Actuator Fault Detection System

for a Mini-Quadrotor

A. Freddi, S. Longhi and A. Monteriù

Dipartimento di Ingegneria Informatica, Gestionale e dell’Automazione

Università Politecnica delle Marche

Via Brecce Bianche, 60131 Ancona, Italy

Email: freddi@diiga.univpm.it, {sauro.longhi, a.monteriu}@univpm.it

Abstract—This paper addresses the problem of actuator faultdetection for a mini-quadrotor. First a model for a four-rotorvehicle, obtained via a Lagrange approach, is presented. In orderto stabilize the quadrotor at low cruise speed, a control strategybased on nested saturation controllers is presented. Using aThau’s observer, a diagnostic system has been developed for thenonlinear model of the quadrotor. Different simulation trials havebeen performed and the analysis of the results proves that thedeveloped diagnostic system represents an effective solution tothe problem of actuator fault detection in mini-flying machines.

I. INTRODUCTION

Unmanned aerial vehicles (UAVs) or flying robots have

attracted enormous interest during the last years. Recent de-

velopments in solid state disks, integrated miniature actuators

and MEMS (Micro Electromechanical Systems) technology

sensors have made autonomous miniaturized flying robots pos-

sible. One type of mini-aerial vehicle with a strong potential is

the four-rotor aerial vehicle, also called quadrotor. This vehicle

has been chosen by many researchers as a very promising

vehicle for indoor/outdoor navigation using multidisciplinary

concepts ([1], [2], [3], [4]).

A quadrotor simply consists in four DC motors on which

propellers are fixed. These motors are arranged to the extrem-

ities of a X-shaped frame, where all the arms make an angle

of 90 degrees with one another. As shown in Figure 1, the

front and rear motors (M1 and M3) spin in the clockwise

direction with angular velocities ω1 and ω3, while the other

pair of rotors (M2 and M4) spin in the counter-clockwise

direction with angular velocities ω2 and ω4. Thus, a quadrotor

is an under-actuated system with four independent inputs and

six coordinate outputs. The quadrotor has some advantages

over conventional mini-helicopters. One of the advantages of

quadrotors is that they have more lift thrusts than conventional

helicopters. Moreover, they are potentially simpler to build and

highly maneuverable.

To ensure the normal operation, and increase the safety and

reliability of such vehicles, the problem of fault detection is

very important. If a fault is rapidly detected, the structure

of the controller can be changed to get the best possible

response of the system, or even the system can be brought to

an emergency manoeuvre ([5]). The information provided by

an efficient diagnostic system could assist in the development

Fig. 1. Quadrotor rotorcraft.

of fault accommodation strategies which would guarantee fail-

safe operation of the control system.

Several methods have been proposed to control a quadrotor

vehicle ([1], [6], [7], [8]), and different approach have been

developed for detecting and isolating sensor faults on a such

vehicle ([9], [10]), however only few researches have been

devoted to the important problem of detecting the vehicle

actuator faults.

The interest of the present work is to successfully develop

and apply a diagnostic observer to the nonlinear quadrotor

model for detecting the rotorcraft actuator faults. Exploiting

the Lipschitzianity of the nonlinear model, a full order Thau

observer ([11]) is obtained and used to generate the residuals

on which the FDI system is based. In faultless situations,

residuals remain around zero, while if an actuator fault occurs,

their values change, permitting to detect the fault. The scheme

proposed in this paper permits to detect actuator faults, and to

isolate which pair of rotors is faulty.

The nonlinearity of the quadrotor model does not permit to

exploit the significant developments which have been made in

the area of fault detection in linear systems ([12]).

The paper is organized as follows. In Section II, the non-

linear model of a mini-quadrotor is presented. The strategy to

control the quadrotor is detailed in Section III. The proposed

actuator fault diagnostic nonlinear observer is described in978-1-4244-6392-3/10/$26.00 ©2010 IEEE 2055

Section IV. Section V is devoted to the presentation of the

simulation results obtained for various fault scenarios when

the proposed scheme is applied to the quadrotor. Conclusions

and future directions are presented at the end of the paper.

II. DYNAMICAL MODELING OF QUADROTOR

The mathematical model developed here is based on some

basic assumptions as given below:

• Design is symmetrical.

