Development of a real-time look-ahead interpolation methodology with spline-fitting technique for high-speed machining

Methodologies for converting short line segments into parametric curves were proposed in the past. However, most of the algorithms only consider the position continuity at the junctions of parametric curves. The discontinuity of the slope and curvature at the junctions of the parametric curve might cause feedrate fluctuation and velocity discontinuous. This paper proposes a look-ahead interpolation scheme for short line segments. The proposed interpolation method consists of two modules: spline-fitting and acceleration/deceleration (acc/dec) feedrate-planning modules. The spline-fitting module first looks ahead several short line segments and converts them into parametric curves. The continuities of the slope and curvature at each junctions of the spline curve are ensured. Then the acc/dec feedrate-planning module proposes a new algorithm to determine the feedrate at the junction of the fitting curve and unfitted short segments, and the corner feedrate within the fitting curve. The chord error and acceleration of the trajectory are bounded with the proposed algorithm. Simulations are performed to validate the tracking and contour accuracies of the proposed method. The computational efforts between the proposed algorithm and the non-uniform rational B-spline (NURBS)-fitting technique are compared to demonstrate the efficiency of the proposed method. Finally, experiments on a PC-based control system are conducted to demonstrate that the proposed interpolation method can achieve better accuracy and reduce machining time as compared to the approximation optimal feedrate interpolation algorithm.     

基于弧长参数的Akima刀位轨迹拟合算法研究

在数控加工中,通常用小线段表达刀位轨迹,往往会导致刀位点庞大且轨迹不平滑。基于Akima曲线具有光顺连接且端点连接处保证G1以上连续等特点,将刀位点拟合成Akima样条曲线,提出了基于弧长参数的保凸Akima拟合刀位轨迹算法。该算法分为刀位点搜索和拟合两个阶段：首先利用拟合刀位轨迹的误差测试（双弦误差测试和弦切误差测试）约束,获得该段的首末刀位点;然后在该段内根据首末刀位点计算切线矢量,用弧长信息对刀位点参数化,生成一段Akima样条。利用UG软件生成了内含75个刀位点的鞋底模型,通过MATLAB编程和仿真实验,设置不同测试阈值,对比了弧长参数化Akima曲线与节点参数化Akima曲线、NURBS曲线之间的拟合效果,验证了算法的有效性。     

Feature extraction from slice data for reverse engineering

A new approach to feature extraction for slice data points is presented. The reconstruction of objects is performed as follows. First, all contours in each slice are extracted by contour tracing algorithms. Then the data points on the contours are analyzed, and the curve segments of the contours are divided into three categories: straight lines, conic curves and B-spline curves. The curve fitting methods are applied for each curve segment to remove the unwanted points with pre-determined tolerance. Finally, the features, which consist of the object and connection relations among them, are founded by matching the corresponding contours in adjacent slices, and 3D models are reconstructed based on the features. The proposed approach has been implemented in OpenGL, and the feasibility of the proposed method has been verified by several cases.     

Modeling of the residual kinematic errors of coordinate measuring machines using LaserTracer system

It is complex and divergent for the conventional motion controllers to process various G codes using different interpolation algorithms. This impairs programming efficiency and robustness of the controller. In this paper, we propose the universal non-uniform rational B-splines (NURBS)-based interpolator which can simplify the architecture of interpolation in spite of interpreting different kinds of inputs. Direct conversion of long G01 and G02/G03 numerical control (NC) segments to NURBS segment is first implemented. The fitting of multiple short segments into a continuous and smooth NURBS segment is then carried out. More importantly, the universal NURBS-based interpolator utilizes the Cox–de Boor algorithm which is highly efficient and can take advantage of the parallel computing scheme to accelerate the processing speed. Furthermore, due to the construction of real-time environment, the proposed algorithm enables interpretation, look-ahead functions, and motion control to work simultaneously. A 2D NC program possessing hundreds of short G01 segments and long segments (i.e., G02, G03, and long G01) is tested on an in-house developed XY-table with a PC-based motion controller. The results have shown the effectiveness and feasibility of the proposed real-time NURBS-based interpolator.     

