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Robust control and modeling a 2-DOF Inertial

Stabilized Platform

Hamed Khodadadi

Department of Technical and EngineeringIslamic Azad University, Khomeini Shahr Branch,

Isfahan, Irankhodadadi@iaukhsh.ac.ir

Mohammad Reza Jahed Motlagh, Mohammad Gorji

Electrical Engineering DepartmentIran University of Science & Technology

Tehran, Iran

jahedmr@iust.ac.ir , mgorji@ee.iust.ac.ir

Abstract This paper presents an uncertain linear model for an

Inertial Stabilization Platform (ISP). The system has a 2 degree of

freedom (DOF) gimbal which will be attached to a moving vehicle

and the optical sensors mounted on stabilized gimbal. The main

purpose of an ISP is to eliminate the various disturbance and body

motion to make the line of sight (LOS), hold steady in an inertial

space. Due to dynamical model of the 2-DOF gimbal, friction, cable

restraint, noise, other disturbances from the outside environment

and the motions of the vehicles body (as a result of maneuvering or

vibration), the pointing and tracking accuracy of the gimbaled

system may strongly degrade. These disturbances are typically

nonlinear. Modeling and controlling the stabilization loop with

these nonlinearities are the main problem. The approach of this

paper is considering a linear doubted model. Linearity makes the

model simple and so linear classic controller can be used to meeting

design requirements. Therefore the cost of the design and

implementation stages will decrease. In addition uncertainty makes

the model practical and so the nonlinear nature of mentioned

disturbance can be interpreted by this model. So this built ISP by a

PI controller which is capable enough to overcome the disturbances

and to meet the implication requirements will be controlled. For

decreasing the effect of these uncertainties and other probable

variation in ISP's parameters on system performance, a determinedsimple structure for robust enhancer compensator is used and

designed.

KeywordsISP, gimbal, LOS, stabilization loop, nonlinearity,

uncertainty

I. INTRODUCTIONThe main purpose of the optical sensing equipment, (such as cameras,television, radars, lasers and navigation instruments) is to point thesensor from a dynamical platform to a fixed or moving surface. For

achieving accurate pointing, it is necessary to control the sensors LOS.In such an environment where the equipment is typically mounted on amovable platform, maintaining sensor orientation toward a target is a

serious challenge. An ISP is an appropriate way that can solve thischallenge [1, 2].

An ISP is a mechanism, which is used for controlling the inertialorientation of it's payload[1]. In fact, a LOS stabilized platform is a system that maintains thesightline of an optical sensor relative to another object or inertial space

while external disturbances such as accidental vehicle motion andmaneuvers and other additional disturbances subject it.In a multi-axis ISP, there are extra nonlinearities that make thestabilization task more difficult [2]. These nonlinearities arise not only

from the dynamics of gimbaled system but also from the nonlinear

behavior of other disturbance torques such as friction, spring or cable

restraint torques, mass imbalance, vehicle motions and maneuversnoise and Environmental torques. Among these undesired factors, the

gimbal friction, restoring torque and mass imbalance have the moseffect [3].Fortunately in the shaping of our implication mass imbalance does no

exist but the gimbal friction and restoring torque should be consideredprecisely. The papers which focus on ISP modeling or disturbancerejection in stabilization loop use various manners for consideringfriction. Some of them such as [4-6] represented a model by assumingfriction only as viscous or coulomb or combination of them while in

presented model by [2, 3, 7-8], friction assumed as nonlinear part thacan be estimated by using the Kalman filter or can be rejected by usingan adaptive multilayer feedforward neural network. Cable restraint has

been investigated in few papers such as [1, 3, 7 and 9].

Generally, studying and probing a linear differential equation with someuncertainties is simpler than a hardly nonlinear precise one. Forachieving to both benefits (simplicity of first group of paper andaccuracy of seconds), in this paper by using the most complete classica

model for friction in the static case (presented by Olsson et al [10]) andconverting it to the sum of coulomb and viscous models withuncertainty about both and generalizing this approach to other nonlinear

parts, an uncertainly linear model for a 2-DOF gimbaled ISP will bepresented. For reducing these uncertainties, a simple structure for robusenhancer compensator which has been introduced by Zhu et al [11] wil

be used and designed.

