modeling and numerical simulation of unmanned aircraft vehicle restricted by non-holonomic...

JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 1, pp. 251-268, Warsaw 2012

50th anniversary of JTAM

MODELING AND NUMERICAL SIMULATION OF

UNMANNED AIRCRAFT VEHICLE RESTRICTED BY

NON-HOLONOMIC CONSTRAINTS

Edyta adyyska-Kozdra

Warsaw University of Technology, Faculty of Mechatronics, Warsaw, Poland

e-mail: [email protected]

The paper presents the modeling of ight dynamics of an unmanned air-craft vehicle (UAV) using the Boltzmann-Hamel equations for mechani-cal systems with non-holonomic constraints. Control laws have been tre-ated as non-holonomic constraints superimposed on dynamic equationsof motion of UAV. The mathematical model containing coupling dy-namics of the aircraft with superimposed guidance have been obtainedby introducing kinematic relationships as the preset parameters of themotion resulting from the process of guidance. The correctness of the de-veloped mathematical model was conrmed by the carried out numericalsimulation.

Key words: automatically steered aircraft vehicle, non-holonomic con-straints, Boltzmann-Hamel equations

1. Introduction

In recent years, unmanned aircraft vehicles (UAV) have been the fastest gro-wing means of recognizing and supporting the armed forces. On the modernbattleeld, light small-sized UAVs perform the mission of ground target de-tection, tracking and illumination. Their modied versions, combat UAVs,are supposed not only to autonomously detect the target, but also destroyit with on-deck homing missiles (Fig. 1). UAVs usefulness has been conr-med unequivocally both by operations carried out by the Americans in Iraqand Afghanistan, as well as Israeli military operations in Lebanon and Ga-za. No wonder that acquiring a UAV for the Polish army has become oneof the priorities of its technical modernization in the years 2009-2018. Forthe same reasons, the problems associated with the study of dynamics and

252 E. adyyska-Kozdra

control of the UAV is the focus of many scientic and research centres inPoland and the world. There are also many methods of modelling these issu-es from the equations of classical mechanics using the principle of changingangular momentum (Dogan and Venkataramanan, 2005; Ducard, 2009; Etkinand Reid, 1996; Maryniak, 2005; Rachman and Razali, 2011; Sadraey andColgren, 2005), to the methods of analytical mechanics for nonholonomic sys-tems. Analytical mechanics provides several methods of generating equationsof motion of ying objects. Boltzmann-Hamel equations, as well as Maggis,Gibbs-Appel, Keyn equations, and the projective method developed by Profes-sor W. Blajer, used in the study, are among them (Blajer, 1998; Blajer et al.,2001; Chelaru et al., 2009; Grastein et al., 1997; Gutowski, 1971; Koruba andadyyska-Kozdra, 2010; adyyska-Kozdra, 2008, 2009, 2011; Maryniak,2005; Sadraey and Colgren, 2005; Ye et al., 2006).

Fig. 1. General view of combat UAV mission performance (Koruba andadyyska-Kozdra, 2010)

The dynamical modeling of UAV forms the heart of its simulation (Sadraeyand Colgren, 2005). The numerical simulation of the aircraft dynamics is themost important tool in the development and verication of the ight controllaws and equations of motion for a UAV. The ability to test autopilot systemsin a virtual (software) environment using a software ight dynamics modelfor UAVs is signicant for development, as shown for example in the work byBlajer et al. (2001), Chelaru et al. (2009), Dogan and Venkataramanan (2005),Jordan et al. (2006), adyyska-Kozdra (2011), Ou et al. (2008), Shim et

Modeling and numerical simulation of UAV... 253

al. (2003). In many cases, testing newly developed autopilot systems in avirtual environment is the only way to guarantee absolute safety. Additionally,the model would allow better repeatability in testing, with controlled yingenvironments.

This study is a continuation and generalization of the article published inthe Journal of Theoretical and Applied Mechanics (Koruba and adyyska--Kozdra, 2010), where was proposed an algorithm of guiding combat UAV,which on having autonomously detected targets, attack them (e.g. radar sta-tions, combat vehicles or tanks) or illuminate them with a laser (Fig. 1). Thesimplied equations of UAV motion developed there were now generalized forthe object with nonholonomic constraints. In the article, a novel method wasproposed based on treating the nonholonomic constraints as laws of control-ling and conjugating them with nonlinear equations of motion of an object andwith kinematic guidance relations (adyyska-Kozdra, 2008, 2009, 2011).

