Augmented Taylor's expansion method for B-spline curve interpolation for CNC machine tools

In computer numerical control (CNC) systems, parametric curves can be used instead of a large amount of linear blocks to describe tool paths for freeform surface or curve machining. However, existing parametric curve interpolation methods may cause large feed fluctuations or even a failure of the machining process near the sharp corners of a parametric curve. Therefore, a parametric curve interpolation method with an error correction and failure prevention scheme is required. In this paper, the augmented Taylor's expansion (ATE) method for computing B-spline curve parameters is proposed. A group of calibrators consisting of the knots and the arc lengths between adjacent knots are pre-computed before the interpolation starts. The parameter is computed based on Heun's method in a prediction–correction manner, and the accumulated errors caused by the cut-off errors of Taylor's expansion are eliminated by the calibrators at the knots. To cope with the extreme cases that usually occur near the sharp corners of a curve, a linear parametric interpolation between the previous parameter and its next calibrator is carried out when Heun's method fails to obtain a parameter in the domain. Simulation and experimental results show that, when the arc length increments are kept small enough near the sharp corners, the ATE method attains high accuracy and robust computation. The proposed method is also applicable to the NURBS curves.     

Machineability in NURBS interpolator considering constant material removal rate

Increasing demands on precision machining of three-dimensional free-form surfaces have necessitated that the tool move smoothly and at varying feedrates. To achieve this, parametric interpolators, such as the Non-Uniform Rational B-Spline (NURBS) interpolator, have been introduced in CNC machining systems. Such interpolators reduce the data burden in the Numerical Control (NC) code, increase data transfer rate into the NC controller, and finally give smooth motion while machining. In this research, a new concept to control cutting load in a NURBS interpolator based on the degree of curvature was tried. This protects the cutting tool and improves machineability. To prove the system, cutting force and surface topography were evaluated. From the experimental results, the interpolator is adequate for machining a free-form surface.     

Fast real-time NURBS path interpolation for CNC machine tools

In this paper, a novel fast real-time non-uniform rational B-spline (NURBS) path interpolation method is presented. This method efficiently integrates the data processing of a NURBS path in a CNC controller, from pre-processing to real-time interpolation. In the calculation of the total length of the NURBS path, the numerical adaptive quadrature method adapts to the integrand, i.e. the first derivative of the length function, automatically, dividing the parameter interval into subintervals with fine or coarse spacing according to the varying condition of the integrand. This new method takes full advantage of the subdivision scheme. The key point is to generate inverse length functions (ILF) for each resulting subinterval. In the real-time NURBS path interpolation, the new setting path parameter can be calculated directly using the ILF without the need for any time-consuming computation of NURBS derivatives and iteration. The proposed method is extremely fast, accurate and suitable for real-time implementation, and simulations and practical tests proved its effectiveness.     

A parametric interpolator with minimal feed fluctuation for CNC machine tools using arc-length compensation and feedback correction

Traditionally, approximation methods are utilized in the parametric interpolation because of the nonanalytic relationship between the spline parameter and the arc length. The approximation error has been considered as the source of the feedrate fluctuation. This paper shows that the discrepancy between the desired tool path and the target trajectory of the motion system is another primary source, and presents a feedback interpolator to eliminate the feedrate fluctuation. To evaluate the initial parameter value for the interpolator, an arc-length based Taylor's expansion with arc-length compensation is proposed, which alleviates greatly the feedrate command error caused by the trajectory deviation. Then, a feedback correction scheme is developed to further reduce the feedrate command error that results from the approximation error. Both computational load analysis and numerical simulations are conducted, and the results show that the present interpolator has very good performance in both efficiency and accuracy, thus is a good choice for high speed and high precision CNC machines.     

Development and implementation of a NURBS interpolator with smooth feedrate scheduling for CNC machine tools

Parametric interpolation for Non-Uniform Rational B-Spline (NURBS) curve has become more important than ever before in the control of CNC machine tools. An effective NURBS interpolator not only can obtain accurate contour trajectories, but also have smooth dynamics performance. This paper proposes a numerically efficient NURBS interpolation scheme which consists of two stages namely preprocessing and interpolating. In the stage of pre-processing, the parameter interval is split into several blocks at breakpoints and an iterative numerical quadrature method is applied for each block. By means of the iterative quadrature method, the initial parameter intervals of each block are divided into several subintervals according to the arc length approximation error. Meanwhile, the curvature of each knot and the cubic polynomial coefficients of each subinterval are obtained. Then the critical points with large curvature of each block are found from the candidate points and the tolerated speed of each critical point is calculated according to the constraints of chord error and centripetal acceleration. Hence, the feedrate scheduling based on the S-shaped acceleration profile for each block can be preplanned via the approximate arc length of each subinterval, the tolerated speed of each critical point and kinematics characteristics such as acceleration/deceleration and jerk limits of the machine tools. In the stage of interpolating, the parameter of the next interpolation point can be calculated directly using the cumulative arc length and the cubic polynomial coefficients of each subinterval. Finally, a series of numerical simulations and real machining experiments are conducted, and the simulation and experimental results have showed the good performance of the proposed NURBS interpolator both in efficiency and accuracy.     

