Optimal Preview Control of Cutting Force in CNC Turning Machines

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Copyright © IFAC System Structure and Control, Prague, Czech Republic, 2001

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OPTIMAL PREVIEW CONTROL OF CUTTING FORCE IN CNC TURNING MACHINES

M.M.Negm*

A. M. Bassiuny **

*Department ofElectrical Engineering, Ain Shams University, Abbasia, Cairo, Egypt, [email protected]

** Department of Production Engineering. Faculty ofEngineering. Helwan University. He/awn. Egypt [email protected]

Abstract: In this paper, a new linearized state-space model for a Computerized Numerical Control (CNC) turning machine is synthesized. The Iinearization is implemented using a factorization method in which the feed is divided into two tenns. The first term is the measurable part of the integer feed and the second is a fractional of feed, which will be calculated. An optimal preview controller for the CNC machine is implemented. A new error system technique is introduced into the control system to cure the adverse phenomena that is caused due to parameter uncertainties, un-mode led dynamics and disturbances. The disturbances comprise material properties, depth of cut and tool conditions while the un-modeled dynamics to dispense with the fractional feed. A preview feed-forward controller utilizing the desired cutting force is designed to improve the transient response of the CNC control system. The robustness of the proposed controller copes with any changes in depth of cut and/or cutting speed during the cutting process where they are embedded in the Iinearized model of the CNC machine and are considered as system parameters. Copyright © 200J JRtl Keywords: CNC machine, turning operation, optimal preview control, factorization.

accomplished by increasing material removal rate. Concurrently, to avoid damaging either the cutting tool or the machine itself, the cutting force must be limited to a prescribed constant level. However, maintaining the cutting force at a constant level during cutting process is hard to achieve due to the complexity of cutting conditions. From control point of view, the cutting process using turning CNC machines exhibit non-linear time-varying parameters. These variations result from variations of material properties, cutting speeds, and variable depth of cut. Furthermore, the dynamics of the chip removal process, the structural dynamics of machine tool and dynamics of drivers are also interacting with each other in turning operations.

I. INTRODUCTION Modem manufacturing systems that utilize computer as an integral part of their control can be classified into two main categories. The first one consists of small standalone CNC machine tools and robots and the second includes comprehensive systems with flexible manufacturing systems, which contain many standalone units. CNC machine tools have greatly reduced the operator input, maintain accuracy of produced parts through the full range of speeds and feeds and make it possible to produce complex shapes. The increasing capital cost of these machines, however, makes their efficient utilization very important. Increasing productivity can be

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order system with time varying gain where the regulator was designed based on the highest process gain. In the second, introducing non-linear transformation linearized the cutting process. The regulator was designed based on a constant process gain, and the time-varying parameters were considered as disturbances. Both regulators included some parameter adaptation by gain scheduling with respect to the variations of spindle speed. The presented results show some instability of the closed loop system when spindle speed changed.

Many research works have addressed this problem and several solutions were suggested. Simple systems based on the use of integral regulator were described in Ulsoy et al.( 1983 )-Lin et al.( 1987). These works were based on the use of variable gain adaptive controller to obtain the desired performance. The total process gain was estimated and the estimate value was used to adjust the regulator integral term. Fassios et al. (1989) developed an algorithm for process mode ling and control. Process mode ling was accomplished using recursive least squares, and the adaptive controller was implemented based on online estimation and pole placement technique. Daneshmend et al. (1986), described a similar approach based on Model Reference Adaptive Controller (MRAC) for regulating the feed rate. The developed MRAC was implemented online using a microprocessor based CNC turning machine. The main goal of the previous research works was to reduce the computational complexity of the developed control systems, which is not a limitation of today's computers. Tomizuka et al.(1988) developed a mathematical model for the dynamics of the cutting process which takes into consideration the variations of chip thickness during the cutting process. The model was used for tuning a PIcontroller utilizing the root locus technique. The main limitation of this scheme is that a PI-controller parameters tuned for one cutting condition are not suited for the others. A slightly improvement in the system response was achieved through the estimation of process gain. Chen et al. (1989) presented another similar approach, where a constant parameter PIcontroller for the mean value of specified operating ranges is described. Chen et al. (1991) proposed a non-linear model comprises PI-controller, servosystem, and dynamics of cutting process. The nonlinear time varying gain of the cutting process is divided into an average value and a perturbation part. Based on functional analysis, an algorithm was developed for synthesizing the PI-controller to tolerate changes of feed and time-varying depth of cut.

