- Email: [email protected]
Guide curve based interpolation scheme of parametric curves for precision CNC machining
International Journal of Machine Tools & Manufacture 46 (2006) 235–242 www.elsevier.com/locate/ijmactool
Guide curve based interpolation scheme of parametric curves for precision CNC machining Yuwen Suna,*, Jun Wangb, Dongming Guoa b
a School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney 2052, Australia
Received 17 May 2005; accepted 26 May 2005 Available online 25 July 2005
Abstract Real-time parametric interpolation has played a key role in the computer control of machine tools. To achieve highest quality parts, generated trajectories not only describe the desired toolpath accurately, but also have smooth dynamics profiles. This paper presents a novel parametric interpolator based on guide curve. The relationships between geometric properties and kinematic properties are firstly discussed. Then, with a consideration of the important effect of the curvature of curvilinear path on the machining dynamics, a corresponding formula, which describes the relation of the maximum allowed feed acceleration/deceleration and the maximum allowed rate of change of curvature radius of paths, is built. Thus, based on a near arc parameterization and through modifying the curvature radius curve to deal with corners, key regions and other cases, adaptive feedrate schedule is completed according to the reconstructed smooth curvature radius curve. Consequently, confined chord errors, corners on the path and the acceleration/deceleration capabilities of the machine tool are simultaneously considered and incorporated into the guide curve based parametric interpolation system without using look-ahead scheme. Simulation results indicate the feasibility and precision of the proposed interpolation method. q 2005 Elsevier Ltd. All rights reserved. Keywords: Machining dynamics; CAD/CAM; Feedrate schedule; NC machining; Toolpath
1. Introduction In high precision and high speed NC machining, kinematic and geometric properties of the moving cutter have important effects on the efficiency and quality of the machined parts. However, machining dynamics is often ignored in current toolpath generation methods and interpolator schemes. In this case, any discontinuities of the path can result in undesirable high frequency harmonics in the reference trajectory, which is extremely harm to the machine tool, the machined surface quality and the servo control system. Meanwhile, conventional CNC machines often support only straight line and circular interpolator. Due to the difficulty of dividing a line segment into an integer multiple of the product of the commanded feedrate and the sampling time, the line * Corresponding author. E-mail address: [email protected] (Y. Sun).
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.05.024
interpolation inevitably exists undesirable weaknesses such as feedrate fluctuation and excessive jerk in the process of machining a curve. Overall, dynamics performance issues become peculiarly important in NC machining especially in high-speed machining. For high-speed and high-quality machining parts, we have to consider the machine dynamics while controlling chord error under certain tolerance. Otherwise, due to the limitation of the machine tool drivers’ ability, failure in maintaining the commanded feedrate will in turn lead to the tool chatter of breakage. Consequently, parametric interpolators are being developed to eliminate the disadvantages of linear/circular interpolators. Currently, uniform interpolation [1] and constant speed interpolation [2] are the most common used methods. Although the uniform interpolation method satisfies the error tolerance requirement, the feedrate and acceleration/deceleration are not under control. On the contrary, the constant speed interpolation focuses on maintaining a constant feedrate, but the error tolerance control is not considered in the interpolation scheme. It is not surprising that the chord error is beyond the allowed machining requirement.
236
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
Parametric curve
Offline preprocessing 1. Near arc length parameterization; 2. Determine maximum and minimum allowable feedrate, allowable acceleration/ deceleration; 3. Derive the curvature radius curve and its maximum allowable slope; 4. Determine corners and key regions on the given toolpath;
Guide curve generation 1. Curvature radius curve modification including corner and key region processing; 2. Curvature radius curve reconstruction;
Feedrate schedule Tool motions
Path geometry and sampling time parameters
Recently, some improved parametric interpolation methods have been developed. To achieve the highest quality parts, Fleisig and Spence [3] proposed a constant feed and reduced angular acceleration interpolation algorithm for multi-axis machining. The position and orientation splines produced by the algorithm are C2 continuous and independent of machine tool kinematics. Using a circular approximation method, Yeh and Hsu [4] provided an adaptive-feed interpolation algorithm for parametric curves with confined chord errors. To produce the smooth kinematic profiles for parametric curves, jerk limited trajectories have also been developed by means of lookahead scheme [5–7]. Farouki suggested variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-Hodograph curves [8]. Later, a time-dependent feedrate schedule, which is especially useful in accommodating the stringent acceleration/deceleration requirements of high-speed machining applications, is also realized by real-time CNC interpolator algorithms for PH curves [9]. Bahr et al developed a computational method for continuously and optimally varying feedrate and spindle speed during CNC machining of NURBS curves [10]. Through changing the feedrate continuously at arbitrary positions according to the curvature of curve, a curvaturecompensated feedrate scheme is proposed for the purpose of keeping a constant cutting load [11]. However, the interpolator does not take the machine’s acceleration/ deceleration capabilities into account. In this regard, the interpolator might increase/decrease feedrate beyond the machine’s capability so that it will result in deteriorated accuracy and excessive jerks. In addition, based on offline detection of feedrate sensitive corners, a speed-error controlled interpolator is provided in [12]. The interpolator not only confines the chord error within a specified tolerance, but also controls the speed and acceleration/ deceleration of machining during the interpolation processes. However, when the cutter approaches the corners of toolpaths, optimal and reasonable deceleration start points are not very obvious. Actually, they are often handled by simply presetting a conservative deceleration distance or numbers of interpolation points in advance. In general, NURBS interpolator has gained wide applications in NC machining and the methods that automatically adjust feedrate according to the curvature property of the curvilinear path have caught researchers’ attention. Nevertheless the existing methods have not simultaneously considered the chord error, feedrate and acceleration/deceleration. In addition, for dealing with corners and the feedrate sensitive region, a look-ahead scheme is often used in the existing methods to avoid the change of the acceleration out of the machine ability. To a great extent it increases the complexity of computation, and also concerns how to determine the deceleration start point so as to get a maximum machining efficiency. Based on the previous researches, this paper presents a new interpolation algorithm with confined chord errors and smooth kinematic
Parametric interpolator
Fig. 1. The developed parametric interpolation scheme.
profiles. The proposed method is illustrated in Fig. 1. Through the offline detection of corners and key regions, this approach schedules feedrate based on a guide curve without using a look-ahead facility. Also in theory it is shown that the proposed method ensures the maximized machining efficiency while simultaneously constraining chord error and feed acceleration.
2. Development of the interpolation scheme This section firstly explains the relationship of the chord error and feedrate, the dynamic characteristics of a milling cuter and the definitions of corners and key regions. Secondly, the modification curve formula is discussed, followed by the curvature radius curve modification and reconstruction. Finally, this section presents guide curve based parametric interpolation method to gain smooth machine dynamics with constrained chord error. 2.1. Chord error and feedrate Currently, a common used method to calculate the chord error shown in Fig. 2 can be expressed as follows [13] Parametric curve
C(ui)
Chord error
Tool path C(ui-1)
C(ui-1) Fig. 2. The chord error of parametric curve.
