# Reconfiguration for the Maximum Dynamic Wrench Capability of a Parallel Robot
## 3 Reconfiguration for the Maximum Allowable DWC
### Wrench Capability Formulation for a Predefined Trajectory  
To maximize the dynamic wrenches, a linear weighting combination of the dynamic wrench force and wrench moment is defined as the cost function  
\[G=G\left(\begin{array}{cc}\mathbf{a}_{1},&\ldots&,\mathbf{a}_{6}\end{array}\right)= \mathbf{\lambda}^{T}\mathbf{\zeta}=\left[\begin{array}{cc}\mathbf{0}_{1\times 6}&\mathbf{ \lambda}_{f}&\mathbf{\lambda}_{n}\end{array}\right]^{T}\mathbf{\zeta} \tag{25}\]  
or equivalently to minimize the following fitness  
\[\overline{G}=-G=-\mathbf{\lambda}^{T}\mathbf{\zeta} \tag{26}\]  
in which the user-defined weighting coefficients \(\mathbf{\lambda}_{f}\) and \(\mathbf{\lambda}_{n}\) are used to weight the relative importance of the considered applied dynamic wrench force and dynamic wrench moment.  
In order to ensure the practical feasibility of the adjustable parallel robot, the reconfiguration must take account of the physical constraints and design specifications imposed by the parallel robot structure as well as the actuator mechanism on the motion of the manipulator. Broadly speaking, the following inequalities are required to model these constraints:  
Each motor \(i\) has a designed maximum output torque value \({}^{i}\tau_{max}\). Accordingly, the motor torque \(\tau_{i}\) is constrained as  
\[\left|\frac{\tau_{i}}{\tau_{max}}\right|\leqslant 1,\;\;\text{for}\;\;i=1, \ldots,6 \tag{27}\]  
In addition, due to the specified moving range of the base points A\({}_{i}\), the distance from the origin of the \(N\)-frame must be confined to \[\boldsymbol{\gamma}_{i}^{min}\leq|a_{i}|\leq\boldsymbol{\gamma}_{i}^{max},\ \text{for}\ \ i=1,\ldots,6 \tag{28}\]  
In the current study, the aim of this optimization problem is to determine the optimal configuration along the given task trajectory considering the maximum allowable dynamic wrench capability. The allowable dynamic wrench capability is defined as the following: firstly, by discretizing the task trajectory under a presumed configuration \(\boldsymbol{a}_{i}\), the dynamic wrench force \(f_{w}\) and wrench moment \(n_{w}\) at each spaced point are obtained by minimizing the fitness function given in Equation (26) subject to the equality constraint Equation (22) and the inequality constraints formulated in Equations (27) and (28). Then, the minimal dynamic wrench force and wrench moment are searched along the trajectory and then substituted into the Equation (25) so that the cost function value is defined as the allowable dynamic wrench capability for the task-dependent trajectory.  
Conventionally, a discretizing and searching method is applied to this constrained optimization problem by dividing each \(\boldsymbol{a}_{i}\) (\(i=1,\,\ldots,\,6\)) into \(n\) discrete points between \(\boldsymbol{\gamma}_{i}^{min}\leq|a_{i}|\leq\boldsymbol{\gamma}_{i}^{max}\), and then for the specified trajectory, the corresponding allowable dynamic wrench capability for all the \(n^{6}\) configurations of the parallel robot can be determined by the constrained optimization method. Accordingly, the reconfiguration with the maximum allowable dynamic wrench capability can be made by comparing the \(n^{6}\) solutions. However, it is very time-consuming and arduous for a large \(n\). Instead, a hierarchical optimization process is proposed in this paper.