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.     

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.     

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.     

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.     

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.     

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.     

一种改进的NURBS曲线插补算法

针对NURBS曲线曲率变化过快或出现曲率不连续点会导致插补进给速率变化过快,超出机床的加减速能力。提出一种利用NURBS曲线曲率特征的改进插补算法。该算法根据NURBS曲线曲率的变化情况将曲线分成曲率平缓段和曲率突变段,在前瞻过程中扫描出曲率突变段,获得该段的起始点、终止点及最低速率点等信息,采用梯形加减速方法对该段进行速度规划,以满足机床动态特性,实现在曲率平缓段以指令速度插补,在曲率突变段以规划速度平滑插补。仿真实验结果表明,在保证加工精度的前提下,该增强算法以较高效率实现了曲率突变段的平滑插补。     

NURBS曲线实时插补中的速度规划算法

大多数现有的NURBS曲线实时插补算法并未考虑速度方向的变化给各运动轴带来的影响,这会导致加工过程中单轴速度的剧烈变化.提出一种能保证各运动轴平稳运行的速度规划算法,它在满足精度要求的前提下,通过控制切向加速度和加加速度进行速度平滑,并根据各运动轴的当前速度和机床的实际性能再次调节进给速度,保证了机床的平稳运行.模拟实验证实了该算法的有效性.     

A greedy algorithm for feedrate planning of CNC machines along curved tool paths with confined jerk

In this paper, the problem of optimal feedrate planning along a curved tool path for 3-axis CNC machines with the acceleration and jerk limits for each axis and the tangential velocity bound is addressed. It is proved that the optimal feedrate planning must be “Bang–Bang” or “Bang–Bang-Singular” control, that is, at least one of the axes reaches its acceleration or jerk bound, or the tangential velocity reaches its bound throughout the motion. As a consequence, the optimal parametric velocity can be expressed as a piecewise analytic function of the curve parameter u. The explicit formula for the velocity function when a jerk reaches its bound is given by solving a second-order differential equation. Under a “greedy rule”, an algorithm for optimal jerk confined feedrate planning is presented. Experiment results show that the new algorithm can be used to reduce the machining vibration and improve the machining quality.     

A parametric interpolator with confined chord errors, acceleration and deceleration for NC machining

Parametric interpolation has many advantages over linear interpolation in machining curves. Real time parametric interpolation research so far has addressed achieving a uniform feed rate, confined chord errors and jerk limited trajectory planning. However, simultaneous consideration of confined chord errors that respect the acceleration and deceleration capabilities of the machine has not been attempted. In this paper, the offline detection of feed rate sensitive corners is proposed. The velocity profile in these zones is planned so that chord errors are satisfied while simultaneously accommodating the machine's acceleration and deceleration limits. Outside the zone of the feed rate sensitive corners, the feed rate is planned using the Taylor approximation. Simulation results indicate that the offline detection of feed rate sensitive corners improves parametric interpolation. For real time interpolation, the parametric curve information can be augmented with the detected feed rate sensitive corners that are stored in 2×2 matrices.     

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 adaptive real-time NURBS curve interpolation for 4-axis polishing machine tool

The non-uniform rational B-spine (NURBS) curve interpolation is a key technology of the advanced computer numerical control (CNC) system. NURBS curve interpolation can realize a high-speed and high-precision machining, and it can also avoid some inevitable deficiencies of the linear and circular interpolation functions which are generally used in traditional NC system. Before the interpolation, some calculation tasks are finished, which will decrease the amount of calculation during interpolation and increase the interpolation efficiency. Further, an adaptive NURBS curve interpolation with real-time and flexible S-shaped curve acceleration/deceleration (ACC/DEC) control method is added to the interpolation algorithms. The NC machining simulation conducted with the MATLAB software and the NURBS curve interpolation experiments performed on the 4-axis polishing machine tool demonstrate the validity and correctness of the adaptive real-time NURBS curve interpolation algorithm in the CNC system.     

基于Adams-Bashforth-Moulton预估校正法的NURBS插补算法研究

针对传统的基于连续微小线段的插补方法在复杂零件型面加工中具有插补误差大和速度波动大等不足,对NURBS曲线的插补原理、插补参数计算、速度波动等方面进行了研究,结合加工弓高误差、法向加速度、法向加加速度等约束条件进行了速度规划,提出了一种基于Adams-Bashforth-Moulton(ABM)预估校正法的NURBS曲...     

Linear and angular feedrate interpolation for planar implicit curves

In multi-axis CNC machining, a cutting tool combines translation and rotational movements with respect to a workpiece. This entails CNC interpolation to take angular feedrates defining the speed of the rotational movement into account, while current interpolations are overwhelmingly based on linear feedrates defining the speed of the translation movement. This paper considers linear and angular feedrate interpolations of 2D tool paths for multi-axis machining. A 2D tool path is identified by a position curve assumed to be a 2D implicit curve and a tool orientation curve based on the implicit curve. The paper will relate the angular feedrate to the linear feedrate along the tool paths. The result is then used to develop the angular feedrate interpolation from the linear feedrate interpolation. Detailed interpolation schemes for such hybrid feedrates as a constant linear feedrate with constrained angular feedrate (CLCA) and a constant angular feedrate with constrained linear feedrate (CACL) have further been constructed. A corrected interpolation scheme is further present to augment the initial interpolation. Example interpolations including a quadric curve, the nodal cubic and a quartic curve are carried out to illustrate the feasibility and effectiveness of the proposed approach.     

