Research ArticleReal-Time Hovering Control of Unmanned Aerial Vehicles

Cuauhtemoc Acosta Lua 12 Claudia Carolina Vaca Garcıa 1 Stefano Di Gennaro 23

B Castillo-Toledo 24 and Marıa Eugenia Sanchez Morales 1

1Technological Sciences Department La Cienega University Center University of Guadalajara Av Universidad 1115 OcotlanJalisco CP 47820 Mexico2Center of Excellence DEWS (Design Methodologies of Embedded Controllers Wireless Interconnect and Systems-on-chip)University of LrsquoAquila Via Vetoio Loc Coppito LrsquoAquila 67100 Italy3Department of Information Engineering Computer Science and Mathematics University of LrsquoAquila Via Vetoio Loc CoppitoLrsquoAquila 67100 Italy4Center for Research and Advanced Studies Campus Guadalajara Av del Bosque 1145 Col El BajıoZapopan CP 45019 Mexico

Correspondence should be addressed to Cuauhtemoc Acosta Lua cuauhtemocacostacuciudgmx

Received 24 April 2020 Accepted 4 June 2020 Published 27 July 2020

Guest Editor Yi Qi

Copyright copy 2020 Cuauhtemoc Acosta Lua et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the original work is properly cited

In this paper the design of a controller for the altitude and rotational dynamics is presented In particular the control problem isto maintain a desired altitude in a fixed position e unmanned aerial vehicle dynamics are described by nonlinear equationsderived using the NewtonndashEuler approache control problem is solved imposing the stability of the error dynamics with respectto desired position and angular references e performance and effectiveness of the proposed control are tested first vianumerical simulations using the Pixhawk Pilot Support Package simulator provided by Mathworks en the controller is testedvia a real-time implementation using a quadrotor Aircraft F-450

1 Introduction

Quadrotors have recently attracted the attention of manyresearchers due to their interesting applications As a matterof fact the potential applications of such devices arecountless Examples of such applications include searchingand surveillance monitoring and rescuing tasks From amethodological point of view the interest relies on the factthat a quadrotor is a complex underactuated system withhigh nonlinearities and strong dynamical couplings Fur-thermore it is affected by aerodynamic disturbancesunmodeled dynamics and parametric uncertaintieserefore the quadrotors represent an interesting testbedfor testing new control techniques

ere are a large number of works dealing with quad-rotors As far as the mathematical model is concerned inBouabdallah et al [1] Zeng and Zhao [2] and Nagaty et al[3] a NewtonndashEuler model was presented FurthermoreMagnussen et al [4] and Valenti et al [5] consideredquaternions to describe the angular kinematics whilst

Antonio-Toledo et al [6] applied the EulerndashLagrangeequations to obtain the whole quadrotor mathematicalmodel Regarding the control of quadrotors many controltechniques have been proposed In Panomruttanarug et al[7] and Pounds et al [8] the linear quadratic regulatorcontrol and the proportional integral derivative controlrespectively were exploited to design a control law How-ever these controllers ensure only local stability In order toenlarge the basin of attraction nonlinear control techniqueshave also been considered Examples are sliding modelcontrol LuquendashVega et al [9] backstepping Bouabdallahand Siegwart [10] and adaptive control Matouk et al [11]Moreover a global fast dynamic terminal sliding modecontrol method was proposed for position and attitudetracking control in Xiong and Zhang [12] An adaptivecommand filtered backstepping control law was designed fortrajectory tracking in Choi and Ahn [13] In Liu et al [14] arobust adaptive attitude tracking control for a quadrotorwith an unknown inertia matrix and bounded externaldisturbances was proposed A command filtered

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 2314356 8 pageshttpsdoiorg10115520202314356

implementation of an adaptive backstepping was proposedin Dong et al [15] and the stability of the closed loop systemwas proved via the Lyapunov direct method Finally in Islamet al [16] an observer-based adaptive fuzzy backsteppingcontroller was designed for trajectory tracking in the case ofa quadrotor undergoing wind gusts and with parametricuncertainties All these aforementioned methods providegood dynamic performance and robust stability and aretested mainly considering numerical simulations

e main contribution of this paper is the design of acontroller for the attitude and altitude of a quadrotor he-licopter is controller has been designed using thebackstepping technique and has been tested using numericalsimulations and real-time experimentation In particularthe focus of this paper is to obtain a controller ensuringhovering so that

(1) e reference attitude is zero for the Euler anglesdescribing the quadrotor angular position

(2) e reference altitude is a constant value

e performance and effectiveness of the proposedcontroller has been first tested with numerical simulationsusing the Pixhawk Pilot Support Package (PSP) en thereal-time implementation has been performed implement-ing the proposed controller on a real F-450 quadrotorequipped of a Pixhawk and tested under environmentalperturbations

e paper is organized as follows Section 2 introducesthe description and the mathematical model of thequadrotor In Section 3 the control problem is solved InSection 4 numerical simulations and real-time tests areprovided to show the effectiveness of the proposed con-troller Finally some concluding remarks are commentedin Section 5

2 Mathematical Model

e quadrotor considered in this work consists of a rigidframe equipped with four rotors e rotors generate thepropeller force Fi bω2

pi proportional to the propellerangular velocity ωpi i 1 2 3 4 e propellers 1 and 3rotate counterclockwise and the propellers 2 and 4 rotateclockwise

Denote RC(O e1 e2 e3) and RΓ(Ω ε1 ε2 ε3) as theframes fixed with the Earth and the quadrotor respectivelywith Ω coincident with the center of mass of the quadrotor(see Figure 1) e quadrotor absolute position in RC isdescribed by p (x y z)T whereas its attitude is describedby the Euler angles α (ϕ θψ)T where ϕ θ

ψ isin (minusπ2n qπh2) are the pitch roll and yaw angles re-spectively e sequence 3ndash2ndash1 has been considered byHuges [17] Moreover v (v1 v2 v3)

T and ω

(ω1ω2ω3)T are the linear and angular velocities of the

center of mass of the quadrotor expressed in RC and in RΓrespectively

e translation dynamics (in RC) and rotation dynamics(in the RΓ) of the quadrotor are

_p v

_v 1mR(α)Fprop +

1m

Fgrav +1m

Fd

_α M(α)ω

_ω Jminus1

minus1113957ωJω + τprop minus τgyro + Md1113872 1113873

(1)

where m is the mass of the quadrotor J is the inertia matrixof the quadrotor and J diag Jx Jy Jz1113966 1113967 (expressed in RΓ)and

1113957ω

0 minusω3 ω2

ω3 0 minusω1

minusω2 ω1 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (2)

is the so-called dyadic representation of ω

Fprop

0

0

11139364

i1Fi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

τprop

τ1τ2τ3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

l F2 minus F4( 1113857

l F3 minus F1( 1113857

c F1 minus F2 + F3 minus F4( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3)

are the forces and moments produced by the propellers(inputs) with l as the distance between the center of mass tothe rotor shaft Moreover Fgrav 0 0 minusmg( 1113857

T is the forcedue to the gravity expressed in RC

e vectors expressed in RΓ are transformed into vectorsin RC by the rotation matrix

e3

e2

e1

RC

O

mg

RГ

3

1

2

Ωωp3

ωp4

ωp2

ωp1ϕ

ψ

θ

F4

F3F2

z y

x

F1

Figure 1 Quadrotor orientation using Euler angles

2 Mathematical Problems in Engineering

R(α)

cθcψ sϕsθcψ minus cϕsψ cϕsθcψ + sϕsψ

cθsψ sϕsθsψ + cϕcψ cϕsθsψ minus sϕcψ

minussθ sϕcθ cϕcθ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (4)

with cc cos(c) sc sin(c) and c ϕ θψ e angularvelocity dynamics are expressed using the matrix

M(α)

1 sϕtgθ cϕtgθ

0 cϕ minussϕ

0 sϕscθ cϕscθ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (5)

with tgc tan(c) and scc sec(c)e rolling torque τ1 is produced by the forces F2 and F4

Similarly the pitching torque τ2 is produced by the forces F1and F3 Due to Newtonrsquos third law the propellers produce ayawing torque τ3 on the body of the quadrotor in theopposite direction of the propeller rotation Moreover

τgyro 1113944

4

i1(minus1)

i+1Jpωp 1113957ωε3 (6)

is the gyroscopic torque due to the propeller rotations withJp the propeller moment of inertia with respect to its ro-tation axis Finally Fd andMd are the forces and torques dueto the external disturbances which are assumed negligiblehere

Under the assumption of small angles matrix M(α)

reduces to the identity matrixis assumption is justified bythe fact that the control objective is to maintain thequadrotor in an hover position Nagaty et al [3]is leads toa simplified mathematical model of the quadrotor given by

eurox 1m

cϕsθcψ + sϕsψ1113872 1113873u1

euroy 1m

cϕsθsψ minus sϕcψ1113872 1113873u1

euroz 1m

cϕcθu1 minus g

euroϕ Jy minus Jz

Jx

_θ _ψ minusJp

Jx

ωp_θ +

l

Jx

τ1

euroθ Jz minus Jx

Jy

_ϕ _ψ +Jp

Jy

ωp_ϕ +

l

Jy

τ2

euroψ Jx minus Jy

Jz

_ϕ _θ +1Jz

τ3

(7)

where the control variables are defined as

u1 11139444

i1Fi

τ1 l F2 minus F4( 1113857

τ2 l F3 minus F1( 1113857

τ3 c F1 + F3 minus F2 minus F4( 1113857

(8)

where l is the distance of the center of mass to the rotor shaftc is the drag factor and ωp ωp1 minus ωp2 + ωp3 minus ωp4 is theso-called rotor relative speede parameters used in (7) andtheir values are defined in Table 1

3 Control Design

31 Attitude Control To design the attitude control law letus define the following variables

ϕ ϕ1

θ θ1

ψ ψ1

_ϕ ϕ2_θ θ2_ψ ψ2

(9)

so that system (7) can be rewritten as_ϕ1 ϕ2

_ϕ2 Jy minus Jz

Jx

θ2ψ2 minusJp

Jx

ωpθ2 +l

Jx

τ1

_θ1 θ2

_θ2 Jz minus Jx

Jy

ϕ2ψ2 +Jp

Jy

ωpϕ2 +l

Jy

τ2

_ψ1 ψ2

_ψ2 Jx minus Jy

Jz

ϕ2θ2 +1Jz

τ3

(10)

where X (ϕ1 ϕ2 θ1 θ2ψ1ψ2) are the measured signalse control objective is to match the UAV attitude X to adesired reference

Xref ϕ1ref ϕ2ref θ1ref θ2ref ψ1ref ψ2ref1113872 1113873 (11)