NURBS-based fast geometric error compensation for CNC machine tools

In this paper, a novel method for the compensation of geometric errors of CNC machine tools is presented. The key idea is to use the basis functions of the setting NURBS path to approximate its error compensation function and to generate a new compensated NURBS path. In this way, both the setting and the compensated NURBS path have the same NURBS form. More importantly, the control points of the error compensation function can be obtained by simply calculating the positioning deviations of the control points of the setting NURBS path using the error model. A high compensation accuracy can be achieved through the systematic insertion of new knots, which creates new control points and raises the flexibility of NURBS in representing the error compensation function. The real-time interpolation of the compensated NURBS path completes the error compensation automatically. Simulations and experiments have shown that the new method delivers the same positioning accuracy as a model-based real-time geometric error compensation method does, but without additional real-time CPU loading. The proposed method can also be implemented in the post-processor of a CAM system for off-line compensation.     

NURBS interpolator for constant material removal rate in open NC machine tools

Conventionally used linear or circular interpolators are undesirable for the precision machining of 3D free-form surfaces for the following reasons: the transmission errors due to the huge number of point data, discontinuity of curve segmentation, and unsmooth motion speed. In this regard, modern CNC machine tools are designed with a function for machining arbitrary parametric curves. However, these systems do not consider controlling feedrate adaptively, which dominates the quality of the machining process. This paper proposes a NURBS interpolator based on the adaptive feedrate control for the constant material removal rate. This is accomplished by varying feedrate using the curvature of a surface. The curvature-compensated feedrate system has important potential applications in ensuring part accuracy and protecting the cutting tool. The simulated and experimental results show it is applicable to real machining.     

NURBS曲线插补算法及加减速控制研究

针对复杂零件高速高精密加工的需求,提出了一种基于阿当姆斯微分方程的NURBS曲线实时插补算法。通过对算法的合理简化与近似,保证了算法的实时性。此算法基于轮廓误差和法向进给加速度控制,使进给速度能随曲线曲率自适应调整。与之相适应,配合此插补算法,利用NURBS曲线的对称性预测减速点,提出了一种新的插补前抛物线-直线-抛物线S形加减速控制方法。该方法具有位置精度高、速度无突变、过渡平滑、计算简便等优点。通过采用MATLAB对插补轨迹仿真和实例分析,证明了插补算法和加减速控制方法的正确、合理、有效性。     

Development of a dynamics-based NURBS interpolator with real-time look-ahead algorithm

In this paper, a dynamics-based interpolator with real-time look-ahead (DBLA) algorithm is proposed to generate a smooth and jerk-limited acceleration/deceleration (ACC/DEC) feedrate profile. The interpolator consists of three modules: geometric, dynamics-based, and jerk-limited modules. The geometric module can detect the local maximum/minimum (max/min) curvatures, and divide a NURBS curve into small segments according to the information of sharp corners. The feedrates at the sharp corners are determined based on confined chord errors and curvatures of the curve. The dynamics-based module utilizes a dynamics feedrate modification equation (DFME) to estimate contour errors at the sharp corners and adjusts the feedrates at the locations of the sharp corners. The jerk-limited module plans the feedrate profile of the curve according to the segments’ length and the given jerk limit. Simulations are performed to verify real-time performance of the look-ahead algorithm. Experiments using a PC-based motion controller and an X–Y table are conducted to demonstrate that high-accuracy can be achieved with the proposed dynamics-based interpolator as compared to the adaptive-feedrate and the curvature-based feedrate interpolation algorithms.     

