Integral Sliding Mode and Second Order Sliding Mode Attitude and

Altitude Tracking of a Quadrotor System

Theory and Experiment

Mouloud Bouchoucha, Abdessamed Boudane, Kamel Ali

and Sofiane Seghour

Laboratoire de Côntrole & Commande (LCC), Ecole Militaire Polytechnique(EMP),

Bordj-El-Bahri, Algiers, 16111, Algeria

Keywords: dsPIC, Dynamic Modeling, Embedded Control System, IMU, Integral Sliding Mode, Quadrotor, Second

Order Sliding Mode, Super- Twisting , UAV, C.

Abstract: In this paper Attitude and Altitude tracking control design of the four rotors helicopter will be considered.

Two robust control algorithms will be designed for the case of stabilization and tacking of attitude and

altitude system’s outputs. The attitude controller is realized using an inertial measurement unit (IMU) based

on MEMS sensors. The altitude control algorithm uses a sonar sensor output. The control algorithms

designed are implemented on an embedded control system based on a dsPIC C. The obtained experimental

results demonstrate high performance of both controllers and robustness against disturbances.

1 INTRODUCTION

Unmanned Aerial Vehicles (UAVs) have been

designed in the military field since more than one

half century. The main objective was to replace

human pilot in a painful tasks and when the

environment became hostile where the security of

pilots is not assured such as: intervention in hostile

environment, management of the natural risks,

exploration of high buildings or contaminated

tunnels, surveillance, rescue missions, movie filming,

which were not possible before.

Nowadays, researches in this field know a very

big progress with the advance development of

electronic and digital systems. This progress has

given birth to low cost very small and accurate

electronic components, a powerful calculators, and

sensors. All these, aimed to product a small

embedded, autonomous and intelligent systems, able

to perform missions with more effectiveness and

reliability.

Miniature Vertical Takeoff and landing (VTOL)

unmanned aerial vehicles (UAV's) offer challenging

benchmark control problems and have been the

focus for many researchers in the past few years

(Brisset, 2004), (Bouabdallah, 2007). The VTOL

UAV four rotor helicopter named X4, OS4, or

known commonly as a quadrotor shown in Figure

1, is a mini-aircraft with four propellers.

Many researches addressed the modeling, the

control, and the design of the quadrotor system

(Bouabdallah, 2007), (Bouabdallah et al., 2004),

(Escareño et al., 2006), (Osmani et al., 2010),

(Hoffmann et al., 2007), (Kroo and Prinz, 2000),

(Derafa, 2006), (Hanford, 2005), (Hamel et al.,

2002), (McGilvray, 2004), (Bouadi et al., 2007),

(Bouchoucha et al., 2008).

Generally speaking, improving the performance

requires a good knowledge of the model as it is the

case with the almost previous aforementioned works.

Nevertheless it is still possible to achieve robustness

and a highly efficient dynamics using a control

techniques that does not need a good knowledge of

the model; this is the case especially where some

dynamics are neglected, the system parameters are

variable or not known exactly (inertia, thrust and

drag coefficients), or the system is subject to a

disturbance like the wind guest. To overcome to that,

robust control techniques have been proposed

(Bouchoucha et al., 2008), (Waslander et al., 2006),

(Bouabdallah et al., 2005), (Bouchoucha et al.,

2011), (Seghour et al. 2011)..etc. Almost the

designed techniques use sliding mode control

technique and/or they implement only the attitude

dynamics.

In this work, two control techniques are designed

266

Bouchoucha M., Boudane A., Ali K. and Seghour S..

Integral Sliding Mode and Second Order Sliding Mode Attitude and Altitude Tracking of a Quadrotor System - Theory and Experiment.

