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Sliding Mode Controller Based on the Sliding Mode Observer for a QBall 2+ Quadcopter with Experimental Validation

Ayoub Daadiâ,1, Houssam Boulebtinaiâ,2, Saddam Hocine Derrouaouiâ,3,∗, Fares Boudjemaâ,4

aEcole Sup´erieure Ali Chabati, Algires, Algeria

1daadiayoub2022@gmail.com;²HoussamBoulebtinai@gmail.com;³derrouaouish@gmail.com;

4fboudjema@yahoo.fr

∗Corresponding Author

ARTICLE INFO

Article History Received April 22, 2022 Revised May 15, 2022 Accepted May 27, 2022

Keywords

QBall 2+ quadcopter;

PID controller;

Sliding Mode Observer;

Sliding Mode Control;

Flight tests

ABSTRACT

This paper studies a particular Unmanned Aerial Vehicle (UAV), called QBall 2+ quadcopter. This vehicle is a complex system, non-linear, strongly coupled, and under-actuated. First, a non-linear model was developed to represent the dynamics of the studied drone. Once the latter is established, the linear model was used to obtain the best gains of the Proportional Integral Derivative (PID) controller. This controller was applied after on the non-linear model of the UAV. Moreover, a Sliding Mode Controller (SMC) based on Sliding Mode Observer (SMO) was designed for retrieving the system unknown variables.

Through these latter, the QBall 2+ was controlled, taking into account the observer errors. The first contribution in this work is to implement the PID regulator on the QBall 2+ flight controller to validate the results obtained by simulation. Secondly, due to the limitations of the Flex 3 cameras, especially when the drone is outside their working environment, the sliding mode observer was implemented to replace the cameras in order to measure the states of the system considered in this work. Simulation results of the different applied controllers were displayed to evaluate their effectiveness.

This is an open access article under theCC-BY-SAlicense.

1. Introduction

Recently, in the field of flying robots, the use of quadrotors has drastically increased. They have many advantages compared to the fixed-wing drones in terms of maneuverability, structure and versa- tility. Moreover, they have a simple mechanical design, a small size and a low cost[1, 2, 3, 4, 5, 6, 7].

Quadrotors are widely exploited in military and civilian missions such as, search and rescue, mapping, protection of sensitive points, and infrastructure inspection[8, 9, 10, 11, 12]. The QBall 2 quadcopter is an innovative platform intended for scientific research to fly indoor using Optitrack Flex-3 cameras[13]. It has a strongly coupled dynamics and an external protective cage[13, 14, 15, 16, 17]. It was built by Quanser, a Canadian robot manufacturing company. It has an open architecture design that allows operators to test a variety of controllers ranging from basic flight dynamics stabilizers to advanced multi-vehicle path planning and navigation algorithms. The drone has an on-board data acquisition system, an on-board computer and a suite of sensors. The Qball 2+ is an innovative rotary wing vehicle platform suitable for a wide variety of research applications on UAVs. However, it has a working environment limitation by the field of view of the Flex 3 cameras, which are sensitive to light resources and ground reflection. It is designed with an outer protective cage. In the real prototype, the cage gives the QBall 2+ a decisive advantage over other vehicles that would cause significant damage

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in the event of contact between the vehicle and an obstacle, or in the event of a failure[13].

The researchers in paper[18], have performed an Extended Kalman Filter (EKF), this non-linear observer was exploited to control the QBall 2 drone. In addition, they have also applied a Linear Quadratic Regulator (LQR) on the same system to evaluate its performance. Moreover, the obtained results of the designed observer are satisfactory in the presence of model uncertainty. The authors in work[19], have designed an optimal PID controller with a Kalman filter. The latter is based on the time integral taking into account the errors of the system. However, they did not consider the different faults of the drone. In the same context, an Observer based on Sliding Mode Control (OSMC) has been considered in[20, 21, 22], to overcome the practical limitations of rotational and translation speed. A nonlinear PID controller with gains tuning has been implemented to follow the desired 3D trajectory of the QBall 2 quadrotor [23]. In addition, the experimental tests were carried out, where the quadrotor is subjected to external disturbances (gust of wind) in flight. An optimal fault tolerant controller was applied to stabilize the QBall 2 drone[24]. This approach is based mainly on a Finite Time Sliding Mode Observer (FTSMO), which estimates the completeness from outputs and allows to identify the disturbances type. In addition, the proposed strategy has proven its robustness under the influence of external disturbances and uncertainties. In reference[25], the researchers have introduced a new Lyapunov-based trajectory tracking approach to control and stabilize a quadrotor UAV. Furthermore, they have diagnosed an actuator failure using an intelligent output estimator. In article[26], two control algorithms have been applied on a quadrotor drone. The first one is designed to detect and diagnose actuator faults, where the second one works online after detecting the actuator faults. Kose et al. [27], have developed a dynamic model and Matlab simulation for the QBall 2.

