Proceedings of ASME DSCC 2015ASME 2015 Dynamic Systems and Control Conference

October 28-30, 2015, Columbus, Ohio, USA

DSCC2015-9900

ROBUST ATTITUDE TRACKING CONTROL FOR A QUADROTOR BASED ON THEUNCERTAINTY AND DISTURBANCE ESTIMATOR

Jiguo Dai, Qi Lu, Beibei Ren∗

Department of Mechanical EngineeringTexas Tech University

Lubbock, Texas, 79409Email: jiguo.dai@ttu.edu, qi.lu@ttu.edu, beibei.ren@ttu.edu

Qing-Chang ZhongDepartment of Electrical and Computer Engineering

Illinois Institute of TechnologyChicago, Illinois 60616

Email: zhongqc@iit.edu

ABSTRACTIn this paper, a robust control method based on the uncer-

tainty and disturbance estimator (UDE) is developed to achievethe attitude tracking control for a quadrotor. To facilitate the con-trol design, the coupled terms in the roll, pitch and yaw dynamicsare lumped into the uncertainty term and the remained dynamicscan be regarded as decoupled subsystems. As a result, for eachsubsystem, the lumped uncertainty term contains all the coupledterms, uncertainties and disturbances, then the UDE method isapplied for the uncertainty compensation. Compared with the ex-isting UDE control works, the introduced filtered tracking errordynamics simplifies the controller design and implementation.Furthermore, the stability analysis of the closed-loop system isestablished and experimental studies are carried out to illustratethe effectiveness of the developed control method.

INTRODUCTIONAs a promising autonomous system, the quadrotor is de-

signed to fly in complex environments to complete specific mis-sions while offering great autonomy, like rescue and search, re-mote inspection, surveillance, military applications [1]. Thequadrotor has advantages of small size, light weight and abil-ity of vertically taking off and landing. Thus, it holds high ma-neuverability and has attracted much attention from the researchcommunities.

In order to achieve the desired performance, there are usu-ally two sub-problems in the control for the quadrotor : (a) the at-

∗Address all correspondence to this author.

titude control; (b) the position control [2]. The dynamics analysisshows that the position and attitude subsystems of the quadrotorare cascade interconnected [2, 3]. Since the position controllergenerates a reference attitude set points for the attitude controller,the control problem mainly focuses on the attitude control [4–6].Furthermore, the complex working environment, various pay-loads, lacking of accurate model knowledge will generate in-ternal uncertainties and external disturbances. The uncertain-ties and disturbances compensation is an challenging problemin engineering applications. In this paper, the quadrotor attitudecontrol problem in the presence of coupled dynamics, modelinguncertainties and external disturbances is considered.

The quadrotor attitude control has been investigated usingvarious control methods [4, 7–12]. The effectiveness of classicalPID (proportional-integral-derivative) controller and LQ (linearquadratic) were investigated in [4]. However, these linear con-trollers are not easy to deal with the highly nonlinearities. Sincethe model-based controllers, such as feedback linearization [13],dynamic inversion [7], require the exact knowledge of the model,they cannot handle the uncertainties. The sliding mode controlis robust to the large uncertainties, but its switching logic willlead to the chattering phenomenon [8, 9]. The robust methodsin frequency domain like H2, H∞ control methods introducedin [11, 12] work well only within a frequency bandwidth limit.

In this paper, in order to achieve the attitude control of thequadrotor in the presence of internal uncertainties and externaldisturbances, a control method based on the uncertainty and dis-turbance estimator (UDE) is adopted. The UDE method, whichwas proposed in [14], has many advantages in both design and

1 Copyright c© 2015 by ASME

implementation respects and its excellent performance in han-dling the uncertainty and disturbance yet with very easy tun-ing and implementation has been demonstrated in recent yearsthrough both theories (including extensions to both linear [14,15]and nonlinear systems [16–20], systems with delays [15,17] etc.)and wide applications (covering mechatronics and robotics [21],electrical machines and drives [14, 20], renewable energy [22],aerospace and automotive systems [23,24] etc). The basic idea ofthe UDE method is that in the frequency domain an engineeringsignal (the lumped system uncertainty and external disturbance)can be approximated by putting it through a filter with the ap-propriate bandwidth. The main contributions of this paper are asfollows:

1) Extending the UDE method to the decoupling control.The key idea is that lumping the coupled dynamics into an un-certainty term, such that the remained dynamics can be regardedas decoupled subsystems. For each subsystem, the UDE methodcan be applied individually for the uncertainties compensation.