DOI: 10.5220/0004038002660273

In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 266-273

ISBN: 978-989-8565-22-8

Copyright

c

2012 SCITEPRESS (Science and Technology Publications, Lda.)

for the stabilization and the tracking of the attitude

and the altitude of the quadrotor system; the integral

sliding mode (ISM) and the second order sliding

mode (SOSM). The benefit is to demonstrate the

ability of both techniques to stabilize the system and

the ability of the second order technique to eliminate

the chattering phenomena while preserving the

performance comparing with the classical sliding

mode. The algorithms of both techniques are

implemented in real time on a developed embedded

control system based on a dsPIC µC to a quadrotor

platform (a modified version of the Draganflyer of

RCTOYS (Figure1)).

The rest of the paper is organized as follows: in

section 2, a mathematic model of the quadrotor is

presented. Section 3 is devoted to the design of both

control approaches for the attitude and the altitude

system’s outputs. Real time implementation results

of both control algorithms are presented in section 4.

Finally, conclusions are made in section 5.

2 DEFINITION AND

DYNAMICAL MODEL

A Quadrotor is an aircraft that is propelled by four

rotors. This model of rotary wing vehicles is very

interesting since the characteristic of taking-off and

landing so the space of their maneuvers is very

limited while comparing with fixed- wing aircraft.

The motion of this vehicle is controlled by

varying the rotating speed of the four rotors to

change the thrust and torques produced by each one.

The front and rear motors rotate counter clockwise,

while the two other motors rotate clockwise in order

to counter the yaw torque produced at the movement

of the aircraft (McGilvray, 2004), (Tayebi and

McGilvray, 2004). The main thrust derives from the

sum of thrusts of each rotor; it creates the vertical

motion of the platform. The pitch and roll torques

are derived respectively from the differences

and

, while

is the thrust force

of the rotor “i”. The roll and pitch inclination create

the translational motion along X and Y axis

respectively.

The yaw torque is the sum of the reaction torques

of each rotor produced by the shaft acceleration and

the blade’s drag

with

,

is the drag coefficient and

is the

propeller speed of the motor i (Osmani et al., 2010),

(Hanford, 2005).

The force

produced by the rotor “i” is

proportional to the square of the propeller speed,

(a)

(b)

Figure 1: (a) Quadrotor helicopter of the LCC laboratory

(EMP), (b) Quadrotor configuration and principles.

with the proportionality constant of the

thrust force.

The dynamics of the quadrotor is described in the

space by six degrees of freedom according to the

fixed inertial frame related to the ground.

To derive the dynamic model of the quadrotor,

the Newton Euler formalism will be used on both

translation and rotation motions; therefore to obtain

the following equations (Hamel et al., 2002),

(Bouabdallah et al., 2004), (McGilvray, 2004),

(Derafa, 2006).

JJ

mFFF

v

gaf

gdtf

(1)

In this work we mainly focus our interest to the

attitude and the altitude dynamics and we consider

the state space model of reduced dynamical model to

simplify the control design as follows (Bouabdallah

et al., 2004), (Derafa, 2006), (Boudane and Kamel,

2011):

(2)

Integral Sliding Mode and Second Order Sliding Mode Attitude and Altitude Tracking of a Quadrotor System - Theory and

Experiment

267

Where

.

And the control inputs:

(3)

and

are the roll, the

pitch, the yaw, the altitude and their variations

respectively.

The parameters

depend on the mass

and the inertia of the system, the inertia of the rotors,

the aerodynamic friction coefficients, the drag

coefficients of translation.

The parameters

depend on the inertia of the

system and the distance between the center of the

system and the center of the rotor; and the parameter

depend on the inertia of the system.

is the total mass of the quadrotor and is the

gravitational acceleration.

S and C represent the Sinus and Co-sinus

functions respectively.

The rotors are driven by DC-motors with the well

known equations (McGilvray, 2004), (Bouabdallah

et al., 2004):

(4)

Where R

a

, I

a

, k

m,

L,

mi

and τ

i

are the motor

resistance, armature current, motor constant,

armature inductance, motor speed and the rotor

torque respectively.

3 CONTROL LAWS DESIGN

This section is focused to the design of both control

techniques proposed in this work i.e. the integral

sliding mode and the second order sliding mode for

the stabilization of the attitude and the altitude

outputs of the quadrotor system.

