## 1 Introduction  
Nowadays, computer-controlled optical surfacing (CCOS) technology has been widely used to manufacture the telescope mirrors, optical lens, and molds, etc. [1]. The processed surface form error can usually reach tens of nanometers or even several nanometers. However, there still exists a significant challenge, which is the control of the part edge combined with the speed of processing to enhance the processing accuracy. Unlike ion beam figuring or fluid jet polishing [2], there are edge effects in most CCOS processes such as small tool polishing [3], magnetorheological finishing [4], bonnet polishing [5], and so on. The edge effect would lead to edge-roll, and it needs to be further corrected with smaller tools. It takes a large part of the processing time. To date, many studies on controlling the edge have been done since the application of CCOS technology. In 1982, [PERSON] [6] firstly adopted the arc raster path to successfully polishing a segmented mirror using the small tool polishing technology, with the initial error 0.0425A (\(\lambda\) = 632.8 nm) to the final error 0.012\(\lambda\) including the edge. [PERSON] et al. [7] studied the edge removal profile when the tool overhangs the workpiece edge in small tool polishing, and also presented an edge control algorithm for edge figuring [8]. [PERSON] et al. [9] proposed a skin model to evaluate the polishing pressure at the edge when polishing with a circular tool and correctly predicted that a greater amount of material was removed from the edge of the workpiece. [PERSON] et al. [10] analyzed the edge effect of the fluid jet polishing based on the Finite Element Analysis (FEA) method and finally proved that the edge effect could be neglected in this technology. [PERSON] et al. [11] provided a new parametric model to accurately predict the edge Tool Influence Function (TIF) for cases of polishing tools that are either spinning or orbiting over the edge of the workpiece. [PERSON] et al. [12] modeled the edge TIF of computer-controlled active lap polishing based on two methods: one is based on the pressure obtained through FEA, and another is based on a parametric equivalent pressure model. [PERSON] et al. [13] determined the dwell time using a space-variant deconvolution algorithm to control the edge error. Recently, [PERSON] et al. [14] presented a novel method to reduce the edge effect in the magnetorheological finishing (MRF) through further analyzing the stability of the MRF edge tool influence function (TIF) and correcting the post-edge algorithm of dwell time. Bonnet polishing is a promising CCOS process that has been successfully used to polish aspheric or freeform surfaces. It also faces the edge effect problem, especially when polishing the large segmented telescope mirrors. There are three main strategies to control the edge of the segmented mirrors presented by [PERSON] et al. [15]: (1) use of sacrificial wasters around the edge; (2) active control of edge-profiles by optimizing polishing process parameters; (3) controlled turning up of the edges during most of the process, followed by a final edge-rectification stage. However, the first strategy was rejected later because of the potential distortion of the surface induced by the adhesive-bonds and the risk with waster attachment and detachment [16]. The combination of strategies (2) and (3) assisting with the modeled edge tool influence function [17] had been used to effectively control the edge in the process-chain for E-ELT segments, which was using the Tool-Lift method generating various sizes of tool influence functions to produce surfaces with gently-sloping peripheral up-stands edges and followed by corrective finishing with smaller tools [18; 19; 20]. [PERSON] et al. [21] also did some experimentations on extreme ultraviolet blanks to optimize the polishing parameters over the edge. The edge specification for the segmented telescope mirrors is very stringent.