The developed model was validated through various flight scenarios. More recently, Wang et al.[28], have designed and applied an adaptive fault tolerant approach to control a small quadcopter. This approach was tested against actuator faults as well as unknown system parameters. However, the different external disturbances were not studied in this work.

The first contribution in this work is to implement the PID regulator on the QBall 2+ flight controller to validate the results obtained by simulation. To achieve this goal, the various ”Optitrack Flex-3” cameras must be calibrated and tested to locate the position of the quadcopter. Secondly, due to the limitations of the Flex 3 cameras, especially when the drone is outside their working environment, the sliding mode observer was implemented to replace the cameras in order to measure the states of the system considered in this work.

The paper is structured as follows: InSection 1, a mathematical model that represents the dynamic behavior of the QBall 2 was developed. Section 2explains the design steps of the different control techniques. Simulation results were displayed inSection 3. Section 4is devoted to validate the PID controller through some experimental tests.Section 5is intended for conclusion and future works.

2. Modeling of QBall 2+

The following assumptions are taken into account for the dynamic model of QBall 2+[28]:

• The structure of the QBall 2+ is assumed to be rigid and symmetrical, which means that the inertia matrix is diagonal.

• The propellers are supposed to be rigid in order to neglect the effect of their deformation during the rotation.

• The inertia matrixJ ∈R^3×3 is assumed to be constant.

• The origin of the body frame is supposed to coincide with the center of gravity (seeFig. 1).

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With aim to simplify the understanding of the multitude of variables and terms of this work, the notation Table is given as follows (seeTable 1).

Table 1.Table of notation.

Symbols Notation

[x y z]^T Position vector

˙

χ= [ ˙x y˙ z]˙^T Translation speeds vector.

[¨x y¨ z]¨^T Translation accelerations vector.

[θ φ ψ]^T Euler angles vector.

Ω = [p q r]^T Rotation speed vector.

Ω = [ ˙˙ p q˙ r]˙^T Vector of angular accelerations.

V = [u υ ω]^T Vector of linear velocities in the body frame.

J Matrix of angular moments.

I_xx Moment of inertia of the quadcopter along the x axis (Roll axes).

Iyy Moment of inertia of the quadcopter along the y axis (Pitch axes).

I_yz Moment of inertia of the quadcopter along the z axis (Yaw axes).

J_r Represents the inertia matrix of each rotor.

Jrx Inertia of the rotor along the x axis.

Jry Inertia of the rotor along the y axis.

J_rz Inertia of the rotor along the z axis.

M Mass of QBall 2+.

P Weight of the QBall 2+.

R₀ Terrestrial landmark.

R1 Landmark linked to the body of the quadcopter.

Rp Crossing matrix ofR0toR1. F_g_R

0 Force of gravity.

Fi Thrust generated by each propeller.

F_P_R

1 Lift created by the propellers in body frame.

D_R₁ Drag Force.

Kd Drag coefficient.

τ_R_I Torques applied to the quadcopter.

L Distance between the center of gravity and the axis of rotation of the rotor.

τi Torque generated by each motor.

τ_gyr Gyroscopic moment.

K_v Voltage transformation constant at the input of each motor to angular velocity.

υ Battery output voltage.

τ_f Friction torque.

kf Coefficient of friction.

U = [U1U2 U3U4]^T Control vector.

u_i Input Pulse Width Modulation (PWM) of actuatori.

Vi Lyapunov’s function.

2.1. Translation Dynamics

Using Newton-Euler’s law, which is the sum of all external forces acting on a rigid body and equal the modulus of its mass per acceleration as following

F_R₀ =M d dt(V_R₀)

F_R₁ =MV˙_R₁+M(Ω∧V_R₁)

(1)

(4)

where

Ω∧VR1 =





qw−rυ ru−pw pυ−qu



 (2)

where(Ω∧V_R₁) is called the Coriolis term indicating the coriolis effects, and M is the quadrotor mass.

Fig. 1.Frames used to describe the movement of the Qball 2+.

The main forces applied on QBall 2+ are defined as

• Force of gravity

The force of gravityF_g_R

0 = [00-Mg]^T in the fixed frame is along theZaxis (yaw axis), its expression in the mobile frameR1is giving as follows

F_g_R

1 =R_p^TF_g_R

0 =R^T_p



 0 0

−M g



=





M gsin(θ)

−M gsin(φ) cos(θ)

−M gcos(φ) cos(θ)



 (3) withR^T_p is the passage matrix from the fixed frame to the body frame, and

Rp=





C(ψ)C(θ) C(ψ)S(θ)S(φ)−S(ψ)C(φ) C(ψ)S(θ)C(φ) +S(ψ)S(φ) S(ψ)C(θ) S(ψ)S(θ)S(φ) +C(ψ)C(φ) S(ψ)S(θ)C(φ)−C(ψ)S(φ)

−S(θ) C(θ)S(φ) C(θ)C(φ)



 (4) with

C(x) = cos(x) andS(x) = sin(x).