Control, that is override aimed at ensuring that the moving object behavesin a desired manner, was brought to testing of dierences, that is deviationsbetween the required and actual value of the realized coordinate. The valueof this dierence, after appropriate strengthening and transforming, resets thedeviation. Developed in this way control laws were treated as non-holonomicconstraints superimposed on the motion of a UAV. Linkage of these equationswith the dynamic equations of motion of the object made it possible, thanksto the use of Boltzmann-Hamel equations for mechanical systems with non--holonomic constraints, to control the ight of a UAV in an eective manner.

2. Physical model of UAV and adopted reference systems

The article presents the process of modeling and numerical simulation of theight of unmanned aircraft vehicle during the mission. Appropriate formula-tion of the physical model is an important element here, which will constitutethe basis for building a mathematical model of motion of the tested object.

The following assumptions of the physical model have been adopted:

UAV is treated as a non-deformable object, with six degrees of freedomresulting from the movement of rudders;

Motion of a UAV is examined in calm weather;

UAV weight varies during the ight;

UAV motion control may be carried out in four channels: in the pitchchannel by means of elevator deections H , in the yaw channel


by means of rudder deections V , in the roll channel by meansof aileron deection L, and in the speed channel V0 by changing theengine thrust T ;

Control units are moving, but they are non-deformable;

Deection of surfaces of rudders aects the forces and moments of aero-dynamic forces;

Constraints superimposed on a UAV resulting from the adopted controllaws have been treated as non-holonomic constraints;

Forces and moments of aerodynamic forces, from the propulsion systemand gravitation, act on the UAV;

The impact of the curvature of the Earth was ignored;

The environment has an impact on the dynamic properties of the drivethrough changes in air temperature tH , air density H , air pressure pH ,kinematic viscosity H , sound speed aH depending on the altitude.

Fig. 2. Adopted reference systems, linear and angular speed of a UAV

The motion of a unmanned aircraft vehicle is examined in the referencesystem Oxyz, rigidly connected with the moving object, with the beginning inthe center of mass of the UAV after burnup of fuel (Fig. 2). The other referencesystem used in the paper includes: system O1x1y1z1, rigidly connected withthe Earth, as well as systems connected with the object, namely: gravitationalOxgygzg parallel to O1x1y1z1 system and velocity Oxayaza connected withthe air ow direction (Fig. 3).


3. Kinematic correlations and guidance correlations

The motion of a unmanned aircraft vehicle during a mission is described withthe use of coordinates and time in the space of events where the location of theobject is uniquely determined with the use of linear and angular coordinatesin the conguration space.

Pursued ight parameters of the UAV are read automatically by the gu-idance system and depend only on the actual behavior of the guided objecton the track, and thus on the changes of its linear and angular position.

The vector of the actual linear velocity of the UAV (Fig. 2) in the Oxyzsystem is

V 0 = Ui+ V j +Wk (3.1)

where: U , V , W are respectively: longitudinal, lateral and vertical velocity;i, j,k unit vectors of the system associated with the Oxyz object.

The angular velocity vector

= P i+Qj +Rk (3.2)

where: P , Q, R are the angular velocity of banking, tilt and deection (Fig. 2).Kinematic correlations between the components of angular velocities and

derivatives of the angles have the following form (Blajer, 1998; Etkin and Reid,1996; adyyska-Kozdra, 2009, 2011)

=

1 sin tan cos tan 0 cos sin0 sin sec cos sec

P

Q

R

(3.3)

Kinematic correlations between the components of the linear velocityx1, y1, z1 measured in the system O1x1y1z1 rigidly connected with the Earthand the components of U, V,W velocity in the reference system Oxyz asso-ciated with the moving objects are as follows (Blajer, 1998; Etkin and Reid,1996; adyyska-Kozdra, 2009, 2011)

x1y1z1

=

cos cos cos sin sin+ cos cos

cos sin cos++sin cos

sin cos sin sin sin+ sin cos

sin sin cos+ sin cos

sin sin cos cos cos

U

V

W

(3.4)

