# A Flexible Localization Method of Multi-Constrained Free-Form Surface Based on the Profile Curves
## 2 Literature Review  
Many researchers have devoted much effort to workpiece localization in the past. Workpiece localization matches the billet and the to-be-cut model before the machining. It is categorized as a nonlinear least-squares problem to keep to-be-cut material uniform. Using a localization optimization algorithm, the appropriate orientation and position of the workpiece can be obtained which satisfy a user-defined set of constrains. In the past, the most popular method was the iterative closest point (ICP) method, which was first proposed by [PERSON] and [PERSON] [1]. [PERSON] et al. [2] analyzed the sensitivity of this method and concluded that the ICP algorithm is efficient and robust. Subsequently, [PERSON] et al. [3] put forward the iterative closest line (ICL) and iterative closest triangle (ICT) algorithms because the ICP algorithm is sensitive to the direction of the initial black model. However, the iteration speed of the algorithm depends on the number of measurement points. Thus, the ICP algorithm is robust, but its efficiency is not very high. To solve this problem, scholars have made further improvements to the algorithm. [PERSON] et al. [4] proposed three algorithms, including the variational algorithm, the tangent algorithm, and the Hong Tan algorithm. A comparison of the reliability and validity of these algorithms shows that the Hong Tan algorithm has higher accuracy and efficiency. Consequently, [PERSON] and [PERSON] [5] proposed the particle swarm optimization (PSO) algorithm to achieve the accurate localization of the workpiece. Compared with the ICP algorithm, the PSO algorithm has a higher computational efficiency. However, the above algorithms easily become divergent if they do not have an appropriate initial value. In order to overcome this issue, [PERSON] et al. [6] developed a novel method whose objective function is the minimum distance. This method does not require an initial value and has a high precision and convergence rate in the calculation process. However, the drawback of the method is that it depends on a correction coefficient. The iterative process becomes divergent if the correction coefficient is chosen incorrectly. Subsequently, [PERSON] et al. [7] proposed a new localization method that is divided into two steps. In this method, the rough alignment is finished according to the Gauss curvature and the mean curvature of the model. The drawbacks of this method lie in the error from the curvature calculation of the nominal model and of the machined model. Besides, the curvature threshold of  
Figure 1: (**a**) The design surface is constructed based on profile curves by the lofting method; (**b**) different tolerance bands on different segments of the profile curve. two corresponding points is always artificially determined. Thus, it fails easy in the rough alignment. Similarly, [PERSON] et al. [8] also proposed two stages in the localization process. The rough localization is finished according to the three corners of the model. It is then combined with a genetic algorithm and simplex method to complete the precise alignment. However, the above algorithm degrades for models with symmetric structure. Subsequently, a unified algorithm for all kinds of symmetrical parts was proposed by [PERSON] et al. [9], which can be expressed by the same optimization objective function if the symmetrical subgroups of the work pieces are known. These studies improve the efficiency of localization. The above research is aimed at the alignment of the unconstrained model. However, due to the existence of multi-source error, it is likely that the design model cannot be wrapped in the billet model during an unconstrained alignment process. To deal with this problem, [PERSON] et al. [10] introduced the machining allowance problem which is defined as hybrid localization. The process is split into a location problem and a tolerance problem by using differential geometry and Lie group.