Control system of unmanned aerial vehicle used for...

Control system of unmanned aerial vehicle used for endurance autonomous monitoring

TEODOR - VIOREL CHELARU, PHD

University Politehnica of Bucharest / Romanian Space Agency, Str. Gheorghe Polizu , no. 1, PC 011061,Sector 1, Bucharest, ROMANIA

[email protected] http://wwww.pub-rcas.ro

VASILE NICOLAE CONSTANTINESCU, PHD University Politehnica of Bucharest,

Str. Gheorghe Polizu , no. 1, PC 011061, Sector 1, Bucharest, ROMANIA [email protected] http://wwww.pub-rcas.ro

ADRIAN CHELARU

Comfrac R&D Project Expert SRL, Str. Decebal nr.1, Sector 3, Bucharest, ROMANIA

[email protected]

Abstract: - The paper purpose is to present some aspects regarding the control system of unmanned aerial vehicle - UAV, used to local observations, surveillance and monitoring interest area. The calculus methodology allows a numerical simulation of UAV evolution in bad atmospheric conditions by using nonlinear model, as well as a linear one for obtaining guidance command. The UAV model which will be presented has six DOF (degrees of freedom), and autonomous control system. This theoretical development allows us to build stability matrix, command matrix and control matrix and finally to analyze the stability of autonomous UAV flight .A robust guidance system, based on Kalman filter will be evaluated for different fly conditions and the results will be presented. The flight parameters and guidance will be analyzed. The paper is inspired by national project SAMO (Autonomous Aerial Monitoring System for Interest Areas of Great Endurance). Key-Words: - UAV, Simulation, Control, Guidance, Endurance, Surveillance, Monitoring, Kalman filter NOMENCLATURE α - Attack angle (tangent definition); β - Sideslip angle (tangent definition);

aδ - Aileron deflection;

eδ - Elevator deflection;

0eδ The balance deflection angle for the elevator;

rδ - Rudder deflection;

Tδ - Thrust command: ψ - Azimuth angle; θ - Inclination angle; φ - Bank angle; ρ - Air density; Ω - Body angular velocity;

ECBA ,,, - Inertia moments; Az

Ay

Ax CCC ;; - Aerodynamic coefficients of force in

the mobile frame;

An

Am

Al CCC ;; - Aerodynamic coefficients of

momentum in the mobile frame; Tz

Ty

Tx CCC ;; - Thrust coefficients in the mobile

frame; Tn

Tm

Tl CCC ;; - Thrust momentum coefficients in the

mobile frame;

SVF2

2

0 ρ= - Reference aerodynamic force;

lFH Ao 0= - Reference aerodynamic couple;

0T - Reference thrust force;

lTH To 0= -Reference couple thrust;

l - Reference length; m – Mass;

rqp ,, - Angular velocity components along the axes of mobile frame;

WSEAS TRANSACTIONS on SYSTEMS and CONTROLTeodor-Viorel Chelaru, Vasile Nicolae Constantinescu, Adrian Chelaru

ISSN: 1991-8763 766 Issue 9, Volume 5, September 2010

S - Reference area; T - Thrust vector; t - Time; V - Velocity vector;

wvu ,, - Gyroplane velocity components in a mobile frame;

zyx VVV ,, -Velocity components in Earth frame;

000 ZYOX - Normal Earth-fixed frame; Oxyz – Body frame (mobile frame);

000 zyx - Coordinates in Earth-fixed frame.

1 Introduction The paper aims to evaluate the modelling and simulation of the performances of an UAV with an original design as shown in Fig. 1, in different atmospheric conditions.

Figure 1 UAV Endurance configuration

The UAV designed will be capable of assuring a great length of video monitoring (8 hours) on an interest area, on a preprogrammed path, or guided, during the mission. The subject approached is a great interest not only in the perspective of commercial and civil applications, such as infrastructure monitoring, search and rescue missions, traffic control, but also in military applications. For achieving this objective, there have been established two major research directions. The first direction consists in designing and achieving of the carrier platform-UAV, of great endurance, capable of transporting equipment required for commanding the aircraft, for communications, data acquisition and data processing. The second major research direction synthesizes and implements the platform’s automated command system for tracking the default trajectories. It is taken into account the attainment of a flexible infrastructure for the command system which will test the alternative algorithms used for the guidance and control of the platform.

