## 5 Results and Discussion  
In this section, the performance of the proposed hybrid GA is validated by examination through parametric numerical simulations. The simple GA and the IGA are also performed to compare the results. To highlight the capability of determining the feasibility, both appropriate and improper IK requests are given to different algorithms for computing IK solutions. The detailed parameters used in the illustrated cases through numerical simulations with the proposed algorithm are shown in Table 3.  
Among the parameters in Table 3, the fitness value of termination dominates the computation time. As exhibited in Table 4, when the termination value increases, it takes less time to compute, while the accuracy of the solution is sacrificed. It also compares the average execution time of the simple genetic algorithm and the performance of the proposed hybrid GA, which shows that the proposed hybrid GA is two times faster than the simple GA. Additionally, it is known that the selection of the fitness value of termination depends on the application scenario. Thus, the termination condition applied in different tasks was varied accordingly. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline GA type & Generational \\ Representation & Binary \\ Chromosome length & 80 \\ Population number & 500 \\ Selection & 10-tournament \\ Crossover & 2-point \\ Crossover rate & 0.8 \\ Mutation & Bit-flip \\ Mutation rate & 1/L \\ Survivor & 1000 \\ Termination & 3 s or fitness lower than 0.6 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters used in the proposed hybrid genetic algorithm. GA, genetic algorithm. In calculating the inverse kinematics, two points were assigned as the target points of the index finger and thumb. Different algorithms were performed to solve the inverse kinematics, and the best fitness values were recorded for the graphical representation to analyze the performance of each algorithm. Table 5 lists the conditions and results for three illustrated cases. The illustrated case #1 and case #2 were those with reachable target points, whereas an unreachable target point was assigned in the illustrated case #3. In the illustrated case #1, the target fingertip point sets were (50, 0, 130) and (75, 30, 125) for the index finger and the thumb, respectively. As shown in Figure 5(a), the two target points were in the workspace of two fingers and they passed the feasibility check. Additionally, the corresponding angle set of the wrist \((\theta_{w1},\ \theta_{w2})=(12^{\circ},\ 15^{\circ})\) was provided to the hybrid GA as the initial condition. By using the solutions computed by the GA, the tips of the index finger and the thumb reached their target points. The simple GA and IGA were also performed to solve the same IK problem as a comparison. As shown in Figure 5(b), the proposed hybrid GA was observed to have a lower fitness value than the other GAs in the beginning. The proposed hybrid GA was found to obtain the solution for the inverse kinematic problem faster than the other GAs with fewer generations. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Termination Fitness** & **Average Number of** & **Average Execution** \\  & **Value** & **Generation** & **Time (s)** \\ \hline Simple GA & & 52 & 4.41 \\ Proposed hybrid GA & 1 & 24 & 2.06 \\ \hline Simple GA & & 18 & 1.68 \\ Proposed hybrid GA & 5 & 10 & 0.72 \\ \hline Simple GA & & 7 & 0.63 \\ Proposed hybrid GA & 10 & 4 & 0.35 \\ \hline \hline \end{tabular}
\end{table}