Modeling & SMC Based Trajectory Tracking for a Tilt-Rotor

Convertible UAV

Mohamed Zakaria Mimouni, Oualid Araar, Abdelkader Ouadda and Moussa Haddad

Ecole Militaire Polytechnique, Algeria

Keywords:

Tilt-Rotor UAV, SMC Control, Convertible UAV, Trajectory Tracking.

Abstract:

Convertible UAVs combine the vertical takeoff and landing (VTOL) capabilities of multi-rotors with the en-

durance and high speed of ﬁxed-wing drones. This work is concerned with a particular category of convertible

UAVs, commonly termed Tilt-rotor UAVs (TRUAVs). First, a detailed dynamic model for a Quad-TRUAV is

developed. This model features strong non-linearities and coupling, making its control a challenging task. The

second contribution of this work is the design of a sliding mode controller (SMC) to ensure trajectory track-

ing. Simulations conducted on the full non-linear model of the famous Zagi-wing UAV show very promising

results.

1 INTRODUCTION

Unmanned aerial vehicles (UAVs) have a wide range

of military and civilian applications. These include

infrastructure inspection, search and rescue, data col-

lection (Li et al., 2021) delivery services, agriculture

(Kim et al., 2019) and surveillance, to name but a few

(Shakhatreh et al., 2019).

Convertible or Hybrid UAVs is a particular class,

which has attracted considerable attention during the

last few years. Hybrid designs target to combine the

advantages of VTOL UAVs and those of the ﬁxed

wing category. These include high payload capacity,

great operational range, high cruise speed, in addition

to the stationary ﬂight and VTOL capabilities.

Depending on their switching mechanism, con-

vertible UAVs can be categorised into three main

categories. The ﬁrst includes tilt-rotors, which tilt

their rotors to switch between VTOL and cruise ﬂight

modes. The second is tilt-wing in which the drone tilts

both rotors and wings for switching. In the third cat-

egory, named tail-sitters or tilt-bodies, the entire air-

frame is tilted for transitioning between ﬂight modes.

The Tiltrotor UAV (TRUAV) is a very popular

class of hybrid UAVs, in which the mounted rotors

gradually tilt in the direction of ﬂight, driving the

aircraft forward until steady ﬂight is reached (Saeed

et al., 2015) (Hegde et al., 2019). This class is char-

acterized by a simple transition manoeuvre and good

controllability and stability (Saeed et al., 2018).

The TRUAV considered in this work is actuated

using only propellers, i.e it contains no control sur-

faces. In such a category, the tilting of the rotors re-

sults in considerable changes in the drone’s center of

gravity and moment of inertia (Su et al., 2019). It also

results in rapid variations in the aerodynamic forces

acting on the vehicle while changing from hover to

cruise ﬂight (Phung and Morin, 2014). These chal-

lenges complicate the dynamic model of the drone

and hence its control (Ducard and Allenspach, 2021).

Among research works which have dealt with the

control of TRUAVs, Papachristos et al. (Papachristos

et al., 2013) developed a linear quadratic (LQ) control

scheme for position control of a Three-Rotor TRUAV.

In the hover phase, they implemented and experimen-

tally tested PD & double Derivative controller (Pa-

pachristos et al., 2012).

In order to stabilize the pitch angle of a Three-

Rotor TRUAV, Ta et al. (Ta et al., 2012) combined

a linear PID controller and a nonlinear saturated sig-

moid function. For the position, the authors employed

a neural network-based adaptive controller. The pro-

posed controller, however, does not account for the

cross-coupling between states.

Flores et al. (Flores et al., 2012) applied a non-

linear backstepping controller for cruise ﬂight. In

hover mode, the TRUAV was controlled as a conven-

tional multirotor using a feedback linearization tech-

nique.

In another work (Flores and Lozano, 2013), Flo-

res et al. proposed a control solution to deal with the

transition phase. The desired altitude was maintained

Mimouni, M., Araar, O., Ouadda, A. and Haddad, M.

Modeling SMC Based Trajectory Tracking for a Tilt-Rotor Convertible UAV.