Before presenting the design of both controls

which is considered as an external loop we will

present the control of each rotor i.e the propeller

speed (internal loop).

The control torque developed by motor is a

proportional controller with compensation of the

drag torque resulting of the rotation of the propeller

(McGilvray, 2004).

3.1 Control Design of the Quadrotor

Dynamic

In order to stabilize the attitude and the altitude of

the qaudrotor system, two robust control approaches

will be designed: the integral sliding mode and the

second order sliding mode. The benefit within the

use of the integral term in the integral sliding mode

is to improve the tracking errors.

To simplifier the demonstration of the design of

both control approaches, the model of the qaudrotor

presented in (2) can be rewritten as:

(5)

With

and

3.2 Integral Sliding-mode Control

Approach

In this section we use Backstepping technique to

design the integral sliding mode control. The benefit

is the systematic choice of the Lyapunov function in

the stability demonstration. The Backstepping

control technique is designed for a system in

triangular feedback form which is the case for the

dynamic model of the quadrotor. In this technique

the control design pass by several steps, in each step

the actual state is controlled by the next state as a

virtual control, until the last state which is controlled

by the real control. The integral sliding mode

(Skjetne and Fossen, 2004) is the well known sliding

mode robust control (Utkin, 1978) augmented by an

integral term to improve the tracking errors.

However this approach suffers from the chattering

phenomena that limit its realization.

The integral sliding mode will be designed for

the stabilization of the attitude and the altitude of the

qaudrotor model (5) in two steps.

The first step in this design is similar to the one

for the Backstepping approach.

The most common way to include integral action

in Backstepping is to use parameter adaptation.

Another method is to augment the plant dynamics

with the integral state

(Skjetne and

Fossen, 2004). The resulting system is still in strict

feedback form; however, the vector relative degree

is increased to 3.

(6)

ICINCO 2012 - 9th International Conference on Informatics in Control, Automation and Robotics

268

Step 1: in this step we consider the subsystem:

And one define a new state

such as:

, and we introduce the first Lyapunov

function candidate :

Its time derivative is give by:

If we apply the Lyapunov theorem, i.e. by

imposing

condition, the stabilization of

and

can be obtained by introducing a new virtual

control input

where:

with

Step 2: Here we define the sliding surface

(Surface) [15]:

(7)

And we consider the augmented Lyapunov

function:

(8)

The chosen law for the attractive surface is the

time derivative of

satisfying (

):

(9)

In the other hand we have:

As for the Backstepping approach, the control

is extracted as follow:

(10)

And the resulting control laws are given by:

(11)

In the implementation, the sign (signe) function is

replaced by the Sat function in a boundary Layer

(Slotine, 1985) to reduce the chattering problem.

3.3 Second Order Sliding Mode Control

(Super-Twisting Algorithm)

The attitude and the altitude dynamic of the

quadrotor in (2 model) or (5) have relative degree

one with respect to the sliding surface defined in

(12) (the control input appears in the first derivative

of the sliding surface (14)). To remedy of the

chattering phenomena in classical integral sliding

mode, the second order sliding mode by using the

super-twisting algorithm will be applied (Levant,

1997), (Emelyanov et al., 1986), (Emelyanov et al.,

1996), (Fridman and Levant, 1996), (Nollet et al.,

2008).

Let us define here a new sliding surface

of

the system based on the model of the form (6)

without the first equation:

, with

(12)

Its time derivative is given by:

(13)

Replacing

by its equation given in (5) or (6),

become:

(14)

Using the principle of second order sliding mode by

the super-twisting algorithm (Bouchoucha et al.,

2011), (Nollet et al., 2008) the control input

is

given by:

(15)

With

With the super-twisting controls

are

given by:

(16)

(17)

Finally the control inputs

are given by:

(18)

Choosing the values of

and

sufficiently large,

allow to the tracking errors

and

to tend to zero

in finite time. The robustness to the parametric

uncertainties can be ensured by increasing the gains

and

(Slotine, 1985).

