Research ArticleVision-Based Nonlinear Control of Quadrotors Using thePhotogrammetric Technique

José Trinidad Guillén-Bonilla ,1 Claudia Carolina Vaca Garcı́a ,2

Stefano Di Gennaro ,3,4 Marı́a Eugenia Sánchez Morales ,2

and Cuauhtémoc Acosta Lúa 2,4

1Centro Universitario de Ciencias Exactas e Ingenieŕıa–UdG, Blvd. Gral. Marcelino Garćıa Barragán 1421, Oĺımpica, 44430,Guadalajara, Jalisco, Mexico2Departamento de Ciencias Tecnológicas, Universidad de Guadalajara, Centro Universitario de La Ciénega,Av. Universidad 1115, 47820, Ocotlán, Jalisco, Mexico3Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via Vetoio, L’Aquila,Coppito, 67100, Italy4Center of Excellence DEWS, University of L’Aquila, Via Vetoio, L’Aquila, Coppito, 67100, Italy

Correspondence should be addressed to Cuauhtémoc Acosta Lúa; cuauhtemoc.acosta@cuci.udg.mx

Received 12 June 2020; Revised 11 August 2020; Accepted 29 September 2020; Published 17 November 2020

Academic Editor: Aimé Lay-Ekuakille

Copyright © 2020 José Trinidad Guillén-Bonilla et al. +is is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work isproperly cited.

+is paper presents a controller designed via the backstepping technique, for the tracking of a reference trajectory obtained via thephotogrammetric technique. +e dynamic equations used to represent the motion of the quadrotor helicopter are based on theNewton–Euler model. +e resulting quadrotor model has been divided into four subsystems for the altitude, longitudinal, lateral,and yaw motions. A control input is designed for each subsystem. Furthermore, the photogrammetric technique has been used toobtain the reference trajectory to be tracked. +e performance and effectiveness of the proposed nonlinear controllers have beentested via numerical simulations using the Pixhawk Pilot Support Package developed for Matlab/Simulink.

1. Introduction

+e technology of unmanned aerial vehicles (UAVs) isdeveloping rapidly, based on the integration of manytechnologies in the mechanical structure, energymanagement, communication, control, etc, [1]. UAVs arewidely used in many applications in civilian as well as inmilitary contexts. For instance, UAVs provide great ad-vantages in geographical information acquisition and safetypurposes, as well as significant benefits when used for in-spection and detection purposes. Furthermore, UAVs arebecoming more and more accessible due to technologicaldevelopments that enable their increasing use in differentand broad application areas.

Different mathematical models can be used to design acontroller for UAVs. In [2–4], a Newton–Euler model was

presented. Furthermore, the studies in [5, 6] consideredquaternions to describe the angular kinematics, whilst thestudy in [7] applied the Euler–Lagrange equations to obtainthe whole quadrotor mathematical model. Regarding thecontrol of quadrotors, many control techniques wereproposed. +e linear quadratic regulator control andproportional integral derivative control were proposed in[8, 9]. However, the stability of these methods cannot beguaranteed when the quadrotor moves away from itsequilibrium configuration. Compared to linear control, thenonlinear control can ensure global stable flight for aquadrotor. Examples of nonlinear control techniques used todesign a controller for a quadrotor are the sliding mode andthe backstepping techniques [10–13]. +e backstepping andadaptive control [14] were applied to the quadrotor flightcontrol. In [15, 16], an observer-based adaptive fuzzy

HindawiMathematical Problems in EngineeringVolume 2020, Article ID 5146291, 10 pageshttps://doi.org/10.1155/2020/5146291

mailto:cuauhtemoc.acosta@cuci.udg.mxhttps://orcid.org/0000-0003-0041-3932https://orcid.org/0000-0003-1279-0567https://orcid.org/0000-0002-2014-623Xhttps://orcid.org/0000-0003-4018-672Xhttps://orcid.org/0000-0002-7398-2629https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/5146291

backstepping controller was designed for trajectory trackingof a quadrotor subject to wind gusts and parametric un-certainties. In [17], a robust adaptive attitude trackingcontrol for a quadrotor was proposed. In [18], a controltrajectory tracking was designed via the sliding modetechnique and using neural networks to optimize the con-troller parameters through a network learning process whichis based on the control process.

