Research Journal of Applied Sciences, Engineering and Technology 4(21): 4216-4226, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 18, 2011
Accepted: April 23, 2012
Published: November 01, 2012
Adaptive Backstepping Control of an Indoor Micro-Quadrotor
Zheng Fang and Weinan Gao
State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University,
Shenyang, Liaoning, China
Abstract: Micro-Quadrotor is one of the most popular VTOL (Vertical Take-Off and Landing) aerial robots
and has enormous potential applications in the field of near-area surveillance and exploration in military and
commercial applications. However, stabilizing and position control of the robot are difficult tasks because of
the nonlinear dynamic behavior and model uncertainties. Backstepping is a widely used control law for underactuated systems including quadrotor. But general backstepping control algorithm needs accurate model
parameters and isn’t robust to external disturbances. In this study, an adaptive integral backstepping control
algorithm is proposed to realize robust control of quadrotor. The proposed control algorithm can estimate
disturbances online and therefore improve the robustness of the system. Both simulation and experiment results
are presented to validate the performance of the proposed control algorithm.
Keywords: Adaptive integral-backstepping, flying robots, micro-quadrotor, model uncertainties
INTRODUCTION
Unmanned flying robots or Vehicles (UAVs) are
gaining increasing interest because of a wide area of
possible applications. While the UAV market has first
been driven by military applications and large expensive
UAVs, recent results in miniaturization, mechatronics and
microelectronics also offer an enormous potential for
small and inexpensive UAVs for commercial use. These
small UAVs would be able to fly either indoor or outdoor,
leading to completely new applications. However, indoor
flight comes up with some very challenging requirements
in terms of size, weight and maneuverability of the
vehicle that rule out most of the aircraft types. One type
of aerial vehicle with a strong potential for both indoor
and outdoor flight is the rotorcraft (Hanford, 2005) and
the special class of micro four-rotor aerial vehciles, also
called Micro-quadrotor. This vehicle, shown in Fig. 1, has
been chosen by many researchers as a very promising
vehicle (Slawomir et al., 2008; He et al., 2008; Hoffmann
et al., 2006; Guenard, 2006; Bouabdallah, 2006).
Micro-quadrotor is a kind of Vertical Take-off and
Landing UAV with simple mechanical structure. Since it
is agile and has excellent hovering ability, it can be
widely used for surveillance and exploration in both
indoor and outdoor environments. The quadrotor is a
mechatronic system with four propellers in a cross
configuration. While the front and the rear motor rotate
clockwise, the left and the right motor rotate
counterclockwise which nearly cancels gyroscopic effects
and aerodynamic torques in trimmed flight. By varying
Fig. 1: A quadrotor flying robot
the speed of the single motors, the lift force can be
changed and vertical and/or lateral motion can be created.
Pitch movement is generated by a difference between the
speed of the front and the rear motor while roll movement
results from differences between the speed of the left and
right rotor, respectively. Yaw rotation results from the
difference in the counter-torque between each pair (frontrear and left-right) of rotors. The overall thrust is the sum
of the thrusts generated by the four single rotors.
However, in spite of simple mechanical structure and
having four actuators, the high performance control of
quadrotor is a difficult task because it is an under-actuated
system with strong nonlinear and coupling characteristics.
In the past few years, different control methods have been
explored for the attitude and position control of quadrotor.
The most used control method is PID, LQR, Sliding Mode
and Backstepping. PID and LQR are classical linear
control method, but not suitable for systems with strong
Corresponding Author: Zheng Fang, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern
University, Shenyang, Liaoning, China
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
nonlinearity and coupling characteristics (Bresciani, 2008;
Bouabdallah and Siegwart, 2008). Therefore, these
controllers only work better when the quadrotor is near
hovering state. Sliding mode is another powerful control
method (Li, 2008; Bouabdallah and Siegwart, 2005),
which is simple and robust. But, Sliding mode control
requires continuous switching logic, which leads to the
chattering phenomenon. Backstepping has also been used
for the control of quadrotor (Bresciani, 2008). However,
general Backstepping control method cannot overcome
model uncertainties and the robustness is weak. Feedback
Linearization method is also used to control quadrotor.
