Journal of Shandong University(Engineering Science) ›› 2023, Vol. 53 ›› Issue (6): 143-151.doi: 10.6040/j.issn.1672-3961.0.2023.009

• Mechanical, Energy and Power Engineering • Previous Articles     Next Articles

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

RichHTML

6

PDF (PC)

555

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

CLC Number: 

  • TH161

Fig.1

Free-form curve real-time contour error algorithm"

Fig.2

Principle diagram of real-time error estimation algorithm forfree curve based on arc length parameters"

Fig.3

Arc approximation contour error estimation algorithm"

Fig.4

Two-axis contour error control"

Fig.5

Two-axis contour error control scheme proposed in this paper"

Fig.6

The command trajectory of contour tracking task"

Table 1

Ellipse trajectory related parameters"

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

Table 2

Bernouli lemniscate trajectory related parameters"

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

Fig.7

Simulation results of different algorithms under ellipse trajectory"

Fig.8

Simulation results of different algorithms at the maximum curvature of ellipse trajectory"

Fig.9

Simulation results of different algorithms under Bernoulli lemniscatetrajectory"

Fig.10

Simulation results of different algorithms at the maxi-mum curvature and intersection of Bernoulli lemniscate trajectory"

Table 3

The indices of contour error estimation method under ellipse trajectory  单位: mm"

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

Table 4

The indices of contour error estimation method under Bernouli lemniscate trajectory  单位: mm"

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

Table 5

Offline comparison of execution time under different contour error algorithms  单位: s"

