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A universal velocity limit curve generator considering abnormal tool path geometry for CNC machine tools
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Technical Paper
A universal velocity limit curve generator considering abnormal tool path geometry for CNC machine tools夽 Mo Chen ∗ , Xue-Cheng Xi, Wan-Sheng Zhao, Hao Chen, Hong-Da Liu State Key Laboratory of Mechanical System Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 17 November 2016 Received in revised form 13 February 2017 Accepted 3 March 2017 Available online xxx Keywords: Velocity limit curves Parametric curves CNC machine tools
a b s t r a c t Non-linear parametric curves, such as B-spline curves, are becoming increasingly available in modern CNC (Computer Numerical Control) systems. The smoothness of parametric curves offers higher order of continuity and thus invokes less vibration in machines as compared to short line segments. Nevertheless, the computations for velocity limit curves, velocity profiles and interpolation points are quite complicated and time-consuming and therefore approximation methods are applied. Unnecessary accelerations and decelerations, which cost additional time of motion, can be caused by inaccurate computations of the velocity limits around tiny corners. To overcome the problem, the unit arc length increment scanning method (UALISM) is proposed to reduce the time of movement and to improve the efficiency. The scanning interval is fixed at 1 BLU (basic length unit) which is irrelevant to the type, size and shape of the tool path curve. The constraints of chord height errors and axis accelerations are considered and the velocity limit for the specified scanning point is computed using the coordinates of multiple scanning points near the specified scanning point. Simulation results show that the unnecessary accelerations and decelerations can be avoided and thus the total motion time can be reduced by UALISM. © 2017 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction In CNC (Computer Numerical Control) machine tools, the velocity of the feed motion is desired to be constant along a given tool path during machining in order to obtain high surface quality [1]. Around sharp corners in tool paths, however, high commanded velocity can cause a sharp turn with excessive axis accelerations that exceed axis driving capabilities. This will not only lower the surface quality and accuracy but also lead to undesirable vibrations and resonances [2]. As setting up speed limit signs on roads, tool path data shall be collected in order to find a velocity limit curve for the feedrate planning for the tool path before the motion starts. The concept of velocity limit curve sources from the timeoptimal trajectory planning for robots [3]. The velocity limit curves are generally obtained through the following steps: 1) a group of points on the given tool path are sampled by scanning, 2) the velocity limit for each scanning point is computed considering constraints such as acceleration, jerk and chord height error, and 3) the velocity limits are connected or interpolated in the sequence of
夽 45th SME North American Manufacturing Research Conference, NAMRC 45, LA, USA. ∗ Corresponding author. E-mail address: [email protected] (M. Chen).
scanning points to form the velocity limit curve for the tool path. For CNC machine tools, the chord height error, which is part of the contour error, and the axis acceleration, which is related to the driving capability of a motor, are the two major issues for velocity limits. The chord height error, which results from approximating a curve segment with line segments, is defined to be the maximum distance between the curve segment and its corresponding line segment [4]. For curves except lines and circular arcs, it is difficult to give a formula to compute the chord height errors directly, and thus approximate methods are used instead [5]. A typical way is approximating a curve segment with a circular arc and compute the chord height error if the movement were to be done along the circular arc [4,6–9]. Liu et al. [10] and Emami and Arezoo [11] pointed out that the circular arc approximation method gives better computational accuracy of chord height error than using the length of the line segment which connects the midpoints of the curve segment and its chord. The circular arc approximation method works well when the curve direction does not change abruptly since the approximation error can be kept low in a large length of segment. For tool path curves with abnormal geometries such as sharp corners and tiny corners, an approximate method with better accuracy shall be developed. When the machine axes move near a sharp corner in a tool path, the magnitude and direction change abruptly, leading to a large
http://dx.doi.org/10.1016/j.jmsy.2017.04.010 0278-6125/© 2017 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Chen M, et al. A universal velocity limit curve generator considering abnormal tool path geometry for CNC machine tools. J Manuf Syst (2017), http://dx.doi.org/10.1016/j.jmsy.2017.04.010
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y
v Velocity limit curve
s + direction Tiny corner Tool path
Velocity limit at the tiny corner
x
O
s
O
Fig. 1. The excessively low velocity limit in a tiny corner.
commanded acceleration which may exceed the driving capabilities of some axes. Lin et al. [12] took the curvature and centripetal acceleration as the indicator of whether a given point is a sharp corner. However, since the acceleration comes from the motors, axis acceleration constraints are more suitable for determining velocity limits than the total acceleration. In the work presented by Otsuki et al. [13], Timar et al. [14], Dong and Stori [15] and Renton and Elbestawi [16], the axis acceleration constraint is considered to be one of the key constraints for feedrate planning. Feng et al. [17] obtained velocity limits by finding feasible tangent acceleration and velocity respecting axis acceleration constraints. Although there are velocity limit curve generation methods considering various constraints, two major problems still exist in existing methods. On the one hand, there is no general rule of choosing the sampling interval for the scan of the given tool path. If the interval is too large, sharp corners may be ignored and large commanded velocity may be applied to the movement through the sharp corners. If the interval is too small, too much time will be wasted on unnecessary computations. On the other hand, the velocity limit on each scanning point is determined only by geometrical features, such as the first- and second-order derivatives, of the scanning point. As a result, the geometrical features of a curve segment near a specified scanning point are approximated by overgeneralization of the geometrical features of the scanning point. If the curve segment contains a tiny corner, the computed velocity limit may be unnecessarily cut down due to an abrupt change of the geometrical features, as shown in Fig. 1. In order to give an adequate scanning interval on the tool path and avoid unnecessary decelerations and accelerations in tiny corners, in Section 2 of this paper, the unit arc length increment scanning method (UALISM) is proposed for generating velocity limit curves. The constraints for chord height errors and axis accelerations are considered. A constant scanning interval of 1 BLU (basic length unit) is selected and multiple scanning points around each specified scanning point are used for computing the velocity limit for the specified scanning point. Simulation results and discussion is given in Section 3.
motion with constant velocity or constant acceleration cannot be ˙ phase plane. In this paper, the s-s ˙ phase shown explicitly on the u-u plane is used, where s is the arc length parameter of the tool path ˙ phase plane, the motion with constant velocity is curve. In the s-s plotted as a straight line and the motion with constant acceleration is plotted as a parabola. The differences between UALISM and existing velocity limit curve generation methods lie in the selection of the scanning interval and the computation of chord height errors and axis accelerations. They will be discussed in detail in this section.
