# Reconfiguration for the Maximum Dynamic Wrench Capability of a Parallel Robot
## 3 Reconfiguration for the Maximum Allowable DWC
### Solution Procedure by a Hierarchical Optimization Process  
In order to determine the position vectors \(\boldsymbol{a}_{i}\) which maximize the allowable dynamic wrench capability of the associated motion, a hierarchical optimization process combining two optimization algorithms is proposed. As aforementioned, firstly for the initial presumed position vectors \(\boldsymbol{a}_{i}\) (\(i=1,\,\ldots,\,6\)) of the base points and at each discretizing point of the designated trajectory, the fitness function, Equation (26), is minimized under the constraint equations. Thus, the generated optimization process can be realized as a typical linear programming such that the Simplex-type linear programming method can be directly applied to determine the maximal dynamic wrench capability at each discrete point of the given trajectory. Next, from the obtained values, the minimum dynamic wrench force and moment are chosen as the allowable wrenches, and then the allowable dynamic wrench capability for this task trajectory is evaluated by substituting the minimum dynamic wrenches into the cost function, Equation (25). Afterwards, the position vectors \(\boldsymbol{a}_{i}\) of the base points \(\text{A}_{i}\) are varied to maximize the allowable dynamic wrench capability. For this optimization process, a population-based optimization technique, differential evolution, is employed. The differential evolution method simulates a natural evolution with a stochastic direct search mechanism according to the distribution of solutions in the current population [26; 27]. As the algorithm proceeds, all the chromosomes of the next generation are as good as or better than their counterparts in the current generation, and then the individual chromosomes gradually converge toward an optimal solution. Once the termination criteria have been attained (e.g., a specified number of iterations have been performed or the fitness error satisfies the minimum error requirement, for example), the algorithm terminates, and the current optimal chromosome is taken as the global optimal solution of the problem of interest. Due to a low computational complexity, differential evolution outperforms many other optimization methods in terms of convergence speed and robustness for some benchmark functions. Furthermore, differential evolution has very few control parameters to adjust and has therefore been applied to many diverse fields [28; 29; 30]. Based on the optimization procedures in the hierarchical optimization process, the differential evolution is a higher-level optimization solver for varying the position vectors of the base points, and the Simplex-type linear programming is designated as the lower-level optimization solver for the determination of the dynamic wrench capability. The procedures for the optimal geometry adjustment for a given trajectory can be described as follows:1. The higher-level optimization process is randomly initialized with \(P\) parent individuals according to a uniform probability distribution in the \(D\) dimensional problem space. Each parent individual is a set of undetermined position vectors of the base points, namely \(\hat{\textbf{X}}_{\mathbf{p}}\triangleq\left[ \begin{array}{cc}x_{p1}&\ldots&x_{pD} \end{array} \right]^{T}=\)[\(\textbf{{a}}_{\mathbf{p},\mathbf{1}}^{T}\),..., \(\textbf{{a}}_{\mathbf{p},\mathbf{6}}^{T}\)]\({}^{T}\), \(p=1\),..., \(P\), and refers to a candidate solution in a \(D\)-dimensional space based on the differential evolution algorithm for the optimization problem. Also, these candidate solutions must respect the constraint Equation (28). 2. In the lower-level optimization process, for each individual \(p=1\),..., \(P\), compute and record \(f_{\mathbf{i},\mathbf{n}}^{p}\) and \(n_{\mathbf{w},\mathbf{n}}^{p}\), \(n=1,\ldots,N\), for \(N\) discrete equally spaced points of the given trajectory by solving the linear programming problem for the minimization of the fitness function, Equation (26) with the equality and inequality constraints Equations (24) and (27). 3.