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2014 11th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)

Mini Airplane: Design, Aerodynamic Modeling and Stability

F. Guerrero, V. Martinez, O. Garcia and D. L. Martinez

Abstract— This paper presents the development of a miniairplane UAV, focusing on the aerodynamic design, modelingand stability. Mathematical model of the vehicle is obtained us-ing the Euler-Lagrange formulation and includes aerodynamicparameters of the design. For the stability, a classic control lawis proposed taking in account the aerodynamic parameters andthe sensors bandwidth implemented in the aircraft. Simulationresults are shown for the closed-loop system of the vehicle.Finally, aerodynamic design and avionics are based on flyingand handling qualities for having a mini airplane that performsa reliable and stable flight.

I. INTRODUCTION

Nowadays, the development of UAVs (Unmanned Aerial

Vehicles) is growing due the applications and scientific-

technological challenges. For this reason, researchers and

engineers, from university and industry, are working on the

UAVs and their systems such as embedded systems, avionics,

aerospace materials and structure, guidance and navigation.

Rotary-wing UAVs, such as classic helicopters and

Quadrotors, have a number of advantages compared to other

configurations. It means, they do not require a runway for

launch and recovery because they possess much greater

operational flexibility and can operate from any small clear

space. However, rotary-wing UAVs suffer from well-known

deficiencies in terms of range, endurance and forward speed

limitations due to the propulsion systems where the thrustforce is directed opposite to the weight. On the other hand,

Fixed-wing UAVs have essential capabilities such as range

and endurance for performing and completing missions in

hostile places at a long distance from the take off site. This

paper proposes a Mini airplane UAV which is designed and

developed considering the flying and handling qualities for

having a reliable flight and range.

Control and modeling of fixed-wing UAVs have been

presented in the literature; however, most of them do not

present a deep aerodynamic analysis. In [7], authors present a

nonlinear model predictive control (NMPC) to design a high-

level controller for a fixed wing UAV. [5] describes a design

of four controllers, based on backstepping and sliding modesapplied to a fixed-wing UAV. An adaptive Backstepping

approach to obtain directional control of a commercial fixed-

wing UAV in presence of unknown crosswind is developed in

[1]. [13] presents a method for modeling the flight dynamics

This work was supported by CONACYT with the project number 204363.F. Guerrero, V. Martinez, O. Garcia and D. L. Mar-

t ine z are with Aerospace Engineering Research and In-novation Center, CIIIA-FIME-UANL, Monterrey NuevoLeon, Mexico. [email protected],[email protected] , [email protected],[email protected]

of a fixed-wing UAV. [8] presents an autonomous vision-

based net-recovery system for a small fixed wing UAV, and

avionics were developed using several sensors, integrated

with a flight control system and vision system.

The main contribution of this paper is to present a mini

airplane whose design is based on aerodynamics properties

in order to perform a reliable flight during cruise. In addition,

the aerodynamic modeling, stability analysis and the avionics

are described for this mini aerial vehicle in order to reach a

good performance in cruise flight. This paper is organized as

follows: Section II discusses the Mini airplane. The dynamic

model, based on the Euler-Lagrange equations, is presented

in Section III. In this section, the aerodynamic effects aredescribed in forward flight for the air vehicle. The controller

and the stability analysis in closed-loop are presented in

Section IV whereas Section IV-D describes simulation results

for forward flight. Section V discusses the aerodynamic

platform, avionics used and integrated on the vehicle. Finally,

conclusions are given in Section VI.

I I . MINI AIRPLANE

The mini aerial vehicle was developed as a multipurpose

experimental platform of high stability and easy handle. To

do this, a conceptual design of high aerodynamic finesse,

easy manufacturing and inexpensive materials was estab-

lished.

When an aircraft has flying and handling qualities deficien-

cies, it becomes necessary to correct them in order to improve

the stability of the vehicle. That is why we focus on the aero-

dynamic properties which give rise to those deficiencies. This

could be achieved by modification of the aerodynamic design

of the aircraft. However, there exist some external factors

which require a flight control system, even with optimized

aerodynamics of the mini aircraft. Therefore, it is essential

to understand the relationship between the aerodynamics of

the vehicle and its stability.

For this purpose, we have proposed a high wing design

with dihedral and no swept, with conventional empennageand primary control surfaces, built in balsa wood and low-

cost electronic devices. The estimated maximum weight of

the structure was fixed to ensure STOL (Short Take Off and

Landing) characteristics and high payload.

