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International Conference on Control, Automation and Systems 2007Oct. 17-20, 2007 in COEX, Seoul, KoreaDisturbance Observer Based Robust Control for Industrial Robotswith Flexible Joints

Sang-Kyun Park, and Sang-Hun LeeElectro-Mechanical Research Institute, Hyundai Heavy Industries Co. Ltd

102-18, Mabuk-dong, Giheung-gu, Yongin-si, Gyeonggi-do, 446-716, Korea(E-ma il: pskl000@hhi.co.kr)Abstract: In this paper, a disturbance observer based control algorithm is proposed for industrial robots having flexiblejoints. The joint flexibility of the robot is modeled as a two mass system. We study on the practical issues forimplementing disturbance observer based control scheme in flexible joint robots. For industrial robots, generally thesensors are located on the motor side. If we construct disturbance observer using motor side dynamics, due to the zerodynam ics, disturbance observer cannot directly reject the disturbance at the link side. To solve this problem, we proposea dual observer that estimates disturbance and states simultaneously. Using the proposed dual observer, we constructfull state feedback controller. The effectiveness of the proposed control scheme for disturbance rejection and robustnessis demonstrated by numerical simulation and experiment using HILS (Hardware In the Loop Simulation) system.Keywords: Flexible Joint Robot, Disturbance Observer, Dual O bserver, State Feedback Control

1 . I N T R O D U C T I O NThe pose variation of a robot or uncertain payloadhandled by the robot causes model parameteruncertainty of industrial robots while the external forceon the end-effe cter such as the operation force of thespot welding gun causes the external disturbance. Inaddition, dynamic interference coupling torques can beconsidered as a disturbance at each robot axis.Therefore disturbance rejection performance androbustness to model uncertainties are very importantfactors in evaluating the dynamic performance of

industrial robots. Since the disturbance observer basedcontrol scheme has simple structure and powerfulperformance, it is widely used for improvingdisturbance rejection performance and robustness invarious mechanical servo systems [1-2, 6]. Umeno &Hori proposed two degree of freedom controller whichhad inner loop disturbance observer and outer looptracking controller [1]. The outer loop controller wasdesigned by controller parameterization technique forthe internal stability. Wa ng and Tomizuka proposeddesign me thod for a disturbance observer using Hoooptimization scheme [2]. Despite the fact that robotmanipulators with high gear ratios have the jointflexibility, almost disturbance observer based controlmethods assume that the robot joint is rigid body [1-2,7] .In this paper, we consider the robot joint flexibilityand use flexible joint robot model which suggested bySpong [4]. For the real implementation, not onlyexternal disturbance but also model uncertainties,unmodeled dynamics, dynamic coupling torques, andgravity are lumped into disturbance term here. So wecan simplify the control problem as an independent join tcontrol with existing disturbance.First, we study on the practical issues forimplementing disturbance observer based controlscheme in flexible joint robots. For industrial robots,generally the angular sensors are only located on the

motor side [6, 8-10]. So it is a general approach toconstruct disturbance observer using motor sidedynamics. We show the performance limitation of thelink side when the motor side disturbance observer isapplied. Due to the zero dynamics, disturbance observerat the motor side cannot compensate the disturbanceeffects at the link side.To solve this problem, we propose a dual observerbased control scheme which estimates disturbance andstates simultaneously. Dual observer has been studiedby several researchers [5, 7, 11-12]. Their methods areapplicable when a rank condition between the outputmatrix and the disturbance m atrix is satisfied. However,our system which has only motor side measurement donot satisfy this rank condition. To remedy this problem,we assume the disturbance dynamics and augmentobserver states with the disturbance state [12]. Thisapproach does not need the plant dynamic inversionwhich amplifies measurement noise.With this dual observer, we propose a robust trackingcontroller based on output regulator control algorithm.To reduce peaking phenomenon which occurs whenestimated disturbance is compensated, we restrict themagnitude of the disturbance feedback [16].This paper is organized as follows. Flexible jointrobot model and decoupled flexible joint model withdisturbance are given in Section 2. Section 3 reviewsconventional disturbance observer approach and theperformance limitation analysis. In Section 4, a newrobust control method which has combined with dualobserver and output regulator control is proposed.Simulation and experimental results are presented inSection 5. Finally, conclusions are drawn in Section 6.2 . M O D E L O F F L E X I B L E J O I N T R O B O TAccording to [4], a flexible joint robot can be

modeled asM(q)q +C(q 9q) + G(q) = K(Olr-q) +Td9 (1 )