• Quadrotor body is rigid.

• Propellers are rigid.

• There are not external effects on quadrotor body such as

air friction, wind pressure, etc.

• Free stream air velocity is zero.

• The four electric motors’ dynamic is relatively fast and

therefore it will be neglected as well as the flexibility of

the blades.

Two frames are used to study the system motion: an

inertial earth frame {RE} (O, x, y, z), and a body-fixed frame

{RB} {OB , xB , yB , zB}, where OB is supposed to be at

the mass center of the quadrotor. {RB} is related to {RE}

by a vector ξ =[

x y z]T

describing the position of

the center of gravity in {RE}, and three independent an-

gles η =[

φ θ ψ]T

, respectively roll, pitch and yaw,

which describe the rotorcraft orientation, with(

−π2 ≤ φ < π

2

)

,(

−π2 ≤ θ < π

2

)

and (−π ≤ ψ < π) (see Figure 1).

The small rotorcraft is supposed to have six degrees of

freedom according to the earth fixed frame given respectively

by quadrotor position and its attitude. The weight force acts

at the center of mass and is always along negative z-axis. If

l is the length of each arm, then each motor can be located

by position vectors: ~L1 = lıB , ~L2 = lB , ~L3 = −lıB and~L4 = −lB , with (ıB , B , kB) as versors of (xB , yB , zB)-axis.

Analogously, all the forces shown in the Figure 1 by ~fi = fi ~kBare located in RB by position vectors ~Li (i = 1, 2, 3, 4).

With these assumptions, the quadrotor is a solid body

evolving in 3D and subject to one force and three torques.

The imbalance of the forces ~fj , where j = 1, 3 or j = 2, 4,

results in torques, along a direction perpendicular to the plane

containing the force ~fj and the vector ~Lj . This torque is

responsible for the rotation of the vehicle along xB-axis and

yB-axis. The rotation about zB-axis is due to imbalance of

clockwise and counter-clockwise torques.

The translational kinetic energy of the rotorcraft is expressed

as ([13])

Ttrans ,1

2mξT ξ (1)

where m denotes the whole mass of the rotorcraft.

The rotational kinetic energy is given by ([14])

Trot ,1

2ηT Jη (2)

where J is the inertia matrix expressed directly in terms of the

generalized coordinates η

J = WTη IWη (3)

with

Wη =

1 0 −Sθ0 Cφ SφCθ0 −Sφ CφCθ

(4)

where S(.) and C(.) represent sin (.) and cos (.) respectively,

and I is the moment of inertia tensor, which is diagonal due

to the symmetry of the quadrotor.

The only potential energy which needs to be considered is

due to the gravitational field. Therefore, potential energy is

expressed as

U = mgz . (5)

Let q =[

ξT ηT]T

= [x, y, z, φ, θ, ψ]T ∈ R6 be the

generalized coordinates vector for the flying machine, the

Lagrangian is given by

L (q, q) = Ttrans+Trot−U =1

2mξT ξ+

1

2ηT Jη−mgz (6)

The model for the full quadrotor aircraft dynamics is

obtained from the Euler-Lagrange equations with external

generalized force

d

dt

∂L

∂q−∂L

∂q= F (7)

where F =[

FTξ τTη]T

. Fξ defines the translational force

applied to the aerial robot due to the control inputs and relative

to the frame {RE}, and τη is the generalized torques vector.

The small body forces are ignored, and we only consider the

principal control inputs uf and τη , where uf represents the

total thrust, and τη is the generalized torque.

Because of each spinning propeller generates vertically

upward lifting force, labeled as f1, f2, f3 and f4, all the motion

of the quadrotor is a consequence of the sum of these forces:

uf = f1 + f2 + f3 + f4 (8)

with

fj = Klω2j (i = 1, 2, 3, 4) (9)

where ωj is the angular speed of motor Mj (j = 1, 2, 3, 4),while Kl > 0 is the lift constant depending on the air density

ρ, the lift coefficient Cz and the blade rotor characteristics S

(diameter, step, profile, ...) and it results as

Kl =1

2ρSCz . (10)

Then, the force applied to the mini-rotorcraft relative to the

frame {RB} is defined as

F =[

0 0 uf]T

. (11)