To design vision-based controllers, researchers used dif-ferent methods for the recognition of objects, trajectories,road lanes, etc. In [19], an INS/vision-based autonomouslanding system on stationary platforms was proposed,whereas in [20], mobile platforms were used for the landing ofquadrotors. In [21], the position of a quadrotor was estimatedby the detection of a set of concentric circles. In [22], acontroller, based on computer vision techniques applied tohelipad recognition, was proposed, in which the visual rec-ognition of a black and white pattern of the helipad wasexploited. In [23], a computer vision systemwas developed forthe automatic estimation of the position and attitude of ahelipad fixed on a mobile platform. In [24], a trajectorytracking problem for a vision-based quadrotor control systemvia super twisting slidingmode controller was proposed. Also,the study in [25] proposed a neuroadaptive integral robustcontroller for the tracking of ground moving targets in thepresence of various uncertainties, via an Image-Based VisualServo (IBVS) framework. +e authors of [26] presented apredictive control algorithm for autonomous approaches ofquadrotor helicopters to a window using a reference imagecaptured with a photographic camera.+e target is selected byan operator in a reference image which is sent to the vehicle.Besides, the authors of [27] proposed an adaptive slidingmode controller based on the backstepping technique for atracking problem using a monocular algorithm to obtain theaccurate location information of the quadrotor and its ref-erence. In the same context, the work in [28, 29] developed avision-based attitude dynamic surface controller, construct-ing an IBVS to ensure that the visual target remains in thecamera’s field of view all the time.

In this paper, a controller for the tracking of a referencetrajectory is designed via the backstepping technique. +equadrotor model is divided into four subsystems for thealtitude, longitudinal, lateral, and yaw motions, and acontrol input is designed for each subsystem. Furthermore,the photogrammetric technique is used to obtain thereference trajectory to be tracked. +e performance andeffectiveness of the proposed nonlinear controllers are testedvia numerical simulations using the simulation softwarecalled Pixhawk Pilot Support Package (PSP), which predictsaccurately the real dynamic quadrotor helicopter behavior.+e main contributions of this article are as follows:

(i) +e backstepping technique is used to design acontroller capable of tracking a reference positionand yaw of a quadrotor

(ii) +e photogrammetric technique is used to obtainthe reference trajectory

(iii) +e performance and effectiveness of the proposedcontroller have been tested in PSP

+e paper is organized as follows: Section 2 introducesthe description and the mathematical model of the quad-rotor. In Section 3, the control problem is formulated, whilstin Section 4, the nonlinear controller is designed. In Section5, the reference trajectory is obtained via the photogram-metric technique, and some numerical simulations imple-mented in PSP are presented. Some comments conclude thepaper.

2. Mathematical Model of a Quadrotor

+e quadrotor considered in this work consists of a rigidframe equipped with four rotors. +e rotors generate thepropeller force Fi � bω2p,i, proportional to the propellerangular velocity ωp,i, i � 1, 2, 3, 4. Propellers 1 and 3 rotatecounterclockwise, and propellers 2 and 4 rotate clockwise.

Let us indicate with RC(O, e1, e2, e3) andRΓ(Ω, ϵ1, ϵ2, ϵ3) the frames fixed with the Earth and thequadrotor, respectively, andΩ is coincident with the centerof mass of the quadrotor (see Figure 1). +e absoluteposition of quadrotor in RC is described by p � (x, y, z)T,whereas its attitude is described by the Euler anglesα � (ϕ, θ,ψ)T, with ϕ, θ,ψ ∈ (−π/2, π/2) being the pitch,roll, and yaw angles, respectively. +e sequence 3–2–1 ishere considered [30]. Moreover, v � (vx, vy, vz)

T and ω �(ω1,ω2,ω3)

T are the linear and angular velocities of thecenter of mass of the quadrotor, expressed in RC and in RΓ,respectively.