• Force of thrust

It is a resultant obtained by the rotation of each rotor, it is parallel to the Z axis expressed in the body frame R₁. The thrust generated by each propeller is modeled using a first order function[13]. Its model is different to the model of the conventional quadrotors, where it is given by

Fi=K w

s+wui, i= 1,2,3,4. (5) withKis a positive gain andωis the actuator bandwidth.

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• Force of drag

It’s expressed in the body frame as[29, 30]

DR1 = −K_dχ˙ (6)

withK_drepresents the drag coefficient.

According to Equations (5) and (6), we get the various forces expressed in the body frame as following

F_T_R

1 =Fg_R₁ +F_P +D_R₁ (7)

Therefore, Equations (1) and (7) become







Mu˙+M(qw−rv) =−M gsin(θ)−Kxx˙ Mv˙+M(ru−pw) =M gsin(φ) cos(θ)−Kyy˙ Mw˙ +M(pv−qu) =M gcos(θ) cos(ψ)−K_zz˙+F_P_R

1

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2.2. Rotation Dynamics

Using Newton’s second law, which is the sum of the external moments equal to the temporal variation of the angular momentum as follows

τ_R₀ =J_dt^d(Ω_R₀)

τR1 =JΩ + Ω˙ ∧JΩ (9)

whereτ_R₀andτ_R₁are the moments applied on the quadrotor expressed in the fixed and mobile frames respectively.

The different moments (roll, pitch, yaw), which cause the rotation of the QBall 2 along the three axes (seeFig. 2) are given as follows





 τ_θ_R

1 =L(F1−F2) τφR1 =L(F3−F4) τ_ψ_R

1 =τ₁+τ₂−τ₃−τ₄

(10) whereLrepresents the distance between the center of gravity and the axis of rotation of the rotor,τi

is the torque generated by each rotor, it is assumed to have the following relation with respect to the PWM input as

τ_i =K_yu_i, i= 1,2,3,4. (11) whereK_yis a positive gain.

• Gyroscopic moment

It is general expression is given by:

τgyr=







(u₃+u₄−u₁−u₂)J_rzqK_vυ (u1+u2−u3−u4)JrzpKvυ

0







(12)

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Fig. 2.Schematic representation of QBall 2+.

• Friction torque

It’s the resultant of the aerodynamic friction couples. It is modeled as

τ_f = kfΩ² (13)

with

k_f =





k_{f x} 0 0 0 k_{f y} 0 0 0 k_{f z}



 (14)

Equation (9) becomes







J_xxp˙+qr(J_zz−J_yy) =L(F₃−F₄)−J_rzqu_g−k_{f x}p² Jyyq˙+pr(Jxx−Jzz) =L(F1−F2) +Jrzpug−kf yq² J_zzr˙+pq(J_yy−J_xx) =τ₁+τ₂−τ₃−τ₄−k_{f z}r²

(15) withug = (u1+u2−u3−u4)Kvυ.

Due to the weak orientations of QBall 2+, the relation which connects the angular speeds expressed in referenceR₁and the derivatives of the angles of Euler expressed in referenceR₀ is given by

( ˙φ,θ,˙ ψ)˙ ≈(p, q, r) (16) From Equations (8), (15) and (16), the dynamics model of QBall 2+ in the fixed frame is represented by the following state equations











¨

x= _M¹ F_P_R

1(C(ψ)S(θ)C(φ) +S(ψ)S(φ))−^K_M^xx˙

¨

y= _M¹FPR1(S(ψ)S(θ)C(φ)−C(ψ)S(φ))−^K_M^yy˙

¨

z= _M¹ F_P_R

1(C(θ)C(φ))−K_z Mz˙−g φ¨= ^L(F_J³^−F⁴⁾

xx −^J^rz_J^qu^g

xx −θ˙ψ(J˙ zz−Jyy) J_xx −k_{f x}

J_xxp² θ¨= L(F1−F2)

J_yy +^J^rz_J^pu^g

yy −φ˙ψ(J˙ xx−Jzz) J_yy −^k_J^{f y}

yyq² ψ¨= ^τ¹^+τ²_J^−τ³^−τ⁴

zz −^φ^˙^θ(J^˙ ^yy_J^−J^xx⁾

zz − ^k_J^{f z}

zzr²

(17)

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3. Control of QBall 2+

3.1. Linear Control Model

In the almost stationary state, it is shown that the total thrust is approximately equal to the weight, i.eFP_R₁ ≈M g[31]. Due to the small amplitude movements of QBall 2+, it’s assumed that

cos(φ) = 1; cos(θ) = 1; cos(ψ) = 1 and sin(φ) =φ; sin(θ) =θ; sin(ψ) =ψ (18) Then, the dynamics of QBall 2 expressed by Equation (17) can be simplified as follows:











