Linear motion commands of multi-axis computer numerical control (CNC) machine tools need to be smoothed at the transition corners, because the velocity discontinuities at corners can result in fluctuations on machine tool motions and lead to poor surface quality. However, no research has been reported on local corner smoothing algorithm for four-axis CNC machine tools with two rotary axes by considering their special kinematic characteristics. To this end, this paper proposes an analytical C3 continuous local corner smoothing algorithm for four-axis CNC machines with two rotary axes. After coordinates transformation, the tool tip positions and tool orientations are smoothed by locally inserting specially designed three-dimensional (3D) quintic B-splines and one-dimensional (1D) quintic B-splines into the corners between linear motion segments, respectively. The smoothing algorithm guarantees C3 continuity of the tool tip position and C3 continuous synchronization of the tool orientation related to the tool tip position, through analytically evaluating control points of the inserted microsplines. The maximum error tolerances of the tool tip position and tool orientation are mathematically constrained. Experiments on an in-house developed four-axis machine verify the efficacy of the proposed algorithm, where maximal errors caused by the local corner smoothing algorithm are constrained, the synchronization of the tool orientation and the tool tip position are achieved, and the proposed C3 continuous corner smoothing algorithm has lower jerk and jounce but higher tracking and contour accuracy than C2 continuous algorithm.

References

1.
Erkorkmaz
,
K.
, and
Altintas
,
Y.
,
2005
, “
Quintic Spline Interpolation With Minimal Feed Fluctuation
,”
ASME J. Manuf. Sci. Eng.
,
127
(
2
), pp.
339
349
.
2.
Erkorkmaz
,
K.
,
2015
, “
Efficient Fitting of the Feed Correction Polynomial for Real-Time Spline Interpolation
,”
ASME J. Manuf. Sci. Eng.
,
137
(
4
), p.
044501
.
3.
Chen
,
Z. C.
, and
Khan
,
M. A.
,
2012
, “
Piecewise B-Spline Tool Paths With the Arc-Length Parameter and Their Application on High Feed, Accurate CNC Milling of Free-Form Profiles
,”
ASME J. Manuf. Sci. Eng.
,
134
(
3
), p.
031007
.
4.
Cheng
,
C.
, and
Tsai
,
M.
,
2004
, “
Real-Time Variable Feed Rate NURBS Curve Interpolator for CNC Machining
,”
Int. J. Adv. Manuf. Technol.
,
23
(
11–12
), pp.
865
873
.
5.
Lei
,
W. T.
,
Sung
,
M. P.
,
Lin
,
L. Y.
, and
Huang
,
J. J.
,
2007
, “
Fast Real-Time NURBS Path Interpolation for CNC Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
47
(
10
), pp.
1530
1541
.
6.
Duan
,
M.
, and
Okwudire
,
C.
,
2016
, “
Minimum-Time Cornering for CNC Machines Using an Optimal Control Method With NURBS Parameterization
,”
Int. J. Adv. Manuf. Technol.
,
85
(
5–8
), pp.
1405
1418
.
7.
Liu
,
M.
,
Huang
,
Y.
,
Yin
,
L.
,
Guo
,
J.
,
Shao
,
X.
, and
Zhang
,
G.
,
2014
, “
Development and Implementation of a NURBS Interpolator With Smooth Feedrate Scheduling for CNC Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
87
, pp.
1
15
.
8.
Chen
,
M.
,
Zhao
,
W.
, and
Xi
,
X.
,
2015
, “
Augmented Taylor's Expansion Method for B-Spline Curve Interpolation for CNC Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
94
, pp.
109
119
.
9.
Tsai
,
M.
, and
Cheng
,
C.
,
2003
, “
A Real-Time Predictor-Corrector Interpolator for CNC Machining
,”
ASME J. Manuf. Sci. Eng.
,
125
(
3
), pp.
449
460
.
10.
Langeron
,
J. M.
,
Duc
,
E.
,
Lartigue
,
C.
, and
Bourdet
,
P.
,
2004
, “
A New Format for 5-Axis Tool Path Computation, Using Bspline Curves
,”
Comput.-Aided Des.
,
36
(
12
), pp.
1219
1229
.
11.
Lu
,
Y.
,
Ding
,
Y.
, and
Zhu
,
L.
,
2016
, “
Smooth Tool Path Optimization for Flank Milling Based on the Gradient-Based Differential Evolution Method
,”
ASME J. Manuf. Sci. Eng.
,
138
(
8
), p.
081009
.
12.
Fleisig
,
R. V.
, and
Spence
,
A. D.
,
2001
, “
A Constant Feed and Reduced Angular Acceleration Interpolation Algorithm for Multi-Axis Machining
,”
Comput.-Aided Des.
,
33
(
1
), pp.
1
15
.