In calm weather the angle of approach and glide are expressed by thefollowing formulas (Blajer, 1998; Etkin and Reid, 1996; adyyska-Kozdra,2009, 2011):


the angle of approach

= arctanW

U(3.5)

the angle of glide

= arcsinV

V0(3.6)

Preset ight parameters of the unmanned aircraft result from the adop-ted guidance method. Assuming that during the observation of the area theaircraft ies along the pre-programmed route, constraints are superimposedon the location and angular and linear velocity of the object, thus creatingthe generator of its programmed motion (Blajer, 1998; Blajer et al., 2001;adyyska-Kozdra, 2011)

K0Ke = s(x1p, y1p, z1p, p, p, p)

Vz(t) = s(x1p, y1p, z1p, p, p, p, p, p, p)(3.7)

where: x1p, y1p, z1p, p, p, p are the preset parameters of motion of thecontrolled object.

When the UAV detects the target, it is possible to trace or attack it. In thiscase, the preset parameters of the controlled object were selected in the paperwith the use of one of self-guidance methods, that is the method of along thecurve of hunt (Koruba and adyyska-Kozdra, 2010; adyyska-Kozdra,2011). It assumes that the preset angular coordinates of the guided object areconsistent with the parameters of the target

p = t p = t p = t (3.8)

The vector of the preset location of UAV in O1x1y1z1 system (Fig. 2) is

rp = x1pi1 + y1pj1 + z1pk1 (3.9)

where

x1p = rot cost cos t y1p = rot sint cos t z1p = rot sin t

and rot is the distance of the actual location of the object to the target,t, t, t angles of banking, tilt and deection of the maneuvering target.


The components of the vector of the preset linear velocity of UAV in theown system are

UpVpWp

=

cost cos t sint cos t sin tsint cost sin t+ sint cost

sint sint sin t++cost cost

sint cos t

cost cost sin t++sint sint

cost sint sin t+ cost sint

cost cos t

x1py1pz1p

(3.10)The components of the vector of the preset angular velocity of UAV in the

own system arePpQpRp

=

1 0 sin t0 cost sint cos t0 sint cost cos t

ttt

(3.11)

During the ight of the unmanned aircraft, the preset parameters resultingfrom the adopted guidance method are compared with the parameters of theight of the aircraft which are read on the ongoing basis. The dierences whichoccur between these parameters are then removed by the automatic controlsystem. In this way, the control mechanism aects motion of the object bymeans of relevant control units, depending on changes in one or, most often,many parameters of its motion.

4. Control laws

The motion of the unmanned aircraft along the preset trajectory results fromthe superimposed kinematic constraints. These constraints are, however, vio-lated due to various external interferences, construction deciencies, etc. It istherefore necessary to equip the object with a set of devices that will determi-ne the degree of violation of these constraints and will generate appropriatecontrol signals so that the aircraft will y along the required track. All thesetasks are pursued by the control system.

In this way, through the appropriate control units, the mechanism of auto-matic guidance of the object aects its movement, depending on one or, mostoften, many parameters such as acceleration, speed, linear and angular posi-tion, track angle. This aects the dynamics of the controlled object, causing achange of control forces which enforce the ight consistent with the adopted


control system and guidance algorithm. The automatic control system, basingon the values of deviations which compare information about the state of thecontrolled object and the vector of the preset state, generates control signals.Kinematic and geometric deviations relationships in the servomechanisms be-come strengthened and then they are transferred to the actuators, such ashydraulic, electrohydraulic or electromechanical cylinders. The delay of thecontrol system has been described by means of an inertial unit of the rstorder.

Automatic control of the unmanned aircraft is carried out in four channels:in the pitch channel by means of elevator deections H , in the yawchannel by means of rudder deections V , in the roll channel bymeans of aileron deection L, and in the speed channel V0 by changing theengine thrust T .

