NURBS FEEDRATE ADAPTATION FOR
3-AXIS CNC MACHINING

By: Elkeran, A. * and El-Baz, M.A**

Abstract

The development of manufacturing technologies to improve machining and obtain high productivity has become a very important goal in modern industry. Therefore, reducing machining time and improving machining through adaptation of feed rates becomes a great demand. This work effort is focused on improving the efficiency of CNC machining by calculating an efficient variable feed rates that keep the instantaneous volumetric removal rate constant during the cut. The different obtained values of feed rates are then interpolated using NURBS approximation curve fitting approach. In addition, a new NURBS feed rate interpolator is incorporated to an open architecture CNC controller. Experimental evaluation shows that the proposed approach results in much smoother CNC movements as well as achieved a reduction of cutting time that satisfy the requirements of today’s machining needs.

Keywords: NURBS, feed rate optimization, CNC machining, open architecture controller.

1. Introduction

Setting accurate feed rates for machining of die, mold and other sculptured surface parts, where the amount of metal removal is constantly changing, has the potential for great benefits. Despite tremendous progress in cutting tool technology during the last century, it is still not easy to determine the optimum feed rates for metal removal processes. The most modern CAD/CAM systems that generate CNC tool paths for these parts still employ the traditional method of assign a fixed and low machining feed rate for a number of cutter paths, based on the worst case cut geometry. This can result in significant tolerance deviation, variation in cutting loads, and more cutting time.

Many of previous research efforts have been discussed the importance of employ variables feedrates that maintain safe cutting in the machining processes. Some of the first work on feedrate planning was by Wang [1], where he used a z-map representation of the workpiece and a simple volumetric model to relate cutting force to the metal removal rate. Takata et al. [2] utilized a z-map approach for the workpiece/cutter geometry description and combined this with the mechanistic model of Kline et al [3] to investigate cutting forces and tool deflection.

Others works have been discussed a number of discrete models used for optimization of cutting conditions during CNC machining [4-7]. The models can be broadly placed into two categories:

1. Volumetric models in which feed rate is proportional to either average or instantaneous volumetric removal rate,

2. Vector force models in which feed rate is set to values which keep either average or instantaneous cutting forces to prescribed values.

While the latter models are more accurate they are also more difficult to implement on the shop floor. The resulting outputs of the two categories often create many tiny line segments. Each line segment in the CNC part program has a calculated fixed value of the feed rate.

For these tiny line segments with different feed rate the following problems will occur [8]:

• Acceleration and deceleration of CNC machine at transition from each line to the next causes a cycle time delay.

• The DNC data transfer burden is increased, the data transfer speed may not catch up with the cutting feed, which may cause the machine to chock or vibrate.

• The real cutting speed is reduced and the cutting time is prolonged.

• More tiny line segments resulting in a larger CNC data file.

This paper describes a feedrate adaptation system to overcome the problems discussed above. The proposed system determines feedrates for milling operations by integrating a z-map geometric model with a NURBS feedrate interpolator. The different values of feedrates are determined from the geometric model, to keep the volumetric metal removal constant during the cut.

These values of feedrates are then interpolated to a NURBS curve using NURBS approximation curve fitting approach. The results of these steps are written in a CNC part program. On the other hand a NURBS software interpolation program is integrated with open loop architecture CNC milling machine.

NURBS has been selected because it has emerged as the standard free-form geometry representation since late 1980s. It is a superset of various models of free form curves have been developed such as Bezier and B-spline for the ability to represent quadric curves exactly. In NC machining, when a NURBS feed rate curve is interpolated, it results in much finer and smoother NC movements compared with the interpolation of traditional line segments. Furthermore, the NC blocks or statements are greatly reduced, making the data transfer much faster.

Compared to NURBS feedrate tool path, chord tool paths with different feedrates, leave the machine to accelerate and decelerate less efficiently and this inefficiency alone can compromise cycle time. To explain this comparison contrast a contour defined by a single NURBS feedrate block with the same contour defined by a series of feedrate of straight chords. In the NURBS version, the feed rate of each axis changes continually and gradually all along the curve. But in the chord version, the feed rate of each individual axis is constant throughout each chord—until it reaches the end of that chord.