2 General movement equations As shown in the papers [2] and [3] the UAV’s dynamic equations are the projection equations of the force, that are achieved from the impulse theorem, and the moment equations, which come from the kinetic moment theorem. In order to obtain the dynamic equation we start by defining the aerodynamic coefficients in the mobile frame:

.;;

;;;000

A0

AAnA

0

AAmA

0

AA

l

AAz

AAy

AAx

HNC

HMC

HLC

FZC

FYC

FXC

===

=== (1)

where:

lFHSVF oT0 =ρ= ;

2

2

0 . (2)

Similarly, if we consider the thrust T and the nominal thrust as reference 0T , we can define axial thrust coefficient:

0TTCTx = (3)

The force equation can be written in mobile frame:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+

+⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

pvqurupw

qwrv

g

CT

CCC

Fm

wvu

i

Tx

Az

Ay

Ax

00

001

00

A

&

&

&

(4)

or in fixed (Earth) frame:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

g

CT

CCC

Fm

VVV T

x

Az

Ay

Ax

i

z

y

x

00

001

00B&

&

&

, (5)

where the matrix iB is defined using the Euler’s angles: [ ]ji

Tii b ,== AB , (6)

with: θψ= coscos1,1b ;

φψ−ψθφ= cossincossinsin2,1b ;

ψθφ+ψφ= cossincossinsin3,1b ;

θψ−= cossin1,2b ;

φθψ−φψ−= sinsinsincoscos2,2b ;



φθψ−φψ= cossinsinsincos3,2b ;

θ= sin1,3b ; φθ−= sincos2,3b ;

θφ−= coscos3,3b . The moment equation around the centre of the mass of the UAV, written in the mobile frame is:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+−

+−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

EqrpqBAprErpAC

EpqqrCB

CCC

Hrqp

An

Am

Al

A

)()()(

)(2211 JJ 0

&

&

&

,

(7) where the inverse matrix for the inertia moment is given by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−=−

AEBEAC

EC

EAC0

0/)(00

1 22

1J . (8)

The kinematical equations are additional equations, which allow us to obtain the linear coordinates in the inertial frame. If we use the component of velocity in mobile frame we have: [ ] [ ]Ti

T wvuzyx B=000 &&& . (9) Equivalent with this equation, if we use the velocity components in fixed frame we can write:

[ ] [ ]TzyxT VVVzyx =000 &&& . (10)

For Euler’s angle when the rotation velocity components are known we have:

[ ] [ ]TAT

rqpW=ψθφ &&& , (11) where

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

θφθφφ−φθφθφ

=seccossecsin0

sincos0tgcostgsin1

AW . (12)

Supplementary, we have mass equation which describes mass UAV’s modification during the flight:

TCm sp−=& (13)

where spC is specific fuel consumption.

3 Guidance command Resuming [4], the guidance commands for UAV flight are start from relatives parameters

uhz~;~;~;~

φψθ which are given by:

;~;~;~φ−φ=φψ−ψ=ψθ−θ=θ ddd

xx V−=λ ; yy V−=λ zz V−=λ ;

00 xxh dx −= ; 00 yyh dy −= ; 00 zzh dz −= ;

uuu D −=~ (14) where

DdddddD uzyx ;;;, 000ψθφ ; (15) are the input reference values. Also we use integral terms, defined as:

xx hI =& ; yy hI =& ; zz hI =& . (16) The guidance commands are applied through the actuators which are approximated in the paper [4] by relations:

;;a

aua

a

aa

T

Tux

T

TT

ukuk

δ

δ

δδ

δ

δ τ+

τδ

−=δτ

+τδ

−=δ &&

r

rur

r

rr

e

eue

e

ee

ukuk

δ

δ

δδ

δ

δ τ+

τδ

−=δτ

+τδ

−=δ && ; (17)

where reaT δδδδ ττττ ;;; are the time constants and ur

ue

ua

uT kkkk δδδδ ;;; are the gain constants.

