# Reconfiguration for the Maximum Dynamic Wrench Capability of a Parallel Robot
## 4 Examples
### Kinematic Parameters and Differential Evolution Variables  
The system parameters used for the simulation are given as the following,  
1. Platform points in the \(B\)-frame: \[B_{2j-1}=r_{B}\left[ \begin{array}{cc}\cos((2j-1)\pi/3-\pi/18&\sin((2j-1)\pi/ 3-\pi/18)&0\end{array} \right]^{T},\] \[B_{2j}=r_{B}\left[ \begin{array}{cc}\cos((2j-1)\pi/3)&\sin((2j-1)\pi/3&0 \end{array} \right]^{T},j=1,\ldots,3\] with the radius of the moving platform \(r_{B}\) = 0.3 m.
2. Base points in the \(N\)-frame: \[A_{2j-1}=r_{A_{2j-1}}\left[ \begin{array}{cc}\cos(2(j-1)\pi/3)&\sin(2(j-1)\pi/ 3&0\end{array} \right]^{T},\] \[A_{2j}=r_{A_{j}}\left[ \begin{array}{cc}\cos(2(j-1)\pi/3+\pi/18)&\sin(2(j -1)\pi/3+\pi/18&0\end{array} \right]^{T},j=1,\ldots,3.\]  
in which \(r_{A_{i}},i=1,\ldots,6\) is the distance of the \(i\)th base point from the origin of the inertial \(N\)-frame. Inasmuch as the disposed base locations may not exceed the specified ranges, the distance of the \(i\)th base point can be assigned as  
\[r_{A_{i}}=(1-\eta_{i})r_{B}+\eta_{i}r_{Amax},\;0\leqslant\eta_{i}\leqslant 1 \tag{32}\]  
where \(r_{Amax}\) is the maximum allowable distance of the base points, and \(\eta_{i}\) (\(i\) = 1, \(\ldots\), 6) are the proportional coefficients such that the positions of each base point will be confined between \(r_{B}\) and \(r_{Amax}\) = 1.5 m. As a result, the optimal geometry for the reconfigurable parallel robot with the maximum dynamic wrench capability is equivalent to finding the proportional coefficients. Furthermore, all the links are considered to be identical and to have a lower segment of mass \(m_{i1}\) = 2 kg as well as an upper segment of mass \(m_{i2}\) = 1 kg. The lower segment of each link incorporates a ball-screw system with a pitch \(p_{i}\) of 5 mm/rev. The unit vector \(\mathbf{s}_{i}\) expressed in the \(i_{1}\)-frame is given by \(\mathbf{s}_{i}\)=\(\left[ \begin{array}{cccc}1&0&0\end{array} \right]^{T}\). The gravity centers of the two segments of each link are respectively \(\mathbf{d}_{\mathbf{i}\mathbf{1}}\) = \(\left[ \begin{array}{cccc}0.5&0&0\end{array} \right]^{T}\) m and \(\mathbf{d}_{\mathbf{i}\mathbf{2}}\) = \(\left[ \begin{array}{cccc}0.5&0&0\end{array} \right]^{T}\) m. The constant length of the upper segment 2 is \(l_{i2}\) = 1.0 m. The moments of inertia of these two segments of each link about their respective gravity centers in the local frame are \(\mathbf{{}^{C}}\mathbf{I}_{\mathbf{i}\mathbf{1}}\) = \(\mathbf{diag}\)[0.001, 1, 1] kg\(\cdot\)m\({}^{2}\), \(\mathbf{{}^{C}}\mathbf{I}_{\mathbf{2}}\) = \(\mathbf{diag}\)[0.0001, 0.4, 0.4] kg\(\cdot\)m\({}^{2}\). The moments of inertia of the segments 1 and 2 of each link about the respective joint points \(\mathrm{A}_{i1}\) and \(\mathrm{A}_{i2}\) in the local frame can be calculated using the parallel axes theorem as \(\mathbf{I}_{ij}\) =\(\mathbf{{}^{C}}\mathbf{I}_{ij}+m_{ij}\widetilde{\mathbf{d}}_{ij}^{T}\widetilde{\mathbf{d}} _{ij}\), \(j\) = 1, 2. The vector of gravity is \(g\) = \(\left[ \begin{array}{cccc}0&0&-9.81\end{array} \right]^{T}\)m/s\({}^{2}\). The trajectory of the center of mass \(B\) of the moving platform is given as \(\mathbf{p}\) = [0.25\(\sin(0.4\pi\;t)\) 0.3\(\cos(0.4\pi t)\) 1.2 + 0.1\(\sin(0.8\pi t)\)]\({}^{T}\) and \(\mathbf{\gamma}_{B}\) =[0.2\(\sin(0.2\pi t)\) 0.3\(\cos(0.4\pi t)\) 0.25\(\sin(0.1\pi t)\)]\({}^{T}\), and the applied directions of the dynamic wrench force and wrench moment are individually specified as (\(\alpha_{f}\), \(\beta_{f}\)) = (\(\pi/6\), \(\pi/3\)), and (\(\alpha_{n}\), \(\beta_{n}\)) = (\(\pi/3\), \(\pi/6\)) with respect to the \(B\)-frame. Moreover, it is assumed that this trajectory satisfies the kinematic constraints of the parallel robot systems, _i.e._, the link lengths, velocities and accelerations. In addition, the maximum motor torques are specified as \(\left[ \begin{array}{ccc}|\tau_{1}|&\ldots&|\tau_{6}|\end{array} \right]\leqslant \left[3.5\;4\;2.5\;2.5\;3\;3.5\right]\) Nm.