At this point, the CNC commands a new feed rate for the axis, appropriate to the next line of code. In essence, the chord program asks the axes to change their feed rates instantaneously at the end of each block of code. Neither the axes nor the CNC can do this, of course, so the acceleration or deceleration to transition from one chord to the next introduces a cycle time delay.

The remainder of this paper is organized as follows: in section 2 feedrate from geometric model calculation is described. The NURBS feed rate interpolation and approximation curve fitting method is illustrated in section 3. The NURBS interpolation is presented in section 4. Result of a sample example is presented in section 5. The conclusions of this research are summarized in section 6.

2. Feedrate Calculation

Our method for estimating the volume removed by a particular tool movement is based on Wang’s method [1]. This model originally breaks the workpiece block into many discrete vectors, as shown in Figure 1.

Then for each small tool movement the intersection of the resulting tool envelope with the block vectors is found through an efficient geometric method. Now, by assigning a surface area to each vector, the volume (VL) removed by the tool movement step (?x) can be calculated by summing the product of the vector area with the length of the vector removed by the tool.

Inverse model is needed to automatically generate the feed rate for a given tool movement. The model requires information on the allowable cutter deflection, as well as the volume of the metal removed by the cutter for a given tool movement.

The process summaries such as follow:

1. The allowable cutter deflection for a small tool movement is used in a cantilever beam equation to generate the allowable force.

2. With the known spindle speed, the power necessary to generate the given force can be calculated such that P=Ft*V/C where P is the cutting horsepower (HP), V is the cutting speed m/min.

3. The metal removal rate is determined such that P=K*MRR where K is the unit power consumption, and MRR is the metal removal rate (mm3/min).

4. The time required to cut the tool movement step (?x) is calculated such that dt=VL/MRR.

5. Finally the feed rate for this tool movement is calculated such that Feed rate = ?x /dt.

6. Calculating successive feed rates for each tool movement step is repeated until the entire part has been cut.

7. The calculated feed rate values (Q0,Q1,….Qm) will be fitted using NURBS least square method (section 3.2).

Fig.1 Z-map

3. Feed rate NURBS Interpolation

Fitting NURBS curve for a given set of points is of two types [9, 10]. The first method is interpolation and second is approximation. In the case of interpolation, curve is passed through the given set of points. In the case of approximation the curve constructed may not necessarily satisfy the given data but only approximates it.

If one tries to fit NURBS curve using interpolation, the resultant number of control points will be equal to the given data points. Therefore it is not desirable to fit a curve by using interpolation especially for large number of data points. In this work the second method i.e., approximation is used.

To explain how well a curve can approximate the given data points, the concept of error distance is used. The error distance is the distance between a data point and its corresponding point on the curve. Thus, if the sum of these error distances is minimized, the curve should follow the shape of the obtained data polygon closely.

3.1 Definition of NURBS

A NURBS curve is a vector-valued rational polynomial defined by the following form:

Where Pi is the control points, wi weights, and Ni,p(u) the normalized B-spline basis function of pth degree with the following notations:

Computation of a set of basis functions requires specification of a knot vector, U that will be presented in section 3.2.

Note that when wi = 1 for i = 1…n, the rational B-spline is reduced to the B-spline:

3.2 Computational Algorithm

Given m+1 data points Q0, Q1, Q2..., Qm, and wish to find a B-spline curve that can follow the shape of the data polygon with minimum number of control points and satisfies a prescribed value of error distance. Figure 2 shows the flow chart of the NURBS fitting process.

Fig. 2. Flow chart of a NURBS fitting process

In the beginning, an initial number of control points equal to the number of data points/20 and the allowable fitting error for a fitting curve are assigned. Using the approximation method, the minimum number of control points is found to fit a NURBS curve that satisfies the error requirement. If the error is exceeded, the number of control points is increased. The above-described process is iterated until the error requirement is met.

The minimum number of control point equals to the p+1. Refinement is an extended process to improve or reduce the error by recalculating more accurate values of for the same number of the obtained control points. If the refinement gets an improvement of the previous error, another iteration of reduction control points and refinement is processed until the best improvement or reduction in error with the minimum number of control points are met.