Many of the control problems inherent to the previous control methods are solved in this paper by introducing an optimal preview controller. This controller is synthesized and implemented in the CNC turning machine. A new error system technique is introduced into the control system to cure the adverse phenomena that is caused due to parameter uncertainties and un-modeled dynamics. In addition, the preview feed-forward steps are utilized to improve the transient response of the control system. These steps represent the desired cutting force, which is monitored in a regulated closed loop. Extensive computer simulations are made to demonstrate the feasibility and robustness of the proposed optimal preview controller.

2. CNC TURNING MACHINE MODEL The dynamic equation of the turning machine is written as follows, Chen et al. (1989): ..

.

2

vr(t)+2~oon vr(t)+oon vr(t) = kmu(t)

f(t)=Tnvr(t)

(1)

O.5Tn Fe (t) + Fe (t) = a(t)krf(t)P where vt{t): u(t): f(t): Fe(t): ~ ,Oln:

feed rate of the cutting tool (mmlsec), input of the servo system (radian), feed (mmlrev), cutting force (N), damping ratio and angular natural frequency (rad/sec), of the servo system, respectively, T n=60/n: revolution period of spindle (sec.), n: spindle speed (rpm), aCt): depth of cut (mm), km: constant of the servo system (mm rad/sec\ kf : specific cutting force (N/mm 2 ), p: a constant fraction <1.

Alien et al. (1994) employed a self-tuning strategy to adapt cutting force controller. The process dynamics was mode led by a second order Autoregressive Moving Average (ARMA) model, and a generalized minimum variance controller was implemented. The results presented indicate a slow dynamic response of the system and a considerable increase in the system overshoot as the depth of cut change. An improvement in the system response was achieved by introducing a PI-controller for the first 18 seconds following every change in depth of cut. Harder et al. (1997) used a first order system for describing the dynamics of the cutting process. The specific cutting force was approximated as 'a non-linear function of feed, while the dynamics of feed drive was neglected. Two approaches to a PI controller design based on Internal Model Control (lMC) were presented. The first approach treats the cutting process as a first

The state-space model of (1) can be derived after factorizing f (t)P , as given in (2). x(t) = Aox(t)+ Bou(t)+ Eo

(2)

Yf(t) = f(t)P = f(t) + M(t)

(3)

where x(t) = [Fe (t) f(t) f(t) Yf (t)]1

712

=Lle(k)+F aLlx(k)+FiAu(k-1 )+Llz(k+ I)

(8)

where Fa = F(A-14)' Fb = FB, F=-C, Llz(k+ 1)= LlR(k+ I) - LlR(k) The error system (9) is constructed from (7) and (8). c" C2, C3 and C4 are constants and ao is a constant depth of cut depend on the operating point of the control system, Llf(t) is the incremental change of the feed such that M(t)=O ifp=l, and the superscript "t", denotes the transposition.

3. DISCRETE STATE-SPACE MODEL

X(k + 1) =[e(k) Lle(k + I) dx(k + I) Au(k)r

<1>=[8 ~ ~~]

(7x7);

e=[z]

0 0

(7xl);

1

G r =[0 1 0 O]t (7xl)

(4)

5. OPTIMAL PREVIEW CONTROL LAW

where, x(k) and u(k) are the state-variable and input variable, respectively, while k denotes kT and T is the sampling period.

To implement the optimal preview control law, the following performance index h is to be minimized subject to the constraints given by (9):

The output vector is: y(k)=k.t Fc(k) = C x(k)

J

~

= ~)X(k+l)tQX(k+l)+rAu(k) d

2

]

k=O

(5)

where, k.t is the dynamometer gain (N''), and

Here the weighting matrix Q (7x7), is given by:

C = [k.t 0 0 0]

Q:[~ ~ 0]

Consider the desired cutting force F/(k), then the output error is given by: e(k) = F~ (k)-y(k)

(9)

where

o0

The discrete state-space model of system (2) is given by: x(k+I)=Ax(k) + Bu(k) + E

X(k + I) = cpX(k) + eLlu(k) + G rLlz(k + I)

(10)

Accordingly the mmlmlzation process gives the following optimal preview CNC control law:

(6) Llu(k) = GX(k)+G} w(k + 1)+ M

L G i[K} ]i-2 w(k + j)

4. ERROR SYSTEM

(I I)

i=2 where

To derive the error system, the first difference of (4) is written in the form, Negm et a\'(2000)

feedback gain:

G=-

tK 3

feedforward gain : G I = --ye t K Llx(k + I) = ALlx(k) + BLlU(k -I)

(7)

G 2 = _-yetcptA.

