Modeling and Robust Adaptive Control of a 3-Axis Motion Simulator

Xie Yue, Mahinda Vilathgamuwa, K. J. Tseng and N. NagarajanSchool of Electrical and Electronic Engineering

Nanyang Technological UniversityNanyang Avenue, Singapore 639798.

e-mail: p148881015@ntu.edu.sg, emahinda@ntu.edu.sg, ekjtseng@ntu.edu.sg, enara@ntu.edu.sg

Abstract-In this paper, the development of a 3-axis motionsimulator is described. The simulator is used to test andcalibrate certain spacecraft instruments within a hardware-in-the-loop environment. A mathematical electromechanical modelof the simulator is developed. Moreover, a novel robust adaptivenonlinear control law for the simulator is developed based onLyapunov stability theory. The controller can be madeadaptable to constant unknown parameters as well as robust tounknown but bounded fast varying disturbances. The motionsimulator actuators are Permanent Magnet Synchronous Motor(PMSM) drives. The simulation and experimental results arepresented to verify the efficacy of the proposed control systemand the validity of the mathematical model of the simulator.

I. INTRODUCTION

The instruments used in a spacecraft are mission criticalreal-time embedded systems, which have to be tested toensure reliable operation in space [1]-[3]. In order to verifytheir operative correctness, an end-to-end system testing mustbe carried out. Towards this end, a 3-axis motion simulatorcan be used to test the operation of sun sensor, star sensor andother gyroscopic instruments used in a spacecraft within aspace-like environment. The motion simulator generates auser defined 3-dimensional motion profile by using 3 PMSMdirect drive motors fixed along roll, pitch and yaw axes. In this paper, the mathematical model of this mechanicalstructure is derived by means of calculating the angularmomentum of each rotating part with respect to its referenceframe. A compact mathematical model of the wholeelectromechanical system is obtained by representing thePMSM motor drives in the field orientation. There arecomplex nonlinear couplings among three axes in themathematical model of the simulator, which may degrade thecontrol performance. In order to realize high performance motion control of thishighly nonlinear electromechanical system, the input-outputlinearization approach can be used [4], [5]. However, inpractice, the model of the simulator is usually imprecise.There are uncertainties in motor parameters due tomeasurement errors and in load inertia due to change of thepayload. In order to overcome the effects of parameteruncertainty on controllability of the plant, a few controlmethods have been proposed, e.g. sliding mode control,adaptive nonlinear control [6]-[12]. However, most of theseworks concentrate on single motor driven actuator systems incontrast to the proposed system which has 3 axes and ishighly nonlinear. The existing complexity of the system canbe aggravated with the appearance of more coupling terms inthe torque equation if the payload centre of gravity deviatesfrom the intersection point of the three axes. These torquecomponents are in addition to Centripetal and Coriolis torquecomponents, which have already been considered in themodeling process. Despite the presence of these torquecomponents which can be categorized as disturbances, the

system can be made effectively insensitive to disturbancesand parameter variations by adopting a robust adaptivecontrol law. The efficiency and correctness of the proposedcontrol system are verified using simulation results andlaboratory experimental results.

II. THE STRUCTURE OF THE 3-AXIS MOTION SIMULATOR ANDHARDWARE-IN-THE-LOOP TESTING

The structure of 3-axis motion simulator is shown in Fig. 1.It is comprised of an outer gimbal, an inner gimbal, a testtable, three motors, counterweights and a base. The payloadis mounted on the top of the test table. The outer gimbalrotates around a vertical yaw axis, it is driven by a motorlocated inside the base. The inner gimbal moves around ahorizontal pitch axis and is driven by another motor, which isfixed on the outer gimbal. The third motor is fixed on theinner gimbal and drives the test table which rotates about theroll axis, which is perpendicular to the pitch axis. The yaw,pitch and roll axes meet at a single point in space. The wholestructure is designed to be axi-symmetric. Thecounterweights fixed in the inner gimbal are used to balancethe pay-load such that the center of gravity of the innergimbal and the payload lies on the pitch axis. The threemotors used are PMSM direct drive rotary (DDR) motors,

