Intelligent control for handling motion nonlinearity in a retrofitted machining table

Intelligent control for handling motion nonlinearity

in

a

retrofitted machining table

S.-J.Huang

C.-Y. Shy

Indexing terms: Intelligent control, Automation, Nonlinearity, Self-organising fuzzy control, Neural network control

Abstract: For low cost automation, a traditional manually operated milling machine with a lead- screw transmission system was retrofitted with an AC servo-motor. This old fashioned machining table has nonlinear time-varying behaviour caused by obvious backlash and irregular coulomb friction of the sliding surfaces. It is difficult to design an appropriate classical controller for this complicated dynamic system. Hence intelligent model-free self-organising fuzzy control and neural network control strategies equipped with learning ability are employed to control this machining table, to improve both the adaptability and the path tracking accuracy. These control approaches can be implemented without the trial and error process for selecting initial parameters and fuzzy rules. The experimental results show that these control strategies achieve satisfactory transient response and tracking accuracy under the influence of -0.4" of backlash on each axis and large stick- slip friction.

1 Introduction

Computer numerically controlled (CNC) machine tools have been widely used in the mechanical manufacturing industry to produce high precision machined parts and improve productivity. Traditional machine tools manipulated by hand wheels are gradually being dis- carded. However, they are not so useless as to be com- pletely obsolete. Their mechanical structures are similar to those of NC machine tools. If the operation and control of traditional machine tools can be converted to computer control, industrial automation will become more economical. During automation conversion, AC servo-motors were chosen, instead of the original hand wheels, to actuate the lead screw transmission system of a traditional milling machine. Other mechanisms of this machine were maintained without alteration. The mechanical transmission system of this converted machine has nonlinear dynamic characteristics such as static coulomb friction and backlash. Hence, a good

OIEE, 1998

IEE Proceedings online no. 199821 10 Paper received 10th October 1997

The authors are with the Department of Mechanical Engineering, National Taiwan University of Science and Technology, No. 43, Keelung Road, Sec. 4, Taipei, Taiwan, 106

IEE Proc.-Control Theory AppL, Vol. 145, No. 4, July 1998

control method needs be designed to overcome this problem.

NC controllers are updated annually because of developments in electronic technology, and the cost- reduction of microproct:ssors. Reduction of tracking error and path error are important factors in increasing the geometric accuracy of machining. Masonry [l], and Tomizuka and Fung [2], proposed a feed forward speed control to improve the accuracy of trajectory tracking. Tomizuka [3] developed a tracking algorithm with zero phase error to achieve perfect tracking control. How- ever, these methods need a well known tracking path and accurate model and control system parameters. If the system parameters have obvious variation, they will induce a larger error. The problem of implementing these model based control strategies on a complex dynamic system is that of establishing a reasonable mathematical model for controller design. Lee and Tomizuka [4, 51 proposed a robust high speed tracking controller with system identification and disturbance estimation schemes to take care of modelling error.

To reduce path error, previous researchers have shown that cross-coupling control can effectively reduce the path error of multi-axis motion control sys- tems [6, 71. Since the path error is approximately pro- portional to the motion speed, the path error can be used to adjust the motion speed in order to obtain the maximum machining speed for maintaining the path error within a prespecified tolerance range. Without a correct mathematical model, this kind of model-based regulation of path error is difficult to implement on a retrofitted milling machine system. Hence, a model-free fuzzy controll strategy [S, 91 has been proposed to con- trol the machining table which has complicated dynamic behaviour and nonlinearity. However, the tra- ditional fuzzy controller needs expert knowledge, or operator exlperience in establishing control rules and membership functions. The implementation of fuzzy control still needs time-consuming adjustments of fuzzy parameters and fuzzy rulles, especially for a system with nonlinear tiime-varying properties. Furthermore, an optimal fuzzy logic controller cannot be achieved by trial and error. In order to solve these problems, Procky and Mamdani [lo] proposed a self-organising fuzzy controller (SOC). This control strategy estab- lishes control rules through on-line learning instead of human thinking based approaches. It simplifies the design processes and facilitates the implementation of a fuzzy controller. Later, Shao [ll], and Zhang and Edmunds [12], proposed modified learning methods for a SOC. The error and error change were used to cali- brate the control inputs, instead of the performance

table being used. Jang [13] and Wang and Mendel [14], employed neural network learning techniques to tune the fuzzy controller. Huang and Tomizuka [15] pro- posed a self-paced tracking controller for motion con- trol. Jee and Koren [16] employed a SOC controller for friction compensation.

