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BacksteppingControlforaQuadrotorHelicopter

CONFERENCEPAPER·NOVEMBER2006

DOI:10.1109/IROS.2006.282433·Source:DBLP

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2AUTHORS,INCLUDING:

AbdelazizBenallegue

UniversitédeVersaillesSaint-Quentin

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Backstepping Control for a Quadrotor Helicopter

Tarek Madani and Abdelaziz Benallegue

Laboratoire d’Ingenierie des Systemes de Versailles

10-12, avenue de l’Europe, 78140 Velizy - FRANCE

E-mail: {madani and benalleg}@lisv.uvsq.fr

Abstract— This paper presents a nonlinear dynamic modelfor a quadrotor helicopter in a form suited for backsteppingcontrol design. Due to the under-actuated property of quadrotorhelicopter, the controller can set the helicopter track threeCartesian positions (x, y, z) and the yaw angle to their desiredvalues and stabilize the pitch and roll angles. The system has beenpresented into three interconnected subsystems. The first onerepresenting the under-actuated subsystem, gives the dynamicrelation of the horizontal positions (x, y) with the pitch and rollangles. The second fully-actuated subsystem gives the dynamics ofthe vertical position z and the yaw angle. The last subsystem givesthe dynamics of the propeller forces. A backstepping control ispresented to stabilize the whole system. The design methodologyis based on the Lyapunov stability theory. Various simulations ofthe model show that the control law stabilizes a quadrotor withgood tracking.

I. INTRODUCTION

Unmanned Aerial Vehicles (UAV) have unmatched qualities

that make them the only effective solution in specialized

tasks where risks to pilots are high, where beyond normal

human endurance is required, or where human presence is

not necessary. UAVs are being used more and more in civil-

ian applications such as traffic monitoring, recognition and

surveillance vehicles, search and rescue operations [1]. They

are highly capable to flown without an on-board pilot. These

robotic aircrafts are often computerized and fully autonomous.

The first reported quadrotor helicopter (Gyroplane No.1)

was built in 1907 by the Breguet Brothers. However, there was

no means of control provided to the pilot other than a throttle

for the engine to change the rotor speed, and the stability of

the machine was found to be very poor. The machine was

subsequently tethered so that it could move only vertically

upward.

The automatic control of a quadrotor helicopter has at-

tracted the attention of many researches in the past few years

[2][3][4][5]. Generally, the control strategies are based on

simplified models which have both a minimum number of

states and minimum number of inputs. These reduced models

should retain the main features that must be considered when

designing control laws for real aerial vehicles.

In this work, we present the model of a quadrotor helicopter

whose dynamical model is obtained via Newton’s laws. A

backstepping control strategy is proposed having in mind that

the quadrotor can be seen as the interconnected of three sub-

systems: under-actuated subsystem, fully-actuated subsystem

and propeller subsystem. The output of the under-actuated

subsystem are the x and y motion. In order to control them,

tilt angles (pitch and roll) need to be controlled. It appeared

judicious to us to use a backstepping technique to solve this

problem. The idea is to stabilize the system in two phases.

Firstly, the positions (x, y) are controlled by a virtual control

based on the tilt angles. Secondly, the tilt angles are controlled

by varying the rotor speeds, thereby changing the lift forces.

The backstepping technique will be used to control the zmotion and the yaw angle of the fully-actuated subsystem.

The paper is organized as follows: in section II, a dynamic

model for a quadrotor helicopter is developed. Based on this

nonlinear model, we design in section III a backstepping

control law. In section IV, some simulations are carried out to

show the performance and stability of the proposed controller.

Finally, in section V, our conclusions are presented.

II. DYNAMIC MODELING OF A QUADROTOR HELICOPTER

The quadrotor helicopter is shown in figure 1. The two pairs

of rotors (1, 3) and (2, 4) turn in opposite direction in order

to balance the moments and produce yaw motions as needed.

On varying the rotor speeds altogether with the same quantity

the lift forces will change affecting in this case the altitude

z of the system. Yaw angle is obtained by speeding up the

clockwise motors or slowing down depending on the desired

angle direction. The motion direction according (x, y) axes

depends on the sense of tilt angles (pitch and roll) whether

they are positive or negative.

