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

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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
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