4 Balance movement

The study of flight stability will be made accordingly to Liapunov theory, considering the system of movement equations perturbed around the balanced movement. This involves a disturbance shortly applied on the balance movement, which will produce deviation of the state variables. Developing in series the perturbed movement equations in relation to status variables and taking into account the first order terms of the detention, we will get linear equations which can be used to analyze the stability in the first approximation, as we proceed in most dynamic non linear problems. To determine basic movement parameters in equations (1) and (3) is considered

0;0 ====== rqpwvu &&&&&& (18) thous:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

+

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000

00

100

pvqurupw

qwrv

gCCC

TCCC

Fm i

Tz

Ty

Tx

Az

Ay

Ax

A

(19)



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+−

+−+

+⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

000

)()()(

)(221

01

EqrBApqprEACrp

EpqCBqr

CCC

UCCC

HTn

Tm

Tl

An

Am

Al

0

J

J

(20)

5 Linear form of the general equations To obtain the general form of linear equations we start from the linear expression between aerodynamic variables and velocity components:

[ ] ,00110

0

Tzza

aV∆

∂∂

−∆=∆ βα VBM (21)

where: T

MM

⎥⎦⎤

⎢⎣⎡ α∆β∆∆

=∆M ;

[ ]Twvu ∆∆∆=∆V

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

γα

γα

−

γβ

−γβ

−

γαγβ−γ

=

∗∗

∗∗

∗∗∗

βα

coscos0

cos22sin

0coscos

cos22sin

costgcostgcos

2

2

B

By definition [8] aerodynamic angles are: )/arctan( uv−=α , )/arctan( uw=β . (22)

and:

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=γ∗

uwv 22

arctan (23)

Similarly, the relation between unstationary variables is:

VBM &∆=∆ βα2ˆ

Vl (24)

where: T

MM

⎥⎦

⎤⎢⎣

⎡α∆β∆

∆=∆ ˆˆˆ

M ;

[ ]Twvu &&&& ∆∆∆=∆V The following notations are used for the undimensional angular velocities and non-dimensional aerodynamic sizes:

VlMMVlVl

VlrrVlqqVlpp

/ˆ;/ˆ;/ˆ

;/ˆ;/ˆ;/ˆ&&&

&&&

=β=βα=α

=== (25)

where l is reference length – body length. Similarly, for un-stationary components we can write:

VAaV Ω−=& , (26) or in linear form:

ΩAVAaV ∆−∆−∆=∆ Ω V& , (27)

where:

[ ]Tzyx aaa=a ; [ ]Trqp=Ω ;

;0

00

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=Ω

pqpr

qrA ,

00

0

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

uvuwvw

VA

In this case, the expression of the aerodynamic force becomes:

,ˆ

ˆˆ

22

00

0

000

0

AFMF

Fpqr

FzoFM

F

FF

F

zFFaa

MMF

δCMC

C

CMC

CF

∆+∆+

+Ω∆+

+∆+∆+

+⎟⎟⎠

⎞⎜⎜⎝

⎛ρρ∆

+∆

+∆

=∆

δαβ

βα

&&&

(28)

where:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

=∆A

A

A

ZYX

F⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

δ∆δ∆δ∆

=∆

r

e

aAδ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=Az

Ay

Ax

F

CCC

C

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=Az

Ay

Ax

Fz

z

z

z

CCC

0

0

0

0C⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=Az

Ay

Ax

FM

M

M

M

CCC

C ;

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

αβ

αβ

αβ

βαAz

Az

AzM

Ay

Ay

AyM

Ax

Ax

AxM

FM

CCMCCCMCCCMC

C

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=Azr

Azq

Azp

Ayr

Ayq

Ayp

Axr

Axq

Axp

Fpqr

CCCCCCCCC

C

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

αβ

αβ

αβ

αβAz

Az

AMz

Ay

Ay

AMy

Ax

Ax

AMx

MF

CCMCCCMCCCMC

&&&

&&&

&&&

&&&C



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

δδδ

δδδ

δδδ

δA

rzA

ezA

az

Ary

Aey

Aay

Arx

Aex

Aax

F

CCCCCCCCC

C

Similarly, for aerodynamic moment we can write:

,ˆ

ˆˆ

22

00

0

000

0

AHMH

Hpqr

HzpHM

H

HH

H

zHHaa

MMH

δCMC

C

CMC

CH

∆+∆+

+Ω∆+

+∆+∆+

+⎟⎟⎠

⎞⎜⎜⎝

⎛ρρ∆

+∆

+∆

=∆

δαβ

βα

&&&

(29)

where:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

=∆A

A

A

NML

H ;⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

δ∆δ∆δ∆

=∆

r

e

aAδ ;