Consequently

Fξ = RB→EF (12)

where RB→E is the rotation transformation from the body

reference frame to the earth reference frame given by

RB→E =

CθCψ CψSθSφ − CφSψ CφCψSθ + SφSψCθSψ SθSφSψ + CφCψ CφSθSψ − CψSφ−Sθ CθSφ CθCφ

.2056

Indicating the generalized torque as τη ,[

τφ τθ τψ]T

,

the Euler-Lagrange equation can be rewritten as

mξ +

00mg

= Fξ (13)

Jη + Jη −1

2

∂

∂η

(

ηT Jη)

= τη . (14)

Defining the Coriolis vector as

Fc (η, η) = J−1

2

∂

∂η

(

ηT J)

(15)

the expression (14) can be rewritten as

Jη + Fc (η, η) η = τη (16)

.

Finally, the equations of motion for the mini rotorcraft are

expressed as

mξ = Fξ +

00

−mg

(17)

Jη = τη − Fc (η, η) η . (18)

Since J is nonsingular, it is possible to simplify the analysis,

without loss of generality, changing the input variables as

follows

τη = Jτ + Fc (η, η) η (19)

with τ =[

τφ τθ τψ]T

.

Then, with the new inputs, it results:

η = τ . (20)

Rewriting (17) and (18) gives

mx = (CφCψSθ + SφSψ)uf (21)

my = (CφSθSψ − CψSφ)uf (22)

mz = (CθCφ)uf −mg (23)

φ = τφ (24)

θ = τθ (25)

ψ = τψ (26)

III. CONTROL STRATEGY

A control strategy for stabilizing the quadrotor near hover

is recalled in this section. The control input uf is essentially

used to make the altitude reach a desired value. The control

input τψ is used to set the yaw displacement to zero. τθ is

used to control the pitch angle and the horizontal movement

in the x-axis. Similarly τφ is used to control the roll angle and

the horizontal displacement in the y-axis ([7]).

A. Altitude and yaw control

The control of the vertical position can be obtained by using

the following control input ([7])

uf = (−az1 z − az2(z − zd) +mg)1

CφCθ(27)

where az1 , az2 are positive constants and zd is the desired

altitude.

The control of the yaw displacement can be obtained by

using the following control input ([7])

τψ = −aψ1ψ − aψ2

(ψ − ψd) (28)

where aψ1and aψ2

are positive constants, ψd is the desired

yaw angle.

B. Attitude control

The control of the attitude can be obtained by using the

nested saturations strategy ([7]). The law for roll control is:

τφ = −σφ1(φ+ σφ2

(ν1 + σφ3(ν2 + σφ4

(ν3))))

ν3 = 3φ+ φ− 3y

g−y − yd

g

ν2 = 2φ+ φ−y

gν1 = φ+ φ

(29)

The law for pitch control is:

τθ = −σθ1(θ + σθ2(ν1 + σθ3(ν2 + σθ4(ν3))))

ν3 = 3θ + θ + 3x

g+x− xd

g

ν2 = 2θ + θ +x

gν1 = θ + θ

(30)

In both equations σa(s) is a saturation function defined as

σa (s) =

a, if s > a

s, if − a ≤ s ≤ a

−a, if s < −a(31)

while xd and yd are the desired longitudinal and lateral

positions.

IV. FDI SYSTEM

A model-based fault diagnosis system consists of a residual

generation module and a residual evaluation module. This

section recalls the theoretical aspects for developing a residual

generation module and a residual evaluation module for the

considered quadrotor system.

A. Residual generation

In literature several model-based methods have been

adopted for generating residual to be used in FDI schemes for

UAVs. Luenberger observers have been used both for sensor

and actuator fault detection (see [5]). Kalman filters (simple,

extended or unscented) have been used for fault adaptive and

tolerant control (see, e.g., [15] and [16]).

Most of the times the easiest and practical solution when

dealing with model-based techniques is to choose a proper2057

working point and linearize the system around it. However

this solution is suitable only for those vehicles operating most

of the time near the considered conditions. In all the other

cases there are two possible alternatives: linearizing the system

around different operating conditions or using a non-linear

model-based approach.