4 REAL TIME

IMPLEMENTATION

4.1 Experimental Setup

In order to validate the control laws developed in the

Integral Sliding Mode and Second Order Sliding Mode Attitude and Altitude Tracking of a Quadrotor System - Theory and

Experiment

269

previous section, we implemented the controllers on

the embedded control unit based on a dsPIC µC. The

attitude outputs are measured using the IMU 3DM-

GX1 of microstrain and the altitude is measured by

the ultrasonic sensor SRF08. The propellers speeds

are measured using a Hall Effect sensor combined

with little magnets mounted under the main rotor

gear. The sampling period is 30ms for the attitude

motion and 65ms for the altitude motion (the

ultrasonic sensor give the output each 65ms). We are

made for both control laws two experiment. In the

first experiment the attitude motion is stabilized with

a fixed trust U1=2.6N and the system is mounted on

fixed base knee-joint. In the second experiment

where we will give it a great interest we have

liberated the system to stabilize its altitude with

attitude is stabilized around an equilibrium point

zero. The controller’s parameters for both controllers

were tuned by trial and error, until obtaining a better

responses performance of the system.

4.2 Attitude Motion

For both control law and for the case of stabilization

around the equilibrium point and for more

convenience the initial values for the roll, pitch, and

yaw angles are taken almost the same for both

controllers.

The results obtained for both controllers are

shown in the figure.2

The following graphs show the obtained

performances:

Figure 2: Attitude stabilization :(left) integral sliding mode

(ISM), (right) second order sliding mode (SOSM).

The results obtained demonstrate the stabilization

of all the system outputs for both controllers.

Figure 3: Control inputs: (left) ISM, (right) SOSM.

However the second order sliding mode

demonstrates more superiority in term of

performance (settling time and accuracy).

To test the robustness of both approaches two

experiments have been performed.

In the first experiment the robustness test have

been realized to deal with the external load

disturbance and for more convenience, we have

maintained the same work conditions; a mass of 50 g

is fixed on the end of the system axis. The test is

made for the roll and the pitch axis and because the

symmetrical nature of the system we will present

only the results of the roll axis.

Figure 4: Roll response with disturbance: (Left) ISM,

(Right) SOSM.

Figure 5: Control inputs with disturbance: (Left) ISM,

(Right) SOSM.

The results obtained (Fig.4) show the ability of

both controllers to handle the effects of disturbance.

However, the integral sliding mode take more time

(13 seconds) to reject the disturbance effect than the

second order sliding mode (4 seconds) which

confirm the invariance property of SOSM to

eliminate the chattering while keeping the system

0 1 2 3 4 5 6 7 8 9 10

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Pitch (degrees)

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Pitch (degrees)

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Yaw(degrees)

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Yaw(degrees)

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Uroll(N.m)

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U3 (N.m)

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Upitch(N.m)

0 1 2 3 4 5 6 7 8 9 10

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U4 (N.m)

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Uyaw(N.m)

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Roll(degrees)

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U2(N.m)

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-1.3

-0.8

-0.3

0

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Time (sec)

U2 (N.m)

ICINCO 2012 - 9th International Conference on Informatics in Control, Automation and Robotics

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performance comparing with integral sliding mode

with a boundary layer even with an integral term.

The second experiment, both controllers are

tested to a desired trajectories tracking. For that a

hybrid cycloid and sinusoidal reference are used.

The results obtained (Fig.6) show that both

controllers ensure the trajectories tracking. However,

the SOSM controller demonstrates better behavior in

term of accuracy, settling time and overshoot

comparing with the ISM controller.

Figure 6: Desired Angles (Red) and real Angles outputs

(Blue): (left) ISM, (right) SOSM.

4.3 Altitude Motion

The altitude of the quadrotor will be considered here

for the case of stabilization and robustness to

external disturbances and desired trajectory tracking.

For the stabilization case, the system is

controlled to stabilize the altitude around 40cm as a

set point. The results obtained for both controllers

are shown in the figure 7, 8 and 9.

Figure 7: Altitude stabilization: (left) ISM, (right) SOSM.

