Five-axis Tri-NURBS Spline Interpolation Method Considering Nonlinear Error Compensation and Correction of CC Paths

In order to improve the accuracy of tool axis vector position and direction in traditional five-axis NURBS interpolation methods and the controlling accuracy of cutter contacting(CC) paths between cutter and work-piece, a five-axis Tri-NURBS spline interpolation method is presented in this article. Firstly, the spline interpolation instruction format is proposed, which includes three spline curves, such as CC point spline, tool center point spline and tool axis point spline. The next interpolation parameter is calculated based on the tool center point spline combined with the conventional parametric interpolation idea. Different from the traditional spline interpolation using the same interpolation parameter for all spline curves, the idea of equal ratio configuration of parameters is proposed in this paper to obtain the next interpolation parameter of each spline curve. The next interpolation tool center point, tool axis point and CC point on the above three spline curves can be obtained by using different interpolation parameters, so as to improve the accuracy of tool axis vector position and direction. Secondly, the producing mechanism of CC paths’ nonlinear error of the traditional spline interpolation is analyzed and the mathematical calculation model of the nonlinear error is established. And then, the nonlinear error compensation and correction method is also put forward to improve the controlling accuracy of CC paths. In this method, the next CC point on the cutter can be firstly obtained according to the next interpolation tool center point, tool axis point and CC point on the three spline curves. And then, the error compensation vector is determined with the two next CC points. To correct the nonlinear error between the next CC point on the cutter and the CC point spline curve, the cutter is translated so that the two next CC points can be coincided. In the end, the new tool center point and tool axis point after translation can be calculated to obtain the motion control coordinates of each axis of machine tool. The MATLAB software is used as simulation of the real machining data. The results show that the proposed method can effectively reduce the CC paths’ nonlinear error. It has high practical value for five-axis machining in effectively controlling the accuracy of CC paths and im-proving the machining accuracy of complex


Introduction
Five-axis CNC machines are widely used for machining precision devices such as impellers, blades, and precision optical parts due to their high speed, high precision, and the ability to machine complex free-form surfaces.
Compared with the three-axis CNC machine, it has two extra axes endowing it with a higher degree of machining freedom. However, a new principle error, the nonlinear error [1][2][3][4] , is caused by the rotary motion of the cutter, which has become an urgent problem to be solved in the field of five-axis machining technology.
Scholars around the world have proposed various solutions to the nonlinear error control problem of the five-axis machine. Wu Jichun et al. [5] established a nonlinear error model based on harmonic functions according to the error distribution in the classical post-processing to achieve the real-time error compensation at the middle interpolation points. Yang Xujing et al. [6] and Zheng Fengmo et al. [7] controlled the nonlinear error by interpolation at the endpoint of the tool axis between adjacent interpolation points, but the method would significantly increase the number of CNC programs. To reduce the nonlinear error, Wu Zhiqing et al. [8] interpolated the tool axis vector according to the error constraints and obtained the new tool axis vector by a projection method. Ming-Che Ho et al. [9] and LIANG et al. [10] reduced the nonlinear error caused by cutter oscillation through controlling the cutter attitude to achieve the smooth transition between cutter vectors. Dong Chaojie et al. [11] gave an interpolation algorithm with the integrated RTCP function and fixed the nonlinear error.
Aiming to ensure the continuity of the interpolation path, Hong et al. [12] proposed an interpolation method that can reduce the rotation error angle by analyzing the nonlinear error between cutter positions during machining. Xu et al. [13] proposed a fairing method for five-axis machining cutter path, which ensures the smooth transition between cutter orientations so as to reduce nonlinear errors. Liu et al. [14] proposed a dual NURBS cutter path interpolation method based on the improved co-evolutionary genetic algorithm and third-order derived Newton-type parameter calculation to reduce the interpolation error and improve the real-time interpolation performance. To obtain a smoother machined surface, Zhou [15] et al. explained the nonlinear error in terms of the cutter path and real machining cutter motion and gave a mathematical model of the error. Li et al. [16] densified the cutter path and data to control the nonlinear error. Zhang et al. [17] proposed a single spherical linear interpolation method and established an optimized algorithm to avoid the nonlinear error.
All of the above methods can effectively reduce the non-linear error of the cutter center point path during the five-axis machining, but they are limited in precisely controlling the accuracy of the cutter-part CC point path. To address this problem, this paper investigates a five-axis Tri-NURBS interpolation method based on the synchronization method of spline interpolation recorded in the literature [18][19][20] (4) At this point, the obtained parameter +1

T(v)
Programming cutter Programming CC point The parameters of the interpolation tool axis point +1 i v and the parameters of the The interpolation parameters are

Generation of Tri-NURBS spline curves
In the traditional five-axis CNC system, where, 1 i A is the unit vector of the cutter axis at the i+1st interpolation moment.  (11) where r is the radius of the cutter.
As shown in Fig. 7, the nonlinear error  of the CC point path can be defined as the minimum distance that the real CC point ii w d w  (as shown in Fig. 8a).
(  (17) The new tool center point  (18) It can be seen from Eq. (18) (19)      The spline curves in Tri-NURBS interpolation Figure 2 Five-axis Tri-NURBS interpolation format Tri-NURBS interpolation with same parameters Figure 4 Schematic diagram of nonlinear error for CC point Schematic diagram of nonlinear error for CC point path  Tri-NURBS interpolation and nonlinear error compensation Figure 10 Tri-NURBS spline after tting Figure 11 The i component of Vector A Figure 12 The j component of Vector A Figure 13 The k component of Vector A Figure 14 The i component of Vector H Figure 15 The j component of Vector H Figure 16 The k component of Vector H Figure 17 Nonlinear error of the path of CC point before compensation and repair Figure 18 Nonlinear error of the path of CC point after compensation and repair Figure 19 Position deviation of tool center point after error compensation and repair