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山东大学学报 (工学版) ›› 2023, Vol. 53 ›› Issue (6): 143-151.doi: 10.6040/j.issn.1672-3961.0.2023.009

• 机械与能动工程 • 上一篇    下一篇

基于弧长参数的自由曲线实时误差估计算法

路勇良1,2(),张伟3,赵军1,2,*(),张振3,张自健1,2   

  1. 1. 山东大学机械工程学院高效洁净机械制造教育部重点实验室，山东 济南 250061
    2. 山东大学机械工程国家级实验教学示范中心，山东 济南 250061
    3. 山东蒂德精密机床有限公司，山东 济宁 272000
  • 收稿日期:2023-01-05 出版日期:2023-12-20 发布日期:2023-12-19
  • 通讯作者: 赵军 E-mail:202134345@mail.sdu.edu.cn;zhaojun@sdu.edu.cn
  • 作者简介:路勇良(1998—)，男，山东济宁人，硕士研究生，主要研究方向为交叉耦合控制、轮廓精度控制。E-mail：202134345@mail.sdu.edu.cn
  • 基金资助:
    山东省重点扶持区域引进急需紧缺人才项目(2022-6);泰山产业领军人才工程资助项目(tscy20221164)

Real-time error estimation algorithm of free curve based on arc length parameters

Yongliang LU1,2(),Wei ZHANG3,Jun ZHAO1,2,*(),Zhen ZHANG3,Zijian ZHANG1,2   

  1. 1. Key Laboratory of High Efficiency and Clean Mechanical Manufacture of MOE, School of Mechanical Engineering, Shandong University, Jinan 250061, Shandong, China
    2. National Demonstration Center for Experimental Mechanical Engineering Education, Shandong University, Jinan 250061, Shandong, China
    3. Shandong Deed Precision Machine Tool Co., Ltd., Jining 272000, Shandong, China
  • Received:2023-01-05 Online:2023-12-20 Published:2023-12-19
  • Contact: Jun ZHAO E-mail:202134345@mail.sdu.edu.cn;zhaojun@sdu.edu.cn

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摘要：

针对复杂型面零件加工精度要求提高以及在自由曲线跟随任务中实时精确计算轮廓误差难度较大等问题，提出一种基于弧长参数的自由曲线实时误差估计算法。在MATLAB/Simulink中利用非均匀有理B样条(non-uniform rational B-spline, NURBS)插补对椭圆轨迹和双纽线轨迹进行规划，将基于弧长参数的自由曲线实时误差估计算法与切线法、圆近似法、平均速度法和三点圆法的轮廓误差估算精度与执行时间等性能评价指标进行对比分析。仿真试验结果表明，基于弧长参数的自由曲线实时误差估计算法较现有常用的轮廓误差估计算法具有更好的轮廓误差估算精度，并且适用于大曲率自由曲线的情况。利用该方法可以更进一步设计出先进的交叉耦合控制(cross-coupled control, CCC)运动控制方案。

关键词: 轮廓误差, 轮廓跟踪, NURBS, 自由曲线, 曲率, CCC

Abstract:

Aiming at the problems that the machining accuracy of complex surface parts was required to be improved and it was difficult to accurately calculated the contour error in real time in the free curve tracking task, a real-time error estimation algorithm of free curve based on arc length parameters was proposed. In MATLAB/Simulink, non-uniform rational B-spline (NURBS) interpolation was used to plan the elliptical trajectory and the double twist trajectory. The real-time error estimation algorithm of free curve based on arc length parameters was compared with the contour error estimation accuracy and execution time of tangent line-based estimation, circular approximation, average velocity estimation and three points circular approximation. The simulation results showed that the real-time error estimation algorithm of free curve based on arc length parameters had better contour error estimation accuracy than the existing commonly used contour error estimation algorithm, and it was suitable for the case of large curvature free curve. The advanced cross-coupled control (CCC) scheme could be further designed by using this method.

Key words: contour error, contour tracking, NURBS, free curve, curvature, CCC

中图分类号: 

  • TH161

图1

自由曲线实时轮廓误差估计算法"

图2

基于弧长参数的自由曲线实时误差估计算法原理图"

图3

圆弧逼近的轮廓误差估计算法"

图4

双轴轮廓误差控制"

图5

本研究提出的双轴轮廓误差控制方案"

图6

轮廓跟踪任务的指令轨迹"

表1

椭圆轨迹相关参数"

顶点序号 控制点坐标/mm 权因子
① (0, 0) 1.0
② (0, 20) 0.5
③ (60, 20) 0.5
④ (60, 0) 1.0
⑤ (60, -20) 0.5
⑥ (0, -20) 0.5
⑦ (0, 0) 1.0

表2

伯努利双纽线轨迹相关参数"

顶点序号 控制点坐标/mm 权因子
① (0, 0) 1
② (50, 10) 20
③ (50, -10) 20
④ (0, 0) 1
⑤ (-50, 10) 20
⑥ (-50, -10) 20
⑦ (0, 0) 1

图7

椭圆轨迹下不同算法的仿真结果"

图8

椭圆轨迹最大曲率处不同算法的仿真结果"

图9

伯努利双纽线轨迹下不同算法的仿真结果"

图10

伯努利双纽线轨迹最大曲率处及交叉处不同算法的仿真结果"

表3

在椭圆轨迹下不同算法轮廓误差估算指标"

轮廓误差估计算法 εMAX εIAE εRMS
实际轮廓误差 0.027 9 11.432 1 0.004 5
切线法 0.060 8 12.836 2 0.005 7
圆近似法 0.054 3 12.320 4 0.003 9
平均速度法 0.042 3 9.834 2 0.003 1
三点圆法 0.032 9 11.701 2 0.003 8
基于弧长参数的自由曲线实时误差估计算法 0.025 8 11.587 9 0.004 2

表4

在伯努利双纽线轨迹下不同算法轮廓误差估算指标"

轮廓误差估计算法 εMAX εIAE εRMS
实际轮廓误差 0.028 4 25.366 5 0.014 5
切线法 0.079 1 25.125 4 0.015 8
圆近似法 0.058 9 25.813 4 0.014 6
平均速度法 0.036 4 25.431 5 0.016 1
三点圆法 0.032 4 25.411 5 0.016 1
基于弧长参数的自由曲线实时误差估计算法 0.025 6 25.842 3 0.014 8

表5

不同轮廓误差估计算法下执行时间的离线比较"

轮廓误差估计算法 执行时间
椭圆轨迹 双纽线轨迹
切线法 0.883 6 1.865 3
圆近似法 0.764 1 1.670 4
平均速度法 0.176 8 0.376 5
三点圆法 0.145 3 0.354 2
基于弧长参数的自由曲线实时误差估计算法 0.162 1 0.398 7
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