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( / ) .K

J r q ur

+ = (2)

where nq R is the vector of joint coordinates,

( )n n

M q R

is symmetric, positive definite inertia matrix,

( , )n

C q q R includes the Coriolis and centrifugal terms,

( )n

G q R is the gravity term. The vector nR is the

motor angle, J is diagonal matrix which stands for themoment of inertia of the motors, K is joint stiffnessbetween motor and link, ris the gear reduction ratio.

If we rewrite Eq. (1),(2) separating each axis i, therobot dynamic equation is expressed as

( , ) ( ) ( ) ,i

j

ii i ij ij i i i i i d M q M q C q q G q K q + + = ++

( / )ii i i i i i

i

KJ r q u

r + = (3)

Nondiagonal terms, Coriolis and gravity term in

Eq.(3) can be lumped into disturbance term di.Eq.(1),(2) can be separately considered for each singleaxis with disturbance :

( / )ii i i i i i i

M q K r q d= +

( / )ii i i i i i

i

KJ r q u

r + = (4)

( , ) ( ).i ij ij i i

jid

d M q C q q G q=

As a result of decoupling with the disturbance term di,we can simplify multi-axis robot manipulator controlproblem to independent joint control problem with link

side disturbance as shown in Fig 1. Controller designwith Eq. (4) has an advantage of controllerimplementation for its simple structure. For furthersimilar description of real robots, both motor side andlink side viscous friction terms Bm,BL are added.

mJu

m

1/ r

K

LJ

L

LB

mB

d

mJu

m

1/ r

K

LJ

L

LB

mB

d

Fig. 1 Two-mass System

According to the notation of Fig. 1, Eq. (4) ischanged as

( / )L L

J q B q K r q d+ = +

( / )m m

KJ B r q u

r + + = (5)

In the Laplace domain Eq. (5) is represented by theblock diagram of Fig. 2.

2 2

( ) /m m ms J s B s K r= + +

,2 2 2 2

( ) / / /l L L

s J r s B r s K r= + + . . (6)

1

( )m

p s

1

( )l

p s

2/K r

2/K r

/d r

q1/ ru

+

++

+1

( )m

p s

1

( )l

p s

2/K r

2/K r

/d r

q1/ ru

1

( )m

p s

1

( )l

p s

2/K r

2/K r

/d r

q1/ ru

+

++

+

Fig. 2 Block diagram of two-mass system

The state space equation of the single-axis flexiblejoint robot can be represented as

x Ax Bu Nd= + + (7)

where [ ]T

x q q = ,

2

1

1 1

0 1 0 0

0

0 0 0 1

0

1

1

0 0

00, ,

0

0

.r

r r

L

BK KL

J J JL L L

BK K m

J J Jm m m

J

mJ

A B N

= = =

3. TYPICAL DISTURBANCE OBSERVER

WITH FLEXIBLE JOINT ROBOT

Disturbance observer technique is widely used inmechanical servo systems for improving disturbancerejection and robust performance [1-2, 6]. But almostworks assumed that the stiffness of the servo system issufficiently large to neglect the joint flexibility.

Fig. 3 shows the block diagram that typical methodsin [1-2, 6] is applied for a flexible joint robot, where v isouter loop controller command. Pn(s) is nominal plantmodel. Q(s) is low-pass filter with a DC gain of one. Inorder to see the behavior of the typical disturbanceobserver loop without considering joint flexibility, it isnecessary to look at the transfer functions from d, and v

to q, and .

