## 4 Multi-Objective Optimization with Equivalent Static Loads for Parallel Mechanism  
### Equivalent Static Loads  
In MSA, the PM in the workspace is considered a series of structures. Therefore, we added the configuration of the PM as a new dimension to the structural optimization procedure. The equivalent static loads were modified for the elastodynamic optimization of the PM. In ESLs, the dynamic response is transferred into a series of equivalent static loads. Therefore, in the sub-optimization procedure, only static equilibrium algebraic equations need to be solved. The modified ESL procedure is shown in Figure 8.  
As shown in Figure 8, we first chose a set of initial design variables \(\mathbf{b}^{k}\). The design variables are usually the dimension parameters of the PM, including the structural parameters, such as the radii of the links, and mechanism parameters, such as the length of the base and platform.  
In the first step, the configuration sample points \(\mathbf{d}^{i}(k=1,2,\cdots,\text{num}_{\text{c}})\) under the current design variables \(\mathbf{b}^{k}\) were sorted from the mechanism workspace. For each configuration point \(\mathbf{d}^{i}\), the vibration equation was solved and the dynamic response \(\mathbf{z}(\mathbf{b}^{k},\mathbf{d}^{i},t^{i})(j=1,2,\cdots,\text{num}_{\text{t}})\) was obtained. The dynamic response \(\mathbf{z}\) was sampled at time sampling points \(t^{j}\). The vibration equation was obtained from Equations (19) and (20). Since the dimension of the vibration equation is relatively large, usually several hundred, the process is very time-consuming, even within a small time span. However, the vibration equation only needs to be solved one time in each cycle. Therefore, the ESLs greatly reduce the optimization time.  
In the next step, a series of equilibrium static loads \(\mathbf{f}_{eq}(\mathbf{b}^{k},\mathbf{d}^{i},s^{j})\) is calculated by multiplying the dynamic response \(\mathbf{z}(\mathbf{b}^{k},\mathbf{d}^{i},t^{i})\) with the stiffness matrix \(\mathbf{K}(\mathbf{b}^{k},\mathbf{d}^{i})\). The sampling points are equal to the time sampling points \(s^{j}=t^{j}\). When this series of loads \(\mathbf{f}_{eq}\) is statically added to the structure, it can create the same dynamic response in the dynamic vibration process.  
Figure 8: The procedure of equivalent static loads by adding mechanism configuration.  
In the next step, namely linear static optimization, \(f_{eq}(\mathbf{b}^{k},\mathbf{d}^{i},s^{j})\) is considered a constant in this sub-optimization procedure. In this procedure, the classical static optimization process is implemented. The objective function is \(D\). The constraint equation is \(\mathbf{K}(\mathbf{b}^{k},\mathbf{d}^{i})\mathbf{z}=f_{eq}(\mathbf{b}^{k},\mathbf{d}^{i},s^{j})\) and \(\mathbf{g}(\mathbf{b},\mathbf{d}^{i},\mathbf{z})\leq\mathbf{0}\), and the boundary of the design parameter is \(\mathbf{b}_{min}\leq\mathbf{b}\leq\mathbf{b}_{max}\). In this sub-optimization procedure, optimized design parameters \(\mathbf{b}\) will be found. Then, they are used in the next iteration process, \(\mathbf{b}_{k+1}=\mathbf{b}\). The iteration converges and stops when the discrepancy of the adjacent two sets of parameters is within an acceptable range, \(|\mathbf{b}_{k}-\mathbf{b}_{k+1}|\leq\epsilon\).