2) Compared with the existing UDE control works, the fil-tered tracking error dynamics is introduced to simplify the con-troller design and implementation.

The rest of this paper is organized as follows. Section 2illustrates the preliminaries about the quadrotor dynamics. Sec-tion 3 is devoted to system decoupling and Section 4 presentsthe attitude tracking controller design. The stability analysis ofthe closed-loop system is presented in Section 5 and the experi-ment results are shown in Section 6. Finally, Section 7 gives theconclusions.

PROBLEM FORMULATION AND PRELIMINARYAs shown in Fig.1, the quadrotor has six degree of freedoms

which are controlled by four rotors installed in its arms. Therotation of these propellers generates upward lifting forces andtorques along their spinning axes. Two of these propellers rotateclockwise and the remaining rotate anticlockwise. The quadro-tor adjusts its height, position, yaw, pitch, roll and completestaking-off, landing actions by changing the speeds of four rotors.The dynamic model is under those assumptions: the structure ofquadrotor is supposed to be rigid and symmetric, thrust and dragare supposed to be proportional to the square of rotor speed andthe origin of the body frame is fixed at the center of mass.

The thrust and torque generated by each propeller is definedby

fk = bω2k (1)

τk = dω2k , k = 1,2,3,4 (2)

where ωk is the angular speed of kth motor, b is the positivethrust constant, and d is the positive drag constant. The torques

𝜏𝜙

𝜏𝜃

𝜏2

𝜏1

𝜏3

𝜏4

𝜔3

𝜔2

𝜔4

𝜔1

𝐹𝑟𝑜𝑛𝑡

𝑥𝐵

𝑦𝐵

𝑚𝑔

𝑓3 𝑓2

𝜏𝜓

𝑥𝐼

𝑦𝐼 𝑧𝐼

𝑧𝐵

𝑓4 𝑓1

Figure 1. Quadrotor configuration

generated by the motors along the three body axes, xB, yB and zB,can be expressed as

τφ = l ( f4 − f2) = bl(ω24 −ω

22) (3)

τθ = l ( f3 − f1) = bl(ω23 −ω

21) (4)

τψ = τ2 + τ4 − τ1 − τ3

= d(ω22 +ω

24 −ω

21 −ω

23) (5)

where l is the distance from the motors to the center of mass.Consequently, the attitude dynamics is modeled as follows [4]

φ = θψ(Iyy − Izz

Ixx)− J

IxxθΩ+

1Ixx

τφ (6)

θ = φψ(Izz − Ixx

Iyy)+

JIyy

φΩ+1

Iyyτθ (7)

ψ = θφ(Ixx − Iyy

Izz)+

1Izz

τψ (8)

where φ,θ,ψ are the roll, pitch and yaw angles and Ixx, Iyy, Izz arethe moments of inertia. J denotes the inertia of propeller andΩ = ω2 +ω4 −ω1 −ω3. The generalized coordinates are cho-sen as p(t) = [φ,θ,ψ]T , q(t) =

[φ, θ, ψ

]T , and the control inputis u(t) =

[τφ,τθ,τψ

]T . Assume that the system has an externalacceleration disturbance d(t) which might be caused by model-ing errors, wind or other environmental factors, then equations(6)-(8) can be written in the following forms

p(t) = q(t) (9)q(t) = f (p,q)+g(p,q)u(t)+d(t) (10)

with f (p,q)=

θψ

(Iyy−Izz

Ixx

)− J

IxxθΩ

φψ

(Izz−Ixx

Iyy

)+ J

IyyφΩ

θφ

(Ixx−Iyy

Izz

), g(p,q)=

1

Ixx0 0

0 1Iyy

00 0 1

Izz

,

which is non-singular.