1

( )mp s

1

( )lp s

2/K r

2/K r

/d r

q1/ rv+

++

+

1( ) ( )n

Q s P s( )Q s+

u

+

Disturbance observer

1

( )mp s

1

( )lp s

2/K r

2/K r

/d r

q1/ rv+

++

+

1( ) ( )n

Q s P s( )Q s+

u

+

Disturbance observer

Fig. 3 Disturbance observer loop at the motor side

The transfer function from d, v to motor angle isgiven by

2(1 ( )) ( ) ( )( / )

( )( ) (1 ( )) ( ) ( ) ( )

n

d

l n

Q s P s P sK rG s

p s r Q s P s Q s P s

=

+

(8)

( ) ( )( )

(1 ( )) ( ) ( ) ( )

n

v

n

P s P sG s

Q s P s Q s P s

=

+

(9)

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whereP(s) is the transfer function of the plant from u to

in Fig 2.

2 2

( )( )

( ) ( ) ( / ) .

l

m l

p sP s

p s p s K r=

(10)

The transfer function from d, v to link angle q isgiven by

2

(1 ( )) ( ) ( ) ( ) ( ) ( )1

( ) (1 ( )) ( ) ( ) ( )

n m

qd

l n

Q s P s P s p s Q s P sG

p s r Q s P s Q s P s

+=

+

.(11)

2( ) ( )/

( ) (1 ( )) ( ) ( ) ( )

n

qv

l n

P s P sK rG

s r Q s P s Q s P s=

+

.(12)

When ( ) 1Q s , Eq. (8) goes to zero and Eq. (9)

becomesPn(s). This indicates that disturbance observerreject the disturbance and compensate modeluncertainty effectively in the view of motor side. Sofrom the view of motor side, motor side dynamicsbehaves as the nominal model. However Eq. (11), (12)become

2

1

( )

1qd

l

Gsr

,

2/

( )( )

qv n

l

K rG P s

p s r . -. (13)

Eq. (13) shows that although disturbance rejectionand robustness is improved at the motor side, at linkside, motor side zero dynamicspl(s) in Eq. (10) remainsthe poles of the whole system including disturbanceobserver. In real plant, becauseBL is much smaller thanK, the poles ofpl(s) are located in the left half plane

near the imaginary axis. These poles generate the linkside vibration.

To remedy link side vibration caused by disturbance,disturbance observer considering joint flexibility mustbe implemented together with the link angle q. But ingeneral, industrial robots can only measure the motorposition. So we need a new approach to estimate linkside states as well as disturbance.

4. ROBUST CONTROLLER DESIGN

4.1 Dual Observer Design

We consider the dynamic system with disturbance todesign dual observer. The state space equation is

x Ax Bu Nd= + + f

y Cx= f (14)

where x, u , d, and y are states, control input,disturbance, and measurement outputs.

There is variety of existing dual observers whichestimates states and disturbances simultaneously[7,13,14]. The unknown input observer(UIO) method,one of the most well known approach, assumes that

disturbance is proportional that output estimation error.The UIO structure[7,13,14] is represented as

( )x Ax Bu Nd L y Cx= + + +

( )d K y Cx= (15)

The UIO as shown in Eq. (15) is applicable when therank condition is satisfied [13-14]. The rank condition is

that CN has a full rank. Unfortunately, since oursystem has only one measurement, which is motor angle,the rank condition cannot be satisfied. In order to designstable dual observer, we need two assumptions [15]:

(A1) disturbance dynamics is known.(A2) The poles of disturbance dynamics do not locateleft-half plane.

According to assumption (A1) and (A2) disturbancemodel is represented by

w Sw=

d Qw= cc (16)

where dnw R , and eigenvalues ofSdont have negative

real part.Plant model Eq.(14) can be augmented with the

disturbance model in Eq. (16). Then, augmented plantmodel is given by

0 0

x A NQ x Bu

w S w = = +

(17)

[ ]0y C =

where is the augmented state. Then the dual observer

can be constructed as follows.

