## 7 Results of Optimization and Discussion  
In Table 1, the search intervals for all design variables are given. A minimum for the objective function (16) was found in the given search intervals for the ten design variables by using the function fmincon [36]. The results were the same for all initial estimates. They were checked by the function ga [37]. Optimization results (optimum values of the design variables) obtained are as follows:  
\(r_{2,\text{opt}}=0.0022\) m, \(\theta_{2,\text{opt}}=0.069813\) rad, \(s_{2,\text{opt}}=0.0038\) m, \(\psi_{2,\text{opt}}=1.090381\) rad,  
\(x_{\text{K,opt}}=-0.07\)m, \(y_{\text{K,opt}}=-0.05\) m, \(x_{\text{M,opt}}=0.18\) m, \(y_{\text{M,opt}}=-0.004\) m, \(c_{\text{d}}=0.15\),  
\(c_{\text{c}}=1.721687\). The minimum value of the objective function (16) is \(f_{\text{op}}=0.111182\) N.  
The lower bounds of the following design variables were reached: \(r_{2}\), \(x_{\text{K}}\), \(y_{\text{M}}\), and \(c_{\text{d}}\). Then, the upper bounds of the following design variables were reached: \(s_{2}\), \(\theta_{2}\), \(y_{\text{K}}\), and \(x_{\text{M}}\), see Table 1. The optimum values of the design variables are presented above. Theoretically, it would be possible to shift the lower and upper bounds of the active design variables to further reduce the value of the objective function \(f_{\text{op}}\), although this would no longer be appropriate from a design point of view. Figure 6 shows the dependence of the force difference \(F_{\text{f}}-F_{\text{gr}}\) on the jaw angular displacement \(\theta\). The difference between these two forces is less than 0.35 N (3.5% of the 10 N grasping force). The force difference in the substantial part of the jaw angular displacement interval (70%, except the 15% intervals with the lowest and highest angular displacements of the grasper jaw) is less than 0.1 N or 1% of the 10 N grasping force. It means that the optimized force transmission from the handle to the jaw of the grasper mechanism is very close to 1. In Figure 7 the dependence of the total elastic potential energy of the driving and compensation springs on the jaw angular displacement \(\theta\) is shown. It can be seen that the constraint on the maximum elastic potential energy is satisfied. Static balancing of mechanisms--e.g., in the works of [PERSON], is based on the assumption of constancy of potential energy (if kinetic energy is zero or a nonzero constant). In the case of the laparoscopic grasper mechanism considered here, although this condition is not satisfied,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variable** & **Units** & **Lower Bound** & **Upper Bound** \\ \hline \(r_{2}\) & m & 0.0022 & 0.0034 \\ \(\theta_{2}\) & \({}^{\circ}\) & 1 & 40 \\ \(s_{2}\) & m & 0.0028 & 0.0038 \\ \(\psi_{2}\) & \({}^{\circ}\) & 20 & 70 \\ \(x_{\text{K}}\) & m & \(-0.07\) & \(-0.02\) \\ \(y_{\text{K}}\) & m & \(-0.03\) & \(-the force ratio between the grasper handle and its jaw is important and, as can be seen above, is sufficiently close to 1. The maximum allowable compression force \(F_{\text{rod,max}}\) in connecting rod 6 was not exceeded, as can be seen in Figure 8. Negative values of this force only mean that there is no tension in this link (as assumed in the free body diagrams of rollers 2 and 4), yet there is compression. Figures 9 and 10 show the courses of the forces of \(T_{\text{A}}\) and \(T_{\text{B}}\) in the flexible steel bands. It can be seen that the \(T_{\text{A}}\) force acts opposite to what was assumed in the free body diagram of roller 2 (Figure 4). The \(T_{\text{B}}\) force changes its direction as the rotation of the movable jaw 3 increases (Figure 5). Figure 8: Dependence of the axial compression force in the connecting rod \(F_{\text{rod}}\) on the jaw angular displacement \(\theta\). Figure 7: Dependence of the total elastic potential energy of the driving and compensation springs on the jaw angular displacement \(\theta\).