Besides these classical control methods, some researchers
also tried intelligent control methods to control the
quadrotor. Holger proposed a nonlinear control method
based on a combination of State-Dependent Riccati
Equations (SDRE) and neural networks. Abhijit apply
Backstepping on the Lagrangian form of the quadrotor
dynamics instead of the state space form and introduces
two neural nets to estimate the aerodynamic components
to deal with unmodeled state-dependent disturbances and
forces.
The above control methods can be divided into two
groups. The first group linearizes the model and use linear
control law to design the controller. But most of the
controllers only work better near hovering state. The
robustness and stability still need to be improved. The
second group use nonlinear control theory to design the
controller. Although these controllers perform well in
simulation experiments, but since they depend on accurate
system model, the real-time control performance generally
even bad than PID controller. Therefore, considering the
limited onboard computing resources of the quadrotor,
how to develop a controller which can not only control
the quadrotor attitude precisely, but also has strong antidisturbance and environmental adaptive abilities is an
important issue for quadrotor control.
In this study, we propose an adaptive control method
to design the controller of quadrotor. The adaptive unit of
this method can effectively estimate the changes of mass
and the value of disturbances. The integral action can
eliminate the steady-state errors of the system. The
proposed method can restrain system overshoot, reduce
response time and enlarge the selection region of control
parameters. Several simulation experiments and first realtime control experiments have validated the proposed
control method.
Fig. 2: The earth inertial frame and the body-fixed frame
C
C
C
C
C
Base on the above assumptions, the quadrotor is a 6DOF rigid body and its throttle is provided by four
motors. So, a quadrotor is a 4-input and 6-output control
system. The kinematics of a quadrotor can be described
by Eq. (1) and (2):
1E = R1B
(1)
SE = T SB
(2)
where, 1E and SE are position vector and angular velocity
vector in the earth inertial frame , 1B and SB are position
vector and angular velocity vector in the body-fixed
frame, R is rotation matrix and T is transfer matrix.
Rotation matrix and transfer matrix are shown in Eq. (3)
and (4):
⎛ cosθ cosψ
⎜
R = ⎜ cosθ sin ψ
⎜
⎝ − sin θ
sin φ sin θ cosψ − cos φ sin ψ
sin φ sin θ sinψ − cos φ cosψ
sin φ cosθ
⎛ 1 sin φ tan θ
⎜
cos φ
T = ⎜0
⎜
⎝ 0 sin φ cosθ
METHODOLOGY
System dynamic modeling: Let us consider earth fixed
frame 'E(OE XE YE ZE) and body fixed frame 'B (OB XB YB
ZB), as shown in Fig. 2. And, we consider following
assumptions:
Elastic deformation and shock of the quadrotor is
ignored.
Inertia matrix is time-invariant.
Distribution of the mass of the quadrotor is
symmetrical which simplify the equations.
Drag factor and thrust factor of the quadrotor is
constant.
Air density around is constant.