2. Velocity limit curve generation by UALISM
The chord height error is one of the sources of the contour error of the motion. It is caused by approximating a curve segment with a line segment connecting the two ends of the curve segment. The chord height error is the maximum distance from the curve seg ment to the line segment. For the curve segment between C uk1
2.1. Selection of the scanning interval As mentioned in Section 1, an inadequate scanning interval may lead to either the ignorance of sharp corners or the waste of computational time. Based on the concept of Unit Generalized Arc Length Increment Method [20], the BLU is the minimum unit of the coordinate of each axis. The projection of a curve segment of 1 BLU long on each axis is less than 1 BLU. This means than a tiny step of 1 BLU can only cause a tiny change of the coordinate of each axis around its minimum unit. In UALISM, the velocity limit of a specified scanning point is computed by using several scanning points near the specified one. One BLU is fine enough as the arc length interval for the scanning. Therefore, the scanning points are selected to be uniformly distributed on the tool path with a fixed arc length parameter interval of 1 BLU. To be specific, for the tool path described by a parametric curve C (u) (u ∈ [u0 , uMk ]), the scanning
d (k, k1 , k2 ) =
uk+1
C (u) du = 1, k = 0, 1, . . .Mk − 1
(1)
uk
Each curve parameter uk can be computed by a parametric curve interpolation method such as the Taylor’s expansion method or the augmented Taylor’s expansion method [21]. 2.2. Velocity limit under the chord height error constraint
In this section, a velocity limit curve generation method, namely the UALISM, is proposed. Constraints of chord height errors and axis accelerations are considered but they do not prevent the generation process from considering other constraints such as jerks. To show how the velocity limit varies along a tool path, the velocity limit curve is plotted on a phase plane [18]. Dong and ˙ Stori [19] uses the u-u phase plane to demonstrate the velocity limit curve, where u is the tool path curve parameter and u˙ is the first-order derivate of u with respect to time. However, since u˙ is generally not proportional to velocity s˙ for non-linear curves, the
points C (u0 ), C (u1 ), . . ., C uMk are selected such that:
and C uk2 , by using the scanning points C uk1 , C uk1 +1 , . . .,
C uk2 , the chord height error is given by: ε(c) (k1 , k2 ) =
max
k=k1 +1,k1 +2,...,k2 −1
d (k, k1 , k2 ) .
(2)
The d (k, k1 , k2 ) in Eq.(2) is the between the point C (uk ) distance and the line segment C uk1 -C uk2 : 2
2
2
C (uk ) − C uk1 C uk2 − C uk1 − C (uk ) − C uk1 , C uk2 − C uk1
C uk2 − C uk1
(3)
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40 TC
SC1
SC2
30
where · is the Euclidean norm and ·, · is the inner product. Eq. (3) can be used for C (u) in any dimension. For C (u) in the threedimensional space, Eq. (3) can be simplified as: d (k, k1 , k2 ) =
C(uk )−C uk
× C uk
1
C uk
2
−C uk
2
1
−C uk
1
33.501 20 Z (mm)
Z (mm)
Fig. 2. An example of computing chord height error by UALISM.
33.499 -0.1
.(4)
-20
0.1
-10
0
10
20
Y (mm)
C uk2 , i.e., ε(c) (k1 , k2 ) = d (k1 + 4, k1 , k2 ). Denote the arc length parameter of curve C (u) at the kth scanning point as sk . The interval of the scanning point is 1 BLU, therefore sk = k. Denote thevelocity the chord height error limit under constraint at point C uk1 as v(c) sk1 . The velocity limit v(c) sk1 can be regarded as the maximum velocity that is allowed to move through C uk1 in a constant velocity without violating the constraints. Since the motion with constant velocity shall be expressed on a curve segment rather than a point, two points, C uk0 and
Velocity (mm/s)
0 Y (mm)
0
distance between the scanning point and the line segment C uk1 -
Fig. 3. The pin-shaped tool path curve.
C uk2 , are introduced to give two parts of motion. Each part of motion lasts for time T, where T is the interpolation period of the CNC system. The velocities of the first and the second part of motion are
v sk0 = sk1 − sk0 /T
(5)
Velocity limit curve
High velocity limit at the tiny corner
100
MCP
10
Fig. 2 shows anexample when the 5 scanning k2 − k1 = 6. Among points between C uk1 and C uk2 , C uk1 +4 gives the maximum
33.5
Commanded velocity profile
50 0
0
20
40
60
80
100
120
140
Velocity (mm/s)
Arc length parameter (mm) (a) Velocity limit curve obtained by UALISM and the commanded velocity profile
Velocity limit curve
Low velocity limit at the tiny corner
100
Commanded velocity profile
50 0
0
20
40
60
80
100
120
140
Arc length parameter (mm) (b) Velocity limit curve obtained by Feng’s method and the commanded velocity profile
ds/du
600 400 200 0
0
20
40
60
80
100
120
140
Arc length parameter (mm) (c) The Euclidean norm of the first-order derivative of the pin-shaped tool path curve
2
2
||dC /du ||
10000
5000
0
0
20
40
60
80
100
120
140
Arc length parameter (mm) (d) The Euclidean norm of the second-order derivative of the pin-shaped tool path curve Fig. 4. Velocity limit curves, commanded velocity profiles and the Euclidean norms of derivatives of the pin-shaped curve.