¨ x=gθ

¨

y=−gφ

¨

z= _M¹ F_P_R

1 −g φ¨= ^L(F_J³^−F⁴⁾

xx

θ¨= ^L(F_J¹^−F²⁾

yy

ψ¨= ^τ¹^+τ²_J^−τ³^−τ⁴

zz

(19)

3.2. Nonlinear Control Model

For the nonlinear control model, the state equations are used as











˙ x1 =x2

˙

x₂ =a₁x₄x₆+a₂x²₂+a₃x₄u_g+b₁U₂

˙ x3 =x4

˙

x₄ =a₄x₂x₆+a₅x²₄+a₆x₂u_g+b₂U₃

˙ x5 =x6

˙

x₆ =a₇x₂x₄+a₈x²₆+b₃U₄

˙ x7 =x8

˙

x₈ =a₉x₈+^U_M^xU₁

˙

x9 =x10

˙

x₁₀=a₁₀x₁₀+^u_M^yU₁

˙

x11=x12

˙

x12=a11x12+_M¹U1c(x1)c(x3))−g

(20)

with

a₁=−^(J^zz_J^−J^yy⁾

xx , a₂=−^k_J^{f x}

xx, a₃ =−^J_J^rz

xx, b₁ = _J^L

xx

a4=−^(J^xx_J^−J^zz⁾

yy , a5 =−^k_J^{f y}

yy, a6 = _J^J^rz

yy, b2 = _J^L

yy

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a7=−^(J^yy_J^−J^xx⁾

zz , a8=−^k_J^{f z}

zz, b3 = _J¹

zz

a₉=−^K_M^x,a₁₀=−^K_M^y,a₁₁=−^K_M^z UX =C(x1)S(x3)C(x5) +S(x1)S(x5)

U_y =C(x₁)S(x₃)S(x₅)−S(x₁)C(x₅)

The control vector is expressed by a thrust force and moments as U = [U1 U2 U3 U4]^T with





 U1

U₂ U₃ U4







=







k k k k

0 0 k −k

k −k 0 0

ky ky −k_y −k_y











 u1

u₂ u₃ u4







(21)

The choice of state vector is given as follows X=n

φ φ˙ θ θ˙ ψ ψ˙ x x˙ y y˙ z z˙ o

(22) The system outputs are given as

Y = [φ θ ψ x y z]^T 3.3. PID Controller Design

Here, the PID controller based on the linear model presented by (19) is designed. This control law will be applied after to the nonlinear control model presented by (20).

The PID control laws are given as

U1 =Kpzez+Kiz

Z +∞

0

ez+K_dzdez

dt (23)

U2=Kpφeφ+Kiφ

Z +∞

0

eφ+Kdφ

de_φ

dt (24)

U₃=K_pθe_θ+K_iθ Z +∞

0

e_θ+K_dθde_θ

dt (25)

U₄ =K_pψe_ψ+K_iψ Z +∞

0

e_ψ+K_dψde_ψ

dt (26)

After choosing the altitude and attitude regulators, a position controller is developed. The PID controllers are used to calculate the desired positions[32], and the desired rollφdand pitchθdangles, which are given by







θ_d=K_px∗e_x+K_ix∗R+∞

0 e_x+K_dx∗ ^de_dt^x

φ_d=Kpy∗ey+Kiy∗R+∞

0 ey +K_dy∗ ^de_dt^y

(27) withez =z_d−z,e_φ=φ_d−φ,e_θ =θ_d−θ,e_ψ =ψ_d−ψ,ex=x_d−x,ey =y_d−y, andKP, K_i andK_dare respectively proportional, integral and derivative gains.

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3.4. Sliding Mode Observer (SMO) Design 3.4.1 Sliding Mode Observer State Model

In this subsection, a nonlinear sliding mode observer is designed. The latter will be tested against uncertain system parameters. The observer state model is given by

Σ :ˆ











xˆ˙1 = ˆx2+ Λ1(z1)

xˆ˙₂ =a₁xˆ₄xˆ₆+a₂xˆ²₂+a₃xˆ₄u_g+b₁U₂+ Λ₂(z₂) xˆ˙3 = ˆx4+ Λ3(z3)

xˆ˙₄ =a₄xˆ₂xˆ₆+a₅xˆ²₄+a₆xˆ₂u_g+b₂U₃+ Λ₄(z₄) xˆ˙5 = ˆx6+ Λ5(z5)

xˆ˙₆ =a₇xˆ₂xˆ₄+a₈xˆ²₆+b₃U₄+ Λ₆(z₆) xˆ˙7 = ˆx8+ Λ7(z7)

xˆ˙₈ =a₉xˆ₈+^U_M^ˆ^xU₁+ Λ₈(z₈) xˆ˙9 = ˆx10+ Λ9(z9)

xˆ˙₁₀=a₁₀xˆ₁₀+^U_M^ˆ^yU₁+ Λ₁₀(z₁₀) xˆ˙11= ˆx12+ Λ11(z11)

xˆ˙₁₂=a₁₁xˆ₁₂+_M¹U₁(c(ˆx₁)c(ˆx₃))−g+ Λ₁₂(z₁₂)

