## 1 Introduction
The points used for the T-spline surface were obtained by using the on-machine measurement, and then the mirror symmetry model was adopted to construct the compensation surface. [PERSON] et al. [7] performed machining surface error compensation through an on-machine measurement system. They performed multiple machining to obtain the error distribution. Based on the relationship between cutting conditions and surface error, an attempt was made to correct the tool path in order to reduce the errors using an iterative algorithm. [PERSON] et al. [8] classified the errors in the machining of free-form surfaces as machine-related errors and process-related errors. The machine-related errors can be compensated offline, and the process-related errors can only be compensated online by re-planning the tool paths. [PERSON] et al. [9] developed the global offset method to compensate for the machining errors of a five-axis machine tool. The method was based on the off-machine measurement of machined parts by using a coordinate measuring machine (CMM), and a compensation processor was used to calculate the parameters for the controller of the machine tool to perform compensation. [PERSON] et al. [10] decomposed the machining errors into systematic errors and random errors by using the empirical mode decomposition method, and the systematic errors can  
Figure 1: Illustration of the surface after semi-finishing machining and after finishing machining; (a) target surface of machining; (b) the profile after semi-finishing compared to the target surface; (c) the profile after finishing compared to the target surface. then be compensated by using the offline measurement data. [PERSON] et al. [11] developed an iterative compensation algorithm to modify the NC code based on the geometric errors, so that the real tool path generated by the compensated NC code and the designed tool path were within the tolerance. [PERSON] et al. [12] established the error of kinematics synthesis model and used the model to compensate the \(x\)-axis and \(z\)-axis positioning errors of a four-axis polishing machine. [PERSON] et al. [13] presented a method of compensation by the insertion of the nodes to reduce the errors caused by the Bspline and Cspline interpolations on the machining surface. [PERSON] [14] used an artificial neural network to predict the thermal error and used a PC-based controller to send the compensation information to the CNC controller to perform real-time error compensation. Because the on-machine measurement depends on the precision of the machine tool, the source of the error on the machine tool is also critical. [PERSON] et al. [15] used the online servo-controlled inclinometers to measure the geometric errors of the motion axes. That information was used by the CNC controller to compensate for the positioning errors. [PERSON] et al. [16] used on-line measurement data for achieving compensation. Dimensional compensation can be achieved by correcting the tool size and the tool path. They proposed a method based on the cutting depth distribution to correct the tool offset error. Regarding manufacturing the free-form surfaces, some novel methods have been proposed. [PERSON] et al. [17] proposed an algorithm to find patches on free-form surfaces that can be machined by the flank of a conical cutting tool, and they extended their work to find the initial path for five-axis flank CNC machining with a general cutting tool [18]. [PERSON] et al. [19] also developed an algorithm to generate the flank milling paths for free-form surface machining with the use of a tapered mill tool. Using the flank of a cutting tool can increase the efficiency of manufacturing the free-form surfaces. Because gear surfaces may be treated as free-form surfaces, we can also find the information related to making free-form surface from gear manufacturing. [PERSON] et al. [20] presented machining strategies and parameters used in different stages for milling the gear surfaces of spiral bevel gears by using NX CAD/CAM software.