To this aim let us define the tracking errors as

eϕ1 ϕ1 minus ϕ1ref

eϕ2 ϕ2 minus ϕ2ref + k1eϕ1

eθ1 θ1 minus θ1ref

eθ2 θ2 minus θ2ref + k3eθ1

eψ1 ψ1 minus ψ1ref

eψ2 ψ2 minus ψ2ref + k5eψ1

(12)

with k1 k3 k5 gt 0 Note that ej1 0 implies ej2 0j ϕ θψ Deriving the error system (12) with respect to(10) is possible to calculate the error dynamics

Mathematical Problems in Engineering 3

_eϕ1 eϕ2 minus k1eϕ1 minus _ϕ1ref + ϕ2ref

_eϕ2 Jy minus Jz

Jx

θ2ψ2 minusJp

Jx

ωpθ2 +l

Jx

τ1

minus _ϕ2ref + k1 eϕ2 minus k1eϕ1 + ϕ2ref minus _ϕ1ref1113872 1113873

_eθ1 eθ2 minus k3eθ1 minus _θ1ref + θ2ref eθ2 minus k3eθ1

minus _θ1ref + θ2ref

_eθ2 Jz minus Jx

Jy

ϕ2ψ2 +Jp

Jy

ωpϕ2 +l

Jy

τ2

minus _θ2ref + k3 eθ2 minus k3eθ1 + θ2ref minus _θ1ref1113872 1113873

_eψ1 eψ2 minus k5eψ2 minus _ψ1ref + ψ2ref

_eψ2 Jx minus Jy

Jz

ϕ2θ2 +1Jz

τ3 minus _ψ2ref

+ k5 eψ2 minus k5eψ1 + ψ2ref minus _ψ1ref1113872 1113873

(13)

Using the following control law [18]

τ1 Jx

lminus

Jy minus Jz

Jx

θ2ψ2 +Jp

Jx

ωpθ2 + _ϕ2ref minus k1 eϕ2 minus k1eϕ111138721113888

+ ϕ2ref minus _ϕ1ref1113873 minus k2eϕ21113889

τ2 Jy

lminus

Jz minus Jx

Jy

ϕ2ψ2 minusJp

Jy

ωpϕ2 + _θ2ref minus k3 eθ2 minus k3eθ111138721113888

+ θ2ref minus _θ1ref1113873 minus k4eθ21113889

τ3 Jz minusJx minus Jy

Jz

ϕ2θ2 + _ψ2ref minus k5 eψ2 minus k5eψ1 + ψ2ref11138721113888

minus _ψ1ref1113873 minus k6eψ21113889

(14)

with gains ki gt 0 i 2 4 6 in (13) the tracking error dy-namics become

_eX

_eϕ1

_eϕ2

_eθ1

_eθ2

_eψ1

_eψ2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusk1 1 0 0 0 0

0 minusk2 0 0 0 0

0 0 minusk3 1 0 0

0 0 0 minusk4 0 0

0 0 0 0 minusk5 1

0 0 0 0 0 minusk6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

eϕ1

eϕ2

eθ1

eθ2

eψ1

eψ2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

AXeX

(15)

where eX eϕ1 eϕ2 eθ1 eθ2 eψ1eψ21113872 1113873

T which converge

exponentially to zero In fact considering the Lyapunovcandidate

VX t eX( 1113857 12e

TXPXeX (16)

and differentiating along the error dynamics (15) oneobtains

_VX t eX( 1113857 eTX A

TXPX + PXAX1113872 1113873eX minuse

TXQXeX

le minus λQX

min eX

2 le minus αVX t eX( 1113857 α

2λQX

min

λQX

min

gt 0

(17)

with PX solution of the equation ATXPX + PXAX minusQX for a

fixed matrix QX QTX gt 0 erefore

VX t eX( 1113857le eminusαt

V 0 eX(0)( 1113857 (18)

so that the tracking error converges globally exponentially tozero [19]

32AltitudeControl For the altitude control let us considerthe altitude dynamics

_z1 z2

_z2 g + cϕ1cθ11m

u1

(19)

e control problem is to maintain the quadrotor at adesired constant altitude z1ref z2ref 0 e tracking errorsare defined as

Table 1 Quadrotor parameters

m Mass of the airframe 11 kgl Distance of the center of mass to the rotor shaft 0223mJx Inertia in the x-axis 6825times 10minus3 kgm2

Jy Inertia in the y-axis 6825times 10minus3 kgm2

Jz Inertia in the z-axis 1239times 10minus3 kgm2

Jp Inertia of the propellers 6times 10minus5 kgm2

g Gravity acceleration 981ms2b rust factor 542times 10minus6

c Drag factor 11times 10minus6 N s2 radminus2

ϕ Roll angle degθ Pitch angle degψ Yaw angle degz1 z-position m

4 Mathematical Problems in Engineering

ez1 z1 minus z1ref

ez2 z2 minus z2ref (20)

Choosing

u1 m

cϕ1cθ1g + _z2ref minus kz1ez1 minus kz2ez21113872 1113873 (21)

with kz1 kz2 gt 0 we obtain_ez Azez (22)

where ez (ez1 ez2)T and

Az 0 1

minuskz1 minuskz21113888 1113889 (23)

e stability proof is similar to that of the previoussection

4 Simulation and Experimental Results

In this section the performance of the attitude and altitudecontrollers (14) and (21) is tested considering an F-450quadrotor First numerical simulations are carried outen real-time experimental tests are performed showingthe effectiveness of the control design

41 Simulation Results For the numerical simulations thequadrotor model provided by the Pixhawk PSP in Simulinkhas been used is model contains the attitude and altitudeflight control model called px4 demo attitude control whichshows a good performance in predicting the dynamicquadrotor behavior very close to the real drone dynamics

Controllers (14) and (21) use the nominal values ofTable 1 and the gains of Table 2 e simulations have beenperformed in two steps In the first one the quadrotor isstabilized in altitude In Figure 2 the altitude z is shown Inthe second step the quadrotor is stabilized in attitude epitch roll and yaw angles are shown in Figure 3 e initialconditions considered are z(0) 0 ϕ(0) minus573degθ(0) 573deg and ψ(0) 573deg e reference values arezref 2 m and ϕref θref ψref 0

42 Experimental Results In this section we describe thephysical setting of the embedded control that allows stabilizingthe quadrotor An embedded control system generally consistsof three elements sensors actuators and a microcontrolleremicrocontroller interacts with the continuous dynamics ofthe plant via the sensors and actuators and itsmajor function isto compute and generate control commands for the actuatorsthat are based on sensor measurements e onboard elec-tronic system consists of a flight controller a hardware setupand some ultrasonic sensors

421 Flight Controller e 3DR-PIXHAWK is a high-performance autopilot-on-module suitable for fixed wingmulti rotors helicopters cars boats and other mobile ro-botic platforms Its processor can run to 168MHz252 MIPS

Cortex-M4F with 256KB in RAM and 2MB flash and has 14PWMServo outputs and abundant connectivity options foradditional peripherals such as 5x UART (serial ports) onehigh-power capable 2x with HW flow control 2x CANSpektrum DSMDSM2DSMndashXreg Satellite compatible inputPPM sum signal input RSSI (PWM or voltage) input I2CSPI 33 and 66V ADC input Moreover the 3DR-Pixhawkhas different sensors such as ST Micro L3GD20H 16 bitgyroscopes ST Micro LSM303D 14 bit accelerometersmagnetometers Invensense MPU 6000 3-axis accelerome-tersgyroscopes and MEAS MS5611 barometers

422 Hardware Setup e structure of the quadrotor iscomposed by an F-450 frame with integrated PCB wiringwhereas the rotors are brushless motors manufactured byEndashmax with 935 rpmV and a 10times 45Prime propellers Turnigyspeed drivers (ESC) are BHC type at 18A max e batteryused in this setup is a 3S 2800mAh 25Ce radio-transmitteris a Turnigy with 9PPM channels working at 24GHz

An LV-MaxSonar-EZ4 sensor with a resolution of254 cm 20Hz reading rate 42 kHz ultrasonic sensormeasures a maximum Range of 645 cm operating in therange of 25ndash55 VDC was used to measure the altitude

For the real-time running the same parameter valuesand gains shown in Table 2 were used e initial conditionfor the altitude was z(0) 216m For the pitch roll andyaw angles the initial condition were chosen at ϕ1(0) 16degθ1(0) 09deg and ψ1(0) minus16deg Figures 4 and 5 show thedynamic behavior of the altitude and the pitch roll and yawangles respectively It is worth noticing that these

Table 2 Gain values

kp1 46 kd1 35 ki1 28kp2 43 kd2 9 ki2 02kp3 45 kd3 7 ki3 02kp4 7 kd4 19 ki4 01

0 1 2 3 4 5Time (s)

z 1 (m

)

6 7 8 9 101

15

2

25

Figure 2 Numerical simulation quadrotorrsquos altitude z1 (solid) andaltitude reference z1ref (dash)

Mathematical Problems in Engineering 5

15105Time (s)

0ndash6

ndash4ϕ 1 (d

egre

e)

ndash2

0

2

(a)

151050ndash2

0

2

4

6

θ 1 (d

egre

e)

Time (s)

(b)

1510

Time (s)

50ndash2

0

2

4

6

ψ 1 (d

egre

e)

(c)

Figure 3 Numerical simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (b) pitch angle θ1 (solid) and pitch anglereference θ1ref (dash) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

15 20 25 30 35 40 45Time (s)

z 1 (m

)

50 55 60 65 701

15

2

25

Figure 4 Experimental implementation quadrotorrsquos altitude z1 (solid) and altitude reference z1ref (dash)

6 Mathematical Problems in Engineering

experimental tests were performed in an outdoor envi-ronment without using the GPS sensor

5 Conclusions

In this paper a controller based on the stabilization tech-nique for the altitude and attitude error has been proposedfor a quadrotor e simulation results have been performedusing the Pixhawk PSP en the controller has beenimplemented on a laboratory quadrotor e simulation andexperimental results show a good performance even inoutdoor environment showing some degree of robustness inthe presence of environmental disturbances

Data Availability

e figures tables and other data used to support this studyare included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is paper has been partially supported by the EuropeanProject ECSELndashJU RIAndash2018 ldquoComp4Dronesrdquo and project

ldquoCoordination of autonomous unmanned vehicles for highlycomplex performancesrdquo Executive Program of Scientific andTechnological Agreement between Italy (Ministry of ForeignAffairs and International Cooperation Italy) and Mexico(Mexican International Cooperation Agency for the De-velopment) SAAP3