Fig. 1. Quadrotor helicopter

1-4244-0259-X/06/$20.00 ©2006 IEEE3255

Proceedings of the 2006 IEEE/RSJInternational Conference on Intelligent Robots and Systems

October 9 - 15, 2006, Beijing, China

The equations describing the attitude and position of a

quadrotor helicopter are basically those of a rotating rigid body

with six degrees of freedom [6] [7]. They may be separated

into kinematic equations and dynamic equations [8].

Let be two main reference frames (see Fig. 1):

• the earth fixed inertial reference frame Ea :(Oa,

−→ea1 ,

−→ea2 ,

−→ea3).

• the body fixed reference frame Em : (Om,−→em1 ,

−→em2 ,

−→em3 )

regidly attached to the aircraft.

Let the vector ζ � [x, y, z]T and η � [φ, θ, ψ]T denote

respectively the altitude positions and the attitude angles of

the quadrotor (frame Em) in the frame Ea relative to a fixed

origin Oa. The attitude angles {φ, θ, ψ} are respectively called

pitch angle (−π2 < φ < π

2 ), roll angle (−π2 < θ < π

2 ) and

yaw angle (−π ≤ ψ < π).The quadrotor is restricted with the six degrees of freedom

according to the reference frame Em: Three translation ve-

locities V = [V1, V2, V3]T and three rotation velocities Ω =[Ω1,Ω2,Ω3]T . The relation existing between the velocities

vectors (V,Ω) and (ζ, η) are:{ζ = RtVΩ = Rrη

(1)

where Rt and Rr are respectively the transformation velocity

matrix and the rotation velocity matrix between Ea and Em

such as:

Rt =

⎡⎣ CφCψ SφSθCψ − CφSψ CφSθCψ + SφSψCθSψ SφSθSψ + CφCψ CφSθSψ − SφCψ−Sφ SφCθ CφCθ

⎤⎦(2)

and

Rr =

⎡⎣ 1 0 −Sθ

0 Cφ CθSφ0 −Sφ CφCθ

⎤⎦ (3)

where S(.) and C(.) are the respective abbreviations of sin(.)and cos(.).

One can write Rt = RtS(Ω) where S(Ω) denotes the skew-

symmetric matrix such that S(Ω)v = Ω × v for the vector

cross-product × and any vector v ∈ R3. In other words, for

a given vector Ω,the skew-symmetric matrix S(Ω) is defined

as follows:

S(Ω) =

⎡⎣ 0 −Ω3 Ω2

Ω3 0 −Ω1

−Ω2 Ω1 0

⎤⎦ (4)

The derivation of (1) with respect to time gives{ζ = RtV + RtV = RtV +RtS(Ω)V = Rt(V + Ω × V )Ω = Rrη +

(∂Rr

∂φ φ+ ∂Rr

∂θ θ)η

(5)

Using the Newton’s laws in the reference frame Em,

about the quadrotor helicopter subjected to forces∑Fext and

moments∑Text applied to the epicenter, one obtains the

dynamic equation motions:{ ∑Fext = mV + Ω × (mV )∑Text = IT Ω + Ω × (ITΩ)

(6)

where m and IT = diag[Ix, Iy, Iz] are respectively the mass

and the total inertia matrix of helicopter,∑Fext and

∑Text

includes the external forces/torques developed in the epicenter

of a quadrotor according to the direction of the reference frame

Em, such as:{ ∑Fext = F − Faero − Fgrav∑Text = T − Taero

(7)

where the forces {F, Faero, Fgrav} and the torques {T, Taero}are explained in the table I where G = [0, 0, g]T is the

gravity vector (g = 9.81m.s−2), {Kt,Kr} are two diagonal

aerodynamic friction matrices.

Model SourceF = [0, 0, F3]T

T = [T1, T2, T3]Tpropeller system

Faero = KtVTaero = KrΩ

aerodynamic friction

Fgrav = mRTt G gravity effect

TABLE I

MAIN PHYSICAL EFFECTS ACTING ON A QUADROTOR

The forces F and torques T produced by the propeller

system of a quadrotor are:

F =

⎡⎣ 0

0∑4i=1 Fi

⎤⎦ , T =

⎡⎣ d (F2 − F4)

d (F3 − F1)c∑4i=1(−1)i+1Fi

⎤⎦ (8)

where d is the distance from the epicenter of a quadrotor to

the rotor axes and c > 0 is the drag factor.