13.
Yuen
,
A.
,
Zhang
,
K.
, and
Altintas
,
Y.
,
2013
, “
Smooth Trajectory Generation for Five-Axis Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
71
, pp.
11
19
.
14.
Yang
,
J.
, and
Altintas
,
Y.
,
2013
, “
Generalized Kinematics of Five-Axis Serial Machines With Non-Singular Tool Path Generation
,”
Int. J. Mach. Tools Manuf.
,
75
, pp.
119
132
.
15.
Yang
,
J.
,
Chen
,
Y.
,
Chen
,
Y.
, and
Zhang
,
D.
,
2015
, “
A Tool Path Generation and Contour Error Estimation Method for Four-Axis Serial Machines
,”
Mechatronics
,
31
, pp.
78
88
.
16.
Pateloup
,
V.
,
Duc
,
E.
, and
Ray
,
P.
,
2010
, “
Bspline Approximation of Circle Arc and Straight Line for Pocket Machining
,”
Comput.-Aided Des.
,
42
(
9
), pp.
817
827
.
17.
Zhao
,
H.
,
Zhu
,
L.
, and
Ding
,
H.
,
2013
, “
A Real-Time Look-Ahead Interpolation Methodology With Curvature-Continuous B-Spline Transition Scheme for CNC Machining of Short Line Segments
,”
Int. J. Mach. Tools Manuf.
,
65
, pp.
88
98
.
18.
Fan
,
W.
,
Lee
,
C.
, and
Chen
,
J.
,
2015
, “
A Realtime Curvature-Smooth Interpolation Scheme and Motion Planning for CNC Machining of Short Line Segments
,”
Int. J. Mach. Tools Manuf.
,
96
, pp.
27
46
.
19.
Sencer
,
B.
,
Ishizaki
,
K.
, and
Shamoto
,
E.
,
2015
, “
A Curvature Optimal Sharp Corner Smoothing Algorithm for High-Speed Feed Motion Generation of NC Systems Along Linear Tool Paths
,”
Int. J. Adv. Manuf. Technol.
,
76
(
9–12
), pp.
1977
1992
.
20.
Farouki
,
R. T.
,
2014
, “
Construction of Rounded Corners With Pythagorean-Hodograph Curves
,”
Comput. Aided Geometric Des.
,
31
(
2
), pp.
127
139
.
21.
Beudaert
,
X.
,
Lavernhe
,
S.
, and
Tournier
,
C.
,
2013
, “
5-Axis Local Corner Rounding of Linear Tool Path Discontinuities
,”
Int. J. Mach. Tools Manuf.
,
73
, pp.
9
16
.
22.
Tulsyan
,
S.
, and
Altintas
,
Y.
,
2015
, “
Local Toolpath Smoothing for Five-Axis Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
96
, pp.
15
26
.
23.
Bi
,
Q.
,
Shi
,
J.
,
Wang
,
Y.
,
Zhu
,
L.
, and
Ding
,
H.
,
2015
, “
Analytical Curvature-Continuous Dual-Bézier Corner Transition for Five-Axis Linear Tool Path
,”
Int. J. Mach. Tools Manuf.
,
91
, pp.
96
108
.
24.
Shi
,
J.
,
Bi
,
Q. Z.
,
Zhu
,
L. M.
, and
Wang
,
Y. H.
,
2015
, “
Corner Rounding of Linear Five-Axis Tool Path by Dual PH Curves Blending
,”
Int. J. Mach. Tools Manuf.
,
88
, pp.
223
236
.
25.
Yang
,
J.
, and
Yuen
,
A.
,
2017
, “
An Analytical Local Corner Smoothing Algorithm for Five-Axis CNC Machining
,”
Int. J. Mach. Tools Manuf.
,
123
, pp.
22
35
.
26.
Piegl
,
L.
, and
Tiller
,
W.
,
1997
,
The NURBS Book
, 2nd ed.,
Springer-Verlag
,
New York
, p.
646
.
27.
Heidenhain,
2013
, “Technical Manual iTNC 530 HSCI,” Heidenhain Corporation, Traunreut, Germany.
28.
Erkorkmaz
,
K.
,
2004
, “Optimal Trajectory Generation and Precision Tracking Control for Multi-Axis Machines,”
Ph.D. thesis
, The University of British Columbia, Vancouver, BC, Canada.
29.
Beudaert
,
X.
,
Pechard
,
P. Y.
, and
Tournier
,
C.
,
2011
, “
5-Axis Tool Path Smoothing Based on Drive Constraints
,”
Int. J. Mach. Tools Manuf.
,
51
(
12
), pp.
958
965
.
30.
Altintas
,
Y.
,
2012
,
Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design
, 2nd ed.,
Cambridge University Press
, New York.
31.
Yang
,
J.
