Geometric Tracking Control of a Quadrotor UAV on SE(3)
Taeyoung Lee
∗
, Melvin Leok
†
, and N. Harris McClamroch
Abstract— This paper provides new results for the tracking
control of a quadrotor unmanned aerial vehicle (UAV). The
UAV has four input degrees of freedom, namely the magnitudes
of the four rotor thrusts, that are used to control the six
translational and rotational degrees of freedom, and to achieve
asymptotic tracking of four outputs, namely, three position
variables for the vehicle center of mass and the direction of
one vehicle body-fixed axis. A globally defined model of the
quadrotor UAV rigid body dynamics is introduced as a basis
for the analysis. A nonlinear tracking controller is developed
on the special Euclidean group SE(3) and it is shown to
have desirable closed loop properties that are almost global.
Several numerical examples, including an example in which the
quadrotor recovers from being initially upside down, illustrate
the versatility of the controller.
I. INTRODUCTION
A quadrotor unmanned aerial vehicle (UAV) consists of
two pairs of counter-rotating rotors and propellers, located
at the vertices of a square frame. It is capable of vertical
take-off and landing (VTOL), but it does not require com-
plex mechanical linkages, such as swash plates or teeter
hinges, that commonly appear in typical helicopters. Due
to its simple mechanical structure, it has been envisaged for
various applications such as surveillance or mobile sensor
networks as well as for educational purposes. There are
several university-level projects [1], [2], [3], [4], and com-
mercial products [5], [6], [7] related to the development and
application of quadrotor UAVs.
Despite the substantial interest in quadrotor UAVs, little
attention has been paid to constructing nonlinear control
systems for them, particularly to designing nonlinear tracking
controllers. Linear control systems such as proportional-
derivative controllers or linear quadratic regulators are widely
used to enhance the stability properties of an equilibrium [1],
[3], [4], [8], [9]. A nonlinear controller is developed for
the linearized dynamics of a quadrotor UAV with saturated
positions in [10]. Backstepping and sliding mode techniques
are applied in [11]. Since these controllers are based on
Euler angles, they exhibit singularities when representing
complex rotational maneuvers of a quadrotor UAV, thereby
fundamentally restricting their ability to track nontrivial
trajectories.
Taeyoung Lee, Mechanical and Aerospace Engineering, Florida Institute
of Technology, Melbourne, FL 39201 taeyoung@fit.edu
Melvin Leok, Mathematics, University of California at San Diego, La
Jolla, CA 92093 mleok@math.ucsd.edu
N. Harris McClamroch, Aerospace Engineering, University of Michigan,
Ann Arbor, MI 48109 nhm@umich.edu
∗
This research has been supported in part by NSF under grants CMMI-
1029551.
†
This research has been supported in part by NSF under grants DMS-
0726263, DMS-1001521, DMS-1010687, and CMMI-1029445.
Geometric control is concerned with the development of
control systems for dynamic systems evolving on nonlinear
manifolds that cannot be globally identified with Euclidean
spaces [12], [13], [14]. By characterizing geometric proper-
ties of nonlinear manifolds intrinsically, geometric control
techniques provide unique insights to control theory that
cannot be obtained from dynamic models represented using
local coordinates [15]. This approach has been applied to
fully actuated rigid body dynamics on Lie groups to achieve
almost global asymptotic stability [14], [16], [17], [18].
In this paper, we develop a geometric controller for
a quadrotor UAV. The dynamics of a quadrotor UAV is
expressed globally on the configuration manifold of the
special Euclidean group SE(3). We construct a tracking
controller to follow prescribed trajectories for the center of
mass and heading direction. It is shown that this controller
exhibits almost global exponential attractiveness to the zero
equilibrium of tracking errors. Since this is a coordinate-
free control approach, it completely avoids singularities and
complexities that arise when using local coordinates.
Compared to other geometric control approaches for rigid
body dynamics, this is distinct in the sense that it controls
an underactuated quadrotor UAV to stabilize six translational
and rotational degrees of freedom using four thrust inputs,
while asymptotically tracking four outputs consisting of its
position and heading direction. We demonstrate that this
controller is particularly useful for complex, acrobatic ma-
neuvers of a quadrotor UAV, such as recovering from being
initially upside down.