(28)

where the dynamic of the observer errors is expressed as follows











˙

z1=z2−Λ1

˙

z₂=a₁∆_x₄_x₆+a₂∆_x2

2 +a₃∆_x₄u_g−Λ₂

˙

z3=z4−Λ3

˙

z₄=a₄∆_x₂_x₆+a₅∆_x2

4 +a₆∆_x₂u_g−Λ₄

˙

z5=z6−Λ5

˙

z₆=a₇∆_x₂_x₄+a₈∆_x²

6 −Λ₆

˙

z7=z8−Λ7

˙

z₈=a₉z₈+_M¹ ∆_U_xU₁−Λ₈

˙

z9=z10−Λ9

˙

z₁₀=a₁₀z₁₀+_M¹ ∆_U_yU₁−Λ₁₀

˙

z11=z12−Λ11

˙

z₁₂=a₁₁z₁₂+_M¹ U₁∆_(c(x₁_)c(x₃₎₎−Λ₁₂

(29)

with











∆_x_i_x_j =x_ix_j−xˆ_ixˆ_j

∆_x²

i =x²_i −xˆ²_i

∆_x_i =x_i−xˆ_i

∆_(c(x₁_)c(x₃₎₎= (c(x₁)c(x₃))−(c(ˆx₁)c(ˆx₃))

∆Ux =Ux−Uˆx

∆_U_y =U_y−U_y_ˆ

(30)

(10)

and 





zi=yi−yˆi if i odd.

z_i=x_i−xˆ_i if i pair.

(31) with

Λ_i(z_i) =K_isign(x_i−xˆ_i)

3.4.2 Observer Design Assumptions

With the aim to design the sliding mode observer, we must define the measured states and the missing states. The measurements of the optitrack cameras are:











φ=x₁ θ=x3

ψ=x₅

φ˙ =x₂ θ˙=x4

ψ˙=x₆

x=x₇ y=x9

z=x₁₁

(32)

Nevertheless, the unmeasured states are given as











˙ x=x₈

˙ y=x10

˙ z=x₁₂

(33)

3.4.3 Observer Stability Analysis

In order to study the stability and the convergence of the observer, it’s necessary to choose the candidate Lyapunov function, which is a function of the error and then the stability has to be proved.

The errors must converge to zero i.e. the estimated states must converge to the actual states after the following steps.

• Step 1: Considering the dynamics of the error of the first observer as

˙

z₁=z₂−K₁sign(z₁) (34) The choose of the Lyapunov function is giving as following

V₁ = 1

2z²₁ (35)

Its derivative is given as

V˙₁ =z₁z˙₁ =z₁(z₂−K₁sign(z₁)) (36) So thatxˆ1→x1it is necessary thatV˙1 <0. For that, theK1is chosen as

K₁ >|z₂|_max (37)

Then, the errorz₁ →0after a finite timet₁, which gives

z2 =K1sign(z1) (38)

After that,x2becomes

x₂= ˆx₂+K₁sign(z₁) (39)

(11)

• Step 2: In this step,z2converges towards zero.

To stay on the surfacez2= 0, it must beK1 >|z₂|_max, but this is verified ifz2decreases after t1.

The dynamics of errorz2becomes

˙

z₂=a₁∆_x₄_x₆ +a₂∆_x²

2 +a₃∆_x₄u_g−K₂sign(z₂) Then, the Lyapunov function is taken as

V₂ = ¹₂z₁²+ ¹₂z₂²

Its derivative is

V˙₂=z₁z˙₁+z₂z˙₂ =z₂(a₁∆_x₄_x₆ +a₂∆_x2

2+a₃∆_x₄u_g−K₂sign(z₂))

To getV˙₂ <0, it’s necessary to choose

K2 >|a₁∆x4x6+a2∆_x²

2 +a3∆x4ug|_max Therefore,z₂→0after a finite timet₂.

The steps 3, 5, 7, 9, and 11 are identical to step 1, where the obtained results are:

• Step 3:K₃ >|z₄|_max, thereforez₃→0after at₃finished, andx₄ = ˆx₄.

• Step 5:K5 >|z₆|_max, thereforez6→0after at5finished, andx6 = ˆx6.

• Step 7:K₇ >|z₈|_max, thereforez₇→0after at₇finished, andx₈ = ˆx₈.

• Step 9:K₉ >|z₁₀|_max, thereforez₉ →0after at₉finished, andx₁₀= ˆx₁₀.