+e translation dynamics, expressed in RC, and rotationdynamics, expressed in RΓ, of the quadrotor are

_p � v,

_v �1mR(α)Fprop +

1m

Fgrav +1m

Fd,

_α � M(α)ω,

_ω � J− 1 −ωJω + τprop − τgyro + Md ,

(1)

where m is the mass of the quadrotor, J � diag Jx, Jy, Jz (expressed in RΓ) is the inertia matrix of the quadrotor, and

ω �

0 −ω3 ω2ω3 0 −ω1

−ω2 ω1 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (2)

is the so-called dyadic representation of ω. Furthermore,

2 Mathematical Problems in Engineering

Fprop �

0

0

4

i�1Fi

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

Fgrav �

0

0

−mg

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

τprop �

τ1τ3τ2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ �

l F2 − F4(

l F3 − F1(

c F1 − F2 + F3 − F4(

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ �

lu3

lu2

u4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

(3)

are the input forces and moments produced by the pro-pellers, where l is the distance between center of mass CG tothe rotor shaft and c is the drag factor. Moreover, Fgrav in (3)is the force due to the gravity, expressed in RC.

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

R(α) �

cθcψ sϕsθcψ − cϕsψ cϕsθcψ + sϕsψ

cθsψ sϕsθsψ + cϕcψ cϕsθsψ − sϕcψ

−sθ sϕcθ cϕcθ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (4)

where c∗ � cos(∗ ), s∗ � sin(∗ ), and ∗ � ϕ, θ,ψ.+e angular velocity dynamics are expressed using the

matrix

M(α) �

1 sϕtgθ cϕtgθ0 cϕ −sϕ0 sϕscθ cϕscθ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (5)

with tg∗ � tan(∗ ), sc◇ � sec(∗ ), and ∗ � ϕ, θ,ψ.Considering small angles, matrix (5) can be approxi-

mated by the identity matrix, i.e., M(α) � I3x3. +is as-sumption is justified by the fact that the movements of thequadrotor are slow and soft [4].

+e rolling torque τ1 is produced by the forces F2 andF4. Similarly, the pitching torque τ3 is produced by theforces F1 and F3. Due to Newton’s third law, the propellersproduce a yawing torque τ2 on the body of the quadrotor,in the opposite direction of the propeller rotation.Moreover,

τgyro � 4

i�1(−1)i+1Jpωp ωϵ3, (6)

is the gyroscopic torque due to the propeller rotations, withJp being the propeller moment of inertia with respect to itsrotation axis, and ωp � ωp,1 − ωp,2 + ωp,3 − ωp,4 is the so-called rotor relative speed. Finally, Fd and Md are the forcesand torques due to the external disturbances, here assumednegligible.

Now, using (1) and (3) and under the small angle as-sumption, the mathematical model of the quadrotor can beexpressed by

€x �1m

cϕsθcψ + sϕsψ u1,

€y �1m

cϕsθsψ − sϕcψ u1,

€z � −g +1m

cϕcθu1,

€ϕ �Jy − Jz

Jxθ.

_ψ −Jp

Jxωpθ

.

+l

Jxu3,

θ..

�Jz − Jx

Jy

_ϕ _ψ +Jp

Jyωp _ϕ +

l

Jyu2,

€ψ �Jx − Jy

Jz

_ϕθ.

+1Jz

u4,

(7)

where the control variables uj, j � 1, 2, 3, 4, are defined as(3). +e values of the parameters used in (7) are defined inTable 1.

3. Formulation of the State FeedbackControl Problem

+e control problem is to ensure the asymptotic converge ofthe variables χ � (x, y, z,ψ) to some reference trajectoriesχref � (xref , yref , zref ,ψref ). In view of the control of thequadrotor, the following coordinate change is considered:_ϕ � ωϕ, θ

.

� ωθ, and _ψ � ωψ (ωϕ≃ω1, ωθ≃ω2, ωψ≃ω3), and thesystem of equations (7) became

F3F2

F1

ε2

ε3

ε1

ωp,3

ωp,4

ωp,1

ωp,2

F4

e3

e2

e1

RΓΩ

zy

x

θ

ψ

mg

ϕ

RC

0

Figure 1: Quadrotor orientation using Euler angles.