This paper is divided in 7 parts. The first is this introduction to thestructure and principle of 2-axis gimbaled ISP for stabilizing LOS. In

the next section, after an introduction of the geometric transformationsused to go from one to another of the coordinate frames, the cinematicrelation between them will be considered. Then dynamical equation o

the 2-DOF system for elevation and azimuth stabilizing gimbaAccording to Yoon and Lundberg [12] will be represented. Thirdsection is devoted to present a real and practical model by notice tomentioned nonlinearities especially friction and restoring torque.

Section four begins with representation a linear and practical model ofstabilization loop with some uncertainty on its parameters. Then, by

doing experimental test for identification of system and by comparingthe result and the presented model with numerical value, the model wil

be validated. Section five assigns to studying the performance ostabilization loop by using the designed PI controller while the externadisturbance and applied to it. Then with considering lower and uppe

band of uncertainty range of uncertain parameters, the performance of

two worse case systems will be shown. For decreasing theseuncertainties, the robustness enhancer feedforward compensator whichhas been introduced by Zhu et al [11] will be used. This compensatornot only attenuates the effect of uncertainty and decrease the existingdifference by a desired factor but also improve disturbance rejection

International Conference on Electrical, Control and Computer EngineeringPahang, Malaysia, June 21-22, 2011

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This compensator will be designed for the desired implication of thisbuilt ISP in section 6. Finally the ability of disturbance rejection ofcompensated systems (nominal and worse case systems) will be

considered. In the seventh section, the conclusion of this paper will berepresented.

II. CINEMATIC AND DYNAMIC EQUATIONS1. kinematics relations

As we can see in fig. 1, to establish a 2-axis gimbal system threedifferent coordinate frames are used: The gimbal platform or base frame(B), followed by the gimbal mediator or outer frame (K), and finally theinner frame (A) [12]. The optical sensors can be placed on the innergimbal which provides LOS stabilization on yaxis or elevation gimbal.

LOS stabilization on zaxis or azimuth gimbal can be achieved by usingouter gimbal.

In this configuration, 1 is defined as the relative azimuth angle

between the mediator and base coordinate frames where 2 is defined

as the relative elevation angle between inner and mediator coordinateframes. Considering these rotations and by using Euler matrix,

transformation matrixes between the base and mediator frames (K

B C )

and (A

KC ) as the mediator and inner frames become

(1)

,

0

010

0

22

22

=

CS

SC

CAK

=

100

0

0

11

11

CS

SC

CKB

where

(2));cos( 11 =C );sin( 11 =S );cos( 22 =C )sin( 22 =S

by introducing the [ ]/

TB

B I

q r =G

,

[ ]/TK

K I k k kp q r =G

and [ ]/TA

A I a a aq r =G

as the

angular velocity of base, outer and inner frame to the inertial

space's angular velocity1and by using transformation matrixes,

/K

K IG

, /A

A IG

and the inertia matrixes of the outer and inner

gimbals are equal to:

(3)

+=

+=

+=

+=

=

1

11

11

///

GGG

rr

qCpSq

qSpCp

C

r

q

p

k

k

k

IB

BK

BBK

K

k

k

k

IK

K

(4)

+=

+=

=

+=

=

22

2

22

///

CrSprqq

SrCpp

Crq

p

kka

ka

kka

IK

KA

KBA

A

A

A

A

IA

A

GGG

2. Dynamical model of Elevation gimbalYoon and Lundberg [14] indicated that the dynamical equation ofelevation gimbal with attention to torque relationship about y-axis

(controllable axis) can be derived as:

1 Where the component ofIB

B

/G

i.e. p,q,r are traditional notations for roll,

pitch, and yaw in flight dynamics

(5)

2 2( ) ( )

( ) ( )

ay a y az ax a a xz a a

yz a a a xy a a a

I q T I I p r I p r

I r p q I p r q

= + +

+

In this formula, Ty is the total external torque about y-axis and can bereferred as the sum ofTelv (motor torque) and Td1 (external disturbance

torque). Suppose that the products of inertia can be neglected.