Fig. 1. 3-axis motion simulator

553

0-7803-7116-X/01/$10.00 (C) 2001 IEEE

which are installed along these three axes and are named roll,pitch and yaw motors respectively. Primarily there are two kinds of 3-axis motion platforms.The first platform is 3-axis air bearing with sensors,actuators, and control electronics mounted on it. The secondis 3-axis servo table platform for sensors with controlelectronics and actuators outside the platform, and the controlloop is closed through a digital computer that simulatessatellite dynamics and in turn drives the servo table. Airbearing simulations are the most common throughout thespace industry for testing the satellite Attitude ControlSystem (ACS) as it is easy to implement. However, it hasmany shortcomings for precise evaluation of the performancedue to undesirable air bearing torques comparable to controltorques and difficulties in simulating actual satellite inertia.In recent times, precise servo table simulations have becomemore common [3]. A block diagram of Hardware-in-the-loop (HWIL)simulation for the satellite ACS systems is shown in Fig. 2.The main components of HWIL simulation include rotationtable, motion table controller, sensors, data acquisition andReal Time (RT) simulator. This rotation table, namely motion simulator is used formounting and calibrating the performance of attitude sensors(like sun, star sensors, and gyros). The motion table controllerincludes motion control processor and electronics, it is drivenby track inputs, ..ei satellite’s attitude, angular rates andangular acceleration, from the satellite RT simulator. Sensorsmeasure the attitude signals excited by the rotation table and,through sensors-flight control processor interface, provide theoutputs to ACS computer for RT simulation. Data acquisitionprocessor logs in the data from sensors, motion controller,and motion table for post analysis. The satellite dynamics aresimulated in real-time, based on a mathematical model of thespace environment, which includes atmospheric drag, solarradiation pressure and magnetic field, the satellite rigid bodydynamics and certain additional ACS components. Thesatellite position, rate and acceleration calculated from the

simulated satellite behavior in space are fed to motioncontroller and they are used to drive the motion simulator toexcite those optical and inertial attitude sensors.

III. DYNAMIC MECHANICAL MODEL OF 3-AXIS MOTION

SIMULATOR

The simulator has three rotating parts, namely roll partincluding payload and test table; pitch part including innergimbal, counterweights and roll motor; and yaw partincluding outer gimbal and pitch motor. As shown in Fig. 3,angles φ ,θ and ψ are denoted as roll, pitch and yaw anglesrespectively and measured from their initial states.Accordingly, the angular velocities about these three axes are

)(φω &r , )(θω &p and )(ψω &y . Note that n is perpendicular to

rp plane. The angular velocity of rpn frame fixed on theinner gimbal is,

nrp θωθωω

ωωω

sincos yyp

yprpni

++=

+=rrr

(1)

The angular velocities of the simulator about three axesexpressed in rpn frame are,

nr

pr

θωθωω

ωωωω

sincos

,

yyy

pprr

+=

==r

rr

(2)

The angular momentum of the roll part is, )( yprrr IH ωωω

rrvr++= (3)

where the roll part is assumed to be axi-symmetric, and itsconstant moment of inertia matrix is of the form

],,[ rpnrpnrrr IIIdiagI = in the rpn frame. Therefore the

external torque irM

racting on the roll part is,

npr rnrprr

rrpnirpn

rir

MMMHHM

++=×+=

rr&rrω (4)

where rrM is produced by the roll motor, while

rpM and rnM are acting as transverse forces on the bearings,

Fig. 3. Axial representation of motion simulator basedon classical Euler angles

X

Y

Z, y(yaw)

p(pitch)

Y1

r(roll)

ψ

θ

ψ

θ

ψ&

θ&

φ&

n

O

Fig. 2. HWIL Simulation for the satellite ACS systems

554

which connect the shaft of the inner motor. Combining (1)with (4), we have, rrrrrr IIM )sincos( θθψθψφ &&&&&& −+= (5)