cos φ tan θ ⎞
⎟
− sin φ ⎟
⎟
cos φ cosθ ⎠
sin φ sin ψ + cos φ sin θ cosψ ⎞
⎟
cosφ sin θ sin ψ − sin φ cosψ ⎟
⎟
cos φ cosθ
⎠
(3)
(4)
Base on the Newton-Euler equation, the dynamics
can be described as follow:
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
⎧⎪ m(v& B + ω B × v B ) = FB
⎨
⎪⎩ I .ω& B + ω B × ( Iω B ) = τ B
⎛ x2
⎜
⎜ U x (U1 + Dx ) / m
⎜
⎜ x4
⎜ U (U + D ) / m
x
⎜ y 1
⎜ x6
⎜
⎜
f ( X ,U ) = ⎜ cos x7 cosx9 U + Dz / m −
⎜ x8
⎜
⎜ x10 x12 a1 + x10a2 Ω r + b1U 2
⎜ x10
⎜
⎜ x8 x12 a3 + x8a4 Ω r + b2U 3
⎜ x12
⎜
⎝ x10 x8a5 + b3U 4
(5)
Thanks to the distribution of the mass of the
quadrotor is symmetrical, the inertia matrix is diagonal,
which can simplify the dynamic model. The model is
defined for hover and hover like flight, in this case, we
can replace TE with TB. Considering the disturbance on
three axes Di(i = x, y, z), the dynamic model of a
quadrotor can be described by Eq. (6):
4
⎧
⎡
⎤
⎪ x&& = ⎢ ( cos φ sin θ cosψ + sin φ sin ψ ) ∑ Ti + Dx ⎥ / m
i =1
⎣
⎦
⎪
4
⎪
⎡
⎤
⎪ &&
y = ⎢ ( cos φ sin θ sin ψ − sin φ cosψ ) ∑ Ti + Dy ⎥ / m
i =1
⎣
⎦
⎪
⎪
4
⎛
⎞
⎪ &&
z = ⎜ cos φ cosθ ∑ Ti − mg + Dz ⎟ / m
⎝
⎠
⎪
i =1
⎪
⎨
I − I zz θψ + J r θ Ω r + bl(Ω 24 + Ω 22 )
⎪ φ&& = yy
⎪
I xx
⎪
2
2
⎪ && ( I zz − I xx )θψ − J r φ Ω r + bl(Ω 3 + Ω 1 )
θ
=
⎪
I yy
⎪
⎪
I xx − I yy φθ + d (Ω 24 + Ω 22 − Ω 23 − Ω 12 )
⎪ &&
ψ
=
⎪
I zz
⎩ 4
(
)
(
[(
(
a 3 = ( I zz − I xx ) / I yy
velocity of a propeller, b is thrust factor, d is drag factor,
l is the center of quadrotor to center of propeller distance,
Jr is total rotational moment of inertia around the propeller
axis, Sr is overall propellers’ speed.
The system can be rewritten in state-space form X =
f(X, U), the state vector X and input vector U are shown
as follows:
x4 =
x5 =
x6 =
y&
z
z&
(7)
x7 = φ
x = φ&
8
x9 = θ
(8)
x10 = θ&
x11 = ψ
x12 = ψ&
(
(
(
(
) ⎞⎟
⎛b Ω2 + Ω2 + Ω2 + Ω2
1
2
3
4
⎜
⎛ U1 ⎞ ⎜
2
2
⎜ ⎟
bl
Ω
Ω
−
4
2
⎜ U2 ⎟ ⎜
U =⎜ ⎟ =⎜
2
2
U3
⎜
bl Ω 3 − Ω 1
⎜ ⎟
⎝ U4 ⎠ ⎜
⎜d Ω2 + Ω2 − Ω2 − Ω2
4
2
3
1
⎝
)
)
⎟
⎟
⎟
⎟
⎟
⎟
⎠
)
Take Eq. (7)-(9) into (6), we can obtain:
)
a 2 = J r / I xx
a 4 = J r / I yy
(
i= 1
x2 = x&
x3 = y
(10)
a1 = I yy − I zz / I xx b1 = l / I xx
where, ∑ Ti is total thrust by four propellers, Si is angular
x1 = x
]
where,
(6)
)
X = ( x , x&, y , y&, z , z&, φ , φ&,θ ,θ&,ψ ,ψ& )
)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
g⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(9)
b2 = l / I yy
b3 = l / I zz
(11)
)
a5 = I xx − I yy / I zz
U x = cos φ sin θ cosψ + sin φ sin ψ
U y = cos φ sin θ sin ψ + sin φ cosψ
(12)
Adaptive integral backstepping controller: From the
mathematical model, the quadrotor is a multivariate,
strong coupling and nonlinear under-actuated control
system. And, the system model also has uncertainties of
mass and inertia. Therefore, general backstepping control
algorithm performs not very well when the mass of the
quadrotor is varying or in winded environments. In order
to improve the robustness, this study adopts adaptive
scheme to design the controller. Adaptive control method
can estimate and compensate model uncertainties and
disturbances, therefore improve the robustness of the
control system.
