A parametric interpolation method based on prediction and iterative compensation

Abstract

Parametric interpolation for spline plays an increasingly important role in modern manufacturing. It is critical to develop a fast parametric interpolator with high accuracy. To improve the computational efficiency while guaranteeing low and controllable feedrate fluctuation, a novel parametric interpolation method based on prediction and iterative compensation is proposed in this article. First, the feedrate fluctuation and Taylor's expansion are analyzed that there are two main reasons to reduce the calculation accuracy including the truncation errors caused by neglecting the high-order terms and discrepancy errors between the original curve and the actual tool path. Then, to reduce these errors, a novel parametric interpolation method is proposed with two main stages, namely, prediction and iterative compensation. In the first stage, a quintic polynomial prediction algorithm is designed based on the historical interpolation knowledge to estimate the target length used in the second-order Taylor's expansion, which can improve the calculation accuracy and the convergence rate of iterative process. In the second stage, an iterative compensation algorithm based on the secondorder Taylor's expansion and feedrate fluctuation is designed to approach the target point. Therefore, the calculation accuracy is controllable and can satisfy the specified value through several iterations. When finishing the interpolation of current period, the historical knowledge is updated to prepare for the following interpolation. Finally, a series of simulations are conducted to evaluate the good performance in accuracy and efficiency of the proposed method.

Innovation of NURBS feedrate direct interpolator in five-axis for CNC high-precision and high-efficiency machining

“…For a tool path, the S-shaped flexible acceleration/deceleration algorithm is firstly used to generate smooth feedrate, and then the feedrate scheduling algorithm is used to obtain the duration τ1-τ7 of the 7 segments acceleration/deceleration curve. With the feedrate, duration time τ1-τ7, the first-order and second-order derivative, the interpolation parameter can be calculated by Taylor expansion formula (12) in the conventional STE interpolator.…”

“…Butterfly and S-shape part are applied to show the details of FDI method and FSI method for dual-NURBS in five-axis, respectively. And the performance of the proposed FDI is compared to second-order Taylor expansion with feedrate round-up (STE-FR) [50] and second-order Taylor expansion with iterative compensation (STE-IC) [12]. In the conventional interpolator for five-machining, the tool orientation and tool tip are the same interpolation parameter (SIP) [51],…”

“…The calculation of higher-order derivative in Taylor expansion greatly reduced real-time performance of NURBS interpolator. The iterative method for calculating the interpolation parameter mainly referred to the "prediction-correction" interpolation algorithm [12][13][14][15][16][17][18][19]. Authors [12][13][20][21], such as Ye in Table 2, also did research about the iterative compensation method on Taylor expansion interpolation.…”

“…The iterative method for calculating the interpolation parameter mainly referred to the "prediction-correction" interpolation algorithm [12][13][14][15][16][17][18][19]. Authors [12][13][20][21], such as Ye in Table 2, also did research about the iterative compensation method on Taylor expansion interpolation. The interpolation parameter was estimated by the Taylor method or the velocity polynomial method, and the deviation between the target speed and the actual speed obtained by using the estimated methods were calculated.…”

An efficient and accurate interpolation method for parametric curve machining

“…Zhao et al 26 proposed an interpolation parameter calculation method with arc length correction and feedback correction, which can improve interpolation efficiency and accuracy. Ni et al 27 proposed a quintic polynomial prediction algorithm, and estimated the target arc length in the second-order Taylor expansion to improve the calculation accuracy and iterative convergence speed. The iteration method needs to be repeated in each interpolation cycle, and the number of iterations is not fixed, which affects the real-time performance of interpolation.…”