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CHAPTER 3
UAV Modeling
This chapter presents the mathematical modeling of an aircrafts. This section is not meant
to give a complete explanation of aircraft dynamics, but rather is a primer intended to enable the
reader to be able to understand the nomenclature and general dynamics of an aircraft. Most of
the information in this chapter are available in the literature [9], [10], [11], [12], [13], [14], [15], [11]
and [16]. Aircraft model trimming and liberalization procedures are also discussed in this
chapter.
3.1
Coordinate Systems (Axis System)
In aerospace modeling, three co-ordinate frames are generally used as follows
{E}
Earth fixed reference frame
{A}
Airplane fixed reference frame
{B}
Body fixed reference frame
3.1.1
Earth Fixed Coordinate Frame {E}
This coordinate system has its < x, y > plane on the surface of the earth. The z−axis points
towards the center of earth. This is illustrated in Figure 3.1, where Ex , Ey and Ez represent
the axes
• The Earth is assumed to be flat.
• An arbitrary point on the Earth’s surface is defined as the origin (Oe ).
• A right handed orthogonal system of axes ( Eo , Ex , Ey , Ez ) is defined at the origin. Where
Ex pointed towards North, Ey forward East and Ez vertically down.
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Figure 3.1: Co-ordinate systems
3.1.2
Airplane Reference Coordinate Frame {A}
The airplane reference coordinate system has axes in the same direction as of the fixed
coordinate system. However the difference is that the airplane reference coordinate system has
its origin located at the center of gravity (COG) of the airplane. See Figure 3.1, where Ax , Ay
and Az represent the axes.
3.1.3
Body Fixed Coordinate System {B}
This coordinate system has its origin in COG of the plane. The x−axis is in the center
line of the airplane (pointing out through the aircraft’s nose). The y axis is orthogonal to the
x−axis and in the direction of the wind tip. The z−axis is orthogonal to the < x, y > plane. See
Figure 3.1 where Bx , By and Bz represent the axes. The difference between the two coordinate
systems A and B can be described as a rotation around the common origin COG. The axis
system was constrained to move with the aircraft as it rolled, pitched and yawed.
3.2
Control Surfaces
The control surfaces used for manoeuvering the airplane are illustrated in Figure 3.2, and
consists of: ailerons, elevator, throttle and rudder.
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Figure 3.2: Conventional control surfaces
• Ailerons are located on both wings of the airplane. The ailerons are used to roll the
airplane. Two ailerons always deflect in opposite direction of each other. When the
right aileron is seen from behind and is deflected upwards, the resulting roll direction is
considered positive.
• Elevator is located on the tail of the airplane and is used to change the airplane pitch.
When the elevator is seen from behind deflects downwards, the resulting pitch direction
is considered negative.
• Rudder is located on the vertical stabilizer on the tail of the airplane. The rudder is used
to change the yaw of the aircraft. When the rudder is seen from behind, deflects to the
right the resulting yaw direction is considered positive.
• Throttle is the engines thrust power controlling mechanism. General thrust line coincide
with x axis of the aircraft.
3.3
3.3.1
The Nonlinear Model of the UAV
6-DOF Aircraft Equations of Motion
The UAV dynamics can be explained with twelve ordinary equations
The state vector of the UAV is as follows,
x = [u v w p q r xe ye ze f uel mass ω h]
(3.1)
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where,
[ u, v, w] =aircraft linear velocity w.r.t body frame (xb ,yb ,zb )(See Fig)
[ p, q, r] =aircraft angular velocity w.r.t body frame (xb ,yb ,zb )
[ xe , ye , ze ] =aircraft position in Earth frame (xe ,ye ,ze )
Figure 3.3: Non-Linear state space aircraft model
The output vector is chosen as,
y = [ xe ye ze φ θ ψ V β α]
(3.2)
where,
[φ θ ψ] =Euler angles which define the attitude of aircraft with relative to the Earth
The aircraft equations of motion are derived from basic Newtonian mechanics [15]. The
general force and moment equations for a rigid body are,
F =m
∂V
+Ω×V
∂t
(3.3)
These equations express the motions of a rigid body relatively to an inertial reference
frame.V = [u v w]T is the velocity vector at the center of gravity, Ω = [p q r]T is the angular velocity vector about the COG. F = [Fx Fy Fz ]T is the total external force vector, and
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M = [L M
N ]T
is the total external moment vector. I is the inertia tensor of the rigid body,
which is defined as:
M=
∂(I.Ω)
+ Ω × (I.Ω)
∂t