From (10), the rotational motions do not depend on
translational motion while the opposite is not true. Thus,
double-loop control architecture is designed for the flying
robot's attitude and position control, as shown in Fig. 3.
The inner control loop was designed for stability and
tracking of desired Euler angles, with an outer control
loop for regulating the robot position.
To control the position of a quadrotor accurately, we
use adaptive integral backstepping method to design
position controller and attitude controller is still designed
using integral backstepping method.
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
⎡ xd ⎤
⎢y ⎥
⎢ d⎥
⎣⎢ z d ⎦⎥
Ref.
Pos.
ψ
φd
Position
Control
d
θd
ψd
U1
U2
Attitude
Control
F1K4
Control
Allocation
U3
U4
Quadrotor
Fig. 3: Double-loop control architecture
Attitude control: Three separate controllers are designed
to track the desired roll, pitch, yaw angles. From Eq. (10),
the attitude control system can be described by Eq. (13):
⎧ x&1 = x2
⎨
⎩ x&2 = au + b
V& = − c1e12 − c2 e22 < 0
[(
[(
[(
(15)
where, c1, 8 > 0, p1 = ∫ e1 (τ ) dτ ,combine Eq. (14) and
(15), we can obtain:
x1d + λp& 1 − &&
x1
⎧ e&2 = c1e&1 + &&
⎨
⎩ e&1 = − c1e1 − λp1 + e2
(16)
)
]
Horizontal position control: Take the x-axis horizontal
position control for example, Suppose D$ x is
online disturbance estimate value of quadrotor. Define the
error between actual and estimated values as:
(22)
~&
&
Dx = − D$ x
Construct a Lyapunov function:
]
~
V = V ( px , ex1 , ex 2 , Dx )
1
(1 − c12 + λ )e1 + (c1 + c2 )e2 − c1λp1 + &&x1d − b (18)
a
(23)
The derivative of V can be written as equation as
followings:
Construct a Lyapunov function as follow:
1 2 1 2 1 2
λp + e + e
2 1 2 1 2 2
(
]
] (21)
Assume constant parameters, then:
when e&2 = − c2 e2 − e1 ( c2 > 0):
V=
)
)
~
Dx = Dx − D$ x
e&2 = e1 ( − c1e1 − λp1 + e2 ) + &&
x1d + λe1 − au − b (17)
[
(
where, (cN1, cN2, c21, c22 , cR1, cR2, 8N, 82, 8R)>0, pN, p2, pR
are integrals of eN1, e21, eR1, respectively:
Put e&1 into e&2 :
u=
)
)
⎧
1
& &a − θ&a Ω
1 − cφ21 + λφ eφ 1 + cφ 1 + eφ 2 eφ 2 − cφ 1λφ pφ + φ&&d − θψ
⎪U 2 =
1
2 r
b1
⎪
⎪⎪
1
& &a − θ&a Ω
1 − cθ21 + λθ eθ 1 + ( cθ 1 + eθ 2 ) eθ 2 − cθ 1λθ pθ + φ&&d − θψ
⎨U 3 =
3
4 r
b2
⎪
⎪
1
1 − cψ2 1 + λψ eψ 1 + cψ 1 + eψ 2 eψ 2 − cψ 1λψ pψ
⎪U 4 =
⎪⎩