2 Copyright c© 2015 by ASME

A set of continuous smooth and bounded reference signalsare given as

rd(t) = [φd ,θd ,ψd ]T ∈C2(R3)

⋂L2(R3) (11)

where C2(·) and L2(·) are the function spaces of second differ-ential and bounded functions.

The objective is to design a robust control algorithm for thequadrotor system (9) and (10) to force the states p(t) = [φ,θ,ψ]T

to asymptotically track the reference signals rd(t) in (11), i.e., toachieve

p(t)→ rd as t → ∞. (12)

SYSTEM DECOUPLINGIn the system dynamics (10),

f (p,q) = fc(t)+∆ f (t) (13)

where fc(t) =

θψ

(Iyy−Izz

Ixx

)φψ

(Izz−Ixx

Iyy

)θφ

(Ixx−Iyy

Izz

) denote the coupled terms and

∆ f (t) =

− JIxx

θΩ

JIyy

φΩ

0

denote the uncertainty terms since the pa-

rameter J is unknown.Then the system dynamics (9), (10) can be written as

p(t) = q(t) (14)q(t) = ( fc(t)+∆ f (t)+d(t))+g(p,q)u(t)

= ud(t)+g(p,q)u(t) (15)

where ud(t)= fc(t)+∆ f (t)+d(t) includes all the coupled states,uncertainties and disturbances.

Since g(p,q) is diagonal and invertible and ud(t) is regardedas the lumped uncertainty term, the system (14), (15) can be de-coupled into a set of second order ODEs. For each subsystem,the system dynamics becomes

pi(t) = qi(t)

qi(t) = udi(t)+giiui(t) (16)

where i = 1,2,3 denote the roll, pitch and yaw, respectively.

CONTROLLER DESIGNOnce the system is decoupled into three second-order sub-

systems (16), the UDE-based controller for each subsystem willbe designed following the procedures in [14] .

The tracking error vector for each subsystem is defined as

ei(t) =[

rdi(t)− pi(t), rdi(t)−qi(t)]T (17)

In order to simplify the controller design and implementation,the filtered tracking error is introduced as follows

esi(t) = λi [rdi(t)− pi(t)]+ [rdi(t)−qi(t)] (18)

where λi ∈ R is chosen such that the polynomial s+λi = 0 hasroot in the open left half of the complex plane. According to (16),the derivative of the filtered tracking error in (18) with respect tothe time t is given by

esi(t) = λi [rdi(t)−qi(t)]+ [rdi(t)−udi(t)−giiui(t)] (19)

The controller for (19) is chosen as

ui(t) = vli(t)+ vdi(t) (20)

where vli(t) is for the feedback linearization and vdi(t) is for theuncertainties compensation. If the term udi(t) in (19) is known,the uncertainties compensation control part can be chosen as

vdi(t) =−g−1ii udi(t) (21)

and the feedback linearization control part vli(t) can be chosenas

vli(t) = g−1ii kmiesi(t)+g−1

ii λi [rdi(t)−qi]+g−1ii rdi(t) (22)

where kmi is a positive design parameter and all the signals in(22) are known or measurable. Then substituting (21) and (22)into (20), there is

ui(t) = g−1ii kmiesi(t)+g−1

ii λi [rdi(t)−qi]+g−1ii rdi(t)−g−1

ii udi(t)(23)Furthermore, substituting (23) into (19) results in

esi(t) =−kmiesi(t) (24)

3 Copyright c© 2015 by ASME

It shows that the closed-loop system is stable and esi(t) will expo-nentially converge to zero when t tends to infinity. Furthermore,it can be concluded from (17), (18) that the tracking error ei(t)also exponentially converges to zero when t tends to infinity.