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=An

Am

Al

H

CCC

C ;

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=Anz

Amz

Alz

Hz

CCC

0

0

0

0C ;⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=AnM

AmM

AlM

HM

CCC

C ;

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=An

An

AnM

Am

Am

AmM

Al

Al

AlM

HM

CCMCCCMCCCMC

αβ

αβ

αβ

βαC ;

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=Anr

Anq

Anp

Amr

Amq

Amp

Alr

Alq

Alp

Hpqr

CCCCCCCCC

C ;

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=An

An

AMn

Am

Am

AMm

Al

Al

AMl

MH

CCMCCCMCCCMC

αβ

αβ

αβ

αβ

&&&

&&&

&&&

&&&C ;

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=A

rnA

enA

an

Arm

Aem

Aam

Arl

Ael

Aal

H

CCCCCCCCC

δδδ

δδδ

δδδ

δC .

Considering the relationship between aerodynamic components and velocity components in body-frame the aerodynamic force becomes:

AFMF

Fpqr

FzpFMF

FM

FV

lFV

lF

zlz

aaM

zF

VF

δCVBC

ΩC

CCC

VBCF

δβααβ

βαβα

+∆+

+∆+

+∆⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

−∂∂ρ

ρ+

+∆=∆

020

0

000

0

0

11

&&&&

(30) Similarly, for aerodynamic moment, we can write:

AH0MH2

0

Hpqr0

HzHMH0

HM0

HV

lHV

lH

zlz

aaM

zH

VH

δCVBC

ΩC

CCC

VBCH

∆+∆+

+∆

+∆⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

−∂∂ρ

ρ+

+∆=∆

δβααβ

βαβα

&&&&

0000

11

(31)

Regarding thrust, it can be put in linear form as follows:

TT

TMTz

TM

T

zza

aM

lT

MVT

δ∆+

+∆⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−+

+∆=∆

δ

βα

C

CC

VbCT

0

00

00

0

1 , (32)

where:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γαγβ−

γ=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

=∆

δ

δ

δ

δ

∗

∗

∗

βα

TTz

TTy

TTx

TTzz

Tyz

Txz

Tz

T

TzM

TyM

TxM

TMT

T

T

CCC

CCCC

CCC

ZYX

0

0

0

0

;costgcostg

cos

C

bCT

Similarly, for thrust moment we can write:



TU0Upqr0

UMUz0

UM0

UV

lU

zza

aM

l1U

MVU

δ∆+∆+

+∆⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−+

+∆=∆

δ

βα

CΩC

CC

VbCU

00

0 (33)

where:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

=∆

δ

δ

δ

δT

Tn

TTm

TTl

UTnr

Tmq

Tlp

Upqr

Tnzp

Tmzp

Tlzp

UzpTnM

TmM

TlM

UMT

T

T

CCC

CC

C

CCC

CCC

NML

CC

CCU

;00

0000

;;;

Finally, if we know the expression of the components of the gravity along the mobile frame:

0gAg i= (34) we express the variation of the gravity along mobile frame axis:

rAg ∆=∆ gR (35) where:

[ ]Tψ∆θ∆φ∆=∆r

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡ψ∂

∂θ∂

∂φ∂

∂=

0

0

0

000000

gg

gAAA

A iiigR

Starting from force equation (4) in mobile frame, we obtain:

,11pVmm

fΩAVAgTFV ∆+∆−∆−∆+∆+∆=∆ Ω&

(36) where:

[ ]Tpppp ZYXm

∆∆∆=∆1f (37)

means perturbation force. Similarly, for momentum equation (7) we can write:

,111pK mΩAJUJHJΩ ∆+∆+∆+∆=∆ −−−& (38)

where:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

qrrp

pqE

pqprqr

BAAC

CB

K

02020

00

0

000000

A

and [ ]Tpppp NML ∆∆∆=∆ −1Jm (39)

means perturbation moment. Starting from these relations we can write:

pT

TAAV

RzV z

ffδFVF

rGfΩFVFV

∆+δ∆+∆+∆+

+∆+∆+∆+∆=∆

δδ

Ω

&

&

&

00 (40)

where:

Ωβαβαβα −+= AbCBCF TMFMV MmVT

mVF 00

VFpqrmVlF

ACF −=Ω0 ; gRR AG = ;

