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16TH INTERNATIONAL CONFERENCE ON MECHATRONICS TECHNOLOGY, OCTOBER 16-19, 2012, TIANJIN, CHINA

The Trajectory Tracking Problem for Quadrotor System

Na Zhang*

and Qing-He WuAutomation Department, Beijing Institute of Technology, Beijing, China

*Corresponding: zhangna419314@163.com

AbstractThis paper presents the design of a nonlinear controller

based on integral backstepping control technique combined with

the super twisting control algorithm for trajectory tracking of a

quadrotor helicopter. A simple version of dynamical system based

on Newton-Euler formalism is used. The performance and

effectiveness of the proposed controller are tested in a simulation

study taking into account external disturbances.

Keywordsquadrotor system, trajectory tracking, integral

backstepping, supertwist algorithm

1 INTRODUCTIONOver the last decade, autonomous unmanned aerial

vehicles (UAV) have gained considerable attention among

researchers. Recent advances in sensor technology, signal

processing and integrated actuators have taken the

development of small UAVs one step further. Quadrotor, as a

popular type of UAV, is a kind of Vertical Take-Off and

Landing (VTOL) vehicle which could be launched from any

place and hover above any target. In comparison with classical

helicopters conventionally equipped with a main rotor and a

tail rotor, quadrotors possess a larger payload capacity and

higher versatility and maneuverability to carry out various

types of tasks such as rescue, near-area surveillance and

exploration. However the dynamics of the quadrotor are

nonlinear, multivariable and under actuated with four control

inputs and six degrees of freedom (DOF). Besides, variables

are strongly coupled with each other. All these characteristics

make the control of quadrotors increasingly challenging. A

great number of efforts have been made by researchers in

designing the control methodologies that will make an

unmanned aerial autonomous. Popular control methods include

input-output feedback linearization [11,12], backstepping

[1,10], sliding mode control [1,4,7,10] and so on.

The quadrotor is a kind of rotorcraft, whose mechatronic

system consists of four rotors in a cross configuration. Rotors

are paired to spin in opposite directions in order to cancel out

gyroscopic effects and aerodynamic torques in trimmed flight.

The motion of a quadrotor can be divided into two parts:

translational subsystem and rotational subsystem. As a

difference of speed is applied between opposing rotors, thehelicopter tilts towards the slowest one, which results in

acceleration along that direction. In other words, the rotational

and translational motions are closely related.

The main work of this paper aims to use nonlinear control

techniques to perform the stabilization and tracking control of

a quadrotor. The controller for translational subsystem is

designed to stabilize the altitude and gives the desired roll and

pitch angles to the rotational subsystem controller in the

meanwhile.

Fig. 1: The quadrotor system

2 QUADROTOR MODELGenerally speaking, the control strategies are based on

simplified models which have both a minimum number of

states and a minimum number of inputs. These reduced models

should retain the main features that must be considered when

designing control laws for real aerial vehicles [5].

The quadrotor model consists of four rotors powered by

electric motors and mounted at each end of an X-shaped frame

as shown in Fig. 1. The collective input is the sum of the

thrusts of each motor. Increasing or decreasing the speeds of

four motors together generates the vertical motion. Pitch

movement is obtained by increasing (reducing) the speed of

the rear motor and reducing (increasing) the speed of the front

motor. The roll movement is generated using the lateral motors

in the similar way. Yaw motion results from the difference in

the counter-torque between each pair of propellers.

Let E denote an inertial frame, and B be a body fixed

frame shown in Fig. 2. The generalized coordinates of the

quadrotor can be described as q=(x, y, z, , , ), where (x, y, z)

denote the position of the center of gravity of the vehicle in

frameEwhile three Euler angles (, , ) (roll, pitch and yaw

angles) represent its angular velocity in frame B and give the

orientation of the quadrotor. Therefore, the model is naturally

divided into translational coordinates =(x, y, z)Tand rotational

coordinates =(, , )T.

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Fig. 2: Coordinate system

The transformation of vectors from the body fixed frame

to the inertial frame is given by the rotation matrixR:

ccscs

sccssccssscs

sscsccsssccc

R

wheres() and c() denotesin() and cos(), respectively.

