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The Moving Least Squares (MLS) and Moving Total Least Squares (MTLS) method are widely used for approximating discrete data in many areas such as surface reconstruction. One of the disadvantages of MLS is that it only considers the random errors in the dependent variables. The MTLS method achieves a better fitting accuracy by taking into account the errors of both dependent and independent variables. However, both MLS and MTLS suffer from a low fitting accuracy when applied to the measurement data with outliers. In this work, an improved method named as α-MTLS method is proposed, which uses the Total Least Square (TLS) method based on singular value decomposition (SVD) to fit the nodes in the influence domain and introduces a geometric characteristic parameter αto associated with the abnormal degree of nodes. The generated fitting points are used to construct the parameter and quantify the abnormal degree of the nodes. The node with the largest parameter value is eliminated and the remaining nodes are used to determine the local coefficients. By trimming only one node per influence domain, multiple outliers of measurement data can be effectively handled. There is no need to set threshold values subjectively or assign weights which avoids the negative influence of manual operation. The performance of the improved method is demonstrated by numerical simulations and measurement experiment. It is shown that the α-MTLS method can effectively reduce the influence of the outliers and thus has higher fitting accuracy and greater robustness than that of the MLS and MTLS method.

Edge effect is unavoidable in polishing process when the polishing pad passes by the workpiece edge, which influences the entire form accuracy of free-form surfaces of optical components in computer-controlled polishing, or even reduces their effective aperture. This article focuses on a theoretical and experimental investigation into the material removal influenced by edge effect for polishing along a certain path. The contact pressure distribution models for polishing the workpiece edge are summarized and modified into four representative models: linear model, skin model, linear skin model and divided skin model, which are adopted to calculate the theoretical material removal profiles orthogonal to the straight or curved polishing path, in this article. So, the material removal models are built in the process of polishing along tool path instead of polishing on a single point. And experiments are carried out to choose the most suitable contact pressure distribution model according to the comparison of experimental profiles and the different theoretical profiles. Experimental results reveal the influences of edge effect on material removal and a modified parameter is introduced into the theoretical material removal profile for the curved path to coincide with the experimental profile better. In addition, some qualitative analyses about how to reduce the edge effect are also given in this article.