fuzzy logic based quadrotor flight controllervigir.missouri.edu/~gdesouza/research/conference... ·...

FUZZY LOGIC BASED QUADROTOR FLIGHT CONTROLLER

Syed Ali Raza and Wail GueaiebSchool of Information Technology and Engineering, University of Ottawa

800 King Edward Avenue, Ottawa, ON, [email protected], [email protected]

Keywords: Quadrotor, Fuzzy Logic, Flight Controller.

Abstract: Quadrotor unmanned aerial vehicles (UAVs) have gained a lotof research interest in the past few years, due tothe clear advantages posed by their vertical take-off and landing (VTOL), hovering capability, and slow precisemovements. These characteristics make quadrotors an idealcandidate for applications that require traversingthrough difficult environments with many obstacles. Belonging to the helicopter rotorcraft class, quadrotorsare highly nonlinear systems that are difficult to stabilize. This paper proposes a fuzzy logic based flight con-troller for an autonomous quadrotor. Two types of fuzzy logic controllers are implemented. The developedflight controllers are tested in a quadrotor simulator and simulation results are presented to demonstrate theperformance of each controller. The controllers performances are also benchmarked against conventional con-trol based techniques such as input-output linearization,backstepping and sliding mode control. In comparisonwith other conventional control techniques mostly designed for indoor applications, the proposed fuzzy logicbased controllers showed satisfactory control of the quadrotor in the presence of various disturbances such assensor noise and high wind conditions.

NOMENCLATURE

The following notations are used in the paper:Fn represents the reference frame where the subscriptn∈ i,b,v,φ,θ.()Fn represents a point or a vector in reference frameFn.(), () represent first and second time derivatives, re-spectively.RF2F1

∈ Rn×n is the rotation matrix that maps frameF1

to frameF2.sθ = sinθ, cθ = cosθ.I is the identity matrix.COG is the quadrotor’s center of gravity.PT = [px py pz] is the position of the quadrotor’sCOG.M is the mass of quadrotor (including the motors).m is the mass of one motor.l is the length of the arms of quadrotor.JT = [ jx jy jz] is the moment of inertia vector of thequadrotor.φ,θ,ψ are the quadrotor’s roll, pitch, and yaw angles,respectively.ΩT = [φ θ ψ] is the quadrotor’s orientation vector.KT andKτ are motor constants.Tmotor,τmotor,PWMmotor are thrust, drag and PWMvalue for motor∈ f , r,b, l, the subscriptsf , r, b,

and l , denote front, right, back, and left, motors re-spectively.

1 INTRODUCTION

The motivation for employing robots in search andrescue operations is multifaceted. However, the tech-nology is still in its infancy and there are many issueswhich need to be addressed. Search and rescue mis-sions, as well as simulations, have demonstrated sev-eral areas in which robot contributions need to be im-proved (Fincannon et al., 2004). Advances in comput-ing, MEMS inertial measurement sensors, and com-munications technology make it possible to achieveautonomous performance and coordination of thesevehicles in test environments (Waslander et al., 2005).But, achieving complete autonomous performance inreal environments is a milestone yet to be reached.

A quadrotor, as depicted in Figure 1, is a rotarywing UAV, consisting of four rotors located at theends of a cross structure. By varying the speeds ofeach rotor, the flight of the quadrotor is controlled.Quadrotor vehicles possess certain essential charac-teristics, which highlight their potential for use insearch and rescue applications. Characteristics that

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Figure 1: Conceptual diagram of the quadrotor.

provide a clear advantage over other flying UAVs in-clude their Vertical Take Off and Landing (VTOL)and hovering capability, as well as their ability tomake slow precise movements. There are also defi-nite advantages to having a four rotor based propul-sion system, such as a higher payload capacity, andimpressive maneuverability, particularly in traversingthrough an environment with many obstacles, or land-ing in small areas.

The quadrotor is an under-actuated system, in thatthe four forces applied on the body result in move-ments achieved in six degrees. Being a non-linearsystem it offers an interesting and challenging con-trol problem. The flight controller is responsible forachieving the desired attitude, i.e. the orientation(roll, pitch and yaw angles) of the quadrotor and alsothe desired position of quadrotor in the space.

