## 5 Results and Discussion

Table 4: Effect of termination value on the convergence speed of GA.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Target Points** & \multirow{2}{*}{**Feasibility**} & \multirow{2}{*}{**Feasibility**} & \multirow{2}{*}{**Seolutions of the Proposed**} & \multirow{2}{*}{**Solutions of the Proposed**} & \multirow{2}{*}{**Error Sum of Two Fingers**} \\  & **(mm)** & & & & **Algorithm** & **(mm)** \\ \hline \multirow{6}{*}{\#1} & \multirow{6}{*}{Index finger: (50, 0, 130)} & & \(\theta_{w1}\) & \(14.4981^{\circ}\) & \\  & & & \(\theta_{w2}\) & \(-5.0000^{\circ}\) & \\  & & & \(\theta_{w1}\) & \(14.9817^{\circ}\) & \\  & & \(\theta_{w1}:12^{\circ}\) & \(\theta_{t1}\) & \(-8.4396^{\circ}\) & \\  & & \(\theta_{w2}:-15^{\circ}\) & \(\theta_{t3}\) & \(-21.9780^{\circ}\) & 0.5795 \\  & & & \(\theta_{t1}\) & \(-45.0001^{\circ}\) & \\  & & & \(\theta_{t2}\) & \(55.2968^{\circ}\) & \\  & & & \(\theta_{t3}\) & \(45.0110^{\circ}\) & \\ \hline \multirow{6}{*}{\#2} & \multirow{6}{*}{Index finger: (45, 52, 172)} & & \(\theta_{w1}\) & \(0.2515^{\circ}\) & \\  & & & \(\theta_{w2}\) & \(-4.9987^{\circ}\) & \\  & & & \(\theta_{t1}\) & \(15.0110^{\circ}\) & \\  & & & \(\theta_{w1}:0^{\circ}\) & \(\theta_{t2}\) & \(-19.6923^{\circ}\) & \\  & & & \(\theta_{w2}:0^{\circ}\) & \(\theta_{t3}\) & \(-8.8571^{\circ}\) & 0.5875 \\  & & & & \(\theta_{t1}\) & \(-19.6374^{\circ}\) & \\  & & & & \(\theta_{t2}\) & \(14.5494^{\circ}\) & \\  & & & & \(\theta_{t3}\) & \(41.4726^{\circ}\) & \\ \hline \multirow{6}{*}{\#3} & \multirow{6}{*}{Index finger: (50, 0, 300)} & \multirow{6}{*}{Drop out} & \multirow{6}{*}{Not} & \multirow{6}{*}{Not Computed} & \multirow{6}{*}{Not Computed} \\  & & & & & & Computed \\ \cline{1-1} \cline{4-4} \cline{5-5}  & & & & & & Computed  
In the illustrated case #3, the target fingertip point sets were (50, 0, 300) and (60, 50, 125) for the index finger and thumb, respectively. As shown in Figure 7a, the target point of the index finger was obviously out of the workspace. In this circumstance, as shown in Figure 7b, the proposed hybrid GA immediately ceased the inappropriate motion command to prevent unnecessary computation. However, the other GAs still conducted computation until meeting certain termination criteria. As a result, Figure 7b shows high fitness values of the simple GA and IGA, while the proposed hybrid GA successfully stopped computing. The hybrid GA, thus, was successful in avoiding the unnecessary computation for a given inappropriate IK request. Results in Table 6 demonstrate the distribution of population by exemplary generations during the evolution process in the illustrated case #1 for the simple GA and the proposed hybrid GA. The blue dots represent all the thumbs' tips of the population in one generation, whereas the red dots serve as those of the index finger. The corresponding cross marks stand for the target point or the answer of the IK problem. In the initial generation,  
Figure 6: (**a**) Schematic illustration showing the solution resulting from an appropriate motion command and (**b**) the corresponding fitness performance of the illustrated case #1. Figure 7: (**a**) Schematic illustration showing the solution resulting from an inappropriate motion command and (**b**) the corresponding fitness performance of the illustrated case #3.  
the population distribution of the simple GA was scattered more widely than that of the proposed hybrid GA. Because of the feedforward wrist angle, the proposed hybrid GA started with a smaller searching scope. When the generation increased, the proposed hybrid GA had more chances to get close to the target points and, therefore, converged faster than the simple GA.  
To surpass the performance of the other GA, the workspace of two fingers plays an important role in the proposed hybrid GA. Initially, the workspace analysis can provide the nearest distance between the target point and the workspace. As a result, the feasibility of the IK request can be checked accordingly.