Control laws of the UAV take the following form: in the pitch channel

TH3 H + TH2 H = K

Hx1(x1 x1p) +KHz1(z1 z1p) +K

HU (U Up)

+KHW (W Wp) +KHQ (QQp) +K

H ( p) + H0

(4.1)

in the yaw channel

T V1 v + TV2 V = K

Vy1(y1 y1p) +KVV (V V1p) +K

VW (W Wp)

+KVR (RRp) +KV ( p) + V 0

(4.2)

in the roll channel

TL1 L + TL2 L = K

L ( p) +K

LP (P Pp) +K

LV (V Vp)

+KLR(RRp) +KL ( p) + L0

(4.3)

in the velocity channel

T T1T + T T2 T = K

Tx1(x1 x1p) +KTU (U Up) +K

TW (W Wp)

+KT ( p) +KTQ(QQp) +K

TR(RRp) +K

T ( p) + T0

(4.4)

where: T ji are time constants, Kji amplication coecients.


The designated control laws are non-integrable and impose restrictionson the motion system, and therefore they were considered as kinematic equ-ations of non-holonomic constraints (Bloch, 2003; adyyska-Kozdra, 2011;Nejmark and Fufajew, 1971). In the paper, the aforementioned equations werelinked with the dynamic equations of motion of the unmanned aircraft vehicle,derived using analytical equations of mechanics in the form of the Bolzmann-Hamel equations with multipliers.

This approach made it possible to remove dierences resulting from thedynamics of motion of the controlled object occurring between geometric, ki-nematic and dynamic preset parameters and their pursuance during the ightof the tested object.

5. General equations of motion of the unmanned aircraft vehicle

Boltzmann-Hamel equations for mechanical systems with

non-holonomic constraints

Description of dynamics of the unmanned aircraft, treated as a non-deformablemechanical system, was made in the reference system rigidly connected withthe Oxyz object. In addition, control laws (4.1)-(4.4) were treated as non-holonomic constraints superimposed on motion of the system. Therefore, inorder to determine the equations of motion, the Boltzmann-Hamel equationsof motion of non-holonomic systems in the generalized coordinates (Grasteinet al., 1997; Gutowski, 1971; adyyska-Kozdra, 2011; Maryniak, 2005) wereused.

From the Boltzmann-Hamel equations, after calculating the value of Bolt-zmann multipliers and determination of kinetic energy in quasi-velocities, asystem of ordinary second-order dierential equations was obtained: the equation of longitudinal motion

d

dt

(T U

)

T

VR+

T

WQ

T

x1cos cos

T

y1sin cos

+T

z1sin +

T

H

(QKHW K

Hx1cos cos +KHz1 sin

TH2TH1

KHU

)

+T

V

(QKVW RK

VV +K

Vy1sin cos

)

T

LRKLV

+T

T

(QKTW K

Tx1cos cos

T T2T T1

KTU

)= QX

(5.1)


the equation of lateral motion

d

dt

(T V

)

T

x1(sin cos sin sin sin)

T

y1(sin sin sin + cos cos)

T

z1sin cos

T

UR+

T

WP +

T

L

TL2TL1

KLV (5.2)

+T

H[PKHW K

Hx1(sin cos sin sin sin)KHz1 sin cos

+RKHU ] +T

V

[PKVW +

T V2T V1

KVV KVy1(sin sin sin + cos cos)

]

+T

T[PKTW K

Tx1(sin cos sin sin sin) +RKTU ] = QY

the equation of climbing motion

d

dt

(T W

)

T

x1(cos cos sin + sin sin)

T

y1(cos sin sin cos sin)

T

z1cos cos +

T

VP

T

UQ+

T

LPKLV

T

H[(cos cos sin + sin sin)KHx1

+ (cos cos )KHz1 TH2TH1

KHW +QKHU ]

+T

V

[PKVW +

T V2T V1

KVV (cos sin sin cos sin)KVy1

]

+(T V2T V1

KTW +QKTU

)T T

= QZ

(5.3)

the equation of roll motion

d

dt

(T P

)

T

+T

U(RKHQ + V K

HW ) +

T

WV

T

VW

+T

RQ

T

QR+

T

V(V KVW WK

VV +QK

VR ) (5.4)

+T

L

(QKLR K

L WK

LV +

TL2TL1

KLP

)+T

T(V KTW RK

TQ) = QL


the equation of pitch motion

d

dt

(T Q

)

T

sin tan

T

cos

T

sincos

T

WU +

T

UW

T

RP +

T

PR+

(WKHU + UK

HW +K

H cos+

TH2TH1

KHQ

)T H (5.5)

+(UKVW PK

VR +K

V

sincos

) V

+ (RKLP +KL sin tan )