In past few years, a lot of research has alreadybeen conducted on the modeling and control of aquadrotor. Many control techniques are proposed inthe literature, however, their primary focus is mostlyfor indoor flight control and therefore does not ac-count for uncertainties. Lyapunov stability theory isused for stabilization and control of the quadrotor in(Bouabdallah et al., 2004a) (Dzul et al., 2004). Con-ventional PD2 feedback, and PID structures are usedfor simpler implementation of control laws and com-parison with LQR based optimal control theory ispresented in (Tayebi and McGilvray, 2006) (Bouab-dallah et al., 2004b). Backstepping control is alsoproposed with the drawback of higher computationalloads in (Guenard et al., 2005). Visual feedback isused in many cases using onboard or offboard cam-eras for pose estimation by (Altug et al., 2002) (Gue-nard et al., 2008). And finally some fuzzy logic basedcontrol techniques (Coza and Macnab, 2006), neuralnetworks (Tarbouchi et al., 2004) and reinforcement

learning (Waslander et al., 2005).Fuzzy logic based control offers a great advan-

tage over conventional control, specifically in deal-ing with nonlinear systems with uncertainties. In thispaper two fuzzy logic based flight controllers for anautonomous quadrotor are proposed. Implementationof the proposed control is carried out using Mamdaniand Sugeno fuzzy inferencing methodologies. Theadvantage of the proposed methodology over otherconventional methods is the ability to achieve sta-ble flight control under disturbances that are com-mon in outdoor environments. Also the implemen-tation is lighter in terms of computational loads ascompared to techniques such as backstepping control.Both controllers are simulated and their performancesare benchmarked against various conventional con-trol techniques. The results are presented to show thesatisfactory performance of the proposed fuzzy logicbased controllers despite the presence of various dis-turbances, such as sensor noise and high wind condi-tions.

2 QUADROTOR’S KINEMATICSAND DYNAMICS

The flight behavior of a quadrotor is determined bythe speeds of each of the four motors, as they varyin concert, or in opposition with each other. Hence,a mathematical representation of the system can beused to predict the position and orientation of thequadrotor, based on its inputs. The same can furtherbe used to develop a control strategy, whereby ma-nipulating the speeds of individual motors results inthe achievement of the desired motion. To derive amathematical model of the quadrotor, we need to de-fine its kinematics and dynamics. The kinematics ofthe quadrotor gives a relation between the position ofthe vehicle in the inertial frame, and its velocity in thebody frame. The dynamics of the quadrotor providesthe relation between the applied body forces and theresulting accelerations.Three frames of reference are adopted;

1. The inertial frame,F i = (−→xi ,−→yi ,

−→zi ), is an earth-fixed coordinate system with the origin located onthe ground, for example, at the base station. Byconvention, the x-axis points towards the north,the y-axis points towards the east, and the z-axispoints towards the center of the earth.

2. The body frame,Fb = (−→xb,−→yb,−→zb), with its ori-gin located at the center of gravity (COG) of thequadrotor, and its axes aligned with the quadrotorstructure such that the x-axis−→xb is along the arm

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

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with front motor, the y-axis−→yb is along the armwith right motor, and the z-axis−→zb = −→xb ×

−→yb.

3. The vehicle frame,Fv =(−→xv ,−→yv ,−→zv ), is the inertial

frame with the origin located at the COG of thequadrotor. The vehicle frame has two variations,Fφ andFθ. Fφ, is the vehicle frame,Fv, rotatedabout its z-axis−→zv by an angleψ so that−→xv and−→yv are aligned with−→xb and−→yb, respectively.Fθis theFφ frame rotated about its y-axis,−→yφ , by apitching angle,θ, such that−→xφ and−→zφ are alignedwith −→xb and−→zb , respectively.