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
d i Z ri K
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2i K ðLi =2Þ2
(1)
where rZ1/Ki, Li Z jjPðuiC1 ÞK Pðui Þjj, Ki and ri are the curvature and the radius of curvature at the parameter value ui, respectively. When the cutter moves with the same velocity through the entire length Li, the velocity is expressed as v Z Li =Ts
(2)
where Ts is sampling time. Substituting Eq. (2) into Eq. (1), then we have [4] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r2 K ðr K dÞ2 vZ (3) Ts In precision machining, it is unexpected that extremely large curvature exists in the toolpath planning, in other words, d/r, then the above equation can be approximately given as pffiffiffiffiffi 2 2d pffiffiffi r (4) vZ Ts It shows that the feedrate with confined chord error has linear approximation relationship with the square root of the curvature radius of paths. If the square root of the curvature gets small, the feedrate should reduce correspondingly in order to keep the chord error constant. For simplification, we call the square root of the curvature radius as the reference variable. 2.2. Corners and key regions Any discontinuities in tangency or curvature will disturb the behaviour of the machine tool during the toolpath follow-up [14]. At these corners, the cutter has to pass through at reduced speed or near to rest in order to confine the contour error and reduce the inertia. Apart from corners, there exist some key regions where the curvature changes dramatically in the curvilinear path. Since the path curvature has natural relationship with the acceleration/deceleration properties of the cutter, some curve segment whose curvature changes dramatically also easily lead to the dramatic change of acceleration/deceleration to some extent. For achieving feedrate schedule with good dynamics we need pay more attention to these corners and key regions. To detect key regions on the given toolpath, we assume that the toolpath is represented by NURBS with a near arc length parameterization. The near arc length parameterization can be obtained by various methods. A simple method is to make a reparameterization based on chord parameterization. Thus, the detection of key regions can be performed according to the relationships between geometric properties and kinematic properties of the curvature radius curve. Assuming that the cutter continuously increases feedrate from vi to viC1 with maximum allowed acceleration amax,
then based on the kinematics equation we have 8 < viC1 K vi Z amax Dt : v Dt C 1 a Dt2 Z L i i 2 max
237
(5)
where Li is the distance so that the current feedrate can be increased up to the desired feedrate with the acceleration amax. Since near arc length parameterization is used here, the distance has an approximated linear relationship with the curve parameter, namely Li Z lðuiC1 K ui Þ. Among the formula, l is a constant, ui and uiC1 are the parameters corresponding to the start point and the end point of this acceleration process. If the parameter u is normalised then the constant l will be equal to the length of the toolpath. Then from the Eq. (5), we have lðuiC1 K ui Þ Z
v2iC1 K v2i 2amax
(6)
The above equation can be further arranged as viC1 K vi 2amax l Z viC1 C vi uiC1 K ui
(7)
When the parameter ui is indefinitely close to uiC1, the feedrate vi also indefinitely approximates viC1. Thus, let the slope of reference variable curve at the parameter ui be zmax. Then we have zmax Z
amax l amax lTs2 Z pffiffiffi vi 8d r
(8)
It shows that the maximum allowed slope changes with the variation of reference variable. If the real slope at some point in reference variable curve is larger than its maximum allowed slope, the acceleration will be out of maximum allowed acceleration when feeding with a constrained chord error. Similarly, for the curvature radius curve of the given path, we have lðuiC1 K ui Þ Z
4dðr1 K r2 Þ amax Ts2
(9)
Then the maximum allowed slope can be expressed as follows zmax Z
r1 K r2 a T 2l Z max s 4d ui K u2
(10)
From the above equation we can see that the maximum allowed slope is a constant. Hence, if the maximum allowed acceleration amax, the sampling time Ts and the chord error d are given in advance, the correspondent maximum allowed slope at this point also is determined. Here, the maximum allowed slope is called reference slope. If the real slope at some point on the given paths is larger than the reference slope, the point is viewed as a key point. Correspondingly, the curve segment is called key region if any point in this curve segment is a key point.
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
2.3. Guide curve based interpolation From the analysis of kinematic and geometric properties, it shows that the geometrical invariants especially curvature have obvious effect on the machine dynamics. In the process of scheduling feedrate, it is very important to consider the geometrical properties such as corners, key regions and curvature variations of the path in advance. In this case, a guide curve based parametric interpolation method is proposed. Once the offline detection of corners and key regions is completed, a curvature radius modification operation is then performed. Consequently, interpolation is completed according to the reconstructed smooth curvature radius curve of the given toolpath. 2.3.1. Modification curve function Due to the corners and key regions on the given toolpath, the original curvature radius curve or reference variable curve is not suitable to schedule the feedrate. Some modification operation should be performed first. Otherwise, the feed acceleration at corners and key regions will be beyond the machine driver’s capacity. So an optimum modification curve has to be used instead of the original curve segment. Without losing generosity, let the start point to modify the original curve be P0(u0) and the feedrate associated with this point is v0. From the above kinematics formula, the modification curve for constant deceleration process can be given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vi Z v20 K 2amax lju0 K ui j (11) where vi is the feedrate associated with the toolpath parameter ui. Similarly, for a constant acceleration process the modification curve can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (12) vi Z v20 C 2amax lju0 K ui j When performing the operation of modifying the reference variable curve, the following equation can be used pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Ts v20 C 2amax lju0 K ui j pffiffiffiffiffi rZ (13) 2 2d It can be seen that the optimum modification curve instead of the key region of the original reference variable curve is a non-linear segment in order to satisfy the acceleration limit. For a curvature radius curve, the above equation can be further arranged as follows 8 T2 > > < r Z Kzmax ui C s ðv20 C 2amax lu0 Þ; if u0 O ui 8d (14) 2 > T > s : p Z zmax ui C ðv20 K 2amax lu0 Þ; if u0 ! ui 8d The above equation shows that the optimum modification curve is a linear segment when a key region of the curvature radius curve is substituted to ensure the acceleration requirement.
2.3.2. Guide curve generation As aforementioned, the path curvature radius is closely related to the chord error and acceleration/deceleration. It is very important to adaptively varies feedrate with the variation of path curvature and simultaneously considerate the influence of the corners and key regions. In theory, the reference variable curve and curvature radius curve can both be selected as an initial guide curve to schedule feedrate. Here, the curvature radius curve is chosen for reducing computation complexity. Given the NURBS parametric equation of toolpaths, a curve between the curvature radius and the parametric variable of paths can be derived as shown in Fig. 3. On this curve, there exist corners and some segments with small and large variation of curvature radius. Also, it has proved that the variation of curvature radius implies the acceleration/deceleration variation. Hence, a dramatic variation of curvature radius will make the acceleration/deceleration out of the machine capacity. So some modifications must be made to deal with the cases of corners and key regions. Additively, with a consideration of the machining efficiency, a commanded feedrate and a minimum allowed feedrate are often given in advance in real NC machining. Correspondingly, the curvature radius has to be fixed within a definite range to match the linear relationship between the reference variable and the feedrate. In fact, if the curvature radius is larger than a given rmax in some curve segments, in theory it is obvious that a larger feedrate than the commanded feedrate can be used while maintaining the chord error under control. Thus, for simplification, as shown in Fig. 3 such a curvature radius in these curve segments can be adjusted as the same value rmax. Similarly, the desired feedrate will be below the minimum allowed feedrate if the curvature radius is smaller than rmin. In this case, the curvature radius that is smaller than rmin can be replaced by the same value rmin. That is ( ri Z rmax if ri R rmax (15) ri Z rmin if ri % rmin Further, there still exists the problem of dramatic variation of curvature radius. To deal with this situation, we assume that the maximum allowed variation of curvature radius is zmax. Thus, in each key region with dramatic variation of curvature radius, a maximum curvature radius point can be found. As shown in Fig. 3, the maximum point must be the beginning point or the end point of the key Curvature radius
238
B pmin
A pmin ua ub
uc
parameter
Fig. 3. Modification of a curvature radius curve.
