Real-time Bezier interpolation satisfying chord error constraint for CNC tool path

G01 code generated by a computer-aided manufacture (CAM) system is the most common form of tool trajectory in computer numerical control (CNC) machining. A tool path composed of short line segments has discontinuous tangency and curvature, generating large fluctuations of feedrate and acceleration, which in turn produces vibration in a machine tool. To obtain a smooth tool path, many methods on tool-path smoothing have been developed. However, the shortcomings in these methods exist when they are employed in a CNC system. It is difficult to simultaneously to guarantee the following requirements of CNC machining: (1) chord error should be rigidly constrained; (2) G01 points should be interpolated; (3) curvature should be continuous (G²); (4) machining should be applicable to spatial cases; (5) real-time performance of computation is required. Based on these various requirements, this study proposes an interpolation scheme using cubic Bezier curves and includes an adjustment strategy to eliminate deficiencies in the tool path. The tool path generated is G², chord-error-constrained, G01-point-interpolated, loop-free, and optimized for both stretch and jerk energy. The method is applicable to 3D cases and involves only simple algebraic computations. Thus, the algorithm can be applied to real-time CNC machining. A simulation is conducted to validate the efficiency of the algorithm. In addition, an experiment reveals its advantage over Hermite interpolation in surface quality and machining efficiency.     

连续加工路径的进给速度规划算法研究

为了实现数控系统连续轨迹的速度规划,本文提出了混合进给速度的规划方法.此方法通过建立转弯处加工路径和加工误差的数学模型,实现了转弯处进给速度的控制.实验结果表明,这种方法不仅可以极大的提高机床的加工效率,减小由于路径转弯剧烈引起的机床震动,从而大大提高工件的加工质量.混合进给速度的规划方法保证了加工精度和加减速的稳定性.     

微线段高速加工的圆锥截线拐角插补算法

针对数控高速加工时微小线段拐角过渡的平滑问题,建立基于二次有理Bézier形式的圆锥截线拐角过渡矢量模型,推导三种类型的园锥截线过渡曲线的最大轮廓偏差和最大曲率计算公式,提出一种微小线段间圆锥截线拐角过渡的插补算法。算法可在满足轮廓加工精度和相邻小线段长度的条件下选择圆锥截线的权因子,确定圆锥截线拐角过渡曲线的类型;根据数控系统的动态响应能力、弓高误差和过渡曲线多步转接的要求,确定拐角曲线过渡的最大加工速度;进一步提出通过修改圆锥截线的权因子实现拐角曲线的加工速度与各直线段路径的进给速度衔接过渡的前瞻处理方法。与圆弧拐角过渡的对比结果表明,算法可有效提高微小线段间的拐角加工速度,缩短数控加工时间。     

NURBS及Hermite混合高速加工插补算法

在考虑工件轮廓的几何特点对加工效率与加工平稳性带来的影响的基础上,提出直线、NURBS及Hermite曲线混合高速加工插补算法.该算法在满足插补精度的基础上,对工件轮廓的线段长度、相邻线段夹角等几何参数进行定量分析,筛选出满足条件的部分进行NURBS曲线拟合.对存在尖角的曲线段间采用三次Hermite曲线进行衔接,改善了加工轮廓的光滑性.在满足最大弓高误差及最大加速度的要求下对直线段、NURBS曲线以及Hermite曲线进行混合插补,从而实现任意轮廓高速高质量的数控加工.对比分析和加工实验表明,采用该算法能够提高轮廓的连续性,有利于改善加工效率和加工平稳性.     

数控加工中的连续多段直线轨迹B-Spline拟合

为了克服直线和圆弧插补的各种不足,实现曲线插补,针对数控加工中的连续多段小直线刀具轨迹,提出了一种简单快捷的B-Spline曲线(NURBS曲线的特殊形式)拟合算法.该算法以标准的B-Spline最小二乘方法为基础,通过简化后的误差计算模块,可以将拟合后的曲线和原有路径的偏离控制在要求的范围内,并提出了可以实现快速连许多段小直线刀具轨迹拟合的搜索方法-最大拟合区间的稳步前进方法.通过实例计算分析,证明了给出的拟合方法可靠高效.     