Real-Time NURBS Interpolator for Distributed Motion Control

Off-line toolpath planning for machine tools inevitably leads to suboptimal use of the given equipment and, even more important, excludes the possibility to accommodate for unforeseen or unmodelled effects. Online path planning however requires the availability of real-time interpolators which are both numerically efficient and deterministic and which do not depend on global information on the path to interpolate. Continuity of the interpolant up to higher derivatives is highly desirable.This paper presents an interpolation scheme that meets all these requirements. The algorithm generates a third-order, C²-continuous Non Uniform Rational B-Spline curve from a series of exactly interpolated position and velocity setpoints. This curve is at all instants completely determined from the first up to the last available setpoint. It is shown that a new setpoint affects only the last control points and the end of the knot vector of the interpolant. A graphical interpretation of the algorithm and a discussion on numerical issues and start conditions are presented.     

CNC machine tool surface interpolator for ball-end milling of free-form surfaces

This paper presents a new type of CNC machine tool interpolator that is capable of generating the cutter path for ball-end milling of a free-form surface. The surface interpolator comprises on-line algorithms for cutter-contact (CC) path scheduling, CC path interpolation, and tool offsetting. The interpolator algorithms for iso-parametric, iso-scallop and iso-planar machining methods are developed, respectively. The proposed surface interpolator method gains the advantages for minimizing the data loaded to the CNC machine tool and maintaining the desired feedrate and position accuracy along the CC path.     

Bézier polygons for the linearization of dual NURBS curve in five-axis sculptured surface machining

Motivated by the excellent performance of three-axis NURBS interpolation, this paper presents a numerically efficient and accuracy controllable five-axis sculptured surface machining method with dual NURBS curve. Unlike the traditional three-axis NURBS interpolation, a dual NURBS format of the five-axis toolpath is developed to accurately and smoothly describe the tool movement in the part coordinate system. Different from the subdivision methods using the Taylor series expansion or inverse function, a piece-wise Bézier curve method is implemented to fast subdivide the NURBS curve within the user-defined tolerance. A generic rotation tool center point management module is also designed to realize the coordinate transformation and adaptive nonlinear error control for major five-axis machine tools. The overall effectiveness of the proposed five-axis NURBS machining scheme is demonstrated by the five-axis machining of an impeller’s flow channel.     

Fast Bezier interpolator with real-time lookahead function for high-accuracy machining

This study presents a real-time fast Bezier interpolation method that solves linearly segmented contour problems that occur during milling using conventional CNC machines. Depending on the length and the CSB criterion, these linearized segments can be regarded as noise, continuous short blocks (CSBs), or G01 blocks. The CSB criterion proposed in this paper is employed to identify CSBs during the NC code interpreting stage. The CSBs are fitted into cubic Bezier curves and interpolated to produce smoother contours in real-time machining. Two different NC programs possessing a large number of short blocks were tested on our PC-based real-time control system. Simulations and experimental results demonstrate that the proposed fast Bezier interpolator (FBI) with real-time lookahead function provides satisfactory performance.     

Analytical curvature-continuous dual-Bézier corner transition for five-axis linear tool path

A novel analytical five-axis path-smoothing algorithm is developed for the high speed machining of a linear five-axis tool path. Segment junctions of the linear tool path in the machine tool coordinate system, which are tangent-discontinuous points, are all blended by two transition cubic Bézier curves. One cubic Bézier curve is used to smooth the segment junction of the translational path, and the other Bézier curve is used to smooth the segment junction of the rotational path. The tangency and curvature continuities are both guaranteed in the new path. The dual-Bézier transition algorithm has three advantages: (1) Compared with the path-smoothing method in the workpiece coordinate system, the new dual-Bézier transition method directly and simultaneously smooths the machine tool axis trajectories of both translational path and rotational path. The feed speed and stability will both be improved because the tool path discontinuities are the most important source of feed fluctuation. (2) The constraints of approximation error and the synchronization of parametrization of two smoothed curves, which are the most challenging problems in the smoothing of 5-axis tool path, are both considered. (3) The transition cubic Bézier curve pair has an analytical solution and can be easily integrated in the real-time interpolator. Computational examples and the cutting experiment of an impeller blade show that the novel path-smoothing method has obvious advantages in both feed smoothness and cutting efficiency over the original linear interpolator.     

Quantitative non-destructive evaluation of wall thinning defect in double-layer pipe of nuclear power plants using pulsed ECT method

In this paper, an inversion algorithm for three-dimensional profile reconstruction of wall thinning defect in a double-layer region of a typical coolant pipe of nuclear power plants from pulsed eddy current testing (PECT) signals has been proposed and experimentally validated, based on a fast simulator of PECT signals and a deterministic optimization strategy. First, the fast simulator developed by authors for PECT signal prediction based on a Fourier-series scheme in addition with interpolation and database approaches is briefly described as a base for the inversion of PECT signals. Then, the formula of the conjugate graduate inversion algorithm for sizing three-dimensional wall thinning from PECT signals is deduced in detail based on that for crack like reconstruction using single frequency ECT signals. The three-dimensional local wall thinning is modeled as a group of planar defects with different length and depth which are reconstructed from two-dimensional scanning PECT signals through inverse analysis. Through conducting PECT experiment for double-layer coolant tube test-piece and reconstructing wall thinning profile from the measured signals, the efficiency and the robustness of the proposed inversion algorithm are demonstrated.     

