A parametric interpolation method based on prediction and iterative compensation

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 second-order 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.

Efficient Feedrate Optimization Method for Spline Toolpath With Curvature-Base Planning and Accurate Interpolating

The well-designed feedrate optimization algorithm can obtain higher machining efficiency with various machining related constraints, thus, it is widely considered in the high-speed and high-precision machining. However, the low computational efficiency still limits the application of the optimization method. For the non-linear optimization problem of spline toolpath with feedrate-, actuator velocity-, acceleration- and jerk-limited, a linear approximation is adopted by a pseudo-jerk method and the efficient linear programming method can be applied to solve the optimization problem. To improve computational efficiency further, curvature-base window technique is presented and the whole spline toolpath is split at the curvature extreme points, which are also named critical points in traditional planning method. Thereafter, a novel feedback interpolation is presented based on Steffensen iterative accelerator method to eliminate the feedrate fluctuation caused by nonanalytic relationship of spline parameter and arc-length. Finally, simulations and experiments validations show that the proposed method is able to reduce computational burden and traversal time notably with multi-constraints.

Fast parametric curve interpolation with minimal feedrate fluctuation by cubic B-spline

Due to the reliable feedrate fluctuation and computation load of the existing parametric curve interpolation, a fast interpolation method by cubic B-spline for parametric curve is presented which results in a minimum feedrate fluctuation and light computation load. As there are many geometry implementation tools and many good properties in the B-spline compared with the polynomial, the relation between the arc length s and curve parameter u can be fitted by the cubic B-spline accurately. Because the feedrate fluctuation of the generally used Taylor approximation method is sensitive to the curvature of the toolpath, its accuracy cannot be controlled. For a given feedrate fluctuation of 0.05%, the proposed interpolation method can guarantee the error requirements by increasing the number of the control points. After the de Boor method is applied in real time, the computation load of the cubic B-spline interpolation is decreased compared with the Taylor approximation method and higher order polynomial-fitting method. In order to save the memory consumption for storing the parameters of the fitted cubic B-spline, an iterative optimization process is applied to obtain the knot vector elements and optimize the control points. Simulations and experiments show that the interpolation method can attain high accuracy and computation efficiency. According to the simulations, for most of the complex curves, the feedrate fluctuation of the proposed interpolation method is decreased by about 50% when the feedrate is scheduled and the computation load of the proposed method is decreased by about 70% compared with the second-order interpolation method.

A parametric interpolation method with minimal feedrate fluctuation by nonuniform rational basis spline

Because the relation between the arc length s and curve parameter u cannot be represented by explicit function for most of the curves, it is difficult to consider the accuracy, robustness, and computational efficiency for most of the parametric interpolation, especially when the curves are complex or extremes. Therefore, an off-line fitting interpolation method by using nonuniform rational basis spline is presented in this paper. As nonuniform rational basis spline has many geometry implementation tools and numerous good properties as compared to the polynomial, the required fitting accuracy can be obtained more easily than with polynomial. After the de Boor method is applied, the computational load of nonuniform rational basis spline is decreased as compared to the Taylor approximation and the higher order polynomial fitting method. In order to obtain the proper s-u fitting nonuniform rational basis spline and reduce the computational load of the fitting process, the sampled s-u data points are divided according to the properties of nonuniform rational basis spline, and in each segment, the knot vectors, control points, and weights are calculated by the iterative-optimization method. Then the s-u nonuniform rational basis spline can be applied in real-time interpolation, and the accuracy, robustness, and computational efficiency are demonstrated by simulations and experiments.