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and
v sk1 = sk2 − sk1 /T
(6)
respectively, v sk0 = v sk1 and sk1 − sk0 = sk2 − sk1 = s. Denote the desired velocity as vdes and the total velocity constraint as vmac , v(c) sk1 is computed by:
v(c) sk1 = sk(c) /T
(7)
1
where (c)
sk = max
(c)
i.e., set s(ac) (k1 ) :=s − 1. The corresponding velocity limit is
v(ac) (k1 ) = s(ac) (k1 ) /T . Step 9. Set k1 := k1 + 1. Step 10. If k1 ≤ Mk , then go to Step 2; otherwise end the procedure. The velocity limit curve can be generated by connecting the velocity limits v(ac) (k1 ) (k1 = 0, 1, . . . Mk ) in turn. Note that Step 5 checks the relationship between the length of curve segment s, length of the line segment C uk1 − C uk0
(c)
(max)
s|ε(c) sk1 , sk2 ≤ εmax , ε(c) sk0 , sk1 ≤ εmax , s ∈ N, 0 ≤ s ≤ sk
1
Step 8. Record the arc length increment limit at point C uk1 ,
1
,(8)
(c)
N is the set of natural numbers, εmax is the chord height error constraint and
(max) sk 1
= min vdes T, vmac T, sk1 , Mk − sk1
.
(9)
2.3. Velocity limit under the axis acceleration constraint Denote the limit velocity under the axis acceleration constraint at point C uk1 as v(a) sk1 . Like the computation of v(c) sk1 ,
C uk0
and C uk2
are introduced to give two parts of motion.
The acceleration vector at C uk1 , which is denoted as ak1 , can be obtained by: C
uk
2
−C
uk
1
T
ak1 =
C
−
−C
uk
1
uk
0
T
=
T
C uk2
− 2C uk1
+ C uk0
v(a) sk1 = sk(a) /T
(11)
1
where (a) sk 1
= max
s|
cj uk2 − 2cj uk1 + cj uk0
(a)
(max)
is the acceleration constraint for the jth axis, cj (uk ) is the component for the jth axis in C (uk ) and (max) sk = min vdes T, vmac T, sk1 , Mk − sk1 . (13) 1
2.4. Procedure of UALISM The entire procedure of UALISM is given in this subsection. First, scanning points are found by the method introduced in Section 2.1. Second, according to Section 2.2 and 2.3, the procedure of velocity limit curve generation can be summarized as follows: Step 1. Let the current point index for computing velocity limit k1 := 0. (max) := min vdes T, vmac T, k1 , Mk − k1 . Step 2. Let s := 1, sk (max) 1
1
, then go to Step 8.
Step 4. Set k0 := k1 − s j = 1, 2, . . ., Mj check if
and k2 := k1 + s, for axis the acceleration constraint (a)
cj uk2 − 2cj uk1 + cj uk0 /T 2 ≤ aj,max is met. If one of the constraints is not met, then go to Step 8.
2
Step 5. If s > s(ac) (k1 − s) and C uk1 − C uk0 <
(c)
s2 − 2εmax
2
(c)
, then check if ε(c) (k1 − s, k1 ) ≤ εmax is met. If (c)
,(12)
from the origin, goes through the sharp corner SC1, tiny corner TC and sharp corner SC2 and ends at the origin. The tool path is zoomed in near TC and it can be seen that the length of the major curved part (MCP) of the tiny corner is about 0.01 mm. The parameters for simulation are list in Table 2. The velocity limit curves, commanded velocity profiles and the Euclidean norms of the first- and second-order derivatives of the pin-shaped tool path curve are given in Fig. 4. Note that the tiny corner lies in the middle of the tool path where the arc length parameter is 70,984 BLU. By comparing Fig. 4(a) with (b), it can be seen that the two velocity limit curves are almost the same except the velocity limit at the tiny corner. As shown in Fig. 4(c) with (d), the Euclidean norms of both the first- and second-order derivatives change abruptly and approach zero. The Feng’s method solves equations and inequalities which contain the first- and secondTable 1 The data of the pin-shaped tool path curve. Item
Data
Control points (mm, mm)
{{0.000, 0.000}, {0.000, 14.100}, {−0.250, 28.800}, {-4.250, 33.400}, {-23.750, 33.600}, {−19.050, 33.580}, {-0.050, 33.500}, {0.000, 33.500}, {0.050, 33.500}, {19.050, 33.580}, {23.750, 33.600}, {4.250, 33.400}, {0.250, 28.800}, {0.000, 14.100}, {0.000, 0.000}} {0, 0, 0, 0, 0.08, 0.17, 0.25, 0.33, 0.42, 0.5, 0.58, 0.67, 0.75, 0.83, 0.92, 1, 1, 1, 1} 141.967
ε(c) (k1 − s, k1 ) ≤ εmax is not met, then go to Step 8.
2
(c)
Step 6. If C uk2 − C uk1 < s2 − 2εmax (c) εmax is met. If ε(c)
2
(k1 , k1 + s) ≤ (k1 , k1 + s) ≤ then go to Step 8. Step 7. Let s := s + 1 and then go to Step 3.