b2
(14)
Set a virtual input x2d:
x2 d = c1e1 + x&1d + λp1
(20)
According to Lyapunov second law, the system is
asymptotically stable. Therefore taking Eq. (10)-(12) into
(18), we can obtain quadrotor system actual control
inputs:
(13)
Define tracking error of x1 and x2 are:
⎧ e1 = x1d − x1
⎨
⎩ e2 = x2 d − x2
The derivative of V is:
∂V
∂V
∂V
& ∂V
V& = p&
+ e&x1
+ e&
− D$ x ~
∂p
∂ex 1 x 2 ∂ex 2
∂Dx
(19)
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
Considering Eq. (10) and (16), we will get:
Ux =
⎛ ∂V
⎛ ∂V
∂V ⎞
∂V ⎞
V& = p& x ⎜
+ λx
⎟ + e& x1 ⎜
+ c x1
⎟
∂ ex2 ⎠
∂ ex2 ⎠
⎝ ∂ px
⎝ ∂ p x1
U x U 1 + Dx ⎞ ∂ V
⎛
& ∂V
− D$ x ~
− ⎜ x&&d −
⎟
⎠ ∂ ex 2
⎝
m
∂ Dx
m⎛
D$
m
U1
⎡ D$ x
+ &&
x d + λx p& x + c x1e& x1 + f ( p x , e x1 , e x 2 )
⎢
⎣ m
]
+ k x ( β x1 p x + β x 2 e x1 + β x 3 e x 2 )
(24)
In order to satisfy Eq. (25), we can define Lyapunov
function as:
⎞
x
Let U x = U x1 + U x 2U x1 = U ⎜⎝ − m + &&xd ⎟⎠
1
~
∂V
Dx &$
∂V
Dx = − γ x
(γ > 0)
~ =
∂ ex2 x
∂ Dx mγ x
(β P e + βx 2 ex1ex 2 + βx 3ex22 )
2 βx 3 x1 x x 2
1 ~2
+
D + (⋅)
2mγ x x
V =
1
The very last term in V is a part that we do not need
to construct explicitly. This explains the name implicit
construction of Lyapunov function.
Similarly for the y-axis disturbance, we can also
use this method to estimate, the result is Eq.(34):
⎛ ∂V
⎛ ∂V
∂V ⎞
∂ V ⎞ U x 2U 1 ∂ V
V& = p& x ⎜
+ λx
⎟ + e& x1 ⎜
+ c x1
⎟ −
m ∂ px2
∂ ex2 ⎠
∂ ex2 ⎠
⎝ ∂ px
⎝ ∂ e x1
(26)
(
)
⎧ D&$ = − γ β p + β e + β e
y
y1 y
y 2 y1
y3 y2
⎪ y
⎪
$
⎡
m − Dy
⎪
⎢
+ &&
y d + λ y p& y + c y1e& y1 + f p y , e y1 , e y 2
⎨U y =
U
1 ⎢
⎪
⎣ m
(34)
⎪
⎪⎩ + k y β y1 p y + β y 2 e y1 + β y 3 e y 2
(
Define Ux2 = Ux3 + Ux4, Ux3 = m, U x 3 = m(λx p& x + cx1e& x1 ) / U 1
where, (cx1, cx2, cy1, cy2, 8x, 8y) > 0, px, py are integrals of
ex1, ey1, respectively.
(27)
Suppose there exists a function f(px, ex1, ex2), such that:
p& x
∂V
∂V
∂V
+ e& x1
= f ( p x , e x1 , e& x 2 )
∂ px
∂ e x1
∂ ex2
(28)
Choose Ux4= f(px, ex1, ex2), + kx . (MV / Mex2) m / U1, where
kx > 0, then:
Altitude control: We need to consider the influence by
mass variation and vertical disturbance when designing
altitude controller. Although when the mass varies, the
system inertia will also change, but the simulation results
show that the effect of inertia variation is negligible.