III. MINI AIRPLANE EQUATIONS

The dynamic model for forward flight of this mini air-

craft, considering the aerodynamic effects, is obtained by

employing the Euler-Lagrange formulation. This formulation

is introduced as follows

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Preliminaries

Consider an inertial fixed frame and a body frame fixed

attached to the center of gravity of the aircraft denoted by

I ={xI , yI , zI } and B ={xB, yB, zB}, respectively. The windframe W ={xW , yW , zW } is considered during the cruise of the airplane, [12].

Assume the generalized coordinates of the mini UAV asq = (x,y,z,ψ,θ,φ)T ∈ R6, where ξ = (x,y,z)T ∈ R3

represents the translation coordinates relative to the inertial

frame, and η = (ψ,θ,φ)T ∈ R3 describes the angularposition and it contains the Euler angles. These angles ψ,θ, and φ are called yaw, pitch and roll, respectively. Assumethe translational velocity and the angular velocity in the body

frame as ν = (u,v,w)T ∈ R3 and Ω = ( p, q, r)T ∈ R3,respectively. Thus, R ∈ SO(3) represents an orthogonalrotation matrix parameterized by the Euler angles from the

body frame to the inertial frame R : B → I

R =

cθcψ sφsθcψ − cφsψ cφsθcψ + sφsψcθsψ sφsθsψ + cφcψ cφsθsψ − sφcψ−sθ sφcθ cφcθ

where the shorthand notation of sa = sin(a) and ca =cos(a) is used. For this matrix, the order of the rotations isconsidered as yaw, pitch and roll (ψ,θ,φ) [11]. The attitudekinematics is described as

Ṙ = R Ω̂

where Ω̂ is a skew-symmetric matrix such that Ω̂a = Ω ×a.For the aerodynamic analysis, a matrix B : B → W

describes the transformation of a vector from the body frame

to the wind frame. This wind frame is represented by as

B = cαcβ cαsβ sα

−sβ cβ 0−sαcβ −sαsβ cα

where α is the angle of attack and β are the sideslip angle

[3], [4], [11], [12].Therefore, the equations of motion of the aircraft are

obtained using the Euler-Lagrange formulation

d

dt

∂ L(q, q̇)

∂ q̇

−

∂ L(q, q̇)

∂ q

= τ (1)

where τ = (F ,Γ )T ∈ R6 denotes the forces and momentsacting on the body frame, L(q, q̇) = K(q, q̇) − U (q)describes the Lagrangian equation which consists of the total

kinetic energy K(q, q̇) and the potential energy U (q) of thesystem. Moreover, K and U are defined as

K = Kt + Kr

U = mgz

Kt = 1

2ξ̇T m ξ̇

Kr = 1

2 η̇T M (η) η̇

where m ∈ R is the mass of the vehicle, Kt is thetranslational kinetic energy and Kr is the rotational kineticenergy with M (η) = X (η)T I X (η). I ∈ R3×3 denotes themoments of inertia of the mini aircraft.

Forces

The forces acting on the aircraft include those of the

propulsion system F p and aerodynamic effects F a. These

forces are described as follows

F = F p + F a

with

F p =

T c0

0

, F a = BT

−DY

−L

where T c is the thrust force of the rotor. The lift force L,sideforce Y and drag force D are defined as aerodynamicforces [9].

Moments

The moments generated on the mini aircraft are due to

actuators (actuator moment Γ act, reaction moment Γ rot and

gyroscopic moment Γ gyro), and the aerodynamic effects Γ a.

These moments are defined as follows

Γ =

Γ LΓ M

Γ N

= Γ act + Γ rot + Γ gyro + Γ a

with

Γ act =

τ φτ θ

τ ψ

, Γ rot =

I rot ω̇r0

0

,

Γ gyro =

0rI rωr

−qI rωr

, Γ a =

L̄M̄

N̄

where τ φ

= a

(f a

1

− f a2),

τ θ =

ef e and

τ ψ =

ef r are the

control inputs with a and e that represent the distance fromthe center of mass to the forces f e1 and f e2. ωri denotes theangular velocity of the rotor, I ri is the inertia moment of the propeller and I roti is the moment of inertia of the rotoraround its axis for i= 1, 2. L̄, M̄ and N̄ are aerodynamicrolling, pitching and yawing moments, respectively [9].

