Certified space curve fitting and trajectory planning for CNC machining with cubic B-splines

Lin, Fengming; Shen, Li-Yong; Yuan, Chun-Ming; Mi, Zhenpeng

doi:10.1016/j.cad.2018.08.001

Cited by 29 publications

(13 citation statements)

References 30 publications

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“…To solve the problem that control points are redundant or inadequate, for the 3D case, Ref. [19] extended the planar case [20], and proposed an adaptive addition and removal process to refine the control points for the B-spline curve. Some articles also discuss the parameterization problems of spatial data points for other applications, while Refs.…”

Section: Previous Workmentioning

confidence: 99%

Computing knots by quadratic and cubic polynomial curves

Zhang

Liu

et al. 2020

Comp. Visual Media

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A new method is presented to determine parameter values (knot) for data points for curve and surface generation. With four adjacent data points, a quadratic polynomial curve can be determined uniquely if the four points form a convex polygon. When the four data points do not form a convex polygon, a cubic polynomial curve with one degree of freedom is used to interpolate the four points, so that the interpolant has better shape, approximating the polygon formed by the four data points. The degree of freedom is determined by minimizing the cubic coefficient of the cubic polynomial curve. The advantages of the new method are, firstly, the knots computed have quadratic polynomial precision, i.e., if the data points are sampled from a quadratic polynomial curve, and the knots are used to construct a quadratic polynomial, it reproduces the original quadratic curve. Secondly, the new method is affine invariant, which is significant, as most parameterization methods do not have this property. Thirdly, it computes knots using a local method. Experiments show that curves constructed using knots computed by the new method have better interpolation precision than for existing methods.

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Section: Previous Workmentioning

confidence: 99%

Computing knots by quadratic and cubic polynomial curves

Zhang

Liu

et al. 2020

Comp. Visual Media

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“…As mentioned above, the number of control points defines the complexity of the fitting curve: typically, a smaller number of control points results in a simpler curve, but also in greater fitting errors. Nevertheless, an elevated number of control points does not assure accurate fitting, as some control points may be redundant or inadequate [10]. In addition, in an approximation problem, the fitting tolerance is defined as the maximum permitted error between the given data points and the resultant approximation curve.…”

Section: Analysis Of the Curve Fitting Errorsmentioning

confidence: 99%

“…Nevertheless, control points may be redundant or inadequate. Redundant control points unnecessarily increase the computational complexity without decreasing the fitting errors, whereas inadequate control points increase the fitting errors and, thus, the curve fails to satisfy the precision requirements [10]. Thus, the challenge of curve fitting, either by interpolation or approximation, lies in obtaining the most accurate trajectory, without compromising the computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

“…Other works [11] have focused on optimizing trajectory generation for a smooth interpolating motion, in order to respect the machine kinematic feed-rate and acceleration limits, as well as to avoid fluctuations due to discontinuities in the first derivatives along the tool path. Spline-type interpolation has been applied in high-speed machining to limit the speed discontinuities while respecting the contour tolerance [10,12]. The target of these works is to achieve a confined error and minimal machining time; however, this task becomes harder for more accurate tolerances.…”

Section: Introductionmentioning

confidence: 99%

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Trajectory Definition with High Relative Accuracy (HRA) by Parametric Representation of Curves in Nano-Positioning Systems

et al. 2019

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Nanotechnology applications demand high accuracy positioning systems. Therefore, in order to achieve sub-micrometer accuracy, positioning uncertainty contributions must be minimized by implementing precision positioning control strategies. The positioning control system accuracy must be analyzed and optimized, especially when the system is required to follow a predefined trajectory. In this line of research, this work studies the contribution of the trajectory definition errors to the final positioning uncertainty of a large-range 2D nanopositioning stage. The curve trajectory is defined by curve fitting using two methods: traditional CAD/CAM systems and novel algorithms for accurate curve fitting. This novel method has an interest in computer-aided geometric design and approximation theory, and allows high relative accuracy (HRA) in the computation of the representations of parametric curves while minimizing the numerical errors. It is verified that the HRA method offers better positioning accuracy than commonly used CAD/CAM methods when defining a trajectory by curve fitting: When fitting a curve by interpolation with the HRA method, fewer data points are required to achieve the precision requirements. Similarly, when fitting a curve by a least-squares approximation, for the same set of given data points, the HRA method is capable of obtaining an accurate approximation curve with fewer control points.

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“…Then the time optimal transition curve is obtained through indirect method based on Pontryagin's minimum principle. Later, another fitting method for linear paths based on optimal control is proposed, which can obtain time optimal fitted curve and feedrate profile simultaneously [15].…”

Section: Introductionmentioning

confidence: 99%

Smooth Trajectory Generation for Predefined Path With Pseudo Spectral Method

2020

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In view of smooth trajectory generation for a 3-axis machine tool, many methods have been presented. Among them, the optimal control based method is increasingly concerned, because it is considered to be able to make full use of kinematic abilities of machine tools. Under the unified framework of optimal control, the feedrate can be adjusted flexibly by adding or removing axial constraints and tangential constraints. But the problem of smooth trajectory generation (PSTG) based on optimal control for a machine tool is not easily to be solved. In this paper, to efficiently solve the PSTG, it is divided into two sub-problems: the problem of minimum time trajectory planning (PMTTP) and the pseudo problem of smooth trajectory generation (PPSTG). Since both sub-problems are convex, the existence of unique solutions can be guaranteed. Then, the PMTTP and the PPSTG are transformed into nonlinear programming (NLP) problems with radau-pseudo-spectral (RPM) method successively. Due to convexity, the two NLP problems can be efficiently solved with mature optimization methods. In addition, the RPM method allows two sub-problems to have different Legendre-Gauss-Radau (LGR) points, thereby further saving computational costs. Finally, three different predefined paths are employed to test the proposed method, and simulation results show the effectiveness of proposed method.

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Certified space curve fitting and trajectory planning for CNC machining with cubic B-splines

Cited by 29 publications

References 30 publications

Computing knots by quadratic and cubic polynomial curves

Computing knots by quadratic and cubic polynomial curves

Trajectory Definition with High Relative Accuracy (HRA) by Parametric Representation of Curves in Nano-Positioning Systems

Smooth Trajectory Generation for Predefined Path With Pseudo Spectral Method

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