[11] developed an observer for a special class of non-linear

systems. This observer has already been applied to the fault

detection and isolation of non-linear dynamic systems and uses

the following non-linear system model ([12]):{

x(t) = Ax(t) + Bu(t) + h(x(t),u(t)) + F 1f(t)y(t) = Cx(t) + F 2f(t)

(32)

where x ∈ Rn is the state vector, u ∈ Rr is the input vector,

y ∈ Rp is the output vector, f ∈ Rh is the fault vector,

A,B and C are known system matrices with appropriate

dimensions, h(x(t),u(t)) represents the nonlinearity, and F 1

and F 2 are known fault entry matrices which represent the

effect of faults on the system.

For developing the Thau observer, the system model (32)

has to satisfy the two following conditions:

C1 - the pair (C,A) must be observable;

C2 - the non-linear function h(x(t),u(t)) must be locally

Lipschitz with constant , i.e.

||h(x1(t),u(t))−h(x2(t),u(t))|| ≤ ||(x1 −x2)||(33)

When these two conditions are satisfied, a stable observer for

the system (32) has the form{

˙x(t) = Ax(t) + Bu(t) + h(x(t),u(t)) + K(y(t) − y(t))y(t) = Cx(t)

(34)

where K is the observer gain matrix defined by

K = P−1δ CT . (35)

The matrix P δ is the solution to the Lyapunov equation

ATP δ + P δA − CTC + δP δ = 0 (36)

where δ is a positive parameter which is chosen such that (36)

has a positive definite solution.

In order to develop the Thau observer for the quadrotor

system considered in this paper, it is necessary to write the

system described by the equations (21)-(26) in a state-space

form. Defining

x =[

x y z φ θ ψ x y z φ θ ψ]T

(37)

u =[

uf τφ τθ τψ]T

(38)

f =[

fx fy fz fφ fθ fψ]T

(39)

y =[

x y z φ θ ψ]T

(40)

as the state, input, fault and output vector, respectively, then

the system described in (21)-(26) can be rewritten as a state

space system of the form (32), where

A =

[

0(12×12)I6

0(6×6)

]

, (41)

B = 012×4, (42)

C =[

I6 0(6×6)

]

, (43)

h(x,u) =

06×1

(CφCψSθ + SφSψ) um

(CφSθSψ − CψSφ)um

(CθCφ)um

− g

J−11 ((τη − Fc (η, η) η))

J−12 ((τη − Fc (η, η) η))

J−13 ((τη − Fc (η, η) η))

(44)

and

F 1 = 0(12×6) , (45)

F 2 =

[

0(6×3)0(3×3)

I3

]

, (46)

where In is the n-by-n identity matrix, 0(n×m) is the n-by-m

matrix of zeros and J−1i is the i− th row of the J−1 matrix.

The observability matrix results

O =[

CT (CA)T (CA2)T · · · (CA11)T]T

(47)

where[

CT (CA)T]T

= I12, while CAi = 0(6×12) (i =2, . . . , 11). Thus, the observability matrix has full rank, and

the pair (C,A) is observable, and condition C1 is satisfied.

h(x(t),u(t)) is a continuously differentiable function and

thus the condition C2 is satisfied as long as u(t) is bounded

([17]).

The parameter δ, and thus the gain matrix K, defines how

much the difference between the real system outputs and

the observer outputs must affect the residuals and should be

chosen as the best trade-off between the magnitude of residuals

due to faults and that due to noise. After several simulations

the gain matrix has been chosen as

K =

[

1.1I6

0.3025I6

]

. (48)

Since the considered quadrotor system is supposed to be

equipped with either a GPS receiver (outdoor operation) or

a vision system (indoor flight) and an Attitude and Heading

Reference System (AHRS), the system outputs are the position

in the earth frame (x, y and z) and the attitude angles (φ, θ

and ψ). The residuals are built as the difference between the

system outputs and the Thau observer outputs, that is to say:

r1(t) = x(t) − x(t) (49)

r2(t) = y(t) − y(t) (50)

r3(t) = z(t) − z(t) (51)

r4(t) = φ(t) − φ(t) (52)

r5(t) = θ(t) − θ(t) (53)

r6(t) = ψ(t) − ψ(t) . (54)

which are supposed to be zero as long as there are no faults

on the actuators and to differ from zero in case of faults.2058

B. Residual evaluation

The residual evaluation module detects a change in the mean

of an observed and distributed random sequence achieved by

sequential change detection algorithms. Based on the residual

evaluation algorithm developed by [18], each residual ri(i = 1, . . . , 6) has been compared to a fixed threshold, and

the following decision is made at time t:

ri(t) ≥ Hi or ri(t) ≤ hi in faulty case (55)

ri(t) < Hi or ri(t) > hi in fault-free case (56)

where Hi and hi are the upper and lower thresholds, respec-

tively, with i = 1, . . . , 6.