II. QUADROTOR DYNAMICS MODEL
Consider a quadrotor vehicle model illustrated in Fig.
1. This is a system of four identical rotors and propellers
located at the vertices of a square, which generate a thrust
and torque normal to the plane of this square.
We choose an inertial reference frame {
~
i
1
,
~
i
2
,
~
i
3
} and a
body-fixed frame {
~
b
1
,
~
b
2
,
~
b
3
}. The origin of the body-fixed
frame is located at the center of mass of this vehicle. The
first and the second axes of the body fixed frame,
~
b
1
,
~
b
2
,
lie in the plane defined by the centers of the four rotors, as
illustrated in Fig. 1. The third body fixed axis
~
b
3
is normal
to this plane, and points downward, opposite to the direction
of the total thrust. Define
m ∈ R the total mass
J ∈ R
3×3
the inertia matrix with respect to the body-
fixed frame
R ∈ SO(3) the rotation matrix from the body-fixed frame
to the inertial frame
Ω ∈ R
3
the angular velocity in the body-fixed frame
49th IEEE Conference on Decision and Control
December 15-17, 2010
Hilton Atlanta Hotel, Atlanta, GA, USA
978-1-4244-7744-9/10/$26.00 ©2010 IEEE 5420
~
i
1
~
i
2
~
i
3
~
b
1
~
b
2
~
b
3
f
1
f
2
f
3
f
4
x
R
Fig. 1. Quadrotor model
x ∈ R
3
the location of the center of mass in the inertial
frame
v ∈ R
3
the velocity of the center of mass in the inertial
frame
d ∈ R the distance from the center of mass to the
center of each rotor in the
~
b
1
,
~
b
2
plane
f
i
∈ R the thrust generated by the i-th propeller along
the −
~
b
3
axis
τ
i
∈ R the torque generated by the i-th propeller about
the
~
b
3
axis
f ∈ R the total thrust, i.e., f =
P
4
i=1
f
i
M ∈ R
3
the total moment in the body-fixed frame
The configuration of this quadrotor UAV is defined by the
location of the center of mass and the attitude with respect to
the inertial frame. Therefore, the configuration manifold is
the special Euclidean group SE(3), which is the semidirect
product of R
3
and the special orthogonal group SO(3) =
{R ∈ R
3×3
| R
T
R = I, det R = 1}.
We assume that the thrust of each propeller is directly
controlled, i.e., we do not consider the dynamics of rotors
and propellers, and the direction of the thrust is normal to
the quadrotor plane. The total thrust is f =
P
4
i=1
f
i
, which
acts along the direction of −
~
b
3
. According to the definition
of the rotation matrix R ∈ SO(3), the direction of the i-th
body fixed axis
~
b
i
is given by Re
i
in the inertial frame, where
e
1
= [1; 0; 0], e
2
= [0; 1; 0], e
3
= [0; 0; 1] ∈ R
3
. Therefore,
the total trust is given by −f Re
3
∈ R
3
in the inertial frame.
We also assume that the torque generated by each propeller
is directly proportional to its thrust. Since it is assumed that
the first and the third propellers rotate clockwise, and the
second and the fourth propellers rotate counterclockwise,
when they are generating a positive thrust f
i
along −
~
b
3
, the
torque generated by the i-th propeller can be written as τ
i
=
(−1)
i
c
τ f
f
i
for a fixed constant c
τ f
. All of these assumptions
are common [19], [4], and the presented control system can
readily be extended to include linear rotor dynamics studied
in [11]. Under these assumptions, the total thrust f and the
total moment M can be written as
f
M
1
M
2
M
3
=
1 1 1 1
0 −d 0 d
d 0 −d 0
−c
τ f
c
τ f
−c
τ f
c
τ f
f
1
f
2
f
3
f
4
. (1)
The determinant of the above 4 × 4 matrix is 8c
τ f
d
2
, so it
is invertible when d 6= 0 and c
τ f
6= 0. Therefore, for a given
f, M, the thrust of each rotor f
i
can be obtained from (1).