Using (5), (6) and (7) allows to give the equation of

the dynamics of rotation of the quadrotor expressed in the

reference frame Ea :⎧⎪⎨⎪⎩

F = mRTt ζ +KtRTt ζ +mRTt G

T = ITRrη + IT

(∂Rr

∂φ φ+ ∂Rr

∂φ θ)η

+KrRrη + (Rrη) × (ITRrη)

(9)

The dynamic model (9) of the quadrotor helicopter has six

outputs {x, y, z, φ, θ, ψ} while it has only four independent

inputs. Therefore the quadrotor is an under-actuated system.

We are not able to control all of the states at the same time. A

possible combination of controlled outputs can be {x, y, z, ψ}in order to track the desired positions, more to an arbitrary

heading and stabilize the other two angles, which introduces

stable zero dynamics into the system [5]. A good controller

should be able to reach a desired position and a desired yaw

angle while guaranteeing stability of the pitch and roll angles.

III. BACKSTEPPING CONTROL DESIGN

The main objective of this section is to design a backstep-

ping controller of a quadrotor helicopter ensuring that the

position {x(t), y(t), z(t), ψ(t)} tracks the desired trajectory

{xd(t), yd(t), zd(t), ψd(t)} asymptotically. This can be done

in two phases:

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• The dynamic model is written in an appropriate form.

Indeed, it is divided into three subsystems: an under-

actuated subsystem, a fully-actuated subsystem and a

propeller subsystem.

• Synthesis of the control law via backstepping technique

in seven steps.

Let u = [F1, F2, F3, F4]T the control input. The dynamics

equations (9) can be rewritten in a state-space form according

to the following state vectors:

x1 =[xy

], x3 =

[φθ

], x5 =

[ψz

],

x2 =[xy

], x4 =

[φ

θ

], x6 =

[ψz

],x7 =

⎡⎢⎢⎣F1

F2

F3

F4

⎤⎥⎥⎦ .(10)

We obtain the stat-space equations of an under-actuated

subsystem S1, a fully-actuated subsystem S2 and a propeller

subsystem S3:

S1 :

⎧⎪⎪⎨⎪⎪⎩

x1 = x2

x2 = f0(x2, x3, x5, x6) + g0(x5, x7)ϕ0(x3)x3 = x4

x4 = f1(x3, x4, x6, x7) + g1(x3)ϕ1(x7)

S2 :{x5 = x6

x6 = f2(x3, x4, x6, x7) + g2(x3)ϕ2(x7)(11)

S3 :{x7 = u

where the matrices gi (i = 0, 1, 2) are1

g0 =P4

i=1 Fi

m

[Sψ Cψ−Cψ Sψ

], g1 =

[1Ix

1IySφTθ

0 1IyCφ

]

g2 =[

1IzCφSeθ 00 1

mCφCθ

](12)

the vectors ϕi (i = 0, 1, 2) are

ϕ0 =[

SφCφSθ

], ϕ1 =

[d(F2 − F4)d(F3 − F1)

]ϕ2 =

[c(F1 − F2 + F3 − F4)F1 + F2 + F3 + F4

] (13)

and the vectors fi (i = 0, 1, 2) are

f0 =[fxfy

], f1 =

[fφfθ

], f2 =

[fψfz

](14)

with⎡⎣ fxfyfz

⎤⎦ = − 1

mRtKtR

Tt ζ −G (15)

⎡⎣ fφfθfψ

⎤⎦ = −(ITRr)−1[ IT (∂Rr

∂φ φ+ ∂Rr

∂φ θ)η−KrRrη − (Rrη) × (ITRrη)]

+

⎡⎢⎣

cIzCφTθ

∑4i=1(−1)i+1Fi

− cIzSφ

∑4i=1(−1)i+1Fi

dIySφSeθ(F3 − F1)

⎤⎥⎦

1Tθ and Se(.) are respectively the abbreviations of tan(.) and 1cos(.)

Using the backstepping approach [9], one can synthesize the

control law forcing the subsystems S1, S2 and S3 to follow the

desired trajectories. In this purpose, we will build the control

law by the seven following steps:

• Step 1: For the first step we consider the virtual system

x1 = v1 (16)

Let the first tracking error

z1 = x1d − x1 (17)

and considering the Lyapunov function positive definite

V1 =12zT1 z1 (18)

Its time derivative is

V1 = zT1 z1 = zT1 (x1d − x1) = zT1 (x1d − v1) (19)

The stabilization of z1 can be obtained by introducing a first

virtual control input v1:

v1 = A1z1 + x1d (20)

with A1 ∈ R2×2 is a positive definite matrix.

