Approximately Arc-Length Parametrized C3 Quintic Interpolatory Splines

1999 ◽  
Vol 121 (3) ◽  
pp. 430-439 ◽  
Author(s):  
F.-C. Wang ◽  
P. K. Wright ◽  
B. A. Barsky ◽  
D. C. H. Yang
Keyword(s):  
High Speed ◽  
Interpolation Method ◽  
Free Form ◽  
Precision Machining ◽  
Arc Length ◽  
Quintic Spline ◽  
Line Segments ◽  
Spline Curve ◽  
Straight Line ◽  
Data Points

A quasi-global interpolation method that fits a quintic spline curve to a set of designated data points is described in this paper. The resultant curve has several important features. First, the curve is smooth with C3 continuity and has no unwanted oscillations. Second, the generated quintic spline is “optimally” parametrized; that is, the curve is parametrized very closely to its arc length. In addition, with the interpolation method, straight line segments can be preserved to generate a quintic spline of hybrid curve segments. The properties of C3 continuity and the “near arc length” parametrization have direct applications to trajectory planning in robotics and the development of new types of machine tool controllers for high speed and precision machining. The encapsulation of straight line segments enhances the capability for the shape designers to design more complicated shapes, including free form curves and straight line segments in a uniform way.

Open Architecture Controllers for Machine Tools, Part 2: A Real Time Quintic Spline Interpolator

1998 ◽  
Vol 120 (2) ◽  
pp. 425-432 ◽  
Author(s):  
Fu-Chung Wang ◽  
P. K. Wright
Keyword(s):  
Real Time ◽  
Tool Path ◽  
Interpolation Method ◽  
Tool Path Generation ◽  
Open Architecture ◽  
Free Form ◽  
Path Generation ◽  
Arc Length ◽  
Quintic Spline ◽  
Data Points

In Part 2 of this two-part paper, we describe a new quintic spline interpolator for real time trajectory generation during free form curve machining on the Open Architecture machine tool platform described in Part I. The research provides new capabilities for advanced CAD/CAM systems. Complicated curves and shapes designed in a CAD system are usually represented by a set of discrete data points. Subsequently, for manufacturing by a CAM system, these data points need to be converted into tool paths for machining. Therefore, the paper presents a new interpolation method that interpolates the designated data points with a quintic spline curve for real-time tool path generation. The resultant curve, generated by the proposed interpolation method, is nearly arc-length parametrized and has C3 continuity. The near arc-length parametrization property makes the real time generation of the reference commands of the cutting tool path easier. The C3 continuity guarantees the smooth motion of continuous speed, acceleration and even jerk during the machining. The combined properties of this new interpolation method enable a new quintic spline interpolator to be developed for real time tool path generation.

Quintic Spline Interpolation With Minimal Feed Fluctuation

Manufacturing ◽  
2003 ◽  
Author(s):  
Kaan Erkorkmaz ◽  
Yusuf Altintas
Keyword(s):  
Interpolation Method ◽  
Spline Interpolation ◽  
Tracking Error ◽  
Arc Length ◽  
Tracking Accuracy ◽  
Structural Vibrations ◽  
Quintic Spline ◽  
The Relationship ◽  
Quintic Spline Interpolation ◽  
Spline Parameter

This paper presents a parameterization and an interpolation method for quintic splines, which result in a smooth and consistent feedrate profile. The discrepancy between the spline parameter and the actual arc length leads to undesirable feed fluctuations and discontinuity, which elicit themselves as high frequency acceleration and jerk harmonics, causing unwanted structural vibrations and excessive tracking error. Two different approaches are presented that alleviate this problem: The first approach is based on modifying the spline toolpath so that it is optimally parameterized with respect to its arc length. The second approach is based on scheduling the spline parameter to accurately yield the desired arc displacement (i.e. feedrate), either by approximation of the relationship between the arc length and the spline parameter with a feed correction polynomial, or by solving the spline parameter iteratively in real-time at each interpolation step. The two approaches are compared to nearly arc length parameterized C3 quintic spline interpolation in terms of feedrate consistency and experimental tracking accuracy.