References

[1] S Bouabdallah P Murrieri and R Siegwart ldquoDesign andcontrol of an indoor Micro quadrotorrdquo in Proceedings of theIEEE International Conference on Robotics and Automationpp 1ndash6 New Orleans LA USA April-May 2004

[2] Y Zeng and L Zhao ldquoParameter identification for unmannedfourndashrotor helicopter with nonlinear modelrdquo in Proceedings ofthe 2014 IEEE Chinese Guidance Navigation and ControlConference pp 922ndash926 Yantai China August 2014

[3] A Nagaty S Saeedi C ibault M Seto and H Li ldquoControland navigation framework for quadrotor helicoptersrdquo Journalof Intelligent amp Robotic Systems vol 70 no 1ndash4 pp 1ndash122013

[4] O Magnussen M Ottestad and G Hovland ldquoExperimentalvalidation of a quaternionndashbased attitude estimation withdirect input to a quadcopter control systemrdquo in Proceedings ofthe International Conference on Unmanned Aircraft Systems(ICUAS) pp 48ndash485 Atlanta GA USA May 2013

[5] R Valenti I Dryanovski and J Xiao ldquoKeeping a good at-titude a quaternion-based orientation filter for IMUs andMARGsrdquo Sensors vol 15 no 8 pp 19302ndash19330 2015

ndash10

0

10

15 20 25 30 35 40 45Time (s)

ϕ 1 (d

egre

e)

50 55 60 65 70

(a)

ndash10

0

10

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

θ 1 (d

egre

e)

(b)

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

ndash10

0

10

ψ 1 (d

egre

e)

(c)

Figure 5 Experimental simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (degree vs s) (b) pitch angle θ1 (solid) andpitch angle reference θ1ref (dash) (degree vs s) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

Mathematical Problems in Engineering 7

[6] M Elena AntoniondashToledo A Y Alanis and E N SanchezldquoRobust neural decentralized control for a quadrotor UAVrdquoin Proceedings of the International Joint Conference on NeuralNetworks (IJCNN) pp 714ndash719 Vancouver Canada July2016

[7] B Panomruttanarug K Higuchi and F Mora-CaminoldquoAttitude control of a quadrotor aircraft using LQR statefeedback controller with full order state observerrdquo in Pro-ceeding of the International Conference on InstrumentationControl and Information Technology (SICE) pp 2041ndash2046Konya Turkey June 2013

[8] P E I Pounds D R Bersak and A M Dollar ldquoStability ofsmall-scale UAV helicopters and quadrotors with addedpayload mass under PID controlrdquo Autonomous Robotsvol 33 no 1-2 pp 129ndash142 2012

[9] L LuquendashVega B CastillondashToledo and A G LoukianovldquoRobust block second order sliding mode control for aquadrotorrdquo Journal of the Franklin Institute vol 349 no 2pp 719ndash739 2012

[10] S Bouabdallah and R Siegwart ldquoBackstepping and sliding-mode techniques applied to an indoor Micro quadrotorrdquo inProceedings of the 2005 IEEE International Conference onRobotics and Automation Barcelona Spain April 2005

[11] D Matouk O Gherouat F Abdessemed and A HassamldquoQuadrotor position and attitude control via backsteppingapproachrdquo in Proceedings of the 8th International Conferenceon Modelling Identification and Control (ICMI) pp 73ndash79Algiers Algeria November 2016

[12] J-J Xiong and G-B Zhang ldquoGlobal fast dynamic terminalsliding mode control for a quadrotor UAVrdquo ISA Transactionsvol 66 pp 233ndash240 2017

[13] Y C Choi and H S Ahn ldquoNonlinear control of quadrotor forpoint tracking actual implementation and experimentaltestsrdquo IEEEASME Transactions on Mechatronics vol 20no 3 pp 1179ndash1192 2015

[14] Y-C Liu J Zhang T Zhang and J-Y Song ldquoRobust adaptivespacecraft attitude tracking control based on similar skew-symmetric structurerdquo Computers amp Electrical Engineeringvol 56 pp 784ndash794 2016

[15] W Dong J A Farrell M M Polycarpou V Djapic andM Sharma ldquoCommand filtered adaptive backsteppingrdquo IEEETransactions on Control Systems Technology vol 20 no 3pp 566ndash580 2012

[16] S Islam J Dias and L D Seneviratne ldquoAdaptive outputfeedback control for miniature unmanned aerial vehiclerdquo inProceedings IEEE International Conference on Advanced In-telligent Mechatronics AIM pp 318ndash322 Banff Canada July2016

[17] P C Hughes Spacecraft Attitude Dynamics Dover Publica-tions Inc Mineola NY USA 1986

[18] A IsidoriNonlinear Control Systems Springer-Verlag BerlinGermany 3rd edition 1995

[19] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 2002

8 Mathematical Problems in Engineering

R(α)

cθcψ sϕsθcψ minus cϕsψ cϕsθcψ + sϕsψ

cθsψ sϕsθsψ + cϕcψ cϕsθsψ minus sϕcψ

minussθ sϕcθ cϕcθ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (4)

with cc cos(c) sc sin(c) and c ϕ θψ e angularvelocity dynamics are expressed using the matrix

M(α)

1 sϕtgθ cϕtgθ

0 cϕ minussϕ

0 sϕscθ cϕscθ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (5)

with tgc tan(c) and scc sec(c)e rolling torque τ1 is produced by the forces F2 and F4

Similarly the pitching torque τ2 is produced by the forces F1and F3 Due to Newtonrsquos third law the propellers produce ayawing torque τ3 on the body of the quadrotor in theopposite direction of the propeller rotation Moreover

τgyro 1113944

4

i1(minus1)

i+1Jpωp 1113957ωε3 (6)

is the gyroscopic torque due to the propeller rotations withJp the propeller moment of inertia with respect to its ro-tation axis Finally Fd andMd are the forces and torques dueto the external disturbances which are assumed negligiblehere

Under the assumption of small angles matrix M(α)

reduces to the identity matrixis assumption is justified bythe fact that the control objective is to maintain thequadrotor in an hover position Nagaty et al [3]is leads toa simplified mathematical model of the quadrotor given by

eurox 1m

cϕsθcψ + sϕsψ1113872 1113873u1

euroy 1m

cϕsθsψ minus sϕcψ1113872 1113873u1

euroz 1m

cϕcθu1 minus g

euroϕ Jy minus Jz

Jx

_θ _ψ minusJp

Jx

ωp_θ +

l

Jx

τ1

euroθ Jz minus Jx

Jy

_ϕ _ψ +Jp

Jy

ωp_ϕ +

l

Jy

τ2

euroψ Jx minus Jy

Jz

_ϕ _θ +1Jz

τ3

(7)

where the control variables are defined as

u1 11139444

i1Fi

τ1 l F2 minus F4( 1113857

τ2 l F3 minus F1( 1113857

τ3 c F1 + F3 minus F2 minus F4( 1113857

(8)

where l is the distance of the center of mass to the rotor shaftc is the drag factor and ωp ωp1 minus ωp2 + ωp3 minus ωp4 is theso-called rotor relative speede parameters used in (7) andtheir values are defined in Table 1

3 Control Design

31 Attitude Control To design the attitude control law letus define the following variables

ϕ ϕ1

θ θ1

ψ ψ1

_ϕ ϕ2_θ θ2_ψ ψ2

(9)

so that system (7) can be rewritten as_ϕ1 ϕ2

_ϕ2 Jy minus Jz

Jx

θ2ψ2 minusJp

Jx

ωpθ2 +l

Jx

τ1

_θ1 θ2

_θ2 Jz minus Jx

Jy

ϕ2ψ2 +Jp

Jy

ωpϕ2 +l

Jy

τ2

_ψ1 ψ2

_ψ2 Jx minus Jy

Jz

ϕ2θ2 +1Jz

τ3

(10)

where X (ϕ1 ϕ2 θ1 θ2ψ1ψ2) are the measured signalse control objective is to match the UAV attitude X to adesired reference

Xref ϕ1ref ϕ2ref θ1ref θ2ref ψ1ref ψ2ref1113872 1113873 (11)

To this aim let us define the tracking errors as

eϕ1 ϕ1 minus ϕ1ref

eϕ2 ϕ2 minus ϕ2ref + k1eϕ1

eθ1 θ1 minus θ1ref

eθ2 θ2 minus θ2ref + k3eθ1

eψ1 ψ1 minus ψ1ref

eψ2 ψ2 minus ψ2ref + k5eψ1

(12)

with k1 k3 k5 gt 0 Note that ej1 0 implies ej2 0j ϕ θψ Deriving the error system (12) with respect to(10) is possible to calculate the error dynamics

Mathematical Problems in Engineering 3

_eϕ1 eϕ2 minus k1eϕ1 minus _ϕ1ref + ϕ2ref

_eϕ2 Jy minus Jz

Jx

θ2ψ2 minusJp

Jx

ωpθ2 +l

Jx

τ1

minus _ϕ2ref + k1 eϕ2 minus k1eϕ1 + ϕ2ref minus _ϕ1ref1113872 1113873

_eθ1 eθ2 minus k3eθ1 minus _θ1ref + θ2ref eθ2 minus k3eθ1

minus _θ1ref + θ2ref

_eθ2 Jz minus Jx

Jy

ϕ2ψ2 +Jp

Jy

ωpϕ2 +l

Jy

τ2

minus _θ2ref + k3 eθ2 minus k3eθ1 + θ2ref minus _θ1ref1113872 1113873

_eψ1 eψ2 minus k5eψ2 minus _ψ1ref + ψ2ref

_eψ2 Jx minus Jy

Jz

ϕ2θ2 +1Jz

τ3 minus _ψ2ref

+ k5 eψ2 minus k5eψ1 + ψ2ref minus _ψ1ref1113872 1113873

(13)

Using the following control law [18]