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
1 KOREN Y . Cross-coupled biaxial computer control for manufacturing systems[J]. Journal of Dynamic Systems, Measurement, and Control, 1980, 102 (4): 265- 272.
doi: 10.1115/1.3149612
2 KOREN Y , LO C C . Variable-gain cross-coupling controller for contouring[J]. CIRP Annals, 1991, 40 (1): 371- 374.
doi: 10.1016/S0007-8506(07)62009-5
3 CHUANG H , LIU C . Cross-coupled adaptive feed rate control for multi axis machine tools[J]. Journal of Dynamic Systems, Measurement, and Control, 1991, 113 (3): 451- 457.
doi: 10.1115/1.2896431
4 TARNG Y S , CHUANG H Y , HSU W T . An optimization approach to the contour error control of CNC machine tools using genetic algorithms[J]. The International Journal of Advanced Manufacturing Technology, 1997, 13 (5): 359- 366.
doi: 10.1007/BF01178256
5 YEH S S, HSU P L. A new approach to bi-axial cross-coupled control[C]//Proceedings of the 2000 IEEE International Conference on Control Applications. Anchorage, AK, USA: IEEE, 2000: 168-173.
6 YEH S S , HSU P L . Adaptive-feedrate interpolation for parametric curves with a confined chord error[J]. Computer-Aided Design, 2002, 34 (3): 229- 237.
doi: 10.1016/S0010-4485(01)00082-3
7 CHEN S L , LIU H L , TING S C . Contouring control of biaxial systems based on polar coordinates[J]. IEEE/ASME Transactions on Mechatronics, 2002, 7 (3): 329- 345.
doi: 10.1109/TMECH.2002.802723
8 ERKORKMAZ K , YEUNG C H , ALTINTAS Y . Virtual CNC system: Part Ⅱ: High speed contouring application[J]. International Journal of Machine Tools and Manufacture, 2006, 46 (10): 1124- 1138.
doi: 10.1016/j.ijmachtools.2005.08.001
9 CHENG M Y , LEE C C . Motion controller design for contour-following tasks based on real-time contour error estimation[J]. IEEE Transactions on Industrial Electronics, 2007, 54 (3): 1686- 1695.
doi: 10.1109/TIE.2007.894691
10 CHEN S L , WU K C . Contouring control of smooth paths for multiaxis motion systems based on equivalent errors[J]. IEEE Transactions on Control Systems Technology, 2007, 15 (6): 1151- 1158.
doi: 10.1109/TCST.2007.899719
11 YANG J , LI Z . A novel contour error estimation for position loop-based cross-coupled control[J]. IEEE/ASME Transactions on Mechatronics, 2010, 16 (4): 643- 655.
12 ZHU L M , ZHAO H , DING H . Real-time contouring error estimation for multi-axis motion systems using the second-order approximation[J]. International Journal of Machine Tools and Manufacture, 2013, 68, 75- 80.
doi: 10.1016/j.ijmachtools.2013.01.008
13 ZHAO G , AN H , ZHAO Q . Contour error coupled-control strategy based on line interpolation and curve interpolation[J]. Journal of Computers, 2013, 8 (6): 1512- 1519.
14 徐赫楠. 数控机床进给伺服系统轮廓控制方法研究[D]. 天津: 天津大学, 2017.
XU Henan. Research on contour control for feeding servo system of CNC machine tools[D]. Tianjin: Tianjin University, 2017.
15 LI B , WANG T Y , WANG P . Cross-coupled control based on real-time double circle contour error estimation for biaxial motion system[J]. Measurement and Control, 2021, 54 (3/4): 324- 335.
16 LYU D , LIU Q , LIU H , et al. Dynamic error of CNC machine tools: a state-of-the-art review[J]. The International Journal of Advanced Manufacturing Technology, 2020, 106, 1869- 1891.
17 CHEN H R , CHENG M Y , WU C H , et al. Real time parameter based contour error estimation algorithms for free form contour following[J]. International Journal of Machine Tools and Manufacture, 2016, 102, 1- 8.
doi: 10.1016/j.ijmachtools.2015.11.009
18 PIEGL L . On NURBS: a survey[J]. IEEE Computer Graphics and Applications, 1991, 11 (1): 55- 71.
doi: 10.1109/38.67702
19 路勇良, 赵军, 李莉莉, 等. 一种进给伺服系统非线性PID交叉耦合控制[J/OL]. 机床与液压. (2022-09-22)[2023-01-03]. https://kns.cnki.net/kcms/detail/44.1259.th.20220919.1854.004.html.
20 WANG T Y , ZHANG Y B , DONG J C , et al. NURBS interpolator with adaptive smooth feed rate scheduling and minimal feed rate fluctuation[J]. International Journal of Precision Engineering and Manufacturing, 2020, 21 (2): 273- 290.
doi: 10.1007/s12541-019-00288-6
21 吴彩成. 面向双轴数控加工系统的交叉耦合轮廓误差补偿综合控制策略研究[D]. 广州: 华南理工大学, 2021.
WU Caicheng. Research on comprehensive control strategy of cross-coupling contour error compensation for two-axis CNC machining system[D]. Guangzhou: South China University of Technology, 2021.
[1] XU Mingsan, WANG Tao, LI Jianfeng, DAI Tengyun. Performance of W6Mo5Cr4V2 laser cladding on 40Cr curved substrate [J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2018, 48(1): 57-64.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] WANG Su-yu,<\sup>,AI Xing<\sup>,ZHAO Jun<\sup>,LI Zuo-li<\sup>,LIU Zeng-wen<\sup> . Milling force prediction model for highspeed end milling 3Cr2Mo steel[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2006, 36(1): 1 -5 .
[2] ZHANG Yong-hua,WANG An-ling,LIU Fu-ping . The reflected phase angle of low frequent inhomogeneous[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2006, 36(2): 22 -25 .
[3] LI Kan . Empolder and implement of the embedded weld control system[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2008, 38(4): 37 -41 .
[4] SHI Lai-shun,WAN Zhong-yi . Synthesis and performance evaluation of a novel betaine-type asphalt emulsifier[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2008, 38(4): 112 -115 .
[5] KONG Xiang-zhen,LIU Yan-jun,WANG Yong,ZHAO Xiu-hua . Compensation and simulation for the deadband of the pneumatic proportional valve[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2006, 36(1): 99 -102 .
[6] LAI Xiang . The global domain of attraction for a kind of MKdV equations[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2006, 36(1): 87 -92 .
[7] YU Jia yuan1, TIAN Jin ting1, ZHU Qiang zhong2. Computational intelligence and its application in psychology[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2009, 39(1): 1 -5 .
[8] LI Liang, LUO Qiming, CHEN Enhong. Graph-based ranking model for object-level search
[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2009, 39(1): 15 -21 .
[9] CHEN Rui, LI Hongwei, TIAN Jing. The relationship between the number of magnetic poles and the bearing capacity of radial magnetic bearing[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2018, 48(2): 81 -85 .
[10] WANG Bo,WANG Ning-sheng . Automatic generation and combinatory optimization of disassembly sequence for mechanical-electric assembly[J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2006, 36(2): 52 -57 .
Copyright 2018 © Editorial office of Journal of Shandong University (Engineering Science)
Supported by Beijing Magtech Co.Ltd