ε(c)
, then check if (c) εmax
1
(a) aj,max
Step 3. If s > sk
In this section, the proposed UALISM is compared with Feng’s velocity limit curve generation method [17]. The pin-shaped tool path is shown in Fig. 3. The tool path is a third-degree B-spline curve on the YZ plane and its data are given in Table 1. The motion starts
/T 2 ≤ aj,max , j = 1, 2, . . ., Mj
s ∈ N, 0 ≤ s ≤ sk
3. Simulation results and discussion
Then
(10)
.
T2
(c)
and the chord height error constraint εmax . If C uk1 − C uk0 approaches s, it means that the curve segment is closed enough to the line segment, and the chord height error will not violate the constraint and there is no need to check whether ε(c) (k1 − s, k1 ) ≤ (c) εmax is met. The case in Step 6 is similar to that in Step 5. From the procedure of UALISM it can be seen that the velocity limit curve is generated only by using the coordinates of scanning points. The accuracy of the velocity limit is not affected by the situation when the derivatives of the tool path curve change abruptly or approach zero. The validation will be given in the next section.
Knot vector
is not met, Total length of the tool path (mm)
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Velocity (mm/s)
80 60 40 20 0
0
0.5
1
1.5
2
2.5
3
2
2.5
3
Time (s) (a) Commanded velocity
Chord height error ( µm)
0.25 0.2 0.15 0.1 0.05 0
0
0.5
1
1.5
2
Axis acclerations (mm/s )
Time (s) (b) Chord height error
1000 800 600 400 200 0 -200 -400 -600 -800
Acceleration of the Y-axis Acceleration of the Z-axis
0
0.5
1
1.5
2
2.5
3
Time (s) (c) Axis accelerations Fig. 5. The commanded velocity, chord height error and axis accelerations of the motion under the constraint of the velocity limit curve generated by UALISM.
Table 2 Parameters for the simulation. Parameter
Value
Scanning interval of the curve parameter for Feng’s method Min {vdes , vmac } (mm/s) Interpolation period (ms) (c) εmax (m) Length per BLU (m) (a) aj,max (j = 1, 2) (mm/s2 )
0.00001 70 2 2 1 0.25
order derivatives to obtain the velocity limit at each scanning point, and the solved velocity limit at the tiny corner is lower than 20 mm/s that there is an obvious process of deceleration and acceleration near the tiny corner. To check whether the chord height error and axis acceleration constraints are met in the motion under the two different velocity limit curves, the commanded velocity, chord height errors and axis accelerations with respect to time are given in Fig. 5 (UALISM) and Fig. 6 (Feng’s method). Neither of the two velocity limit curve generation methods violates any constraint of the chord height error or the axis acceleration. Therefore, the over-deceleration to a velocity
Table 3 Total time of motions under the constraints of velocity limit curves generated by two methods. Velocity limit curve generation method
Total time of motion (s)
Feng’s method UALISM
3.142 3.028
under 20 mm/s is unnecessary. By using UALISM, the fluctuation of curve derivatives near the tiny corners can be “filtered out”. The total time of motions are compared in Table 3. From Fig. 3, Tables 1 and 3 it can be found that, even the length of the major curved part of the tiny corner is only about 0.01 mm (about 0.007% of the total length of the tool path), 4% of the total time of motion is saved by carrying out the motion in terms of the velocity limit curve generated by UALISM. If the number or the density of tiny corners increases, the time saving can be remarkable. 4. Conclusion In order to eliminate the time wasted on the over-deceleration and acceleration near the tiny corner of the tool path, a univer-
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Velocity (mm/s)
80 60 40 20 0
0
0.5
1
1.5
2
2.5
3
2
2.5
3
Time (s) (a) Commanded velocity
Chord height error ( µm)
0.2 0.15 0.1 0.05 0
0
0.5
1
1.5
2
Axis acclerations (mm/s )
Time (s) (b) Chord height error
1000 800 600 400 200 0 -200 -400 -600 -800
Acceleration of the Y-axis Acceleration of the Z-axis
0
0.5
1
1.5
2
2.5
3
Time (s) (c) Axis accelerations Fig. 6. The commanded velocity, chord height error and axis accelerations of the motion under the constraint of the velocity limit curve generated by Feng’s method.
sal velocity limit curve generation method named unit arc length increment scanning method (UALISM) is proposed. Issues including the selection of scanning interval, computations of chord height error and axis acceleration have been discussed. The features of UALISM can be summarized as follows:
• The scanning interval is determined by the arc length instead of the curve parameter. The scanning interval is fixed to be 1 BLU (basic length unit) which is irrelevant to the type, size and shape of the tool path curve. Therefore, UALISM is a universal method. By using the scanning interval around the minimum unit of the coordinate of each axis, the constraints can be checked without ignoring abnormal geometries such as sharp corners and tiny corners. • The velocity limit for the specified scanning point is computed in terms of the coordinates of multiple scanning points near the specified scanning point instead of using only the derivatives of the specified scanning point. As a result, the computational accuracy of the velocity limit curve does not deteriorate even if the curve derivatives change abruptly or approaches zero.
Simulation results show that the computational accuracy of the velocity limit can be improved by using UALISM. Unnecessary decelerations and acceleration can be avoided and the total time of motion can be reduced. In the future research, more constraints such as the jerk constraints can be introduced to improve the smoothness of the velocity profile. Acknowledgements The authors thank the financial supports from National Natural Science Foundation of China (Grant No. 51675340 and 51421092), National Major Scientific and Technological Special Project (2014ZX04001061), Gas Turbine Research Institute, Shanghai Jiao Tong University (Grant No. AF0200088/015), and Industry-Academia-Research Project of Shanghai Municipal Education Commission (Grant No. 15CXY03). References [1] Dong H, Chen B, Chen Y, Xie J, Zhou Z. An accurate NURBS curve interpolation algorithm with short spline interpolation capacity. Int J Adv Manuf Technol 2012;63(9–12):1257–70.