With the development of neural network theory, neu- ral network strategy is increasingly applied to engineer- ing. Hopfield [17] proposed a neural network model by introducing the concept of energy function, which pro- vided the basis for the judgement of network stability. Rumelhart and McClelland [ 181 proposed a multi-layer neural network learning structure with a back propaga- tion algorithm to attain predictive learning ability. It has been successfully employed in some practical appli- cations. In this paper, the controlled plant is an AC servo-motor converted milling machine with a lead screw transmission system. This retrofitted system has serious coulomb friction, and its lead screw transmis- sion system has -0.4" of backlash on each axis. In order to compensate for the backlash, and other non- linear effects of this machining table, and improve the trajectory tracking accuracy, the self-organising fuzzy control and neural network control intelligent algo- rithms with learning ability were employed to obtain adaptability and tracking accuracy.

2 Self-organising fuzzy controller

Since the control input of a fuzzy logic control system is calculated from the fuzzy inference, it does not need any system mathematical model. This model-free fea- ture eliminates the difficult modelling process when designing controllers for complicated dynamic systems. Fuzzy logic control has been successfully employed in many industrial applications. However, there have been no definite procedures for designing an optimal fuzzy controller. The parameters of the membership func- tions and fuzzy rules need to be suitably planned by an expert, or based on experience [8, 91. Usually these parameters are designed by trial and error to obtain appropriate performance, which is often a time-con- suming process. Therefore, the intelligent fuzzy control- ler with learning ability is proposed to facilitate implementation.

The main difference between a SOC and a traditional fuzzy controller is the properties of their database and fuzzy rules. The database and fuzzy rules of a tradi- tional fuzzy controller are fixed after the design step.

However, the database and fuzzy rules of the SOC are accumulated or modified continuously, based upon learning strategies during the control processes to improve the system output precision. The SOC was proposed by Procyk and Mamdani [lo] and modified by Shao [ll]. The learning rule was based on a per- formance decision table; however, the design of a per- formance decision table is as difficult as the design of a fuzzy rule table. Then the output error and error change were employed directly to modify the linguistic fuzzy rules table [19]. The fuzzy rules table of this SOC can be started with zero initial fuzzy rules.

The self-organising part is introduced into a tradi- tional fuzzy controller to constitute a SOC as shown in Fig. 1. The self-organising part consists of three steps: performance measure, model estimation and rule modi- fication. The measure of system performance is critical in producing successful learning controllers. Usually, two physical features, such as system output error and error change, are measured as performance index to establish a performance decision table, which is just like establishing a fuzzy rules table. The model estima- tion is used to find the relationship between the system output performance and the control input. The per- formance measure is then employed to calculate the correction value of each fuzzy rule based on the estima- tion model. However, the appropriate performance decision table is difficult to establish for each control system. Here a real-time linguistic self-organising fuzzy control strategy is proposed by using two parameters to take care of the function of the performance meas- ure, instead of using the performance decision table.

For the rules modification, the dimension of the rules table is limited to that of the original fuzzy rules table, the correction value of each fuzzy rule is introduced into the original fuzzy rules as the new control rule. This approach can improve the database expansion defect of the Procky scheme and increase the comput- ing speed. In addition, the system output characteristic may be monitored by the definite design parameters. The system dynamic response feature can be repre- sented as an auto-regression and moving average (ARMA) model: X ( n T ) = A ( z - ' ) X ( n T - T )

+

M u ( n T - m T ) A ( K 1 ) = uo

+

~ 1 z - l

+

. .

*

+

U Z , - ~ Z - ( ~ - ~ ) B ( 2 - I ) = bo

+

biz-'

+

*

-

8

+

b,-,-lz

+

B(z-')u(nT - mT - T ) ₍₁₎ - (s-m- 1) ... ... Fig. 1 404

Control block diagram of the SOC

where mT is the system time delay and M is the direct

forward system gain for this position control system. The values of r , s and m depend on the dynamic char-

acteristics of the machining table. Due to system non- linearity and uncertainty they are difficult to estimate for this retrofitted old fashioned milling machine. For- tunately, the fuzzy control has model-free features. It does not require definite mathematical model and sys- tem parameters and so they are not employed in the following controller design. If the system is excited with a different control input u'(nT - mT) at time step n T -

mT, it will obtain a different corresponding output value X'(nT) at time step nT. The voltage difference Au

of the servo-motor control input will cause a system position output deviation AX. If the deviations AX and

A X are small, then the relationships between control input and corresponding output deviations are

AX = M .