The equation (19) is then V1 = −zT1 A1z1 < 0.Step 2: For the second step we consider the following new

virtual system

x2 = f0(x2, x3, x5, x6) + g0(x5, x7)v2 (21)

where v2 is a second virtual control input.

Let us proceed to variable change by making

z2 = v1 − x2 = A1z1 + x1d − x1︸ ︷︷ ︸z1

(22)

hence z1 = −A1z1 + z2.For the second step we consider the augmented Lyapunov

function:

V2 =12

∑2

i=1zTi zi (23)

The time derivative of (23) is

V2 = zT1 z1 + zT2 z2 = zT1 (−A1z1 + z2) + zT2 (v1 − x2)

= −zT1 A1z1 + zT2 (z1 + v1 − f0 − g0v2) (24)

The stabilisation of z2 can be obtained by introducing the

following augmented virtual control

v2 = g−10 (z1 +A2z2 + v1 − f0) (25)

with A2 ∈ R2×2 is a positive definite matrix.

Remark 1: It is worthy noting that the determinant of the

matrix g0 is(

1m

∑4i=1 Fi

)2

> 0, if∑4i=1 Fi �= 0. Therefore,

the matrix g0 is nonsingular in general operation condition

because∑4i=1 Fi represents the total thrust on the body in the

z-axis and is generally nonzero to overcome the gravity.

By using the equation (24) is comes V2 =−∑2

i=1 zTi Aizi < 0.

• Step 3: For the third step we consider the virtual system

x3 = v3 (26)

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Let

z3 = v2−ϕ0(x3) = g−10 (z1 +A2z2 + v1−f0)−ϕ0(x3) (27)

then

g0z3 = z1 +A2z2 + v1 − g0ϕ0︸ ︷︷ ︸z2

hence z2 = −z1 −A2z2 + g0z3.For the third step we consider the Lyapunov function:

V3 =12

∑3

i=1zTi zi (28)

The derivatives with respect to time of (28) is

V3 = zT1 z1 + zT2 z2 + zT3 z3 (29)

= zT1 (−A1z1 + z2) + zT2 (−z1 −A2z2 + g0z3)+zT3 (v2 − ϕ0)

= −zT1 A1z1 − zT2 A2z2 + zT3 (gT0 z2 + v2 − J0v3)

where J0 is the Jacobian matrix of ϕ0 such as:

J0 =∂ϕ0(x3)∂x3

=[

Cφ 0−SφSθ CφCθ

](30)

Remark 2: It should be noted that the determinant of the

jacobian matrix J0(x3) is C2φCθ. Therefore, J0(x3) is nonsin-

gular when (−π2 < φ < π

2 ) and (−π2 < θ < π

2 ), which is

satisfied generally.

The stabilization of z3 can be obtained by introducing a

virtual control:

v3 = J−10 (gT0 z2 +A3z3 + v2) (31)

with A3 ∈ R2×2 is a positive definite matrix.

Using (29), we get V3 = −∑3i=1 z

Ti Aizi < 0.

• Step 4: we consider the final virtual system of an under-

actuated subsystem S1:

x4 = f1(x3, x4, x6, x7) + g1(x3)v4 (32)

Putting

z4 = v3 − x4 = J−10 (gT0 z2 +A3z3 + v2) − x4 (33)

we get

J0z4 = gT0 z2 +A3z3 + v2 − J0x4︸ ︷︷ ︸z3

(34)

hence z3 = −gT0 z2 −A3z3 + J0z4.The global Lyapunov function of a subsystem S1 is

V4 =12

∑4

i=1zTi zi (35)

Its time derivative is

V4 = zT1 z1 + zT2 z2 + zT3 z3 + zT4 z4 (36)

= zT1 (−A1z1 + z2) + zT2 (−z1 −A2z2 + g0z3)+zT3 (−gT0 z2 −A3z3 + J0z4) + zT4 (v3 − x4)

= −zT1 A1z1 − zT2 A2z2 − zT3 A3z3+zT4 (JT0 z3 + v3 − f1 − g1v4)

The stabilization of subsystem S1 can be obtained by

introducing a following virtual control law:

v4 = g−11 (JT0 z3 +A4z4 + v3 − f1) (37)

with A4 ∈ R2×2 is a positive definite matrix.