Here Ll=(I-q·I), where q-I is the backward shift operator and u(k) is delayed by one sampling period to compensate for the microprocessor's execution time. Substitution from (5) into (6), and by considering R(k)=F/(k), gives:

G; =G;_1.K 1 ; i=3,4, ... ,M KI=KlcptA., and K, y and A. are the steady-state solution of the following Riccati equation: K(i) = Q +cptA.(i + I)cp A.(i + I) = K(i + 1)[17 - 8y(i + l)e'K(i + I)] y(i + I) = [R +etK(i + I)er l w(k + I) = G rLlz(k + I)

Lle(k+ 1)=LlR(k+ 1)-Lly(k+ 1)=LlR(k+ 1)-CLlx(k+ I) Then, the substitution from (7), Lle(k+ 1)=Lle(k)+FALlx(k)+FBLlu(k-l)

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The real time optimal preview controller u(k), can be derived by induction from (11), such that: k

u(k) = gl

L e(i)+(gl - g2 )e(k) + g3 x (k)+ ;=0

M

g4 u(k-I)+ LFrj[M(k+i)-M(i)] (12) ;=1

Fig. 2. ProfIle of work-piece used in simulation.

where: G==[g, g2 g3 ~], Frj == GjG, ,j-I.2", M M > I, is the preview feed-forward step.

__

~

__

~

______

~

__

~

~

-

-

~

o

2

Fig. I. Optimal preview CNC turning machine control system.

o

The control system structure is implemented using (12) as shown in Fig. l.

o

The simulation results shown in Figs. 3-10, are obtained on the basis of (12), using a sample time T=(1.02 sec. The effect of optimal preview controller is indicated with preview steps M=2 for Figs. 3-8, and M=O and M==2 in Figs. 9- 10. The weight factors are selected as q==1 and r=107. The spindle speed is selected either constant at 600 and 900 rpm. or variable up to 1800 rpm. Figs. 3,5,7 and 9, show the results when p=O.85, while Figs. 4,6,8 and 10, display the results when p==l , (M(t)=O) . The horizontal line in these figures represents the time in samples. In Figs. 3-8, the vertical lines from up to down represent the spindle speed net) (rpm.), the desired cutting force Fc'=900Kd (N) in dotted line and its response Fe(t) (N) in solid line. The next vertical line indicates the control input u.(t)=u(t) (radian) and the feed f(t) in mmlrev is illustrated in the last line. ____- .

:l~f : : : , : : i rmE • =--:1 o

6. RESULTS AND DISCUSSIONS The proposed optimal preview controller is used in this paper to control the CNC turning machine with the following parameters: K.t=4N·', ~=O.42, 0ln==36 rad/sec, kFI732 N/mm2, k.n==1296mm.rad/sec 3, n==600 rpm, p=O.85 and Fc'=900k.! (N). The applicability of the proposed controller is investigated in two cases. In the first case, M(t) in (3) is taken into consideration and the process is treated as a non-linear system. In the second case, the incremental change of feed M(t), is neglected, i.e. yt
200

~ 1;:

400

600

800

1000

. -; .-

1200

1400

1600

.~-

!':~'~'g200

400

600 SOO 1000 TIme (•• mp•• )

1200

1400

1600

Fig.3. Response of the proposed CNC control system, (n(t)= 600 rpm, p=O.85 and M==2).

::::f

:

200

:

200

400

lOO

0

200

400

lOO

0

200

400

ll§P ~ 1 ': ~ \

The control system is subjected to variable changes in depth of cut. The depth of cut is increased stepwise from 2 mm to 4 mm, then decreased to 3 mm. After 12 seconds, the depth of cut is decreased gradually to I mm and maintained constant until 24 seconds, at which the depth is increased in step to 2.5 mm. The profile of the work-piece is shown in Fig.2, where the dashed area depicts the cut. The timing diagram of these variations is shown in the upper part of Figs. 9 or 10. No limitations were made on the control signal, i.e. the feed.