V. SIMULATION RESULTS

The non-linear quadrotor system along with the FDI system

have been developed using the Matlab and Simulink R© soft-

ware. The system outputs are supposed to be the linear posi-

tions in the earth frame (x, y and z) and the euler angles (φ, θ

and ψ). Noise is included in the model, affecting the sensors

measurment with an uncertainity of ±15cm on the linear

position and ±2 deg on the attitude position. The residuals

(six) are built as the difference between the nonlinear system

outputs and the Thau observer outputs. They are supposed to

be almost zero as long as there are no faults and to differ from

zero in case of faults.

The controller is set in order to allow the quadcopter to

reach a height of 25m and to remain in hover flight thereafter.

During the hover flight, a fault is introduced into one of the

actuators: indicating with T the instant time when a stuck fault

occurs, the behaviour of the actuator fi can be modelled as:

fj (t) =

{

fj (t) , t < T

fj(

T)

, t ≥ Tfor j=1, . . . ,4 (57)

The faulty actuator can no longer change its value once that

the fault has occurred. For the simulation a time T = 20s has

been chosen (fig. 2).

0 5 10 15 20 25 30 35 401.15

1.2

1.25

1.3

1.35

time [s]

f1 [N]

f1

Fig. 2. f1 in case of fault.

Once the actuator is stuck, the controller can no longer

control the position of the vehicle, due to the yaw displacement

that differs from zero, while it can still stabilize the attitude

angle on the axis in which the faulty actuator is present (that

is to say the angle θ if the fault is on f1 or f3, and φ if the

fault is on f2 or f4). Thus all the residuals should differ from

zero when a fault occurs on f1 or f3 with the exception of

the residual of θ, or with the exception of the residual of φ in

case of fault on f2 or f4.

Due to the symmetry of the system, a fault on f1 or f3produces the same results as a fault on f2 or f4: this implies

that a fault can always be detected, but the isolation logic can

only isolate the couple on which the fault has occurred and

not the single faulty actuator. These assumptions are confirmed

by the simulation results. Due to the symmetry of the problem

they are reported only for a fault on the actuators f1 and f2.

In absence of faults all the residuals show the same be-

haviour, and they are affected only by noise and slightly differ

from zero.

When the actuator f1 get stuck the dynamic of the real

system behaves in a different way then that of the observer:

the residual raises above the maximum value it had in the

faulty-free case (fig.3).

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

20

time [s]

r1 [m]

r1 [m]

threshold

Fig. 3. Residual 1 in case of fault on the 1st actuator.

A similar behaviour can be seen in all the other residuals,

with the exception of the residual calculated as (φ − φ), as

shown in fig. 4.

As already stated before, a similar behaviour can be seen if

the fault occurs on f3 (see table I).

In case of faults on f2 all the residual are similar to that of

fig. 5 with the exception of the residual calculated as (θ− θ),as shown in fig. 6.

As already stated before, a similar behaviour can be seen if

the fault occurs on f4 (see table I).

VI. CONCLUDING REMARKS

In this paper, the FDI problem for a quadrotor vehicle has

been faced. Using a Thau’s observer, a set of residuals have2059

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time [s]

r4 [rad]

r4 [rad]

threshold

Fig. 4. Residual 4 in case of fault on the 1st actuator.

0 5 10 15 20 25 300

1

2

3

4

5

6

time [s]

r1 [m]

r1 [m]

threshold

Fig. 5. Residual 1 in case of fault on the 2nd actuator.

been generated for FDI tasks. The Thau observer estimates

the system states using a simple algorithm and it is accurate

enough to provide a reliable signal for the FDI system, thus

representing an effective alternative to Kalman-based filters.

