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.

Triple parametric tool path interpolation for five-axis machining with three-dimensional cutter compensation

Parametric interpolation obtains a great success in three-axis surface machining with smooth motion, high accuracy, and high machining efficiency, but does not go well in five-axis surface machining due to lack of appropriate and efficient methods of tool path generation, interpolation, and three-dimensional cutter compensation. This article proposes a triple parametric tool path interpolation method for five-axis machining with three-dimensional cutter compensation, which proposes an appropriate triple parametric tool generation method for realizing the three-dimensional cutter compensation in five-axis parametric interpolation. A triple parametric interpolation algorithm is also proposed to realizing the simultaneous interpolation of the source data, which ensures the primitivity and maintains the accuracy. The proposed three-dimensional cutter compensation can compensate the errors caused by minor changes in cutter size, thus machining accuracy can be improved. Finally, illustrated example verifies the feasibility and applicability of the proposed methods.

Regionalizing non-parametric precipitation amount models on different temporal scales

Parametric distribution functions are commonly used to model precipitation amounts. Precipitation amounts themselves are crucial for stochastic rainfall generators or weather generators. Non-parametric kernel density estimates (KDEs) offer a more flexible way to model precipitation amounts. As it is already stated in their name, these models do not exhibit a parameter which can easily be regionalized to run rainfall generators at ungauged locations as well as at gauged locations. To overcome this deficiency we present a new interpolation scheme for non-parametric models and evaluate it for different temporal resolutions ranging from hourly to monthly. During the evaluation the non-parametric methods are compared to commonly used parametric models like the two parameter gamma and the mixed exponential distribution. As water volume is considered to be an essential parameter for applications like flood modeling, a Lorenz-curve based criterion is also introduced. To add value to the estimation of data at sub-daily resolutions, we incorporated the plentiful daily measurements in the interpolation scheme and this idea was evaluated. The study region is the federal state of Baden-Württemberg in the southwest of Germany with more than 500 rain gauges. The validation results show that the newly proposed non-parametric interpolation scheme works, and additionally seems to be more robust compared to parametric interpolation schemes, and that the incorporation of daily values in the regionalization of sub-daily models is very beneficial.

A segment feedrate profile fitting method for parametric interpolation

There are many researches in scheduling an optimal feedrate profile under various constraints by numerical calculation. A large number of discrete feedrate data points are obtained. They are inconvenient for the parametric interpolator. Therefore, these discrete feedrate data points need to be fitted by parameter curves. Different from the regular curve fitting, the inappropriate feedrate fitting method can generate larger acceleration and jerk that seriously affect the machining accuracy and stability, although the feedrate satisfies the error requirements. In order to generate a suitable feedrate profile, a segment feedrate profile fitting method using B-spline is proposed in this article. The discrete feedrate data points are segmented in the jerk discontinuous points. In each segment, the feedrate profile is fitted by the linear least squares method. These fitted feedrate profiles are combined to generate a unified feedrate profile. The unified fitted feedrate profile and the tool path trajectory are used in the controller to command the axis. In this article, the process of parametric interpolation is separated into the arc-length calculation process and the curve parameter calculation process. Using parallel computation, the two processes are calculated simultaneously in the controller, and the computational efficiency is improved. Both simulation and experiment are carried out to verify that the fitted feedrate profile satisfies the error requirements, and the novel interpolation can be applied to the controller appropriately.