Remark 3: Since the condition (−π2 < φ < π

2 ) is generally

satisfied, then the matrix g1 given by (12) is nonsingular.

While introducing the control (37) in equation (36) one ob-

tains V4 = −∑4i=1 z

Ti Aizi < 0. Consequently, the subsystem

S1 is asymptotically stable with the virtual control inputs v1,

v2, v3 and v4.

After having stabilized an under-actuated subsystem S1,

there remains to us the stabilization of a fully-actuated sub-

system S2. To carry out this task, we will devote the two next

steps.

• Step 5: In this step we consider the virtual system

x5 = v5 (38)

Let the tracking error

z5 = x5d − x5 (39)

and the Lyapunov function:

V5 =12zT5 z5 (40)

The time derivative of (40) is

V5 = zT5 z5 = zT5 (x5d − x5) = zT5 (x5d − v5) (41)

The stabilization of z5 can be obtained by introducing a

virtual control:

v5 = A5z5 + x5d (42)

with A5 ∈ R2×2 is a positive definite matrix.

By using (42) in (41) one obtains V5 = −zT5 A5z5 < 0.• Step 6: For this step we consider the system

x6 = f2(x3, x4, x6, x7) + g2(x3)v6 (43)

Let

z6 = v5 − x6 = A5z5 + x5d − x5︸ ︷︷ ︸z5

(44)

Then z5 = −A5z5 + z6.The augmented Lyapunov function for this step is

V6 =12(zT5 z5 + zT6 z6) (45)

Its time derivative is

V6 = zT5 z5 + zT6 z6 = zT5 (−A5z5 + z6) + zT6 (v5 − x6)

= −zT5 A5z5 + zT6 (z5 + v5 − f2 − g2v6) (46)

The stabilization of S2 can be obtained by introducing a

following virtual control:

v6 = g−12 (z5 +A6z6 + v5 − f2) (47)

with A6 ∈ R2×2 is a positive definite matrix.

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Remark 4: Knowing that (−π2 < φ < π

2 ) and (−π2 < θ <

π2 ), it is easy to show in (12) that the matrix g2 is nonsingular.

The introducing the control (47) in equation (46) one obtains

V6 = −zT5 A5z5 − zT6 A6z6 < 0. One can thus conclude that

the subsystem S2 is asymptotically stable.

Now, in the following step, we will develop the real control

which stabilizes the whole system.

• Step 7: For the final step we consider the propeller

subsystem

x7 = u (48)

Let

z7 =[v4 − ϕ1(x7)v6 − ϕ2(x7)

](49)

=

⎡⎢⎢⎣g−11 (JT0 z3 +A4z4 + v3 − f1 − g1ϕ1︸ ︷︷ ︸

z4

)

g−12 (z5 +A6z6 + v5 − f2 − g2ϕ2︸ ︷︷ ︸

z6

)

⎤⎥⎥⎦

hence z4 = g∗1z7 − JT0 z3 −A4z4 and z6 = g∗2z7 − z5 −A6z6where g∗1 = [g1, 02×2] and g∗2 = [02×2, g2] with 02×2 is a null

matrix in R2×2.

The Lyapunov function candidate of the whole system

{S1, S2, S3} is

V7 =12

∑7

i=1zTi zi (50)

Its time derivative is given

V7 =∑7i=1 z

Ti zi (51)

= zT1 (−A1z1 + z2) + zT2 (−z1 −A2z2 + g0z3)+zT3 (−gT0 z2 −A3z3 + J0z4)+zT4 (−JT0 z3 −A4z4 + g∗1z7) + zT5 (−A5z5 + z6)

+zT6 (−z5 −A6z6 + g∗2z7) + zT7

([v4v6

]−

[ϕ1

ϕ2

])

= −∑6i=1 z

Ti Aizi + zT7

([g1 02×2

02×2 g2

]T [z4z6

]

+[v4v6

]−

[J1

J2

]x7︸︷︷︸u

⎞⎠

where J1 = ∂ϕ1(x7)∂x7

and J2 = ∂ϕ2(x7)∂x7

are the Jacobianmatrices of ϕ1 and ϕ2 such as:

J1 =[

0 d 0 −d−d 0 d 0

], J2 =

[c −c c −c1 1 1 1

](52)

Therefore, the stabilization of the whole system can be

obtained by introducing a following control law:

u =[J1

J2

]−1([

g1 02×2

02×2 g2

]T [z4z6

]+

[v4v6

]+A7z7

) (53)

with A7 ∈ R4×4 is a positive definite matrix.

