zeta}=\left[ \begin{array}{cc}\mathbf{\tau}^{T}&\mathbf{Q}_{w}^{T}\end{array}  \right]^{T}\) include the input torques \(\mathbf{\tau}\), and the dynamic wrench magnitudes \(Q_{w}=[\![\tau_{w}\;n_{w}]^{T}\).