Translational and rotational dynamics

Since the Lagrangian equation contains no cross-terms in

the kinetic energy combining ξ̇ with η̇, the Euler-Lagrangeequation (1) can be partitioned into dynamics for ξ coordi-

nates and η coordinates [2].The translational motion can be obtained using the follow-

ing expression

d

dt

∂ Lt

∂ ξ̇

−

∂ Lt∂ ξ

= F (2)

with

Lt = 1

2ξ̇T m ξ̇ − mgz (3)

Thus, after some computations, the translation motion of this

vehicle is described as mẍmÿ

mz̈

= R exT c + RBT

−DY

−L

+ mgez (4)

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where ex = (1, 0, 0)T and ez = (0, 0, 1)

T are the unit

vectors.

Similarly, the rotational motion is given as

d

dt

∂ Lr∂ η̇

−

∂ Lr∂ η

= Γ (5)

where

Lr = 12

η̇T M (η) η̇ (6)

From (5) and (6)

M (η)η̈ + Ṁ (η) η̇ − 1

2

∂

∂ η

η̇T M (η) η̇

= Γ (7)

From equation (7), the Coriolis and Centrifugal vector is

defined as

C (η, η̇) η̇ =

Ṁ (η) −

1

2

∂

∂ η

η̇T M (η)

η̇ (8)

Thus, the dynamic model for the rotational motion is rewrit-

ten as

M (η)η̈ + C (η, η̇) η̇ = Γ (9)

IV. STABILITY OF THE MINI AIRCRAFT

In order to design a controller that stabilizes the attitude

dynamics of the mini UAV, we simplify the equation (9) as

η̈ = M (η)−1 [Γ − C (η, η̇) η̇] (10)

then, we take a control input as Γ̄ =M (η)−1 [Γ − C (η, η̇) η̇], it yields

η̈ = Γ̄ (11)

In order to stabilize the attitude dynamics (11), we propose

a the following Lyapunov function

V = 1

2 η̇2 + ks1 ln(cosh(η)) (12)

where V > 0 and ks1 > 0. The corresponding time derivativeis given as

V̇ = η̇η̈ + ks1 tanh(η) η̇ (13)

the previous equation can be rewritten as

V̇ = η̇ (η̈ + ks1 tanh(η)) (14)

now, in order to render V̇ negative definite, we propose thefollowing control input

Γ̄ = −ks1 tanh(η) − ks2 tanh( η̇) (15)where ks2 > 0, then substituting the control input in (14)leads to the following expression

V̇ = −η̇ks2 tanh( η̇) (16)

finally, it follows that V̇ < 0, it proves that the proposedcontrol input (15) asymptotically stabilizes the attitude of

the vehicle.

On the other hand, the range dynamics from translation

motion (4) is described as

mẍ = r11T c + f 1(t) (17)

where r11 is term of the first column and first row from R ,and f 1(t) is the term of the vector F (t) = RB

T (D , Y , L)T .

Proposing a thrust control in the Laplace domain as

T c(s) = C (s)(X d(s) − X (s))

A. System

Now we are concern about the control and stability of or

system in x, in the Fig. 1 we shown a block diagram of

the control loop considering the simplifications mentioned

before. Transforming the equation (17) in to Laplace domain

we got.

ms2X (s) = r11T c(s) + F 1(s)

X (s) = r11T c(s) + F 1(s)

ms2

X (s) = G(s)(T c(s) + δu)

X (s) = G(s)(C (s)X d(s) + δu)

1 + G(s)C (s) (18)

Where X d(s) is the desired trajectory, C (s) is the transferfunction of the control, δu = F 1(s) is the perturbation atthe input, G(s) = 1

ms2 is the transfer function of the plant,

X (s) is the output trajectory and δs is the sensor noise.

Fig. 1: Block diagram.

B. Control Requirements

Based on prior analysis, we have the following control

requirements:

• The system must meet E ss = 0 in stationary estateunder a step input.

• The system must reject turbulence perturbation frequen-

cies up to 1 Hz.

• The system must reject the sensor noise frequencies

greater than 10 Hz.

• The response time will be established at convenience.

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C. Control Design

Our control design methodology is based on the frequency

response. In this methodology, it is important to remark that

there exist two terms such as Phase Margin (PM) and Gain

Margin (GM).

First, in order to find the E ss = 0 requirement, we pro-

pose, as an approximation of our control, a simple integratorand obtain the bode response of the system in open-loop to

analyze the system stability.

−60

−40

−20

0

20

40

60

M a g n i t u d e ( d B )

10−1

100

101

−271

−270.5

−270

−269.5

−269

P h a s e ( d e g )

G(s) System no compensated

Frequency (rad/s)

Fig. 2: Bode G(s) not compensated.