Similarly, the angular momentums of the pitch part andyaw are given by )( ypprp IHH ωω

rrrr++= (6)

yyypy IHH ωrrr

+= (7)

where constant moment of inertia matrix of the pitch part is],,[ pnppprp IIIdiagI = and yyI is yaw part moment of

inertia about the yaw axis. The rotation matrix from rpn frame to the inertia frame isgiven by

−

−=

θθθψθψψ

θψθψψθψ

csscccs

sscscRirpn

0),( (8)

where c indicates cosine and s indicates sine. Using aformation similar to (4), we can obtain the external torque

components ipM

rand i

yMr

, which act on the pitch part andyaw part, in rpn frame and inertia frame respectively.Among these torque components, we have

θθψ

θψφθ

csIIII

sIIIM

pnrpnprrr

rrpprpnpp2)(

)(

&

&&&&

−−+

+++= (9)

)2)((

)()(

2 θθψθψ

θθφθφψ

scIIII

scIIIIM

pnrpnprrr

rrpnrpnyyyz

&&&&

&&&&&&

−−−++

−+++= (10)

where ppM and yzM are produced by the pitch and yawmotors respectively. The mathematical model of the motion simulator can beobtained by combining (5), (9) and (10): MT f =−τ (11)

where Temyempemr TTT ][=τ is demanded torque vector of

the three motors; Tyzpprr MMMM ][= is the load torque

vector. Disturbance torque vector Tfyfpfrf TTTT ][= is

comprised of friction, external disturbances and additionalcoupling torques due to unstructured uncertainties inmodeling process, e.g. moment of inertia matrix rI being notdiagonal because of the payload’s uncertainty, both in itsshape and location on the test table and the non-orthogonalityof yaw and roll axes about the pitch axis. The mathematical model (11) gives the relationshipsbetween motor torques and rotational angles about three axes,which are directly associated with the payload’s attitudevariables expressed with Euler’s angles [13]. This makes itconvenient to control the motion simulator within a satellitebased reference frame.

IV. MATHEMATICAL MODEL OF ELECTROMECHANICALSYSTEM

In synchronous d-q rotor reference frame, the PMSMmotor equations can be written as,

( )[ ]qiiqidiqidiiemi

iiiidiidiqiqiqiiqi

iqiiqidididiidi

iiiLLPT

PiPLiLiRu

iPLiLiRu

λ

ωλω

ω

+−=

+++=

−+=

23

&

&

(12)

where ypri ,,= ( ypr ,, denote roll, pitch and yaw motors);

diu , qiu are stator voltages; iR , diL and qiL represent stator

resistance and motor d and q axis inductances; qidi ii , are dand q axis currents; iλ is flux linkage of the rotor magnet;

iω , iP and emiT are rotor speed, number of pole pairs andelectromagnetic torque of the motor respectively. Rearranging (11), we have τ=++ fTqqqCqqH &&&& ),()( (13)

where Tq ][ ψθφ= is the position vector. )(qH is the

simulator inertia matrix which is a symmetric positivedefinite matrix.

+=

θθ

θ

2341

2

11

000

0)(

cIIcII

cIIqH

−−−

−=

θθθθθψθθθθψθψ

θψ

csIcsIsIcsIsI

sIqqC

&&&&&

&

&

331

31

10

00),(

pnrpnyy

pnrpnprrr

pprpnrr

IIIIIIIII

IIIII

++=

−−+=

+==

4

3

21

;

;

The mathematical model of the electromechanical systemcan be obtained by combining (12) and (13) as

u000010001

TqCFF

xH000A000A

f

q

d

q

d

+

+−−

=

τ&

& (14)

where

=

dy

dp

dr

dL

LL

A00

0000

,

=

qy

qp

qr

qL

LL

A00

0000

Tdydpdrd

Tqd

Tqyqpqrq

Tdydpdrd

Tqd

uuuu

Ruuuu

qiiiiiiii

Rxiix

][

,]0[

,][,][

,][

9

9

=

∈=

===

∈=

&ω

ω

Tqyqpqrq uuuu ][=

Tqyqpqrq

Tdydpdrd

fffF

fffF

][

][

=

=

555

iiiidiidiqiiqi

iqiiqidiidi

PiPLiRfiPLiRf

ωλω

ω

−−−=

+−=

By differentiating (13), the input-output linearization ofMIMO system (14) can be carried out as follows,

