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.

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.

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.

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