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MICROPROCESSOR-BASED FORCE CONTROL FOR MANIPULATOR USING VARIABLE STRUCTURE WITH SLIDING MODE Y. Dote*, T. Manabe* and S. Murakami** *Department of Electronic Engineering, Muroran Institute of T echnology, Mi:.umoto 27-1 , Muroran, Hokkaido, japan **Yaskawa Electric Mfg. Co. Ltd., Otemachi Building, Chiyodaku, Tokyo , japan

Abstract. The desi gn principles of sliding mode control a re studied and then are applied to force control of a manipulator hand powered by a P\~ transistor converter-fed dc servomotor. Each step in the formulation of the control algorithm is clearly defined and explained. The feasibilit y of this control scheme is experimentally verified. The control strateg y is implemented by using a Motorola 6809 Microprocessor. The proposed c ontroller is robust in both dynamic and steady state performances and provides extremely smooth force control independent of s ystem parameter variat ions, nonlinearities and external disturbances. In addition to this, the desi gn procedure of the controller is very simple, since it does not require accurate modeling, but it is sufficient to know only the bounds of the model parameters and disturbances. The designer can easily determine a desired time constant by choosing a sliding curve. Keywords. Force control; manipulator;nonlinear system;va riable structure; sliding mode;robustness~ P~ transistor converter-fed dc servomotor. INTRODUCTION Drazenovic's(1969) work and is simple to be designed, since it does not require ac cur a te modeling, but it is sufficient to know onl y the bounds of the model parameters and disturbences. First, a dc servomotor s ystem has been represented by a set of state variables. After presenting the derived model, the paper examines the exsistence and reachability conditions of the sliding mode and the sta bility in the sliding mode. Then the c ontrol al gorithm is defined. The fe as ibility o f this control strate gy is verified by the experime nts. It is implemented by usin g a Motorola 6809 Microprocessor. A mecha ni c al hand powered by a Pw}! transistor converterfed dc servomotor is used. The satisfact or y results are obtained. Usuall y a PI controller is utilized t o f orce the system t o the desired final va l ues. However, an integrator represents an inertial effect and causes the system to overshoot . The response for the return transition is slow. The proposed controller provides much f a ster responses than the conventiona l linear PI controller.

Motor:. drives have traditionally benefitted from developments in other fields, such as control theory, electronics and instrumentation, etc .. Recently, a new method for control of nonlinear systems has become available f or practical applications by investigators (Itkis;1976,Utkin;1977, Young ;1978,Klein;1979,Sabanovic;1979,Dote;1980). The method, called sliding mode control, belongs to a wider class of variable structure systems. It is applied to force control of a mechanical hand powered by a P~ transistor converter-fed dc servomotor. In the over all system there exsist ma ny nonlinearities including differential efficiencies depending on direction of motor and stiction in entering and leaving the lock state, variable torque loads, gear train backlashes and well-known nonlinearities in the converter-fed dc servo drive systems. When applying the sliding mode control, however, the nonlinear system can be made to operate as if it were linear with a time constant determined by the designer. Thus the controller causes the system to follow a prescribed straight line (sliding curve) path in phase space by a discontinuous switching between different control laws along the sliding curve. This controller is insensitive to system parameter variations, nonlinearities and external disturbances according to

DERIVATION OF MODEL A 100 Yaskawa Servopack supply ing a 100 Yaskawa dc servomotor with a fixed field is used her for powering a mechanical hand with a dedicated M~torola 6809 microprocessor 145

Y. Dote, T. Manabe and S. Murakami

146

ontroller. A block diagram of the control system is shown in Fig .l. A semiconductor strain gauge and a small dc tchometer provide a force anf a speed feedback signals respe ctively . The state variable representation for this system may be written as w

= k. Thus, for ss <0, the following inequlity must be satisfied. sF

err

2 ( -c k - ac + bcjJ

Therefore i f 2 -c k 2 if -c k -

sFe r> 0, ac .+ ba < sF < 0, e r a c .+ b 8 >

<0

(10)

choose a 0 choose 8 0

such that (ll)

such that (12)

(1)

w

-aw +

y

f( S)

bu - f(t)

(2)