However, udi(t) in (19) is unknown due to uncertaintiesand disturbances, so the uncertainties compensation control partvdi(t) in (21) should be redesigned. Substituting (20), (22) into(19) leads to

esi(t) =−kmiesi(t)+ [−giivdi −udi(t)] (25)

Solving udi(t) gives

udi(t) =−esi(t)− kmiesi(t)−giivdi(t) (26)

It indicates that the lumped uncertainty term can be observed bythe filtered tracking error and the control signal. As shown in[14], adopting a strictly proper low-pass filter G f i(s) with unitygain and zero phase shift in the spectrum of udi(t), the estimationof udi(t) is obtained as

udi(t) = L−1G f i(s)∗ [−esi(t)− kmiesi(t)−giivdi(t)] (27)

where udi(t) is the estimation of udi(t) and L−1 is the inverse-Laplace transform operator. Selecting

vdi(t) =−g−1ii udi(t) (28)

and using (27) lead to

vdi(t) = −g−1ii L−1G f i(s)

∗ [−esi(t)− kmiesi(t)−giivdi(t)]

(29)

Solving (29) for vdi(t) results in

vdi(t) =−g−1ii

(L−1

G f i(s)

1−G f i(s)

∗ [−esi(t)− kmiesi(t)]

)(30)Substituting (22) and (30) into (20) leads to the UDE-based con-troller

ui(t) = g−1ii kmiesi(t)+g−1

ii λi [rdi(t)−qi]+g−1ii rdi(t)

−g−1ii

(L−1

G f i(s)

1−G f i(s)

∗ [−esi(t)− kmiesi(t)]

)(31)

STABILITY ANALYSISThe overall control system is depicted in Fig. 2 and the per-

formance is analyzed in the following theorem.

Theorem 1. Consider the closed-loop system consisting of theplant (16), the reference signals (11) and the control law (31).Then for any initial value, the filtered tracking error will remainwithin the compact set Ωes :=

es ∈ R|‖es‖2 ≤ maxγ,e2

si(0)

,

with γ =µ2‖udi(t)‖2

L∞

2µkmi−1 , which can be adjusted by parameters µ, kmi

and the uncertainty estimation error udi (t) = udi (t)−udi (t).

Proof. Substituting (20), (22), (28) into (19) results in the fol-lowing closed-loop system error dynamics

esi(t) =−kmiesi(t)+ udi (t) (32)

Choose the following Lyapunov function candidate

Vi(t) =12

e2si(t) (33)

Taking the derivative of V (t) with respect to time t and using(32), there is

Vi(t) = esi(t)esi(t)

= esi(t) [−kmiesi(t)+ udi(t)] (34)

By using the Young’s inequality, we deduce the following results

Vi(t) ≤ −kmi ‖esi(t)‖2 +‖esi(t)‖‖udi(t)‖

≤ −kmi ‖esi(t)‖2 +

(‖esi(t)‖2

2µ+

µ‖udi(t)‖2

2

)

= −(

kmi −12µ

)‖esi(t)‖2 +

µ2‖udi(t)‖2

= −β1 ‖esi(t)‖2 +β2 ‖udi(t)‖2 (35)

where µ > 0, β2 =µ2 > 0 and β1 = kmi − 1

2µ > 0.According to (33), there is

Vi(t)≤−2β1Vi(t)+β2 ‖udi(t)‖2 (36)

Let c1 = 2β1 and c2 = β2 ‖udi(t)‖2, there is

Vi(t)≤−c1Vi(t)+ c2 (37)

4 Copyright c© 2015 by ASME

-1iig ( ),if p q

mik

∫

id t

ie t iu t ,p q( ),ig p q

,d dr r

ddt

fiG

div t

liv t

ˆdiu t

dir

-1iig

[ ]1iλ sie t

[ ]0iλ

mik

Figure 2. UDE-based control framework

then

Vi(t)≤Vi(0)e−c1t +c2

c1

(1− e−c1t) (38)