一种基于最小二乘法的离散点螺旋线式拟合算法

针对数控加工中,小线段刀路轨迹数据量较大的问题,提出了一种利用阿基米德螺线拟合小线段轨迹（离散点）的算法,分析了阿基米德螺线参数与切矢、径矢夹角的关系,给出了拟合误差的计算方法。基于最小二乘原理,通过前寻、回溯离散点集,在满足精度要求的前提下,用较少段螺旋线拟合该离散点集。仿真结果表明,阿基米德螺线拟合离散点具有数据量小、拟合精度较高的优势。     

基于精度控制的刀具轨迹自适应拟合算法

为解决传统数控系统在进行曲线曲面离散连续微小线段刀具轨迹加工中频繁加减速、加工速度缓慢、加工质量不高等问题,提出了一种基于精度控制的刀具轨迹自适应NURBS曲线拟合算法,该算法通过提取连续微小线段刀具轨迹的主要特征点,进行基于精度控制的自适应添加特征点的迭代拟合。仿真测试结果表明,该算法可以在保证拟合精度的条件下有效提高计算效率,压缩数据量。     

一种基于插值拟合的复杂刀具轨迹在线平滑压缩算法

提出了一种基于插值拟合的在线复杂刀具轨迹平滑压缩算法,该算法依据主导点的选取策略,对原始数据点进行离线预处理,然后进行主导点的在线插值拟合以及非主导点的误差检测,进而生成一条满足拟合精度要求的B样条曲线。主导点依据离散数据点的曲率阈值、曲率极大值、曲线拐点,以及分段Bezier曲线逼近拟合后的误差最大值点进行选取。在具有C2连续性的分段Bezier曲线逼近拟合前,需要利用长度均分策略,提取长度突变点作为新增的主导点,以保证拟合的准确性。对主导点进行B样条插值拟合后,利用轮廓误差跟随法对非主导点到拟合曲线的误差进行检测。该方法与牛顿迭代法相比,其计算速度更快且能提高算法效率。仿真结果表明,提出的算法可对复杂刀具轨迹进行平滑压缩,且误差检测的精度能够满足要求。     

LOOK-AHEAD ALGORITHM FOR VELOCITY CONTROL BASED ON PARAMETERIZED CURVE INTERPOLATOR

To avoid suffering gouge and transient overshooting in high speed cutting machining, a novel parametefized curve interpolator model with velocity look-ahead algorithm is proposed. Based on a prearrangement step interpolation algorithm for parameterized curves and considering high curvature points, parameterized curve tool path is divided into acceleration segments and deceleration segments by look-ahead algorithm. Under condition of characteristics of acceleration and deceleration stored in control system, deceleration before high curvature points and acceleration after high curvature points are realized in real-time in high speed cutting machining. Based on new parameterized curve interpolator model with velocity look-ahead algorithm, a real cubic spline is machined simulativly. The simulation results show that velocity look-ahead algorithm improves velocity changing more smoothly.     

曲面笔式加工刀位点曲线的实时拟合

针对曲面高速高精度加工问题,给出一种刀位点轨迹实时生成算法。该方法的实现包括曲面上离散刀触点的生成和基于三次非均匀B样条曲线的刀位点轨迹实时高精度拟合两部分。第一部分,由导动曲线和刀触点轨迹的运动学关系,通过计算导动曲线参数,间接得到投影在曲面上的离散刀触点;第二部分,通过合理参数化、构建模长因子等实现刀位点参数曲线的分段实时拟合。仿真实例表明该算法简单易于编程,曲线拟合精度高,适用于自由曲面笔式加工中刀具路径计算与生成,从而满足复杂曲面高性能数控加工需求。     

自由曲线轮廓加工的圆弧样条刀具路径研究

给出了一种平面曲线轮廓的圆弧样条拟合及刀具路径生成算法，该算法面向零件轮廓的光顺性刀具路径生成，通过应用曲线的曲率关系，对以NURBS表示的被拟合自由曲线按参数递增的方向用G^1连续的圆弧或直线段逐段拟舍，并生成相应的圆弧样条刀具路径，从而实现零件曲线轮廓表面的光顺加工。实例计算和分析结果证明了该算法具有计算稳定性好、计算效率高的优点，易于在数控加工中实现。     