According to the formalism of Newton-Euler, the dynamic

equations are written in the following form:

ext

ext

TJJ

RSR

Fm

V

)(

Where V=(u, v, w)T is the absolute velocity of the vehicle

expressed with respect to the inertial frame, m is the mass of

the structure, =(p, q, r)Tis the angular velocity expressed in

B, the relationship between and Euler angles is as follows:

coscossin0

sincoscos0sin01

(1)

S() is a skew-symmetric matrix for a given vector=(1,

2, 3)T, which is defined as:

.

0

0

0

)(

12

13

23

S

(2)

Jis the inertia matrix given by

.00

00

00

z

y

x

I

I

I

J (3)

The coupling inertia is assumed to be zero because of the

symmetry of the geometric form of the quadrotor. The

notations Fextand Textstand for the vector of external forces

and that of external torque, respectively.

Based on the force analysis of the system, we obtain

,0

0

0

0

mgT

RFext

)(

)(

)(

2

3

2

1

2

4

2

2

2

1

2

3

2

2

2

4

d

lb

lb

Text

T is the total thrust produced by all four rotors, g is the

gravity constant (g = 9.81m/s2), i denotes the rotational

velocity of rotori. lrepresents the distance from the rotors to

the center of gravity of the vehicle. b>0 is thrust factor and

d>0 is the drag factor. The thrust generated by rotor i is given

by Ti=bi2. So the total thrust can be described as

)(

2

4

2

3

2

2

2

1 bT .Consequently, complete dynamic model of the quadrotor is

as follows:

AII

IIpqr

AI

l

I

II

prq

AI

l

I

IIqrp

Am

Tgw

Am

Tv

Am

Tu

zz

yx

yy

xz

xx

zy

z

y

x

1)(

)(

)(

)cos(cos

)cossinsinsin(cos

)sinsincossin(cos

(4)

B

x

z

y

mg

T4

T3

T1

T2

X

Z

Y

E

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where (Ax, Ay, Az, A, A, A) represent disturbances in 6

degrees of freedom.

From the equation above, we can observe that the tilt

angles and the motion of the helicopter are closely related.

That is, the six degrees of freedom are strongly coupled with

each other and there are only four independent variables in

dynamic model equation, which makes trajectory tracking

control possible.

3 STATEMENT OF THE CONTROL PROBLEMThe control problem considered in this paper is to make

the system position variables x, y, z and the orientation

variable asymptotically track the desired references (the yaw

angle is set to be maintained at zero during flight) by designing

control in terms of torque commands to the propeller under the

assumption that orientation variables and always stay

inside the open set (-/2, /2) and T > 0.

4 CONTROL STRATEGYThe dynamic model of the quadrotor helicopter has six

dependent outputs while it has four independent inputs, so we

cannot control all the states at the same time. This problem can

be solved by backstepping method, which is a good way to

deal with underactuation problem. As a matter of fact, the

backstepping is well suited for the cascaded structure of the

quadrotor dynamics.

The overall control system has two loops and is depicted

in Fig. 3. The external inputs to the system are assumed to be

the desired position d and the desired yaw d. According tothe difference between the current quadrotor position and the

desired position, the position controller generates the desired

d and d to ensure d. Subsequently, Euler angular

controller uses the information provided by position controller

to calculate the desired angular velocity d which ensures

d. Then, the angular velocity controller calculates the

actual control inputs (T, , , ) which ensures d when

they are applied to the UAV system. The output vector is then

fed back into the controllers.

4.1 Translational subsystem controller designThe translational subsystem of the quadrotor is

represented by (4) and can be rewritten as follows:

mTgw

m

Tv

m

Tu

wz

vy

ux

)cos(cos

)cossinsinsin(cos

)sinsincossin(cos

(5)

Now, we define a virtual control u1 as:

.

)(

)(

)(

3,1

2,1

1,1

1

u

u

u

m

Tccg

m

Tcsssc

m

Tsscsc

u

(6)

Then the original subsystem is transformed into a simpler

second order linear system. Backstepping is a recursive

procedure that interlaces the choice of a Lyapunov functionwith the feedback control. It breaks a design problem for the

full system into a sequence of design problems for lower order

subsystems. By exploiting the extra flexibility that exists with

lower order subsystems, backstepping can often solve

stabilization, tracking, and robust control problems.