With the knowledge of the inertial frame positionstate variablesPi and the body frame speed state vari-ablesPFb, the translational motion relationship is de-rived as

pxpy−pz

F i

=[

RFbF v

]T

pxpypz

Fb

where[

RFbF v

]T∈ R

3×3 is the rotation that maps frameFb to

frameFv and is defines by

[

RFbF v

]T=

cθcψ sφsθcψ−cφsψ cφsθcψ+sφsψcθsψ sφsθsψ+cφcψ cφsθsψ−sφcψ−sθ sφcθ cφcθ

The rotational motion relationship is also derived us-ing the state variables involved:

φθψ

Fv

=

1 sφ tanθ cφ tanθ0 cφ −sφ0 sφ/cθ cφ/cθ

φθψ

Fb

.

The quadrotor’s dynamics is obtained by applyingNewton-Euler formulation.[

MI3×3 00 I3×3

][

PFb

ΩFb

]

+

[

ΩFbMPFb

ΩFbIΩFb

]

=

[

FFb

τFb

]

whereFT = [ fx fy fz] andτT = [τφ τθ τψ] are ex-ternal force and torque vectors applied on the quadro-tor’s COG.τφ, τθ, andτψ are the roll, pitch and yawtorques respectively. Hence, the models for the trans-lational and rotational motion are

pxpypz

Fb

=

ψpy− θpz

φpz− ψpx

θpx− φpy

Fb

+1M

fxfyfz

Fb

φθψ

Fb

=

jy− jzjx

θψjz− jx

jyφψ

jx− jyjz

φθ

Fb

+

1jx

τφ1jy

τθ1jz

τψ

Fb

where jx = jy = jz = 2Mr2

5 + 2l2m. The momentof inertia is calculated by assuming the COG of thequadrotor as a sphere of radiusr and massM. Also,

each motor is represented by a point massmwith armslength l . The relationship between the quadrotor’soverall thrust and drag produced by individual thrustof the motors is defined by

T = Tf +Tr +Tb+Tl , τψ = (τ f + τb)− (τr + τl )

τφ = l(Tl −Tr) , τθ = l(Tf −Tb)

with

Tmotor = KT ×PWMmotor andτmotor = Kτ ×PWMmotor

for motor∈ f , r,b, l. In matrix form, we get

PWM fPWMrPWMbPWMl

= G×

Tτφτθτψ

where

G =

KT KT KT KT0 −l ×KT 0 l ×KT

l ×KT 0 −l ×KT 0−Kτ Kτ −Kτ Kτ

−1

.

Including the forces and torques acting on the system,the equations of motion become as defined in (1), withg being the gravitational force.

3 FLIGHT CONTROLLERDESIGN

Using the knowledge of the quadrotor’s behavior fromits equations of motion, a rule base is developed,which serves as a starting point for developing thefuzzy logic control strategy. TheseIF-THEN ruleswill later be combined to form a fuzzy controller,where fuzzy logic is used in a direct control schemeapplied for the flight control of the quadrotor.

Figure 2: Block diagram of fuzzy logic based flight con-troller.

For attitude control, three fuzzy controllers aredesigned to control the quadrotor’s roll (φ), pitch (θ)


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pxpypz

Fb

=

ψpy− θpz

φpz− ψpx

θpx− φpy

Fb

+

−gsθgcθsφgcθcφ

+

00− fzM

,

φθψ

Fb

=

jy− jzjx

θψjz− jx

jyφψ

jx− jyjz

φθ

Fb

+

1jx

τφ1jy

τθ1jz

τψ

Fb

(1)

and yaw (ψ) angles. Here FLCφ and FLCθ fuzzycontrollers are implemented in order to achieveattitude stabilization. Three fuzzy controllers arefurther designed for controlling the quadrotor’sposition. All the six fuzzy controllers, as depicted inFigure 2, have the same structure with two inputs andone output. The inputs are the errore, which is thedifference between the desired and the actual statenormalized to the interval[−1,+1], and the errorrate e normalized to the interval[−3,+3]. Threemembership functions are used to fuzzifye, e, andthe output U : µN(e) = trapezoid(−1,−0.15,0),µZ(e) = triangle(−0.15,0,0.15),µP(e) = trapezoid(0,0.15,1), µN(e) =trapezoid(−3,−1.5,0), µZ(e) =triangle(−1.5,0,1.5), µP(e) = trapezoid(0,1.5,3),µN(U) = trapezoid(−1,−0.85,0), µZ(U) =triangle(−0.1,0,0.1), µP(U) = trapezoid(0,0.85,1).