(60=== yzxzxy III

Figure 1.A two-axis gimbal typical configuration [9].

Assuming further that

(7)azax II =

Then Eq. (5) becomes

(81delvyaay TTTqI ==

So if Eq. (6) and Eq. (7) are met, the block diagram of elevationequation can be represented as fig. 2. It can be seen that the elevationangular velocity of inner gimbal is only a function of torque motor and

external disturbance and is not affected by the body or elevation gimbamotions.

3. Dynamical model of Azimuth gimbalAccording to Yoon and Lundberg [12], the dynamical equation for

azimuth gimbal can be derived:

(9)321 dldldlzkk TTTTrJ +++=

In this formulaJk is the total inertia of two-axis gimbal about z-axis(Jk= Iaz+ Jkz), Tz is the total external torque about azimuth gimbal andcan be referred as the sum of Taz (motor torque) and Td2 (externadisturbance torque), and finally Tdl1, Tdl2, and Tdl3 represent differen

gimbal inertia disturbances.The simplified model of Eq. (9) with two previous section

assumptions (i.e. Eq. (6) and Eq. (7)) and supposing that the products ofinertia of the azimuth gimbal can be put aside, i.e.

(100=== yzxzxy JJJ

can be represented as:

(11)(

)( 2

kyaxkxkk

akaydazkkzaz

JIJqp

qpITTrJI

++

=+

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The block diagram of azimuth equation can be represented as fig. 3.Since the output which should be controlled is ra and not rk, the desiredoutput can be illustrated by using Eqs. (2) and (3) as Fig. 3.

III. REAL AND PRACTICAL MODELAs it was mentioned in pervious sections, total external implementedtorque to each axis can be modeled as sum of the desired torque for

stabilization (Telv or Taz) and disturbance torque (Td1 or Td2). In thissection, disturbance torque and it's various component that might affect

the stabilization loop will be explained. Among these undesired factors,the most complex nonlinear phenomenon is gimbal friction, restoringtorque and mass imbalance [3]. The complete and real model of gimbal

system can be achieved after studying these factors.

1. FrictionDue to the nonlinear friction in the gimbal, the disturbance torque

which has been introduced by Lin and Hsiao [3] is described by usingthe following formula

(12))( hf kfT

=

Modeling this nonlinear function has been mentioned in manyreferences. Olsson et al [10] present four classical models in the staticfriction case. The simplest of them considers the friction only ascoulomb friction (see fig. 4a) while the second of them combines viscousfriction (which is proportional to relative velocity between two movingsurfaces) with coulomb friction as shown in fig. 4b.

The more complete model can be defined by paying attention to thefact that friction force at rest is higher than coulomb friction level. As itcan be seen in fig. 4c, stiction is the static term that must be added to theformer terms.

The real and practical model is given in fig. 4d. Stribeck observedthat friction dose not decrease discontinuously and the velocitydependence is continuously.

As it mentioned briefly in introduction, the papers which focus oninertial stabilized platform modeling or disturbance rejection instabilization loop use various manners for considering friction. Some of

them such as Seong et al [4] represented a model by assuming frictiononly as viscous while [5] considered it as coulomb. In represented model

by Kwon et al [6] coulomb and viscous are the components of friction.Kennedy et al [7] divided friction in two parts: a linear part proportionalto velocity or viscous model and a nonlinear one. Li et al [8] and Orejas[2] used the coulomb friction model and dynamically estimated andidentified this value by using the Kalman filter. Lin and Hsiao [3]introduced a nonlinear friction model and tried to reject the disturbancetorque resulted from it by using an adaptive multilayer feedforwardneural network. Ji and Li [13] believe that the mechanical friction, as atypical nonlinear factor, can hardly realize compensate with the precisemodel, especially under the low velocity.