Jerk-limited feedrate scheduling and optimization for five-axis machining using new piecewise linear programming approach

In this paper, a new computation method and an optimization algorithm are presented for feedrate scheduling of five-axis machining in compliance with both machine drive limits and process limits. Five-axis machine tool with its ability of controlling tool orientation to follow the sculptured surface contour has been widely used in modern manufacturing industry. Feedrate scheduling serving as a kernel of CNC control system plays a critical role to ensure the required machining accuracy and reliability for five-axis machining. Due to the nonlinear coupling effects of all involved drive axes and the saturation limit of servo motors, the feedrate scheduling for multi-axis machining has long been recognized and remains as a critical challenge for achieving five-axis machine tools' full capacity and advantage. To solve the nonlinearity nature of the five-axis feedrate scheduling problems, a relaxation mathematical process is presented for relaxing both the drive motors' physical limitations and the kinematic constraints of five-axis tool motions. Based on the primary optimization variable of feedrate, the presented method analytically linearizes the machining-related constraints, in terms of the machines' axis velocities, axis accelerations and axis jerks. The nonlinear multi-constrained feedrate scheduling problem is transformed into a manageable linear programming problem. An optimization algorithm is presented to find the optimal feedrate scheduling solution for the five-axis machining problems. Both computer implementation and laboratorial experiment testing by actual machine cutting were conducted and presented in this paper. The experiment results demonstrate that the proposed method can effectively generate efficient feedrate scheduling for five-axis machining with constraints of the machine tool physical constraints and limits. Compared with other existing numerical methods, the proposed method is able to find an accurate analytical solution for the nonlinear constrained five-axis feedrate scheduling problems without compromising the efficiency of the machining processes.     

Smoothness-oriented path optimization for robotic milling processes

Industrial robots are increasingly used for five-axis machining operations, where the rotation of the end effector along the toolaxis direction is functionally redundant. This functional redundancy should be carefully resolved when planning the robot path according to the tool path generated by a computer-aided manufacturing(CAM) system. Improper planning of the redundancy may cause drastic variations of the joint motions, which could significantly decrease the machining efficiency as well as the machining accuracy. To tackle this problem, this paper presents a new optimization-based methodology to globally resolve the functional redundancy for the robotic milling process. Firstly, a global performance index concerning the smoothness of the robot path at the joint acceleration level is proposed. By minimizing the smoothness performance index while considering the avoidance of joint limits and the singularity and the constraint of the stiffness performance, the resolution of the redundancy is formulated as a constrained optimization problem. To efficiently solve the problem, the sequential linearization programming method is employed to improve the initial solution provided by the conventional graph-based method. Then, simulations for a given tool path are presented. Compared with the graph-based method, the proposed method can generate a smoother robot path in which a significant reduction of the magnitude of the maximum joint acceleration is obtained, resulting in a smoother tool-tip feedrate profile. Finally, the experiment on the robotic milling system is also presented. The results show that the optimized robot path of the proposed method obtains better surface quality and higher machining efficiency, which verifies the effectiveness of the proposed method.     

剃齿刀修形砂轮斜截面法数控精密加工机理研究

剃齿刀修形砂轮侧面突起是弦长弦高比很大的圆弧,现有数控插补算法无法加工出符合精度要求的修形砂轮。针对这一缺陷,提出斜截面法放大弦高从而减小弦长弦高比的方法,推导加工曲线的方程,并结合算例,获得离散点坐标和加工刀具轨迹。结果表明通过斜截面法放大弦高的加工方式可以获得较高精度的插补轨迹。同时也为大曲率小圆弧曲面的数控加工奠定了一定的理论基础。     