Considering the uncertainties and disturbances of quadrotor

model, we choose the IB (integral backstepping) to perform

the tracking command.

Reference

trajectoryQuadrotor

dynamic

PositionController

Euler

Angular

Controller

Angular

Velocity

Controller

d

dd .,

d

d

d

r

q

p

d

d

d

z

y

x

T

z

y

x

r

q

p

Fig. 3: The overall control strategy of the quadrotor system

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Firstly, we introduce the desired trajectory d=(xd, yd, zd)T

for the position variable and define the tracking error as

.1 de (7)Using (5) and (7) we obtain its dynamics:

.1 Vdt

ded

(8)

We consider V as an input of the upper dynamics. Tostabilize the dynamics, the virtual control input Vis chosen as:

,)(0

1011

t

aadd deKeKV (9)

where Ka1 and Ka0 are diagonal matrices with positive

constants.

Then the velocity tracking error is defined by

VVe d 2 It results in:

.10112

VeKeKe aad (10)So we rewrite the tracing error dynamics ofe1:

20

0111 )( edeKeK

dt

de taa

(11)After we apply (5), (6) and (11) to (10), the stabilization of

e2can be obtained by selecting the input u1 as

.)(

)()(1

001

21110

2

1

deKK

eKKeKKIu

t

aa

abaad

The control input of the position subsystem is a function of

Euler angles and the total thrust T. As a result, we need toinvert u1 to compute the thrust and the Euler angle commandsthat will be used as reference for the following subsystem.

As for (6), by squaring left and right sides of all threeequations in (6) and adding them together, we get the thrust T;

by subtracting the second equation multiplied by cos fromthe first equation multiplied bysin, we get the desired pitchangle d; the desired roll angle can be easily competed from

the first equation. Then the overall inversion is presented as:

(note that the yaw angle d is set to be maintained at zeroduring flight)

.

)coscos

sinsinarcsin(

))cossin(sin(

)(

1,1

2,11,1

2

3,1

2

2,1

2

1,1

dd

dd

d

ddd

uT

m

uuT

mar

uguum

T

(12)

4.2 Rotational subsystem controller designThe rotational subsystem of the quadrotor is denoted by (4)

and (1) which can be rewritten as:

r

q

p

c

c

c

s

sc

tcts

0

0

1 , M

(13)

z

y

x

z

yx

y

xz

x

zy

I

I

l

Il

I

IIpq

I

IIpr

IIIqr

r

q

p

1)(

)(

)(

.)( EUf

(14)where t() denotes tan().

Attitude control is the heart of the control system; it keepsthe 3D orientation of the quadrotor to the desired value, hence,prevents the vehicle from flipping over and crashing. As weknow, sliding mode controller is widely used due to its

attractive characteristics of finite-time convergence androbustness to uncertainties, and super-twisting algorithm is animportant class of second order sliding modes. It ensuresrobustness with respect to modeling errors and externaldisturbances while reducing the chattering phenomenon causedby all first order sliding mode controllers. [8] In this section,were going to combine the super-twisting algorithm andbackstepping technique, which turns out to get a satisfactoryresult in attitude stabilization.

In the previous section, desired Euler angles have beencalculated to act as the reference input of the rotationalsubsystem. We set

dz1 ,then its first derivative becomes

.1 Mz dd

(15)

To stabilize the above dynamics, the virtual control input d

could be:

).)((0

1011

1 dzKzKMt

ccdd

(16)

Again we define

)(2 dMz ,and use (16) and (14). It yields

).)(()( 10112 EUfMzKzKz ccd

(17)

Then the dynamics ofz1becomes:

.)( 20

10111 zdzKzKzt

cc

(18)

We choose the sliding variable as s1 = z2. Now we design

the control inputs applying the super twisting control algorithm

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)(

)(

12

212

1

11

ssigns

sssignss

(19)

with control gains > 0 and > 0. The proof of finite time

convergence to zero of the variable s1ands2 is explicitly given

in [6].