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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Technical Paper
A universal velocity limit curve generator considering abnormal tool path geometry for CNC machine tools夽 Mo Chen ∗ , Xue-Cheng Xi, Wan-Sheng Zhao, Hao Chen, Hong-Da Liu State Key Laboratory of Mechanical System Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 17 November 2016 Received in revised form 13 February 2017 Accepted 3 March 2017 Available online xxx Keywords: Velocity limit curves Parametric curves CNC machine tools
a b s t r a c t Non-linear parametric curves, such as B-spline curves, are becoming increasingly available in modern CNC (Computer Numerical Control) systems. The smoothness of parametric curves offers higher order of continuity and thus invokes less vibration in machines as compared to short line segments. Nevertheless, the computations for velocity limit curves, velocity profiles and interpolation points are quite complicated and time-consuming and therefore approximation methods are applied. Unnecessary accelerations and decelerations, which cost additional time of motion, can be caused by inaccurate computations of the velocity limits around tiny corners. To overcome the problem, the unit arc length increment scanning method (UALISM) is proposed to reduce the time of movement and to improve the efficiency. The scanning interval is fixed at 1 BLU (basic length unit) which is irrelevant to the type, size and shape of the tool path curve. The constraints of chord height errors and axis accelerations are considered and the velocity limit for the specified scanning point is computed using the coordinates of multiple scanning points near the specified scanning point. Simulation results show that the unnecessary accelerations and decelerations can be avoided and thus the total motion time can be reduced by UALISM. © 2017 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction In CNC (Computer Numerical Control) machine tools, the velocity of the feed motion is desired to be constant along a given tool path during machining in order to obtain high surface quality [1]. Around sharp corners in tool paths, however, high commanded velocity can cause a sharp turn with excessive axis accelerations that exceed axis driving capabilities. This will not only lower the surface quality and accuracy but also lead to undesirable vibrations and resonances [2]. As setting up speed limit signs on roads, tool path data shall be collected in order to find a velocity limit curve for the feedrate planning for the tool path before the motion starts. The concept of velocity limit curve sources from the timeoptimal trajectory planning for robots [3]. The velocity limit curves are generally obtained through the following steps: 1) a group of points on the given tool path are sampled by scanning, 2) the velocity limit for each scanning point is computed considering constraints such as acceleration, jerk and chord height error, and 3) the velocity limits are connected or interpolated in the sequence of
夽 45th SME North American Manufacturing Research Conference, NAMRC 45, LA, USA. ∗ Corresponding author. E-mail address: [email protected] (M. Chen).
scanning points to form the velocity limit curve for the tool path. For CNC machine tools, the chord height error, which is part of the contour error, and the axis acceleration, which is related to the driving capability of a motor, are the two major issues for velocity limits. The chord height error, which results from approximating a curve segment with line segments, is defined to be the maximum distance between the curve segment and its corresponding line segment [4]. For curves except lines and circular arcs, it is difficult to give a formula to compute the chord height errors directly, and thus approximate methods are used instead [5]. A typical way is approximating a curve segment with a circular arc and compute the chord height error if the movement were to be done along the circular arc [4,6–9]. Liu et al. [10] and Emami and Arezoo [11] pointed out that the circular arc approximation method gives better computational accuracy of chord height error than using the length of the line segment which connects the midpoints of the curve segment and its chord. The circular arc approximation method works well when the curve direction does not change abruptly since the approximation error can be kept low in a large length of segment. For tool path curves with abnormal geometries such as sharp corners and tiny corners, an approximate method with better accuracy shall be developed. When the machine axes move near a sharp corner in a tool path, the magnitude and direction change abruptly, leading to a large
http://dx.doi.org/10.1016/j.jmsy.2017.04.010 0278-6125/© 2017 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
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y
v Velocity limit curve
s + direction Tiny corner Tool path
Velocity limit at the tiny corner
x
O
s
O
Fig. 1. The excessively low velocity limit in a tiny corner.
commanded acceleration which may exceed the driving capabilities of some axes. Lin et al. [12] took the curvature and centripetal acceleration as the indicator of whether a given point is a sharp corner. However, since the acceleration comes from the motors, axis acceleration constraints are more suitable for determining velocity limits than the total acceleration. In the work presented by Otsuki et al. [13], Timar et al. [14], Dong and Stori [15] and Renton and Elbestawi [16], the axis acceleration constraint is considered to be one of the key constraints for feedrate planning. Feng et al. [17] obtained velocity limits by finding feasible tangent acceleration and velocity respecting axis acceleration constraints. Although there are velocity limit curve generation methods considering various constraints, two major problems still exist in existing methods. On the one hand, there is no general rule of choosing the sampling interval for the scan of the given tool path. If the interval is too large, sharp corners may be ignored and large commanded velocity may be applied to the movement through the sharp corners. If the interval is too small, too much time will be wasted on unnecessary computations. On the other hand, the velocity limit on each scanning point is determined only by geometrical features, such as the first- and second-order derivatives, of the scanning point. As a result, the geometrical features of a curve segment near a specified scanning point are approximated by overgeneralization of the geometrical features of the scanning point. If the curve segment contains a tiny corner, the computed velocity limit may be unnecessarily cut down due to an abrupt change of the geometrical features, as shown in Fig. 1. In order to give an adequate scanning interval on the tool path and avoid unnecessary decelerations and accelerations in tiny corners, in Section 2 of this paper, the unit arc length increment scanning method (UALISM) is proposed for generating velocity limit curves. The constraints for chord height errors and axis accelerations are considered. A constant scanning interval of 1 BLU (basic length unit) is selected and multiple scanning points around each specified scanning point are used for computing the velocity limit for the specified scanning point. Simulation results and discussion is given in Section 3.
motion with constant velocity or constant acceleration cannot be ˙ phase plane. In this paper, the s-s ˙ phase shown explicitly on the u-u plane is used, where s is the arc length parameter of the tool path ˙ phase plane, the motion with constant velocity is curve. In the s-s plotted as a straight line and the motion with constant acceleration is plotted as a parabola. The differences between UALISM and existing velocity limit curve generation methods lie in the selection of the scanning interval and the computation of chord height errors and axis accelerations. They will be discussed in detail in this section.