0

x x

w w

A L C NQ Lx Bxu y

L Q S Lww

= + +

(18)

In Eq. (18), the observer gains Lx and Lw can bedesigned by pole placement method.

4.2 Robust Controller Design

The design of the robust controller is based on Eq. (7)together with assumption Eq. (16). Because the controlobjective is that the link side angle is tracking thereference trajectory, the control problem is representedas

x Ax Bu Nd= + +

dz Hx q= + .(19)

where [ ]1 0 0 0T

H = ,and control objective is

lim 0t

z

.

We design the robust controller dividing two parts.The first part is feedforward controller for the tracking

of reference trajectory. The second part is feedbackcontroller for the stability of the system and disturbance

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rejection. The controller structure is represented as

ff fb ff x x wu u u u K e K w= + = + + (20)

whereff

u is feedforward term andfb

u is feedback term.

(A) Feedforward controller

For the design of the feedforward controller, weassume that disturbance and initial state error are zero.Then Eq. (19) is

d d ff x Ax Bu= +

0d dz Hx q= + = -(21)

where the notation dmeans the desired value.From Eq. (19), we can obtain the feedforward

controller :

1 2 3 4 5ff f d f d f f d f du K q K q K q K q K q= + + + +

1

2

3

4

5

2

2

0

/

( )

/

T

T

m Lf

m Lf

m L

ff

L m m L

f

f

m L

B B rK

B BKJ J

KKKB J B J

KK

KJ J

K

r

+

+ +

= =

+

(22)

From Eq. (22), for the feedforward control, it isnecessary that the reference trajectory is 4th order

differentiable.

(B) Feedback controller

As a result of feedforward control in Eq. (22), Eq.(19) is changed to error dynamics,

fbx xe Ae Bu Nd = + + ,

xHe= . (23)

where ex= xd x is state error. With the assumption inEq. (16), this formula is appropriate to apply the outputregulation control algorithm in [15]. The feedbackcontroller consists of state feedback part and

disturbance feedback part. The control structure is givenby

fb x wxu K e K w= (24)

Together with Eq. (17), Eq. (23) can be augmented as

fbx xe Ae Bu NQw= + +

w Sw= (25)For the output zregulation, the tracking errorex shouldbe located in output zeroing manifold.

xe Xw= (26)

From Eq. (17),(24),(25), and (26), the stable output

regulation controller can be designed by followingtheorem.

Theorem 1 suppose that A-BKx is Hurwitz, and thereexists a matrixXand feedback gainKw satisfying

( )

0

x wA BK X XS BK NQ

HX

=

=

.(27)

Then lim 0t

z

is satisfied.

proof : Define new state variablex x

e e Xw= , then

system (25) can be rewritten as

{ }( ) ( )x x x x w

e A BK e A BK X XS BK NQ w= + +

xz He HXw= + (28)

Substituting Eq. (27) into Eq. (28), the above equivalentsystem description is

( )x x xe A BK e=

xz He= (29)

Since A-BKx is Hurwitz, the whole closed loop system

statex

e goes to zero. Therefore, the output regulation

condition lim 0t

z

is satisfied.

Q.E.D

The state feedback gain Kx can be designed by manymethods. In this paper we designed Kx using poleplacement technique. The remaining term of feedback

controller is disturbance feedback gain Kw. For thesimplicity of the problem, we can assume that thedisturbance is step function, then the disturbance modelin Eq.(16) is chosen as

0, 1S Q= = (30)

Using the system model in Eq.(7) with the assumption(30) and given Kx, we can obtainXandKw, the solutionof Eq. (27). The result is

2

3

2

( / )1

/

x

w

K r KK

r K r

+=

[ ]0 0 / 0T

X r K= (31)For the practical implementation, we restrict

disturbance feedback torque in some range withsaturation scheme. With this scheme, we prevent thepeaking phenomenon, which is occurred when the

estimated disturbance change rapidly [16].