τ1 Jx

lminus

Jy minus Jz

Jx

θ2ψ2 +Jp

Jx

ωpθ2 + _ϕ2ref minus k1 eϕ2 minus k1eϕ111138721113888

+ ϕ2ref minus _ϕ1ref1113873 minus k2eϕ21113889

τ2 Jy

lminus

Jz minus Jx

Jy

ϕ2ψ2 minusJp

Jy

ωpϕ2 + _θ2ref minus k3 eθ2 minus k3eθ111138721113888

+ θ2ref minus _θ1ref1113873 minus k4eθ21113889

τ3 Jz minusJx minus Jy

Jz

ϕ2θ2 + _ψ2ref minus k5 eψ2 minus k5eψ1 + ψ2ref11138721113888

minus _ψ1ref1113873 minus k6eψ21113889

(14)

with gains ki gt 0 i 2 4 6 in (13) the tracking error dy-namics become

_eX

_eϕ1

_eϕ2

_eθ1

_eθ2

_eψ1

_eψ2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusk1 1 0 0 0 0

0 minusk2 0 0 0 0

0 0 minusk3 1 0 0

0 0 0 minusk4 0 0

0 0 0 0 minusk5 1

0 0 0 0 0 minusk6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

eϕ1

eϕ2

eθ1

eθ2

eψ1

eψ2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

AXeX

(15)

where eX eϕ1 eϕ2 eθ1 eθ2 eψ1eψ21113872 1113873

T which converge

exponentially to zero In fact considering the Lyapunovcandidate

VX t eX( 1113857 12e

TXPXeX (16)

and differentiating along the error dynamics (15) oneobtains

_VX t eX( 1113857 eTX A

TXPX + PXAX1113872 1113873eX minuse

TXQXeX

le minus λQX

min eX

2 le minus αVX t eX( 1113857 α

2λQX

min

λQX

min

gt 0

(17)

with PX solution of the equation ATXPX + PXAX minusQX for a

fixed matrix QX QTX gt 0 erefore

VX t eX( 1113857le eminusαt

V 0 eX(0)( 1113857 (18)

so that the tracking error converges globally exponentially tozero [19]

32AltitudeControl For the altitude control let us considerthe altitude dynamics

_z1 z2

_z2 g + cϕ1cθ11m

u1

(19)

e control problem is to maintain the quadrotor at adesired constant altitude z1ref z2ref 0 e tracking errorsare defined as

Table 1 Quadrotor parameters

m Mass of the airframe 11 kgl Distance of the center of mass to the rotor shaft 0223mJx Inertia in the x-axis 6825times 10minus3 kgm2

Jy Inertia in the y-axis 6825times 10minus3 kgm2

Jz Inertia in the z-axis 1239times 10minus3 kgm2

Jp Inertia of the propellers 6times 10minus5 kgm2

g Gravity acceleration 981ms2b rust factor 542times 10minus6

c Drag factor 11times 10minus6 N s2 radminus2

ϕ Roll angle degθ Pitch angle degψ Yaw angle degz1 z-position m

4 Mathematical Problems in Engineering

ez1 z1 minus z1ref

ez2 z2 minus z2ref (20)

Choosing

u1 m

cϕ1cθ1g + _z2ref minus kz1ez1 minus kz2ez21113872 1113873 (21)

with kz1 kz2 gt 0 we obtain_ez Azez (22)

where ez (ez1 ez2)T and

Az 0 1

minuskz1 minuskz21113888 1113889 (23)

e stability proof is similar to that of the previoussection

4 Simulation and Experimental Results

In this section the performance of the attitude and altitudecontrollers (14) and (21) is tested considering an F-450quadrotor First numerical simulations are carried outen real-time experimental tests are performed showingthe effectiveness of the control design

41 Simulation Results For the numerical simulations thequadrotor model provided by the Pixhawk PSP in Simulinkhas been used is model contains the attitude and altitudeflight control model called px4 demo attitude control whichshows a good performance in predicting the dynamicquadrotor behavior very close to the real drone dynamics

Controllers (14) and (21) use the nominal values ofTable 1 and the gains of Table 2 e simulations have beenperformed in two steps In the first one the quadrotor isstabilized in altitude In Figure 2 the altitude z is shown Inthe second step the quadrotor is stabilized in attitude epitch roll and yaw angles are shown in Figure 3 e initialconditions considered are z(0) 0 ϕ(0) minus573degθ(0) 573deg and ψ(0) 573deg e reference values arezref 2 m and ϕref θref ψref 0

42 Experimental Results In this section we describe thephysical setting of the embedded control that allows stabilizingthe quadrotor An embedded control system generally consistsof three elements sensors actuators and a microcontrolleremicrocontroller interacts with the continuous dynamics ofthe plant via the sensors and actuators and itsmajor function isto compute and generate control commands for the actuatorsthat are based on sensor measurements e onboard elec-tronic system consists of a flight controller a hardware setupand some ultrasonic sensors

421 Flight Controller e 3DR-PIXHAWK is a high-performance autopilot-on-module suitable for fixed wingmulti rotors helicopters cars boats and other mobile ro-botic platforms Its processor can run to 168MHz252 MIPS

Cortex-M4F with 256KB in RAM and 2MB flash and has 14PWMServo outputs and abundant connectivity options foradditional peripherals such as 5x UART (serial ports) onehigh-power capable 2x with HW flow control 2x CANSpektrum DSMDSM2DSMndashXreg Satellite compatible inputPPM sum signal input RSSI (PWM or voltage) input I2CSPI 33 and 66V ADC input Moreover the 3DR-Pixhawkhas different sensors such as ST Micro L3GD20H 16 bitgyroscopes ST Micro LSM303D 14 bit accelerometersmagnetometers Invensense MPU 6000 3-axis accelerome-tersgyroscopes and MEAS MS5611 barometers

422 Hardware Setup e structure of the quadrotor iscomposed by an F-450 frame with integrated PCB wiringwhereas the rotors are brushless motors manufactured byEndashmax with 935 rpmV and a 10times 45Prime propellers Turnigyspeed drivers (ESC) are BHC type at 18A max e batteryused in this setup is a 3S 2800mAh 25Ce radio-transmitteris a Turnigy with 9PPM channels working at 24GHz

An LV-MaxSonar-EZ4 sensor with a resolution of254 cm 20Hz reading rate 42 kHz ultrasonic sensormeasures a maximum Range of 645 cm operating in therange of 25ndash55 VDC was used to measure the altitude

For the real-time running the same parameter valuesand gains shown in Table 2 were used e initial conditionfor the altitude was z(0) 216m For the pitch roll andyaw angles the initial condition were chosen at ϕ1(0) 16degθ1(0) 09deg and ψ1(0) minus16deg Figures 4 and 5 show thedynamic behavior of the altitude and the pitch roll and yawangles respectively It is worth noticing that these

Table 2 Gain values

kp1 46 kd1 35 ki1 28kp2 43 kd2 9 ki2 02kp3 45 kd3 7 ki3 02kp4 7 kd4 19 ki4 01

0 1 2 3 4 5Time (s)

z 1 (m

)

6 7 8 9 101

15

2

25

Figure 2 Numerical simulation quadrotorrsquos altitude z1 (solid) andaltitude reference z1ref (dash)

Mathematical Problems in Engineering 5

15105Time (s)

0ndash6

ndash4ϕ 1 (d

egre

e)

ndash2

0

2

(a)

151050ndash2

0

2

4

6

θ 1 (d

egre

e)

Time (s)

(b)

1510

Time (s)

50ndash2

0

2

4

6

ψ 1 (d

egre

e)

(c)

Figure 3 Numerical simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (b) pitch angle θ1 (solid) and pitch anglereference θ1ref (dash) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

15 20 25 30 35 40 45Time (s)

z 1 (m

)

50 55 60 65 701

15

2

25

Figure 4 Experimental implementation quadrotorrsquos altitude z1 (solid) and altitude reference z1ref (dash)

6 Mathematical Problems in Engineering

experimental tests were performed in an outdoor envi-ronment without using the GPS sensor

5 Conclusions

In this paper a controller based on the stabilization tech-nique for the altitude and attitude error has been proposedfor a quadrotor e simulation results have been performedusing the Pixhawk PSP en the controller has beenimplemented on a laboratory quadrotor e simulation andexperimental results show a good performance even inoutdoor environment showing some degree of robustness inthe presence of environmental disturbances

Data Availability

e figures tables and other data used to support this studyare included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is paper has been partially supported by the EuropeanProject ECSELndashJU RIAndash2018 ldquoComp4Dronesrdquo and project

ldquoCoordination of autonomous unmanned vehicles for highlycomplex performancesrdquo Executive Program of Scientific andTechnological Agreement between Italy (Ministry of ForeignAffairs and International Cooperation Italy) and Mexico(Mexican International Cooperation Agency for the De-velopment) SAAP3

References

[1] S Bouabdallah P Murrieri and R Siegwart ldquoDesign andcontrol of an indoor Micro quadrotorrdquo in Proceedings of theIEEE International Conference on Robotics and Automationpp 1ndash6 New Orleans LA USA April-May 2004

[2] Y Zeng and L Zhao ldquoParameter identification for unmannedfourndashrotor helicopter with nonlinear modelrdquo in Proceedings ofthe 2014 IEEE Chinese Guidance Navigation and ControlConference pp 922ndash926 Yantai China August 2014

[3] A Nagaty S Saeedi C ibault M Seto and H Li ldquoControland navigation framework for quadrotor helicoptersrdquo Journalof Intelligent amp Robotic Systems vol 70 no 1ndash4 pp 1ndash122013

[4] O Magnussen M Ottestad and G Hovland ldquoExperimentalvalidation of a quaternionndashbased attitude estimation withdirect input to a quadcopter control systemrdquo in Proceedings ofthe International Conference on Unmanned Aircraft Systems(ICUAS) pp 48ndash485 Atlanta GA USA May 2013

[5] R Valenti I Dryanovski and J Xiao ldquoKeeping a good at-titude a quaternion-based orientation filter for IMUs andMARGsrdquo Sensors vol 15 no 8 pp 19302ndash19330 2015

ndash10

0

10

15 20 25 30 35 40 45Time (s)

ϕ 1 (d

egre

e)

50 55 60 65 70

(a)

ndash10

0

10

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

θ 1 (d

egre

e)

(b)

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

ndash10

0

10

ψ 1 (d

egre

e)

(c)

Figure 5 Experimental simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (degree vs s) (b) pitch angle θ1 (solid) andpitch angle reference θ1ref (dash) (degree vs s) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

Mathematical Problems in Engineering 7

[6] M Elena AntoniondashToledo A Y Alanis and E N SanchezldquoRobust neural decentralized control for a quadrotor UAVrdquoin Proceedings of the International Joint Conference on NeuralNetworks (IJCNN) pp 714ndash719 Vancouver Canada July2016

[7] B Panomruttanarug K Higuchi and F Mora-CaminoldquoAttitude control of a quadrotor aircraft using LQR statefeedback controller with full order state observerrdquo in Pro-ceeding of the International Conference on InstrumentationControl and Information Technology (SICE) pp 2041ndash2046Konya Turkey June 2013

[8] P E I Pounds D R Bersak and A M Dollar ldquoStability ofsmall-scale UAV helicopters and quadrotors with addedpayload mass under PID controlrdquo Autonomous Robotsvol 33 no 1-2 pp 129ndash142 2012

