## 2 Maximum Performance Evaluation
### Numerical Results  
In this study, RecurDyn (FunctionBay, Inc., Seongnam, Korea, commercial multibody dynamics software) was used to evaluate the dynamic behavior of the DAM-3R and JAM-3R. To solve (1), the optimization framework was constructed based on the analysis module (RecurDyn) and the optimization module (in-house code written in MATLAB) (Figure 4). As shown in Figure 4, RecurDyn imports the CAD model of a target manipulator to create an initial model for the multibody dynamics analysis. The updated design variables are transferred to RecurDyn to translate and/or rotate the model. Then, RecurDyn performs multibody dynamics analysis and sends the simulation results to the optimization module. The optimization module evaluates the objective and constraint function values (end-effector velocity and thrusting force, respectively) and then calculates the sensitivity of each design variable by using the finite difference method (FDM). If the convergence criterion is satisfied, the optimization process is terminated. Otherwise, the above process continues. Sequential quadratic programming in the MATLAB optimization toolbox [26] was selected as the optimization algorithm. The computing environment for the simulation was as follows: AMD Ryzen 7 5800X 8-Core Processor (CPU), NVIDIA GeForce GTX 1080 Ti (GPU), and DDR4 64 GB (RAM). The parameters used in the simulation are summarized in Tables 2 and 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Design Variables**} & \multicolumn{2}{c}{**Joint 1**} & \multicolumn{2}{c}{**Joint 2**} & \multicolumn{2}{c}{**Joint 3**} \\ \cline{2-7}  & **Lower Bound** & **Upper Bound** & **Lower Bound** & **Upper Bound** & **Lower Bound** & **Upper Bound** \\ \hline \(\theta_{1}\) [\({}^{\circ}\)] & \(\theta_{1}\) (\({}^{\circ}\)) & \(\theta_{1}\) (\({}^{\circ}\)) & \(\theta_{2}\)(\({}^{\circ}\)) & \(\theta_{2}\)(\({}^{\circ}\)) & \(\theta_{3}\)(\({}^{\circ}\)) & \(\theta_{3}\)(\({}^{\circ}\)) \\ \(\delta_{j}\) [mm] & \(37.0\) & \(77.0\) & \(37.0\) & \(77.0\) & \(37.0\) & \(77.0\) \\ \(\delta_{j}\) [mm/s] & \(-1.8\) & \(7.1\) & \(-7.1\) & \(7.1\) & \(-7.1\) & \(7.1\) \\ \(\delta_{j}^{b}\) [mm/s] & \(-1.8\) & \(7.1\) & \(-7.1\) & \(7.1\) & \(-7.1\) & \(7.1\) \\ \hline
**Design Constants** & **Joint 1** & **Joint 2** & **Joint 3** & \\ \hline \(L_{j}\) [mm] & \(114.0\) & \(114.0\) & \(129.0\) & \(\\c_{j}\) [mm] & \(80.0\) & \(80.0\) & \(80.0\) & \(80.0\) \\ \(F_{\text{max}}\) [N] & \(195.9\) & \(55.8\) & \(55.8\) & \(55.8\) \\ \(s_{\text{max}}\) [mm/s] & \(1.8\) & \(7.1\) & \(7.1\) & \(7.1\) \\ \(M\) & & \(16\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design and control parameters of the DAM-3R. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Design Variables**} & \multicolumn{2}{c}{**Joint 1**} & \multicolumn{2}{c}{**Joint 2**} & \multicolumn{2}{c}{**Joint 3**} \\ \cline{2-7}  & **Lower Bound** & **Upper Bound** & **Lower Bound** & **Upper Bound** & **Lower Bound** & **Upper Bound** \\ \hline \(\theta_{1}\) [\({}^{\circ}\)] & \(\theta_{1}\) (\({}^{\circ}\)) & \(\theta_{1}\) (\({}^{\circ}\)) & \(\theta_{2}\)(\({}^{\circ}\)) & \(\theta_{2}\)(\({}^{\circ}\)) & \(\theta_{3}\)(\({}^{\circ}\)) & \(\theta_{3}\)(\({}^{\circ}\)) \\ \(\hat{\theta}_{j}\) [\({}^{\circ}\)/s] & \(-3.3\) & \(3.3\) & \(-13.2\) & \(13.2\) & \(-13.2\) & \(13.2\) \\ \hline
**Design Constants** & **Joint 1** & **Joint 2** & **Joint 3** & \\ \hline \(L_{j}\) [mm] & \(114.0\) & \(114.0\) & \(129.0\) & \(\\\tau_{\text{max}}\) [Nm] & \(12.3\) & \(3.5\) & \(3.5\) \\ \(\hat{\theta}_{\text{max}}\) [\({}^{\circ}\)/s] & \(3.3\) & \(13.2\) & \(13.2\) & \(13.2\) \\ \(M\) & & & \(16\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design and control parameters of the JAM-3R.