2. Velocity limit curve generation by UALISM
The chord height error is one of the sources of the contour error of the motion. It is caused by approximating a curve segment with a line segment connecting the two ends of the curve segment. The chord height error is the maximum distance from the curve seg ment to the line segment. For the curve segment between C uk1
2.1. Selection of the scanning interval As mentioned in Section 1, an inadequate scanning interval may lead to either the ignorance of sharp corners or the waste of computational time. Based on the concept of Unit Generalized Arc Length Increment Method [20], the BLU is the minimum unit of the coordinate of each axis. The projection of a curve segment of 1 BLU long on each axis is less than 1 BLU. This means than a tiny step of 1 BLU can only cause a tiny change of the coordinate of each axis around its minimum unit. In UALISM, the velocity limit of a specified scanning point is computed by using several scanning points near the specified one. One BLU is fine enough as the arc length interval for the scanning. Therefore, the scanning points are selected to be uniformly distributed on the tool path with a fixed arc length parameter interval of 1 BLU. To be specific, for the tool path described by a parametric curve C (u) (u ∈ [u0 , uMk ]), the scanning
d (k, k1 , k2 ) =
uk+1
C (u) du = 1, k = 0, 1, . . .Mk − 1
(1)
uk
Each curve parameter uk can be computed by a parametric curve interpolation method such as the Taylor’s expansion method or the augmented Taylor’s expansion method [21]. 2.2. Velocity limit under the chord height error constraint
In this section, a velocity limit curve generation method, namely the UALISM, is proposed. Constraints of chord height errors and axis accelerations are considered but they do not prevent the generation process from considering other constraints such as jerks. To show how the velocity limit varies along a tool path, the velocity limit curve is plotted on a phase plane [18]. Dong and ˙ Stori [19] uses the u-u phase plane to demonstrate the velocity limit curve, where u is the tool path curve parameter and u˙ is the first-order derivate of u with respect to time. However, since u˙ is generally not proportional to velocity s˙ for non-linear curves, the
points C (u0 ), C (u1 ), . . ., C uMk are selected such that:
and C uk2 , by using the scanning points C uk1 , C uk1 +1 , . . .,
C uk2 , the chord height error is given by: ε(c) (k1 , k2 ) =
max
k=k1 +1,k1 +2,...,k2 −1
d (k, k1 , k2 ) .
(2)
The d (k, k1 , k2 ) in Eq.(2) is the between the point C (uk ) distance and the line segment C uk1 -C uk2 : 2
2
2
C (uk ) − C uk1 C uk2 − C uk1 − C (uk ) − C uk1 , C uk2 − C uk1
C uk2 − C uk1
(3)
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40 TC
SC1
SC2
30
where · is the Euclidean norm and ·, · is the inner product. Eq. (3) can be used for C (u) in any dimension. For C (u) in the threedimensional space, Eq. (3) can be simplified as: d (k, k1 , k2 ) =
C(uk )−C uk
× C uk
1
C uk
2
−C uk
2
1
−C uk
1
33.501 20 Z (mm)
Z (mm)
Fig. 2. An example of computing chord height error by UALISM.
33.499 -0.1
.(4)
-20
0.1
-10
0
10
20
Y (mm)
C uk2 , i.e., ε(c) (k1 , k2 ) = d (k1 + 4, k1 , k2 ). Denote the arc length parameter of curve C (u) at the kth scanning point as sk . The interval of the scanning point is 1 BLU, therefore sk = k. Denote thevelocity the chord height error limit under constraint at point C uk1 as v(c) sk1 . The velocity limit v(c) sk1 can be regarded as the maximum velocity that is allowed to move through C uk1 in a constant velocity without violating the constraints. Since the motion with constant velocity shall be expressed on a curve segment rather than a point, two points, C uk0 and
Velocity (mm/s)
0 Y (mm)
0
distance between the scanning point and the line segment C uk1 -
Fig. 3. The pin-shaped tool path curve.
C uk2 , are introduced to give two parts of motion. Each part of motion lasts for time T, where T is the interpolation period of the CNC system. The velocities of the first and the second part of motion are
v sk0 = sk1 − sk0 /T
(5)
Velocity limit curve
High velocity limit at the tiny corner
100
MCP
10
Fig. 2 shows anexample when the 5 scanning k2 − k1 = 6. Among points between C uk1 and C uk2 , C uk1 +4 gives the maximum
33.5
Commanded velocity profile
50 0
0
20
40
60
80
100
120
140
Velocity (mm/s)
Arc length parameter (mm) (a) Velocity limit curve obtained by UALISM and the commanded velocity profile
Velocity limit curve
Low velocity limit at the tiny corner
100
Commanded velocity profile
50 0
0
20
40
60
80
100
120
140
Arc length parameter (mm) (b) Velocity limit curve obtained by Feng’s method and the commanded velocity profile
ds/du
600 400 200 0
0
20
40
60
80
100
120
140
Arc length parameter (mm) (c) The Euclidean norm of the first-order derivative of the pin-shaped tool path curve
2
2
||dC /du ||
10000
5000
0
0
20
40
60
80
100
120
140
Arc length parameter (mm) (d) The Euclidean norm of the second-order derivative of the pin-shaped tool path curve Fig. 4. Velocity limit curves, commanded velocity profiles and the Euclidean norms of derivatives of the pin-shaped curve.