Using this equation, the total thrust f ∈ R and the moment
M ∈ R
3
are considered as control inputs in this paper.
The equations of motion of this quadrotor UAV can be
written as
˙x = v, (2)
m ˙v = mge
3
− fRe
3
, (3)
˙
R = R
ˆ
Ω, (4)
J
˙
Ω + Ω × JΩ = M, (5)
where the hat map
ˆ
· : R
3
→ so(3) is defined by the condition
that ˆxy = x × y for all x, y ∈ R
3
.
III. GEOMETRIC TRACKING CONTROL ON SE(3)
We develop a controller to follow a prescribed trajectory
of the location of the center of mass, x
d
(t), and the desired
direction of the first body-fixed axis,
~
b
1
d
(t).
The overall controller structure is as follows. The trans-
lational dynamics of a quadrotor UAV is controlled by the
total thrust −fRe
3
, where the magnitude of the total thrust
f is directly controlled, and the direction of the total thrust
−Re
3
is along the third body-fixed axis −
~
b
3
. For a given
translational command x
d
(t), we select the total thrust f,
and the desired direction of the third body-fixed axis
~
b
3
d
to
stabilize the translational dynamics.
Once the desired direction of the third body-fixed frame
~
b
3
d
is chosen, there is one remaining degree of freedom in
selecting the desired attitude, R
d
∈ SO(3). This corresponds
to the heading direction of the quadrotor UAV in the plane
normal to
~
b
3
d
, and it is determined by the desired direction
of the first body-fixed axis,
~
b
1
d
(t). More explicitly, we
assume that
~
b
1
d
is not parallel to
~
b
3
d
, and we project it
onto the plane normal to
~
b
3
d
to obtain the complete desired
attitude: R
d
= [
~
b
2
d
×
~
b
3
d
,
~
b
2
d
,
~
b
3
d
] ∈ SO(3), where
~
b
2
d
=
(
~
b
3
d
×
~
b
1
d
)/k
~
b
3
d
×
~
b
1
d
k. The control moment M is selected
to follow this desired attitude.
In short, four-dimensional control inputs are designed to
follow a three-dimensional translational command, and a
one-dimensional heading direction, i.e., we guarantee that
x(t) → x
d
(t) and Proj[
~
b
1
(t)] → Proj[
~
b
1
d
(t)] as t → ∞,
where Proj[·] denotes the normalized projection onto the
plane orthogonal to
~
b
3
d
. The overall controller structure and
the definition of the heading direction are illustrated in Fig.
2 and 3.
Trajectory
tracking
Attitude
tracking
-
-
-
-
-
Quadrotor
Dynamics
-
f
M
~
b
3
d
x
d
~
b
1
d
x, v, R, Ω
6
-q
q
Controller
Fig. 2. Controller structure
5421
~
b
3
d
~
b
1
d
~
b
2
d
=
~
b
3
d
×
~
b
1
d
k
~
b
3
d
×
~
b
1
d
k
Proj[
~
b
1
d
] =
~
b
2
d
×
~
b
3
d
Plane
normal to
~
b
3
d
Fig. 3. Definition of the desired heading direction
We develop this controller directly on the nonlinear con-
figuration Lie group and thereby avoid any singularities and
complexities that arise in local coordinates. As a result, we
are able to achieve almost global exponential attractiveness
to the zero equilibrium of tracking errors. Due to page limit
constraint, all of the proofs are relegated to [20].