• Step 11:K11>|z₁₂|_max, thereforez11→0after at11finished, andx12= ˆx12. Also for the steps 4, 6, 8, 10, and 12 will be identical to step 2.

• Step 4: The dynamics of errorz₄will be:

˙

z₄=a₄∆_x₂_x₆+a₅∆_x²

4 −K₄sign(z₄) Involved

K₄ >|a₄∆_x₂_x₆+a₅∆_x²

4|_max To ensure the convergence ofz4 →0after a finite timet4.

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• Step 6: The dynamics of errorz6will be

˙

z₆ =a₈∆_x2

6−K₆sign(z₆) Involved

K6>|a₈∆_x²

6|_max

To ensure the convergence ofz6 →0after a finite timet6.

˙

z₈=−K₈sign(z₈)

(U_x −U_x_ˆ) = 0 becauseU_x and U_x_ˆ are modeled according to (x₁, x₃, x₅) and(ˆx₁,xˆ₃,xˆ₅) respectively.

Here, the choice ofK₈is given as

K8 >0

To ensure the convergence ofz₈ →0after a finite timet₈.

• Step 10: The dynamics of errorz₁₀will be

˙

z₈ =−K₁₀sign(z₁₀)

(Uy −Uyˆ) = 0because Uy and Uyˆ are modeled according to(x1, x3, x5)and (ˆx1,xˆ3,xˆ5) respectively.

The choice ofK₁₀is given as

K₁₀>0

To ensure the convergence ofz10→0after a finite timet10.

˙

z12=−K₁₂sign(z12)

∆_(c(x₁_)c(x₃₎₎= 0are modelled according to(x₁, x₃)and(ˆx₁,xˆ₃).

In this step, the choice ofK₁₂is given as

K12>0

To ensure the convergence ofz₁₂→0after a finite timet₁₂.

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3.5. Sliding Mode Control Law Design

The choice of the sliding surface is given by a general form as

S = ˙Z_i+α_iZ_i, Z_i=x_id−x_i (40) Its derivative is obtained as

S˙_i =−k_isign(S_i) = ¨Z_i+α_iZ˙_i (41) The control lawU₁is developed as follows.

We assume that

z11=z_d−x11=⇒z˙11= ˙z_d−x˙11= ˙z_d−x12 (42) We replacex₁₂byxˆ₁₂+e₁₂

¨

z₁₁= ¨z_d−x˙₁₂= ¨z_d−(ˆ˙x₁₂+ ˙e₁₂) (43) Therefore

¨

z11= ¨z_d−e12−g−k12sat(e11)− C(x1)C(x2)

M U1 (44)

Now, we suppose thate₁₂ande˙₁₂are bounded which gives:







e12≤|e12|_max=β12e

˙

e₁₂≤|e˙₁₂|_max= ˙β₁₂.

(45) Thus

S˙11=g+ ¨zd−e˙12−k₁₂sat(e11)+α11( ˙zd−xˆ12−e₁₂)−C(x1)C(x2)

M U1−a₁₁xˆ12=−k₁₁sign(S11) (46) We use the upper terms, we find

U₁ = M

C(x1)C(x2)(k₁₁sign(S₁₁)+α₁₁( ˙z_d−ˆx₁₂−β₁₂)+g+ ¨z_d−a₁₁xˆ₁₂−β˙₁₂−K₁₂sat(e₁₁)) (47) Adopting the same steps of development, we obtain the others controllers represented by the system of equations as











U2 = _b¹

1[k1sign(S_φ) + ¨φ_d+α1( ˙φ_d−x2)−a1x4x6

−a₂x²₂−a₃x₄u_g] U3 = _b¹

2[k3 sign(S_θ) + ¨θ_d+α3 ( ˙θ_d−x4)−a4x2x6

−a₅x²₄−a₆x₂u_g] U4 = _b¹

3[k5sign(S_ψ) + ¨ψ_d+α5( ˙ψ_d−x6)−a7x2x4

−a₈x²₆]

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and







Ux= _U^M

1(k7sign(S7) + ¨xd−a9xˆ8−K8sat(e7)−β˙8+α7( ˙xd−xˆ8−β8)) U_y = ^M_U

1(k₉sign(S₉) + ¨y_d−a₁₀xˆ₁₀−K₁₀sat(e₉)−β˙₁₀+α₉( ˙y_d−xˆ₁₀−β₁₀))

(49) Fig. 3shows the block diagram of the sliding mode observer developed in this work.

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Fig. 3.Block diagram of the sliding mode observer.

The Qball 2+ control architecture is composed of three main blocks, the position control block to control the translations (x, y, z), the orientation control block to control the attitude (ϕ, θ, ψ) and the sliding mode observer block (seeFig. 3). The latter is used in order to replace the Flex 3 cameras when the drone is outside the working environment of the cameras. In this case, the states of the system are not measurable and the control of the drone will be lost.