region. Then, regarding the point as the initial point, a straight line whose slope is given according to the maximum allowed variation rate z can be drawn. If the variation of curvature radius is positive in that key region, the slope of the line is zmax. Otherwise, the slope is Kzmax. The intersectional point of the line and the curvature radius curve r(u) must satisfy the following equation dðuÞ Z zmax ðu K umax Þ K rðuÞ C Pmax Z 0
ukC1 Z uk K dðuk Þ=d 0 ðuk Þ;
d 0 ðuÞ Z zmax K r 0 ðuÞ
parameter Fig. 4. Reconstructing curvature radius curve.
(16)
Finally the intersectional point can be solved by an iterative algorithm
( Ni;0 ðuÞ Z
(17)
Actually, the slope of the line implies the information of the maximum allowed acceleration/deceleration. In other words, to some extent the acceleration/deceleration will not be out of the machine capacity if the cutter moves according to the given linear segment instead of the original curve of curvature radius in that key region. The parameter range of the curve segment that needs to be modified can be given according to the initial point and the intersectional point between the line and the original curvature radius curve. For example, as shown in Fig. 3, the intersectional point is A, and its correspondent parameter is ua. Thus, the original curvature radius curve segment within the parameter range [ua,ub] can be substituted by the line segment AB. Apart from key regions, corners on the curve also need to be further considered. Due to the discontinuities in curvature at those corners on the path, the minimum allowed feedrate is often used when the cutter approaches the corners. Corresponding to the preset minimum allowed feedrate, at each corner the curvature radius is selected as rmin, and then two lines whose slopes are Gzmax can be drawn as shown in Fig. 3. Similar to the method used to deal with key regions, the original curve segment between two intersectional points can be substituted by the above two line segments. Through the above operation, a modified curvature radius curve can be derived. However, such a curve maybe is not smooth and sometimes can not be used as a final guide curve to schedule feedrate, so if necessary the modified curve can be resampled to construct a smooth 1D NURBS curve r(u). Currently, there are various methods to fit NURBS curve to the given sampled points, and details on NURBS are clearly described in [15,16]. The most important thing we should notice here in the operation is that some modified key points must be included in the set of sampled points to ensure accuracy. As a result, the reconstructed curvature radius curve shown in Fig. 4 can be expressed by a NURBS curve as follows Pm wi bi Ni;k ðuÞ rðuÞ Z PiZ0 (18) m iZ0 wi Ni;k ðuÞ where wi is weight factors, bi is 1D control points, and mC1 is the number of control points. Basis function Ni,k(t) is given by the recursive formula
239
Curvature radius
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
Ni;k ðuÞ Z
1
mi % u% miC1
0
otherwise
(19)
t K mi N ðuÞ miCkK1 K mi i;kK1 C
miC1 K t N ðuÞ miCk K miC1 iC1;kK1
(20)
where mi is knot vector and u is the parameter value. 2.3.3. Parametric interpolator For a given curvilinear path expressed by a NURBS curve, the first-order approximation interpolation algorithm is adequately applied to generate commands in general motion systems. Suppose C(u) is the parametric curve function and the time function u is the curve parameter with uðti ÞZ ui and uðtiC1 ÞZ uiC1 . By using Taylor’s expansion of the parameter u with respect to time t to obtain the first order approximation interpolation algorithm, the approximation up to the first derivative is [16] uiC1 Z ui C
du j ðt K ti Þ C H:O:T: dt tZti iC1
(21)
The curve speed v(ui) can be given as vðui Þ Z k
dCðuÞ dCðuÞ du ktZti Z k kuZui $ jtZti dt du dt
(22)
and the first derivative of u with respect to t is du vðu Þ j Z dCðuÞ i dt tZti k dt kuZui
(23)
Further Eq. (23) can be rearranged as follows uiC1 Z ui C
k
vðui ÞTs dCðuÞ du kuZui
(24)
where T is sampling time in interpolation, that is T Z tiC1 K ti . v(ui) can be the feedrate command or any desired speed in a general machining process. Although the above first-order approximation interpolation algorithm makes the curve speed almost equal to the desired value v(ui), the chord error and machining dynamics may become unacceptable if an improper curve speed v(ui) is given. Since the proposed method adaptively changes feedrate according to the guide curve r(u), the velocity v(ui) at some point C(ui) on the given toolpath can be simply given as follows
240
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi vðui Þ Z 2 2d rðui Þ=Ts
method. The maximum chord error of the proposed method is 0.001026mm. Compared with the maximum chord error 0.004708 mm of the constant feedrate method, the error reduces 78.21%. Meanwhile, the chord error of the constant feedrate method at the corner point is 0.1990 mm, but the chord error of the proposed method at this point is only 0.0115 mm. Results in terms of chord errors and machine dynamics show that the proposed algorithm performs well. Simulation also shows that the chord error of constant feedrate interpolation fluctuates without restraint. Due to the reduced speed at high curvature point on the toolpath, the proposed method has smaller inertia than the existing method. In addition, using the proposed method the machining time is 261 sampling times. Compared with the 224 sampling times of the constant feedrate method, although the machining time increases a little because of the reduced feedrate at corners and key regions, better machining precision and machine dynamics can be achieved even under high speed machining. Machining of an oval shaped toolpath segment was simulated for feed rates in the range of 500–1000 mm/s. Fig. 6(b) shows the feedrate variation with the proposed method. The chord errors and acceleration/deceleration profile are also given in Fig. 6(c) and (d). Results show that the feedrate is reduced at corners and key regions. Meanwhile the chord errors are also within the preset tolerance. Need to mention that, to some extent the chord errors can exceed the set limit slightly because of the circular approximation in scheduling the feedrate and the Taylor approximation in computing the next parameter value. Results also show that the proposed guide curve algorithm achieves higher accuracy while varying feedrate within a constrained acceleration or deceleration rate as seen in Table 1. Although there are some dramatic curvature regions in the original toolpath segment, the maximum acceleration/deceleration is still limited to 20,000 mm/s2 and is not out of the preset rate through modifying and reconstructing the curvature radius curve. Additively, the number and the curve length of key regions increase with
(25)
Using such a guide curve based parametric interpolation scheme, the chord error and acceleration/deceleration are simultaneously considered. Meanwhile, if the maximum allowed curvature radius and the minimum allowed curvature radius are not specified, they can be determined according to Eq. (4).