βααβ= BCF&&&& MFV mV

lF2

0 ; δδ = FA

mF

CF 0

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

−∂∂ρ

ρ=

TMTz

FzFMFz

za

aM

lmT

lza

aM

zmF

CC

CCCf

00

0

000

00

1

11

δδ = TT

mT

Cf 0 ;

Starting from moment equation (3) we obtain:

pTTAA

V

zV z

mmδMVM

mΩMVMΩ

∆+δ∆+∆+∆+

+∆+∆+∆=∆

δδ

Ω

&

&

&

00 (41)

where:

βα−

βαβα− += bCJBCJM UMHMV M

VU

VH 1010

KUpqrHpqr VlU

VlH

AJCJCJM 11010 −−−Ω ++=

βααβ−= BCJM

&&&& MFV VlH 1

20 ; δ

−δ = H0A H CJM 1 ;

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

−∂∂ρ

ρ=

−

−

UMUzp

HzHMHz

za

aM

l1U

lza

aM

zH

CCJ

CCCJm

0

10

000

100

11

δ−

δ = UT U CJm 1

0 ; Starting from cinematic equation (9) we obtain:

rPVPp ∆+∆=∆ RV& (42) where:

[ ]Tzyx 000 ∆∆∆=∆p

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

∂∂

=V

VV

ψB

θB

φB

P00

0000

iiiR

iV BP = and from equation (11):

rRΩRr ∆+∆=∆ Ω R& (43) where:



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

∂φ∂

=Ω

ΩΩ

WWWR00

0000

ψθAAA

R

AWR =Ω Using linear equations (40)…(43) we can put the system in regular form:

BuAxx +=& (44) where: [ ] 0

11 AAIA −−= ; [ ] 0

11 BAIB −−=

We can also highlight the stability and control matrixes as: Table 1 The stability matrix with stationary variables 0A

R

RV

zV

RzV

RRr

PPp

mMMΩ

GfFFV

rpΩV

Ω

Ω

Ω

000000000000

0

0

Table 2 The stability matrix with nonstationary variables 1A

r

p

MΩ

FV

rpΩV

V

V

&

&

&&&&

Table 3 Control matrix 0B

r

p

mMΩ

fFV

δ

TA

TA

TA

δδ

δδ

δ

Finally, we express the perturbation vector:

Table 4 The perturbation vector 0p

r

p

mΩ

fV

p

p

∆

∆

4. Extended stability and control matrixes

Besides the general motion equations in linear form as outlined above, UAVs needs other relationships to be added. Among them, the most important and which can not be neglected are the actuator equations and the guidance equations. For the autonomous flight, as is case of UAV's, the guidance equation is necessary to introduce integrated terms specific to PID-type controllers.



Starting from (17) linear form of the actuator equation became:

[ ][ ] uDD ∆+δ∆δ∆δ∆δ∆=

=δ∆δ∆δ∆δ∆

δ uT

Trea

TTrea&&&&

(45)

where:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

τ−τ−

τ−τ−

=

δ

δ

δ

δ

δ

T

r

e

a

1000010000100001

D ;

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ττ

ττ

=

δδ

δδ

δδ

δδ

TuT

rur

eue

aua

u

kk

kk

000000000000

D ,

Similarly, linear form of auxiliary equation (16) became:

[ ] [ ] f∆+∆∆∆−=∆∆∆ Tppp

Tzyx zyxIII &&&

(46)

where f∆ means reference values as input function. Using linear relation (45 and (46) we can build extended stability and control matrixes.

Table 5 Stability extended matrix A

II

δ

RRr

PPp

mMmMMΩ

fFGfFFV

IδrpΩV

−

Ω

δδΩ

δδΩ

R

RV

TAzV

TARzV

000000000000

0

0

.

