# A Flexible Localization Method of Multi-Constrained Free-Form Surface Based on the Profile Curves
## 3 The Localization Problem of the Near-Net Shape Blade Surface  
### Establishment of the Localization Optimization Model  
For the free-form surface of a blade, the profile curves on the design blade sections are the basis of controlling the surface shape (Figure 1a). The blade surface is constructed by lofting these profile curves. That is to say, the to-be-cut blade surface can be reconstructed in the same way. Thus, it is important to find a group of appropriate profile curves. The to-be-cut blade surface is represented by \(r(u,v)\). The blade billet shape is used to judge the material allowance for the to-be-cut blade surface. The blade billet shape is always recognized by on-machine measurement. The to-be-measured points are planned on the designed blade profiles. The measured result represents the real material boundary on the profile curves section (Figure 2). They are represented by \(P_{M}=\{p_{i}|i=1,2,\cdots n\}\). The blade surface is usually divided into the suction-surface segment, the pressure-surface segment, the leading-edge segment, and the trailing-edge segment. Different tolerances are required for these surface areas. After the blade surface localization, the profile of the to-be-cut blade surface segment needs to be within the respective profile tolerance zone (Figure 3). The to-be-cut material on the blade billet is more uniform, and the cutting status is more stable. Therefore, the objective of the proposed localization is to make the material allowance more uniform. To obtain a favorable to-be-cut blade surface, we establish a flexible localization implemented by a multi-constraint optimization. **Optimization variable:** the translation and the rotation matrices of the profile curves on the designed blade sections. The translation matrix is denoted as **T**, and the rotation matrix is denoted as **R**. Each profile curve is localized by its translation matrix and realized by the rotation matrix. The to-be-cut blade surface is reconstructed by the lofting method in terms of the respective profile curves that were localized. Figure 3: The profile curves of the to-be-cut blade surface need to be within the tolerance band. **Optimization Objective**: The objective of the uniform to-be-cut material is substituted by the distance between the measured points of the blade billet and the to-be-cut blade surface. It is represented by  
\[\text{min}f\left(\mathbf{R}_{j},\mathbf{T}_{j}\right)=\text{min}\left(\sum_{i=0 }^{n}\|d_{i}\|^{2}\right)=\text{min}\left(\sum_{i=0}^{n}\|r(u,v)-p_{i}\|\right) \tag{1}\]  
where \(d_{i}\) are the distances from the measured points to the closest points on the to-be-cut blade surface. **Material allowance constraint:** The first term is the material allowance constraint. The to-be-cut blade surface should be completely covered by the blade billet. It is represented by  
\[h1\left(\mathbf{R}_{j},\mathbf{T}_{j}\right)=\left(p_{i}-r(u,v)\right)\cdot \mathbf{n}_{i}\geq 0 \tag{2}\]  
where \(\mathbf{n}_{i}\) is the normal of the closest point on the to-be-cut blade surface corresponding to the measured point. **Profile tolerance constraint:** The different segments of the profile curves that belong to the optimized blade surface need to be within corresponding profile tolerances (Figure 4). They are represented by  
\[h2\left(\mathbf{R}_{j},\mathbf{T}_{j}\right)=\left(\mathbf{C}_{p,k}-\mathbf{ S}_{p,k}\right)\cdot\mathbf{n}_{p,k}\geq 0\ \ \ \ \ p=1\cdots m,k=1\cdots l \tag{3}\]  
\[h3\left(\mathbf{R}_{j},\mathbf{T}_{j}\right)=\left(\mathbf{C}_{q,k}-\mathbf{ S}_{q,k}\right)\cdot\mathbf{n}_{q,k}\leq 0\ \ \ \ \ q=1\cdots m,k=1\cdots l \tag{4}\]  
where \(\mathbf{S}_{p,k}\) represent the closest points on the suction segment and the pressure segment of the profile curves, \(\mathbf{S}_{q,k}\) represent the closest points on the leading-edge segment and the trailing-edge segment of the profile curves. \(\mathbf{C}_{p,k}\) and \(\mathbf{C}_{q,k}\) are the scatter points on the tolerance boundaries.