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and
v sk1 = sk2 − sk1 /T
(6)
respectively, v sk0 = v sk1 and sk1 − sk0 = sk2 − sk1 = s. Denote the desired velocity as vdes and the total velocity constraint as vmac , v(c) sk1 is computed by:
v(c) sk1 = sk(c) /T
(7)
1
where (c)
sk = max
(c)
i.e., set s(ac) (k1 ) :=s − 1. The corresponding velocity limit is
v(ac) (k1 ) = s(ac) (k1 ) /T . Step 9. Set k1 := k1 + 1. Step 10. If k1 ≤ Mk , then go to Step 2; otherwise end the procedure. The velocity limit curve can be generated by connecting the velocity limits v(ac) (k1 ) (k1 = 0, 1, . . . Mk ) in turn. Note that Step 5 checks the relationship between the length of curve segment s, length of the line segment C uk1 − C uk0
(c)
(max)
s|ε(c) sk1 , sk2 ≤ εmax , ε(c) sk0 , sk1 ≤ εmax , s ∈ N, 0 ≤ s ≤ sk
1
Step 8. Record the arc length increment limit at point C uk1 ,
1
,(8)
(c)
N is the set of natural numbers, εmax is the chord height error constraint and
(max) sk 1
= min vdes T, vmac T, sk1 , Mk − sk1
.
(9)
2.3. Velocity limit under the axis acceleration constraint Denote the limit velocity under the axis acceleration constraint at point C uk1 as v(a) sk1 . Like the computation of v(c) sk1 ,
C uk0
and C uk2
are introduced to give two parts of motion.
The acceleration vector at C uk1 , which is denoted as ak1 , can be obtained by: C
uk
2
−C
uk
1
T
ak1 =
C
−
−C
uk
1
uk
0
T
=
T
C uk2
− 2C uk1
+ C uk0
v(a) sk1 = sk(a) /T
(11)
1
where (a) sk 1
= max
s|
cj uk2 − 2cj uk1 + cj uk0
(a)
(max)
is the acceleration constraint for the jth axis, cj (uk ) is the component for the jth axis in C (uk ) and (max) sk = min vdes T, vmac T, sk1 , Mk − sk1 . (13) 1
2.4. Procedure of UALISM The entire procedure of UALISM is given in this subsection. First, scanning points are found by the method introduced in Section 2.1. Second, according to Section 2.2 and 2.3, the procedure of velocity limit curve generation can be summarized as follows: Step 1. Let the current point index for computing velocity limit k1 := 0. (max) := min vdes T, vmac T, k1 , Mk − k1 . Step 2. Let s := 1, sk (max) 1
1
, then go to Step 8.
Step 4. Set k0 := k1 − s j = 1, 2, . . ., Mj check if
and k2 := k1 + s, for axis the acceleration constraint (a)
cj uk2 − 2cj uk1 + cj uk0 /T 2 ≤ aj,max is met. If one of the constraints is not met, then go to Step 8.
2
Step 5. If s > s(ac) (k1 − s) and C uk1 − C uk0 <
(c)
s2 − 2εmax
2
(c)
, then check if ε(c) (k1 − s, k1 ) ≤ εmax is met. If (c)
,(12)
from the origin, goes through the sharp corner SC1, tiny corner TC and sharp corner SC2 and ends at the origin. The tool path is zoomed in near TC and it can be seen that the length of the major curved part (MCP) of the tiny corner is about 0.01 mm. The parameters for simulation are list in Table 2. The velocity limit curves, commanded velocity profiles and the Euclidean norms of the first- and second-order derivatives of the pin-shaped tool path curve are given in Fig. 4. Note that the tiny corner lies in the middle of the tool path where the arc length parameter is 70,984 BLU. By comparing Fig. 4(a) with (b), it can be seen that the two velocity limit curves are almost the same except the velocity limit at the tiny corner. As shown in Fig. 4(c) with (d), the Euclidean norms of both the first- and second-order derivatives change abruptly and approach zero. The Feng’s method solves equations and inequalities which contain the first- and secondTable 1 The data of the pin-shaped tool path curve. Item
Data
Control points (mm, mm)
{{0.000, 0.000}, {0.000, 14.100}, {−0.250, 28.800}, {-4.250, 33.400}, {-23.750, 33.600}, {−19.050, 33.580}, {-0.050, 33.500}, {0.000, 33.500}, {0.050, 33.500}, {19.050, 33.580}, {23.750, 33.600}, {4.250, 33.400}, {0.250, 28.800}, {0.000, 14.100}, {0.000, 0.000}} {0, 0, 0, 0, 0.08, 0.17, 0.25, 0.33, 0.42, 0.5, 0.58, 0.67, 0.75, 0.83, 0.92, 1, 1, 1, 1} 141.967
ε(c) (k1 − s, k1 ) ≤ εmax is not met, then go to Step 8.
2
(c)
Step 6. If C uk2 − C uk1 < s2 − 2εmax (c) εmax is met. If ε(c)
2
(k1 , k1 + s) ≤ (k1 , k1 + s) ≤ then go to Step 8. Step 7. Let s := s + 1 and then go to Step 3.
ε(c)
, then check if (c) εmax
1
(a) aj,max
Step 3. If s > sk
In this section, the proposed UALISM is compared with Feng’s velocity limit curve generation method [17]. The pin-shaped tool path is shown in Fig. 3. The tool path is a third-degree B-spline curve on the YZ plane and its data are given in Table 1. The motion starts
/T 2 ≤ aj,max , j = 1, 2, . . ., Mj
s ∈ N, 0 ≤ s ≤ sk
3. Simulation results and discussion
Then
(10)
.