Ixx
−Jxy


I = −Jyx Iyy

−Jzx −Jzy
Jxz
(3.4)



−Jyz 

Izz
(3.5)
UAV model can be broken down into a number of different sub modules [15], [17] most of
which are commonly applicable to all the airframes. These sub systems in the model as follows,
• Aerodynamics forces and moments
• Gravity force and momentum (F,M)
• Thrust forces and engine model
• Atmosphere model
• Equations of motions
Figure 3.4: Nonlinear state space aircraft model
These sub models are illustrated in Figure 3.4. The axis forces equations are,
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m.(u˙ + q.w − r.v) = X − m.g.sθ + T.cε
m.(v˙ + r.u − p.w) = Y − m.g.cθ..sφ
(3.6)
m.(w˙ + p.v − q.u) = Z − m.g.cθ.cφ + T.sε
And moment equations are
Ix .p˙ + Ixx .r˙ + (Iz − Iy).q.r + Ixz .p.q = L
Iy .q˙ + (Ix − Iz ).p.r + Ixz .(r2 − p2 ) = M
(3.7)
Iz .r˙ + Ixz .p˙ + (Iy − Ix ).p.q − Ixz .q.r = N
Rotational rates of the aircraft is given by
φ˙ = p + sφ.tθ.q + cφ.tθ.r
θ˙ = cφ.q − sφ.r
(3.8)
sφ
cφ
ψ˙ =
.q + .r
cθ
cθ
It is possible to combine these equations to form a nonlinear state space system as follows
x˙ = f (x, Ftot (t), Mtot (t))
(3.9)
The aerodynamic moments and forces are functions of all the status and all of the control
inputs, for instance x = (u, v, w, p, q, r, δa , δe , δr , δt ), where δa = aileron deflection, δe = elevator
deflection, δr = rudder deflection, δt = throttle deflection. The aerodynamics forces and moments are obtained from the dimensional aerodynamics coefficient at a given flight condition as
follows.
XA = q¯SCX , YA = q¯SCY , YA = q¯SCY ,
(3.10)
¯ = q¯SbCl , M
¯ = q¯SbCm , N
¯ = q¯SbCn
L
The analysis of the aerodynamics behavior of the aircraft is better understood along the
stability axes, or wind axes, system. In particular, the aerodynamics forces are easily handled
in the wind axes, which is given in terms of the angle of attack α and the sideslip angle β.
Hence, the force equations in wind-axes are introduced as follows,
p
u2 + v 2 + w 2
p
α = tan−1 (w/ u2 + v 2 )
V =
(3.11)
β = tan−1 (v/u)
Aerodynamic coefficients can be described [15] as in the Figure 3.12 where a significant cross
coupling can be observed within some coefficients.
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q¯
c
+ CXδr δr + CXδf δf + CXαδf αδf
V
˙
pb
βb
rb
CYa = CY0 + CYβ β + CYp
+ CYr
+ CYδa δa + CYδr δr + CYδr α δr α + CYβ˙
2V
2V
2V
q¯
c
3
2
CZa = CZ0 + CZα α + CZα3 α + CZq + CZδe δe + CZδ β2 δe β + CZδf δf + CZαδf αδf
e
V
pb
rb
Cla = Cl0 + Clβ β + Clp
+ Clr
+ Clδa δa + Clδr δr + Clδaα δa α
2V
2V
q¯c
rb
Cma = Cm0 + Cmα α + Cmα2 α2 + Cmq + Cmδe δe + Cmβ2 β 2 + Cmr
+ Cmδf δf
V
2V
pb
q¯
c
rb
Cna = Cn0 + Cnβ β + Cnp
+ Cnr
+ Cnδa δa + Cnδr δr + Cnq + Cnβ3 β 3
2V
2V
V
CXa = CX0 + CXα α + CXα2 α2 + CXα3 α3 + CXq
(3.12)
Gravity force can be identify as in Figure 3.12, where W is the aircraft weight, θ is the pitch
angle, and ϕ is the roll angle of the vehicle, Resultant Fgrav force is
Fgrav


