Development of an integrated look-ahead dynamics-based NURBS interpolator for high precision machinery

Methodologies for planning motion trajectory of parametric interpolation such as non-uniform rational B-spline (NURBS) curves have been proposed in the past. However, most of the algorithms were developed based on the constraints of feedrate, acceleration/deceleration (acc/dec), jerk, and chord errors. The errors caused by servo dynamics were rarely included in the design process. This paper proposes an integrated look-ahead dynamics-based (ILD) algorithm which considers geometric and servo errors simultaneously. The ILD consists of three different modules: a sharp corner detection module, a jerk-limited module, and a dynamics module. The sharp corner detection module identifies sharp corners of a curve and then divides the curve into small segments. The jerk-limited module plans the feedrate profile of each segment according to the constraints of feedrate, acc/dec, jerk, and chord errors. To ensure that the contour errors are bounded within the specified value, the dynamics module further modifies the feedrate profile based on the derived contour error equation. Simulations and experiments are performed to validate the ILD algorithm. It is shown that the ILD approach improves tracking and contour accuracies significantly compared to adaptive-feedrate and curvature-feedrate algorithms.     

基于NURBS曲线直接插补算法的嵌入式数控系统设计与实现

提出了一种结合486SX级别的X86微处理器和可编程逻辑器件CPLD两级控制的嵌入式数控系统设计方案,阐述了该系统的硬件接口电路设计;提出了基于改进S形加减速的NURBS曲线直接插补算法,在满足最大弦高误差、最大法向加速度以及最大进给速度要求的情况下,对插补曲线的加速段和减速段进行速度规划;最后采用基于该插补算法的嵌入式数控系统,在半圆形毛坯上进行了五角星NURBS曲线的实际加工,验证了所设计嵌入式数控系统的可行性和有效性,具有一定的工程应用价值。     

An accurate adaptive NURBS curve interpolator with real-time flexible acceleration/deceleration control

Parametric interpolation has been widely used in CNC machining because of its advantages over the traditional linear or circular interpolation. Many researchers focused on this field and have made great progress in the specific one, NURBS curve interpolation. These works greatly improved the CNC machining with constant feedrate, confined chord error and limited acceleration/deceleration. However, during CNC machining process, mechanical shocks to machine tool caused by the undesired acceleration/deceleration profile will dramatically deteriorate the surface accuracy and quality of the machined parts. This is, in most occasions, very harmful to machine tools. In this paper, an accurate adaptive NURBS curve interpolator is proposed with consideration of acceleration–deceleration control. The proposed design effectively reduces the machining shocks by constraining the machine tool jerk dynamically. Meanwhile, the constant feedrate is maintained during most time of machining process, and thus high accuracy is achieved while the feedrate profile is greatly smoothed. In order to deal with the sudden change of the acceleration/deceleration around the corner with large curvature, a real-time flexible acceleration/deceleration control scheme is introduced to adjust the feedrate correspondingly. Case study has been taken to verify the feasibility and advantages of the proposed design.     

运行时间周期化工业机器人模型迭代寻优NURBS轨迹插补

为满足工业机器人高精度复杂曲线运动的需求,本文提出运行时间周期化工业机器人模型迭代寻优NURBS轨迹插补算法.首先,根据轨迹最大轮廓误差和机器人动力学特性对曲线分段.随后,提出优化回溯算法,使各子曲线段均可用S曲线加减速规划.之后,为保证机器人在进给速度极小值处不超速,将各加减速阶段运行时间调整为插补周期的整数倍,并对子曲线段衔接处速度平滑处理.最后,提出模型迭代寻优曲线插补,大大降低了速度波动率.仿真试验表明,该方法插补轨迹的各项指标均满足要求且最大速度波动率仅为0.000099%.真机试验也验证了该方法可有效减小轨迹误差.     

多轨迹段平滑过渡的前瞻插补算法

针对轨迹规划时采用首尾速度为零的加减速控制方法中存在的频繁启停,以及末端执行器在插补过程中加速度过渡不平滑等问题,提出了一种基于非对称S形加减速控制的多轨迹段平滑过渡的前瞻插补算法.该算法在相邻轨迹段间采用圆弧模型对衔接拐角处平滑过渡,在给定轨迹衔接点坐标和过渡圆弧半径等参数的情况下,规划出衔接圆弧处的最优速度.对插补算法中归一化因子的求解,采用一种新型柔性加减速控制算法,该算法由余弦加减速曲线在直线形加减速曲线上拟合而成,减少了余弦加减速算法的运算量,保证了加速度控制的平稳性.试验结果表明,该算法可以实现多轨迹段衔接处的圆滑过渡,保证运动速度的平滑度与连续性,有效提升了末端执行器的运行效率.     