Au

and A X =

-Au

( 2 )

If a system at time step n T has position output error

AX and error change AX, the theoretical corresponding control input correction values are Aue and Auce,

respectively. Then M T ( 3 ) TAX and Au,, = - AX

au,

= - M M

Since the system has only one control input U , the

control input correction must be an appropriate combi- nation of the above two terms. In general cases, the fol- lowing form is chosen:

where

5

is a design parameter representing the weight- ing distribution between Aue and Auce. If there is a big difference between the system output X(nT) and desired value Xd, the appropriate design choice is to select a

value X'(nT) between X(nT) and X,. Then the system output X will approach the desired value Xd gradually

with a weighting parameter 7

Then the output and output change deviations become

(6)

au

= (1 -

<)Au,

+

gau,,

₍₄₎

X ' ( n T ) = (1 - y ) X ( n T )

+

y x d , 0

<

y

<

1 (5) A X ( n T ) = y [ X , - X ( n T ) ] = y e ( n T )

A X ( n T ) = re(nT) = -ce(nT)

From eqns. 3 and 7, the correction value of the con-

( 7 )

i/

T

trol input can be represented as

Y

M

Au

= - [(l - ( ) e ( n T )

+

( c e ( n T ) ] ₍₈₎

The output error e and error change ce are divided into 11 fuzzy subsets with an integer value from -5 to

+5. For each control step, the singleton fuzzy inputs of

the output error and error change will stimulate two fuzzy subsets of the e and ce universe of disclose,

respectively. Since the control input U is derived from the fuzzy rules inference, the rules modification will influence four fuzzy rules for each control step. The correction value of each fuzzy rule is proportional to its excitation strength

w.

The excitation strength is designed as a triangular membership function and cal- culated with a linear interpolation algorithm. Then the control input of the ith rule is

ui(nT

+

T ) ui(nT)

+

A u ~

The term 1/lM in the above equation can be consid- ered as the correction weighting. In this paper, M is chosen as 1 in order to eliminate the identification pro- cedure and reduce computer time during implementa- tion. The correction weighting is only regulated by the parameter y A larger value of y will introduce a large correction of fuzzy rules and system output oscillation. This parameter only influences the transient response but not the steady state performance. According to experiment this parameter can be initially selected as a large value (e.g. 0.9), and it can be adjusted to a smaller value (e.g. 0.3) after the learning procedure. It is not crucial for this control strategy. Generally, a y

value of between 0.3 and 0.9 will achieve stable conver- gent SOC fiizzy control systems. The appropriate value for the desired dynamic performance can be found from experimental tests.

3 Neural1 network control

The multi-layer feed forward neural network is cur- rently the most popular neural network application. Here it is clombined with a back propagation learning algorithm to modify the weighting of the neural net- work. It has learning and emergent abilities which reflect the basic characteristics of the human brain's neural network. A multi-layer feed forward neural net- work consis,ts of many processing elements which are interconnected with data weighting. If the weighting between processing elements is larger, the influence of that connection is strong. The summation of those input signals multiplied by their corresponding weight- ing is used to determine the activation value.

nett =:

C

w,h7,0:--l

(10)

0: = f ( n e t t ) (11)

where net: is the net input function of thejth process- ing element on the kth layer, W j is the weighting between the interconnection of ith and jth processing elements, and Of-' is the output of the ith processing element on the (k - 1)th layer. f is the activation func-

tion, which must be differentiable and not reducing. Each processing element can be interconnected arbi- trarily although in general they are interconnected sequentially The complete connection consists of an input layer, a hidden layer and an output layer.

The back propagation learning method employs the steepest descent scheme to minimize the error function, which is defined as the summation of the error square between the real output of the neural network and the desired value, by adjusting the weighting values in the network. Diuring the neural network training process, an inertia term was introduced into the learning equa- tions to improve the oscillation behaviour of the con- trol performance. The correction value of the weighting can be represented as

(12) (13) where 0 < CI < 1 is the momentum coefficient, q is the learning rate parameter, and

qk

is the negative partial differential of the output error function with respect to

net;.

If the processing element j is on the output layer, then

AW,",(n) = qd;O;-'

+

aAW:(n - 1) = -qd:

+

aAO,k:,(n - 1)

6; = (6, - O , ) f ' ( n e t t ) (14) 405 IEE Proc-Control Theory Appl., Vol. 145, No. 4, July 1998

The activation functions used in this research are a lin- ear functionfix) = x for the output layer and a hyper- bolic tangent function for the hidden layer:

x ~ e r i ~ e ~ t a l results and discussion

The controlled plant is an AC servo-motor converted milling machine with a lead screw transmission system. This retrofitted system has significant coulomb friction and its lead screw transmission system has -0.4" of backlash on each axis. An optical scale is installed on the machining table to feed the position data of each axis back. The resolution of the optical scale is 5 p .