As a result, control input U should be designed as

)())(

)()(()(

1

222

1

2

200

1

011

2

10

1

fEszsignz

zKdz

KKzKKMEUc

t

ccccd

5 SIMULATION RESULTSIn order to verify the performance of the proposed

controller, simulations with and without external disturbances

were made. As already noted, the controllers task is to make

the system position variables x, y, z asymptotically track the

desired references (the yaw angle is set to be maintained at

zero during flight). As we know, sinusoidal signal is one of the

typical input signals in control systems, and especially gives usmore challenges in tracking control problems. Therefore we

set the desired position trajectory as x = cos(t), y = sin(t) andz

= t. Initially, the quadrotor is placed at the origin and in a

horizontal position, that is, (x, y, z) = (0, 0, 0) and (, , ) =

(0, 0, 0), keeping constant.

5.1 Position tracking without external disturbancesFirstly we consider the situation of position tracking without

external disturbances. Fig. 4 shows the tracking of desired

trajectory, which clearly reflects the effectiveness of the

controller. Fig. 5 and Fig. 6 give a more distinct process of

tracking in 3 directions.

-1-0.5

00.5

11.5

-1

0

1

20

5

10

15

20

refernce

real trajectory

Fig. 4: Position trajectory of the quadrotor without external

disturbances

Fig. 5: Position (x, y, z) tracking of the quadrotor without external

disturbances

Fig. 6: Orientation (, ) of the quadrotor without external

disturbances

5.2 Position tracking with external disturbancesThe disturbances in position subsystem areAx = 0.02,Ay =

0.02, Az = 0.02. Moreover, the moment disturbances in

rotational subsystem are A = 0.1sin(0.1t), A = 0.1cos(0.1t),

A = 0.1sin(0.1t). The obtained results are depicted in Fig. 7,

Fig. 8 and Fig. 9. Its concluded from the simulation that the

good performance of the proposed control approach is

maintained.

-1-0.5

00.5

11.5

-1

0

1

20

5

10

15

20

reference

actual trajectory

Fig. 7: Position trajectory of the quadrotor with external

disturbances

Fig. 8: Position (x, y, z) tracking of the quadrotor with external

disturbances

Fig. 9: Orientation (, ) of the quadrotor with external

disturbances

0 2 4 6 8 10 1 2 14 1 6 1 8 2 0-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

theta

0 2 4 6 8 10 1 2 14 1 6 1 8 2 0-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

phi

0 2 4 6 8 10 1 2 1 4 1 6 1 8 2 0-1

-0.5

0

0.5

1

1.5

refernce

y

0 2 4 6 8 10 1 2 1 4 1 6 18 2 0-1

-0.5

0

0.5

1

1.5

refernce

x

0 2 4 6 8 10 1 2 1 4 1 6 1 8 2 00

2

4

6

8

10

12

14

16

18

20

refernce

z

0 2 4 6 8 10 12 14 1 6 18 2 0-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

phi

0 2 4 6 8 10 1 2 1 4 1 6 1 8 2 0-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

theta

0 2 4 6 8 10 1 2 1 4 1 6 18 2 0-1

-0.5

0

0.5

1

1.5

refernce

x

0 2 4 6 8 10 1 2 14 1 6 18 2 00

2

4

6

8

10

12

14

16

18

20

refernce

z

0 2 4 6 8 10 1 2 14 1 6 18 2 0-1

-0.5

0

0.5

1

1.5

refernce

y

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Although the simulation results under both situations are

satisfactory, we cannot achieve perfect tracking of sinusoidal

signal with zero steady-state error according to internal model

control theory. But perfect tracking of step signal can be

implemented by proposed algorithm, which has been proved

by simulations. Due to the limit of space, we wont show the

results here.

6 CONCLUSIONSIntegral backstepping control technique combined with the

super twisting control algorithm has been used to stabilize and

perform output tracking control on a simple version of

dynamical system. The super twisting control algorithm is

applied so as to improve robustness to external disturbances

and model uncertainties. Simulations performed on Matlab

Simulink show the ability of the controller to perform output

tracking control even external disturbances and model

uncertainties exist.

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