A universal velocity limit curve generator considering abnormal tool path geometry for CNC machine tools

Non-linear parametric curves, such as B-spline curves, are becoming increasingly available in modern CNC (Computer Numerical Control) systems. The smoothness of parametric curves offers higher order of continuity and thus invokes less vibration in machines as compared to short line segments. Nevertheless, the computations for velocity limit curves, velocity profiles and interpolation points are quite complicated and time-consuming and therefore approximation methods are applied. Unnecessary accelerations and decelerations, which cost additional time of motion, can be caused by inaccurate computations of the velocity limits around tiny corners. To overcome the problem, the unit arc length increment scanning method (UALISM) is proposed to reduce the time of movement and to improve the efficiency. The scanning interval is fixed at 1 BLU (basic length unit) which is irrelevant to the type, size and shape of the tool path curve. The constraints of chord height errors and axis accelerations are considered and the velocity limit for the specified scanning point is computed using the coordinates of multiple scanning points near the specified scanning point. Simulation results show that the unnecessary accelerations and decelerations can be avoided and thus the total motion time can be reduced by UALISM.     

Adaptive-feedrate interpolation for parametric curves with a confined chord error

Recently, modern manufacturing systems have been designed which can machine arbitrary parametric curves while greatly reducing data communication between CAD/CAM and CNC systems. However, a constant feedrate and chord accuracy between two interpolated points along parametric curves are generally difficult to achieve due to the non-uniform map between curves and parameters. A speed-controlled interpolation algorithm with an adaptive feedrate is proposed in this paper. Since the chord error in interpolation depends on the curve speed and the radius of curvature, the feedrate in the proposed algorithm is automatically adjusted so that a specified limit on the chord error is met. Both simulation and experimental results for non-uniform rational B-spline (NURBS) examples are provided to verify the feasibility and precision of the proposed interpolation algorithm.     

Estimation of NC machining time using NC block distribution for sculptured surface machining

The estimation of NC machining time is of importance because it provides manufacturing engineers with information to accurately predict the productivity of an NC machine, as well as its production schedule. NC programs contain various machining information, such as tool positions, feed and speed rates, and other machine instructions. Nominal NC machining time can easily be obtained based on the NC program data. Actual machining time, however, cannot simply be found due to the dynamic characteristics of a NC machine controller, such as acceleration and deceleration effect. Hence, this study presents an NC machine time estimation model for machining sculptured surfaces, considering such dynamic characteristics of the machine. The proposed estimation model uses several factors, such as the distribution of NC blocks, angle between the blocks, federates, acceleration and deceleration constants, classifying tool feed rate patterns into four types based on the acceleration and deceleration profile, NC block length, and minimum feed rate. However, there exists an error for the actual machining time due to the lack of the measurement equipment or tools to gauge an exact minimum feed rate. Thus, this paper proposes a machining time estimation model using NC block distributions, lowering down the error caused by the inaccurate minimum feed rate. The proposed machining time estimator performs at around 10% of mean error.     

Isophote interpolation

The isophote is an important class of characteristic curves on a parametric surface. Accordingly, as a kind of feature based path generation method, isophote based tool paths has been proposed as an important path pattern for parametric surface machining. In this paper, an approach has been proposed for the isophote interpolation on a parametric surface. The paper has related the arc-length derivatives to time derivatives of parameters along the isophote. The results are then used to derive the parametric interpolation. The isophote curve interpolation is developed based on parametric interpolation. The proposed interpolation guarantees that interpolated points always stay on the parametric surface. An improvement interpolation has been presented to alleviate inclination errors and point deviation from the isophote. Simulations of isophote interpolation have been carried out to verify the effectiveness of the proposed algorithm. The proposed algorithm has applications in real time tool path interpolation for the machining of parametric surfaces.     

A look-ahead command generator with control over trajectory and chord error for NURBS curve with unknown arc length

There are currently no analytical methods available which determine the exact arc length for NURBS curves and for this reason, a smooth feedrate profile with desired trajectory cannot be achieved. Numerical methods used to calculate the arc length are time-consuming processes which make generating a feedrate profile with desired accelerations difficult in real-time. This paper introduces a look-ahead trajectory generation method which determines the deceleration stage according to the fast estimated arc length and the reverse interpolation of each curve at every sampling time. This results in a feedrate trajectory generation with jerk-limited acceleration profiles for the NURBS curves. The feedrate profile is adjusted dynamically according to the geometrical path constraint determined by chord error for the curved path. A NURBS curve by two different kinematics conditions was used as a means to test the feasibility of the developed interpolation scheme and command generator.     