3.2.1 Least Squares Curve Approximation

In this section, an algorithm to compute the numbers and the values of control points that satisfy minimization of the sum of the square error distance is presented. The distance between Qk and the corresponding point of on the curve is |Qk – C ( )|. Hence, the sum of all squared error distances is :

Our goal is to find those control points P1, ..., Pn-1 such that the function f( ) is minimized Let m+1 is the number of the given data points Q0, Q1, .., Qm. p is the degree of the obtained B-spline curve and n+1 is number of the required control points Pi. The following illustrates the solution procedure:

1- Calculate a set of parameters , ..., using one of the following schemes:
i- Chord length method: let d be the total chord length

Then

This method is generally adequate. It gives a good parameterization to the curve, and it is most widely used.
ii- Centripetal method: let

Then

This method gives better results than the chord length method when the data takes very sharp turns.

2- Calculate knot vector U, usually the following Eqs. are used to calculate U .

This method generates wiggle at the start of the curve. So, a developed equation of calculating U is presented such that:

Figure 3 shows an example to compare the effect of the two methods of calculating the knot vector on the NURBS curve. This figure illustrates that the second method provides better results of both oscillation and error for the same number of control points.

3. Compute Rk using the following formula:

4. Compute the following and save it to the i-th column of matrix R;

5. Compute Ni,p(uk) and save to row k and column i of N;
6. Compute
7. Solving for P from M*P = R;
Where N is the (m-1) (n – 1) matrix scalars

R is the vector of (n-1) points
,and
And Row i of P is control point Pi;

LU-decomposition method is used to solve M*P = Q. While other methods such as Gaussian elimination method and Cholesky method can do the job well.

Fig.3 Knot vector calculation.
(a) Modified method (n=9, p=3, and error=3.668),
(b) Piegl’s method (n=9, p=3, and error=9.527)

Calculating of using chord length method gives very rough approximation. Therefore, a refinement process of these values is required. Schneider [11] proposed an iterative process to improve parameterization for Bezier curve. Fig. 4 shows the effect of successive refinement of values on the normal distance from the data points to the curve.

Newton-Raphson method is iterated to get a better values. refinement reduces the error in such a way that permits another reduction of the number of control points in iterative manner as shown in Fig. 2. Fig. 5 shows an improvement of the fit of the curve through reduction of the number of control points for the same error using refinement.

Fig. 4 Iterative refinement process. (a) Chord length parameterization,
(b) First iteration, (c) Second iteration, and (d) Third iteration

(a) (b)
Fig.5 refinement reduces number of control points (a) n=11, (b) n=9

4. CNC Feed Rate NURBS Interpolator

The proposed NURBS feed rate interpolator module is integrated to a Vertical Machining Center (ElettronicaVeneta FCN 500/PLUS). The machine was previously retrofitted with a PC based open architecture CNC controller. The feed rate interpolator uses the advantage of the open architecture CNC system that allows feed rate changes during cutting motion. The DDA motion interpolator deals with axis positional control The imbedded CNC feed rate interpolator is illustrated in Fig. 6.

Fig. 6 NURBS interpolator flowchart

5. Experimental Evaluation

As an example to evaluate the proposed approach, slot milling operation was performed on an Aluminum workpart as shown in Fig. 7. The spindle speed was set at 2000 rev/min. 10-mm cutter diameter, and four tooth HSS end mill cutter was used. The desired volumetric metal removal rate was determined to keep at 20000 mm3/min. The cutter passed through the workpart, and met variations in the depth of cut during the path of the cut.

Three methods to assign feed rate for the example are considered to evaluate the proposed approach. The first method assigns a constant feed rate according the worst depth of cut. The second method divides the cut path into different NC blocks. Each block has a determined feed rate value according to the worst depth of cut during each path. The third method assigns variable feed rate according to the proposed NURBS feed rate approach.

Table 1 Shows the results of different 10 experimental sets. The actual feedrates of the machining were measured and shown in Fig. 8. The results show that the proposed approach, experiment 10, uses 1 NC block with 38.94 sec cutting time. Experiment 1, which assigns a fixed feedrate according to the worst depth of cut to keep volumetric metal removal rate not exceed 20000mm3/min. It uses one NC block, but the cutting time expanded to 77.43 sec. The increasing ratio is normally 198%, which is significant.

Experiments 2-9 divide the cutting path to different distances, and assign feed rate value for each division. Experiment 2 for example, assigns feedrate value for each 40 mm cut path. Experiments 2-6 presented that the cutting time decreased and the number of NC blocks increased for those experiments. Due to the increasing division paths in experiments 7-9, and the changes in the value of feedrate assigned for each block, the actual feedrate oscillate and could not reach to the determined feedrate value in most cases.