A. Tracking Errors
We define the tracking errors for x, v, R, Ω as follows. The
tracking errors for the position and the velocity are given by:t
e
x
= x − x
d
, (6)
e
v
= v − v
d
. (7)
The attitude and angular velocity tracking error should be
carefully chosen as they evolve on the tangent bundle of
the nonlinear space SO(3). The error function on SO(3) is
chosen to be
Ψ(R, R
d
) =
1
2
tr
I − R
T
d
R
. (8)
This is locally positive-definite about R = R
d
within the
region where the rotation angle between R and R
d
is
less than 180
◦
[14]. This set can be represented by the
sublevel set of Ψ where Ψ < 2, namely L
2
= {R
d
, R ∈
SO(3) | Ψ(R, R
d
) < 2}, which almost covers SO(3). When
the variation of the rotation matrix is expressed as δR = Rˆη
for η ∈ R
3
, the derivative of the error function is given by
D
R
Ψ(R, R
d
) · Rˆη = −
1
2
tr
R
T
d
Rˆη
=
1
2
(R
T
d
R − R
T
R
d
)
∨
· η, (9)
where the vee map
∨
: so(3) → R
3
is the inverse of the
hat map. We used the fact that −
1
2
tr[ˆxˆy] = x
T
y for any
x, y ∈ R
3
. From this, the attitude tracking error e
R
is chosen
to be
e
R
=
1
2
(R
T
d
R − R
T
R
d
)
∨
. (10)
The tangent vectors
˙
R ∈ T
R
SO(3) and
˙
R
d
∈ T
R
d
SO(3)
cannot be directly compared since they lie in different
tangent spaces. We transform
˙
R
d
into a vector in T
R
SO(3),
and we compare it with
˙
R as follows:
˙
R −
˙
R
d
(R
T
d
R) = R
ˆ
Ω − R
d
ˆ
Ω
d
R
T
d
R = R(Ω − R
T
R
d
Ω
d
)
∧
.
We choose the tracking error for the angular velocity as
follows:
e
Ω
= Ω − R
T
R
d
Ω
d
. (11)
We can show that e
Ω
is the angular velocity of the rotation
matrix R
T
d
R, represented in the body-fixed frame, since
d
dt
(R
T
d
R) = (R
T
d
R) ˆe
Ω
.
B. Tracking Controller
For given smooth tracking commands x
d
(t),
~
b
1
d
(t), and
some positive constants k
x
, k
v
, k
R
, k
Ω
, we define
~
b
3
d
= −
−k
x
e
x
− k
v
e
v
− mge
3
+ m¨x
d
k−k
x
e
x
− k
v
e
v
− mge
3
+ m¨x
d
k
, (12)
where we assume that
k−k
x
e
x
− k
v
e
v
− mge
3
+ m¨x
d
k 6= 0. (13)
We also assume that
~
b
1
d
is not parallel to
~
b
3
d
. The desired
attitude is given by R
d
= [
~
b
2
d
×
~
b
3
d
,
~
b
2
d
,
~
b
3
d
] ∈ SO(3), where
~
b
2
d
= (
~
b
3
d
×
~
b
1
d
)/k
~
b
3
d
×
~
b
1
d
k. The desired trajectory satisfies
k − mge
3
+ m¨x
d
k < B (14)
for a given positive constant B. The control inputs f, M are
chosen as follows:
f = −(−k
x
e
x
− k
v
e
v
− mge
3
+ m¨x
d
) · Re
3
, (15)
M = −k
R
e
R
− k
Ω
e
Ω
+ Ω × JΩ
− J(
ˆ
ΩR
T
R
d
Ω
d
− R
T
R
d
˙
Ω
d
). (16)
The control moment M given in (16) corresponds to a
tracking controller on SO(3). For the attitude dynamics of a
rigid body described by (4), (5), this controller exponentially
stabilizes the zero equilibrium of the attitude tracking errors.
Similarly, the expression in the parentheses in (15) corre-
sponds to a tracking controller for the translational dynamics
on R
3
. The total thrust f and the desired direction
~
b
3
d
of
the third body-fixed axis are chosen such that if there is
no attitude tracking error, the thrust vector term −fRe
3
in the translational dynamics of (3) becomes this tracking
controller in R
3
. Therefore, the translational tracking error
will converge to zero provided that the attitude tracking error
is identically zero.
