# A Flexible Localization Method of Multi-Constrained Free-Form Surface Based on the Profile Curves
## 4 Experiment  
To find a favorable to-be-cut surface before machining, we considered three optimization methods: unconstrained optimization, material-constraint optimization, and multi-constraint optimization. They are explained in detail as follows. **Unconstrained optimization**: The to-be-cut blade surface is localized in terms of minimizing the distance from the blade billet. **Material-constraint optimization**: In addition to minimizing the distance between to-be-cut blade surface and the blade billet, the to-be-cut blade surface is localized to ensure there is sufficient material. **Multi-constraint optimization**: In comparison with the material-constraint optimization, a profile tolerance constraint is added into the optimization model. In our experiment, a compressor blade was used to verify the advantage of a proposed localization model that could guarantee sufficient material allowance and blade profile tolerances. The measured points and the blade design surface before the localization are shown in Figure 7.  
Figure 7 demonstrates that there was an absence of material allowance in some areas of the design blade model. If the design blade model was used directly to generate the toolpath, the machining would fail. Thus, it was necessary to implement the localization before the subsequent machining. The above three optimizations of localization were implemented. The convergence rates of the three optimizations are shown in Figure 8. In the unconstrained optimization, the square sum of the unsigned distances that only let the to-be-cut blade surface be in the center of the blade biblet was minimized. In the material-constraint optimization, the square sum of the signed distances with all positive distances was minimized. In the multi-constraint optimization, the objective function was achieved with fewest iterations. Figure 8: Comparison of three kinds of optimization models. Figure 7: The design blade model and the blade biblet surface constructed by the measured points. The optimized to-be-cut blade surfaces after optimization are shown in Figure 9. The chart color represents the material allowance left on the optimized to-be-cut blade surface. In the case of the unconstrained optimization, the profile curves were adjusted to put the to-be-cut blade surface in the middle of the blade billet (Figure 9). However, there was not enough material because this was be ensured in the optimization model. In the case of material-constraint optimization, there was sufficient material for subsequent machining (Figure 10). There was enough material because of the material constraint in the optimization model. In the case of the multi-constraint optimization, the material allowance was satisfied for subsequent machining (Figure 11). This achieved the same effect as the previous optimization method. The distances between the optimized profile curves and the design profile curves for the unconstrained optimization are plotted in Figure 12. It can be observed that the profile curves were outside the profile tolerance. That is to say, the to-be-cut blade surface was not a qualified surface for generating a toolpath. Figure 13 shows the distances between the optimized profile curves and the design profile curves for the material-constraint optimization. Similarly, the to-be-cut blade surface is not a qualified surface due to the off-tolerance profile curves. The distances between the optimized profile curves and the design profile curves are plotted in Figure 14 for the multi-constraint optimization. In contrast to the results of the other two methods, they are within the profile tolerances. Thus, the to-be-cut blade surface reconstructed by the optimized profile curves is qualified. Figure 11: Material distribution on the to-be-cut blade surface by multi-constraint optimization. Figure 12: The distances from the profile curves to the tolerance bands with unconstrained optimization.