:

0

:

lOO

:

aoo

1200

1000

1200

1000

'200

~

: lOO

.00

,==:3

1000

1

1100

'400

:~ '400

'I 0:

f 0

\ 200

lOO

• 00

1100

I

~ 400

1100

1000

n.n. (..flip".)

~,a..

1200

'4"

Fig.4. Response of the proposed CNC control system, (n(t)=600 rpm, p=1.0 and M~2).

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2000 E

e-

I~~~~~_

_

,

~~_

1000 "--- _ _

,

~

Co OL.~~'.~.-~I~••~-~~~~~~~~~~~~~' 6 ••

0 ••

1 000

! lm ~~,----~-~--

1 200

1400

11500 1

..-~

~ ~ggL_ 10 o

200

1 400

6 •• __

~~~~_~~

• ~

.• ,..

0 •• _

~

_1000 __

11500 1400 -----, _1200_ __

5~ '

•

E E

~

I ••

0 ' ,_ - _ - -

. ~~ , •

..

I ••

6 ••

0 ••

1000

1200

1600

1400

-~

•••

0 ••

1000

1400

12 00

1600

Fig.5. Response of the proposed CNC control system, (n(t)= 900 rpm, p=0.85 and M=2) . 2000 r---~------~-------,

: 1··: ~f---~--------------4 ..

o

200

400

600

800

1000

1200

1400

1600

,.1200 ~ '~gg ----_~---~---A_l ~ j I ••

, ••

"s .

'-____,:-:-:--~:-___:_:_=_____,o:_:_-_:_:':_:___:_:'::___,_:_o:___:_

~ ':~ 1.:b~ o

200

....

400

~

I •••

I •••

600 800 1000 Tim. (I am pl.l)

".~

••

1200

'400

Figs. 9-10, indicate the robustness of the tracking capability of the regulated optimal preview controlIer with the CNC turning machine. In this respect, different changes are made in depth of cut, spindle speed and the reference cutting force . The desired response of cutting force is achieved with a minimal transient response . This response is rapidly improved with only 2-steps ahead preview controlIer (M=2).

1600

Fig.6. Response of the proposed CNC control system, (n(t)=900 rpm, p= 1.0 and M=2)

As demonstrated from these figures coincidental regulating response is achieved in case ofp=O.85 and p= 1, but at the expense of the control input and feed in the latter. Furthermore, the proposed optimal preview controlIer adapts welI for various changes in depth of cut with minimum overshoot, minimum rise and settling time and zero steady-state regulating error. However, the controlIer sustains the cutting force at the desired level by adjusting the feed to compensate for the changes in depth of cut. The results reveal that fast dynamic response is achieved every step change in depth of cut.

E2000 200 400 600 800 1000 1200 1400 1600 ~ 100:[1 : : : :: ~ :~F; ~o 600 M,o~;:;;;;=~r

q

0

200

~::l:

.00

600

,. 0.50

200

.00

200

400

u..

,.

y

'000 I-_-_--_,=======::;--~

~,··: f---L-. o

200

.00

600

800

1000

1200

1400

1600

!

800

1000

1200

1400

1600

600

800

1000

1200

1400

1600

600

800

1000

1200

1400

1600

M~

0

-

0

Time (nmples)

.oo

i :::1 Fig.7. Response of the proposed CNC control system, (n(t): variable, p=O.85 and M=2).

:: : fr------

!~ lmEI L--.------.y=---4 ~gg oL----:'O:-.O:-----:-I.:-:.--.:-:.~. o

2 00

400

600

800

1000

..

StO

~

10.

~

Time (ump .. s)

1200

9

1400

...

r

'so r .so ~~-~-~---~-~-~~---:~-~~ Magnified cutting force (-- desired; _ measured)

16 0 0

Fig.9. Response of the proposed CNC control system, (n(t): variable, p=O.85, M=O and 2).

--':00:-••:---::,.:'::••:---:,7,.:-::c.- -,-,1"'••:---:,0-:.,••

'· I,------------~ ---------,j

I':b o

200

400

&00

/'~ lOO

1 000

1200

1400

1800

, ,----------------~---,

1·:!== k~~~~;;;;J~. o

2GG

400

600 100 1000 Tim. (s.mpl.s)

1200

1400

11500

Fig.8. Response of the proposed CNC control system, (n(t):variable, p=1.0 and M=2).

715

The feasibility of the proposed optimal preview controller of the CNC turning system in comparison with the previous research work is investigated when the cutting speed is changed. For instance, Harder and Isakson (1997) indicated that the closed loop system becomes unstable when the spindle speed changes. This is because the self-tuning regulator was designed based on a first order system, where the dynamics of the servo drive was neglected. Also the