5. SIMULATION AND EXPERIMENT

5.1 Simulation Result

Two different controllers are simulated to compare

the tracking performance and robustness. The firstcontroller is observer based state feedbackcontroller(OSC) in [9], the second is proposed controller.

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Two controllers which are designed by pole placementmethod have the almost same closed loop poles. Plant ismodeled with the parameters of two-mass systemexperiment equipment in HHI, and the plant modelincludes the link side Coulomb friction.

Fig.4 shows that two controller has similar tracking

performance when the plant model parameters areknown exactly. But at the steady state, while theOSC(red) has the steady state error at link angle,proposed controller does not have steady state error.This means that the Coulomb friction term iscompensated well.

3 3.1 3.2 3.3 3.4 3.5

198.5

198.7

198.9

199.1

199.3

time(sec)

Angle(rad)

DOB

OSC

(a) Link angle

1.1 1.3 1.5 1.7 1.897

101

105

time(sec)

Angula

rVelocity(rad/sec)

OSC

DOB

(a) Link Velocity

Fig. 4 Simulation results for proposed control(DOB)

and observer based state feedback control(OSC)

We simulate the situation where there are 30%perturbations on the link side inertia. Fig. 5 shows thatthe tracking error for the two controllers. The proposed

controller(DOB) shows more robust performance thanobserver based state feedback controller(OSC).

0.5 1 1.5 2 2.5 3 3.5 4 4.5

-0.2

0

0.2

0.4

0.6

time(sec)

Angle(rad)

100%

70%

130%

(a) DOB

0.5 1 1.5 2 2.5 3 3.5 4 4.5-0.2

0

0.2

0.4

0.6

time(sec)

An

gle(rad)

100%

70%

130%

(b) OSC

Fig. 5 Tracking error plots for DOB and OSC control

5.2 Experimental results

To evaluate control performance, we use the

HyRoHILS(Hyundai Robot Hardware In the LoopSimulation) system as shown in Fig 6. It consists of ahost station, a prototyping device(dSPACE equipment),drive units and a 6-DOF HA006 robot manipulator.

Fig. 6 HyRoHILS system

Short pitch motion trajectory is used for theevaluation of controller. Short pitch motion is thegeneral performance index of industrial robot in termsof vibration suppression. The position of end-effecter ismeasured by 3D-position measurement unit.

Proposed controller is compared with conventionalPPI controller. PPI control consists of inner velocity

feedback PI controller and outer position feedback Pcontroller. The proposed control algorithm onlyimplemented in base axis. Two controllers are designedwith the standard position parameter of HA006.Experiment is performed for the case that robot has 40%larger load inertia than nominal value in base axis. Theexperimental results are shown in Fig 7. The proposedcontroller(DOB) shows good performance in short pitchmotion. Despite of 40% perturbation of load inertia, theoscillations of all the direction x, y, and z are muchsmaller than PPI controller.

6. CONCLUSION

A disturbance observer based control algorithm wasproposed for flexible joints of industrial robots. The

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multi axis robot control problem was considered as theindependent single axes of two mass system with linkside disturbance. For the estimation state anddisturbance simultaneously, dual observer was designedwith the assumption that disturbance dynamic wasknown. The controller was implemented with output

regulation control. To prevent peaking phenomenon, thedisturbance feedback torques was restricted by somerange. The HyRoHILS system was used for theevaluating control algorithm. The proposed algorithmwas found to have good tracking performance androbustness against model uncertainty.

-120 -70 -20 30878.2

878.6

879

879.4

x (mm)

z(mm)

Short Pitch motion

PPI

DOB

(a) Short pitch motion

0 0.5 1 1.5 2 2.5 3 3.5 4

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

time (sec)

x(mm)

PPI

DOB

(b) plot of position x

0 1 2 3 41124

1126

1128

1130

1132

1134

1136

time (sec)

y(mm)

PPI

DOB

(c) plot of position y

Fig. 7 Experimental results for proposed

control(DOB) and PPI control(PPI)

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