Where a=(k kb)/R + f/J, b= k k I(JR ), f(t)=T IJ, a a amp a a = angular displacement , y=output w = angular velocity , (force) u= input to PWM power amplifier, k = power amp. gain, kb emf constant, k =t~&ue constant, J=inert1a, R = armature a a resistance, T =external torque 1 Let e 1 be ( 6 - Sd)' Sd is a desired commana position satisfY1ng f( Sd) - F f=O, where F re is a desired force(constanEj. • Let F e ( f( S) - F f) and e =e =w. l 2 Then {If a~d (2) becom~e

e

t

(3)

y = f( e + Sd ) (4) l is applied such A force feedback control that

become~

e 2 -ae

~l

e

2

(6)

+ bcjJ Ferr - f (t) 2

Sliding Mode Control The sliding curve s=O is defined a s e

2

= 0,

c >O

(7)

A variable structure control law with sliding mode is of the form u = cjJ F + Where m is a consl~fit.

:

[

m sgn s if s F if s F

err err

(8)

>0 ( 9)

<0

Existence Condition of Sliding Mode We select a and S such that sand have opposite signs. s s

=

s

= s

+ e Let

f(t)/b

s

c Ferr + e 2 c df(el+Sd)/de

< m and

2

l

e

Stability of Sliding Mode When system (6) is in the sliding mode, its trajectories in the plane (F ,e ) satisfy s= cF +e =0 which implies,err 2 err 2

~l = -c(f(e l + Sd) - f( Sd»

(l3)

For example, if f(e +S )=(e +s )3 l d l d 3

2

2

(14)

e l = -c ( e l + 3 Sdel + 3Sdel

In order to check the stability of (14i' generate a Liapunov function V(e ) = e and l l

·22

To illustrate some of the fundamental design c oncepts o f variable structure systems with sliding mode, consider equation (3).

c Ferr +

Reachability Condition of Sliding Mode For the nonlinear system (6) , some Liapunov technique in respect to s has to be utilized for obtaining a reachable region. We must consider the input constrain which also yields a finite reachable region in the phase plane.

2

V(el)=2elel=-2cel(el + 3S e + 3S ) < 0 d l d

SLIDING MODE CONTROL DESIGN

s =

This is shown in Fig.2. The sliding mode occurs on s=O, and the motion continues along the switching curve (7). Notice that the sliding mode is not a part of trajectories which belong to either of the two feedback systems.

l

) < 0

max. df(el+Sd)/de

l

Thus, (13) is globally asymptotically stable. The setting time ts can be estimated as •

2

2

~ -V(e )/V(e ) = 1/(2c(e +3 S e +3 S » d l d~15) l s 1 1 The larger c and el(O) are, the smaller td is. Equation (15) does not depend on system parameters. The response speed in the sliding mode is determined by e (0) and by c which we can choose arbitrarily . However, in reality, the control switches at finite frequency. Therefore the system trajectory does not follow the ideal sliding curve, but it becomes very close to this curve when the switching frequency is sufficiently high. If the switching frequency is not sufficiently high, the effects of chattering must be evaluated. Let the magnitude of the chattering be ~ , and the switching delay time be t , d then for sufficiently small t , d (16) ~ "' s td EXPERIMENTAL RESULTS

t

The control algorithm developed in the previous section was practically implemented. The microprocessor system used to control a gripping force of a mechanical hand is summarized in Fig.3. The heart of the system is a FM-8 (Fujitsu) Microprocessor, which

