Non-uniform Rational B-spline Curve Interpolation Research Articles

A NURBS curve interpolator with small feedrate fluctuation based on arc length prediction and correction

Non-uniform rational B-spline (NURBS) curve has been widely used in manufacturing systems. A good interpolator can help the system to improve the contour accuracy and get smooth dynamics performance, but it is hard to get a balance between the interpolation performance and computational load. As the derivative and curvature of NURBS curves used in manufacturing systems are high-order continuous, it is possible to predict a desired interpolation arc length based on the relationship of historical feed chord length and its corresponding arc length in one interpolation cycle. Therefore, this paper proposes a novel interpolation method, which consists three stages. Firstly, a NURBS curve is split into several high-order continuous segments based on its degrees and control points. Secondly, a prediction model based on Newton’s divided differences interpolation equation is derived from the relationship of interpolated chord length and its corresponding arc length, so that the target arc length of the next interpolation cycle can be predicted. Finally, target parameter u of every interpolation cycle is calculated with Taylor’s expansion, whose values are corrected by iteration, and then the position of the target interpolation point can be achieved. Performance of the proposed algorithm is tested and compared with other methods, and the simulation results show the proposed method can achieve smaller velocity fluctuation with low computational load.

Research on Acceleration/Deceleration in NURBS Curve Interpolation

A non-uniform rational B-spline (NURBS) parametric curve interpolation and S-shape acceleration/deceleration (ACC/DEC) for CNC machine tool were studied. In the proposed algorithm, the commanded velocity is maintained at most of the time and adaptively reduced in large curvature areas to meet the demand of the machining accuracy requirement, while at the same time, a S-shape look ahead algorithm is proposed to limit the acceleration and jerk values within the machine’s capabilities during the whole interpolation process. It reduces mechanical shock to the machine tools. The real time performance of this interpolator is also measured to demonstrate its practical application. A NURBS curve interpolation experiments is provided to verify the feasibility and advantages of the proposed algorithm.

An accurate NURBS curve interpolation algorithm with short spline interpolation capacity

In this paper, a novel and accurate real-time non-uniform rational B-spline curve interpolation algorithm is proposed. This algorithm not only considers chord errors, feedrate fluctuations, jerk-limited, and acceleration/deceleration (Acc/Dec) capabilities of the machine, but also optimizes the look-ahead process. In the meanwhile, it improves machining efficiency by adding the circular buffer and pre-interpolation (non-off-line) and enhances the real-time performance by removing the time-consuming calculation from the interrupt service routine. Furthermore, the proposed interpolation algorithm can interpolate both the long spline and the short spline with uniform method. The advantages of the proposed method were confirmed by the simulation results.

The Research of the NURBS Curve Interpolation Algorithm

In this article, the definition, the nature and the parameter of the NURBS (Non-Uniform Rational B-spline) curves are defined, the more thorough analysis and research of the real-time NURBS curve interpolation algorithm and the feed rate adaptive control are carried out.

A practical real-time non-uniform rational B-spline curve interpolator for computer numerical control machining

Non-uniform rational B-spline (NURBS) interpolator, a key technique in computer numerical control (CNC) machining, has many advantages over linear and circular interpolator. A practical real-time NURBS interpolator based on an off-line algorithm, which is developed to handle the limitation of velocity and acceleration, is presented in this article. Two processing modules are employed to realize the real-time interpolator. In one module, a NURBS curve is split into shorter NURBS curves or approximated as continuous line segments according to the minimum splitting length first, and then the velocity planning of a short curve is applied. Inthe other module, the interpolation according to the velocity planning results is implemented. The proposed algorithm realizes the real-time velocity planning and interpolation of a NURBS curve regardless of the curve length or the continuity. It is also applicable to three-dimensional surface designed by computer-aided design system, which is composed of multiple NURBS curves or both NURBS curves and continuous line segments. Both the simulation and experiment results have proved the effectiveness of the proposed practical real-time NURBS interpolator.

Real-time NURBS curve interpolator based on section

Parameter interpolation is more capable of modern computer numerical control (CNC) than traditional linear/circular interpolation with higher speed and higher precision. Most of non-uniform rational B-spline (NURBS) interpolation algorithms were developed based on the chord error and machines capability, where interpolation points are calculated beforehand to overcome acceleration/deceleration (acc/dec) and jerk problem, which needs large memory. In this paper, a NURBS interpolator based on feedrate section is proposed. Instead of a single interpolation point, this interpolator aims to feedrate section, which makes it possible to run on a digital signal processor or field programmable gate array (FPGA) whose memory is limited. Experiment on FPGA showed the performance of interpolation. A mould experiment verifies the feasibility of application.

Robust Control of an LUSM-Based $X\hbox{--}Y\hbox{--}\theta$ Motion Control Stage Using an Adaptive Interval Type-2 Fuzzy Neural Network

The robust control of a linear ultrasonic motor based X-Y-thetas motion control stage to track various contours is achieved by using an adaptive interval type-2 fuzzy neural network (AIT2FNN) control system in this study. In the proposed AIT2FNN control system, an IT2FNN, which combines the merits of an interval type-2 fuzzy logic system and a neural network, is developed to approximate an unknown dynamic function. Moreover, adaptive learning algorithms are derived using the Lyapunov stability theorem to train the parameters of the IT2FNN online. Furthermore, a robust compensator is proposed to confront the uncertainties including the approximation error, optimal parameter vectors, and higher order terms in Taylor series. To relax the requirement for the value of lumped uncertainty in the robust compensator, an adaptive lumped uncertainty estimation law is also investigated. In addition, the circle and butterfly contours are planned using a nonuniform rational B-spline curve interpolator. The experimental results show that the contour tracking performance of the proposed AIT2FNN is significantly improved compared with the adaptive type-1 FNN. Additionally, the robustness to parameter variations, external disturbances, cross-coupled interference, and frictional force can also be obtained using the proposed AIT2FNN.

Implementation of a CNC NURBS curve interpolator based on control of speed and precision

An efficient real-time Non-Uniform Rational B-Spline (NURBS) curve interpolator for high-speed and high-accuracy curve machining is developed and implemented, which considers the confined contour error, acceleration/deceleration (ACC/DEC) planning and the machine dynamic response simultaneously. A practical method is proposed to adjust the feed rate to the curvature radius of NURBS curves to achieve higher machining precision. However, at the sharp corner of the curves, the feed rate fluctuation might result in large impacts to machine tools. The offline pre-interpolation for sharp corners is presented. The feed rate is reduced in advance according with the machine's acceleration capability as well as satisfying the confined contour error. The velocity profile can be chosen dynamically to meet the requirements of the machining process by means of flexible ACC/DEC planning. In addition, a new strategy based on the geometrical properties of NURBS curves is suggested to predict the deceleration point. Therefore, the process of acceleration and deceleration is so smooth that it can satisfy the need of the machining process. This NURBS interpolator has been validated on a CNC system, which greatly improves machining accuracy for both position and velocity control.