Implementation of online NURBS curve fitting process on CNC machines

The study aims to develop a method for implementing an online non-uniform rational B-spline (NURBS) curve fitting process on CNC machines for improving the quality and efficiency of machining. Conventional CAD/CAM/CNC systems usually induce some machining difficulties and limit the machining results in real applications. Therefore, some researchers have proposed various methods for improving the machining results. An online NURBS curve fitting process consists of the fetching and fitting stages for CNC machines is developed in this paper. The fetching stage fetches a block of NC code and generates motion commands for obtaining data points. The fitting stage executes the NURBS curve fitting method along with a given set of data points. Moreover, the optimal search method is designed for obtaining good results in the NURBS curve fitting process. Simulations and machining tests carried out on a vertical machining center show that the proposed approach reduces the machining time to about 23% while maintaining the machining quality for the sample test.     

Unconstrained and constrained curve fitting for reverse engineering

Curve fitting is commonly used in reverse engineering for the reconstruction of curves from measured points, and it is critically important to provide various kinds of curve-fitting algorithms to acquire curves that satisfy different constraint conditions. We divide the curve-fitting problem into unconstrained and constrained types. For the unconstrained type, three curve-fitting algorithms are investigated: general, smooth and extended curve fitting. The general curve fitting considers only the accuracy of the fitted curve; the smooth curve fitting can control both the accuracy and the fairness of the fitted curve, while the extended curve fitting can acquire a curve longer than the range of the measured points. For the constrained type, we propose three curve-fitting conditions: fixed end-points, closed curve and continuity to adjacent curves. Detailed discussion for each of the above cases is presented. Associated examples are also provided to illustrate the feasibility of the proposed algorithms.     

A local smoothing interpolation method for short line segments to realize continuous motion of tool axis acceleration

In traditional processing, a large number of G01 blocks are adopted to discretize free surface or curve for NC machining. But, the continuity of G01 line segments is only C⁰, which may lead to discontinuity of axis acceleration, resulting in the frequent fluctuation of tool motion at the junctions in high-speed machining, deteriorating the quality of work piece, and reducing processing efficiency. To solve this problem, a local smoothing interpolation method is proposed in this paper. At first, the analytic relationship between the continuity of the trajectory and the continuity of the axes motion is first systematically described by formula. Based on this relationship, a local smoothing algorithm and a feed-rate scheduling method are proposed to generate a C² continuous tool path motion with axis-acceleration continuity. The local smoothing algorithm smoothes the corners of G01 blocks by the cubic B-spline according to the cornering error tolerance specified by the user. After the feed rate at critical points of smoothed tool path was determined by a modified bidirectional scanning algorithm by considering constrains of chord error and kinematic property, an iterative S-shape feed rate scheduling is employed to minimize residual distance caused by round of time while ensuring the continuity of feed rate and acceleration. Then, a look-ahead interpolation strategy combined with smoothing algorithm and feed-rate scheduling as mentioned is proposed for real-time interpolation of short line segments. At last, simulations are conducted to verify the effectiveness of the proposed methods. Compared with the traditional G01 interpolation, it can significantly improve the processing efficiency and shorten the processing time within error tolerance.     

NURBS curve and surface fitting for reverse engineering

Reverse engineering is an approach for constructing a computer-aided design (CAD) model from a physical part through dimensional measurement and surface modelling. This paper presents alternatives for reverse engineering of free-form parts using Non-Uniform Rational B-Spline (NURBS) surfaces. A two-step linear approach is employed for fitting NURBS curves and surfaces using the measured points. During the first step, the weights of the control points are identified from a homogeneous system using symmetric eigenvalue decomposition. The control points are further processed in a way similar to B-spline curve and surface fitting. Some examples are presented to illustrate both the steps of reverse engineering and the process of NURBS curve and surface fitting.     