DOI: 10.5220/0012121700003546

In Proceedings of the 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2023), pages 97-103

ISBN: 978-989-758-668-2; ISSN: 2184-2841

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

𝑻

𝟐

𝑻

𝟏

𝜸

𝟏

𝜸

𝟏

𝑻

𝟒

𝑻

𝟑

𝜸

𝟐

ൗ

𝝅

𝟐

𝜸

𝟐

𝒙

𝒛

𝒚

Figure 1: Illustration of the Quad-TRUAV considered in

this work.

using nested saturation control. The input control vec-

tor includes elevator deﬂection, tilt angle, total rotor

thrust, and difference in thrust between the front and

rear rotors.

Hernandez-Garcia et al.(Hernandez-Garcia and

Rodriguez-Cortes, 2015) applied a gain scheduling

controller to a dual TRUAV. Based on the Jacobian

linearisation of the model of the drone, a set of linear

controllers was designed for takeoff, vertical, hover-

ing, and horizontal ﬂight conditions.

Govdeli et al. (Govdeli et al., 2019) presented a

Quad-Tilt-Rotor with two conﬁgurations. The authors

used a PID controller to generate the control inputs for

the longitudinal plan. The parameters of the forward

transition controller were different from those of the

backward transition controller. The control scheme

did not guarantee to maintain the altitude during the

transition.

Chen et al. (Chen et al., 2021) designed Cascaded

controller for a Three-Rotor TRUAV. During the tran-

sition, the drone was controlled by mixing control sur-

faces and rotors depending on the airspeed. A sliding

mode controller was used for all channels.

To the best of the author’s knowledge, all previous

works employ two controllers for each ﬂight phase. In

this paper, a control architecture employing a single

controller for both VTOL and cruise ﬂight phases is

considered. A sliding mode controller (SMC) is then

derived to ensure trajectory tracking. The proposed

solution is validated on the full nonlinear model of

the famous Zagi wing drone.

The remainder of this paper is organised into four

sections. Section II presents the development of the

TRUAV’s kinematic and dynamic model. The control

architecture and associated sliding mode controller

are discussed in section III. Obtained results are pre-

sented and analysed in section IV.

2 DYNAMIC MODELING OF THE

Quad-TRUAV

The dynamic model of the Quad-TRUAV is derived

based on the Newton-Euler formalism

= R

IΩ + Ω× IΩ = M

(1)

Using the blade element theory, the thrust and re-

active torque generated by a rotor are modelled as

(Prouty, 2002)



= K

· ω

= K

· ω

(2)

with K

and K

the thrust and drag coefﬁcients.

After development, the total thrust forces gener-

ated by the four rotors are expressed in the body frame













1+2

Cγ

+ T

3+4

Cγ

−T

1+2

Sγ

− T

3+4

Sγ





(3)

where T

1+2

= T

+ T

and T

3+4

= T

+ T

The total moment generated by the four rotors is

expressed w.r.t the body frame as









(4)

with





cγ

− l

sγ

1−2

+ (K

cγ

− l

sγ

3−4

1+2

− l

3+4

−(K

sγ

+ l

cγ

1−2

− (K

sγ

+ l

cγ

3−4





(5)

With T

1−2

= T

− T

, T

3−4

= T

− T

, l

and ly are

the position coordinates of rotors, and K

= K

denotes the constant of proportionality between the

two drag and the thrust

Assuming a small angle of attack and that the

airﬂow remains laminar and attached, the aerody-

namic forces F



L D f



and moments M



l m n



expressed in the wind frame, R

, can be

approximated as (Beard and McLain, 2012)

L =

ρV

D =

ρV

(6)

l =

ρV

SbC

m =

ρV

ScC

n =

ρV

SbC

(7)

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

with V

the resultant air mass velocity, calculated

as the difference between the drone’s ground speed

and the wind speed:

= V

− V

wind

(8)

Parameter S refers to the wing area, c is the mean

chord, and b is the wing span. The aerodynamic

coefﬁcients C

, C

and C

are respectively the lift,

drag, and pitch moment coefﬁcients for the longitudi-

nal plan. For the lateral plan, C

, C

and C

are the lat-

eral forces, roll and yaw moments coefﬁcients. These

parameters are approximated to (Beard and McLain,

2012)

= C

α +C

= C

α +C

= C

α +C

= C

β +C

p +C

= C

β +C

p +C

= C

β +C

p +C

(9)

with α the angle of attack, and β the side slip angle.