Mathematical Problems in Engineering 3

_x � vx,

_vx �1m

cϕsθcψ + sϕsψ u1,

_y � vy,

_vy �1m

cϕsθsψ − sϕcψ u1,

_z � vz,

_vz � −g +1m

cϕcθu1,

_ϕ � ωϕ,

_ωϕ �Jy − Jz

Jxωθωψ −

Jp

Jxωpωθ +

l

Jxu3,

θ.

� ωθ,

_ωθ �Jz − Jx

Jyωϕωψ +

Jp

Jyωpωϕ +

l

Jyu2,

_ψ � ωψ ,

_ωψ �Jx − Jy

Jzωϕωθ +

1Jz

u4.

(8)

Clearly, model (8) is composed of rotational andtranslational dynamics. Figure 2 shows that the input controlu1 does not influence the rotational dynamics but only thetranslational ones and can be used to impose z⟶ zref .Furthermore, in the spirit of the backstepping technique, the(ϕ, θ,ψ) can be used to impose x⟶ xref , y⟶ yref . In

turn, (ϕ, θ,ψ) are influenced by the inputs u3, u2, and u4,which can be used to impose ϕ⟶ ϕref , θ⟶ θref , andψ⟶ ψref . Hence, the following four subsystems will beconsidered: the altitude subsystem S1, the latitudinal sub-system S2, the longitudinal subsystem S3, and the yawsubsystem S4:

S1 �

_z � vz

_vz � −g +1m

cϕcθu1,

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

S2 �

_x � vx

_vx �1m

cϕsθcψ + sϕsψ u1

θ.

� ωθ

_ωθ �Jz − Jx

Jyωϕωψ +

Jp

Jyωpωϕ +

l

Jyu2,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

S3 �

_y � vy

_vy �1m

cϕsθsψ − sϕcψ u1

_ϕ � ωϕ

_ωϕ �Jy − Jz

Jxωθωψ −

Jp

Jxωpωθ +

l

Jxu3,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

S4 �

_ψ � ωψ

_ωψ �Jx − Jy

Jzωϕωθ +

1Jz

u4.

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(9)

ϕ

θ

ψ

x

y

z

u1

u2

u3

u4

Rotatio

nal

dynamics T

ranslational

dynamics

Figure 2: Connection between rotational and translationaldynamics.

Table 1: Quadrotor’s coefficients and variables.

Variable Valuem 1.1 kgl 0.223mJx 6.825×10− 3 kgm2Jy 6.825×10− 3 kgm2Jz 12.39×10− 3 kgm2Jp 6×10− 5 kgm2g 9.81m/s2c 1.1×10− 6 N s2 rad− 2

x my mz mvx m/svy m/svz m/sϕ degθ degψ degωϕ deg/sωθ deg/sωψ deg/s

4 Mathematical Problems in Engineering

+e controller design for each subsystem will be carriedout in the following section.

4. Design of the Nonlinear Controller

+e translation dynamics in equation (7) are dependent onthe ϕ, θ, and ψ angles. In the next sections, a nonlinearcontrol is used for each subsystem to solve the controlproblem.

4.1. Altitude Control. In this section are established the al-titude control and its stability proof via Lyapunov function.For this purpose, it is considered that the altitude control S1subsystem is

_z � vz,

_vz � −g +1m

cϕcθu1.

(10)

+e control aim is to maintain the quadrotor at a desiredconstant altitude zref , with vz,ref � _zref . To this aim, settingthe tracking error for the altitude control as

ez,1 � z − zref ,

ez,2 � vz − vz,ref + kz,1ez,1,(11)

the dynamic errors are

_ez,1 � −kz,1ez,1 + ez,2,

_ez,2 � −g +1m

cϕcθu1 − _vz,ref

+ kz,1 ez,2 − kz,1ez,1 + vz,ref − _zref .

(12)

+e proposed altitude control u1 is

u1 �m

cϕcθg + _vz,ref − kz,1 ez,2 − kz,1ez,1 + vz,ref − _zref

− k3ez,1 − k4ez,2,

(13)

where cϕ ≠ 0 and cθ ≠ 0.+erefore, using altitude control (13)in (12), the altitude error dynamics became

_ez,1

_ez,2 � Az

ez,1

ez,2 ,

Az �−kz,1 1

−k3 −k4 .