4. Simulation Results

Before displayed the obtained results, the parameters of Qball 2+ are presented inTable 2. The PID and SMO observer parameters are given inTable 3,Table 4andTable 5respectively. Fig. 4,Fig.

5,Fig. 6, andFig. 7show the simulation results of the PID controller.Fig. 8,Fig. 9andFig. 10show the simulation results of the SMO.

Table 2.QBall 2+ parameters.

Parameter Definition Value Unit

Kt Lift coefficient 12 N

K_d Drag coefficient 0.4 N.m

K_υ Transformation constant 106.8 rad.sV⁻¹

M Mass 1.79 Kg

L Distance between the center of gravity and the center of each rotor 0.2 m I_x Moment of inertia of QBall 2 with respect to X axis 0.03 Kg.m² Iy Moment of inertia of QBall 2 with respect to Y axis 0.03 Kg.m² I_Z Moment of inertia of QBall 2 with respect to Z axis 0.04 Kg.m² J_rz Moment of inertia of rotor with respect to Z axis 0.0018 Kg.m²

ω Actuator bandwith 15 rad/s

V_max Maximum input voltage 12 V

4.1. Result and Discussion

Fig. 4andFig. 5show that the evolution curves of the positionsx, y andz follow the desired states with some errors. These latter appear when the quadrotor changes its positions. For the attitude dynamics, it’s observed fromFig. 5andFig. 6that, the Euler angles (φ,θ,ψ) follow the reference angles with less accuracy compared to the positions (x,y,z). The control signals displayed inFig.

7 vary according to the change of their corresponding positions (see Fig. 5). Overall, the results obtained by the PID controller are acceptable.

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Table 3.PID controller parameters used in simulations.

z K_pz = 0.5, K_iz = 0.02, K_dz = 0.15 φ K_pφ= 2, K_iφ= 0, K_dφ = 0.3 x K_px= 0.5, K_ix = 0, K_dx= 0.4 θ K_pθ= 2, K_iθ = 0, K_dθ = 0.3 y K_py= 0.5, K_iy= 0, K_dy= 0.4 ψ K_pψ = 0.45, K_iψ = 0, K_dψ = 0.3

Table 4.SMO parameters used in simulations.

k₁ k₃ k₅ k₇ k₉ k₁₁ β₈ β₁₀ β₁₂ β˙₈ β˙₁₀ β˙₁₂

5 5 4.5 9 9 15 0.3 0.35 0.9 1.4 1.2 1.3

α1 α3 α5 α7 α9 α11 K8 K10 K12

6 7 7 5 5 12 1 1.5 1.3

Table 5.Observer gains.

Parameter K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11 K12

Values 15 17 10 13 12 16.5 14 15 11 13 14.5 17

Fig. 4.Evolution of 3D trajectory for the PID controller.

Fig. 8 and Fig. 9 display the evolution of the desired trajectory, realized trajectory and the observed trajectory of the Qball 2+. It’s clear that the trajectory realized by the QBall 2+ follows the reference trajectory with some tracking errors in the beginning (seeFig. 8andFig. 9). It can be concluded that the observed states converge rapidly towards the desired states in a finite time, whether for the positions or the Euler angles.Fig. 10shows the evolution of the control signals. The controllerU1 starts from a maximum value and then stabilizes around a constant value. It is clear that the evolution ofU₂,U₃andU₄depends on the evolution of the translation and the rotation of the quadrotor (seeFig. 10). The obtained results (seeFig. 8,Fig. 9andFig. 10), have shown that the sliding mode observer has proven its efficiency against uncertain parameters of the system.

4.2. Comparative Study

In this section, a comparative study between two control laws simulated previously (PID and SMO) will be carried out. With the aim to facilitate the comparison between the applied control laws,

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Fig. 5.Evolution of positions and roll angles for the PID controller.

Fig. 6.Evolution of pitch and yaw angles for the PID controller.

Fig. 7.Input signals for the PID controller.

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Fig. 8.Evolution of 3D trajectory for SMO.

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-0.6 -0.4 -0.2 0 0.2 0.4

x [m]

x d x

obs x

0 0.2 0.4 0.6

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

y [m]

yd yobs y

0 0.2 0.4 0.6

0 0.1 0.2 0.3

28.2 28.4 28.6 28.8 0.48

0.5 0.52

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

z [m]

z d z

obs z

0 0.1 0.2 0.3 0.4 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

[rad]

10^-4

d obs

0 0.1 0.2 0.3

0 0.5 1 1.5 2 2.5 10^-4

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-1.5 -1 -0.5 0 0.5 1 1.5

[rad]

10^-4

d obs

0 0.1 0.2 0.3

-10 -5 0 5 10-5

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-5 0 5 10 15 20

[rad]