After developing the total force and moment act-

ing on the Quad-TRUAV, the dynamics of the trans-

lational motion of the drone are expressed w.r.t the

inertial frame as





¨x

¨y

¨z









1+2

cγ + F

−T

1+2

sγ − T

3+4

+ F













(10)

with F

, F

, and F

the aerodynamic forces ex-

pressed in the body frame. The rotational dynamics of

the Quad-TRUAV are expressed w.r.t the body frame





˙p

˙q

˙r





= J

−1







−









× J













+ m











(11)

with m

, m

, and m

the components of the

aerodynamic moments expressed in the body frame.

This gives, after development





˙p

˙q

˙r









pq + I

qr + I



+ l



+ I



+ m



pr + I

(



− p



+ τ

+ m

)

pq + I

qr + I



+ m



+ I



+ m







(12)

with:

= I

− I

, (13)

− I

+ I

)), (14)



− I



, (15)

, (16)

, (17)

− I

)

, (18)

, (19)



+ I

− I



, (20)

− I

), (21)

. (22)

3 CONTROL SCHEME & SMC

DESIGN

The control scheme adopted in this work is outlined

in Fig.2. Unlike conventional quadrotors our Quad-

TRUAV has an extra control input, which is the tilt

angle of the front rotors. In our control scheme, this

input is exploited to decouple the forward position

control from that of the pitch angle. The TRUAV’s

trajectory is, thus, speciﬁed by ﬁve independent vari-

ables: the 3D position (x

, y

, z

), as well as the pitch

and yaw angles (θ

, ψ

In our control scheme, the output of the forward

position controller, U

, is mapped to the tilt angle γ.

The loss in vertical thrust force caused by the tilting

of the front rotors is compensated for by raising the

rotation speed of these rotors such that the projection

of the generated forces on the z axis meets the control

output of the altitude controller, U

The virtual control output U

is mapped to the roll

angle φ in the same way as it is in a conventional mul-

tirotor. The heading angle, ψ, is also controlled like a

conventional multi-rotor. The pitch angle controller,

on the other hand, takes into account the variation in

the tilt angle of the front rotors.

To simplify the controller design, the following

assumptions are considered:

- The roll φ and pitch θ angles are small, so Euler

angle rate are approximated to the angular rates in

the body frame

















(23)

Modeling SMC Based Trajectory Tracking for a Tilt-Rotor Convertible UAV

x-position control block

</>a

desired roll

and tilt

tf>-angle

control b lock

u,,

y-position control block

angles

generation

block

'l'a

calculator

</>d-yd

</>

z-position control block

8-angle control block

' y

)

fP.angle control block

Tip

Allocation Matrix

tT1 tT2tT3tT4

"--'/

.----

</>

........•. ...

𝜙

∗

& 𝛾

∗

𝜙

∗

𝛾

∗

𝜓

∗

𝜃

∗

𝑧

∗

𝑦

∗

𝑥

∗

Quad-TRUAV Plant

Figure 2: Proposed Control scheme of the Quad-TRUAV.

- Given the symmetry of the Quad-TRUAV, the

term I

in the inertia matrix is neglected. This

simpliﬁes equation (12) to















−I

ψ +

−I

ψ +

−I

θ +







(24)

The control input vector is chosen as

U =





(25)

with

= −T

1+2

sinγ − T

3+4

(26)

= T

1+2

cosγ (27)

= (K

cosγ − l

sinγ)T

1−2

− l

3−4

(28)

= sinγl

1+2

− l

3+4

(29)

= −(K

sinγ + l

cosγ)T

1−2

+ K

3−4

(30)

The corresponding allocation matrix is

M =







−(sθcγ+cθcφsγ) −cθcφ 0 0

0 0 (K

l d

cγ−l

sγ) −l

sγl

−l

0 0

0 0 −(K

l d

sγ+l

cγ) K

l d







(31)

The thrust forces are hence calculated as







1+2

3+4

1−2

3−4







= M

−1













(32)

For the sake of clarity, the following intermediate

control variables are considered

= cψcθ · U

+ (cψsθcφ + sψsφ)U

(33)

= sψcθ · U

+ (cφsθsψ − sφcψ)U

(34)

= −sθU

+ (cφcθ)U

(35)

In what follows, we present the design of an SMC

controller for each variable of the Quad-TRUAV. Slid-

ing mode control belongs to the family of discontinu-

ous control systems. It is an ideal instrument for ad-

dressing the control of complex dynamic plants, such

as that of the Quad-TRUAV.