(14)

For the stability analysis of system (14), the followingLyapunov function candidate is proposed:

Vz ez( �12e

Tz Pzez,

ez �

ez,1

ez,2

⎛⎝ ⎞⎠,

Pz �

pz,11 pz,12

pz,12 pz,22

⎛⎝ ⎞⎠,

(15)

where Pz � PTz > 0. Deriving Lyapunov function (15), oneobtains

_Vz ez( � eTz A

Tz Pz + PzAz ez, (16)

where Pz is taken as solution of the Lyapunov equation

ATz Pz + PzAz � −Qz,

Qz �q1,z 0

0 q1,z ,

(17)

with Qz � QTz > 0 being fixed. Using (17) into (16), one finallyobtains

_Vz ez( � −eTz Qzez ≤ λ

Qzmin ez

��������2 ≤ − αVz ez( , (18)

with α � 2λQzmin/λPzmaz and Pz as solution of (17). +erefore, by

[31], Vz(ez)≤ e− αtV(ez(0)), so that the tracking errorsconverge globally exponential to zero.

4.2. Longitudinal Motion Control. In this section, the lon-gitudinal motion is studied making use of the subsystem S2and using the backstepping technique. One considers thetracking error ex,1 � x − xref , where vx,ref � _xref , and theerror dynamics

_ex,1 � vx − _xref , (19)

or, equivalently,

_ex,1 � −kx,1ex,1 + vx − _xref + kx,1ex,1

� −kx,1ex,1 + ex,2, kx,1 > 0,(20)

where

ex,2 � vx − vx,ref , (21)

is the velocity error and

vx,ref � _xref − kx,1ex,1, (22)

is the reference velocity. Successively, one works out thevelocity error dynamics as

Mathematical Problems in Engineering 5

_ex,2 �1m

cϕsθcψ + sϕsψ u1 − €xref + kx,1 _ex,1

� −kx,2ex,2 +1m

cϕsθcψu1 +1m

sϕsψu1 − €xref − k2x,1ex,1

+ kx,1 + kx,2 ex,2,

(23)

with kx,2 > 0, where u1 is given by (13). For the stabilizationof ex,2, the reference value for sθ can be fixed as follows:

sθ,ref �m

cϕcψu1€xref + k

2x,1ex,1 − kx,1 + kx,2 ex,2 −

sϕsψ

cϕcψ,

(24)

where cϕ ≠ 0 and cψ ≠ 0, so that

_ex,2 � −kx,2ex,2 +1m

cϕcψu1eθ,1,

eθ,1 � sθ − sθ,ref ,

θref � arcsinsθ,ref .

(25)

To impose such a reference for θ, one considers the trackingerror eθ,1 and its derivative:

_eθ,1 � −kθ,1eθ,1 + cθωθ − θ.

refcθref + kθ,1eθ,1 � −kθ,1eθ,1 + cθeθ,2,

(26)

with kθ,1 > 0, where

ωθ,ref �1cθ

θ.

refcθref − kθ,1eθ,1 , (27)

is the reference angular velocity and

eθ,2 � ωθ − ωθ,ref , (28)

is the angular velocity error. +erefore, one works out thevelocity error dynamics as

_eθ,2 �Jz − Jx

Jyωϕωψ +

Jp

Jyωpωϕ +

l

Jyu2 − _ωθ,ref . (29)

Finally, one determines the input control u2 as

u2 � −Jz − Jx

lωϕωψ −

Jp

lωpωϕ +

Jy

l_ωθ,ref − kθ,2

Jy

leθ,2, (30)

so that the velocity error dynamics become

_eθ,2 � −kθ,2eθ,2, (31)

with kθ,2 > 0.

