NURBS Curve Interpolation And Acceleration-Deceleration Control Method

The number of interpolation functions of a CNC system, core of the CNC equipment,and the performance of the interpolation algorithm are important indexes to evaluate the performance of a CNC control system. Traditional CNC systems, most of which have the interpolation function of only line, circle and parabola, cannot directly machine the free-form curves and surfaces. The proposal of NURBS interpolation provides a solution for CNC equipment to machine the components and parts with complex curves and surfaces in a high speed and high precision, and it is also one of the key problems for developing a high-speed and high-precision CNC system.In this paper, the research work is carried out from two aspects of improving the real-time performance of the NURBS curve interpolation algorithm and the speed variation stability of the acceleration and deceleration control method, as follows.Firstly, the basic theory of NURBS curve is introduced; the interpolation principle,iterative process, iterative initial value and iteration termination condition of the NURBS curve interpolation algorithm based on the Taylor expansion method and the NURBS curve interpolation algorithm based on the Newton iteration method are elaborated.Moreover, the problems of these two interpolation algorithms are analyzed.Secondly, the NURBS curve interpolation algorithm based on the secant method is proposed on the basis of the NURBS curve interpolation algorithm based on the Newton iteration method. In this algorithm, the secant method replaces the Newton iteration method to get the parameters of the next interpolation point, which avoids the complex derivation process of the Newton iteration method and improves the real-time performance of the interpolation without loss of the interpolation precision.Thirdly, the control models of the T-type and the S-type acceleration and deceleration control methods and the respective advantages and disadvantages of the two are analyzed.Based on the above, the 5-segment acceleration and deceleration control method based on polynomials is proposed. In this method, the change of jerk varies with the trend described by the given polynomial, which makes the change continuous and smooth, and avoids the problems in the S-type method, like the jerk mutation and the instable change of acceleration.Finally, the simulation platform with the tool of C# language and the software of MATLAB is built. Moreover, the real-time performance of NURBS curve interpolation algorithm based on the secant method and the speed variation stability of the 5-segment acceleration and deceleration control method based on polynomials are both validated by simulation.