We start with the design of the roll angle con-

troller. The tracking error of the roll angle is deﬁned

as,

= φ

∗

− φ (36)

The corresponding sliding surface is chosen such that

= ˙e

+ λ

(37)

with λ

> 0.

This gives the following derivative for the sliding

surface:

= ¨e

+ λ

˙e

(38)

∗

−

φ + λ

(

∗

−

φ) (39)

A Lyapunov function, V , is then chosen, such that

V =

(40)

To ensure the sliding condition we have

= −η

sign(σ

) (41)

with η

> 0.

By deriving the sliding surface in equation (39)

and replacing

φ by its deﬁnition from equation (24),

the control input τ

of the SMC controller is obtained



−a

ψ − b

l +

∗

+ λ

(

∗

−

φ) + η

signσ



(42)

with η

> 0.

Following the same steps as for the roll controller,

the control inputs for the pitch angle τ

, the heading

angle τ

, and the 3D position U

, U

, and U

are cal-

culated as



−a

ψ − b

m +

∗

+ λ

(

∗

−

θ) + η

signσ



(43)



−a

θ − b

n +

∗

+ λ

(

∗

−

ψ) + η

signσ



(44)

= m



−g −

+ ¨z

∗

+ λ

(˙z

∗

− ˙z) + η

signσ



(45)

= m



−

+ ¨x

∗

+ λ

( ˙x

∗

− ˙x) + η

signσ



(46)

= m

−

+ ¨y

∗

+ λ

( ˙y

∗

− ˙y) + η

signσ

(47)

with η

, η

and η

real positive

parameters.

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

100

Table 1: Parameters of the Zagi ﬂying wing used in this work.

Parameter Value

Longitudinal

Coef.

Value

Lateral

Coef.

Value

m 1.56 kg CL

0.09167 CY

0.1147 kg m

0.01631 Cl

0.0576 kg m

-0.02338 Cn

0.1712 kg m

3.5016 CY

-0.07359

0.0015 kg m

0.2108 Cl

-0.02854

S 0.2589 m

-0.5675 Cn

-0.00040

b 1.4224 m CL

2.8932 CY

c 0.3302 m CD

0 Cl

-0.3209

ρ 1.2682 kg/m

-1.3990 Cn

-0.01297

0.03066

-0.00434

Figure 3: 3D trajectory followed by the Quad-TRUAV.

4 SIMULATION RESULTS AND

DISCUSSION

To validate the proposed control scheme and the asso-

ciated SMC controller, simulations are conducted on

the full non-linear model of the Quad-TRUAV. The

model considered in this work is that of the famous

Zagi ﬂying wing (Beard and McLain, 2012). The geo-

metrical, inertial, and aerodynamic parameters of this

drone are summarised in table 1.

We consider the saturation of the four actuators to

make our model even more realistic. The saturation

is selected so that the four rotors can only provide a

lift force equal to double the drone’s weight. The tra-

jectory under consideration contains three segments.

The initial phase consists of a takeoff and ascent to

a height of 7.5 meters. During the second phase, the

drone pitches up and speeds forward until it reaches 7

m/s, which it sustains for 20 seconds. The drone then

begins to decelerate until it comes to a stop, levels out,

and begins to land.

Fig.3 depicts the 3D reference trajectory and the

trajectory followed by the drone. As the ﬁgure shows,

the proposed control scheme and associated sliding

mode controller were capable of smoothly tracking

the reference trajectory in all its phase. This solution

also ensured a smooth transition, between VTOL and

ﬁxed wing mode.

Fig. 4a shows in more detail the temporal varia-

tion of the 3D trajectory followed by the drone and

the corresponding tracking errors. The tracking er-

rors along the x axis are much higher compared to the

y and z axes. This is due to the higher manoeuvres

that the drone makes along this axis.