[9] L LuquendashVega B CastillondashToledo and A G LoukianovldquoRobust block second order sliding mode control for aquadrotorrdquo Journal of the Franklin Institute vol 349 no 2pp 719ndash739 2012

[10] S Bouabdallah and R Siegwart ldquoBackstepping and sliding-mode techniques applied to an indoor Micro quadrotorrdquo inProceedings of the 2005 IEEE International Conference onRobotics and Automation Barcelona Spain April 2005

[11] D Matouk O Gherouat F Abdessemed and A HassamldquoQuadrotor position and attitude control via backsteppingapproachrdquo in Proceedings of the 8th International Conferenceon Modelling Identification and Control (ICMI) pp 73ndash79Algiers Algeria November 2016

[12] J-J Xiong and G-B Zhang ldquoGlobal fast dynamic terminalsliding mode control for a quadrotor UAVrdquo ISA Transactionsvol 66 pp 233ndash240 2017

[13] Y C Choi and H S Ahn ldquoNonlinear control of quadrotor forpoint tracking actual implementation and experimentaltestsrdquo IEEEASME Transactions on Mechatronics vol 20no 3 pp 1179ndash1192 2015

[14] Y-C Liu J Zhang T Zhang and J-Y Song ldquoRobust adaptivespacecraft attitude tracking control based on similar skew-symmetric structurerdquo Computers amp Electrical Engineeringvol 56 pp 784ndash794 2016

[15] W Dong J A Farrell M M Polycarpou V Djapic andM Sharma ldquoCommand filtered adaptive backsteppingrdquo IEEETransactions on Control Systems Technology vol 20 no 3pp 566ndash580 2012

[16] S Islam J Dias and L D Seneviratne ldquoAdaptive outputfeedback control for miniature unmanned aerial vehiclerdquo inProceedings IEEE International Conference on Advanced In-telligent Mechatronics AIM pp 318ndash322 Banff Canada July2016

[17] P C Hughes Spacecraft Attitude Dynamics Dover Publica-tions Inc Mineola NY USA 1986

[18] A IsidoriNonlinear Control Systems Springer-Verlag BerlinGermany 3rd edition 1995

[19] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 2002

8 Mathematical Problems in Engineering

_eϕ1 eϕ2 minus k1eϕ1 minus _ϕ1ref + ϕ2ref

_eϕ2 Jy minus Jz

Jx

θ2ψ2 minusJp

Jx

ωpθ2 +l

Jx

τ1

minus _ϕ2ref + k1 eϕ2 minus k1eϕ1 + ϕ2ref minus _ϕ1ref1113872 1113873

_eθ1 eθ2 minus k3eθ1 minus _θ1ref + θ2ref eθ2 minus k3eθ1

minus _θ1ref + θ2ref

_eθ2 Jz minus Jx

Jy

ϕ2ψ2 +Jp

Jy

ωpϕ2 +l

Jy

τ2

minus _θ2ref + k3 eθ2 minus k3eθ1 + θ2ref minus _θ1ref1113872 1113873

_eψ1 eψ2 minus k5eψ2 minus _ψ1ref + ψ2ref

_eψ2 Jx minus Jy

Jz

ϕ2θ2 +1Jz

τ3 minus _ψ2ref

+ k5 eψ2 minus k5eψ1 + ψ2ref minus _ψ1ref1113872 1113873

(13)

Using the following control law [18]

τ1 Jx

lminus

Jy minus Jz

Jx

θ2ψ2 +Jp

Jx

ωpθ2 + _ϕ2ref minus k1 eϕ2 minus k1eϕ111138721113888

+ ϕ2ref minus _ϕ1ref1113873 minus k2eϕ21113889

τ2 Jy

lminus

Jz minus Jx

Jy

ϕ2ψ2 minusJp

Jy

ωpϕ2 + _θ2ref minus k3 eθ2 minus k3eθ111138721113888

+ θ2ref minus _θ1ref1113873 minus k4eθ21113889

τ3 Jz minusJx minus Jy

Jz

ϕ2θ2 + _ψ2ref minus k5 eψ2 minus k5eψ1 + ψ2ref11138721113888

minus _ψ1ref1113873 minus k6eψ21113889

(14)

with gains ki gt 0 i 2 4 6 in (13) the tracking error dy-namics become

_eX

_eϕ1

_eϕ2

_eθ1

_eθ2

_eψ1

_eψ2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusk1 1 0 0 0 0

0 minusk2 0 0 0 0

0 0 minusk3 1 0 0

0 0 0 minusk4 0 0

0 0 0 0 minusk5 1

0 0 0 0 0 minusk6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

eϕ1

eϕ2

eθ1

eθ2

eψ1

eψ2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

AXeX

(15)

where eX eϕ1 eϕ2 eθ1 eθ2 eψ1eψ21113872 1113873

T which converge

exponentially to zero In fact considering the Lyapunovcandidate

VX t eX( 1113857 12e

TXPXeX (16)

and differentiating along the error dynamics (15) oneobtains

_VX t eX( 1113857 eTX A

TXPX + PXAX1113872 1113873eX minuse

TXQXeX

le minus λQX

min eX

2 le minus αVX t eX( 1113857 α

2λQX

min

λQX

min

gt 0

(17)

with PX solution of the equation ATXPX + PXAX minusQX for a

fixed matrix QX QTX gt 0 erefore

VX t eX( 1113857le eminusαt

V 0 eX(0)( 1113857 (18)

so that the tracking error converges globally exponentially tozero [19]

32AltitudeControl For the altitude control let us considerthe altitude dynamics

_z1 z2

_z2 g + cϕ1cθ11m

u1

(19)

e control problem is to maintain the quadrotor at adesired constant altitude z1ref z2ref 0 e tracking errorsare defined as

Table 1 Quadrotor parameters

m Mass of the airframe 11 kgl Distance of the center of mass to the rotor shaft 0223mJx Inertia in the x-axis 6825times 10minus3 kgm2

Jy Inertia in the y-axis 6825times 10minus3 kgm2

Jz Inertia in the z-axis 1239times 10minus3 kgm2

Jp Inertia of the propellers 6times 10minus5 kgm2

g Gravity acceleration 981ms2b rust factor 542times 10minus6

c Drag factor 11times 10minus6 N s2 radminus2

ϕ Roll angle degθ Pitch angle degψ Yaw angle degz1 z-position m

4 Mathematical Problems in Engineering

ez1 z1 minus z1ref

ez2 z2 minus z2ref (20)

Choosing

u1 m

cϕ1cθ1g + _z2ref minus kz1ez1 minus kz2ez21113872 1113873 (21)

with kz1 kz2 gt 0 we obtain_ez Azez (22)

where ez (ez1 ez2)T and

Az 0 1

minuskz1 minuskz21113888 1113889 (23)

e stability proof is similar to that of the previoussection

4 Simulation and Experimental Results

In this section the performance of the attitude and altitudecontrollers (14) and (21) is tested considering an F-450quadrotor First numerical simulations are carried outen real-time experimental tests are performed showingthe effectiveness of the control design

41 Simulation Results For the numerical simulations thequadrotor model provided by the Pixhawk PSP in Simulinkhas been used is model contains the attitude and altitudeflight control model called px4 demo attitude control whichshows a good performance in predicting the dynamicquadrotor behavior very close to the real drone dynamics

Controllers (14) and (21) use the nominal values ofTable 1 and the gains of Table 2 e simulations have beenperformed in two steps In the first one the quadrotor isstabilized in altitude In Figure 2 the altitude z is shown Inthe second step the quadrotor is stabilized in attitude epitch roll and yaw angles are shown in Figure 3 e initialconditions considered are z(0) 0 ϕ(0) minus573degθ(0) 573deg and ψ(0) 573deg e reference values arezref 2 m and ϕref θref ψref 0

42 Experimental Results In this section we describe thephysical setting of the embedded control that allows stabilizingthe quadrotor An embedded control system generally consistsof three elements sensors actuators and a microcontrolleremicrocontroller interacts with the continuous dynamics ofthe plant via the sensors and actuators and itsmajor function isto compute and generate control commands for the actuatorsthat are based on sensor measurements e onboard elec-tronic system consists of a flight controller a hardware setupand some ultrasonic sensors

421 Flight Controller e 3DR-PIXHAWK is a high-performance autopilot-on-module suitable for fixed wingmulti rotors helicopters cars boats and other mobile ro-botic platforms Its processor can run to 168MHz252 MIPS

Cortex-M4F with 256KB in RAM and 2MB flash and has 14PWMServo outputs and abundant connectivity options foradditional peripherals such as 5x UART (serial ports) onehigh-power capable 2x with HW flow control 2x CANSpektrum DSMDSM2DSMndashXreg Satellite compatible inputPPM sum signal input RSSI (PWM or voltage) input I2CSPI 33 and 66V ADC input Moreover the 3DR-Pixhawkhas different sensors such as ST Micro L3GD20H 16 bitgyroscopes ST Micro LSM303D 14 bit accelerometersmagnetometers Invensense MPU 6000 3-axis accelerome-tersgyroscopes and MEAS MS5611 barometers

422 Hardware Setup e structure of the quadrotor iscomposed by an F-450 frame with integrated PCB wiringwhereas the rotors are brushless motors manufactured byEndashmax with 935 rpmV and a 10times 45Prime propellers Turnigyspeed drivers (ESC) are BHC type at 18A max e batteryused in this setup is a 3S 2800mAh 25Ce radio-transmitteris a Turnigy with 9PPM channels working at 24GHz

An LV-MaxSonar-EZ4 sensor with a resolution of254 cm 20Hz reading rate 42 kHz ultrasonic sensormeasures a maximum Range of 645 cm operating in therange of 25ndash55 VDC was used to measure the altitude

For the real-time running the same parameter valuesand gains shown in Table 2 were used e initial conditionfor the altitude was z(0) 216m For the pitch roll andyaw angles the initial condition were chosen at ϕ1(0) 16degθ1(0) 09deg and ψ1(0) minus16deg Figures 4 and 5 show thedynamic behavior of the altitude and the pitch roll and yawangles respectively It is worth noticing that these

Table 2 Gain values

kp1 46 kd1 35 ki1 28kp2 43 kd2 9 ki2 02kp3 45 kd3 7 ki3 02kp4 7 kd4 19 ki4 01

0 1 2 3 4 5Time (s)

z 1 (m

)

6 7 8 9 101

15

2

25

Figure 2 Numerical simulation quadrotorrsquos altitude z1 (solid) andaltitude reference z1ref (dash)

Mathematical Problems in Engineering 5

15105Time (s)

0ndash6

ndash4ϕ 1 (d

egre

e)

ndash2

0

2

(a)

151050ndash2

0

2

4

6

θ 1 (d

egre

e)

Time (s)