Suppose m$ is the online estimated value of quadrotor
and D$ z is the online estimated value of vertical
disturbance, define the parameter estimation error as:
~
~ = m − m$ , D
$
m
z = Dz − Dz
2
⎛ ∂V ⎞
⎟ <0
V& = − k x ⎜
⎝ ∂e x 2 ⎠
)
)]
(
then:
∂V
∂ V U x 4 u1 ∂ V
+ e& x1
−
∂ px
∂ e x1
m ∂ ex 2
(33)
(25)
So Eq. (24) can be simplified by Eq. (26):
V& = p& x
(32)
(35)
(29)
From Eq. (21), we can obtain input variable U1:
∂V ∂e x2 can be chosen as:
∂ V ∂ e x 2 = β x 1 p x + β x 2 e x1 + β x 3 e x 3
(30)
&
D$ x = − γ x ( β x1 p x + β x 2 e x1 + β x 3 e x 2 )
(31)
[
m$
(1 − cz21 + λz )ez1 + (cz1 + cz 2 )ez2
Az
D$
− cz1λz pz + &&
zd + g − z
Az
U1 =
]
(36)
So,
where, Az = cosN cos2, (cz1, cz2, 8z) > 0 , pz = ∫ ez1 (τ ) dτ ,
in order to make system stable, let e&z 2 = − cz 2 ez 2 − ez1 , so
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
Table 1: Parameters of quadrotor
Parameters
Mean
m
Mass of quadrotor
l
Arm length
R
Propeller radius
Inertia on x axis
Ixx
Iyy
Inertia on y axis
Izz
Inertia on z axis
A
Area of propeller
Jr
Rotor inertia
sp
Sample time
In
I nput r(k)
S ampling y(k)
r (k)- y( k) =e( k)
V& = − cz1ez21 − cz 2 ez22 < 0
No
|e(k )|<A
Yes
AIB Control
AB Contr ol
Fig. 4: Integral-separated adaptive integral backstepping method
~
m
(1 − cz21 + λz )ez1 + (cz1 + cz 2 )ez 2
m
~
D
− cz1λz pz + &&
zd + g − z
m
[
]
(37)
Construct a Lyapunov function:
V =
1
1
1
1 ~2
1 ~2
λ p2 + e2 + e2 +
m +
D (γ , γ > 0) (38)
2 z z 2 z1 2 z 2 2γ z1m
2γ z 2 m z z1 z 2
The derivative of V is:
[
]
[
&
D$ Z as:
&$ = γ e (1 − c 2 + λ )e + (c + c )e
⎧m
z1 e 2
z1
1
z1
z1
z1
z2
⎪
⎪
zd + g
⎨ − cz1λz pz + &&
⎪ &
⎪⎩ D$ = − γ z 2 ez 2
]
Take Eq. (40) into (39), we can obtain:
(41)
Integral-separated adaptive integral backstepping
controller: As it is known to all, integral action can
reduce the steady-state error and improve the control
accuracy. However, when the system errors are large,
integral action should be removed to avoid big overshoot.
Therefore, we introduce Integral-separated scheme into
our adaptive integral backstepping control law as shown
in Fig. 4. If the system error is beyond threshold A, then
the integral term is deleted. Otherwise, adaptive integral
backstepping control method is used.
Integral-separated adaptive integral backstepping
method can not only reduce overshoot and reaction time,
but also enlarge the selection region of parameters to find
best adaptive control rule to improve the estimation and
control accuracy.
EXPERIMENTAL RESULTS
The proposed control algorithm is firstly
implemented in Matlab/Simulink for simulation
experiments. Firstly, we test the effect of the integralseparated action. Then, we will compare our Adaptive
Integral Backstepping (AIB) with PID controller and
Integral Backstepping controller (IB) in different test
conditions. The constant parameters of the model are
illustrated in Table 1.
~
m
&$ Dz D&$
V& = λz pz p& z + ez1e&z1 + ez 2 e&z 2 −
m
γ z1m γ z 2m z
~
m
= − cz1ez21 − cz 2 ez22 + {ez 2 (1 − cz21 + λz )ez1 + (cz1 + cz 2 )ez 2 (39)
m
&
m&$ ⎫ ~ ⎛⎜ ez 2
D$ ⎞⎟
−
− cz1λz pz + &&
zd + g −
⎬ + Dz ⎜ −
⎟
γ z1 ⎭
⎝ m γ z 2m ⎠
&$ and
In order to make system stable, choose m
Unit
kg
m
m
kg .m2
kg .m2
kg .m2
m2
kg .m2
s
As a result, we can get an altitude controller which
can make system stable.