Xgr




=  Ygr  = W


Zgr

− sin θ





cos θ cos ϕ


cos θ cos ϕ
(3.13)
Hence, the force equipments in wind-axes are introduced as follows,
mV˙T = T cos(α + αT )cosβ − D + mg1
˙ T = −T cos(α + αT )sinβ − Cw + mg2 − mVT rs ,
mβV
(3.14)
mαV
˙ T cosβ = −T sin(α + αT ) − L + mg3 − mVT (qcosβ − ps sinβ),
where, VT is total air speed, ps and rs are the pitch and yaw rates projected on to the
stability axes, m is the mass of the vehicle, and g1 , g2 and g3 are the gravitational accelerations
on the wind axes system. The aerodynamic forces are lift (L), drag (D), and cross wind force
(Cw ) while T is the thrust force exerted on the aircraft.
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3.3.2
The non-Linear Simulink Aerosonde UAV Model
The aerosonde UAV shown in Figure 3.5 flies between 15ms−1 to 30ms−1 , and maximum
fuel capacity is 2Kg. The maximum allowed bank angle is 300 and this aircraft doesn’t have flap
actuator. Thus, there are four main trim parameters; airspeed, bank angle, fuel and altitude.
AeroSim toolbox [17] includes the simulink model and its initialization file aerosondecfg.mat
which contains all the required aerosonde data. A sample simulink file of the simulator is shown
in Figure 3.7 and Figure 3.8.
Figure 3.5: Aersonde UAV
The MATLAB AeroSim [17] simulation environment features a 6 degrees-of-freedom nonlinear nonlinear aircraft model together with earth and atmospheric models, propulsion model
propulsion and modeling of actuators and sensors. In addition the von Karman [17] wind turbulence model already built into the aircraft model in Matlab. Model coefficients in this view
of aerosonde UAV are made available. Due to the easy access to the aerosonde model, the
controllers in this thesis have been developed and tested with respect to aerosonde model. Atmosphere, earth, aerodynamic and propulsion, inertia and equation of motion major sub models
are within UAV model can be identified in Figure 3.6.
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Figure 3.6: Aircraft model with sub models
Figure 3.7: Simple open loop simulink diagram of the UAV simulation
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Figure 3.8: Simulink sub-models of the UAV
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3.4
Linear Decoupled Model
Flight dynamics is decoupled into lateral and longitudinal modes to simplify the control
analysis as in [16]. If the aircraft is assumed to be at equilibrium, or trim, condition then
the equations of motion can be linearized and the states can be resolved into their lateral and
longitudinal states. The assumption made for decoupling the linearized model is that the crosscoupling effects between two modes is negligible. This assumption is made for the aerosonde
UAV platform for the following reasons:
• It is designed with conventional aileron, rudder and elevator control surfaces that do
not give significant cross-coupling control between the lateral-directional and longitudinal
modes.
• The aircraft is symmetrical about the xz plane in which the inertia cross-coupling (in xy
and xz axes) resulting to cross-coupling between the lateral-directional and longitudinal
modes is minimum.
3.4.1
Trimming
In the trim process, all the state derivatives are fixed to zero, except for the first derivative
which corresponds to Vx . Steady-state flight can be defined in terms of the remaining nine state
variables of the flat-earth equations:
˙ p˙ = 0 u˙ = constant
p,
˙ q,
˙ r,
˙ V˙ , α,
˙ β,
(3.15)
Major steady states can be identified as,