改进的NURBS曲线预估校正插补方法

现行的NURBS曲线参数预估校正插补方法不能预估最开始的两个参数,而且不能控制预估参数迭代修正的次数。提出改进的NURBS预估校正插补方法,该方法预估插补点参数,根据最大进给步长约束、轮廓误差约束以及法向加速度约束,判定预估参数与实际参数的误差,当误差在允许范围内,则对参数步长增量进行调整,否则进行迭代修正。仿真实验结果表明,新方法通过对参数步长增量的调整,使得预估参数更加接近实际参数,较现有方法有效减少了参数迭代修正的次数,利于实时插补的实现。     

磨料水射流切割速率平滑处理算法的仿真优化

在S型曲线加减速算法的基础上,设计开发了一种新的速度变化率与插补位置的非线性算法,得到一种新的速率平滑处理方法.并针对水射流切割速度对切割质量的主要影响因素,进行了一系列的速率处理方式变化、速率平滑处理下加减速距离变化的模拟实验,从而找到水射流切割速率与切割质量的关系和规律.     

凸轮轴磨床工件旋转轴转速预测算法的研究

采用三次B样条曲线插补的凸轮磨削加减速能力不足,易产生过磨和少磨。根据凸轮磨削的数学模型,分析了砂轮进给轴运动的理论与实际速度、加速度、加加速度,提出了一种预测工件旋转轴转速的加工方法,在速度变化剧烈处自动降低工件旋转轴转速,以避免加速度和加加速度的变化对伺服系统造成的机械冲击,利用Matrix VB控件编程技术,设计了凸轮轴磨削软件,并将其移植到YTMK-CNC8326全数控高速凸轮轴磨床中。测试表明,采用该方法磨削的凸轮轴型线误差小于±0.01mm,工件表面粗糙度得到明显改善,实现了凸轮轴的精密加工。     

An adaptive parametric interpolator for trajectory planning

A real-time interpolation algorithm for trajectory planning is studied in this paper. The NURBS interpolation algorithm is proposed to confine contour errors and feedrate fluctuations. The feedrate is adjusted adaptively according to the specified acceleration/deceleration values and jerk value. A direct digital convolution method is also introduced into velocity planning for NURBS interpolator, and it is more efficient than the traditional method of polynomial functions. The feedrate at the sharp corner is smoothed by imposing limitations on the acceleration and jerk values generated in the machining process. Since the computation of the total length of NURBS curve is required for the digital convolution method, a numerical adaptive quadrature algorithm is used to approximate the integrand. Simulation results demonstrate the effectiveness of the proposed interpolator for machining curved paths.     

NURBS曲线参数插补的实时前瞻算法

针对引入自适应插补算法而使机床在进给过程中产生的进给速度突变,进给加速度超出机床最大允许值的问题进行了研究,提出了一种利用回溯和重插补的前瞻方法。当进入曲率敏感区域时,向前追溯已插补点,找到一个符合进给速度校验的点,从该点进行重新插补,将所得的插补速度代替之前自适应存储的速度,达到自行调整进给速度的目的,使得插补加速度在机床的允许范围内变化,保证机床平稳运行。仿真和实验结果验证了所提算法的有效性和实时性。     

基于精化曲线累加弦长的角点检测技术

提出基于曲线累加弦长来检测角点，可以使得检测结果具有旋转不变性，隐式精化曲线策略可以显著改善角点检测算法的计算精度，从而增强角点检测结果的可靠性．具体给出了一种基于精化曲线累加弦长的角点检测方法，并对其可行性进行数学论证，同时提出了一种区别圆角和尖角的实用技术．实验结果表明，使用文中技术和方法能够有效地解决实际的角点检测问题．     

Curve interpolation based on the canonical arc length parametrization

We use the canonical equations (CE) of differential geometry, a local Taylor series representation of any smooth curve with parameter the arc length, as a unifying framework for the development of new CNC algorithms, capable of interpolating 2D and 3D curves, represented parametrically, implicitly or as surface intersections, with accurate feedrate control. We use a truncated form of the CE to compute a preliminary point, at an arc distance from the last interpolation point selected to achieve a desired feedrate profile. The next interpolation point is derived by projecting the preliminary point on the curve. The coefficients in the CE involve the curve’s curvature, torsion and their arc length derivatives. We provide computing procedures for them for common Cartesian representations, demonstrating the generality of the proposed method. In addition, our algorithms admit corrections, which render them more accurate in terms of the programmed feedrate, compared to existing parametric algorithms of the same order.     

Angular interpolation of bi-parameter curves

This paper presents an approach to the interpolation of angular feedrate for bi-parameter curve paths in multi-axis machining. A bi-parameter curve is the intersection of a parametric surface and an implicit surface. A tool path is identified by a position curve and an orientation curve, both of which are generated based on the bi-parameter curve. The angular feedrate interpolator calculates the tool position and orientation at each sampling cycle according to the specified angular feedrates and the given tool path. The paper analytically relates the angular arc-length derivatives to the time derivatives of the parameters along the path making use of both angular feedrates and angular feed acceleration. The results are then used to interpolate the parameters of the bi-parameter curve leading to an accurate calculation of the position and orientation of the cutting tool. A general parametric surface has been used to verify the effectiveness of the algorithm. The bi-parameter curves of the surface have been computed for arbitrarily selected intersecting cylinders.