T

L

+(UKTW +WK

TU +K

T cosK

T

sincos

+T T2T T1

KTQ

)T T

= QM

the equation of yaw motion

d

dt

(T R

)

T

cos tan

T

sin+

T

sincos

+T

VU

T

UV

T

PQ+

T

QP +

T

H(V KHU +K

H sin)

+T

V

(UKVV +

T V2T V1

KVR KV

sincos

)

+T

L

(QKLP +

T V2T V1

KLR KL cos tan K

L

sincos

)

+T

T

(PKTQ +K

T sinK

TR cos tan K

T

sincos

)= QN

(5.6)

the equation of elevator deections

d

dt

(T H

)

T

H= QH (5.7)

the equation of direction rudder

d

dt

(T V

)

T

V= QV (5.8)

the equation of aileron motion

d

dt

(T L

)

T

L= QL (5.9)

the equation of drive unit

d

dt

(T T

)

T

T= QT (5.10)


This set of equations constitutes, after the determination of kinetic ener-gy and components of forces and moments of generalized forces, the generalmathematical model of motion of the unmanned aircraft vehicle.

Kinetic energy of the ying object moving in any spatial motion, in thereference system Oxyz associated with it, has been expressed in linear andangular quasi-velocities. Since the UAV is treated as a non-deformable objectwith movable control systems, its total kinetic energy is the sum of the ki-netic energy of the object without surfaces of rudders and kinetic energy ofparticular control units.

Fig. 3. Vectors of force and the moment of the external force as well as theircomponents in the reference system Oxyz

Right sides of the equations create forces F and moments of forces Mgenerated by external loads acting on the object in the smooth congu-ration (Fig. 3), composed of forces and gravitational moments Qg, genera-ted by the drive QT , control Q and aerodynamic Qa forces (Koruba andadyyska-Kozdra, 2010; adyyska-Kozdra, 2011; Nejmark and Fufajew,1971). When determining the moments of forces acting on movable surfaces ofrudders, the occurrence of moments from active forces generated by the driveof control elements (electric motor) (MHN , MV N , MLN ), hinge moments(MHZ , MV Z , MLZ) and moments of forces generated by the inhibitory re-sponse resulting from the inertia of the system (MHR,MV R,MLR) are takeninto account (Grastein et al., 1997; adyyska-Kozdra, 2011).

The equations of motion of the unmanned aircraft, derived with the useof Boltzmann-Hamel equations (5.1)-(5.10) together with equations of non-holonomic constraints (4.1)-(4.4) and equations of kinematic correlations andguidance correlations (3.1)-(3.11), constitute the set of ordinary dierentialequations, with the use of which one may, at given initial conditions, determine


16 unknowns being a function of time determining the components of linearand angular velocity of the object U , V , W , P , Q, R, its temporary locationon the route during the guidance x1, y1, z1, , , and angles of deectionsof aerodynamic rudders H , V , L, T .

The preset ight parameters of UAV resulting from the adopted guidan-ce method, (3.7)-(3.11), were included in equations of dynamics (5.1)-(5.10)through suitable amplication coecients Kji in particular control channels.This way, they were included in the mathematical model of the UAV for con-jugating parameters preset with the realized state of the ight. This revealed aclose relationship of dynamic equations of motion of a controlled ying objectwith the laws of control and kinematic relations, which as a whole, form asystem of nonholonomic constraints.

The equations derived in such a way make it possible to conduct mathema-tical calculations for controlled aircrafts: airplanes, unmanned aircrafts, roc-kets and aerial bombs (Koruba and adyyska-Kozdra, 2010; adyyska-Kozdra, 2008, 2011).

A separate issue is the proper selection of amplication coecients in thecontrol laws, which problem, still remaining in research (Blajer et al., 2001;Grastein, 2009), has been examined by the author, among others, in ady-yska (2009). When selecting the reinforcement coecients of the autopilot,an integral, quadratic criterion of quality control was used in the study, whichcomplemented the assessment of transitional processes in all control channels

J =4i=1

tk

0

[yi(t) yzi(t)yimax

]2dt (5.11)

where: yi(t) stands for the actual course of the variable, yzi(t) denotes thepredetermined course of the variable, yimax is the maximum preset range ofthe i-th state variable or the preset value yzi of the i-th state variable, if ittakes a non-zero value.