Remark 5: It should be noted that the determinant of the

matrix

[J1

J2

]is 8cd2. Therefore, this matrix is nonsingular

when c > 0 and d > 0.

While introducing the control (53) in equation (51) one

obtains V7 = −∑7i=1 z

Ti Aizi < 0. Consequently, by using

(17, 20, 22, 25, 27, 31, 33, 37, 39, 42, 44, 47 and 53), the

whole system (11) is asymptotically stable with the following

control law:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

v1 = A1(x1d − x1) + x1d

v2 = g−10 [(x1d − x1) +A2(v1 − x2) + v1 − f0]

v3 = J−10 [gT0 (v1 − x2) +A3(v2 − ϕ0) + v2]

v4 = g−11 [JT0 (v2 − ϕ0) +A4(v3 − x4) + v3 − f1]

v5 = A5(x5d − x5) + x5d

v6 = g−12 [(x5d − x5) +A6(v5 − x6) + v5 − f2]

u =[J1

J2

]−1([

g1 02×2

02×2 g2

]T [v3 − x4

v5 − x6

]+

[v4v6

]+A7

[v4 − ϕ1

v6 − ϕ2

])(54)

IV. SIMULATION RESULT

In order to verify the effectiveness and the efficiency of

the proposed backstepping control law, an application to

quadrotor helicopter is conducted by simulation using Rung-Kutta’s method with variable step. The nominal parame-

ters for quadrotor helicopter are: m = 2kg, Ix = Iy =Iz/2 = 1.2416Nm.s2/rad, d = 0.2m, c = 0.01m, g =9.81m/s2, Kt = diag[10−2, 10−2, 10−2]N.s/m and Kr =diag[10−3, 10−3, 10−3]Nm.s/rad. The following controller

parameters are used: A1 = . . . = A6 = diag[3, 3] and A7 =diag[3, 3, 3, 3]. The proposed backstepping control law (54)

requires the knowledge of v1, v2, v3, v4, v5 and v6. In order

to avoid analytical derivation difficulties, we estimate them by

using the following finite difference time approximation:

vi =ΔviΔt

for i = 1, . . . , 6 (55)

where Δvi is the change in input value and Δt is the change in

time since the previous simulation time step. The helicopter is

initially in hover flight and the initial conditions are: x1(0) =. . . = x6(0) = [0, 0]T and x7(0) = mg

4 [1, 1, 1, 1]T . The

reference trajectory chosen for xd(t), yd(t), zd(t) and ψd(t)is that of the step response of the following transfer function:

H(s) =1

(s+ 1)6(56)

with s is the Laplace variable.

Figures 2 and 3 show the positions and the tracking errors

of a quadrotor helicopter respectively. It can be seen the good

tracking of the desired trajectory. Moreover, we can notice

in figure 2 an optimization of tilt angles (pitch and roll)

and consequently the use of minimal energy. The behavior

of all virtual controls is given in figure 4. We note that

their evolutions are smooth and derivable. The obtained input

control signals (see figure 5) are acceptable and physically

realizable.

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Fig. 2. Position outputs

Fig. 3. Tracking errors for x, y, z and ψ

Fig. 4. Evolution of the virtual controls: solid line with the left y-axis denotethe first element and the dashed line with the right y-axis denote the secondelement.

Fig. 5. Force control inputs

V. CONCLUSION

In this paper, we have presented the dynamic modeling

of quadrotor helicopter and introduced a new approach of a

backstepping control. This process is under-actuated system

because it has six outputs while it has only four inputs.

The whole system of the quadrotor helicopter is divided into

three subsystems: an under-actuated subsystem, fully-actuated

subsystem and a propeller subsystem. A backstepping control

algorithm is proposed to stabilize the whole system and is

able to drive a quadrotor to the desired trajectory of Cartesianposition and yaw angle. The simulation results show the good

performance of the proposed control approach.

REFERENCES

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