A real-time configurable NURBS interpolator with bounded acceleration,jerk and chord error

Advances in manufacturing technologies and in machine tools allow for unprecedented quality and efficiency in production lines, but also ask for new and increasing requirements on the motion planning and control systems. The increase of CPU processing power has permitted, in traditional CNC systems, the introduction of NURBS interpolation capabilities, thus determining a further increase in machining quality and efficiency. This has posed new and still unsolved issues, such as the need to satisfy multiple opposite constraints like limiting chord error, acceleration and jerk and offering real-time guarantees. In addition, the ability of privileging the production throughput by relaxing one or more of the previous constraints in a simple way, has emerged as another requirement of modern manufacturing plants. Nevertheless, none of the existing NURBS interpolators have these characteristics. In this work, we propose a NURBS interpolator that is able to satisfy all the manufacturing technology requirements and is able to respect, thanks to its bounded computational complexity, the position control real-time constraints. Such an interpolator is easily reconfigurable, i.e., it can relax some of the constraints while maintaining performances better than previously proposed solutions, and can be adapted in order to include constraints that were not originally considered. Performances of the proposed algorithm have been evaluated both by simulations and by real milling experiments.     

High speed interpolation for micro-line trajectory and adaptive real-time look-ahead scheme in CNC machining

CNC machining plays an important role in mechanical manufacturing.A key issue is to improve the machining feedrate while keeping the machining precision and satisfying the acceleration constraints of the CNC machine.For the consecutive micro-line segments interpolation,the velocities at the junction of two segments are the bottlenecks for the machining efficiency.This paper proposes a multi-period turning method to improve the feedrate at the junctions using the linear acceleration and deceleration mode,whi...     

A newly developed corner smoothing methodology based on clothoid splines for high speed machine tools

Continuous linear commands are widely executed in computer numerical control (CNC) machining. The tangential discontinuity at the junction of consecutive segments restricts the machining efficiency and deteriorates the surface quality. Corners of linear segments have been successfully blended by inserting parametric splines. There still exists challenges when the common methods are employed in the line-segment commands due to part of the following restrictions: (1) the stringent computation for iteratively calculating the arc-length; (2) the unwanted feedrate fluctuation; (3) the oversize contour deviation for separately completing curve fitting and velocity planning.A novel smoothing method based on a clothoid pair to synchronously accomplish planning of geometry blending and speed scheduling is proposed, the spline parameter of which is arc-length-parameterized. The arc-length, curvature extreme, and geometric shape of the transition curve are analytically expressed by the transition length. On these bases, the transition curve and the velocity profile are concurrently constructed based on the predefined approximation error, the reachable velocity, and normal kinematic constraints in the look-ahead stage. Then, a real-time interpolation scheduling is developed to overcome the crossing difficulties between the linear and parametric segments. Compared with existing methods, the proposed method can analytically calculate the length of transition curves for the arc-length-parameterized expression form. Furthermore, the feedrate fluctuation is eliminated in the fine interpolation. Moreover, the overlarge contour derivation produced by corner smoothing is significantly avoided. It is friendlier to the CNC system for the on-line executing smooth motion since more computing resources can be released to handle other tasks, smoother motion can be achieved and higher contour accuracy can be obtained. The experimental results also demonstrate its practicability and reliability.     

CNC系统中NURBS曲线插补及相关算法研究

为提高计算机数字控制（CNC）系统的轮廓控制精度,需解决系统控制软件样条直接输出的问题。基于Taylor公式得到了非均匀有理B样条（NURBS）曲线上两个相邻插补点参数间的递推关系。对于NURBS曲线插补过程中需要频繁计算的B样条基函数及其任意阶导数提出了一种分块矩阵连乘积形式的统一计算方法。应用数值计算方法解决了插补过程中曲线长度等的相关计算问题。实例运算表明所提出的方法可以应用到实际CNC系统中。     

参数曲线插补自适应加减速控制方法研究

CNC系统中参数曲线插补广泛地应用于复杂轨迹的生成.为了弥补参数曲线插补时加减速能力的不足,提出自适应的加减速控制方法.在插补预处理中,对轮廓误差引起的速度变化曲线进行分析,在加加速度满足要求的同时,对加速度的变化进行控制.应用该方法,在保证系统加工精度的基础上,可以根据曲线的形状,自适应的调整进给速度,使进给速度变化趋于平滑,有效地避免了加速度和加加速度的变化对伺服系统造成的机械冲击.文中给出了B样条曲线插补的实例对控制方法进行了说明和验证.仿真结果表明,该方法切实可行而且有效.