The approach that is chosen in this paper is to use the most completeclassical model (i.e. Stribeck friction) and to convert it to the sum ofcoulomb and viscous models with uncertainty about both. As it is shown

in fig. 5-d this or any method for continuously decreasing and thenaccelerating friction value can be defined as a combination of a coulombvalue and a velocity- related one with a variant value as:

Figure 2.Dynamical model of elevation gimbal.

Figure 3.Dynamical model of azimuth gimbal.

(13)vFvFT vcf

~

)sgn(

~+=

For example in low speed, the constant value is relatively high

where velocity-related one is negative and increasing in speed leads toreduction in friction quantity. Reversely in high speed, the constant valueis relatively low and velocity-related one is positive. Consequently speedgrowth results in increase of friction quantity. According to Li et al [8]"Friction between moving components not only varies with time but isnonlinear". This statement can encourage us to use an uncertain model.

2. Restoring torqueCable restraint or spring force is the other factor that may affect and

degrade the stabilization. Kennedy et al [7] introduced a model with alinear part related to angle position and a nonlinear one as:

(14)ncrpcrcr FFT +=

Similar to the approach which was represented in previous sectionTcr is expressed as the sum of a constant and a relative part withuncertainty on both parameters.

(15)pcrccrcr FFT~~

+=

3. Elevation practical modelBy combination of Eqs (13), (15) and (7), the dynamical equation ofelevation gimbal can be represented as:

(16)

1 1 1

1 10

sgn( ) ( )

( )

ay a elv di c a k v a k

t

ccr pcr a k

I q T T F q q F q q

F F q q d

=

whereTdi1 = other disturbance torque on elevation axis

1cF = uncertain coulomb part of friction on inner gimbal

1vF = uncertain viscous part of friction on inner gimbal

1ccrF = uncertain constant part of cable restraint on inner gimbal

1pcrF = uncertain relative part of cable restraint on inner gimbal

Figure 4.Four classical model of friction [10].

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Figure 5.Practical model of elevation gimbal.

This model can be presented as fig. 5. Similarly, the dynamical equation

of azimuth gimbal can be represented.

IV. STABILIZATION LOOPIn an ISP the goal is to hold the payloads orientation steady in inertial

space [1]. In this case, a direct drive PMDC torque motor for generating

the required torque for each axis has been used. As it can be seen inmany references such as Dorf and Bishop [14], in an armature-controlled DC motor, the armature current generates productive torque.This torque can be spent for gimbal motion and disturbance rejection as

the Eq. (16) for elevation stabilization. The block diagram of speedcontrol (for example in elevation stabilization axis) is presented in fig.6.To simplify the block diagram, it is useful to obtain the transfer function

from motor speed to three inputs (armature voltage, base motion andexternal disturbance).

(17)

1

~)

~~()

~(

)()(

)(

111

2

1

312

den

sK

RFsFRLFKKsIRLFsIL

sKsH

sV

sq

t

apcrvaapcrtbayaavaya

tgs

a

a

=

++++++

==

(18)

1

)(

~)

~~()

~(

)()(

111

2

1

3

1

den

sRsL

RFsFRLFKKsIRLFsIL

sRsL

T

sq

aa

apcrvaapcrtbayaavaya

aa

d

a

+=

++++++

+=

(19)

1

)~~

)((

~)

~~()

~(

)~~

)(()(

11

111

2

1

3

11

den

FsFRsLsKK

RFsFRLFKKsIRLFsIL

FsFRsLsKK

q

sq

pcrvaatb

apcrvaapcrtbayaavaya

pcrvaatb

k

a

+++

=

++++++

+++

=

Since the main purpose of the implicated ISP is stabilization the optical

sensor's speed in order to inertial space, the angular velocity of sensorshould be measured and used in feedback loop. So, two rate gyros (onefor each axis) are installed on the inner gimbal in elevation and outergimbal in azimuth axes.