Real-time NURBS interpolator: application to short linear segments

This study proposes the use of a real-time non-uniform rational B-spline (NURBS) interpolator with a look-ahead function to handle numerous short linear segments. The short linear segments conforming to the continuous short block (CSB) criterion can be fitted into NURBS curves in real time. A modified maximum feedrate equation based on the geometric characteristics of the fitting curves and the dynamics of the servo control system has been derived in this paper. Taking advantage of the multi-thread design and the look-ahead function, the real-time NURBS interpolator can process enough G01 block information and complete feedrate planning before interpolation. In addition, the S-shaped jerk-limited acceleration method is adopted for smoother feedrate profiles. Two part shapes, which possess more than 1,000 short linear segments, are tested on our PC-based real-time control system. Both simulation and experimental results verify the feasibility and precision of the proposed interpolation algorithm.     

凸轮列表点曲线数控编程方法研究

针对平面凸轮轮廓列表点曲线的数控加工,提出了一种采用三次样条等误差双圆弧拟合法实现数控编程的计算方法.首先求出凸轮轮廓列表点的三次样条函数曲线方程,用等误差法计算插值点的坐标.再用双圆弧拟合插值点,得到各分段圆弧的圆心坐标,取圆弧的起点、终点坐标,作为编程所需的刀具中心轨迹的数据,并用相应的数控机床指令写出数控加工程序.该方法既能满足精度要求又能使程序段最少,可以保证数控加工时曲线的光滑性及加工精度,提高加工效率.     

数控加工中定曲线方程轮廓的等弦高误差拟合算法研究

针对指定曲线方程轮廓,提出了等弦高误差拟合算法,通过指定弦高误差,应用非均匀递归插值方法获得刀触点,通过拟合轮廓等距偏置获得刀位点,算法简单可靠且效率相对较高,对削步长具有沿加工方向的曲率适应性,加工效率高。可应用于2轴及2.5轴轮廓加工以及线切割加工。     

基于能量最优的局部优化光顺处理

在对CAD/CAM生成的大量微小直线段进行样条曲线拟合过程中,会出现拟合出的样条曲线不光顺的情况,进而影响数控加工出的工件的表面质量。针对该种情况,提出了一种同时考虑相邻弦间夹角和曲率的＂坏点＂判定方法,并采用能量最优的局部优化法同时对＂坏点＂的位置和切向量进行修正。然后选取一组数据点进行仿真实验,仿真结果表明,经＂坏点＂修正处理后的样条曲线的曲率更均匀了,从而验证了该算法的有效性。     

Constant cusp toolpath generation in configuration space based on offset curves

Constant cusp is a common strategy for generating tool paths in many NC machining applications. Cusps need to be regulated to ensure high precision without wasting machining efforts. Constant cusp strategies frequently operate on NURBS surfaces or triangular meshes and, thus, have to deal with the issues of patch-boundary oscillations or long, stretched triangles. To avoid these issues, one can operate in a pre computed configuration space (c-space). The c-space is given in form of a regular quadrilateral heightfield mesh, which may be adaptively subdivided, where the slope is large. This simple data structure is memory efficient and is widely used in CAD/CAM frameworks. In this paper we introduce an algorithm for creating a constant cusp tool path with the help of a given c-space. The constant cusp algorithm iteratively produces curves in the c-space by fitting a tube around the current curve and intersecting the tube with the c-space mesh to detect the subsequent curve. As tool paths are handed to the machine controller in form of point sequences, it suffices to operate on piecewise linear curves. The tube becomes a concatenation of cylinders, which we derive using geometric considerations. In each iteration of the constant cusp algorithm, intersection points of the cylinders with the not yet traversed part of the mesh are detected and checked for their validity. The validity check can efficiently remove global or local self-intersections of the new curve by just deleting the respective points. In a final step, the detected intersection points are connected to form constant cusp tool paths. Dealing with piecewise linear curves, we achieve low computation times for real-world data sets.     

基于MLS改进算法的曲线拟合技术研究

针对散乱点拟合曲线的算法问题,分析了移动最小二乘逼近法（MLS）的特点,提出了用MLS改进算法对散乱点进行曲线拟合,并结合实例进行了验证。结果表明,MLS改进算法在拟合曲线时,选取不同阶的基函数和不同的权函数,在保证精度和光滑度的基础上,解决了传统曲线拟合方法在拟合曲线时出现的运算不稳定现象,显示了MLS改进算法在曲线拟合中的优越性。