Simulation trials have shown that faults on the actuators can be

correctly detected and that the couple on which the fault occurs

can be correctly isolated. A strategy for disturbance reduction

and fault tolerant control is currently under investigation.

REFERENCES

[1] E. Altug, J. Ostrowski, and R. Mahony, “Control of a quadrotor heli-copter using visual feedback,” in Proc. of IEEE Int. Conf. on Robotics

and Automation, 2002.[2] A. Mokhtari and A. Benallegue, “Dynamic feedback controller of euler

angles and wind parameters estimation for a quadrotor unmanned aerialvehicle,” 2004.

[3] A. Tayebi and S. McGilvray, “Attitude stabilization of a vtol quadrotoraircraft,” IEEE Transactions on Control Systems Technology, vol. 14,no. 3, pp. 562–571, 2006.

[4] B. Bethke, M. Valenti, and J. How, “Uav task assignment,” IEEE Trans.

on Robotics and Automation, vol. 15, no. 1, pp. 39–44, 2008.

0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time [s]

r5 [rad]

r5 [rad]

threshold

Fig. 6. Residual 5 in case of fault on the 2nd actuator.

TABLE IEFFECTS OF THE ACTUATOR FAULTS ON THE RESIDUALS

f1 f2 f3 f4

r1 1 1 1 1r2 1 1 1 1r3 1 1 1 1r4 0 1 0 1r5 1 0 1 0r6 1 1 1 1

[5] G. Heredia, A. Ollero, M. Bejar, and R. Mahtani, “Sensor and actuatorfault detection in small autonomous helicopters,” Mechatronics, 2007.

[6] S. Bouabdallah, P. Murrieri, and R. Siegwart, “Design and control of anindoor micro quadrotor,” 2004, pp. 4393–4398.

[7] P. Castillo, R. Lozano, and A. Dzul, Modelling and control of mini-flying

machines. Springer-Verlag, 2005.[8] A. Das, K. Subbarao, and F. Lewis, “Dynamic inversion with zero-

dynamics stabilisation for quadrotor control,” IET Contr. Theory &

Applications, pp. 303–314, 2009.[9] C. Berbra, S. Lesecq, and J. Martinez, “A multi-observer switching

strategy for fault-tolerant control of a quadrotor helicopter,” in Control

and Automation, 2008 16th Mediterranean Conference on, 2008, pp.1094–1099.

[10] H. Rafaralahy, E. Richard, M. Boutayeb, and M. Zasadzinski, “Si-multaneous observer based sensor diagnosis and speed estimation ofunmanned aerial vehicle,” 2008, pp. 2938–2943.

[11] F. Thau, “Observing the state of non-linear dynamic systems,” Int. Jour.

of Control, vol. 17, no. 3, pp. 471–479, 1973.[12] J. Chen and R. Patton, Robust model-based fault diagnosis for dynamic

systems, 1999.[13] S. Salazar, H. Romero, R. Lozano, and P. Castillo, “Modeling and

real-time stabilization of an aircraft having eight rotors,” Journal of

Intelligent and Robotic Systems, vol. 54, no. 1, pp. 455–470, 2009.[14] G. Raffo, M. Ortega, and F. Rubio, “An integral predictive/nonlinear

control structure for a quadrotor helicopter,” Automatica, pp. 29–39,2009.

[15] J. Qi, Z. Jiang, X. Zhao, and J. Han, “UKF-based rotorcraft UAV Faultadaptive control for actuator failure,” in IEEE International Conference

on Robotics and Biomimetics, 2007. ROBIO 2007, 2007, pp. 1545–1550.[16] C. Rago, R. Prasanth, R. Mehra, R. Fortenbaugh, S. Inc, and M. Woburn,

“Failure detection and identification and fault tolerant controlusing theIMM-KF with applications to the Eagle-Eye UAV,” in Proc. of the 37th

IEEE Conf. on Decision and Control, 1998.[17] H. Khalil, Nonlinear systems (3rd edition). Prentice Hall, 2002.[18] A. Monteriù, P. Asthana, K. Valavanis, and S. Longhi, “Real-Time

Model-Based Fault Detection and Isolation for UGVs,” Journal of

Intelligent and Robotic Systems, vol. 56, no. 4, pp. 425–439, 2009.

2060