+

++−−+−

=

q

d

qdf

d

dd

uu

DBDFBFTqCqCHF

qi

HA

01)(

00

&&&&&&

&&&

&

(15)

where B and D are diagonal matrices and are described by

=

dyqyy

dpqpp

drqrr

LiaLia

LiaB

/00

0/000/

++

+=

qyydyy

qppdpp

qrrdrr

LbiaLbia

LbiaD

/)(000/)(000/)(

iiiqidiii PbLLPa λ23),(

23 =−=

As the total relative degree is the same as the order of thesystem (14), there is no internal dynamics.

V. ROBUST ADAPTIVE NONLINEAR TRACKING CONTROLDESIGN

Now the control task is to get the output q to track a

specific time-varying 9-dimensional vector, Tqqq ][ ∗∗∗∗ = &&&q ,i.e. a desired trajectory based on satellite attitude dynamics.Assume motor parameters and simulator’s moment of inertiamatrices are unknown, disturbance vector fT is unknown but

has known bounds. For the outputs di and q of the system (14), two slidingsurfaces 1s and 2s are defined as follows,

∫+=t

d dtidtds

011

~)( λ (16)

qdtds ~)( 2

22 λ+= (17)

where ∗−= ddd iii~ , ∗−= qqq~ ; ∗di and ∗q are reference

values of di and q ; 1λ and 2λ are strictly positive constantdiagonal matrices. In order to select a suitable Lyapunov function for entiresystem and therefore to obtain a suitable control law, thefollowing two functions are defined,

111 21)( sAstV d

T= (18)

222 21)( HsstV T= (19)

where dA and )(qH are symmetric positive definitematrices. Differentiating both sides of (18) and (19), andusing (15), yields

][)(3

111 d

iii

T uaYstV +−= ∑=

& (20)

][)(6

422 qf

iii

T DuTaYstV +−−= ∑=

&& (21)

where 61 ~ YY are known matrices based on ∗dqd iii ,, , ∗qq,

, ∗di& and ∗q&&& (see the Appendix). 6Y contains du and is

assumed to be known as it is based on (22). 61 ~ aa areparameter vectors of uncertain parameters and theircombinations,

Typr RRRa ][1 = , T

qyqpqr LLLa ][2 =T

dydpdr LLLa ][3 = , TIIIIa ][ 43214 =6

518

55555 ,,][ RaRaaaaa iT

ypr ∈∈=

rqi

di

di

qii R

LL

LL

a )(15 −−= , iqi

i2i5 R

La λ

=

qidi

qidii L

LLL

a)(

35−

−= , diqi

qidii L

LLL

a)(

45−

=

iqi

qidii L

LLa λ

)2(55

−= ,

qi

ii L

a2

5 6λ

=

T

dy

qydy

dp

qpdp

dr

qrdrL

LLL

LLL

LLa ]

)(,

)(,

)([6

−−−=

Taking the control law to be 11332211 ˆˆˆ sKaYaYaYu Dd −++= (22)

)]sgn(ˆ[ˆ2322

6

4

1 sKsKaYDu DDi

iiq −−= ∑=

− (23)

where 1DK and 2DK are constant positive definite diagonalmatrices; 3DK is a positive diagonal matrix; )6~1(ˆ =ja j

and D are the estimated parameter matrices. By substituting the control law in (20) and (21), we have

]~[)( 11

3

111 sKaYstV D

iii

T −= ∑=

& (24)

)](sgn)](sgn

ˆ[ˆ~~[)(

232223

22

6

4

16

422

sKsKTsK

sKaYDDaYstV

DDfD

Di

iii

iiT

−−−−

−+= ∑∑=

−

=&

& (25)

where the known function fiD TjjK &−≥),(3 , ( 1=j , ri = ;