Noting (33) and Vi(0) = 12 e2

si(0), it can be derived that

‖esi(t)‖2 ≤ e2si(0)e

−c1t +2c2

c1

(1− e−c1t) (39)

As c1 > 0, the first term on the right hand side of (39) will decayto 0. The second term on the right hand side is bounded by 2c2

c1,

which can be further expressed as

2c2

c1≤ µ2 ‖udi(t)‖2

2µkmi −1

≤µ2 ‖udi(t)‖2

L∞

2µkmi −1(40)

:= γ

Therefore, the filtered tracking error remains within the compactset Ωes :=

esi ∈ R|‖esi‖2 ≤ maxγ,e2

si(0)

. The bound radiusγ is determined by control parameters µ, kmi and the uncertaintyestimation error ‖udi(t)‖.

Corollary 1. Under the same conditions of Theorem 1, thetracking error of the system will be bounded, i.e. ‖ei(t)‖ ≤max

‖ei (0)‖ ,

√γ,‖esi(0)‖

.

Corollary 2. While under the conditions β1 = kmi − 12µ > 0

and the estimation of the uncertainty is accurate enough, that

Figure 3. AscTec Hummingbird Quadrotor

is, limt→∞ udi(t) = 0, then the system states will asymptoticallytrack the reference, i.e. limt→∞ pi(t) = rdi(t).

EXPERIMENTAL VALIDATIONIn this section, experiments are carried out to verify the ef-

fectiveness of the developed control algorithm. The experimen-tal platform is the Hummingbird from Ascending TechnologiesGmbH(Asctec), as shown in Fig. 3. According to [14], thechoice of a first-order low pass filter in the form of

G f i(s) =1

τis+1(41)

is reasonable, where τi > 0 is the time constant and covers thespectrum of the lumped uncertainty term udi(t). Then the con-

5 Copyright c© 2015 by ASME

troller (31) is simplified as

ui(t) = g−1ii kmiesi(t)+

g−1iiτi

esi(t)+g−1ii

kmi

τi

ˆ t

0esi(ξ)dξ

+g−1ii λi [rdi(t)−qi]+g−1

ii rdi(t)

= g−1ii

[kmi +

1τi

]esi(t)+g−1

iikmi

τi

ˆ t

0esi(ξ)dξ

+g−1ii λi [rdi(t)−qi]+g−1

ii rdi(t) (42)

The time constant τi, controller parameters λi and kmi are chosendifferently for each subsystem which are listed in the Table 1. Inaddition, the thrust constant b = 6.11× 10−8N/rpm2 , the drugconstant d = 1.5×10−9Nm/rpm2 [25].

Table 1. UDE based controller parameters

Dof λi τi kmi

Roll φ 0.5 1 4

Pitch θ 0.5 1 4

Yaw ψ 5 2.5 3.5

For comparison, a PID controller

upidi (t)= g−1

ii

[KPi (rdi − pi)+KDi (rdi −qi)+KIi

ˆ(rdi − pi)dt

](43)is also implemented with the control parameters shown in theTable 2.

Table 2. PID controller parameters

Dof KPi KDi KIi

Roll φ 3 5 1

Pitch θ 3 5 1

Yaw ψ 8 5 1

Case 1: Nominal CaseThe goal of this case is to achieve the roll and pitch angles

stabilization, while the yaw angle tracking. The results usingthe developed UDE-based controller and the PID controller areillustrated in Fig. 4.

The whole experiment period is about 100 seconds. Thequadrotor is set to take off at t = 10s and start landing at t = 90s.The references of the roll and pitch are set to zero. The yawreference signal is chosen as follows,

rd3 =

180 t = 0s ∼ 20s210 t = 20s ∼ 40s180 t = 40s ∼ 50s150 t = 60s ∼ 80s180 t = 80s ∼ 90s

(44)

From Fig. 4 (a), (c), it can be seen that both the UDE basedcontroller and the PID controller could stabilize the roll and pitchangles in the range of ±5, which is a moderate steady state error.For the yaw subsystem, in Fig. 4 (e), it shows that the yaw statesuccessfully tracks the yaw reference and the steady state errorof UDE based controller is within the range of ±2, the PIDcontroller is within the range of ±5. And the control inputs ofboth controllers are shown in Fig. 4 (b), (d), (f).