PI-regulator used for a specified period after every change in the depth of cut is not appropriate for higher order systems. This serious problem is cured using the optimal preview controller as shown in results of Figs. 4-10. These figures show the response of cutting force and the command feed when the spindle speed is changed from 600 rpm to 900 rpm and also in the case of variable cutting speeds from 600 to 1800 rpm during the cutting process. A zero steady state regulated error is achieved over the entire period of cutting process, however a slight increase in the transient response is occurred during the following change in depth of cut. The results shown in these figures investigate the robustness of the controlled system as the cutting speed change, which is the case for machining soft materials.

Time (umP'es)

0 ••

••• ••• 75. ~ 7 00

:!. jf Wo.

6 50 800

••• I 5•• r 450

r 1~ ••~.--~.5~.--'=.~.--="~.~".~.=.--='.=5.--'~'.=.--'~"=.~ Time (lemp' •• )

Magnified cutting force ( ... desired; _ measured)

Fig. 10. Response of the proposed CNC control system, (n(t): variable, p= 1.0, M=O and 2).

7. CONCLUSIONS A synthesized method for regulating the cutting force in the CNC turning machine, based on optimal preview control theory, is proposed. The preview feed-forward steps are introduced in the control law to improve the transient response of the system. The robustness of the controlled system is indicated by changes made in both of depth of cut, spindle speed and the desired cutting force during the cutting operation. Coincidental response between the desired cutting force and its measured value is achieved, in

7]6

spite of changes in depth of cut and/or spindle speed. A few preview controller steps are used to improve the transient response. Extensive simulation results are made to show the feasibility of applying the proposed optimal preview controller to improve the performance of the cutting process. The results demonstrate the robustness and feasibility of the system in the face of variable cutting conditions.

REFERENCES Ulsoy, A., Y. Koren and F. Rasmussen (1983). Principle developments in the adaptive control of machine tools, Trans. ASME, J. Dvnamic Sys., Meas. and Control, 105, pp. ]07-] ]2. Masory, O. (1984). Real time estimation of cutting process parameters in turning, Trans. ASME, 1. Eng. ForIndustry,106,pp. 2]8-22] . Masory, O. and Y . Koren (1985). Stability analysis of a constant force adaptive control system for turning, Trans. ASME, 1. Eng. For Industry, 107, pp. 295-300. Lin, S. and O. Masory (]987). Gain selection for a variable gain adaptive control system for turning, Trans. ASME, 1. Eng. For Industry, 109, pp. 399-403, ]987. Fassois, S., K. Eman and S. Wu (1989) A fast algorithm for on-line machining process mode ling and adaptive control, Trans. ASME, 1. Eng. For Industry, HI, pp. 133-139. Daneshmend, I., and H. Pak (1986). Model reference adaptive control of feed force in turning, Trans. ASME, 1. Dvnamic Sys.. Measurement and Control, 108, pp. 2] 5-222 . Tomizuka, M., and S. Zhang (1988) Modeling and conventionaV adaptive PI control of a lathe cutting process, Trans. ASME, J. Dvnamic Sys .. Measurement and Control, 110, pp. 350-354. Chen, B. and Y. Chang (1989). Constant turning force adaptive controller design, Trans. ASME, J. Eng. ForIndustry, 111,pp. ]25-]32. Chen, B., and Y. Chang (199]) Robust PI controller design for a constant turning force system, Int. 1. Mach. Tools Manufact., 31, pp. 257-272, Allen, R.and K. Huang (] 994). Self-tuning control of cutting force for rough turning operations, Proc. Instn. Mech. Engrs. Part B, 208, pp. ]57-]65. Harder, L. and A. Isakson (1997). Force control in turning based on robust PI controller design, Proc. Instn. Mech. Engrs. Part B, 2H, pp. ]65175. Negm, M. M. (2000). Preview and stochastic control for motion control of robotics manipulator with control input constraints, Proc. of the 2000IEEE, ICRA2000, San Frans., pp. 30203027.