2=j , pi = ; 3=j , yi = ), thus 3DK can be used tocompensate time-variable disturbances. Parameter estimatederror iii aaa −=ˆ~ . Note that D is a diagonal matrix, we have

88772322

6

4

1 ~~)](sgnˆ[ˆ~ aYaYsKsKaYDD DDi

ii +=−−∑=

− (26)

556

where 7Y and 8Y are known matrices (see the Appendix).Parameter vectors 7a and 8a are given by,

T

qy

qydy

qp

qpdp

qr

qrdrL

LLL

LLL

LLa ][7

−−−=

T

qy

y

qp

p

qr

rLLL

a ][8λλλ

=

Now let Lyapunov function candidate to be

∑=

−Γ++=8

1

12211 ]~~[

21

)(i

iiTi

Td

T aaHsssAstV (27)

where iΓ is a symmetric positive definite matrix. Differentiating (27) and using (24) and (25), we have,

∑

∑∑

=

−

==

Γ+−

−−+−=

8

1

123

22

8

4211

3

11

~ˆ)](sgn

~[]~[)(

iii

TiD

Dfi

iiT

Di

iiT

aasK

sKTaYssKaYstV

&

&&

(28)

The parameter estimation update rules can be described asfollows

1ˆ sYa Tiii Γ−=& , 3,2,1=i (29)

2ˆ sYa Tiii Γ−=& , 8,7,6,5,4=i (30)

Thus, we have 0)( 222111 ≤−−≤ sKssKstV D

TD

T& (31) This means that system output errors converge to thesliding surfaces 01 =s and 02 =s as defined in (16) and

(17). Moreover ia is bounded. Therefore, both globalstability of the system and convergence of the tracking errorare guaranteed by the developed robust adaptive control law.

VI. SIMULATION RESULTS

The performance of the developed control law isinvestigated with respect to tracking error and robustnessusing computer simulations. The desired trajectories are )sin( tAy iii ω= and their

derivatives, where θφ,=y andψ ; 1.0=iA rad; 0.2=rω rad/s;

5.1=pω rad/s; 0.1=yω rad/s, and 0=∗di .

Also assume the initial positions and velocities are3.0)0( =φ rad; 2.0)0( =θ rad; 1.0)0( =ψ rad; 3.0)0( −=φ&

rad/s; 2.0)0( −=θ& rad/s, 1.0)0( −=ψ& rad/s and the initialaccelerations are zero. A constant disturbance

TfT ]202010[= Nm is introduced at =t 5 s and it lasts for

1 s . Actual plant and control parameters used in thesimulation are listed in the Appendix.

The output q track the desired trajectories ∗q as shown inFig. 4 to Fig.7.

Fig. 4. Three axes position tracking

Fig. 5. Three axes velocity tracking

Fig. 6. Three axes acceleration tracking

Simulation results show that the simulator can track thedesired trajectories about three axes perfectly, even withdisturbances and parameter uncertainty.

0 2 4 6 8 10-0.5

0

0.5

Rol

l Axi

s

0 2 4 6 8 10-0.2

0

0.2

0 2 4 6 8 10-0.2

0

0.2

Pitc

h A

xis

Yaw

Axi

s

Time (sec)

Posi

tion

(rad)

0 2 4 6 8 10-0.5

0

0.5

0 2 4 6 8 10-0.5

0

0.5

0 2 4 6 8 10-0.2

0

0.2

Rol

l Axi

s

Vel

ocity

(rad

/sec

)

Pitc

h A

xis

Yaw

Axi

s

Time (sec)

0 2 4 6 8 10-4

-2

0

2

0 2 4 6 8 10-2

-1

0

1

0 2 4 6 8 10-0.5

0

0.5

Rol

l Axi

s

P

itch

Axi

s

Y

aw A

xis

Acc

eler

atio

n (r

ad/s

ec2 )

Time (sec)