Certainly, the attitude tracking error may not be zero at any
instant. As the attitude tracking error increases, the direction
of the control input term fRe
3
of the translational dynamics
deviates from the desired direction R
d
e
3
. This may cause
instability in the complete dynamics. In (15), we carefully
design the magnitude of the total thrust f so that it is reduced
when there is a large attitude tracking error. The expression
of f includes the dot product of the desired third body-fixed
axis
~
b
3
d
= R
d
e
3
and the current third body-fixed axis
~
b
3
=
Re
3
. Therefore, the magnitude of f is reduced when the
angle between those two axes becomes larger. These effects
are carefully analyzed in the stability proof for the complete
dynamics in [20].
In short, this control system is designed so that the
position tracking error converges to zero when there is no
attitude tracking error, and it is properly adjusted for non-
zero attitude tracking errors to achieve asymptotic stability
of the complete dynamics.
C. Exponential Asymptotic Stability
We first show exponential stability of the attitude dynamics
in the sublevel set L
2
= {R
d
, R ∈ SO(3) | Ψ(R, R
d
) <
2}, and based on this result, we show exponential stability
5422
of the complete dynamics in the smaller sublevel set L
1
=
{R
d
, R ∈ SO(3) | Ψ(R, R
d
) < 1}
Proposition 1: (Exponential Stability of Attitude Dynam-
ics) Consider the control moment M defined in (16) for any
positive constants k
R
, k
Ω
. Suppose that the initial condition
satisfies
Ψ(R(0), R
d
(0)) < 2, (17)
ke
Ω
(0)k
2
<
2
λ
max
(J)
k
R
(2 − Ψ(R(0), R
d
(0))), (18)
where λ
min
(J) denotes the minimum eigenvalue of the
inertia matrix J. Then, the zero equilibrium of the attitude
tracking error e
R
, e
Ω
is exponentially stable. Furthermore,
there exist constants α
2
, β
2
> 0 such that
Ψ(R(t), R
d
(t)) ≤ min
2, α
2
e
−β
2
t
. (19)
Proof: See [20].
In this proposition, (17), (18) represent a region of attrac-
tion for the attitude dynamics. This requires that the initial
attitude error should be less than 180
◦
. Therefore, the region
of attraction for the attitude almost covers SO(3), and the
region of attraction for the angular velocity can be increased
by choosing a larger controller gain k
R
in (18).
Since the direction of the total thrust is the third body-
fixed axis, the stability of the translational dynamics depends
on the attitude tracking error. More precisely, the position
tracking performance is affected by the difference between
~
b
3
= Re
3
and
~
b
3
d
= R
d
e
3
. In the proceeding stability
analysis, it turns out that for the stability of the complete
translational and rotational dynamics, the attitude error func-
tion Ψ should be less than 1, which states that the initial
attitude error should be less than 90
◦
. For the stability of
the complete system, we restrict the initial attitude error to
obtain the following proposition.
Proposition 2: (Exponential Stability of the Complete Dy-
namics) Consider the control force f and moment M defined
in (15), (16). Suppose that the initial condition satisfies
Ψ(R(0), R
d
(0)) ≤ ψ
1
< 1 (20)
for a fixed constant ψ
1
. Define W
1
, W
12
, W
2
∈ R
2×2
to be
W
1
=
c
1
k
x
m
−
c
1
k
v
2m
(1 + α)
−
c
1
k
v
2m
(1 + α) k
v
(1 − α) − c
1
, (21)
W
12
=
k
x
e
v
max
+
c
1
m
B 0
B 0
, (22)
W
2
=
"
c
2
k
R
λ
max
(J)
−
c
2
k
Ω
2λ
min
(J)
−
c
2
k
Ω
2λ
min
(J)
k
Ω
− c
2
#
, (23)
where α =
p
ψ
1
(2 − ψ
1
), e
v
max
= max{ke
v
(0)k,
B
k
v
(1−α)
},
c
1
, c
2
∈ R. For any positive constants k
x
, k
v
, we choose
positive constants c
1
, c
2
, k
R
, k
Ω
such that
c
1
< min
k
v
(1 − α),
4mk
x
k
v
(1 − α)
k
2
v
(1 + α)
2
+ 4mk
x
,
p
k
x
m
,
(24)
c
2
< min
k
Ω
,
4k
Ω
k
R
λ
min
(J)
2
k
2
Ω
λ
max
(J) + 4k
R
λ
min
(J)
2
,
p
k
R
λ
min
(J),
r
2
2 − ψ
1
k
R
λ
max
(J)
,
(25)
λ
min
(W
2
) >
4kW
12
k
2
λ
min
(W
1
)
. (26)
Then, the zero equilibrium of the tracking errors of the com-
plete system is exponentially stable. The region of attraction
is characterized by (20) and
ke
Ω
(0)k
2
<
2
λ
max
(J)
k
R
(1 − Ψ(R(0), R
d
(0))). (27)
Proof: See [20].