(b)

1510

Time (s)

50ndash2

0

2

4

6

ψ 1 (d

egre

e)

(c)

Figure 3 Numerical simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (b) pitch angle θ1 (solid) and pitch anglereference θ1ref (dash) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

15 20 25 30 35 40 45Time (s)

z 1 (m

)

50 55 60 65 701

15

2

25

Figure 4 Experimental implementation quadrotorrsquos altitude z1 (solid) and altitude reference z1ref (dash)

6 Mathematical Problems in Engineering

experimental tests were performed in an outdoor envi-ronment without using the GPS sensor

5 Conclusions

In this paper a controller based on the stabilization tech-nique for the altitude and attitude error has been proposedfor a quadrotor e simulation results have been performedusing the Pixhawk PSP en the controller has beenimplemented on a laboratory quadrotor e simulation andexperimental results show a good performance even inoutdoor environment showing some degree of robustness inthe presence of environmental disturbances

Data Availability

e figures tables and other data used to support this studyare included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is paper has been partially supported by the EuropeanProject ECSELndashJU RIAndash2018 ldquoComp4Dronesrdquo and project

ldquoCoordination of autonomous unmanned vehicles for highlycomplex performancesrdquo Executive Program of Scientific andTechnological Agreement between Italy (Ministry of ForeignAffairs and International Cooperation Italy) and Mexico(Mexican International Cooperation Agency for the De-velopment) SAAP3

References

[1] S Bouabdallah P Murrieri and R Siegwart ldquoDesign andcontrol of an indoor Micro quadrotorrdquo in Proceedings of theIEEE International Conference on Robotics and Automationpp 1ndash6 New Orleans LA USA April-May 2004

[2] Y Zeng and L Zhao ldquoParameter identification for unmannedfourndashrotor helicopter with nonlinear modelrdquo in Proceedings ofthe 2014 IEEE Chinese Guidance Navigation and ControlConference pp 922ndash926 Yantai China August 2014

[3] A Nagaty S Saeedi C ibault M Seto and H Li ldquoControland navigation framework for quadrotor helicoptersrdquo Journalof Intelligent amp Robotic Systems vol 70 no 1ndash4 pp 1ndash122013

[4] O Magnussen M Ottestad and G Hovland ldquoExperimentalvalidation of a quaternionndashbased attitude estimation withdirect input to a quadcopter control systemrdquo in Proceedings ofthe International Conference on Unmanned Aircraft Systems(ICUAS) pp 48ndash485 Atlanta GA USA May 2013

[5] R Valenti I Dryanovski and J Xiao ldquoKeeping a good at-titude a quaternion-based orientation filter for IMUs andMARGsrdquo Sensors vol 15 no 8 pp 19302ndash19330 2015

ndash10

0

10

15 20 25 30 35 40 45Time (s)

ϕ 1 (d

egre

e)

50 55 60 65 70

(a)

ndash10

0

10

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

θ 1 (d

egre

e)

(b)

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

ndash10

0

10

ψ 1 (d

egre

e)

(c)

Figure 5 Experimental simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (degree vs s) (b) pitch angle θ1 (solid) andpitch angle reference θ1ref (dash) (degree vs s) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

Mathematical Problems in Engineering 7

[6] M Elena AntoniondashToledo A Y Alanis and E N SanchezldquoRobust neural decentralized control for a quadrotor UAVrdquoin Proceedings of the International Joint Conference on NeuralNetworks (IJCNN) pp 714ndash719 Vancouver Canada July2016

[7] B Panomruttanarug K Higuchi and F Mora-CaminoldquoAttitude control of a quadrotor aircraft using LQR statefeedback controller with full order state observerrdquo in Pro-ceeding of the International Conference on InstrumentationControl and Information Technology (SICE) pp 2041ndash2046Konya Turkey June 2013

[8] P E I Pounds D R Bersak and A M Dollar ldquoStability ofsmall-scale UAV helicopters and quadrotors with addedpayload mass under PID controlrdquo Autonomous Robotsvol 33 no 1-2 pp 129ndash142 2012

[9] L LuquendashVega B CastillondashToledo and A G LoukianovldquoRobust block second order sliding mode control for aquadrotorrdquo Journal of the Franklin Institute vol 349 no 2pp 719ndash739 2012

[10] S Bouabdallah and R Siegwart ldquoBackstepping and sliding-mode techniques applied to an indoor Micro quadrotorrdquo inProceedings of the 2005 IEEE International Conference onRobotics and Automation Barcelona Spain April 2005

[11] D Matouk O Gherouat F Abdessemed and A HassamldquoQuadrotor position and attitude control via backsteppingapproachrdquo in Proceedings of the 8th International Conferenceon Modelling Identification and Control (ICMI) pp 73ndash79Algiers Algeria November 2016

[12] J-J Xiong and G-B Zhang ldquoGlobal fast dynamic terminalsliding mode control for a quadrotor UAVrdquo ISA Transactionsvol 66 pp 233ndash240 2017

[13] Y C Choi and H S Ahn ldquoNonlinear control of quadrotor forpoint tracking actual implementation and experimentaltestsrdquo IEEEASME Transactions on Mechatronics vol 20no 3 pp 1179ndash1192 2015

[14] Y-C Liu J Zhang T Zhang and J-Y Song ldquoRobust adaptivespacecraft attitude tracking control based on similar skew-symmetric structurerdquo Computers amp Electrical Engineeringvol 56 pp 784ndash794 2016

[15] W Dong J A Farrell M M Polycarpou V Djapic andM Sharma ldquoCommand filtered adaptive backsteppingrdquo IEEETransactions on Control Systems Technology vol 20 no 3pp 566ndash580 2012

[16] S Islam J Dias and L D Seneviratne ldquoAdaptive outputfeedback control for miniature unmanned aerial vehiclerdquo inProceedings IEEE International Conference on Advanced In-telligent Mechatronics AIM pp 318ndash322 Banff Canada July2016

[17] P C Hughes Spacecraft Attitude Dynamics Dover Publica-tions Inc Mineola NY USA 1986

[18] A IsidoriNonlinear Control Systems Springer-Verlag BerlinGermany 3rd edition 1995

[19] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 2002

8 Mathematical Problems in Engineering

ez1 z1 minus z1ref

ez2 z2 minus z2ref (20)

Choosing

u1 m

cϕ1cθ1g + _z2ref minus kz1ez1 minus kz2ez21113872 1113873 (21)

with kz1 kz2 gt 0 we obtain_ez Azez (22)

where ez (ez1 ez2)T and

Az 0 1

minuskz1 minuskz21113888 1113889 (23)

e stability proof is similar to that of the previoussection

4 Simulation and Experimental Results

In this section the performance of the attitude and altitudecontrollers (14) and (21) is tested considering an F-450quadrotor First numerical simulations are carried outen real-time experimental tests are performed showingthe effectiveness of the control design

41 Simulation Results For the numerical simulations thequadrotor model provided by the Pixhawk PSP in Simulinkhas been used is model contains the attitude and altitudeflight control model called px4 demo attitude control whichshows a good performance in predicting the dynamicquadrotor behavior very close to the real drone dynamics

Controllers (14) and (21) use the nominal values ofTable 1 and the gains of Table 2 e simulations have beenperformed in two steps In the first one the quadrotor isstabilized in altitude In Figure 2 the altitude z is shown Inthe second step the quadrotor is stabilized in attitude epitch roll and yaw angles are shown in Figure 3 e initialconditions considered are z(0) 0 ϕ(0) minus573degθ(0) 573deg and ψ(0) 573deg e reference values arezref 2 m and ϕref θref ψref 0

42 Experimental Results In this section we describe thephysical setting of the embedded control that allows stabilizingthe quadrotor An embedded control system generally consistsof three elements sensors actuators and a microcontrolleremicrocontroller interacts with the continuous dynamics ofthe plant via the sensors and actuators and itsmajor function isto compute and generate control commands for the actuatorsthat are based on sensor measurements e onboard elec-tronic system consists of a flight controller a hardware setupand some ultrasonic sensors

421 Flight Controller e 3DR-PIXHAWK is a high-performance autopilot-on-module suitable for fixed wingmulti rotors helicopters cars boats and other mobile ro-botic platforms Its processor can run to 168MHz252 MIPS

Cortex-M4F with 256KB in RAM and 2MB flash and has 14PWMServo outputs and abundant connectivity options foradditional peripherals such as 5x UART (serial ports) onehigh-power capable 2x with HW flow control 2x CANSpektrum DSMDSM2DSMndashXreg Satellite compatible inputPPM sum signal input RSSI (PWM or voltage) input I2CSPI 33 and 66V ADC input Moreover the 3DR-Pixhawkhas different sensors such as ST Micro L3GD20H 16 bitgyroscopes ST Micro LSM303D 14 bit accelerometersmagnetometers Invensense MPU 6000 3-axis accelerome-tersgyroscopes and MEAS MS5611 barometers

422 Hardware Setup e structure of the quadrotor iscomposed by an F-450 frame with integrated PCB wiringwhereas the rotors are brushless motors manufactured byEndashmax with 935 rpmV and a 10times 45Prime propellers Turnigyspeed drivers (ESC) are BHC type at 18A max e batteryused in this setup is a 3S 2800mAh 25Ce radio-transmitteris a Turnigy with 9PPM channels working at 24GHz

An LV-MaxSonar-EZ4 sensor with a resolution of254 cm 20Hz reading rate 42 kHz ultrasonic sensormeasures a maximum Range of 645 cm operating in therange of 25ndash55 VDC was used to measure the altitude

For the real-time running the same parameter valuesand gains shown in Table 2 were used e initial conditionfor the altitude was z(0) 216m For the pitch roll andyaw angles the initial condition were chosen at ϕ1(0) 16degθ1(0) 09deg and ψ1(0) minus16deg Figures 4 and 5 show thedynamic behavior of the altitude and the pitch roll and yawangles respectively It is worth noticing that these

Table 2 Gain values

kp1 46 kd1 35 ki1 28kp2 43 kd2 9 ki2 02kp3 45 kd3 7 ki3 02kp4 7 kd4 19 ki4 01

0 1 2 3 4 5Time (s)

z 1 (m

)

6 7 8 9 101

15

2

25

Figure 2 Numerical simulation quadrotorrsquos altitude z1 (solid) andaltitude reference z1ref (dash)

Mathematical Problems in Engineering 5

15105Time (s)

0ndash6

ndash4ϕ 1 (d

egre

e)

ndash2

0

2

(a)

151050ndash2

0

2

4

6

θ 1 (d

egre

e)

Time (s)

(b)

1510

Time (s)

50ndash2

0

2

4

6

ψ 1 (d

egre

e)