3. Simulations and results Simulation was performed using the constant feedrate method and the proposed parametric interpolation method. All simulation programs were written in Visual CCC and executed on a personal computer. As shown in Figs. 5(a) and 6(a), the toolpath segments used for test are a curvilinear path with a corner and an oval shaped curve. The maximum allowed feedrate are given as 1000 mm/s and the allowed feedrate at corner is 100 mm/s. The chord error tolerance is set at 0.001 mm and the sampling time is set at 0.001 s. The acceleration and deceleration limits were set at 30,000 and 20,000 mm/s2, respectively. According to Eq. (4), the maximum allowed reference variable value pffiffiffiffiffiffiffiffi and allowed reference variable value at corner are 125 and pffiffiffi 5=2, respectively. The maximum allowed slope of the curvature radius curve is calculated according to the formula (10). In the process of reconstructing of curvature radius curve, the curve was modelled using an endpoint interpolating B-spline. For simplification, the initial acceleration and the end deceleration are not considered in this paper. Machining of a curvilinear toolpath segment with a corner was simulated for feed rates in the range of 100– 1000 mm/s. The maximum allowed slope of curvature radius curve is determined and then this profile is adjusted to satisfy the acceleration and deceleration limits. To compare with the conventional method, the constant feedrate method is also tested in this paper. Fig. 5(b) shows the chord error curves of the proposed method and the constant feedrate (a) 120
(b)
adaptive feedrate
0.2
0.8
0.006 0.005
chord error (mm)
y axis (mm)
100
constant feedrate
80 60 40
0.004 0.003 0.002 0.001
20
0
0 0
20
40
60 80 x axis (mm)
100
120
0
0.4 0.6 curve parameter u
Fig. 5. Tool path segment (a) and chord errors of the two interpolation methods (d).
1
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
(a)
(b) 500
241
1200 1000
feedrate (mm/s)
y axis (mm)
400 300 200
800 600 400
100 0
200 0 0
100
200
300
400
500
0
600
0.2
(d)
0.6
0.8
1
0.8
1
0.0012 0.001
0.2
0.4
0.6
0.8
1
chord error (mm)
acceleration (mm/s2)
(c)
25000 20000 15000 10000 5000 0 –5000 0 –10000 –15000 –20000 –25000
0.4
curve parameter u
x axis (mm)
0.0008 0.0006 0.0004 0.0002 0 0
curve parameter u
0.2
0.4
0.6
curve parameter u Fig. 6. Tool path segment (a); feedrate curve (b) acceleration curve (c) and chord error (d).
decreasing maximum deceleration.
allowed
acceleration
and
4. Conclusions Parametric interpolation has good advantages over linear interpolation. It is very important to get superior machine dynamics especially in high speed machining. In this case, the chord error, the feedrate and the acceleration should be simultaneously better controlled for machining high quality parts. The main contributions of our work lie in establishing the relations of geometrical properties and kinematics properties and proposing a new guide curve based parametric interpolation approach. The new parametric interpolation scheme is based on offline detection and modification of corners and key regions in a given curvature radius curve. Using this approach, the feedrate naturally varies with a guide curve associated with the path curvature. Thus, chord errors are confined to predetermined tolerances.
At the same time, the acceleration/deceleration is also under control and is not out of the machine capabilities so that there are no overshoot and undershoot errors. In the developed guide curve based interpolator, it is unnecessary to determine the starting parameter value for deceleration and acceleration. The given relation formula between the maximum allowed slope of the guide curve and the maximum allowed acceleration of the cutter has made it possible to incorporate corners and key regions processing into the system of feedrate schedule. Thus, the NURBS interpolator can be automated performed according to the given NURBS path and the guide curve. Furthermore, the toolpath curve and the guide curve can also be merged into a 4D NURBS curve. The proposed interpolation scheme greatly reduces the complexity of interpolation computation and has simultaneously satisfied the requirement of chord error and machining dynamics without using a look-ahead strategy. Meanwhile, in theory it is also ensured that the machining efficiency is exploited to the utmost extent with the constraints of tolerance and machining dynamics.
Table 1 Simulation results for tool path segments Simulation
Example 1 Example 2
Interpolation method
Constant feedrate method The proposed method The proposed method
Feedrate (mm/s)
Max chord error (mm)
Max
Min
Curve
Corner
700 1000 1000
100 554.72
0.0047 0.0010 0.001063
0.1990 0.0115
Max aec/dec (mm/s2)
Machining time (s)
0 30,000 20,000
0.224 0.261 1.417
242
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
Acknowledgements This research is supported by the National Science Foundation of China under grant number 50405044. The authors would like to thank Professor Liu Jian for the valuable discussions and suggestions.
References [1] S. Bedi, I. Ali, N. Quan, Advanced interpolation techniques for CNC machines, ASME Journal of Engineering for Industry 115 (1993) 329–336. [2] M. Shpitalni, Y. Koren, C.C. Lo, Reat-time curve interpolators, Computer-Aided Design 26 (1994) 832–838. [3] R.V. Fleisig, A.D. Spence, A constant feed and reduced angular acceleration interpolation algorithm for multi-axis machining, Computer Aided Design 33 (2001) 1–15. [4] S.-S. Yeh, R.-L. Hsu, Adaptive-feedrate interpolation for parametric curves with a confined chord error, Computer Aided Design 34 (2002) 229–237. [5] K. Erkorkmaz, Y. Altintas, High speed CNC system design. Part I: Jerk limited trajectory generation and quintic spline interpolation, International Journal of Machine Tools and Manufacture 41 (2001) 1323–1345. [6] S.-H. Nam, M.-Y. Yang, A study on a generalized parametric interpolator with real-time jerk-limited acceleration, Computer Aided Design 36 (2004) 27–36.
[7] X. Liu, F. Ahmad, K. Yamazaki, et al., Adaptive interpolation scheme for NURBS curves with the integration of machining dynamics, International Journal of Machine Tools and Manufacture 45 (2005) 433–444. [8] R.T. Farouki, J. Manjunathaiah, D. Nicholas, G.-F. Yuan, et al., Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean–Hodograph curves, Computer Aided Design 30 (1996) 631–640. [9] Y.F. Tsai, R.T. Farouki, B. Feldman, Performance analysis of CNC interpolators for time-dependent federates along PH curves, Computer Aided Geometric Design 18 (2001) 245–265. [10] K.K. Krishnan, J. Kappen, B. Bahr, Calculation of variable federate and spindle speed for NURBS based CNC machining, Transactions of NAMRI/SME 24 (2001) 429–435. [11] M. Tikhon, T.J. Ko, S.H. Lee, et al., NURBS interpolator for constant material removal rate in open NC machine tools, International Journal of Machine Tools and Manufacture 44 (2004) 237–245. [12] T. Yong, R. Narayanaswami, A parametric interpolator with confined chord errors, acceleration and deceleration for NC machining, Computer Aided Design 35 (2003) 1249–1259. [13] J.J. Stoker, Differential Geometry, Wiley-Interscience, New York, 1969. [14] V. Pateloup, E. Duc, P. Ray, Corner optimization for pocket machining, International Journal of Machine Tools and Manufacture 44 (2004) 1343–1353. [15] L. Piegl, W. Tiller, The NURBS Book, second ed., Springer, Berlin, 1997. [16] R.T. Farouki, Y.F. Tsai, Exact Taylor series coefficient for variablefeedrate CNC curve interpolators, Computer Aided Design 33 (2001) 155–165.