Quintic Spline Interpolation With Minimal Feed Fluctuation

2005 ◽  
Vol 127 (2) ◽  
pp. 339-349 ◽  
Author(s):  
Kaan Erkorkmaz ◽  
Yusuf Altintas
Keyword(s):  
Real Time ◽  
Feed Rate ◽  
Tool Path ◽  
Interpolation Method ◽  
Spline Interpolation ◽  
Tracking Error ◽  
Arc Length ◽  
Quintic Spline ◽  
Rate Profile ◽  
Spline Parameter

This paper presents a parameterization and an interpolation method for quintic splines, which result in a smooth and consistent feed rate profile. The discrepancy between the spline parameter and the actual arc length leads to undesirable feed fluctuations and discontinuity, which elicit themselves as high frequency acceleration and jerk harmonics, causing unwanted structural vibrations and excessive tracking error. Two different approaches are presented that alleviate this problem. The first approach is based on modifying the spline tool path so that it is optimally parameterized with respect to its arc length, which allows it to be accurately interpolated in real-time with minimal complexity. The second approach is based on scheduling the spline parameter to accurately yield the desired arc displacement (hence feed rate), either by approximation of the relationship between the arc length and the spline parameter with a feed correction polynomial, or by solving the spline parameter iteratively in real-time at each interpolation step. This approach is particularly suited for predetermined spline tool paths, which are not arc-length parameterized and cannot be modified. The proposed methods have been compared to approximately arc-length C3 quintic spline parameterization (Wang, F.-C., Wright, P. K., Barsky, B. A., and Yang, D. C. H., 1999, “Approximately Arc-Length Parameterized C3 Quintic Interpolatory Splines,” ASME J. Mech. Des, 121, No. 3., pp. 430–439) and first- and second-order Taylor series interpolation techniques (Huang, J.-T., and Yang, D. C. H., 1992, “Precision Command Generation for Computer Controlled Machines,” Precision Machining: Technology and Machine Development and Improvement, ASME-PED 58, pp. 89–104; Lin, R.-S. 2000, “Real-Time Surface Interpolator for 3-D Parametric Surface Machining on 3-Axis Machine Tools,” Intl. J. Mach. Tools Manuf., 40, No.10, pp. 1513–1526) in terms of feed rate consistency, computational efficiency, and experimental contouring accuracy.

scholarly journals A Feedrate-Constraint Method for Continuous Small Line Segments in CNC Machining Based on Nominal Acceleration

2021 ◽  
Vol 11 (19) ◽  
pp. 8837
Author(s):  
Peng Guo ◽  
Ronghua Wang ◽  
Zhebin Shen ◽  
Haorong Zhang ◽  
Peng Zhang ◽  
...  
Keyword(s):  
Machine Tool ◽  
High Speed ◽  
Cnc Machining ◽  
Machining Process ◽  
Free Form ◽  
Cnc Machine ◽  
Line Segments ◽  
Machining Quality ◽  
Adjacent Line ◽  
Constraint Method

When the CNC machining of continuous small line segments is performed, the direction of the machine tool movement will change abruptly at the corner of adjacent line segments. Therefore, a reasonable constraint on the feedrate at the corner is the prerequisite for achieving high-speed and high-precision machining. To achieve this goal, a feedrate-constraint method based on the nominal acceleration was proposed. The proposed method obtains the predicted value of acceleration during the machining process by the machining trajectory prediction and acceleration filtering. Then, the feedrate at the corner is constrained, according to the predicted acceleration. Specifically, for any corner of adjacent line segments, the proposed method assumes that the CNC machining of a short path centered on the corner is carried out at a constant feedrate. First, the actual machining trajectory is predicted according to the transfer function of the servo system. Then, the nominal acceleration, when the CNC machining is carried out to the corner, is calculated and processed by a low-pass FIR filter. Last, the feedrate-constraint value at the corner is obtained according to the nominal acceleration and the preset normal acceleration. The advantage of the proposed method is that it can be used for different machining paths consisting of long segments or continuous small segments and it has no special requirement for the accuracy of the machining path. As a result, the feedrate-constraint value obtained is reasonable and the smooth machining process can be ensured. The simulation results in both 2D and 3D machining paths show that the proposed method is insensitive to the length of the line segment and the angle of the corner, and the calculated feedrate-constraint value is close to the theoretical value, which has good stability and versatility. In contrast, the feedrate-constraint values obtained by conventional methods change abruptly along the machining path, especially in the 3D simulation, which will damage the machining quality. The experiment was performed on a three-axis CNC machine tool controlled by a self-developed controller, and a free-form surface workpiece was machined by a conventional feedrate-constraint method and the proposed method, respectively. The experimental results showed that the proposed method can make the feedrate of the machining process higher and more stable. Then, machining defects such as overcutting and undercutting can be avoided and the machining quality can be improved. Therefore, the article proposes a new method to constrain the feedrate at the corner of continuous small line segments, which can improve the machining efficiency and quality of the CNC machining.

