# Reconfiguration for the Maximum Dynamic Wrench Capability of a Parallel Robot
## 4 Examples
### Results and Discussion  
In solving the maximum dynamic wrench capability problem of the reconfigurable parallel robot, the positive weighting numbers in Equations (25) and (26) are specified as [\(\lambda_{f}\;\lambda_{n}\)] = [1.0 2.0] considering the desired relative magnitudes of the dynamic wrench force and wrench moment for this given trajectory. Then, an initial determination of the maximum DWC was carried out in which the proportional coefficients for the base point locations were set to be \(\eta_{i}\) = 0.1, (\(i\) = 1,..., 6), or the corresponding distances of the base points are \(r_{A_{i}}\) = 0.42 m. The trajectory is discretized into 51 points with the time step \(\Delta t\) = 0.1 s. The maximum dynamic wrench capability at each point is calculated by the Simplex-type linear programming method, and hence the dynamic wrench force and wrench moment at each point of the prescribed trajectory is shown in Figure 3. It is shown that the minimum dynamic wrench force is 189.6962 N at time \(t\) = 2.3 s and the minimum dynamic wrench moment is 38.3488 Nm at time \(t\) = 5.0 s. Thus, the allowable dynamic wrench capability is 266.3938 for the specified initial base points. Figure 3: Variations of dynamic wrenches at each point of prescribed trajectory. As in the aforementioned, the proposed hierarchical optimization approach combining the differential evolution and the Simplex-type linear programming is used to perform a series of simulations in which the optimal base point locations for an improvement of the allowable dynamic wrench capability are obtained. Furthermore, the performance of a differential evolution algorithm for the formulated maximum dynamic wrench capability problem usually depends on three variables: the population size \(P\), the mutation factor \(f_{m}\), and the recombination rate \(CR\). Thus, in accordance with the results of a few trial and error simulations, in which \(P=25,f_{m}=0.5\) and \(CR=0.8\) are specified here for a trade-off consideration. In the example, the base points are displaced equidistantly such that only one proportional coefficient \(\eta\) is determined for the maximum allowable dynamic wrench capability. Figure 4 shows that the proposed hierarchical optimization algorithm for the maximum allowable dynamic wrench capability leads to a high convergence rate. The obtained convergent proportional coefficients in Figure 5 are \(\eta_{i}=0.3684\), (\(i=1,\,\ldots,6\)), or the corresponding distances of the base points are \(r_{A_{i}}=0.7421\) m. The corresponding allowable maximum dynamic wrench capability, the dynamic wrench force and wrench moment are 408.4895, 224.8770 N and 91.8063 Nm, respectively. It is found that the allowable dynamic wrench capability has been improved. Figure 6 presents the variations of the dynamic wrenches along the prescribed trajectory for this configuration. The minimum dynamic wrench force/moment are still at time \(t=2.3\) s and at time \(t=5.0\) s, respectively. Figure 4: Allowable dynamic wrench capability _versus_ iteration number for base point locations being displaced equidistantly. Figure 5: Base point position _versus_ iteration number for base point locations being displaced equidistantly. Moreover, Table 1 lists the convergence analyses for different population size \(P\) = 15, 25, 35. As shown, the population size \(P\) = 35 goes through fewer iterations to reach the specified convergence criteria because of the wider and more random coverage of the search space; however, the computation time increases with an increase in the required function evaluation. Therefore, the population size \(P\) = 25 is adopted for the differential evolution algorithm. Finally, the base points of the parallel robot are relocated individually to further improve the maximum allowable dynamic wrench capability using the proposed hierarchical optimization. Figures 7 and 8 respectively present the convergence process for the maximum allowable dynamic wrench capability and the six base point positions.