A unified rule base comprising nineIF-THEN

rules is developed for all the controllers. The devel-oped rules are presented in Table 1.

Table 1: The rule base of the fuzzy controller.

eN Z P

N N N Ze Z N Z P

P Z P P

For the fuzzy controller to be independent ofthe quadrotor’s parameters, the input and output sig-nals of the designed fuzzy logic controllers are pre-processed and post-processed, respectively. The pre-processing process calculates the errore and errorrate e and normalize them accordingly. The post-processing process calculates the individual motorPWM values by combining the outputs from the fuzzylogic controllers and adding a priori-defined offsetto counter balance the weight of the quadrotor. ThePWM value of each motor is then computed as fol-lows:

PWMf = Sat(UZ +UXθ −Uψ +Offset)

PWMr = Sat(UZ +UYφ +Uψ +Offset)

PWMb = Sat(UZ−UXθ −Uψ +Offset)

PWMl = Sat(UZ−UYφ +Uψ +Offset)

Being independent of the plant’s parameters setsthe pure fuzzy controllers apart from the conventional

control systems, which depends on the plant’s param-eters. The fuzzy controllers are designed in light ofthe behaviors extracted from the mathematical modelof the quadrotor, whereas conventional control tech-niques are designed using the mathematical model ofthe plant. Any change in the plant dynamics or param-eters results in the failure of such conventional controlsystems. In such situations, fuzzy systems are advan-tageous over conventional control techniques.

Two different fuzzy inference procedures are im-plemented: (i) a Mamdani fuzzy model, and (ii) aTakagi-Sugeno-Kang (TSK) fuzzy model. The Mam-dani fuzzy inference method uses a min-max opera-tor for the aggregation and centroid of area methodfor defuzzification. One problem with the designedMamdani fuzzy control is the high computational bur-den associated to it when implemented on an embed-ded system. To alleviate this problem, a zero orderSugeno fuzzy model is implemented. In this model,the output membership functions of the Mamdanitype FLC are replaced with fuzzy singletonsN = −1,Z = 0 andP = +1.

4 NUMERICAL RESULTS

To test the proposed fuzzy flight controllers, and studytheir performances, a simulation environment is de-veloped. The desired inputs from the user are thetranslatory coordinatesPF i with respect to the inertialframe, and the yaw angleψ. The pitch and roll anglesare set to zero for achieving the desired attitude stabi-lization, as shall be shown later.

To make the simulations more realistic, sensorynoise and environmental disturbances such as windsare also taken into account. Different wind conditionsare generated based on the actual data from CanadaWeather Statistics, representing high, medium andlow wind conditions (Statistics, 2009). The angularaccelerationsφ, θ, andψ are degraded with a whitenoise and then used as a feedback to the fuzzy con-troller.

Several experiments are conducted on the two de-signed fuzzy controllers. Due to the space limita-tion, only three experiments are presented here. Thecontrollers performances are benchmarked againstthose of similar control schemes found in the liter-ature. The values used as the quadrotor’s dynamic


108

0 5 10 15 20 25 30−2

0

2

4

6

8

10

12

14

16

X−

axis

mot

ion

(m)

time (sec)

No noise, no windNoise, no windNoise and wind

(a)

0 5 10 15 20 25 30−5

0

5

10

15

20

25

30

Y−

axis

mot

ion

(m)

time (sec)

(b)

0 5 10 15 20 25 30−5

0

5

10

15

20

25

Z−

axis

mot

ion

(m)

time (sec)

(c)

0 5 10 15 20 25 30−10

−5

0

5

10

15

20

25

30

Pitc

h an

gle

(deg

)

time (sec)

(d)

0 5 10 15 20 25 30−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

Rol

l ang

le (

deg)

time (sec)

(e)

0 5 10 15 20 25 30−5

0

5

10

15

20

25

30

35

Yaw

ang

le (

deg)

time (sec)

(f)

Figure 3: Experiment 1, Mamdani FLC’s performance with and without disturbances. Quadrotor states: (a) x-axis; (b) y-axis;(c) z-axis (altitude); (d) pitch (θ); (e) roll (φ); and (f) yaw (ψ).

parameters are:M = 1.56 Kg, l = 0.3 m, Jx =Jy = 0.08615 Kg·m2, Jz = 0.1712 Kg·m2, KT = 5.45,Ktau = 0.0549, Offset= Mg/(4KT) = 0.7018.