基于Kirov定理的曲线拟合方法

基于Kirov逼近定理,建立一种新的数据拟合方法,研究一类带有附加导数条件的多结点样条插值和B样条拟合．该方法可以在型值点处再给出切线条件,这与通常的多结点样条插值和B样条拟合相比有更大的自由度．此方法有助于CAGD领域的工程人员更好地控制设计曲线的形状．     

Circular Interpolation Algorithms of 5-Axis Simultaneous CNC System

Spatial circular arc can be machined conveniently by a 5-axis NC machine tool.Based on the data sampling method,circular interpolation in two-dimensional plane is discussed briefly.The key is to solve the problem of circular center ex- pressed in the workpiece coordinate system by means of the transformation matrix.Circular interpolation in three-dimensional space is analyzed in detail.The method of undetermined coefficient is used to solve the center of the spatial circle and the method of coor- dinate transformation is used to transform the spatial circle into the XY-plane.Circular arc in three-dimensional space can be ma- chined by the positional 5-axis machining and the conical surface can be machined by the continuous 5-axis machining.The velocity control is presented to avoid the feedrate fluctuation.The interpolation algorithms are tested by a simulation example and the inter- polation algorithms are proved feasible.The algorithms are applied to the 5-axis CNC system software.     

大型叶片五轴联加工的几何误差控制

误差控制和减少加工误差是大型叶片曲面类零件五轴联动数控加工中的重要任务之一。本文作者基于分析和建立加工曲面的型面几何偏差与加工刀位规划参数的关系 ,提出几何误差的控制方法。基于刀具与工件的五维包络运动分析 ,对五轴联动加工中影响较大的的非线性误差进行较为严格的计算 ,提出自适应步长方法以有效控制刀触点轨迹误差。对五轴联动加工中的残余高度误差进行较为严格的分析 ,并提出有效控制几何误差的方法和措施。文中的方法结合CAM软件进行刀位轨迹计算 ,已成功地用于大型轴流式和混流式水轮机叶片五轴联动数控加工中 ,在满足精度要求下 ,加工叶片的效率有明显的提高 ,表明提出的方法和措施能有效控制大型塑曲面加工的几何误差。     

大型叶片五轴联加工的几何误差控制

误差控制和减少加工误差是大型叶片曲面类零件五轴联动数控加工中的重要任务之一。作者基于分析和建立加工曲面的型面几何偏差与加工刀位规划参数的关系,提出几何误差的控制方法。基于刀具与工件的五维包络运动分析,对五轴联动加工中影响较大的非线性误差进行较为严格的计算,提出自适应步长方法以有效控制刀触点轨迹误差。对五轴联动加工中的残余高度误差进行较为严格的分析,并提出有效控制几何误差的方法和措施。文中的方法结合CAM软件进行刀位轨迹计算,已成功地用于大型轴流式和混流式水轮机叶片五轴联动数控加工中,在满足精度要求下,加工叶片的效率有明显的提高,表明提出的方法和措施能有效控制大型塑曲面加工的几何误差。     

大型叶片五轴联加工的几何误差控制

误差控制和减少加工误差是大型叶片曲面类零件五轴联动数控加工中的重要任务之一.本文基于分析和建立加工曲面的型面几何偏差与加工刀位规划参数的关系,提出几何误差的控制方法.基于刀具与工件的五维包络运动分析,对五轴联动加工中影响较大的的非线性误差进行较为严格的计算,提出自适应步长方法以有效控制刀触点轨迹误差.对五轴联动加工中的残余高度误差进行较为严格的分析,并提出有效控制几何误差的方法和措施.文中的方法结合CAM软件进行刀位轨迹计算,已成功地用于大型轴流式和混流式水轮机叶片五轴联动数控加工中,在满足精度要求下,加工叶片的效率有明显的提高,表明提出的方法和措施能有效控制大型塑曲面加工的几何误差.     