Table 6 Control extended matrix B

I

Dδ

r

p

Ω

V

u

Trea uuuu

5 Optimal control using state vector

Supposing to have access to extend state vector x , we can obtain directly the controller K for optimal command:

Kxu −= (47)

In order to satisfy the linear quadratic performance index (cost function):

tJ TT d)(min0

RuuQxx += ∫∞

, (48)

where the extended pair ( )BA, is controllable and the state weighting matrix Q is symmetric and quasi positive:

;0≥Q TQQ = . (49)

while the control weighting matrix R is symmetric and positive:

;0>R TRR = ; (50) In this case, the following relation gives the optimal controller

PBRK 1 T−= (51)

where the matrix P is the solution of the algebraic Riccati equation:



0QPBPBRPAPA T1T =+−+ − (52)

6 Optimal control using Kalman filter Using the optimal controller designed above requires access to all system states, very difficult in view of the limited number of sensors. In this case, for a complete description of the system we use a linear state estimator constructed as a Kalman filter. For this purpose we start from the regular relations:

vDuCxyGwBuAxx

++=++=& (53)

where w is the external noise and v is the internal noise introduced by the sensors, where the matrixes DC,G, are considerate corrected with the stability matrix with non-stationary variables

1A

[ ] 0GAIG 11

−−= ; [ ] 0CAIC 11

−−= ; [ ] 0DAID 11

−−= , (54)

The idea of estimator operation is if that the deliver system )(:1 DC,B,A,Σ with state x, can be "predicted" by system )(:2 DC,B,A,Σ that uses state z, which is accessible in this case to be controlled. In order that the system 2Σ follows the system 1Σ we calculate a regulator L which brings the difference between actual read states 1y and estimated states 2y as a correction into the system 2Σ . In this case we can write:

⎩⎨⎧

++=+δ++=

ΣvDuCxy

GwxBuAxx

1

01 :

& (55)

⎩⎨⎧

+=−+δ++=

ΣDuCzy

yyLzBuAzz

2

2102

)(:& (56)

where initial conditions are introduced by 0x , respectively 0z . Tracking error, including the initial conditions, is given by:

zxx −=~ ; 000~ zxx −= (57)

If we decrease 2Σ from 1Σ and neglect the noise is obtained:

0LCA xx ~~ )( the −= . (58)

Hence if L is dimensioned such that A-LC have eigenvalues with negative real part, the estimation error tends to zero. Since z is provided by the estimator, we have access to all states to make control of the form:

Kzu −= (59) In this case the system 1Σ is described by the equation:

δδ oo xxBKBK)x(AxBKzAxx ++−=+−= ~& (60)

which has the solution: )~( )()(

0LCA

0BKA xBKxx tt ehhe −− +δ= (61)

The process of calculating the estimator is similar to that described above for the optimal regulator. This is based on the dual system:

uCxAx TT +=& (62) for which is considered performance index:

∫∞

+′=0

dmin tJ TT ]uPux)GQ(Gx[ (63)

By solving the matrix Riccati equation: 0GQGCRPRCRAAR 1 =+−+ − TTT (64)

matrix estimator is obtained: 1PRCL −= T (65)

where R is the solution of Riccati equation. 7 Input data, calculus algorithm and results 7.1 Input data for the model

7.1.1 Geometrical data As input data we use the geometrical elements of the UAV from Fig. 2.

Figure 2 a UAV geometry

Figure 2 b UAV geometry



Figure 2 c UAV geometry

Geometrical characteristics for the model are: Reference length – body length: ml 15.2= ; Reference area – cross body area: 2116.0 mS = ;.

7.1.2 Mechanical data Mass characteristics of the model are: kgmi 70= ; Corresponding to initial mass, we have: Centre of mass position: mxcm 3.1= .

Inertial moments: 210kgmA = ; 220kgmB = ; 230kgmC = 25.0 kgmE =

7.1.3 Aerodynamic data For the configuration from Figure 2, considering a Taylor series expanding around the origin, taking into account the parity of the terms, we obtain the following polynomial form of the aerodynamic coefficients in a body frame:

414

313109

28

27

26

222

2211

ˆ α+α+δ+α+

+δ+δ+δ+β+α+=

aaqaa

aaaaaaC

e

raeAx

pbbbbrbbC arAy ˆˆˆ 10926524212 +β+δ+δ++β=

314

213915141110 ˆˆ α+α+α+δ++α+= bbbbqbbbC e

Az

β+δ+δ++= 137653 ˆˆ cccrcpcC raA

l α+δ++α+= ˆˆ 915141110 ddqdddC e

Am

pddddrddC arAn ˆˆˆ 10926524212 +β+δ+δ++β= ,

(66) where the coefficients 1a , 21a … generally are depending on Mach number.

