-
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
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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
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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
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254 E. adyyska-Kozdra
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).
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Modeling and numerical simulation of UAV... 255
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):
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256 E. adyyska-Kozdra
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.
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Modeling and numerical simulation of UAV... 257
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
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258 E. adyyska-Kozdra
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.
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Modeling and numerical simulation of UAV... 259
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)
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260 E. adyyska-Kozdra
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
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Modeling and numerical simulation of UAV... 261
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)
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262 E. adyyska-Kozdra
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
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Modeling and numerical simulation of UAV... 263
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
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264 E. adyyska-Kozdra
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.
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Modeling and numerical simulation of UAV... 265
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
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266 E. adyyska-Kozdra
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.
References
1. Bloch A.M., 2003, Nonholonomic Mechanics and Control. Systems
and Con-trol, Springer, New York
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Modeling and numerical simulation of UAV... 267
2. Blajer W., 1998, Metody dynamiki ukadw wieloczonowych,
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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