In order to improve the accuracy of position and direction of the tool axis vector and the controlling accuracy of cutter contacting (CC) paths between the cutter and workpiece in the traditional five-axis NURBS interpolation method, a five-axis Tri-NURBS spline interpolation method is proposed in this paper. Firstly, the spline interpolation instruction format is proposed, which includes three spline curves, the CC point spline, the tool center point spline, and the tool axis point spline. The next interpolation parameter is calculated based on the tool center point spline combined with the conventional parametric interpolation method. 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 position and direction of the tool axis vector. Secondly, the producing mechanism of the nonlinear error of CC paths of the traditional spline interpolation is analyzed and the mathematical calculation model of the nonlinear error is established. And then, the compensation and correction method of nonlinear error is also proposed to improve the controlling accuracy of CC paths. In this method, the next CC point on the cutter can be first 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 coincide. 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 the machine tool. The MATLAB and VERICUT software are used as a simulation of the real machining data. The results show that the proposed method can effectively reduce the nonlinear error of CC paths. It has high practical value for five-axis machining in effectively controlling the accuracy of CC paths and improving the machining accuracy of complex surfaces.

：In order to solve the problem of deviation between actual and theoretical machining paths due to the presence of rotation axis in five-axis machining, an interpolation algorithm based on the optimization of swing cutter trajectory and the method of corresponding nonlinear error compensation are proposed. Taking A-C dual rotary table five-axis machine tool as an example, the forward and reverse kinematic model of the machine tool is established according to the kinematic chain of the machine tool. Based on the linear interpolation of rotary axis, the generation mechanism of nonlinear error is analyzed, the modeling methods of cutter center point and cutter axis vector trajectory are proposed respectively, and the parameterized model of swing cutter trajectory is formed. The formula for the nonlinear error is obtained from the two-dimensional cutter center point trajectory. According to the established model of swing cutter trajectory, the synchronous optimization method of cutter center point trajectory and cutter axis vector trajectory is proposed, and the nonlinear error compensation mechanism is established. First, pre-interpolation is performed on the given cutter location data to obtain a model of the swing cutter trajectory for each interpolated segment.Then the magnitude of the nonlinear error is calculated based on the parameters of the actual interpolation points during formal interpolation, and the interpolation points with large errors are compensated for the nonlinear error. The simulation results show that the proposed method can effectively reduce the impact of nonlinear errors on machining, and is of high practical value for improving the accuracy of cutter position and the quality of complex free-form machining in five-axis machining.

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 surfaces.

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