Fig. 7 schematically shows the components of a stabilization loop. In

this fig, Hg(s) is the transfer function of rate fiber optic gyro, ng is thegyroscope noise and Celv(s) is the elevation controller.By reaching to this point, the most complete and practical model of a

gimbaled 2-DOF ISP with considering some uncertainty on system'sparameters are presented. The most prominent advantage of the

presented model is that this model is completely linear. Linearity makesthe model simple and therefore the linear classic design algorithms can

be used for designing the controller for stabilizing the ISP.Consequently, the costs of design, building and implementation ofcontroller decrease considerably.For example in the next section a conventional PI controller will be

designed for the ISP. Therefore, if the controller is able to satisfy thedesign requirements despite the existent uncertainties, we achieve ourgoals.

Similar to the most ISP control configuration, in this implicationcontrol systems contains two loops: an inner rate or stabilization loopinside an outer track loop. The stabilization loop (which is focused in

this paper) compensates for disturbances and minimizes unnecessarymotion of vehicle. Meanwhile, the role of track loop is remaining thesensor LOS pointed toward the target [1].

For validating the presented model and numerical value of systemparameters, an experimental test has been implemented. In this test byusing an infrared receiver-transmitter and a scaled plate (as an

incremental shaft encoder which has been mounted on the rotating axisa voltage command apply to system. The infrared receiver-transmittersenses and measures the pulses which are result of turning of scaled

plate and send to a FPGA. After counting the pulses, every one micro

second, the counted value transfer to a PC and store in it. In this case astep order with +40v amplitude applied to system.As it can be seen in fig. 8 the recorded data indicate the angular positionand not angular velocity, so for having a fairly comparison between

model and plant, after putting the numerical value of implementedsystem's parameter in Eq. (17), an integrator must be added to it. Theangular position of model and plant due to +40v step command

illustrated in fig. 8. The conformity between them indicates that thepresented model is practical and can be used in modeling 2-DOF

system.

Figure 6.Real model of elevation axis by using a DC motor.

Figure 7.Complete model for elevation stabilization Loop.

Figure 8.Comparison between step response of model and plant.

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V. PERFORMANCE OF STABILIZATION LOOPThe main purpose of an ISP is rejecting the various disturbance andfuselage motion to make the LOS, hold steady in inertial space.Moreover, because of the implication of this ISP, the stabilization loopshould have a good bandwidth. In other words, the performance of

stabilization loop as inner loop must be so fast that in inspection oftrack or outer loop the dynamic of inner loop can be ignored. This

requirement leads to having an ISP with maximum 35ms settling time.For a large class of high-precision stabilization loops, classical PI and

PID compensators are usually adequate [1]. As it was mentioned before,due to the achieved linearity of this modeling, a PI controller can beused. If a PI compensator with the transfer function as

( ) (331.8 6.48 ) /C s s s= + is used, a 350-Hz servo bandwidth will

obtained on the inner (elevation) axis with the nominal quantity of ISP's

parameters that have been introduced in previous section.Step response and Bode diagram of nominal system indicates that thestabilized system has the settling time less than 35ms and bandwidthhigher than 350Hz (see fig. 9).

To test the ability of ISP in disturbance rejection, the sum of a 10-Hzpulse with 0.06 amplitude (worse case amplitude of the coulombtorque), a 0.02 constant value (worse case constant part of cablerestraint), a 0.1 random signal as the external disturbance and the sum

of three sinusoidal values with the amplitude of10 rad/s and 0.1, 1 and

10 Hz frequencies (as 10 [sin(10 ) sin( ) sin(0.1 )]t t t + + ) as the

outer gimbal speed in y-axis (with considering these frequencies, theeffect of low, medium and high frequencies of body motion can be

pondered) and a 0.1 random signal as fuselage vibration was

implemented to the elevation stabilization loop. The gyroscope noisecan be assumed to be normally distributed with a zero mean and astandard deviation of0.01 rad/s.As it can be comprehended from fig. 10, the effect of the disturbance

and motion has been attenuated in angular velocity of LOS (system'soutput) considerably.By assuming the lower and higher band for parameters with uncertainty,and calculating the two worse cases that have the most differences with

nominal system, the step and disturbance response of them (with thesame controller of nominal system) are illustrated in figures 11 and 12respectively. Also the bandwidth of system in this case is less than 350

Hz. Comparisons between these figures indicate that disturbanceresponses are still pleasing, but the step responses although have a goodbehavior and this point can be considered as a confirmation of linearmodeling of the 2-DOF gimbal ISP, are different to nominal system andexceed the desired settling time.