Steady wings-level flight
˙ ψ˙ = 0
ϕ, ϕ,
˙ θ,
(i.e. p, q, r = 0)
Steady turning flight
ϕ,
˙ θ˙ = 0
ψ˙ =turn rate
Steady pull-up
ϕ, ϕ,
˙ ψ˙ = 0
θ˙ = pull-up rate
Steady roll
˙ ψ˙ = 0
θ,
ϕ˙ =roll rate
The conditions p,
˙ q,
˙ r˙ = 0 require the angular rates and the aerodynamic and thrust moments to be zero or constant. The conditions V˙ , α,
˙ β˙ = 0 require the aerodynamic forces to be
zero or constant. For this reason, the steady-state pull-up/push-over and steady roll conditions
can only exist instantaneously. Still, it can be useful to trim the aircraft dynamics in such flight
conditions (and use the resulting trim values of x and u to linearize the aircraft model for these
flight conditions) because control systems must operate at all times.
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The fuel mass essentially determines the center of gravity of the aircraft. The trim settings
required to maintain the flight condition hence consisted of the airspeed, altitude, bank angle,
Fuel mass and flap setting. Table 3.1 gives the desired flight envelop for the aersonde UAV.
Therefore given a flight condition, the trim routine needed to solve a four variable minimization
problem to find the trim settings that best maintained the aircraft in the flight condition. With
a Simulink model this could be performed using the Matlab trim.m routine.
Airspeed
Altitude
Fuel mass
Bank angle
15−28m/s
0−3000m
0−2Kg
+-30deg
Table 3.1: Aerosonde flight envelop
For some flight conditions the trim routine produced good trim settings and when fed back
into the full non-linear simulation trimmed the aircraft out well and for some that fails. Therefore, custom made simulink model and iteration trim procedure were used find good stable trim
values.
Figure 3.9: Non-Linear state space aircraft model for trimming
From a Matlab script it is possible to manually setup all the state variables within and
inputs to a model. It is then possible to ask Simulink to calculate the instantaneous rates of
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change of the states that would, these parameters are the initial guesses. This was done by
running non linear simulink model for 10s and if the trim error are within thresholds then move
on to iteration process.
The initial values of the states are taken from the Matlab file aerosondecfg.mat. A longitudinal trim of the dynamic system is expected, which means a steady level flight at some fixed
heading and bank angle. Therefore, the trim is calculated at some fixed values of outputs: The
trim algorithm will converge to the corresponding values of the incidence α, the sideslip β, and
the controls deflections. After that, proceed with iteration process till meeting optimization
parameters. The model was trimmed around straight and level flight in the 1000ft, 25ms−1 ,
2kg of fuel flight condition using as below,
Trim flight condition:
Initial guesses for trim inputs are:
Trim airspeed
:
25 m/s
Elevator
=
-0.0702
Trim altitude
:
1000m
Aileron
=
-0.0093
Trim bank angle
:
0rad
Rudder
=
0.0000
Fuel mass
:
2Kg
Throttle
=
0.7349
Flap setting
:
0
Finished trim results are:
INPUTS:
Elevator
=
-0.0759
Aileron
=
-0.0083
Rudder
=
-0.0010
Throttle
=
0.7075
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STATES:
OUTPUTS:
Airspeed
=
25.00 m/s
u
=
24.96 m/s
Sideslip
=
0.02 deg
v
=
0.01 m/s
AOA
=
3.24 deg
w
=
1.41 m/s
Bank
=
-0.02 deg
p
=
-0.00 deg/s
Pitch
=
3.24 deg
q
=
0.00 deg/s
Heading
=
359.89 deg
r
=
0.00 deg/s
Altitude
=
1000.00 m
phi(φ)
=
-0.02 deg
theta(θ)
=
3.24 deg
psi(ψ)
=
-0.11 deg