As we can see in the Fig. 2, the system is unstable in

closed-loop due to the MP = −90. In order to obtain thesystem stable, we must add the phase to our system at least

until M P = 55 which can be achieved by using a lead

compensator.To design the compensator, it is necessary to consider

the system bandwidth, which is bounded by the Pitot Tube

bandwidth (10 Hz) and the turbulence spectrum (0.1 Hz).This means that the control must have hight gain in lower

frequencies and low gain at frequencies higher than 10 Hz.According to the data, the suitable system bandwidth must

be around 1 Hz. The compensator was proposed as follows:zeros (s + 0.75) and (s + 0.945), poles (s + 48), (s + 52)and (s + 20), therefore the transfer function is

C (s) = s2 + 1.695s + 0.7087

s4 + 120s3 + 4496s2 + 49920s

As we can see in the control bode response, in the Fig. 3,we have obtained a hight gain at low frequencies to find the

turbulence rejection; however, we could not achieve the low

gain at 10 Hz, therefore we must filter the signal from the

sensor in order to reject the noisy signals from the sensor.

In the bode response, we can remark that our system, in

order to have a good perturbs rejection, is setup width K =103510, BW = 4.21 rad/s, with a GM= 16.8 dB and PM=46. (See Fig. 4).

Analyzing the step response of the system, we found a

time response around 2.5 s, the response is suitable for thedynamics system.

10−2

10−1

100

101

102

103

−180

−135

−90

−45

0

45

90

P h a s e ( d e g )

C(s) Control

Frequency (rad/s)

−20

0

20

40

60

System: CFrequency (rad/s): 451Magnitude (dB): 0.0252

System: CFrequency (rad/s): 0.842Magnitude (dB): 17


Fig. 3: Bode diagram of C(s).

C(s)G(s) System Compensated

Frequency (rad/s)

−150

−100

−50

0

50

100

150

System: untitled1Frequency (rad/s): 0.913Magnitude (dB): 18.5 System: untitled1

Frequency (rad/s): 18.9Magnitude (dB): −17.2

System: untitled1Frequency (rad/s): 4.21Magnitude (dB): −0.0432


10−2

10−1

100

101

102

103

−360

−270

−180

−90

System: untitled1Frequency (rad/s): 18.7Phase (deg): −180



P h a s e ( d e g )

Fig. 4: Bode diagram of G(s)C(s) compensated.

D. Simulation results

The simulation was implemented in Matlab/Simulink by

considering the perturbations in the process design in order

to test the control performance. The wind turbulence was

simulated as a wave with frequency of 0.1 Hz and amagnitude of 7 dB.

The sensor noise was simulated as a wave with frequencyof 500 Hz and a magnitude of 2 dB. The output in thesimulation was established to follow a step of magnitude

5 dB, and the system response is shown in the Fig. 5. Aswe can see the control has a good performance despite the

wind turbulence.

V. EXPERIMENTAL PLATFORM

A. Aerodynamic platform

The first objective in the preliminary design was to de-

termine the wing airfoil for our airplane. Several types of

airfoils were analyzed using the DesignFoil software in order

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0 5 10 15 20 25 30−5

0

5Error

0 5 10 15 20 25 30

−4

−2

0Perturbation

0 5 10 15 20 25 300

5

10Output

Fig. 5: Performance.

to obtain the lift (C L) and drag coefficients (C D), consideringa low Reynolds number (Re = 97, 934). From the analyzedairfoils, Goettingen 256 (GOE256) airfoil was proposed due

to its high stall angle (15◦), high lift coefficient (C L = 0.83obtained through DesignFoil) corresponding to the angle of

attack at steady flight condition, and its aerodynamic finesse

(L/D) greater than the rest of the airfoils.

Due to the design features required for the aircraft, lift

equation was used to calculate the required wing area (S ) togenerate enough lift for a takeoff velocity 7 m/s with an airdensity of ρ = 1.1549 kg/m3.

With wing area S , and establishing an Aspect Ratio AR =6, other aerodynamic parameters, such as wingspan b, wingchord C w were easily calculated. For dihedral and incidenceangle, we consider the design features and reference tables

for our type of airplane as follows [10]:

TABLE I: Aerodynamic parameters.

Parameters ValuesMin. Theoretical Lift (L) 6.867N

Wing Area (S ) 0.29m2

Wingspan (b) 1.32mWing Chord (C W ) 0.22mAspect Ratio (AR) 6

Incidence Angle (αi) 1.63◦

Dihedral Angle (αd) 3.66◦

A stress simulation was made using SolidWorks consid-

ering a static charge in the edge of the wing to analyze

the effect of Lift force for two dihedral configurations on

the wing based on the attachment points. We have chosen

dihedral at the tips due to their lower stress concentration

showing values under the stress limits for the balsa wood

(see Fig. 6).