To solve this problem, a simple structure robustness enhancerfeedforward compensator will be used. The compensator which has

been introduced by Zhu et al [11] has the ability of uncertainty anddisturbance reduction and can be augmented to main controller.

According to [11] for the design of the robustness enhancer, one can usemany advanced methods. But since the approach of this paper is

presenting a simple and linear model that decrease the design andimplementation cost without degrading in ISP performance, the simple

structure which has no extra cost is appropriate. Another reason forusing robust controller is the typical hazardous environment for most

ISP application. According to [13], the ISP usually works in atrociousconditions, which have many random disturbance factors. So having a

fast dynamic and high stabilization precision ISP which has theadaptability and robustness regarding the model parameter changing, isrequested. Thus in the next section, a robust enhancer controller will bedesigned for the ISP.

VI. ROBUST COMPENSATOR AND SIMULATION RESULTSAccording to the proposed structure in [12], W as the weight of

injected input can be considered as

Figure 9.Step response of nominal system.

Figure 10.Disturbances attenuations of nominal system.

Figure 11.Step response of 2 worse cases.

Figure 12.Disturbances attenuations of worse case System.

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(20)12

)1025.0/(100 += mGsW

As it can be seen, Whas the gainK=100, and two fast dynamic for

pre-filter which make the Wproper multiplied to1

mG

as the inverse of

model of system. With the enhanced compensator, step responses of twoworse cases in comparison to nominal system are illustrated in fig. 13.

As it can be seen in fig. 13, the settling times of two worse cases areless than desired value and consequently bandwidths of two systems areas sufficient as high. Besides, the performance of worse case systems isvery similar to nominal system. This means that if the ISP parametersvary in different conditions, the total performance will not be verydifferent. In fact, the ability of uncertainty reduction of the enhancedrobustness compensator, justifies the act of assuming a linear uncertainmodel for stabilization loop.

Figure 13.Step response of nominal and 2 worse Cases.

Figure 14.Disturbances attenuations of compensated system

Taking the disturbance and fuselage motion similar to previoussection and implementing them to the compensated system, it can beseen that the disturbance and motion effects on the system same asuncertainty output are decreased (see fig. 14)

VII. CONCLUSIONSThe details of a 2-DOF gimbal ISP modeling have been considered inthis paper. The presented model is an uncertain linear model that by

considering all nonlinearities torque such as 2-DOF gimbal inertiadisturbances, friction, cable restraint, noise, other disturbances from theoutside environment and vehicle body motion, was represented. Since

the model is completely linear, linear classic controller can be used tosatisfy desired stabilization performance. Therefore the design andimplementation's cost will decrease. Validating the present model with

experimental data and results of testing the performance of the ISP fornominal and worse cases systems with the designed PI controllerindicated that assuming linear uncertain model for 2-DOF gimbal ISP isreasonable. Although a simple structure robustness enhance

compensator (which imposes little extra costs) for decreasing theseuncertainties, improving the performance of system and guarantyingthat act of ISP in hazardous conditions which will be used is pleasingyet has been enhanced to main controller. With this structure, effect o

any undesired variation of ISP's parameters on system performancewill degrade. In fact this paper presents an appropriate solution fodesigning and controlling ISPs. A PI controller that because of linearity

of the presented model can be response of design requirement plus asimple structure robust enhancer that can be decrease the effect of

uncertainties of model parameters and other disturbance on totaperformance.

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