, and
Altintas
,
Y.
,
2015
, “
A Generalized on-Line Estimation and Control of Five-Axis Contouring Errors of CNC Machine Tools
,”
Int. J. Mach. Tools Manuf.
,
88
, pp.
9
23
.
32.
Erkorkmaz
,
K.
, and
Altintas
,
Y.
,
2001
, “
High Speed CNC System Design—Part I: Jerk Limited Trajectory Generation and Quintic Spline Interpolation
,”
Int. J. Mach. Tools Manuf.
,
41
(
9
), pp.
1323
1345
.
33.
Huo
,
F.
, and
Poo
,
A. N.
,
2013
, “
Precision Contouring Control of Machine Tools
,”
Int. J. Adv. Manuf. Technol.
,
64
(
1–4
), pp.
319
333
.
34.
Tang
,
L.
, and
Landers
,
R. G.
,
2013
, “
Multiaxis Contour Control—the State of the Art
,”
IEEE Trans. Control Syst. Technol.
,
21
(
6
), pp.
1997
2010
.
35.
Davis
,
T. A.
,
Shin
,
Y. C.
, and
Yao
,
B.
,
2015
, “
Adaptive Robust Control of Circular Machining Contour Error Using Global Task Coordinate Frame
,”
ASME J. Manuf. Sci. Eng.
,
137
(
1
), p.
014501
.
36.
Xi
,
X.
,
Chen
,
H.
, and
Zhao
,
W.
,
2017
, “
Simplification of a G-Code Feeding Path in Roughing Multi-Axis Electrical Discharging Machining for Shrouded Blisks With a Contour Error Constraint
,”
ASME J. Manuf. Sci. Eng.
,
139
(
11
), p.
111013
.
37.
El Khalick
,
M. A.
, and
Uchiyama
,
N.
,
2013
, “
Estimation of Tool Orientation Contour Errors for Five-Axis Machining
,”
Rob. Comput.-Integr. Manuf.
,
29
(
5
), pp.
271
277
.
38.
Yang
,
J.
,
Ding
,
H.
,
Zhao
,
H.
, and
Yan
,
S.
,
2016
, “
A Generalized Online Estimation Algorithm of Multi-Axis Contouring Errors for CNC Machine Tools With Rotary Axes
,”
Int. J. Adv. Manuf. Technol.
,
84
(
5–8
), pp.
1239
1251
.
39.
Huo
,
F.
, and
Poo
,
A. N.
,
2012
, “
Improving Contouring Accuracy by Using Generalized Cross-Coupled Control
,”
Int. J. Mach. Tools Manuf.
,
63
, pp.
49
57
.
40.
Zhang
,
D.
,
Yang
,
J.
,
Chen
,
Y.
, and
Chen
,
Y.
,
2015
, “
A Two-Layered Cross Coupling Control Scheme for a Three-Dimensional Motion Control System
,”
Int. J. Mach. Tools Manuf.
,
98
, pp.
12
20
.
41.
Tang
,
L.
, and
Landers
,
R. G.
,
2012
, “
Predictive Contour Control With Adaptive Feed Rate
,”
IEEE/ASME Trans. Mechatronics
,
17
(
4
), pp.
669
679
.
42.
Yang
,
J.
,
Zhang
,
H.
, and
Ding
,
H.
,
2017
, “
Contouring Error Control of the Tool Center Point Function for Five-Axis Machine Tools Based on Model Predictive Control
,”
Int. J. Adv. Manuf. Technol.
,
88
(
9–12
), pp.
2909
2919
.
43.
Yang
,
S.
,
Ghasemi
,
A. H.
,
Lu
,
X.
, and
Okwudire
,
C. E.
,
2015
, “
Pre-Compensation of Servo Contour Errors Using a Model Predictive Control Framework
,”
Int. J. Mach. Tools Manuf.
,
98
, pp.
50
60
.
44.
Xi
,
X. C.
,
Zhao
,
W. S.
, and
Poo
,
A. N.
,
2015
, “
Improving CNC Contouring Accuracy by Robust Digital Integral Sliding Mode Control
,”
Int. J. Mach. Tools Manuf.
,
88
, pp.
51
61
.
45.
Sencer
,
B.
,
Altintas
,
Y.
, and
Croft
,
E.
,
2009
, “
Modeling and Control of Contouring Errors for Five-Axis Machine Tools—Part I: Modeling
,”
ASME J. Manuf. Sci. Eng.
,
131
(
3
), p. 031006.
46.
Sencer
,
B.
, and
Altintas
,
Y.
,
2009
, “
Modeling and Control of Contouring Errors for Five-Axis Machine Tools—Part II: Precision Contour Controller Design
,”
ASME J. Manuf. Sci. Eng.
,
131
(
3
), p. 031007.
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