6. Numerical simulation of guided UAV

Test numerical simulation of a guided UAV for newly designed Wale unman-ned aircraft, which is to be used as an airborne target, was carried out in theTechnical Institute of Air Force (Fig. 4).

According to the diagram shown in Fig. 1, it has been assumed that a UAVmoving along the preset trajectory in the rst phase of simulation performs a


Fig. 4. Reference drawing of Wale unmanned aircraft

steady ight rectilinear (at the speed V0 = 50m/s at an altitude of 500m),then bypasses the obstacle by climbing U-turn (no slip) with a change ofaltitude about 300m, after which it returns to steady ight on the new altitude.

The coecients of amplication appearing in control laws (4.1)-(4.4), tookthe following values:

KHx1 = 0.0029 KHz1= 0.008 KHU = 0.0201

KHW = 1.036 KHQ = 2.3 K

H = 0.3

KVy1 = 0.0007 KVV = 0.00054 K

VW = 0.0231

KVR = 1.1 KV = 0.074

KL = 0.0009 KLP = 0.0007 K

LV = 0.0011

KLR = 1.5 KL = 3.3

KTx1 = 1.06 KTU = 3.004 K

TW = 0.22

KT = 4.1 KTQ = 0.06 K

TR = 0.07

KT = 0.003

The results of simulation have been presented in a graphical way in Figs. 5--Figs. 8.

The correctness of performing the manoeuvre has been ensured throughthe coordinated use of deections and by making a slip angle impossibleto occur. The analysis of graphs shows that the automatic control systemensures the maintenance of the desired ight trajectory, bringing the UAV onthe right course . For this purpose, it has proved necessary to increase thespeed of ight depending on the angle of roll and to increase the angle ofattack . After performing the U-turn, the UAV maintains the values of ightparameters by returning to steady ight on the new course with the set newheight.


Fig. 5. Diagrams of the real and preset UAV ight altitudes (a) and lateralprojections (b)

Fig. 6. History of the angles of roll, pitch, yaw (a) and the angle of attack (b)

Fig. 7. History of the elevator (a) and rudder (b) deection


Fig. 8. History of the aileron deection (a) and the changing engine thrust (b)

7. Conclusions

The use of the Boltzmann-Hamel equations for mechanical systems with non-holonomic constraints made it possible to develop the model of dynamics of theight of unmanned aircraft. Using the control laws as kinematic correlationsof deviations from the preset ideal guidance parameters, the control laws havebeen linked with dynamic equations of motion of unmanned aircraft vehicle.

The obtained simulation results show the correctness of the developed ma-thematical model. The ight of the unmanned aircraft takes place in a propermanner. It maintains the preset parameters resulting from the adopted gu-idance method throughout the entire ight.

The developed mathematical model and simulation program are universaland can be easily adapted to simulation, guidance and calculations of anyunmanned aircraft vehicle (after proper parametric identication of the testedobject).

A reliable UAV simulation process which can be adapted for dierent air-crafts would provide a platform for developing autopilot systems with reduceddependence on expensive eld trials. In many cases, the testing of newly deve-loped autopilot systems in a virtual environment is the only way to guaranteeabsolute safety. Additionally, the model would allow better repeatability intesting, with controlled ying environments.

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Modelowanie i symulacja numeryczna automatycznie sterowanego

bezzaogowego statku powietrznego skrpowanego wizami

nieholonomicznymi

Streszczenie

W pracy zaprezentowano modelowanie dynamiki lotu automatycznie sterowane-go bezzaogowego statku powietrznego z zastosowaniem rwna Boltzmanna-Hameladla ukadw mechanicznych o wizach nieholonomicznych. Prawa sterowania potrak-towano jako wizy nieholonomiczne naoone na dynamiczne rwnania ruchu BSP.Uzyskano model matematyczny zawierajcy sprzenie dynamiki statku powietrzne-go z naoonym sterowaniem, wprowadzajc zwizki kinematyczne jako parametryzadane ruchu wynikajce z procesu naprowadzania. Poprawno opracowanego mode-lu matematycznego potwierdzia przeprowadzona symulacja numeryczna.

Manuscript received December 28, 2010; accepted for print June 14, 2011

modeling and numerical simulation of unmanned aircraft vehicle restricted by non-holonomic...

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