This is occurred due to the acceleration and deceleration of the NC machine motor drives. The result is machining time start again to increase as clearly shown in Table 1 and Fig. 8 Beside the reduction in the cutting time, the cutting process in the proposed approach became more stable due to the fact of keep the metal removal rate constant during the cut. In addition the machining itself became much smother due to the fact of deleting the effect of acceleration and deceleration of the machine motor drives when transfer data from block to another.

Fig. 8. Measured feed rate (mm/min) versus time for experimental sets.

Fig. 8. Measured feed rate (mm/min) versus time for experimental sets (continued).

6. Conclusion

This paper describes a feedrate adaptation system to improve the efficiency of the CNC machining process. The proposed system overcomes the most problems appeared from other effort has been treated with the problem of optimize feedrate during cut. Most of these efforts and the deficiency of controller functions have prevented the real power of CNC machine tools from being fully utilized. An algorithm is proposed to determine feedrates for milling operation by integrating a z-map geometric model with a NURBS feedrate interpolator.

Efficient variable feed rates that keep the instantaneous volumetric removal rate constant are determined. The different obtained values of feed rates are then interpolated using NURBS approximation curve fitting approach. The results of these steps are written to a CNC part program. On the other hand, a new interpolation NURBS feed rate module is incorporated to an open architecture CNC controller.

Experimental results indicate that the proposed system is effective for improving the performance of the cutting process. The advantages of the proposed system have demonstrated in reducing cutting time, keeping the cutting load constant, smoothing cutting process due to delete the effect of acceleration and deceleration of the machine drives, and reducing NC part program.

7. References

1. Wang, K. K.,”Solid modeling for optimizing metal removal rate of three-dimensional NC end milling”, Journal of manufacturing systems, Vol.7, No.1, pp.57-65.year.

2. Takata, S., Tsai, M. D., Inui, M. and Sata, T., “A cutting simulation for machinability evaluation using a workpiece model”, Annals of the CIRP, Vol. 38/1, pp.417-420, 1989.

3. Kline, W. A., Devor, R.E, and Lindberg, J. R, “The prediction of cutting forces in end milling with application to cornering cuts”, Int. journal of machine tool design and research, Vol.22, No.1, pp 7-22, 1982.

4. Jerard, R. B., Fussell, B. K., and Ercan, M. T, “On-line optimization of cutting conditions for NC machining”, 2001 NSF design, Manufacturing & Industrial Innovation research conference, Jan 7-10, 2001, Tempra, Florida.

5. Hemmett, J. G., Fussell, B. K., and Jerard, R. B.,”A robust and efficient approach to federate selection for 3-axis machining” 2000 ASME IMECE, Dynamic and control of material removal process.

6. Robert B. Jerard, Barry K. Fussell, Mustafa T. Ercan, Jeffrey G. Hemmett, "Integration of Geometric and Mechanistic Models of NC Machining into an Open-Architecture Machine Tool Controller", ASME International Mechanical Engineering Congress and Exposition Nov. 5-10, 2000, Walt Disney World Dolphin, Orlando, FL.

7. Donggo Jang, Kwangsoo Kim, and Jungmin Jung “Voxel-based Virtual Muti-Axis Machining”, http://www.mcadcafe.com/MCADVision/article/VirtualMultiAxis.html

8. Yau, H., and Kue, M.,”NURBS machining and federate adjustment for high-speed cutting complex sculptured surfaces” Int. J. Prod. Res. Vol.39, No.1, pp21-41, 2001.

9. Piegle, L.,”The NURBS Book. Springer-Verlag Berlin Heidelberg 1997.

10. Yan Z., ” Surface Interpolation and Approximation”, 

11. Schneider, P., "An algorithm for automatically fitting digitized curves", GRAPHICS GEMS, Academic Press, pp. 612-626, 1990.

* Elkeran, A. Assit. Prof. Prod. Eng. Faculty of Engineering, Mansoura University, elkeran@mum.mans.eun.eg

**El-Baz, M.A Assit. Prof. Industrial & System Eng. Dept. Faculty of Engineering, Zagazig University, elbaza@yahoo.com