557

Fig. 7. Three axes tracking error vectors q~,q~,q~ &&&

VII. EXPERIMENTAL RESULTS

Initial laboratory experiments are carried out on a single-axis PMSM motor drive system, using a 32-bit floating-pointTMS320C31 digital signal processor (DSP) based motioncontroller. The block diagram of the experiment system isshown in Fig. 8. As this is a single-axis drive system, theparameters of the robust adaptive control law developed for3-axis motor drive system should be modified. The controllaws with modified parameters for the single-axis system areas follows: 11332211 ˆˆˆ sKaYaYaYu Dd −++= (32)

)]sgn(ˆ[ˆ2322

6

4

1 sKsKaYDu DDi

iiq −−= ∑=

− (33)

1ˆ sYa Tiii Γ−=& , 3,2,1=i (34)

2ˆ sYa Tiii Γ−=& , 8,7,6,5,4=i (35)

where parameters 81 ~ aa and 81 ~ YY have forms similar totheir respective counterparts in the 3-axis system. (seeAppendix). The only exception is 4Y , which is associatedwith coupling issues. However, the robust adaptive controllaw of the single-axis motor system is similar to that of the 3-axis motor drive system. Therefore preliminary investigationon the performance of the proposed control law can becarried out using the single-axis drive system The 4-pole permanent magnet synchronous motor shaft isdirectly coupled to a position sensor and a dynamometerwhich is used as a programmable load. The motor is suppliedby a three-phase voltage-source PWM inverter with aswitching frequency of 5 kHz. Actual motor parameters, their initial estimated values andcontrol parameters are listed in the Appendix. The desired trajectories are )3.0(sin40 t=θ and its

derivatives and 0=∗di . Assume that the initial position

0.0)0( =θ rad and the motor is initially at standstill. That

means initial velocity error 12)0( −=θ& rad/s. Position, velocity and acceleration have a well-definedrelationship. Therefore, velocity and acceleration can be

derived from position signal measured by a high-resolutionencoder. A lowpass filter is used for acceleration calculation.A third order Butterworth filter is used for this purpose andthe cut-off frequency is 5 rad/s. Fig. 9 shows the motor shaft position, velocity andacceleration tracking the desired trajectories. Fig. 10 showsthe control voltages du and qu . The motor control systembegins tracking at time =t 0 s . After lapse of few seconds,output q starts following the desired trajectories

∗q satisfactorily. At =t 36 s , a constant positive disturbancetorque 6.0=fT Nm is introduced by the dynamometer and itlasts 12 s . Fig. 11 shows an expanded view of the motoroutput q . The currents di and qi are shown in Fig.12. Thecontrol voltages and currents are modified during thedisturbance but position, velocity and acceleration trackingsare unaffected. This shows the efficacy of the proposedcontrol law.

0 2 4 6 8 10-0.5

0

0.5

0 2 4 6 8 10-1

-0.5

0

0.5

0 2 4 6 8 10-4

-2

0

2

Time (sec)

Vel

ocity

(rad/

sec)

Acc

eler

atio

n

(rad

/sec

2 )Po

sitio

n (

rad)

Fig. 8. The block diagram of the experiment system

Fig. 9. Position, velocity and acceleration tracking

Robus t Adap t ive t r ack

Cont ro l le r

P W M

Genera t ion

Voltage

S o u r c e

Inver ter

P M S M

abc

qd

↓

qd

abc

↓

Pos i t ion

Sensordtd

D C

P o w e r S u p p l y

θ

θ

0i d =∗

qi

di

∗qu

∗du

*av*bv*cv

ai

bi

ci

av bv cv

∗q

ω

D y n a m o m e t e r

0 20 40 60 80 100-40

-20

0

20

40

0 20 40 60 80 100-20

-10

0

10

20

0 20 40 60 80 100-20

0

20

40

Time (sec)

Posi

tion

(rad

)V

eloc

ity (r

ad/s

ec)

Acc

eler

atio

n (r

ad/s

ec2 )

558

VIII. CONCLUSIONS

The electromechanical model of a 3-axis motion simulator isdeveloped in this paper. Subsequently, using Lyapunovstability method, a robust-adaptive control law, which adaptsto constant unknown parameters and behaves robustly againstunknown but bounded fast varying disturbances, is

developed. The results obtained from simulations haveproven the accuracy and effectiveness of the proposed controllaw. Preliminary experiments have been carried out using asingle-axis drive system to show the efficacy of the proposedcontrol system. The experiment results have proven thecorrectness and robustness of the control law.