Microprocessor-based Force Control is based on the Motorola 6809. The system includes a keyboard input device, a CRT display, a disket memory and its driver and a printer which have been used for program development and for storing a~d diSplaYing the measured values of e and e. Twelve bit AID and DIA converters provide sufficient accuracy. Programs are written in a assembler. For sensors a semiconductor strain gauge and a dc tachometer are used. A PWM transistor converter-fed dc servomotor is utilized to power a mechanical hand. The control microprocessor has read the force and the motor speed, then compared these quantities to the command values, and finally computed and outputed motor armature voltages which would decrease the error between measured and command quantities. A control program whose flow chart is given in Fig.4. was written for the 6809 microprocessor system. This program has several modes of operation including 1) a data entry mode which the user can change program parameters (background) 2) A display mode where the measured and stored values are displayed on the CRT and are printed out (back ground) 3) An automatic control mode where a sliding mode control is performed and the measured variables are stored( fore ground). The microprocessor finishes the complete set of computations to control the gripping force every 4 ms and simply waits for the next cycle. Thus the ideal infinite switching frequency is limited by the microprocessor speed. The phase plane trajectories are obtained from actual measurements. These are stored in the computer memory and displayed on the CRT. The printed outputs are shown in Fig.S and Fig.6 shows step responses corresponding to the trajectories in Fig.S. A very smooth force control has been achieved. As the slopes decrease and hence the time cons~ tants increase, more chatter is noticeable. However it is more insensitive to non1inearities, parameter variations and external disturbances. CONCLUSION Measurements verify that the sliding mode method is an effective means of forcing an intrinsically nonlinear device to act as a linear system. Sliding mode has been found to be useful for and applicable to important force control of a general manipulator. Implementation of this method using a microprocessor has demonstrated that it can be simple, practical and inexpensive. Several things need to be considered when applying sliding mode control to a system. First,the e ffects of chatter must be evaluated. While in this system chatter produced no difficulties, it could in some applications. Lags in the control path can produce excessive chatter, so sensor and power circuitry must be well designed. Compliance of the mechanical hand may be an important factor in the ability to control gripping forces as done here. The design procedure can be outlined as

J47

follows. First, the desired sliding mode is formed by a choice of parameter c, which determines the dynamic response in the sliding mode. Second, a discontinuous control law is found which garantees the existence of sliding modes at every point on the curve s=O. Therefore the design method is simpler compared with that for a conventional PI controller. Although a PI controller has been traditionally used in order to make the steady state error zero, it is very difficult to tune controller parameters with both "on-line" and "off-line". Even if they are tuned for a specific disturbance, steady state and dynamic errors result for another disturbance including nonlinearities. ACKNOWLEDGEMENTS The authors wish to ackowlege the Yaskawa Electric Mfg. Co. Ltd. for their stimulation and financial support of this project. REFERENCES Itkis,I. (1976). Control Systems of Variable Structure. Wiley, New York. Utkin,V.I.(1977). Variable Structure Systems with Sliding Mode. ~~ Trans, Autom, Control, 22, 212-222. Young,K.D.(1978). Controller Design for a Manipulator Using Theory of Variable Structure Systems. IEEE Trans.Systems, Man and Cybernetics,8,101-109. Klein,C.A. and Maney,J.J.(1979). Real-Time Control of a Multiple-Element Mechanical Linkage with a Microcomputer. IEEE Trans. ,Industrial Electonics and Control Instrumentation, 26,227-234. Sabanovic,A. and Izosimov,D.B.(1979). Application of Sliding Modes to Induction Motor Control. IEEE IAS Conference Record,Cleveland U.S.A.,793-80l. Dot;:Y. and Hoft,R.G.(1980). MicroprocessorBased Sliding Mode Controller for DC Motor Drives. IEEE IAS Conference Record ,Cincinnati U.S.A.,64l-64S. Drazenovic,B.(1969). The Invariant Condition s in Variable Structure Systems.Aptomatica,S,287-29S.

Y. Dote, T. Manabe and S. Murakami

148

Speed reference

---------

Manipulator ' system

DC Servo motor

Tacho-meter Output

*- ..... --.- .. ---

'If

__2> __ ~ITp ---. --~ __-_-_1_ _ Fig.l

Transistor PWM Amp.

Block diagram of control system

Region 1 -------~~-------Ferr

Line S Slope C

1 2

Fig.2

Phase plane(force, angular velocity) showing how sliding mode occurs by different feedback laws depending on value of Ferr and e 2 ( w)

Transister Pwtl Amp.

DC Motor >r-_ _.... _....

Tachometer ~

Fig.)

Hardware details of conputer control system

Microprocessor-based Force Control

-------1 ._--lI..-_...

No

u= ex

err +m sgn s

No

:_________ - -1 Fig.4

Flowchart of sliding mode control program

.... .

Fig.S

Example of state trajectories in phase plane (e,e)

force

300 ms/d1v

• Fig.6

Example of step responses ( Force versus time)

149

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