## 4 Design  
In normal cutting conditions, the mechanism should have high stiffness to ensure high cutting precision. On the other hand, when the burr is too high the mechanism should exploit its compliance to reduce the cutting forces and, as a consequence, a second cutting cycle is necessary to completely remove the burr. The desired regressive behavior of the mechanism is related to its geometrical parameters and to the springs stiffness. If the link lengths (\(l_{1}\) and \(l_{2}\)) are defined by the scale of the application (related to the workpiece dimensions and maximum burr height), there are 5 other parameters to be chosen (see Figure 5): \(k_{1}\), \(k_{2}\), \(k_{3}\), \(a\), and \(b\). It is not straightforward to find a good set of values by trial-and-error, so an optimization algorithm has been used. The algorithm aims at finding the parameters whose resulting mechanism forces \(F\) fit a desired (regressive) curve. In this work the desired force profile is set to \(F=150-300x\) [\(N\)] (blue dotted line in Figure 7) with \(l_{1}=0.15\) m and \(l_{2}=0.3\) m. The _fmincon_ MATLAB function has been used to find the optimal parameters that minimize the error between the actual and desired force profiles. The optimization algorithm yields the values listed in Table 2. It is worth to notice from Figure 7 that the system is stable: the normal component of the deburring force (\(F_{b}\)) decreases with \(x\) faster than the elastic force on the slider (\(F\)):  
\[\frac{\partial F_{b}}{\partial x}<\frac{\partial F}{\partial x} \tag{20}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Mechanism \#1** & **Mechanism \#2** \\ \hline \(l_{1}\) & & 0.15 m \\ \(l_{2}\) & & [0.15, 0.2, 0.3, 0.4] m \\ \(a\) & 0.05 m & 0.14 m \\ \(b\) & 0.15 m & 0.03 m \\ \(c\) & & 0.05 m \\ \(k_{1}\) & & 1500 N/m \\ \(k_{2}\) & 3000 N/m & 0 \\ \(k_{3}\) & 200 N/m & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main mechanism parameters. Figure 7: The optimization algorithm finds the mechanism parameters whose \(F(x)\) (red line) fits at best the objective curve (blue dotted line). Black line shows how \(F_{b}(x)\) decreases with the movement of the mechanism (with \(h=5\) mm). The black line of Figure 7 is the deburring force (normal component, as discussed in Section 2). This curve moves to the right for higher burr heights (with higher cutting forces for \(x=0\)), and vice versa. The intersection between this line and the optimized curve (red line in figure) yields the equilibrium point, i.e., the \(x\) value at which the mechanism moves to balance the deburring force and the elastic force of the mechanism. For very low values of \(h\) the black line does not intersect the optimized curve, so the mechanism does not move. In this case, a precision cut is ensured, with cutting forces below the force threshold that activates the mechanism compliance. For the parameters of Table 2 and Figure 7 (see Table 3 in Section 5.3 for the complete list of parameters) the burr height (steel) that activates the mechanism compliance is 2.5 mm. It is important to notice that it is possible to perform the optimization starting from a different desired force curve and, in particular, different force thresholds can be set depending on the application. This will result in a different set of springs and anchor points. From a mechanical point of view, it is important to include a mechanical end stop for the slider to avoid possible changes in configuration of the slider-crank (i.e., to keep the rocker arm link always above the slider axis) and to avoid that the mechanism works in the TDC. Indeed, the mechanism is designed to work near the TDC, where it can generate a finite elastic force, but not exactly in the TDC, which could be dangerous for the structural integrity of the mechanism if the burr height is very large. The detailed model of the end stop will be presented in Section 5.2.