APPENDIX

1. Actual plant parameters used in the simulation are3=iP , Ω= 513.0iR , mHLdi 74.4= , mHLqi 51.9= ,

turnWbi ⋅= 278.1λ , ypri ,,= ;2

1 3.3 mKgI ⋅= , 22 2.57 mKgI ⋅=

23 0.45 mKgI ⋅= , 2

4 3.154 mKgI ⋅= Initial estimated parameters are 04321 ==== IIII ,

Ω= 1.0iR , mHLL qidi 1.0== , turnWbi ⋅= 1.0λ Control parameters are chosen as

1),(1 =jjλ , 2),(2 =jjλ ; 0),(3 =jiKD , 3,2,1;3,2,1 == ji]300,300,400[1 diagKD = ; ]700,800,1000[2 diagKD = ;

)8~1( =Γ ii is the identity matrix.

2. Parameters of the PMSM motor used in the experiment are2=P , Ω= 1.17R , HLd 278.0= , HLqi 369.0= ,

turnWb ⋅= 21.1λ , 2337.8 mKgeJ ⋅−= ,VVrated 400= , Hzf rated 50= .

Initial estimated parameters are21.0 mKgJ ⋅= , Ω= 1.0iR , mHLL qidi 1.0== ,

turnWbi ⋅= 1.0λ Control parameters are chosen as

51 =λ , 82 =λ ; 101 =DK , 12 =DK , 01.03 =DK ;)6~1(,41 =−=Γ iei ; )13~7(,51 =−=Γ iei .

3. Matrices 81 ~ YY for the 3-axis motion system are

=

dy

dp

dr

ii

iY

000000

1 ,

−−

−

=ψ

θ

φ

&

&

&

yqy

pqp

rqr

PiPi

PiY

0000

00

2

=∗

∗

∗

dry

drp

drr

ii

iY

&

&

&

00

0000

3 , )(1∗∗∗ −−= dddidri iiii λ&&

For matrix 4Y , the non-zero components are listed below:

θψθ

θψφλθθψθψθ

s

ccsqcqqY rpyrrr

&&&

&&&&&&&&&&&&&&&

2

)(2)1,1( 112

114

−

+−−−+= ∗∗∗

θψφθψθφθψ scsqY rr &&&&&&&&& ++= ∗)1,2(4

θθφθφθθφλθθφθθθ ssccsqcqY rrrr&&&&&&&&&&&&&&&& −−−−−= ∗∗

332

14 2)1,3(

θλ &&&&& 2214 2)2,2( −= ∗prqY

Fig. 10. Control voltages du and qu

0 20 40 60 80 100-40

-20

0

20

40

0 20 40 60 80 100-40

-20

0

20

40

Time (sec)

Uq (V

)U

d (V

)

Fig. 11. Position, velocity and acceleration tracking

10 20 30 40 50 60-40

-20

0

20

40

10 20 30 40 50 60

-10

0

10

10 20 30 40 50 60-5

0

5

Time (sec)

Acc

eler

atio

n (r

ad/s

ec2 )

Vel

ocity

(ra

d/se

c)Po

sitio

n (r

ad)

Fig. 12. Currents di and qi

10 20 30 40 50 60-0.4

-0.2

0

0.2

0.4

10 20 30 40 50 60-2

-1

0

1

2

Time (sec)

i q (A

)i d

(A)