The servo-motor and driver are from Japan's WAC0 Giken Company and are of the voltage control type. The optical scale is an AT2N type from Japan's Mitu- toyo Company. A multi-function interface card PCL- 812 was used to control the servo-mechanism on the decoder chip HCTL2020 to read the displacement of the machining table. A multi-function interface card PCL-726 was used to control the servo-on/off of the motor drivers and to monitor the safety limit switch position of the machining table. The control system of this converted milling machine consists of a microcom- puter (IBM PC 486) as the controller CPU, a Turbo C software control program and multi-function interface cards. The sampling frequency of the following experi- ments is 200Hz. 0.20 0.1 5 0.10 0.05 0 -0.05 -0.1 0 5 10 15 20 25 30 35 40 45 time,s

Fig. 2 Circular path tracking error with a PID controller

Since this converted old fashioned milling machine has obvious nonlinear behaviour, a traditional control- ler providing satisfactory performance is difficult to design. Hence, two intelligent control strategies with learning ability were implemented on this plant to achieve the desired control performance. In order to evaluate the system performance of these model-free intelligent controllers, a PID control and a self-tuning feedforward and cross-coupling control [20] were implemented on this machining table for comparison. Since this retrofitted machining table has -0.4" of backlash on each axis, the key point of the retrofitted automation is how to compensate or reduce this effect. Here a circular path on the x-y plane was planned for the trajectory tracking control to investigate the path error and tracking error of each axis. The error history will exhibit the path tracking error and the impulse

error and impulse error caused by the backlash effect. The radius of the planned circle is 60 mm. Since the output response trajectory and the planned path are too close to distinguish, the path tracking error history, instead of the output response, was plotted. The track- ing error of a PID control is shown in Fig. 2. The tra- jectory tracking error is within CtO.O3mm, except for the 0.15" impulse oscillation when backlash occurs. Since the influence of backlash in motion control has dynamic characteristics, the peak values of these impulse oscillations have random feature instead of a constant value. In addition, the PID control scheme needs to find appropriate trial and error gains for each operating condition. Huang and Chen [20] verified experimentally the implementation of a self-tuning feed forward cross-coupling control on this plant. The tra- jectory tracking error was within 10.025mm, except for

0.075" impulse oscillation when backlash occurs. The control block diagram of the SOC is shown in Fig. 1. The design procedures of its traditional fuzzy control structure were similar to that of a traditional fuzzy controller. The scaling factors GE, GC and GU are employed to adjust the appropriate relationship between system variables and fuzzy variables. The uni- verse of disclosures of the error and error change are divided into 11 fuzzy subsets with integer values from -

5 to +5. The parameter M in the rule modification

equation is chosen to be 1 for simplification. Its effect was replaced and included in the parameter y The value of the learning speed parameter yis selected to be 0.5, to achieve moderate rule correction. The weighting parameter E is chosen to be 0.5 with equal weighting to output error and error change. The adjusting factors are set at 1, 3 and 0.5 for GE, GC and GU, based upon experimental response. Since the SOC can start to operate with zero initial fuzzy rule and then establish and modify the appropriate fuzzy rules in each sam- pling step, it has good adaptability and robustness to face various working conditions and eliminate the ini- tial trial and error search for appropriate parameters and rules for fuzzy control implementation. Fig. 3 exhibits the sinusoidal trajectory tracking error for the SOC with zero initial fuzzy rules. The system output

-1

-2

0 2 4 6 8 10 12 14 16 18 20

time$

Fig.3

fuzzy rule Sinusoidal trajectory tracking error of

SOC with zero initial

response converges to Ct0.03" after a quarter cycle of the sinusoidal wave. The circular path tracking error is shown in Fig. 4. The contouring error is within 0.015 mm except for the 0.04mm impulse oscillation when backlash occurs. The trajectory tracking error in the x and y directions for tracking a desired circular path are shown in Figs. 5 and 6. The trajectory tracking error of

the x and y axes are within f0.02 and fO.Olmm, respectively, except for the impulse error caused by backlash. The backlash effect is reduced to T0.04". This path tracking error is about one half of that of a PID controller, and that is better than a self-tuning feed forward cross-coupling controller.