4.3. Lateral Motion Control. In this section is designed thecontroller for the lateral motion along the y-axis of thequadrotor, using the same procedure followed in Section 4.2.One starts with the tracking error

ey,1 � y − yref , (32)

and the dynamics

_ey,1 � vy − _yref � −ky,1ey,1 + vy − _yref + ky,1ey,1 , ky,1 > 0,

� −ky,1ey,1 + ey,2, ey,2 � vy − vy,ref ,

(33)

with vy,ref � _yref − ky,1ey,1. +e dynamics of ey,2 are

_ey,2 � _vy − _vy,ref � −ky,2ey,2 −1m

cψu1eϕ,1,

sϕref �m

cψu1− €yref − k

2y,1ey,1 + ky,1 + ky,2 ey,2 +

sψ

cψcϕsθ,

eϕ,1 � sϕ − sϕref ,

ϕref � arcsinsϕref ,

(34)

with ky,2 > 0, cψ ≠ 0, and u1 given by (13). Furthermore, theangular error dynamics are

_eϕ,1 � −kϕ,1eϕ,1 + cϕωϕ − _ϕrefcϕref + kϕ,1eϕ,1� −kϕ,1eϕ,1 + cϕeϕ,2,

(35)

with kϕ,1 > 0 and

eϕ,2 � ωϕ − ωϕ,ref ,

ωϕ,ref �1cϕ

_ϕrefcϕref − kϕ,1eϕ,1 .(36)

Finally, the angular velocity error dynamics are

_eϕ,2 �Jy − Jz

Jxωθωψ −

Jp

Jxωpωθ +

l

Jxu3 − _ωϕ,ref , (37)

so that the control input is

u3 � −Jy − Jz

lωθωψ +

Jp

lωpωθ +

Jx

l_ωϕ,ref − kϕ,2

Jx

leϕ,2, (38)

with kϕ,2 > 0.

4.4. YawMotionControl. Considering the subsystem S4, andthe errors

eψ,1 � ψ − ψref ,

eψ,2 � ωψ − ωψ,ref ,(39)

one works out the error dynamics as

_eψ,1 � ωψ − _ψref � −kψ,1eψ,1 + eψ,2,

ωψ,ref � _ψref − kψ,1eψ,1,

_eψ,2 �Jx − Jy

Jzωϕωθ +

1Jz

u4 − _ωψ,ref ,

_ωψ,ref � €ψref + k2ψ,1eψ,1 − kψ,1eψ,2,

(40)

6 Mathematical Problems in Engineering

with kψ,1 > 0. +erefore, with the control

u4 � − Jx − Jy ωϕωθ + Jz _ωψ,ref − kψ,2Jzeψ,2, kψ,2 > 0,

(41)

one finally obtains the yaw error dynamics as

_eψ,1

_eψ,2

⎛⎝ ⎞⎠ �−kψ,1 1

0 −kψ,2⎛⎝ ⎞⎠

eψ,1

eψ,2

⎛⎝ ⎞⎠. (42)

+e global exponential stability can be inferred analo-gously to what is done in Section 4.1.

5. Simulation Results

+e photogrammetric technique has been used for mea-suring the important physical quantities in both ground andflight testing, including attitude, position, and shape ofobjects. To generate the trajectory reference

pref(t) � xref(t)yref(t)zref(t)( , (43)

a photogrammetric technique was applied to analyze anatural scenery, with

xref(t) � f1(x(t), y(t), z(t)),

yref(t) � f2(x(t), y(t), z(t)),

zref(t) � f3(x(t), y(t), z(t)),

(44)

and with fi, i � 1, 2, 3, functions of (x(t), y(t), z(t)). +eselected scenario is a serial array of trees, where all trees haveapproximately the same separation. Figures 3 and 4 show thetrees’ frontal and lateral views, respectively. Using thephotogrammetry, the following estimations were deter-mined: Wf � 0.85m for the trees’ width, Tt � 2.72m for thedistance between the two trees, and Ws,1 � 0.75m andWs,2 � 0.522m for the widths of the two trees.

For this application, the reference trajectory has twoimportant characteristics:

(1) +e altitude is constant(2) +e reference trajectory is periodic over the x–y

plane(3) +e reference trajectory avoids the collision between

the quadrotor and the trees

+erefore, the reference position was set equal to

pref(t) �

xref(t)

yref(t)

zref(t)

⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

�

vx,ref t

Aref sin 2πfrefxref(t)( 1

⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

�

0.5t4.25 sin 0.250πxref(t)(

1

⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠,

(45)

with vx,ref � 0.5m/s, Aref being the amplitude, andfref beingthe frequency of the periodical function, such as

fref �1

Tref�

12 Ws,1 + Ws,2 + Tt

� 0.125 cycles/m, (46)

where Tref is the period. Figure 5 shows the reference tra-jectory for xref ∈ [0, 18]m.