In our case, for low subsonic flow, the coefficients 1a , 21a … 10d are practically constant, having the following values:

53.01 −=a ; 7.421 =a ; 83.022 −=a ; 5.66 −=a ;

6.197 −=a ; 5.68 −=a ; 5.29 −=a ; 7.610 =a ;

1.3213 −=a ; 5.96314 −=a 11412 .b = ; 5.442 =b ;

5.652 −=b ; 83.16 =b ; .092 =b ; 57.710 −=b ;

34.10 −=b ; 6.9911 −=b ; 8.141 −=b ; 47.651 =b ;

0.091 =b ; 2.3513 =b 0.95214 =b 43.263 −=c ;

2.05 −=c ; 62.16 =c ; ;054.013 =c 21.07 −=c ;

179.00 =d ; 29.511 −=d ; 6.1141 −=d ;

63.651 =d ; 0.091 =d ; 63.312 −=d ; 5.542 −=d ;

63.652 =d ; 7.06 −=d ;

.092 =d ; 3.610 =d . (67)

7.1.4 Thrust

The propeller thrust is determined by the relation:

TxCTT 0= (68)

Where 0T is the nominal value at ground, a fix

point, and TxC and axial gas -dynamic coefficient.

Fashioning experimental results indicated in work [9] we obtain the following approximate relation:

)()()( 321 TpTx fzfMfC δ= (69)

where the influence of the main parameters ware separated:

Mach number M :

MMf 0.31)(1 −= ; (70)

Altitude pz :

pp zzf 52 101.91)( −⋅−= ; (71)

Thrust command Tδ :

TTf δ+−=δ 677.1677.0)(3 . (72)

Thrust command is limited between: 15.0 <δ< T . It is obvious that for the null velocity (fixed point flight) at ground level with maximum command the thrust takes the nominal value: ][5630 NT =

Similarly, we can obtain specific fuel consumption, by using relation:



)()()( 3210 Tpspsp gzgMgCC δ= , (73)

where the Mach dependence is:

21 966.0133.0026.1)( MMMg +−= ; (74)

Altitude dependence:

pp zzg 62 10.81)( −⋅−= ; (75)

Thrust command influence:

23 )1(3.01)( TTg δ−+=δ . (76)

The nominal specific consume at ground level with maximum command, corresponding with 0T has the value:

]//[1059.1 70 sNKgCsp

−×= . (77)

Using specific fuel consumption, we can evaluate fuel debit that coincides with mass variation, as we see in the equation (8).

6.1.5 Guidance parameters

For flight control system applying Kalman filter relations, we obtain the following guidance gains:

- Controller K values:

01,1 =k ; 21.12,1 =k …. 181.019,4 −=k (78)

- Estimator L values:

4.121,1 =l ; .02,1 =l … 05.1019,19 =l (79)

7.2 Calculus algorithm

The calculus algorithm consists in multi-step method Adams' predictor-corrector with variable step integration method: [1] [11]. Absolute numerical error was 1.e-12, and relative error was 1.e-10.

7.3 Calculus test case

We will consider as a calculus test the situation when the UAV takes-off, makes a rectangular path with four turns maintaining velocity and altitude

flight, followed by a descending phase. During the flight, after the second turn, it attends a turbulent zone. Crossing the turbulent zone, UAV uses the guidance command system, in order to maintain flight parameters. Turbulence zone was designed accordingly with work [5]. Flight parameters are typical for surveillance activity: altitude

mz pd 200= , and axial velocity smud /44= .

7.4 Results

In Figure 3 we showed the flight-path diagram, the test situation when the UAV made a rectangular path with four turns. Fig. 4 shows the velocity diagram during the test flight described above.

0.00.10.10.10.2

z[Km

]

-4-3

-2-1

01

23

45

x[Km]-4

-3-2

-10

y[Km]

X Y

Z

Figure 3 UAV flight-path diagram

0 100 200 300 400 500t[s]

45

50

55

60

V[m

/s]

Figure 4 Velocity diagram



Fig. 5 and Fig. 6 show the ruder deflection and elevator deflection necessary to obtain desired trajectory.

0 100 200 300 400 500t[s]

-2

-1

0

1

2

3

4

5

6

7

8

δ r[d

eg]

Figure 5 Ruder deflection diagram

0 100 200 300 400 500t[s]

-8

-6

-4

-2

0

2

4

6

8

δ e[d

eg]

Figure 6 Elevator deflection diagram

In Fig. 7 and 8 are shown the attitude angles: bank angle and azimuth angel, during the flight.