Guide curve based interpolation scheme of parametric curves for precision CNC machining Yuwen Suna,*, Jun Wangb, Dongming Guoa b
a School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney 2052, Australia
Received 17 May 2005; accepted 26 May 2005 Available online 25 July 2005
Abstract Real-time parametric interpolation has played a key role in the computer control of machine tools. To achieve highest quality parts, generated trajectories not only describe the desired toolpath accurately, but also have smooth dynamics profiles. This paper presents a novel parametric interpolator based on guide curve. The relationships between geometric properties and kinematic properties are firstly discussed. Then, with a consideration of the important effect of the curvature of curvilinear path on the machining dynamics, a corresponding formula, which describes the relation of the maximum allowed feed acceleration/deceleration and the maximum allowed rate of change of curvature radius of paths, is built. Thus, based on a near arc parameterization and through modifying the curvature radius curve to deal with corners, key regions and other cases, adaptive feedrate schedule is completed according to the reconstructed smooth curvature radius curve. Consequently, confined chord errors, corners on the path and the acceleration/deceleration capabilities of the machine tool are simultaneously considered and incorporated into the guide curve based parametric interpolation system without using look-ahead scheme. Simulation results indicate the feasibility and precision of the proposed interpolation method. q 2005 Elsevier Ltd. All rights reserved. Keywords: Machining dynamics; CAD/CAM; Feedrate schedule; NC machining; Toolpath
1. Introduction In high precision and high speed NC machining, kinematic and geometric properties of the moving cutter have important effects on the efficiency and quality of the machined parts. However, machining dynamics is often ignored in current toolpath generation methods and interpolator schemes. In this case, any discontinuities of the path can result in undesirable high frequency harmonics in the reference trajectory, which is extremely harm to the machine tool, the machined surface quality and the servo control system. Meanwhile, conventional CNC machines often support only straight line and circular interpolator. Due to the difficulty of dividing a line segment into an integer multiple of the product of the commanded feedrate and the sampling time, the line * Corresponding author. E-mail address: [email protected] (Y. Sun).
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.05.024
interpolation inevitably exists undesirable weaknesses such as feedrate fluctuation and excessive jerk in the process of machining a curve. Overall, dynamics performance issues become peculiarly important in NC machining especially in high-speed machining. For high-speed and high-quality machining parts, we have to consider the machine dynamics while controlling chord error under certain tolerance. Otherwise, due to the limitation of the machine tool drivers’ ability, failure in maintaining the commanded feedrate will in turn lead to the tool chatter of breakage. Consequently, parametric interpolators are being developed to eliminate the disadvantages of linear/circular interpolators. Currently, uniform interpolation [1] and constant speed interpolation [2] are the most common used methods. Although the uniform interpolation method satisfies the error tolerance requirement, the feedrate and acceleration/deceleration are not under control. On the contrary, the constant speed interpolation focuses on maintaining a constant feedrate, but the error tolerance control is not considered in the interpolation scheme. It is not surprising that the chord error is beyond the allowed machining requirement.
236
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
Parametric curve
Offline preprocessing 1. Near arc length parameterization; 2. Determine maximum and minimum allowable feedrate, allowable acceleration/ deceleration; 3. Derive the curvature radius curve and its maximum allowable slope; 4. Determine corners and key regions on the given toolpath;
Guide curve generation 1. Curvature radius curve modification including corner and key region processing; 2. Curvature radius curve reconstruction;
Feedrate schedule Tool motions
Path geometry and sampling time parameters
Recently, some improved parametric interpolation methods have been developed. To achieve the highest quality parts, Fleisig and Spence [3] proposed a constant feed and reduced angular acceleration interpolation algorithm for multi-axis machining. The position and orientation splines produced by the algorithm are C2 continuous and independent of machine tool kinematics. Using a circular approximation method, Yeh and Hsu [4] provided an adaptive-feed interpolation algorithm for parametric curves with confined chord errors. To produce the smooth kinematic profiles for parametric curves, jerk limited trajectories have also been developed by means of lookahead scheme [5–7]. Farouki suggested variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-Hodograph curves [8]. Later, a time-dependent feedrate schedule, which is especially useful in accommodating the stringent acceleration/deceleration requirements of high-speed machining applications, is also realized by real-time CNC interpolator algorithms for PH curves [9]. Bahr et al developed a computational method for continuously and optimally varying feedrate and spindle speed during CNC machining of NURBS curves [10]. Through changing the feedrate continuously at arbitrary positions according to the curvature of curve, a curvaturecompensated feedrate scheme is proposed for the purpose of keeping a constant cutting load [11]. However, the interpolator does not take the machine’s acceleration/ deceleration capabilities into account. In this regard, the interpolator might increase/decrease feedrate beyond the machine’s capability so that it will result in deteriorated accuracy and excessive jerks. In addition, based on offline detection of feedrate sensitive corners, a speed-error controlled interpolator is provided in [12]. The interpolator not only confines the chord error within a specified tolerance, but also controls the speed and acceleration/ deceleration of machining during the interpolation processes. However, when the cutter approaches the corners of toolpaths, optimal and reasonable deceleration start points are not very obvious. Actually, they are often handled by simply presetting a conservative deceleration distance or numbers of interpolation points in advance. In general, NURBS interpolator has gained wide applications in NC machining and the methods that automatically adjust feedrate according to the curvature property of the curvilinear path have caught researchers’ attention. Nevertheless the existing methods have not simultaneously considered the chord error, feedrate and acceleration/deceleration. In addition, for dealing with corners and the feedrate sensitive region, a look-ahead scheme is often used in the existing methods to avoid the change of the acceleration out of the machine ability. To a great extent it increases the complexity of computation, and also concerns how to determine the deceleration start point so as to get a maximum machining efficiency. Based on the previous researches, this paper presents a new interpolation algorithm with confined chord errors and smooth kinematic
Parametric interpolator
Fig. 1. The developed parametric interpolation scheme.
profiles. The proposed method is illustrated in Fig. 1. Through the offline detection of corners and key regions, this approach schedules feedrate based on a guide curve without using a look-ahead facility. Also in theory it is shown that the proposed method ensures the maximized machining efficiency while simultaneously constraining chord error and feed acceleration.
2. Development of the interpolation scheme This section firstly explains the relationship of the chord error and feedrate, the dynamic characteristics of a milling cuter and the definitions of corners and key regions. Secondly, the modification curve formula is discussed, followed by the curvature radius curve modification and reconstruction. Finally, this section presents guide curve based parametric interpolation method to gain smooth machine dynamics with constrained chord error. 2.1. Chord error and feedrate Currently, a common used method to calculate the chord error shown in Fig. 2 can be expressed as follows [13] Parametric curve
C(ui)
Chord error
Tool path C(ui-1)
C(ui-1) Fig. 2. The chord error of parametric curve.