On the Modeling and Analysis of an Improved CNC Interpolation Algorithm

2009 ◽  
Vol 626-627 ◽  
pp. 459-464 ◽  
Author(s):  
Lei Luo ◽  
L. Wang ◽  
Jun Hu
Keyword(s):  
Interpolation Method ◽  
Control Points ◽  
Interpolation Algorithm ◽  
Modeling And Analysis ◽  
B Spline ◽  
Spline Curve ◽  
Spline Surface ◽  
Cnc Interpolation ◽  
Data Points ◽  
Back Calculation

An improved interpolation method is presented based on B-spline curve back calculation which regards data points as control points. First, a B-spline surface reconstruction is done, and a favorable condition for real-time interpolation can be provided for NC machining. Then, by prejudging the trajectory feedrate, the tangent vectors of spline curve junction can be calculated, which can be used to establish the spline curve equations based on time. At last, with the equations mentioned above, the trajectory and feedrate profile can be generated simultaneously by the improved interpolation algorithm. An error analysis is also discussed and the feasibility of the improved algorithm is verified by the simulation results.

scholarly journals Generation Method of Cutting Tool Paths for High-Speed and High-Quality Machining of Free-Form Surfaces

2021 ◽  
Vol 15 (4) ◽  
pp. 521-528
Author(s):  
Yuki Takanashi ◽  
Hideki Aoyama ◽  
Song Cheol Won ◽  
◽  
Keyword(s):  
High Speed ◽  
Processing Time ◽  
Free Form ◽  
Machining Accuracy ◽  
Cnc Machine ◽  
Block Processing ◽  
Tool Paths ◽  
Nc Program ◽  
Straight Line ◽  
Free Form Surfaces

In general, NC programs for machining free-form surfaces using a computer numerical control (CNC) machine tool are generated using a computer-aided manufacturing (CAM) system. The tool paths (CL data) generated by a CAM system are approximated straight-line segments based on tolerance (allowable error). As a result, the tolerance affects the machining accuracy and time. If the tolerance is set to a small value, the lengths of the segments are shortened, and the machining accuracy is improved. The process in which a CNC machine tool reads and analyzes an NC program and controls the motors requires a minimum processing time of an NC program block (block-processing time). Therefore, if the lengths of the approximated straight-line segments are too small, it will be impossible to reach the indicated feed speed, and the machining time will be longer. In this study, by identifying the block-processing time of a CNC controller and deriving the appropriate length of the approximated straight-line segment based on the block-processing time, a CL data creation method that is capable of high-speed and high-accuracy free-form surface machining is proposed. In addition, experimental verification tests of the method are conducted.

A Novel Adaptive GA-based B-spline Curve Interpolation Method

2019 ◽  
Vol 13 (3) ◽  
pp. 289-304
Author(s):  
Maozhen Shao ◽  
Liangchen Hu ◽  
Huahao Shou ◽  
Jie Shen
Keyword(s):  
Genetic Algorithm ◽  
Traditional Method ◽  
Interpolation Method ◽  
Tangent Vector ◽  
Adaptive Genetic Algorithm ◽  
B Spline ◽  
Spline Curve ◽  
Curve Interpolation ◽  
Data Points ◽  
Data Point

Background: Curve interpolation is very important in engineering such as computer aided design, image analysis and NC machining. Many patents on curve interpolation have been invented. Objective: Since different knot vector configuration and data point parameterization can generate different shapes of an interpolated B-spline curve, the goal of this paper is to propose a novel adaptive genetic algorithm (GA) based interpolation method of B-spline curve. Method: Relying on geometric features owned by the data points and the idea of genetic algorithm which liberalizes the knots of B-spline curve and the data point parameters, a new interpolation method of B-spline curve is proposed. In addition, the constraint of a tangent vector is also added to ensure that the obtained B-spline curve can approximately satisfy the tangential constraint while ensuring strict interpolation. Results: Compared with the traditional method, this method realizes the adaptive knot vector selection and data point parameterization. Therefore, the interpolation result was better than the traditional method to some extent, and the obtained curve was more natural. Conclusion: The proposed method is effective for the curve reconstruction of any scanned data point set under tangent constraints. Meanwhile, this paper put forward a kind of tangent calculation method of discrete data points, where users can also set the tangent of each data point in order to get more perfect interpolation results.

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