Fig. 5 plots the evolution of the drone’s attitude in

terms of Euler angles. It is to note that the variations

of the pitch angle at 25 sec and 70 sec did not affect

the drone’s position despite the coupling between the

x position and pitch angle. This is achieved thanks to

the non-linearity of the SMC controller, which explic-

itly compensates for this coupling.

With regard to the control efforts, the thrust forces

generated by the four rotors are depicted in Fig. 6a.

As the drone moves forward, its wings generate more

lift and less thrust is needed to maintain its altitude.

The average value of the rotors’ thrust is about

4N in the VTOL phase. This value is decreased to

about 2.5N in the cruise ﬂight phase. This results in

less power consumption and hence more endurance as

mentioned earlier.

The main drawback of the SMC controller is the

chattering which appears in all the control inputs.

This effect would be less pronounced in practice, be-

cause it will be ﬁltered by the actuators’ dynamics.

Fig. 6b shows the evolution of the tilting angle.

The pitch up action, which happens at 25 sec, results

in a decrease of the tilt angle by about 70

◦

. During

the transition, the tilt angle continues to decrease to

about 60

◦

From these results, it is concluded that all control

inputs remain within the saturation limits we set for

them. The maximum tilt angle achieved was about

◦

from vertical, while its saturation was set to 60

◦

It is worth noting that this saturation is necessary for

the Quad-TRUAV, since a minimum component of

Modeling SMC Based Trajectory Tracking for a Tilt-Rotor Convertible UAV

101

0 10 20 30 40 50 60 70 80 90 100

Time(s)

100

150

200

X(m)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

-1.5

-1

-0.5

0.5

Y(m)

-3

0 10 20 30 40 50 60 70 80 90 100

Time(s)

-8

-6

-4

-2

Z(m)

(a)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

0.05

0.1

0.15

(m)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

-1.5

-1

-0.5

0.5

(m)

-3

0 10 20 30 40 50 60 70 80 90 100

Time(s)

-5

(m)

-3

(b)

Figure 4: Trajectory tracking results: (a) desired and actual

horizontal positions and altitude; (b) corresponding track-

ing errors.

thrust on the z

is required to allow the control of the

pitch angle.

5 CONCLUSION

This paper dealt with the problem of trajectory track-

ing for a tilt-rotor convertible UAV. Since the drone

contains no control surfaces, only rotors were relied

on to ensure the control of its 6 DOF. A control

scheme which takes into account this particularity and

a sliding mode controller were designed to ensure tra-

jectory tracking.

0 10 20 30 40 50 60 70 80 90 100

Time(s)

-0.4

-0.2

0.2

0.4

(°)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

(°)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

-0.15

-0.1

-0.05

(°)

(a)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

-0.5

0.5

e (°)

0 10 20 30 40 50 60 70 80 90 100

e (°)

-3

0 10 20 30 40 50 60 70 80 90 100

Time(s)

0.05

0.1

0.15

e (°)

(b)

Figure 5: Evolution of the drone’s attitude : (a) From top to

bottom roll, pitch, and yaw angles (b) corresponding track-

ing errors.

The proposed control scheme and associated SMC

were validated on the full non-linear model of the

drone, including all couplings and actuators satura-

tion. Simulation results were very promising. Future

works will investigate solutions for the estimation of

the aerodynamic forces and moments and the imple-

mentation of the proposed control solution on a real

plateform.

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

102

0 10 20 30 40 50 60 70 80 90 100

Time(s)

T(N)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

L(N)

(a)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

100

(°)

0 10 20 30 40 50 60 70 80 90 100

Time(s)

(m/s)

(b)

Figure 6: Control inputs: (a) thrust and lift forces; (b) tilt

angle and airspeed.

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Aircraft Theory and Practice. Princeton University

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Chen, C., Zhang, J., Wang, N., Shen, L., and Li, Y.

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Ducard, G. J. and Allenspach, M. (2021). Review of de-

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Flores, G. and Lozano, R. (2013). Transition ﬂight

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pages 789–794. IEEE. a nonlinear control strat-

egy based on saturations and Lyapunov design is

given¡br/¿¡br/¿stabilize the altitude z with a bounded

control input, we will use the nested saturation control

approach.

Flores, G. R., Escare

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(2012). Quad-tilting rotor convertible mav: Modeling

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