Case 2: Disturbance RejectionThe goal of this case is to test the robustness of the UDE-

based controller and PID controller. The experiment environ-ment and the controller parameters are same with Case 1. Thequadrotor is set to take off at t = 10s and start landing at t = 60s.The roll and pitch references are set to zero and there are no dis-turbance applied. The yaw reference is set as 180o and two spikedisturbances d = 50(t −20) and d = −50(t −40) are applied att = 20s and t = 40s respectively which can be seen in Fig. 5 (e).

After applying the disturbances at 20s and 40s, both theUDE-based controller and PID controller could stabilize the yawsubsystem. However, it could be seen that the UDE based con-troller is more robust and has a better disturbance rejection abil-ity than the PID controller. Furthermore, the figures also showthat the pitch and roll control performance by the PID is degradeddue to the coupling effects of the yaw, while the developed UDE-based control can handle the uncertainty, disturbance, and cou-pling simultaneously and effectively. Fig. 5 (b), (d), (f) illustratethe control inputs of both controllers for each subsystem.

From both Case 1 and Case 2, the experiment results showthe excellent performance of the developed UDE-based con-troller in handling the uncertainty and disturbance yet with veryeasy tuning and implementation advantages over the PID con-troller.

CONCLUSIONSIn this paper, an UDE-based controller for a quadrotor has

been developed to achieve the attitude control in the presence

6 Copyright c© 2015 by ASME

10 20 30 40 50 60 70 80 90−15

−10

−5

0

5

10

t (s)

φ (d

eg)

UDE PID

10 20 30 40 50 60 70 80 90−0.03

−0.02

−0.01

0

0.01

0.02

t (s)

τ φ (N

⋅m)

UDE PID

(a) Roll (b) Roll control

10 20 30 40 50 60 70 80 90−15

−10

−5

0

5

10

t (s)

θ (d

eg)

UDE PID

10 20 30 40 50 60 70 80 90−0.03

−0.02

−0.01

0

0.01

0.02

t (s)τ θ (

N⋅m

)

UDE PID

(c) Pitch (d) Pitch control

10 20 30 40 50 60 70 80 90100120140160180200220

t (s)

ψ (

deg)

UDE PID REF

10 20 30 40 50 60 70 80 90−0.1

−0.05

0

0.05

0.1

t (s)

τ ψ (

N⋅m

)

UDE PID

(e) Yaw (f) Yaw control

Figure 4. Experimental results for nominal case

of coupled dynamics, internal uncertainties and external distur-bances. By regarding the coupled terms in the system as a part ofthe lumped uncertainty, the nonlinear MIMO system with threedegree of freedoms (roll, pitch, yaw) can be decoupled into threedecoupled subsystems. Following the UDE-based control designprocedure, controllers have been developed for each subsystemto track reference signals. The filtered tracking error dynamicshas been introduced to simplify the controller design and im-plementation. Furthermore, the stability of the closed-loop sys-tem has been analyzed and the experiments based on the AscTecHummingbird platform have verified the effectiveness of the de-veloped method.

ACKNOWLEDGMENTThe authors would like to thank Dr. Pedro Garcia, Universi-

dad Politecnica de Valencia, Spain, for his valuable opinions and

suggestions with the sonar sensor integration.

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[3] Ryll, M., Bulthoff, H. H., and Robuffo, G. P., 2012. “Mod-eling and control of a quadrotor UAV with tilting pro-pellers”. In Proceedings of International Conference onRobotics and Automation, IEEE, pp. 4606–4613.

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10 20 30 40 50 60−15

−10

−5

0

5

10

t (s)

φ (d

eg)

UDE PID

10 20 30 40 50 60−0.03

−0.02

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UDE PID

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10 20 30 40 50 60140150160170180190200

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0

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0.1

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