Figure 8: Altitude control input (Trust U1): (left) ISM,

(right) SOSM.

The results obtained (fig.7, fig.8, fig.9) show the

stabilization of the altitude of the two controllers

with a superiority of the SOSM controller in relation

to the ISM controller in term of accuracy, settling

time and overshoot.

To verify the robustness of the proposed

approaches for the altitude output; external

disturbance rejection and trajectory tracking tests are

realized.

Figure 9: Propellers speeds (Trust U1): (left) ISM, (right)

SOSM.

The external disturbance realized by adding a

15% of the value of the actual control input U1 to its

next value.

The results obtained (fig.10 and fig.11) show that

even the deviation of the altitude output from its

stable value in the instance of the application of the

disturbance; both controllers damp the effect of the

disturbance in finite time. However and like it seems

clearly the SOSM controller is largely better in term

of the time (4sec) take it to handle the effect of the

disturbance than the ISM (15 sec) controller.

Figure 10: Altitude response with disturbance: (Left) ISM,

(Right) SOSM.

Figure 11: Altitude control input (Trust U1) with

disturbance: (Left) ISM, (Right) SOSM.

In the second robustness test, the system is

submitted to a cycloidal reference trajectory. The

altitude outputs and the corresponding control inputs

U1 of the quadrotor for the both controllers are

given in the figure 12 and 13 respectively.

0 20 40 60 80 100 120

-10

0

10

Time (sec)

Roll (degrees)

0 20 40 60 80

-16

-12

-8

-4

0

4

8

12

16

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Roll(degrees)

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10

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Pitch (degrees)

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-12

-8

-4

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8

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16

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Roll(degrees)

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20

Time (sec)

Yaw (degrees)

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-30

-20

-10

0

10

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30

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Yaw(degrees)

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10

20

30

40

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Altitude z (cm)

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10

20

30

40

50

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Altitude z(cm)

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2

3

4

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6

7

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Trust U1 (N)

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0

1

2

3

4

5

6

7

8

Time(sec)

Trust U1(N)

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100

120

140

160

180

190

200

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Propellers speeds (rd/s)

0 5 10 15 20

100

120

140

160

180

200

220

Time (sec)

Propellers speeds (rd/s)

rotor1

rotor2

rotor3

rotor4

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20

25

30

35

40

45

50

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Altitude z (cm)

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20

30

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50

Time(sec)

Altitude z(cm)

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5.5

6

6.5

7

7.5

Time (sec)

Trust U1(N )

0 5 10 15 20 25 30

6

6.5

7

7.5

8

8.5

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Trust U1(N)

Integral Sliding Mode and Second Order Sliding Mode Attitude and Altitude Tracking of a Quadrotor System - Theory and

Experiment

271

Figure 12: Desired Altitude (Red) and real Altitude (Blue):

(left) ISM, (right) SOSM.

Figure 13: Control inputs: (left) ISM, (right) SOSM.

These results show that both approaches ensure the

tracking of the cycloidal reference trajectory.

However the SOSM approach shows better tracking

performance than the ISM approach.

5 CONCLUSIONS

This paper presents the real time implementation of

two robust controllers on a realized embedded

control system for the stabilization and the tracking

of the quadrotor system. The embedded control

system is based on a dsPIC C. A 3DM-GX1 IMU,

SRF08 sonar and Hall Effect sensors with a little

magnet are used to measure the attitude, the altitude

and the propellers speeds of the quadrotor

respectively. The robust approaches used are the

integral sliding mode with a boundary layer method

and the second order sliding mode. The

experimental results obtained demonstrate the

superiority of the SOSM controller comparing with

ISM controller in term of performance (accuracy,

settling time and overshoot) for the case of

stabilization and tracking and robustness to external

disturbances while cancelling the chattering

phenomena. These results validate theoretical results

and confirm that the SOSM keep the invariance

property in term of performance while reducing the

effect of the chattering which is not the case of the

sliding mode (with a boundary layer method) even

with additional integral term.

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