自由曲线、曲面的等距算法及其在数控加工中的应用

推导了NURBS曲线、曲面及其等距曲线、曲面的各阶导矢以及曲率计算,分析了参数曲线、曲面的线性插值误差,讨论了插值等距曲线、曲面上点的采样原则及B样条形式表达的插值曲线、曲面的求解。最后提出了三轴联动行切(NC)加工的刀位计算方法。     

水轮机叶片的五轴数控加工工艺规划及轨迹优化

针对大型水轮机叶片的五轴联动数控加工要求,制定出适合国内技术条件的工艺规划。提出在计算叶片五轴联动加工刀位时采用等残余高度刀位规划,并考虑三维非线性误差来计算走刀步长的刀位轨迹生成策略。为了避免刀具后跟角干涉的要求,提出根据叶片的曲面性态分析来确定叶片五轴联动加工中的刀轴控制和刀位轨迹优化计算方法。将该方法与CAM软件结合,用于大型混流式和轴流式叶片数控加工,结果表明：该方法既能严格控制加工误差,又要能提高加工效率。     

参数方程曲线的直接插补算法研究

插补技术是数控系统的核心技术,插补算法的优劣直接影响数控系统的加工效率.通常渐开线、螺旋线等非圆复杂曲线在数控系统中用小段圆弧或直线来逼近,这使得数控代码的数据量极大,且逼近误差较大,计算机辅助编程也很复杂.为此,本文提出一种非圆复杂曲线的插补算法,该算法表达精确、运算速度快;此外,还分析了该插补算法的理论误差,并给出了通用的刀具半径补偿方法.     

Study of Quintic Spline Interpolation and Generated Velocity Profile for High Speed Machining

Modern high speed machining (HSM) machine tools often operates at high speed and high feedrate with high ac- celerations,in order to deliver the rapid feed motion.This paper presents an interpolation algorithm to generate continuous quintic spline toolpaths,with a constant travel increment at each step,while the smoother accelerations and jerks of two-order curve are obtained.Then an approach for reducing the feedrate fluctuation in high speed spline interpolation is presented.The presented ap- proach has been validated to quickly,reliably and effective with the simulation.     

样条插值的MATLAB实现

插值法是工程实践中最常用的函数逼近方法,其方法就是利用有限个数据点来实现对整个函数的拟合.本文介绍了插值法的概念,进而对样条插值的概念和条件进行了阐述.三次样条插值和B样条插值是最常用的两种样条插值方法.本文着重对这两种方法进行了数学分析并基于MATLAB工具箱对其进行仿真实现.     

三次多项式型段内加减速控制新方法

传统的加减速控制方法由于加速度的不连续性,易使机床产生冲击,影响零件加工质量和机床使用寿命.为此提出了一种三次多项式加减速控制模型.针对任意路径段在插补前加减速过程中需要预测减速点的问题,进一步提出了段内加减速控制新方法,实现了对实际减速点的确定.针对理论减速点与实际减速点不重合而导致当减速阶段结束时仍存在低速运行段的问题,实现了对减速点的误差补偿.仿真和试验结果表明,提出的方法能够实现在加工过程中实际减速点的动态、智能化判断,并能够在较高速度下补偿减速点误差,消除了低速运行时间,提高了加工效率     

七自由度番茄收获机械手的轨迹规划与仿真

为了探究番茄果实串收获机械手的轨迹规划方法,以七自由度番茄收获机械手为研究对象,运用D-H法建立运动学模型,得到位置和速度的正、反解.采用多项式插值法、抛物线过渡的线性插值法、摆线运动法和B样条插值法,对关节约束条件下的机械手,从复位点到采摘点的运动进行关节空间轨迹规划与仿真试验.分析关节运动的位移、速度和加速度,对不同规划方法的关节最大速度进行定量分析.利用五次多项式法和摆线运动法,研究末端执行器的位移和速度. 结果表明,利用五次多项式插值法能够满足关节路径点的位移、速度和加速度边界条件,规划的关节运动轨迹连续平滑,关节最大速度较摆线运动法平均降低了6.25%,提高了机械安全性能,对应的末端执行器运动轨迹平滑,无跳变点.