Trajectory generation in high-speed,high-precision micromilling using subdivision curves

Motion control in high-speed micromilling processes requires fast, accurate following of a specified curvilinear path. The accuracy with which the path can be followed is determined by the speed at which individual trajectories can be generated and sent to the control system. The time required to generate the trajectory is dependent on the representations used for the curvilinear trajectory path. In this study, we introduce the use of subdivision curves as a method for generating high-speed micromilling trajectories. Subdivision curves are discretized curves which are specified as a series of recursive refinements of a coarse mesh. By applying these recursive properties, machining trajectories can be computed very efficiently. Using a set of representative test curves, we show that with subdivision curves, trajectories can be generated significantly faster than with NURBS curves, which is the most common method currently used in generating high-speed machining trajectories. Trajectories are computed efficiently with subdivision curves as they are natively discretized, and do not require additional evaluation steps, unlike in the case of NURBS curves. The reduced trajectory generation time allows for improved performance in high-speed, high-precision micromilling. We discuss the use of several metrics to quantify the quality of the subdivision interpolation, and apply them in calculating the error during trajectory generation for the test curves.     

A novel adaptive-feedrate interpolation method for NURBS tool path with drive constraints

Feedrate scheduling is crucial for CNC systems to generate a smooth movement which is able to satisfy increasing requirements on machining quality and efficiency. In this paper, a novel adaptive feedrate interpolation method is proposed for NURBS tool path with drive constraints. The tool path is first expressed in NURBS form, and then the satisfaction conditions of drive constraints are derived according to the kinematic and geometric characteristics of the NURBS tool path. On this base, a proportional adjustment algorithm, which can quantitatively reduce the accelerations and jerks of drive axes at the sensitive regions of feed profile, is proposed to achieve the new positions of violated sampling points. After each adjustment, a curve evolution strategy is used to ensure the feed profile is locally or globally deformed to the target positions with a good smoothness of path curve and the avoidance of re-interpolation. Through the iterative adjustment, a smooth feed profile with limited velocities, accelerations and jerks of drive axes is thus yielded along the entire tool path. Finally, performances of the proposed method are validated by performing both simulations and experiments on two freeform NURBS curves. The results show the effectiveness and reliability of the proposed feedrate interpolation method.     

逆向驱动焊接机器人NURBS轨迹规划

针对焊接机器人逆向运动学不易求解且多解或无解的缺点,提出一种基于ADAMS和MATLAB的逆向驱动轨迹规划方法,并采用非均匀有理B样条(non-uniform rational B-splines,NURBS)进行关节曲线拟合,并进行焊接机器人的运动学仿真.结果表明,与正弦曲线和五次多项式等传统规划算法相比,逆向驱动NURBS轨迹规划算法,获得的各关节角位移曲线光顺性较好,机械手各方向的位移误差均较小,误差小于1 mm,从而快速准确实现焊接机器人的预期轨迹,为焊接机器人轨迹规划提供了一种新思路.     

Real-time generation and control of cutter path for 5-axis CNC machining

This paper presents a new approach to real-time generation and control of the cutter path for 5-axis machining applications. The cutter path generation method comprises real-time algorithms for cutter-contact path interpolation, cutter offsetting, and coordinate conversion. In addition, a global feedback loop is closed by the CNC interpolator so as to augment the controlled accuracy in practical cutter path generation. An error compensation algorithm and a feedrate adaptation algorithm for the control loop are developed, respectively.     

Inverse kinematics of five-axis machines near singular configurations

In five-axis milling, singular configurations of the machine axes may cause tool path errors or collisions between the tool and parts of the milling machine. This paper presents an algorithm for calculating the inverse kinematics of five-axis machines close to singular configurations. The algorithm modifies the exact inverse kinematics in order to give robustness to singularities at the expense of a small tool orientation deviation. The kinematics of a five-axis machine with non-orthogonal rotary axes is analyzed. The forward kinematics is developed, and a closed form solution of the inverse kinematics is presented. The kinematics and the singularity algorithm are implemented in a postprocessor, and machining tests are conducted to verify the algorithms.