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
d i Z ri K
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2i K ðLi =2Þ2
(1)
where rZ1/Ki, Li Z jjPðuiC1 ÞK Pðui Þjj, Ki and ri are the curvature and the radius of curvature at the parameter value ui, respectively. When the cutter moves with the same velocity through the entire length Li, the velocity is expressed as v Z Li =Ts
(2)
where Ts is sampling time. Substituting Eq. (2) into Eq. (1), then we have [4] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r2 K ðr K dÞ2 vZ (3) Ts In precision machining, it is unexpected that extremely large curvature exists in the toolpath planning, in other words, d/r, then the above equation can be approximately given as pffiffiffiffiffi 2 2d pffiffiffi r (4) vZ Ts It shows that the feedrate with confined chord error has linear approximation relationship with the square root of the curvature radius of paths. If the square root of the curvature gets small, the feedrate should reduce correspondingly in order to keep the chord error constant. For simplification, we call the square root of the curvature radius as the reference variable. 2.2. Corners and key regions Any discontinuities in tangency or curvature will disturb the behaviour of the machine tool during the toolpath follow-up [14]. At these corners, the cutter has to pass through at reduced speed or near to rest in order to confine the contour error and reduce the inertia. Apart from corners, there exist some key regions where the curvature changes dramatically in the curvilinear path. Since the path curvature has natural relationship with the acceleration/deceleration properties of the cutter, some curve segment whose curvature changes dramatically also easily lead to the dramatic change of acceleration/deceleration to some extent. For achieving feedrate schedule with good dynamics we need pay more attention to these corners and key regions. To detect key regions on the given toolpath, we assume that the toolpath is represented by NURBS with a near arc length parameterization. The near arc length parameterization can be obtained by various methods. A simple method is to make a reparameterization based on chord parameterization. Thus, the detection of key regions can be performed according to the relationships between geometric properties and kinematic properties of the curvature radius curve. Assuming that the cutter continuously increases feedrate from vi to viC1 with maximum allowed acceleration amax,
then based on the kinematics equation we have 8 < viC1 K vi Z amax Dt : v Dt C 1 a Dt2 Z L i i 2 max
237
(5)
where Li is the distance so that the current feedrate can be increased up to the desired feedrate with the acceleration amax. Since near arc length parameterization is used here, the distance has an approximated linear relationship with the curve parameter, namely Li Z lðuiC1 K ui Þ. Among the formula, l is a constant, ui and uiC1 are the parameters corresponding to the start point and the end point of this acceleration process. If the parameter u is normalised then the constant l will be equal to the length of the toolpath. Then from the Eq. (5), we have lðuiC1 K ui Þ Z
v2iC1 K v2i 2amax
(6)
The above equation can be further arranged as viC1 K vi 2amax l Z viC1 C vi uiC1 K ui
(7)
When the parameter ui is indefinitely close to uiC1, the feedrate vi also indefinitely approximates viC1. Thus, let the slope of reference variable curve at the parameter ui be zmax. Then we have zmax Z
amax l amax lTs2 Z pffiffiffi vi 8d r
(8)
It shows that the maximum allowed slope changes with the variation of reference variable. If the real slope at some point in reference variable curve is larger than its maximum allowed slope, the acceleration will be out of maximum allowed acceleration when feeding with a constrained chord error. Similarly, for the curvature radius curve of the given path, we have lðuiC1 K ui Þ Z
4dðr1 K r2 Þ amax Ts2
(9)
Then the maximum allowed slope can be expressed as follows zmax Z
r1 K r2 a T 2l Z max s 4d ui K u2
(10)
From the above equation we can see that the maximum allowed slope is a constant. Hence, if the maximum allowed acceleration amax, the sampling time Ts and the chord error d are given in advance, the correspondent maximum allowed slope at this point also is determined. Here, the maximum allowed slope is called reference slope. If the real slope at some point on the given paths is larger than the reference slope, the point is viewed as a key point. Correspondingly, the curve segment is called key region if any point in this curve segment is a key point.
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
2.3. Guide curve based interpolation From the analysis of kinematic and geometric properties, it shows that the geometrical invariants especially curvature have obvious effect on the machine dynamics. In the process of scheduling feedrate, it is very important to consider the geometrical properties such as corners, key regions and curvature variations of the path in advance. In this case, a guide curve based parametric interpolation method is proposed. Once the offline detection of corners and key regions is completed, a curvature radius modification operation is then performed. Consequently, interpolation is completed according to the reconstructed smooth curvature radius curve of the given toolpath. 2.3.1. Modification curve function Due to the corners and key regions on the given toolpath, the original curvature radius curve or reference variable curve is not suitable to schedule the feedrate. Some modification operation should be performed first. Otherwise, the feed acceleration at corners and key regions will be beyond the machine driver’s capacity. So an optimum modification curve has to be used instead of the original curve segment. Without losing generosity, let the start point to modify the original curve be P0(u0) and the feedrate associated with this point is v0. From the above kinematics formula, the modification curve for constant deceleration process can be given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vi Z v20 K 2amax lju0 K ui j (11) where vi is the feedrate associated with the toolpath parameter ui. Similarly, for a constant acceleration process the modification curve can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (12) vi Z v20 C 2amax lju0 K ui j When performing the operation of modifying the reference variable curve, the following equation can be used pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Ts v20 C 2amax lju0 K ui j pffiffiffiffiffi rZ (13) 2 2d It can be seen that the optimum modification curve instead of the key region of the original reference variable curve is a non-linear segment in order to satisfy the acceleration limit. For a curvature radius curve, the above equation can be further arranged as follows 8 T2 > > < r Z Kzmax ui C s ðv20 C 2amax lu0 Þ; if u0 O ui 8d (14) 2 > T > s : p Z zmax ui C ðv20 K 2amax lu0 Þ; if u0 ! ui 8d The above equation shows that the optimum modification curve is a linear segment when a key region of the curvature radius curve is substituted to ensure the acceleration requirement.
2.3.2. Guide curve generation As aforementioned, the path curvature radius is closely related to the chord error and acceleration/deceleration. It is very important to adaptively varies feedrate with the variation of path curvature and simultaneously considerate the influence of the corners and key regions. In theory, the reference variable curve and curvature radius curve can both be selected as an initial guide curve to schedule feedrate. Here, the curvature radius curve is chosen for reducing computation complexity. Given the NURBS parametric equation of toolpaths, a curve between the curvature radius and the parametric variable of paths can be derived as shown in Fig. 3. On this curve, there exist corners and some segments with small and large variation of curvature radius. Also, it has proved that the variation of curvature radius implies the acceleration/deceleration variation. Hence, a dramatic variation of curvature radius will make the acceleration/deceleration out of the machine capacity. So some modifications must be made to deal with the cases of corners and key regions. Additively, with a consideration of the machining efficiency, a commanded feedrate and a minimum allowed feedrate are often given in advance in real NC machining. Correspondingly, the curvature radius has to be fixed within a definite range to match the linear relationship between the reference variable and the feedrate. In fact, if the curvature radius is larger than a given rmax in some curve segments, in theory it is obvious that a larger feedrate than the commanded feedrate can be used while maintaining the chord error under control. Thus, for simplification, as shown in Fig. 3 such a curvature radius in these curve segments can be adjusted as the same value rmax. Similarly, the desired feedrate will be below the minimum allowed feedrate if the curvature radius is smaller than rmin. In this case, the curvature radius that is smaller than rmin can be replaced by the same value rmin. That is ( ri Z rmax if ri R rmax (15) ri Z rmin if ri % rmin Further, there still exists the problem of dramatic variation of curvature radius. To deal with this situation, we assume that the maximum allowed variation of curvature radius is zmax. Thus, in each key region with dramatic variation of curvature radius, a maximum curvature radius point can be found. As shown in Fig. 3, the maximum point must be the beginning point or the end point of the key Curvature radius
238
B pmin
A pmin ua ub
uc
parameter
Fig. 3. Modification of a curvature radius curve.