(c)

Figure 3 Numerical simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (b) pitch angle θ1 (solid) and pitch anglereference θ1ref (dash) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

15 20 25 30 35 40 45Time (s)

z 1 (m

)

50 55 60 65 701

15

2

25

Figure 4 Experimental implementation quadrotorrsquos altitude z1 (solid) and altitude reference z1ref (dash)

6 Mathematical Problems in Engineering

experimental tests were performed in an outdoor envi-ronment without using the GPS sensor

5 Conclusions

In this paper a controller based on the stabilization tech-nique for the altitude and attitude error has been proposedfor a quadrotor e simulation results have been performedusing the Pixhawk PSP en the controller has beenimplemented on a laboratory quadrotor e simulation andexperimental results show a good performance even inoutdoor environment showing some degree of robustness inthe presence of environmental disturbances

Data Availability

e figures tables and other data used to support this studyare included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is paper has been partially supported by the EuropeanProject ECSELndashJU RIAndash2018 ldquoComp4Dronesrdquo and project

ldquoCoordination of autonomous unmanned vehicles for highlycomplex performancesrdquo Executive Program of Scientific andTechnological Agreement between Italy (Ministry of ForeignAffairs and International Cooperation Italy) and Mexico(Mexican International Cooperation Agency for the De-velopment) SAAP3

References

[1] S Bouabdallah P Murrieri and R Siegwart ldquoDesign andcontrol of an indoor Micro quadrotorrdquo in Proceedings of theIEEE International Conference on Robotics and Automationpp 1ndash6 New Orleans LA USA April-May 2004

[2] Y Zeng and L Zhao ldquoParameter identification for unmannedfourndashrotor helicopter with nonlinear modelrdquo in Proceedings ofthe 2014 IEEE Chinese Guidance Navigation and ControlConference pp 922ndash926 Yantai China August 2014

[3] A Nagaty S Saeedi C ibault M Seto and H Li ldquoControland navigation framework for quadrotor helicoptersrdquo Journalof Intelligent amp Robotic Systems vol 70 no 1ndash4 pp 1ndash122013

[4] O Magnussen M Ottestad and G Hovland ldquoExperimentalvalidation of a quaternionndashbased attitude estimation withdirect input to a quadcopter control systemrdquo in Proceedings ofthe International Conference on Unmanned Aircraft Systems(ICUAS) pp 48ndash485 Atlanta GA USA May 2013

[5] R Valenti I Dryanovski and J Xiao ldquoKeeping a good at-titude a quaternion-based orientation filter for IMUs andMARGsrdquo Sensors vol 15 no 8 pp 19302ndash19330 2015

ndash10

0

10

15 20 25 30 35 40 45Time (s)

ϕ 1 (d

egre

e)

50 55 60 65 70

(a)

ndash10

0

10

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

θ 1 (d

egre

e)

(b)

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

ndash10

0

10

ψ 1 (d

egre

e)

(c)

Figure 5 Experimental simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (degree vs s) (b) pitch angle θ1 (solid) andpitch angle reference θ1ref (dash) (degree vs s) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

Mathematical Problems in Engineering 7

[6] M Elena AntoniondashToledo A Y Alanis and E N SanchezldquoRobust neural decentralized control for a quadrotor UAVrdquoin Proceedings of the International Joint Conference on NeuralNetworks (IJCNN) pp 714ndash719 Vancouver Canada July2016

[7] B Panomruttanarug K Higuchi and F Mora-CaminoldquoAttitude control of a quadrotor aircraft using LQR statefeedback controller with full order state observerrdquo in Pro-ceeding of the International Conference on InstrumentationControl and Information Technology (SICE) pp 2041ndash2046Konya Turkey June 2013

[8] P E I Pounds D R Bersak and A M Dollar ldquoStability ofsmall-scale UAV helicopters and quadrotors with addedpayload mass under PID controlrdquo Autonomous Robotsvol 33 no 1-2 pp 129ndash142 2012

[9] L LuquendashVega B CastillondashToledo and A G LoukianovldquoRobust block second order sliding mode control for aquadrotorrdquo Journal of the Franklin Institute vol 349 no 2pp 719ndash739 2012

[10] S Bouabdallah and R Siegwart ldquoBackstepping and sliding-mode techniques applied to an indoor Micro quadrotorrdquo inProceedings of the 2005 IEEE International Conference onRobotics and Automation Barcelona Spain April 2005

[11] D Matouk O Gherouat F Abdessemed and A HassamldquoQuadrotor position and attitude control via backsteppingapproachrdquo in Proceedings of the 8th International Conferenceon Modelling Identification and Control (ICMI) pp 73ndash79Algiers Algeria November 2016

[12] J-J Xiong and G-B Zhang ldquoGlobal fast dynamic terminalsliding mode control for a quadrotor UAVrdquo ISA Transactionsvol 66 pp 233ndash240 2017

[13] Y C Choi and H S Ahn ldquoNonlinear control of quadrotor forpoint tracking actual implementation and experimentaltestsrdquo IEEEASME Transactions on Mechatronics vol 20no 3 pp 1179ndash1192 2015

[14] Y-C Liu J Zhang T Zhang and J-Y Song ldquoRobust adaptivespacecraft attitude tracking control based on similar skew-symmetric structurerdquo Computers amp Electrical Engineeringvol 56 pp 784ndash794 2016

[15] W Dong J A Farrell M M Polycarpou V Djapic andM Sharma ldquoCommand filtered adaptive backsteppingrdquo IEEETransactions on Control Systems Technology vol 20 no 3pp 566ndash580 2012

[16] S Islam J Dias and L D Seneviratne ldquoAdaptive outputfeedback control for miniature unmanned aerial vehiclerdquo inProceedings IEEE International Conference on Advanced In-telligent Mechatronics AIM pp 318ndash322 Banff Canada July2016

[17] P C Hughes Spacecraft Attitude Dynamics Dover Publica-tions Inc Mineola NY USA 1986

[18] A IsidoriNonlinear Control Systems Springer-Verlag BerlinGermany 3rd edition 1995

[19] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 2002

8 Mathematical Problems in Engineering

15105Time (s)

0ndash6

ndash4ϕ 1 (d

egre

e)

ndash2

0

2

(a)

151050ndash2

0

2

4

6

θ 1 (d

egre

e)

Time (s)

(b)

1510

Time (s)

50ndash2

0

2

4

6

ψ 1 (d

egre

e)

(c)

Figure 3 Numerical simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (b) pitch angle θ1 (solid) and pitch anglereference θ1ref (dash) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

15 20 25 30 35 40 45Time (s)

z 1 (m

)

50 55 60 65 701

15

2

25

Figure 4 Experimental implementation quadrotorrsquos altitude z1 (solid) and altitude reference z1ref (dash)

6 Mathematical Problems in Engineering

experimental tests were performed in an outdoor envi-ronment without using the GPS sensor

5 Conclusions

In this paper a controller based on the stabilization tech-nique for the altitude and attitude error has been proposedfor a quadrotor e simulation results have been performedusing the Pixhawk PSP en the controller has beenimplemented on a laboratory quadrotor e simulation andexperimental results show a good performance even inoutdoor environment showing some degree of robustness inthe presence of environmental disturbances

Data Availability

e figures tables and other data used to support this studyare included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is paper has been partially supported by the EuropeanProject ECSELndashJU RIAndash2018 ldquoComp4Dronesrdquo and project

ldquoCoordination of autonomous unmanned vehicles for highlycomplex performancesrdquo Executive Program of Scientific andTechnological Agreement between Italy (Ministry of ForeignAffairs and International Cooperation Italy) and Mexico(Mexican International Cooperation Agency for the De-velopment) SAAP3

References

[1] S Bouabdallah P Murrieri and R Siegwart ldquoDesign andcontrol of an indoor Micro quadrotorrdquo in Proceedings of theIEEE International Conference on Robotics and Automationpp 1ndash6 New Orleans LA USA April-May 2004

[2] Y Zeng and L Zhao ldquoParameter identification for unmannedfourndashrotor helicopter with nonlinear modelrdquo in Proceedings ofthe 2014 IEEE Chinese Guidance Navigation and ControlConference pp 922ndash926 Yantai China August 2014

[3] A Nagaty S Saeedi C ibault M Seto and H Li ldquoControland navigation framework for quadrotor helicoptersrdquo Journalof Intelligent amp Robotic Systems vol 70 no 1ndash4 pp 1ndash122013

[4] O Magnussen M Ottestad and G Hovland ldquoExperimentalvalidation of a quaternionndashbased attitude estimation withdirect input to a quadcopter control systemrdquo in Proceedings ofthe International Conference on Unmanned Aircraft Systems(ICUAS) pp 48ndash485 Atlanta GA USA May 2013

[5] R Valenti I Dryanovski and J Xiao ldquoKeeping a good at-titude a quaternion-based orientation filter for IMUs andMARGsrdquo Sensors vol 15 no 8 pp 19302ndash19330 2015

ndash10

0

10

15 20 25 30 35 40 45Time (s)

ϕ 1 (d

egre

e)

50 55 60 65 70

(a)

ndash10

0

10

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

θ 1 (d

egre

e)

(b)

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

ndash10

0

10

ψ 1 (d

egre

e)

(c)

Figure 5 Experimental simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (degree vs s) (b) pitch angle θ1 (solid) andpitch angle reference θ1ref (dash) (degree vs s) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

Mathematical Problems in Engineering 7

[6] M Elena AntoniondashToledo A Y Alanis and E N SanchezldquoRobust neural decentralized control for a quadrotor UAVrdquoin Proceedings of the International Joint Conference on NeuralNetworks (IJCNN) pp 714ndash719 Vancouver Canada July2016

[7] B Panomruttanarug K Higuchi and F Mora-CaminoldquoAttitude control of a quadrotor aircraft using LQR statefeedback controller with full order state observerrdquo in Pro-ceeding of the International Conference on InstrumentationControl and Information Technology (SICE) pp 2041ndash2046Konya Turkey June 2013

[8] P E I Pounds D R Bersak and A M Dollar ldquoStability ofsmall-scale UAV helicopters and quadrotors with addedpayload mass under PID controlrdquo Autonomous Robotsvol 33 no 1-2 pp 129ndash142 2012

[9] L LuquendashVega B CastillondashToledo and A G LoukianovldquoRobust block second order sliding mode control for aquadrotorrdquo Journal of the Franklin Institute vol 349 no 2pp 719ndash739 2012

[10] S Bouabdallah and R Siegwart ldquoBackstepping and sliding-mode techniques applied to an indoor Micro quadrotorrdquo inProceedings of the 2005 IEEE International Conference onRobotics and Automation Barcelona Spain April 2005