10^-5

d obs

0 0.1 0.2 0.3

0 0.5 1 1.5 2 2.5 10-4

Fig. 9.Evolution of positions and Euler angles for the SMO.

two criteria will be adopted as follows:

J(e) = _k¹Pk

i=1(e²_x+e²_y+e²_z+e²_θ+e²_φ+e²_ψ)

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0 10 20 30 40 50 60 70 80 90 100

Time [s]

15.5 16 16.5 17 17.5 18 18.5 19 19.5

U1 [N]

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-3 -2 -1 0 1 2 3

U2 [N]

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-4 -3 -2 -1 0 1 2 3 4

U3 [N]

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-4 -3 -2 -1 0 1 2 3 4

U4 [N.m]

10^-8

Fig. 10. Input signals for the SMO.

J(u) = ¹_kPk

i=1(u²₁+u²₂+u²₃+u²_θ+u²_x+u²_y)

whereJ(e)represents the quadratic tracking error,J(u)represents the energy andkis the number of samples.

4.2.1 Results Interpretation

In this part, the energies of the control signals and their quadratic errors in a simulation time of 100sand a sampling step10⁻³swill be calculated. According to the criteria of the squared errors and energies (seeTable 6), it can be observed that, the SMO has proven its effectiveness in terms of accuracy and consumed energy compared to the results obtained by the conventional PID. Unlike the obtained results in works[33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], where the initial conditions are zero, the SMO has presented good performance.

Table 6.Energies and quadratic errors of the controllers

PID SMO

J(e) J(u) J(e) J(u)

20.4182310⁻⁶ 3.085210² 7.28010⁻⁶ 3.083610²

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5. Experimental validation

In this Section, a PID controller will be implemented in the control flight of QBall 2+ with the same parameters that we have used in simulation. Before launching the flight scenario, the different cameras ”Optitrack Flex-3” must be calibrated and tested to locate the drone position as shown inFig.

13,Fig. 14,Fig. 15andFig. 16.

Fig. 11.To the left, QBall 2+ equipments. To the right, QBall 2+ in flight tests.

5.1. Cameras Flex-3 calibration steps

1. Launching the motivates software to determine the limits of the field of view.

2. Checking if there are no stray objects in the field of vision.

3. Definition of the origin or center of the workplace.

4. Definition of the traceable objective by selecting reflective markers.

5. Determination of the center of the traceable objective to follow it.

Fig. 12.Optihub and cameras optitrack.

Fig. 13.Cameras Flex-3 calibration.

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Fig. 14.Definition of the traceable objective.

Fig. 15.Created the rigid body.

Fig. 16. Change of the geometric center of the body.

5.2. Flight Scenario

The QBall 2+ is controlled to follow the desired points in a frame of−0.3m to 0.3m. After reaching a safety height of 1 meter, the quadrotor moves in the plane (X-Y) passing through the desired points (way points) as shown inFig. 17. The experiment results (seeFig. 17,Fig. 18,Fig. 19 andFig. 20) have shown similarities with those obtained by simulation (seeFig. 4,Fig. 5andFig.

7). Effectively, the difference between the obtained results (simulation and experiment) is mainly due to the neglected parameters of the model used in simulation.

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Fig. 17.Evolution of 3D trajectory for PID controller in experimental tests.

Fig. 18.Evolution of positions and roll angle in experimental tests.

Fig. 19.Evolution of pitch and yaw angles in experimental tests.

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Fig. 20. Evolution of control signals in experimental tests.

6. Conclusion and Future Works

Within this article, a dynamic model of the QBall 2+ quadrotor has been developed. In addition, its linear model has been exploited to obtain the appropriate gains of the PID controller. Further- more, to estimate the uncertain parameters of the system, a nonlinear SMC based on SMO has been developed. The latter has proven its efficiency and robustness against these parameters. Finally, to validate the obtained simulation results, the PID controller has been implemented in the flight control of the QBall 2+. The obtained results of the flight tests have shown a similitude to those obtained by simulation. In future works, the designed observer will be implemented and tested in the real system.

In addition, the fault tolerant control problem will also be investigated.

Author Contribution: All authors contributed equally to the main contributor to this paper. All authors read and approved the final paper.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

References

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[2] S. H. Derrouaoui, Y. Bouzid, M. Guiani, and I. Dib, “A comprehensive review on reconfigurable drones:

Classification, characteristics, design and control technologies,”Unmanned Systems, vol. 10, no. 01, pp.

3-29, 2022,https://doi.org/10.1142/S2301385022300013.

[3] S. H. Derrouaoui, Y. Bouzid, and M. Guiani, “PSO Based Optimal Gain Scheduling Backstepping Flight Controller Design for a Transformable Quadrotor,”Journal of Intelligent & Robotic Systems, vol. 102, no. 3, pp. 1-25, 2021,https://doi.org/10.1007/s10846-021-01422-1.