Out
e&z 2 = − c2 ez 2 − ez1 +
Value
0.530
0.232
0.150
6.228 ×10!3
6.228 ×10!3
1.125 ×10!2
0.005
154 ×10!7
0.010
(40)
Integral-separated experiment: In order to test the
effect of integral-separated action, a couple of
experiments are designed. The initial position of every
experiment is (x, y, z, N, 2, R) = (0, 0, 0, 0.3, 0.3, 0) and
destination (x, y, z, N ,2, R) = (1, 1, 1, 0, 0, 0). The
simulation result of integral-separated action is shown in
Fig. 5. It is obvious that the system works better when
using Integral-Separated scheme. With the integralseparated action, the quadrotor changes to a new hovering
position more quickly. Thus, we add the integralseparated action to every adaptive integral backstepping
controller.
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
Experiments without disturbance and mass variation:
In this experiment, we focus on testing control
performance of our Adaptive Integral Backstepping
Controller with PID controller and Integral Backstepping
controller when there are no disturbances on each axis and
mass variation. We assume that the quadrotor starts at the
same initial position as above. The simulation result is
shown in Fig. 6. From the simulation results, the control
performances of the three control laws are almost the
same.
Mass variation: Moreover, we investigate the robustness
of three controllers when the mass of the quadrotor is
varying. At this stage, a 200 g object is hung on the
quadrotor at 10 sec. The quadrotor will inevitably deviate
from the destination when the mass is varying no
Integral-seperated AIB
Ordinary AIB
2.0
1.5
1.5
1.0
1.0
y(m)
x(m)
Integral-seperated AIB
Ordinary AIB
2.0
0.5
0.0
0.5
0.0
-0.5
5
0
10
15
t (s)
20
25
-0.5
30
0
5
10
(a)
15
t (s)
20
25
30
(b)
Fig. 5: Effect of integral-separated action
1.4
0.4
PID
IB
AIB
1.2
0.2
Pitch(rad)
x(m)
1.0
0.8
0.6
0.1
0.0
-0.1
0.4
-0.2
0.2
-0.3
0.0
5
0
10
15
t (s)
20
PID
IB
AIB
0.3
-0.4
25
0
30
5
10
(a)
15
t (s)
20
25
30
(b)
2.0
PID
IB
AIB
1.5
0.4
PID
IB
AIB
0.2
Roll (rad)
Y(m)
1.0
0.5
0.0
0.0
-0.2
-0.4
-0.5
-1.0
0
5
10
15
t (s)
20
25
-0.6
30
0
(c)
5
10
15
t (s)
(d)
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
PID
IB
AIB
1.5
0.05
Yaw (rad)
1.0
Z(m)
PID
IB
AIB
0.10
0.5
0
-0.05
0.0
-0.10
-0.5
5
0
10
15
t (s)
25
20
-0.15
30
0
5
10
(e)
15
t (s)
20
25
30
(f)
Fig. 6: Attitude and position control without disturbance and mass variation
2.0
1.5
1.0
Z(m)
Z(m)
1.5
Consider both mass and inertia
Consider only mass
2.0
PID
IB
AIB
0.5
1.0
0.5
0.0
0.0
-0.5
0
5
10
15
t (s)
20
25
-0.5
30
0
5
10
15
t (s)
20
25
30
(a) Altitude control result when the mass is changed
Fig. 8: Influence of inertia
1.1
Real
Estimated
1.0
methods. In order to test the accuracy of estimation,
another experiment is designed. We assume that the mass
of the quadrotor increases of 50 g every 10 sec and the
tracking performance is shown in Fig. 7b.
Because the inertia of quadrotor changes when the
mass is varying, we design the following experiment to
discuss the influence of inertia variation. The first takes
into account the impact of the mass and inertia and the
second only consider the impact of the mass. Both of two
experiments use adaptive integral backstepping method.
From Fig. 8, the inertia has little influence, so it can be
neglected.
Z(m)
0.9
0.8
0.7
0.6
0.5
0.4
0
20
40
t (s)
60
80
(b) Mass estimation
Fig. 7: Experimental results with mass variation
matter which control law is adopted. From Fig. 7a, we can
obtain some conclusions. Since the absence of adaptive
estimation unit, PID controller makes a larger dynamic
landing. Integral backstepping controller can reduce
system error to zero, but it takes a long recovery time.