[11] D Matouk O Gherouat F Abdessemed and A HassamldquoQuadrotor position and attitude control via backsteppingapproachrdquo in Proceedings of the 8th International Conferenceon Modelling Identification and Control (ICMI) pp 73ndash79Algiers Algeria November 2016

[12] J-J Xiong and G-B Zhang ldquoGlobal fast dynamic terminalsliding mode control for a quadrotor UAVrdquo ISA Transactionsvol 66 pp 233ndash240 2017

[13] Y C Choi and H S Ahn ldquoNonlinear control of quadrotor forpoint tracking actual implementation and experimentaltestsrdquo IEEEASME Transactions on Mechatronics vol 20no 3 pp 1179ndash1192 2015

[14] Y-C Liu J Zhang T Zhang and J-Y Song ldquoRobust adaptivespacecraft attitude tracking control based on similar skew-symmetric structurerdquo Computers amp Electrical Engineeringvol 56 pp 784ndash794 2016

[15] W Dong J A Farrell M M Polycarpou V Djapic andM Sharma ldquoCommand filtered adaptive backsteppingrdquo IEEETransactions on Control Systems Technology vol 20 no 3pp 566ndash580 2012

[16] S Islam J Dias and L D Seneviratne ldquoAdaptive outputfeedback control for miniature unmanned aerial vehiclerdquo inProceedings IEEE International Conference on Advanced In-telligent Mechatronics AIM pp 318ndash322 Banff Canada July2016

[17] P C Hughes Spacecraft Attitude Dynamics Dover Publica-tions Inc Mineola NY USA 1986

[18] A IsidoriNonlinear Control Systems Springer-Verlag BerlinGermany 3rd edition 1995

[19] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 2002

8 Mathematical Problems in Engineering

experimental tests were performed in an outdoor envi-ronment without using the GPS sensor

5 Conclusions

In this paper a controller based on the stabilization tech-nique for the altitude and attitude error has been proposedfor a quadrotor e simulation results have been performedusing the Pixhawk PSP en the controller has beenimplemented on a laboratory quadrotor e simulation andexperimental results show a good performance even inoutdoor environment showing some degree of robustness inthe presence of environmental disturbances

Data Availability

e figures tables and other data used to support this studyare included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is paper has been partially supported by the EuropeanProject ECSELndashJU RIAndash2018 ldquoComp4Dronesrdquo and project

ldquoCoordination of autonomous unmanned vehicles for highlycomplex performancesrdquo Executive Program of Scientific andTechnological Agreement between Italy (Ministry of ForeignAffairs and International Cooperation Italy) and Mexico(Mexican International Cooperation Agency for the De-velopment) SAAP3

References

[1] S Bouabdallah P Murrieri and R Siegwart ldquoDesign andcontrol of an indoor Micro quadrotorrdquo in Proceedings of theIEEE International Conference on Robotics and Automationpp 1ndash6 New Orleans LA USA April-May 2004

[2] Y Zeng and L Zhao ldquoParameter identification for unmannedfourndashrotor helicopter with nonlinear modelrdquo in Proceedings ofthe 2014 IEEE Chinese Guidance Navigation and ControlConference pp 922ndash926 Yantai China August 2014

[3] A Nagaty S Saeedi C ibault M Seto and H Li ldquoControland navigation framework for quadrotor helicoptersrdquo Journalof Intelligent amp Robotic Systems vol 70 no 1ndash4 pp 1ndash122013

[4] O Magnussen M Ottestad and G Hovland ldquoExperimentalvalidation of a quaternionndashbased attitude estimation withdirect input to a quadcopter control systemrdquo in Proceedings ofthe International Conference on Unmanned Aircraft Systems(ICUAS) pp 48ndash485 Atlanta GA USA May 2013

[5] R Valenti I Dryanovski and J Xiao ldquoKeeping a good at-titude a quaternion-based orientation filter for IMUs andMARGsrdquo Sensors vol 15 no 8 pp 19302ndash19330 2015

ndash10

0

10

15 20 25 30 35 40 45Time (s)

ϕ 1 (d

egre

e)

50 55 60 65 70

(a)

ndash10

0

10

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

θ 1 (d

egre

e)

(b)

15 20 25 30 35 40Time (s)

45 50 55 60 65 70

ndash10

0

10

ψ 1 (d

egre

e)

(c)

Figure 5 Experimental simulation (a) roll angle ϕ1 (solid) and roll angle reference ϕ1ref (dash) (degree vs s) (b) pitch angle θ1 (solid) andpitch angle reference θ1ref (dash) (degree vs s) (c) yaw angle ψ1 (solid) and yaw angle reference ψ1ref (dash)

Mathematical Problems in Engineering 7

[6] M Elena AntoniondashToledo A Y Alanis and E N SanchezldquoRobust neural decentralized control for a quadrotor UAVrdquoin Proceedings of the International Joint Conference on NeuralNetworks (IJCNN) pp 714ndash719 Vancouver Canada July2016

[7] B Panomruttanarug K Higuchi and F Mora-CaminoldquoAttitude control of a quadrotor aircraft using LQR statefeedback controller with full order state observerrdquo in Pro-ceeding of the International Conference on InstrumentationControl and Information Technology (SICE) pp 2041ndash2046Konya Turkey June 2013

[8] P E I Pounds D R Bersak and A M Dollar ldquoStability ofsmall-scale UAV helicopters and quadrotors with addedpayload mass under PID controlrdquo Autonomous Robotsvol 33 no 1-2 pp 129ndash142 2012

[9] L LuquendashVega B CastillondashToledo and A G LoukianovldquoRobust block second order sliding mode control for aquadrotorrdquo Journal of the Franklin Institute vol 349 no 2pp 719ndash739 2012

[10] S Bouabdallah and R Siegwart ldquoBackstepping and sliding-mode techniques applied to an indoor Micro quadrotorrdquo inProceedings of the 2005 IEEE International Conference onRobotics and Automation Barcelona Spain April 2005

[11] D Matouk O Gherouat F Abdessemed and A HassamldquoQuadrotor position and attitude control via backsteppingapproachrdquo in Proceedings of the 8th International Conferenceon Modelling Identification and Control (ICMI) pp 73ndash79Algiers Algeria November 2016

[12] J-J Xiong and G-B Zhang ldquoGlobal fast dynamic terminalsliding mode control for a quadrotor UAVrdquo ISA Transactionsvol 66 pp 233ndash240 2017

[13] Y C Choi and H S Ahn ldquoNonlinear control of quadrotor forpoint tracking actual implementation and experimentaltestsrdquo IEEEASME Transactions on Mechatronics vol 20no 3 pp 1179ndash1192 2015

[14] Y-C Liu J Zhang T Zhang and J-Y Song ldquoRobust adaptivespacecraft attitude tracking control based on similar skew-symmetric structurerdquo Computers amp Electrical Engineeringvol 56 pp 784ndash794 2016

[15] W Dong J A Farrell M M Polycarpou V Djapic andM Sharma ldquoCommand filtered adaptive backsteppingrdquo IEEETransactions on Control Systems Technology vol 20 no 3pp 566ndash580 2012

[16] S Islam J Dias and L D Seneviratne ldquoAdaptive outputfeedback control for miniature unmanned aerial vehiclerdquo inProceedings IEEE International Conference on Advanced In-telligent Mechatronics AIM pp 318ndash322 Banff Canada July2016

[17] P C Hughes Spacecraft Attitude Dynamics Dover Publica-tions Inc Mineola NY USA 1986

[18] A IsidoriNonlinear Control Systems Springer-Verlag BerlinGermany 3rd edition 1995

[19] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 2002

8 Mathematical Problems in Engineering

[6] M Elena AntoniondashToledo A Y Alanis and E N SanchezldquoRobust neural decentralized control for a quadrotor UAVrdquoin Proceedings of the International Joint Conference on NeuralNetworks (IJCNN) pp 714ndash719 Vancouver Canada July2016

[7] B Panomruttanarug K Higuchi and F Mora-CaminoldquoAttitude control of a quadrotor aircraft using LQR statefeedback controller with full order state observerrdquo in Pro-ceeding of the International Conference on InstrumentationControl and Information Technology (SICE) pp 2041ndash2046Konya Turkey June 2013

[8] P E I Pounds D R Bersak and A M Dollar ldquoStability ofsmall-scale UAV helicopters and quadrotors with addedpayload mass under PID controlrdquo Autonomous Robotsvol 33 no 1-2 pp 129ndash142 2012

[9] L LuquendashVega B CastillondashToledo and A G LoukianovldquoRobust block second order sliding mode control for aquadrotorrdquo Journal of the Franklin Institute vol 349 no 2pp 719ndash739 2012

[10] S Bouabdallah and R Siegwart ldquoBackstepping and sliding-mode techniques applied to an indoor Micro quadrotorrdquo inProceedings of the 2005 IEEE International Conference onRobotics and Automation Barcelona Spain April 2005

[11] D Matouk O Gherouat F Abdessemed and A HassamldquoQuadrotor position and attitude control via backsteppingapproachrdquo in Proceedings of the 8th International Conferenceon Modelling Identification and Control (ICMI) pp 73ndash79Algiers Algeria November 2016

[12] J-J Xiong and G-B Zhang ldquoGlobal fast dynamic terminalsliding mode control for a quadrotor UAVrdquo ISA Transactionsvol 66 pp 233ndash240 2017

[13] Y C Choi and H S Ahn ldquoNonlinear control of quadrotor forpoint tracking actual implementation and experimentaltestsrdquo IEEEASME Transactions on Mechatronics vol 20no 3 pp 1179ndash1192 2015

[14] Y-C Liu J Zhang T Zhang and J-Y Song ldquoRobust adaptivespacecraft attitude tracking control based on similar skew-symmetric structurerdquo Computers amp Electrical Engineeringvol 56 pp 784ndash794 2016

[15] W Dong J A Farrell M M Polycarpou V Djapic andM Sharma ldquoCommand filtered adaptive backsteppingrdquo IEEETransactions on Control Systems Technology vol 20 no 3pp 566ndash580 2012

[16] S Islam J Dias and L D Seneviratne ldquoAdaptive outputfeedback control for miniature unmanned aerial vehiclerdquo inProceedings IEEE International Conference on Advanced In-telligent Mechatronics AIM pp 318ndash322 Banff Canada July2016

[17] P C Hughes Spacecraft Attitude Dynamics Dover Publica-tions Inc Mineola NY USA 1986

[18] A IsidoriNonlinear Control Systems Springer-Verlag BerlinGermany 3rd edition 1995

[19] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 2002

8 Mathematical Problems in Engineering