region. Then, regarding the point as the initial point, a straight line whose slope is given according to the maximum allowed variation rate z can be drawn. If the variation of curvature radius is positive in that key region, the slope of the line is zmax. Otherwise, the slope is Kzmax. The intersectional point of the line and the curvature radius curve r(u) must satisfy the following equation dðuÞ Z zmax ðu K umax Þ K rðuÞ C Pmax Z 0
ukC1 Z uk K dðuk Þ=d 0 ðuk Þ;
d 0 ðuÞ Z zmax K r 0 ðuÞ
parameter Fig. 4. Reconstructing curvature radius curve.
(16)
Finally the intersectional point can be solved by an iterative algorithm
( Ni;0 ðuÞ Z
(17)
Actually, the slope of the line implies the information of the maximum allowed acceleration/deceleration. In other words, to some extent the acceleration/deceleration will not be out of the machine capacity if the cutter moves according to the given linear segment instead of the original curve of curvature radius in that key region. The parameter range of the curve segment that needs to be modified can be given according to the initial point and the intersectional point between the line and the original curvature radius curve. For example, as shown in Fig. 3, the intersectional point is A, and its correspondent parameter is ua. Thus, the original curvature radius curve segment within the parameter range [ua,ub] can be substituted by the line segment AB. Apart from key regions, corners on the curve also need to be further considered. Due to the discontinuities in curvature at those corners on the path, the minimum allowed feedrate is often used when the cutter approaches the corners. Corresponding to the preset minimum allowed feedrate, at each corner the curvature radius is selected as rmin, and then two lines whose slopes are Gzmax can be drawn as shown in Fig. 3. Similar to the method used to deal with key regions, the original curve segment between two intersectional points can be substituted by the above two line segments. Through the above operation, a modified curvature radius curve can be derived. However, such a curve maybe is not smooth and sometimes can not be used as a final guide curve to schedule feedrate, so if necessary the modified curve can be resampled to construct a smooth 1D NURBS curve r(u). Currently, there are various methods to fit NURBS curve to the given sampled points, and details on NURBS are clearly described in [15,16]. The most important thing we should notice here in the operation is that some modified key points must be included in the set of sampled points to ensure accuracy. As a result, the reconstructed curvature radius curve shown in Fig. 4 can be expressed by a NURBS curve as follows Pm wi bi Ni;k ðuÞ rðuÞ Z PiZ0 (18) m iZ0 wi Ni;k ðuÞ where wi is weight factors, bi is 1D control points, and mC1 is the number of control points. Basis function Ni,k(t) is given by the recursive formula
239
Curvature radius
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
Ni;k ðuÞ Z
1
mi % u% miC1
0
otherwise
(19)
t K mi N ðuÞ miCkK1 K mi i;kK1 C
miC1 K t N ðuÞ miCk K miC1 iC1;kK1
(20)
where mi is knot vector and u is the parameter value. 2.3.3. Parametric interpolator For a given curvilinear path expressed by a NURBS curve, the first-order approximation interpolation algorithm is adequately applied to generate commands in general motion systems. Suppose C(u) is the parametric curve function and the time function u is the curve parameter with uðti ÞZ ui and uðtiC1 ÞZ uiC1 . By using Taylor’s expansion of the parameter u with respect to time t to obtain the first order approximation interpolation algorithm, the approximation up to the first derivative is [16] uiC1 Z ui C
du j ðt K ti Þ C H:O:T: dt tZti iC1
(21)
The curve speed v(ui) can be given as vðui Þ Z k
dCðuÞ dCðuÞ du ktZti Z k kuZui $ jtZti dt du dt
(22)
and the first derivative of u with respect to t is du vðu Þ j Z dCðuÞ i dt tZti k dt kuZui
(23)
Further Eq. (23) can be rearranged as follows uiC1 Z ui C
k
vðui ÞTs dCðuÞ du kuZui
(24)
where T is sampling time in interpolation, that is T Z tiC1 K ti . v(ui) can be the feedrate command or any desired speed in a general machining process. Although the above first-order approximation interpolation algorithm makes the curve speed almost equal to the desired value v(ui), the chord error and machining dynamics may become unacceptable if an improper curve speed v(ui) is given. Since the proposed method adaptively changes feedrate according to the guide curve r(u), the velocity v(ui) at some point C(ui) on the given toolpath can be simply given as follows
240
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi vðui Þ Z 2 2d rðui Þ=Ts
method. The maximum chord error of the proposed method is 0.001026mm. Compared with the maximum chord error 0.004708 mm of the constant feedrate method, the error reduces 78.21%. Meanwhile, the chord error of the constant feedrate method at the corner point is 0.1990 mm, but the chord error of the proposed method at this point is only 0.0115 mm. Results in terms of chord errors and machine dynamics show that the proposed algorithm performs well. Simulation also shows that the chord error of constant feedrate interpolation fluctuates without restraint. Due to the reduced speed at high curvature point on the toolpath, the proposed method has smaller inertia than the existing method. In addition, using the proposed method the machining time is 261 sampling times. Compared with the 224 sampling times of the constant feedrate method, although the machining time increases a little because of the reduced feedrate at corners and key regions, better machining precision and machine dynamics can be achieved even under high speed machining. Machining of an oval shaped toolpath segment was simulated for feed rates in the range of 500–1000 mm/s. Fig. 6(b) shows the feedrate variation with the proposed method. The chord errors and acceleration/deceleration profile are also given in Fig. 6(c) and (d). Results show that the feedrate is reduced at corners and key regions. Meanwhile the chord errors are also within the preset tolerance. Need to mention that, to some extent the chord errors can exceed the set limit slightly because of the circular approximation in scheduling the feedrate and the Taylor approximation in computing the next parameter value. Results also show that the proposed guide curve algorithm achieves higher accuracy while varying feedrate within a constrained acceleration or deceleration rate as seen in Table 1. Although there are some dramatic curvature regions in the original toolpath segment, the maximum acceleration/deceleration is still limited to 20,000 mm/s2 and is not out of the preset rate through modifying and reconstructing the curvature radius curve. Additively, the number and the curve length of key regions increase with
(25)
Using such a guide curve based parametric interpolation scheme, the chord error and acceleration/deceleration are simultaneously considered. Meanwhile, if the maximum allowed curvature radius and the minimum allowed curvature radius are not specified, they can be determined according to Eq. (4).