In our first experiment the system’s initial statesare set to zero and the desired quadrotor’s positionand orientation arePT

F i= [10,10,25] m andΩT

F i=

[0,0,30] degrees. The purpose of the experiment is toshow the performance of both controllers under dif-ferent disturbance conditions. The experiment is re-peated three times for each controller, first time with-out any disturbances, second time with sensor noiseonly, and third time with sensor noise and mediumnorth and east wind of 10m/s. The experimental re-sults presented in Figures 3 and 4, demonstrate theability of both controllers to perform satisfactorilydespite the presence of sensor noise and wind dis-turbances. The Mamdani fuzzy controller convergesto the desired states relatively faster than the Sugenofuzzy controller, however Sugeno shows better atti-tude stabilization, specially in the presence of the sen-sory noise and wind disturbances. The yaw angledrift under wind disturbance is clearly visible in theSugeno results.

In second experiment, the position and attitudestabilization of the proposed controllers are bench-marked against feedback linearization based conven-

tional control technique, where a series of mode-based, feedback linearizing controllers are imple-mented for the stabilization and control of a quadro-tor (Altug et al., 2002). The system’s initial statesare considered to bePT

F i= [40,20,60] m andΩT

F i=

[45,−45,0] degrees. In the presence of sensor noiseand winds, the system is supposed to stabilize all thestates back to zero. The experimental results are de-picted in Figures 5 and 6. The stabilization achievedby both proposed fuzzy controllers converge to the de-sired state faster than the feedback linearization tech-nique. It is worth pointing out that the controller pro-posed in (Altug et al., 2002) did not consider any typeof noise or external disturbances.

The third experiment benchmarks the proposedfuzzy controllers with simulation results of a back-stepping controller tuned for optimized nonlinearcontrol of a tethered quadrotor (Bouabdallah et al.,2004b). The backstepping control offers high robust-ness against large disturbances as compared to thefeedback linearization technique which is clear fromthe results. To test the attitude stabilization of theproposed fuzzy controllers, the system is initializedwith severe rotational anglesΩT

F i= [45,45,45] de-

grees. Again, the sensory noise and winds are intro-duced as disturbances. The experimental results show


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0 5 10 15 20 25 30−5

0

5

10

15

20

X−

axis

mot

ion

(m)

time (sec)

No noise, no windNoise, no windNoise and wind

(a)

0 5 10 15 20 25 30−5

0

5

10

15

20

25

Y−

axis

mot

ion

(m)

time (sec)

(b)

0 5 10 15 20 25 30−5

0

5

10

15

20

25

Z−

axis

mot

ion

(m)

time (sec)

(c)

0 5 10 15 20 25 30−5

0

5

10

15

20

25

30

Pitc

h an

gle

(deg

)

time (sec)

(d)

0 5 10 15 20 25 30−30

−25

−20

−15

−10

−5

0

5

10

15R

oll a

ngle

(de

g)

time (sec)

(e)

0 5 10 15 20 25 30−5

0

5

10

15

20

25

30

35

40

Yaw

ang

le (

deg)

time (sec)

(f)

Figure 4: Experiment 1, Sugeno FLC’s performance with and without disturbances. Quadrotor states: (a) x-axis; (b) y-axis;;(c) z-axis (altitude); (d) pitch (θ); (e) roll (φ); and (f) yaw (ψ).