559

θθψψθθψθθψ csccsqY ry &&&&&&&& ++= ∗ 2)3,2( 24

θψθθθθψθψλ

θθψθθθψθ

2*2/32

22)()3,3(

233

2214

scsc

ccsqqcqY ryrpyr

&&&&&&&&

&&&&&&&&&&&

−−−

++−= ∗∗∗

ψλ &&&&& 2314 2)4,3( −= ∗yrqY

qqqqr~~2 2

22 λλ −−= ∗∗ &&&&&

qqqqr&&&&&&&&& ~2 2

221 λλ −+= ∗∗∗

∗−= qqq~

=

y

p

r

YY

YY

5

5

5

5

000000

]23,

23

,23

,23

,23

,23

[

22

22225

iiidii

idiiiqiiqiiqidiii

qPqiP

qiPqiPiPiiPY

&&

&&=

−

−−

=

2/300

02/30002/3

6

dyqyy

dpqpp

drqrr

uiPuiP

uiPY

=

qyydy

qppdp

qrrdr

uPi

uPi

uPi

Y

23

00

0230

0023

7

=

qyy

qpp

qrr

uP

uP

uP

Y

2300

023

0

0023

8

4. 81 ~ YY for the single-axis motion system are

diY =1

θ&PiY q−=2

)(13∗∗ −−= ddd iiiY λ&

∗+−= 124 2 rY θθλ &&&&&

]23,

23,

23,

23,

23,

23[ 222222

5 θθθθ &&&& PiPiPiPPiiPiY ddqqqd=

2/36 dquPiY −= ; qrdr uPiY23

7 = ; qruPY23

8 =

5. Parameters of the single-axis motion system areRa =1 , qLa =2 , dLa =3 , Ja =4

RLL

LL

aq

d

d

q )(51 −−= , RL

aq

λ=52

qd

qd LL

LLa

)(53

−−= , d

q

qd LL

LLa

)(54

−=

λq

qd

LLL

a)2(

55−

= , qL

a2

56λ

=

d

qd

LLL

a)(

6−

= ,q

qd

LLL

a−

=7 , qL

a λ=8

REFERENCES

[1] Frangos, C., “Control system analysis of a hardware-in-the-loop simulation,” Aerospace and Electronic Systems,IEEE Trans., Vol. 26 4, July 1990 , pp. 666 -669

[2] Ptak, A. and Foundy, K., “Real-time spacecraftsimulation and hardware-in-the-loop testing,” Real-timeTechnology and Applications Symposium, 1998,Proceedings. Fourth IEEE. 1998 , pp.230 –236,

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[6] In-Cheol Baik, Kyeong-Hwa Kim and Myung-JoongYoun, “Robust nonlinear speed control of PMsynchronous motor using boundary layer integral slidingmode control techniques,” Control Systems Technology,IEEE Trans., Vol. 81, Jan. 2000, pp. 47 -54

[7] Iwasaki, M., Shibata, T. and Matsui, N., “Disturbance-observer-based nonlinear friction compensation in tabledrive system,” Mechatronics, IEEE/ASME Trans., Vol.4. 1, March 1999, pp. 3 –8.

[8] Faa-Jeng Lin, Sheng-Lyin Chiu and Kuo-Kai Shyu,“Novel sliding mode controller for synchronous motordrive,” Aerospace and Electronic Systems, IEEE Trans.Vol. 34. 2, April 1998. pp. 532 -542

[9] Bogosyan, O.S., Gokasan, M. and Jafarov, E.M., “Asliding mode position controller for a nonlinear time-varying motion control system.” Industrial ElectronicsSociety, 1999. IECON '99 Proceedings, the 25th AnnualConference of the IEEE, Vol. 2, pp. 1008 –1013.

[10]Hace, A., Jezernik, K., Curk, B. and Terbuc, M., “Robustmotion control of XY table for laser cutting machine,”Industrial Electronics Society, IECON'98. Proceedingsof the 24th Annual Conference of the IEEE, Vol. 2, 1998,Pp. 1097 -1102.

[11]Weiping Li and Xu Cheng, “Adaptive high-precisioncontrol of positioning tables-theory and experiments,”Control Systems Technology, IEEE Trans., Vol. 3, Sept.1994, pp. 265 -270

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