+e behavior of the controllers designed in Section 4 hasbeen tested with numerical simulations on the quadrotordescribed by equation (1). +e parameters used are given inTable 1, whilst the variables and gains used in the controllers

Wf = 0.85m

Figure 3: Frontal view of the natural scene.

Figure 4: Lateral view of the natural scene.

05

1015

20

x–5

0

5

y

0

0.5

1

1.5

2

z

Figure 5: Reference trajectory pref(t).

Table 2: Controllers’ gains values.

kz,1 � 0.5 k3 � 120 k4 � 120 q1,z � 1kx,1 � 5 kx,2 � 2 kθ,1 � 25 kθ,2 � 46ky,1 � 5 ky,2 � 2 kϕ,1 � 25 kϕ,2 � 32kψ,1 � 5 kψ,2 � 9

Mathematical Problems in Engineering 7

are given in Table 2. In order to simulate the controller foreach subsystem, the software developed by the PSP forSimulink was used.+e reason for choosing the PSP softwarewas due to the performance of this software in predicting thedynamic quadcopter behavior, which is very close to the realbehavior. In fact, the PSP software is proven to representaccurately the quadrotor dynamics. In order to appropriatelyimplement the controllers, the reference trajectory wassampled as in Figure 6.

+e simulations results of the closed-loop system areshown in Figures 7 and 8, where the quadrotor’s initialconditions have been set equal to p(0) � (0, 0, 0)T,

v(0) � (0, 0, 0)T, α(0) � (0, 0, 0)T, and ω(0) � (0, 0, 0)T.Figure 7 shows the quadrotor altitude and yaw, z(t) andψ(t), ensured by the controllers (13) and (18). Regarding thelatter, the most significative tracking error is 0.5 rad im-mediately after the initial time.

Figure 8 shows the quadrotor x and y positions en-sured by the controllers (30) and (38). +e x positionachieves its reference value in 3 s, and the y positionachieves its reference value in 6 s. +e overall controllerperformance is accurate and fast. Moreover, this per-formance could be improved using also integral terms inthe controllers.

05

1015

20

–5

0

5

xy

0

0.5

1

1.5

2

z

Figure 6: Sampled reference trajectory pref .

0

0.5

1

z (m

)

10 20 30 40 50 60 700Time (s)

(a)

10 20 30 40 50 60 700Time (s)

–0.5

0

0.5

ψ (r

ad)

(b)

Figure 7: (a) Quadrotor’s altitude z(t) (solid) and reference altitude zref(t) (dashed); (b) quadrotor’s yaw angle ψ(t) (solid) and yaw anglereference ψref(t) (dashed).

10 20 30 40 50 60 700Time (s)

0

10

20

30

x (m

)

(a)

10 20 30 40 50 60 700Time (s)

–5

0

5

y (m

)

(b)

Figure 8: (a) Longitudinal motion of the UAV x (solid) and the reference x1,ref (dash); (b) latitudinal motion of the UAV y (solid) and thereference y1,ref (dash).

8 Mathematical Problems in Engineering

6. Conclusion

In this paper, a controller based on the backsteppingtechnique has been designed for the position and yawcontrol of a quadrotor helicopter. +e quadrotor dynamicshave been divided into four subsystems, each with a controlinput and an output variable. +e overall controller leads tosatisfactory results. +e photogrammetric technique on ascenario with fixed obstacles, due to some trees, has beenused to determine the scene geometry and to fix the positionreference trajectory. +e numerical simulations of theproposed controllers have been implemented using the PSP,ensuring an accurate approximation of the real quadrotordynamics, and the results show a good performance andeffectiveness of the proposed control law. Future work willregard the real implementation of the proposed controller.

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 work was partially supported by the European ProjectECSEL–JURIA–2018 “Comp4Drones” and by the Project“Coordination of Autonomous Unmanned Vehicles forHighly Complex Performances,” Executive Program ofScientific and Technological Agreement between Italy(Ministry of Foreign Affairs and International Cooperation,Italy) and Mexico (Mexican International CooperationAgency for the Development), SAAP3.

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10 Mathematical Problems in Engineering