0 10 20 30 40 50−10

0

10

20

30

40

50

60

70

80

X−

axis

mot

ion

(m)

time (sec)

Noise, no windNoise and wind

(a)

0 10 20 30 40 50−10

0

10

20

30

40

50

Y−

axis

mot

ion

(m)

time (sec)

(b)

0 10 20 30 40 50−10

0

10

20

30

40

50

60

70

Z−

axis

mot

ion

(m)

time (sec)

(c)

0 10 20 30 40 50−50

−40

−30

−20

−10

0

10

20

30

40

Pitc

h an

gle

(deg

)

time (sec)

(d)

0 10 20 30 40 50−40

−30

−20

−10

0

10

20

30

40

50

Rol

l ang

le (

deg)

time (sec)

(e)

0 10 20 30 40 50−12

−10

−8

−6

−4

−2

0

2

4

Yaw

ang

le (

deg)

time (sec)

(f)

Figure 5: Experiment 2, Mamdani FLC’s performance with a different initial condition. Quadrotor states: (a) x-axis; (b)y-axis; (c) z-axis (altitude); (d) pitch (θ); (e) roll (φ); and (f) yaw (ψ).


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0 10 20 30 40 50−10

0

10

20

30

40

50

60

X−

axis

mot

ion

(m)

time (sec)


(a)

0 10 20 30 40 50−10

0

10

20

30

40

50

Y−

axis

mot

ion

(m)

time (sec)

(b)

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0

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axis

mot

ion

(m)

time (sec)

(c)

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−40

−30

−20

−10

0

10

20

30

Pitc

h an

gle

(deg

)

time (sec)

(d)

0 10 20 30 40 50−40

−30

−20

−10

0

10

20

30

40

50

Rol

l ang

le (

deg)

time (sec)

(e)

0 10 20 30 40 50−20

−15

−10

−5

0

5

10

Yaw

ang

le (

deg)

time (sec)

(f)

Figure 6: Experiment 2, Sugeno FLC’s performance with a different initial condition. Quadrotor states: (a) x-axis; (b)y-axis;;(c) z-axis (altitude); (d) pitch (θ); (e) roll (φ); and (f) yaw (ψ).

0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Pitc

h an

gle

(rad

)

time (sec)


(a)

0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Rol

l ang

le (

rad)

time (sec)

(b)

0 2 4 6 8 10−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Yaw

ang

le (

rad)

time (sec)

(c)

Figure 7: Experiment 3, Mamdani FLC’s attitude stabilization. Quadrotor orientation: (a) pitch angle (θ); (b) roll angle (φ);and (c) yaw angle (ψ).

0 2 4 6 8 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Pitc

h an

gle

(rad

)

time (sec)


(a)

0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Rol

l ang

le (

rad)

time (sec)

(b)

0 2 4 6 8 10−0.1

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0.1

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0.4

0.5

0.6

0.7

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Yaw

ang

le (

rad)

time (sec)

(c)

Figure 8: Experiment 3, Sugeno FLC’s attitude stabilization. Quadrotor orientation: (a) pitch angle (θ); (b) roll angle (φ); and(c) yaw angle (ψ).


111

the performance of the fuzzy controllers as comparedto a robust backstepping technique for nonlinear con-trol in Figures 7, 8. Both Mamdani and Sugeno fuzzycontrollers are able to stabilize the quadrotor fromthe critical initial conditions. The settling times areslightly longer than the backstepping controller, spe-cially for theψ angle which can be improved with fur-ther tuning of the FLC’s membership functions. How-ever, this is overlooked because the fuzzy controllersare stabilizing both position and attitude despite thepresence of strong wind conditions as high as 30m/sand sensory noise of−55dB.

5 CONCLUSIONS

A fuzzy logic approach was proposed for the au-tonomous control of quadrotors without the need for amathematical model of their complex and ill-defineddynamics. The fuzzy technique was implementedthrough Sugeno and Mamdani inference engines forcomparison. The controller comprises of six indi-vidual fuzzy controllers designated for the control ofthe quadrotor’s position and orientation. The investi-gation on the two types of control methodologies isconducted in a simulation environment, where distur-bances such as wind conditions and sensor noise areincorporated for a more realistic simulation. The re-sults demonstrated a successful control performanceof the quadrotor with both Mamdani and Sugenofuzzy controllers despite the disturbances. Whencompared with other control techniques presentedin the literature for the same purpose, the proposedmethod showed a higher robustness and faster con-vergence, where satisfactory attitude stabilization isachieved despite stringent initial conditions and se-vere disturbances. The future work will be directedtowards a real-world implementation of the proposedfuzzy controllers. Furthermore, the online adaptationof controllers will be investigated.

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