0 04 I

I

- ' 1 ' I -0.04

'

0 5 10 15 20 25 30 35 40 45 time,s Fig.4 Circular path tracking error with SOC

0.04 I ! " ,

. -

- 0.02

I

I i E

5

g o

8

0 5 10 15 20 25 30 35 40 45 time,s Fig.5 Trajectory tracking error of x-axis with SOC

E

e

b o

5-

8

I -0.04 0 5 10 15 20 25 30 35 40 45 time,s Fig. 6 Trajectory tracking error of y-axis with SOC

Since the neural network learning ability depends crucially on the quality of training data, the neural net- work was off-line trained with the I/O data of a SOC.

The neural network has gone through one cycle of off- line training with the input and corresponding output data of a self-organising fuzzy control process. The size of the training data comprises 9000 sets. The neural network was then employed to on-line control the machining table. This neural network has two input nodes, 25 hidden nodes and two output nodes. During the on-line control, the parameters

A

and a are chosen

as 0.001 and 0.005, respectively. The learning rate parameter q is selected as 0.5 for off-line, and 0.0001

for on-line operations, respectively. The initial weight- ing values of the neural network were chosen as arbi- IEE Proc-Control Theory AppL, Vol. 145, No. 4, July 1998

trary random real values between -1 and + l . The trajectory error histories of x and y directions for tracking a desired circle path on the x-y plane are shown in Figs. 7 and 8, respectively. The trajectory tracking error is within kO.Olmm for both axes. The backlash efffect is reduced to within f0.02". The experimental results show that the neural network has the best tralcking controll performance.

. , - I [ ,

0 5 10 15 20 25 30 35 40 45

time,s

Trajectory tracking error of x-axis with neural network control Fig. 7 L

e

b 5 . 0

8

' / I j ! , , , , , , 0 5 10 15 20 25 30 35 40 45 time,s

Trajectory tracking error of y-axls with neural network control Fig.8 L b a o 8 111 t ' l ' l L

-

c I ' 11 I L - I , -0 02

e l -

; I - - :

T

I I I c

j

0 5 10 15 20 25 30 35 40 45 time,s

Fig. 9 Circui'ar path tracking error for selforganising fuzzy control In order to investigate the influence of tracking speed and nonlinear behaviour of this plant, a circle tracking path with a different radius (25") and different tracking speed is planned to verify the adaptability/ robustness of these two intelligent controllers. The path tracking error of the self-organising fuzzy control and neural network control are shown in Figs. 9 and 10, respectively. The path tracking error of a SOC is within

kO.01 mm, except for the f0.03" impulse oscillation from backlash. The path tracking error of the neural network control is within f0.007mm, except for the f0.02" impulse oscillation due to backlash. In addi- tion, this machining table has been used in milling operations for machining aluminum and acrylic plastic sheets under various milling conditions in our labora-

tory. The path tracking accuracy can almost be main- tained at the same level. The intelligent controllers can tolerate the environmental noise due to cutting condi- tion variations; e.g. the contouring error for machining a circular path with a 30” radius on an acrylic plas- tic sheet by using a neural network controller is shown in Fig. 11 The path tracking error of the neural net- work control is within iO.Olmm, except for the L0.025” impulse oscillation due to backlash.

E

7 o o l /

-1

I;/

1’;

y:~[-;;

j

-0 02 I

0 5 10 15 20 25 30 35 40 45

time,s

Circular path tracking error for neural network control

Fig. 10 0.04 E P b

E:

E-

0.02

E

o

L -0.02 0 5 10 15 20 25 30 35 40 45 times Fig. 11

plastic sheet with neural network control Contouring error of machining a circular path on an acrylic

5 Conclusion

Self-organising fuzzy control and neural network con- trol strategies were employed to manipulate a retrofit- ted machining table. Both intelligent controllers have model-free features to overcome the difficulty of system modelling and the nonlinear characteristics of the con- verted plant. The fuzzy rules table of the SOC was not established by relying on expert experience. The appro- priate fuzzy rules table can be established and modified for each control situation, based on its self-learning capability. This feature has effectively reduced the time consuming trial and error process for finding appropri- ate fuzzy control parameters and fuzzy control rules during implementation. The experimental results of these approaches have better robustness and control performance than that of PID and self-tuning feed for- ward cross-coupling control methods. The path track-

ing errors are within k0.04 and k0.02” when using self-organising fuzzy control and neural network con- trol, respectively, in spite of the influence of about

0.4” of backlash on each axis. If the mechanism of the milling machine has improved properties, the con- trol performance will be better. In addition, both con- trol schemes have learning abilities to adjust their control structures on-line to compensate for system time-varying properties. This all verified the feasibility of these control schemes for industrial applications.

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