Also, we conducted a numerical simulation of the whole

wing, using the 3D analysis tool of DesignFoil for the

proposed Reynolds Number, Wing Chord and Wingspan , as

well as the theoretical take-off velocity. The lift that supports

Fig. 6: Low stress concentration in diedral configuration.

the weight of the airplane is obtained at α =5◦, which is farfrom stall angle (αstall =15

◦).

For designing the Horizontal stabilizer (HS) and Vertical

stabilizer (VS), we use the NACA 0005 − 93 airfoil due

to its symmetrical and low thickness. We follow the sameprocedure as in the wing in order to obtain the lift generated

by the HS and VS. After calculate the HS area and ensure

that HS Lift force counteract the wing pitching moment

(moment between wing aerodynamic center that must be

counteract by HS-VS aerodynamic center), we obtain the

following parameters showed in Table II and III:

TABLE II: Horizontal stabilizer parameters.

Parameters ValuesHS Chord (C HS ) 0.1099mHS Span (BHS ) 0.4397m

HS Arm Distance (X HS ) 0.66mAspect Ratio (ARHS ) 4Elevator Chord (C EL) 0.022mElevator Span (BEL) 0.44m

TABLE III: Vertical stabilizer parameters.

Parameters ValuesVS Chord (C V S) 0.119mVS Span (BV S) 0.1976m

VS Arm Distance (X V S) 0.66mAspect Ratio (ARV S) 1.66Elevator Chord (C RD) 0.029mElevator Span (BRD) 0.197m

The central body of the aircraft was chosen to be a

structured fuselage due to its inner space, light weight and

high structural resistance. It was designed in SolidWorks (see

Fig. 7) to obtain the inertia moments used in Section III.

B. Avionics

The mini aerial vehicle Braver consists of the low cost

materials, sensors and electronics such as a GPS module

which is based on an uBlox chip and provides an estimated

of the latitude, the longitude and altitude coordinates. This

module also delivers an estimated of the inertial velocity

vector. The IMU module is a CHR-6dm which is an Attitude

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Fig. 7: Final Design of Braver Mini Airplane.

and Heading Reference System (AHRS) that uses a 32-bit

STM32F103T8 ARM Cortex M3 to process data from an

three-axis accelerometer, a pitch/roll rate gyro, a yaw rate

gyro, and a three-axis digital magnetic compass. An on-

board Extended Kalman Filter produces estimates for yaw,

pitch, and roll angles and angle rates, which are sent to the

microcontroller through a serial interface, (see Fig. 8), [6].

Fig. 8: Avionics of the Mini Aerial Vehicle.

The use of the airspeed Micro Sensor with a Pitot tube

could enable measurements of the incoming flow on an

airframe with a temporal resolution of up to WB=10 Hz,

according to the test developed in a Wind Tunnel of the

Aerodynamics Lab (CIIIA-FIME), we have characterized the

sensor performance as shown in Fig, 9.

0 10 20 30 40 505

10

15

20

25

30

35

Time (s)

W i n d S p e e d ( m

/ s )

Pitot Tube (Red)

Fig. 9: Pitot Tube characteristic.

The turbulence spectrum describes all the frequency vari-

ations in the wind speed. There exist models to describe

the turbulence behavior such as the Von Karman Spectrum,

which gives a good description of turbulence in wind tunnel;

however, the Karman Spectrum gives a good approximation

to the atmospheric turbulence. In the frequency spectrum of

the wind speed, the most common frequencies are in the

range between 0.1 Hz and 0.01 Hz.

VI . CONCLUSIONS

We have presented the aerodynamic design, modeling and

stability of a low-cost mini airplane UAV. Euler-Lagrange

formulation was used to obtain the equations of motion

considering the aerodynamic parameters in order to ensure

good handling qualities of the mini airplane. For the flight

control system, a classic control law based on frequency

was proposed, considering the bandwidth sensors and per-

turbations in order to have a high performance during the

cruise flight. As it was shown in the simulation results,

the proposed control fulfills the main requirements of the

airplane system. For a better performance of vehicle, a pre-

filter noise treatment is required due to the limited bandwidth

of the sensors. Finally, a low-cost avionics is presented and

integrated for mini airplane UAV.

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