Adaptive integral backstepping controller can estimate
mass drift. As a result, it performs better than the
other two control
External disturbance: Assume a disturbance is imposed
on the each axes at 10 sec (Dx, Dy, Dz) = (1N, 1N ,1N) and
is removed at 30 sec. The simulation results show that the
anti-disturbance ability of adaptive integral backstepping
controller is much better than PID and Integral
Backstepping controllers as shown in Fig. 9a.
In order to test the estimation accuracy, we impose
disturbances on the quadrotor in the form of piecewise
function. In Fig. 9, each axis’s disturbance changes with
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
6
3
5
4
2
Y(m)
X(m)
PID
IB
AIB
4
PID
IB
AIB
7
3
1
2
1
0
0
-1
-1
0
10
30
20
40
0
50
10
30
20
PID
IB
AIB
2.0
0.5
0.4
1.5
0.3
Roll (rad)
Z(m)
50
40
t (s)
t (s)
1.0
0.5
0.2
0.1
0
0
-0.5
-0.1
0
10
30
20
40
-0.2
50
0
t (s)
30
20
40
50
t (s)
0.3
Real value of Dx
Estimate of Dx
0.2
0.2
0.0
0.1
-0.2
0.0
Dx (N)
Pitch (rad)
10
-0.1
-0.2
-0.4
-0.6
-0.8
-0.3
-0.1
-0.4
0
10
30
20
40
50
-0.2
t (s)
0
20
60
40
80
100
t (s)
(a) Position control result when the axis disturbances are imposed
Real value of Dy
Estimate of Dy
1.0
1.5
Real value of Dz
Estimate of Dz
1.0
Dz (N)
Dy (N)
0.5
0
0.5
0.0
-0.5
-0.5
-1.0
-1.0
-1.5
-1.5
0
20
60
40
80
0
100
20
60
40
t (s)
t (s)
(b) Disturbance estimations
Fig. 9: Experiment with external disturbance
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Res. J. Appl. Sci. Eng. Technol., 4(21): 4216-4226, 2012
0.8
Roll
0.6
Pitch (rad)
Roll (rad)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
20
0.15
40
60
Time (s)
100
80
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
Yaw
0.05
Attitude
Yaw (rad)
0.10
0.00
-0.05
-0.15
-0.20
0
20
40
60
Time (s)
80
Roll
100
40
50
100
60
Time (s)
80
100
200
180
160
140
120
100
80
60
40
20
0
0
(a) Attitude control
20
150
Time (s)
200
300
(b) Altitude control
Fig. 10: Real-time attitude and altitude control on real system
time as shown in the blue line, estimations calculated by
adaptive
integral backstepping are shown in the green line.
The actual and estimated values of the curve are almost
the same.
Real-time control experiment: The control law is also
implemented on the real quadrotor flying robot as shown
in Fig. 1. Till now, we just have completed the attitude
control and altitude control on the prototype. Figure 10a
shows some first test results obtained with the prototype,
which is stabilized around 0 = 0 . At t = 40 a disturbance
is added to the robot and from the results, the robot can go
back to equilibrium quickly. However, since the
resolution of our gyroscopes and accelerometers are not
very high, therefore the angular control accuracy still has
to be improved. In the future, we will use a more accurate
IMU to realize more accurate attitude control. Figure 10b
shows the altitude control works very well. The task was
to climb to 1.8 m, hover and then land. Altitude control
has a maximum of 5 cm deviation from the reference.
unit of this method can effectively estimate the changes of
mass and the value of disturbances. The integral action
can eliminate the steady-state errors of the system. The
proposed method can restrain system overshoot, reduce
response time and enlarge the selection region of control
parameters. Several simulation experiments and first realtime control experiments have validated the proposed
control method.
ACKNOWLEDGMENT
This study was supported in part by the Fundamental
Research Funds for the Central Universities under Grant
No. N100408003, National Science Foundation of China
under Grant 61040014, the Applied Basic Research Fund
of Shenyang Municipal Science and Technology Project
under Grant No. F10-205-1-50 and the Ningbo Municipal
Natural Science Foundation under Grant No.
2010A610134.
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CONCLUSION
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