0 100 200 300 400 500t[s]

0

0.5

1

1.5

2

φ[d

eg]

Figure 7 Bank angle diagram

0 100 200 300 400 500 600t[s]

0

50

100

150

200

250

300

350ψ

[deg

]

Figure 8 Azimuth angle diagram

In all diagrams, except flight –path diagram and azimuth diagram on can observe the influence of turbulence zone on flight parameters.

7 Conclusions The conclusions are structured in two points as the following.

Guidance scheme: A first conclusion regarding the guiding scheme consists in the fact that the UAV will have robust command structure, based on Kalman filter which is capable to lead the UAV on desired flight-path in different atmospheric conditions. From the diagram, previously presented, one can observe that the flight parameters after the turbulence zone came back to the normal values.



From this point of view we must define a number of evolutions that will make several guiding structures, which will be dedicated to each kind of evolutions. This part of the command structure will be able to evolve in the same time with the development of the project, when the experimental results will be available.

Technical solution: Regarding the adopted solution, using unusual tail having two consoles arranged in the shape of the letter "V" inverted, instead of regular vertical and horizontal tail, we can obtain a better stability and at the same time a better control without increasing the deflection attach angle due to upstream wing. References: [1] Bakhvalov, N. Methodes Numeriques –

Analyse, algebre, equations differentielles ordinaires Ed. Mir Moscou ,1976.

[2] Bandu N. P. Performance, Stability, Dynamics and Control of Airplanes Second Edition, AIAA Education Series, ISBN 1-56347-583-9, 2003.

[3] Boiffier, J.L. The Dynamics of Flight – The Equations, John Wiley & Sons , Chichester, New York, Weinheim, Brisbane, Singapore, Toronto , ISBN 0-471-94237-5, 1998.

[4] Chelaru T.V. Dinamica Zborului – Proiectarea avionului fără pilot (Dynamic Flight – the UAV design)., Ed. Printech, Bucureşti, ISBN 973-652-751-4, 2003.

[5] Etkin, B., Dinamics of Atmospheric Flight, John Wiley & Sons, Inc., New York, 1972.

[6] Kuzovkov, N.T., Sistemi stabilizatii letatelnih appararov ,balisticeschih i zenitnih raket, Ed. Vîssaia Skola , Moskva, 1976.

[7] Lebedev, A.A., Gerasiota, N.F., Balistika raket, Ed. Masinostroenie, Moskva, 1970.

[8] Nielsen, J.N., Missile Aerodynamics, McGraw-Hill Book Company, Inc., New-York, Toronto, London, 1960.

[9] Nita, M.M., Moraru, Fl., Patraulea, R., Avioane şi rachete, concepte de proiectare, Ed. Militara, Bucuresti, 1985.

[10] ISO 1151 -1:1988; -2:1985-3:1989 [11] SLATEC Common Mathematical Library,

Version 4.1, July 1993 [12] Piotr Kulczycki- Nonparametric Estimation

for Control Engineering, Proceedings of the 4th WSEAS/IASME International Conference DYNAMICAL SYSTEMS and CONTROL

(CONTROL'08) Corfu, Greece, October 26-28, 2008.

[13] Jerzy Stefan Respondek- Observability of 4th Order Dynamical Systems Proceedings of the 8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

[14] Zheng Jiewang1,Wei Li and Shijun Guo - Lateral Control for an Aircraft of Folding Wing Proceedings of the 8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

[15] R. Roy, J-F. Chatelain, R. Mayer , S. Chalut and S. Engin - Programming of a Machining Procedure for Adaptive Spiral Cutting Trajectories Proceedings of the 10th WSEAS International Conference on AUTOMATIC CONTROL, MODELLING & SIMULATION (ACMOS'08).Istanbul, Turkey, May 27-30, 2008.

[16] Anton V. Doroshin - Synthesis of Attitude Motion of Variable Mass Coaxial Bodies, WSEAS TRANSACTIONS on SYSTEMS AND CONTROL Volume 3, 2008 ISSN: 1991-8763.

[17] Tain-Sou Tsay - High Accurate Positioning Technique for AUV WSEAS TRANSACTIONS on SYSTEMS AND CONTROL Volume 4, 2009 ISSN: 1991-8763.



Date post:	26-Feb-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Control system of unmanned aerial vehicle used for...

Documents