Alt
=
1000.00 m
Fuel
=
2.00 kg
Engine
=
5209 rot/min
3.4.2
Linearization of the Nonlinear Model
linearization is necessary to obtain the state space linear model [16]. As the scenario for
the airplane is stable level flight, a limited operating area does not pose a problem; hence a
linearization in an operating point combined with a taylor approximation can be used. The
operating point must be specified for all degrees of freedom. Therefore the velocity is specified
for the three translation degrees of freedom in the body coordinate system and for the three
rotation degrees of freedom in the reference coordinate system as follows;
U = U0 + ∆U
V = V0 + ∆V = ∆V
W = W0 + ∆W = ∆W
(3.16)
p = p0 + ∆p = ∆p
q = q0 + ∆q = ∆q
p = r0 + ∆r = ∆r
where,U0 , V0 and W0 are the initial velocities in Bx , By and Bz directions. The ( 3.16) is
only valid for: V0 = W0 = p0 = qo = ro = 0, dV and dW are the changes in velocities in the
directions of Bx , Bz and By accordingly. U , V and W are the resulting velocities along the
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above mentioned directions. This selection of operating points results in a steady state vector
for both the longitudinal and the lateral systems that is the null vector:
xlong
˙ = xlat
˙ =0
(3.17)
During the modeling process small angle approximations [16] has been utilized to simplify
the model. The most significant equations derived from the approximations are:
u=
∆U
U0
(3.18)
∆α =
∆V
U0
(3.19)
∆β =
∆W
U0
(3.20)
The linearization of the model about the trimmed equilibrium brings us to a state-space
representation of the form:
3.4.3
x˙t = Axt + But
(3.21)
yt = Cxt + Dut
(3.22)
Decoupling of Linearized Model
Linerised state space aircraft model can be de-coupled into longitudinal and lateral models by assuming that the coupling derivatives are negligible and this is only valid for steady
state condition. A significant cross coupling can occurred due to abnormal maneuvering that
happens seldom. De-coupled longitudinal and lateral state space models were used to analyze
dynamic stability of aircraft. The model was linearized around trimmed straight and level
flight at 1000ft, 25ms−1 , 2kg of fuel flight condition using Matlab linmod routine. The resulting longitudinal state space model has been derived as follows Equations ( 3.23), ( 3.24), ( 3.25).
The longitudinal model comprises of the body axis x and z direction velocities (u, w), pitch
angular rate q, pitch angle θ, altitude h and propeller angular rotation ω state. Control inputs
for the longitudinal model are elevator δe and throttle δT control. The measurement output
variables for the longitudinal model are airspeed Va , angle of attack α, pitch rate q, pitch angle
θ and altitude h.
x˙ long = Along Xlong + Blong Ulong
(3.23)
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ylong = Clong Xlong + Dlong Ulong
(3.24)
xlong = [ u v q ze ω ]T
ulong = [ δe δT ]T
(3.25)
ylong = [ vα α q ze ]T
The lateral model comprises of the body axis y direction velocities v, roll and yaw angular
rate (p, r), roll and yaw angle φ, ψ. Control inputs for the longitudinal model are aileron δa and
rudder δr control. The measurement output variables for the lateral model are sideslip angle β,
roll and yaw angular rate (p, r), roll and yaw angle φ, ψ.
x˙ lateral = Along Xlateral + Blateral Ulateral
(3.26)
ylateral = Clateral Xlateral + Dlateral Ulateral
(3.27)
xlateral = [ v p r φ ψ ]T
ulateral = [ δa δr ]T
ylateral = [ β p r φ ψ ]T
(3.28)

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