3. Simulations and results Simulation was performed using the constant feedrate method and the proposed parametric interpolation method. All simulation programs were written in Visual CCC and executed on a personal computer. As shown in Figs. 5(a) and 6(a), the toolpath segments used for test are a curvilinear path with a corner and an oval shaped curve. The maximum allowed feedrate are given as 1000 mm/s and the allowed feedrate at corner is 100 mm/s. The chord error tolerance is set at 0.001 mm and the sampling time is set at 0.001 s. The acceleration and deceleration limits were set at 30,000 and 20,000 mm/s2, respectively. According to Eq. (4), the maximum allowed reference variable value pffiffiffiffiffiffiffiffi and allowed reference variable value at corner are 125 and pffiffiffi 5=2, respectively. The maximum allowed slope of the curvature radius curve is calculated according to the formula (10). In the process of reconstructing of curvature radius curve, the curve was modelled using an endpoint interpolating B-spline. For simplification, the initial acceleration and the end deceleration are not considered in this paper. Machining of a curvilinear toolpath segment with a corner was simulated for feed rates in the range of 100– 1000 mm/s. The maximum allowed slope of curvature radius curve is determined and then this profile is adjusted to satisfy the acceleration and deceleration limits. To compare with the conventional method, the constant feedrate method is also tested in this paper. Fig. 5(b) shows the chord error curves of the proposed method and the constant feedrate (a) 120
(b)
adaptive feedrate
0.2
0.8
0.006 0.005
chord error (mm)
y axis (mm)
100
constant feedrate
80 60 40
0.004 0.003 0.002 0.001
20
0
0 0
20
40
60 80 x axis (mm)
100
120
0
0.4 0.6 curve parameter u
Fig. 5. Tool path segment (a) and chord errors of the two interpolation methods (d).
1
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
(a)
(b) 500
241
1200 1000
feedrate (mm/s)
y axis (mm)
400 300 200
800 600 400
100 0
200 0 0
100
200
300
400
500
0
600
0.2
(d)
0.6
0.8
1
0.8
1
0.0012 0.001
0.2
0.4
0.6
0.8
1
chord error (mm)
acceleration (mm/s2)
(c)
25000 20000 15000 10000 5000 0 –5000 0 –10000 –15000 –20000 –25000
0.4
curve parameter u
x axis (mm)
0.0008 0.0006 0.0004 0.0002 0 0
curve parameter u
0.2
0.4
0.6
curve parameter u Fig. 6. Tool path segment (a); feedrate curve (b) acceleration curve (c) and chord error (d).
decreasing maximum deceleration.
allowed
acceleration
and
4. Conclusions Parametric interpolation has good advantages over linear interpolation. It is very important to get superior machine dynamics especially in high speed machining. In this case, the chord error, the feedrate and the acceleration should be simultaneously better controlled for machining high quality parts. The main contributions of our work lie in establishing the relations of geometrical properties and kinematics properties and proposing a new guide curve based parametric interpolation approach. The new parametric interpolation scheme is based on offline detection and modification of corners and key regions in a given curvature radius curve. Using this approach, the feedrate naturally varies with a guide curve associated with the path curvature. Thus, chord errors are confined to predetermined tolerances.
At the same time, the acceleration/deceleration is also under control and is not out of the machine capabilities so that there are no overshoot and undershoot errors. In the developed guide curve based interpolator, it is unnecessary to determine the starting parameter value for deceleration and acceleration. The given relation formula between the maximum allowed slope of the guide curve and the maximum allowed acceleration of the cutter has made it possible to incorporate corners and key regions processing into the system of feedrate schedule. Thus, the NURBS interpolator can be automated performed according to the given NURBS path and the guide curve. Furthermore, the toolpath curve and the guide curve can also be merged into a 4D NURBS curve. The proposed interpolation scheme greatly reduces the complexity of interpolation computation and has simultaneously satisfied the requirement of chord error and machining dynamics without using a look-ahead strategy. Meanwhile, in theory it is also ensured that the machining efficiency is exploited to the utmost extent with the constraints of tolerance and machining dynamics.
Table 1 Simulation results for tool path segments Simulation
Example 1 Example 2
Interpolation method
Constant feedrate method The proposed method The proposed method
Feedrate (mm/s)
Max chord error (mm)
Max
Min
Curve
Corner
700 1000 1000
100 554.72
0.0047 0.0010 0.001063
0.1990 0.0115
Max aec/dec (mm/s2)
Machining time (s)
0 30,000 20,000
0.224 0.261 1.417
242
Y. Sun et al. / International Journal of Machine Tools & Manufacture 46 (2006) 235–242
Acknowledgements This research is supported by the National Science Foundation of China under grant number 50405044. The authors would like to thank Professor Liu Jian for the valuable discussions and suggestions.
References [1] S. Bedi, I. Ali, N. Quan, Advanced interpolation techniques for CNC machines, ASME Journal of Engineering for Industry 115 (1993) 329–336. [2] M. Shpitalni, Y. Koren, C.C. Lo, Reat-time curve interpolators, Computer-Aided Design 26 (1994) 832–838. [3] R.V. Fleisig, A.D. Spence, A constant feed and reduced angular acceleration interpolation algorithm for multi-axis machining, Computer Aided Design 33 (2001) 1–15. [4] S.-S. Yeh, R.-L. Hsu, Adaptive-feedrate interpolation for parametric curves with a confined chord error, Computer Aided Design 34 (2002) 229–237. [5] K. Erkorkmaz, Y. Altintas, High speed CNC system design. Part I: Jerk limited trajectory generation and quintic spline interpolation, International Journal of Machine Tools and Manufacture 41 (2001) 1323–1345. [6] S.-H. Nam, M.-Y. Yang, A study on a generalized parametric interpolator with real-time jerk-limited acceleration, Computer Aided Design 36 (2004) 27–36.
[7] X. Liu, F. Ahmad, K. Yamazaki, et al., Adaptive interpolation scheme for NURBS curves with the integration of machining dynamics, International Journal of Machine Tools and Manufacture 45 (2005) 433–444. [8] R.T. Farouki, J. Manjunathaiah, D. Nicholas, G.-F. Yuan, et al., Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean–Hodograph curves, Computer Aided Design 30 (1996) 631–640. [9] Y.F. Tsai, R.T. Farouki, B. Feldman, Performance analysis of CNC interpolators for time-dependent federates along PH curves, Computer Aided Geometric Design 18 (2001) 245–265. [10] K.K. Krishnan, J. Kappen, B. Bahr, Calculation of variable federate and spindle speed for NURBS based CNC machining, Transactions of NAMRI/SME 24 (2001) 429–435. [11] M. Tikhon, T.J. Ko, S.H. Lee, et al., NURBS interpolator for constant material removal rate in open NC machine tools, International Journal of Machine Tools and Manufacture 44 (2004) 237–245. [12] T. Yong, R. Narayanaswami, A parametric interpolator with confined chord errors, acceleration and deceleration for NC machining, Computer Aided Design 35 (2003) 1249–1259. [13] J.J. Stoker, Differential Geometry, Wiley-Interscience, New York, 1969. [14] V. Pateloup, E. Duc, P. Ray, Corner optimization for pocket machining, International Journal of Machine Tools and Manufacture 44 (2004) 1343–1353. [15] L. Piegl, W. Tiller, The NURBS Book, second ed., Springer, Berlin, 1997. [16] R.T. Farouki, Y.F. Tsai, Exact Taylor series coefficient for variablefeedrate CNC curve interpolators, Computer Aided Design 33 (2001) 155–165.