T2
(c)
and the chord height error constraint εmax . If C uk1 − C uk0 approaches s, it means that the curve segment is closed enough to the line segment, and the chord height error will not violate the constraint and there is no need to check whether ε(c) (k1 − s, k1 ) ≤ (c) εmax is met. The case in Step 6 is similar to that in Step 5. From the procedure of UALISM it can be seen that the velocity limit curve is generated only by using the coordinates of scanning points. The accuracy of the velocity limit is not affected by the situation when the derivatives of the tool path curve change abruptly or approach zero. The validation will be given in the next section.
Knot vector
is not met, Total length of the tool path (mm)
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Velocity (mm/s)
80 60 40 20 0
0
0.5
1
1.5
2
2.5
3
2
2.5
3
Time (s) (a) Commanded velocity
Chord height error ( µm)
0.25 0.2 0.15 0.1 0.05 0
0
0.5
1
1.5
2
Axis acclerations (mm/s )
Time (s) (b) Chord height error
1000 800 600 400 200 0 -200 -400 -600 -800
Acceleration of the Y-axis Acceleration of the Z-axis
0
0.5
1
1.5
2
2.5
3
Time (s) (c) Axis accelerations Fig. 5. The commanded velocity, chord height error and axis accelerations of the motion under the constraint of the velocity limit curve generated by UALISM.
Table 2 Parameters for the simulation. Parameter
Value
Scanning interval of the curve parameter for Feng’s method Min {vdes , vmac } (mm/s) Interpolation period (ms) (c) εmax (m) Length per BLU (m) (a) aj,max (j = 1, 2) (mm/s2 )
0.00001 70 2 2 1 0.25
order derivatives to obtain the velocity limit at each scanning point, and the solved velocity limit at the tiny corner is lower than 20 mm/s that there is an obvious process of deceleration and acceleration near the tiny corner. To check whether the chord height error and axis acceleration constraints are met in the motion under the two different velocity limit curves, the commanded velocity, chord height errors and axis accelerations with respect to time are given in Fig. 5 (UALISM) and Fig. 6 (Feng’s method). Neither of the two velocity limit curve generation methods violates any constraint of the chord height error or the axis acceleration. Therefore, the over-deceleration to a velocity
Table 3 Total time of motions under the constraints of velocity limit curves generated by two methods. Velocity limit curve generation method
Total time of motion (s)
Feng’s method UALISM
3.142 3.028
under 20 mm/s is unnecessary. By using UALISM, the fluctuation of curve derivatives near the tiny corners can be “filtered out”. The total time of motions are compared in Table 3. From Fig. 3, Tables 1 and 3 it can be found that, even the length of the major curved part of the tiny corner is only about 0.01 mm (about 0.007% of the total length of the tool path), 4% of the total time of motion is saved by carrying out the motion in terms of the velocity limit curve generated by UALISM. If the number or the density of tiny corners increases, the time saving can be remarkable. 4. Conclusion In order to eliminate the time wasted on the over-deceleration and acceleration near the tiny corner of the tool path, a univer-
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Velocity (mm/s)
80 60 40 20 0
0
0.5
1
1.5
2
2.5
3
2
2.5
3
Time (s) (a) Commanded velocity
Chord height error ( µm)
0.2 0.15 0.1 0.05 0
0
0.5
1
1.5
2
Axis acclerations (mm/s )
Time (s) (b) Chord height error
1000 800 600 400 200 0 -200 -400 -600 -800
Acceleration of the Y-axis Acceleration of the Z-axis
0
0.5
1
1.5
2
2.5
3
Time (s) (c) Axis accelerations Fig. 6. The commanded velocity, chord height error and axis accelerations of the motion under the constraint of the velocity limit curve generated by Feng’s method.
sal velocity limit curve generation method named unit arc length increment scanning method (UALISM) is proposed. Issues including the selection of scanning interval, computations of chord height error and axis acceleration have been discussed. The features of UALISM can be summarized as follows:
• The scanning interval is determined by the arc length instead of the curve parameter. The scanning interval is fixed to be 1 BLU (basic length unit) which is irrelevant to the type, size and shape of the tool path curve. Therefore, UALISM is a universal method. By using the scanning interval around the minimum unit of the coordinate of each axis, the constraints can be checked without ignoring abnormal geometries such as sharp corners and tiny corners. • The velocity limit for the specified scanning point is computed in terms of the coordinates of multiple scanning points near the specified scanning point instead of using only the derivatives of the specified scanning point. As a result, the computational accuracy of the velocity limit curve does not deteriorate even if the curve derivatives change abruptly or approaches zero.
Simulation results show that the computational accuracy of the velocity limit can be improved by using UALISM. Unnecessary decelerations and acceleration can be avoided and the total time of motion can be reduced. In the future research, more constraints such as the jerk constraints can be introduced to improve the smoothness of the velocity profile. Acknowledgements The authors thank the financial supports from National Natural Science Foundation of China (Grant No. 51675340 and 51421092), National Major Scientific and Technological Special Project (2014ZX04001061), Gas Turbine Research Institute, Shanghai Jiao Tong University (Grant No. AF0200088/015), and Industry-Academia-Research Project of Shanghai Municipal Education Commission (Grant No. 15CXY03). References [1] Dong H, Chen B, Chen Y, Xie J, Zhou Z. An accurate NURBS curve interpolation algorithm with short spline interpolation capacity. Int J Adv Manuf Technol 2012;63(9–12):1257–70.
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Please cite this article in press as: Chen M, et al. A universal velocity limit curve generator considering abnormal tool path geometry for CNC machine tools. J Manuf Syst (2017), http://dx.doi.org/10.1016/j.jmsy.2017.04.010