D. Almost Global Exponential Attractiveness
Proposition 2 requires that the initial attitude error is
less than 90
◦
to achieve exponential stability of the com-
plete dynamics. Suppose that this is not satisfied, i.e. 1 ≤
Ψ(R(0), R
d
(0)) < 2. From Proposition 1, we are guaranteed
that the attitude error function Ψ exponentially decreases, and
therefore, it enters the region of attraction of Proposition
2 in a finite time. Therefore, by combining the results of
Proposition 1 and 2, we can show almost global exponential
attractiveness.
Definition 1: (Exponential Attractiveness [21]) An equi-
librium point z = 0 of a dynamic systems is exponentially
attractive if, for some δ > 0, there exists a constant α(δ) > 0
and β > 0 such that kz(0)k < δ ⇒ kz(t)k ≤ α(δ)e
−βt
for
any t > 0.
This should be distinguished from the stronger notion of
exponential stability, in which the constant α(δ) in the above
bound is replaced by α(δ) kz(0)k.
Proposition 3: (Almost Global Exponential Attractiveness
of the Complete Dynamics) Consider a control system de-
signed according to Proposition 2. Suppose that the initial
condition satisfies
1 ≤ Ψ(R(0), R
d
(0)) < 2, (28)
ke
Ω
(0)k
2
<
2
λ
max
(J)
k
R
(2 − Ψ(R(0), R
d
(0))). (29)
Then, the zero equilibrium of the tracking errors of the
complete dynamics is exponentially attractive.
Proof: See [20].
Since the region of attraction given by (28) for the
attitude almost covers SO(3), this is referred to as almost
global exponential attractiveness in this paper. The region
of attraction for the angular velocity can be expanded by
choosing a large gain k
R
in (29).
E. Properties and Extensions
One of the unique properties of the presented controller
is that it is directly developed on SE(3) using rotation ma-
trices. Therefore, it avoids the complexities and singularities
associated with local coordinates of SO(3), such as Euler
angles. It also avoids the ambiguities that arise when using
quaternions to represent the attitude dynamics. As the three-
sphere S
3
double covers SO(3), any attitude feedback con-
troller designed in terms of quaternions could yield different
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control inputs depending on the choice of quaternion vectors.
The corresponding stability analysis would need to carefully
consider the fact that convergence to a single attitude implies
convergence to either of the two disconnected, antipodal
points on S
3
[22]. This requires a continuous selection of the
sign of quaternions or a discontinuous control system, which
are shown to be sensitive to small measurement noise [23].
Without these considerations, a quaternion-based controller
can exhibit an unwinding phenomenon, where the controller
unnecessarily rotates the attitude through large angles [15]. In
this paper, the use of rotation matrices in the controller design
and stability analysis completely eliminates these difficulties.
Another novelty of the presented controller is the choice
of the total thrust in (15). This is designed to follow position
tracking commands, but it is also carefully designed to
guarantee the overall stability of the complete dynamics by
feedback control of the direction of the third body-fixed axis.
This consideration is natural as each column of a rotation
matrix represents the direction of each body-fixed axis.
Therefore, another advantage of using rotation matrices is
that the controller has a well-defined physical interpretation.
In Propositions 1 and 3, exponential stability and exponen-
tial attractiveness are guaranteed for almost all initial attitude
errors, respectively. The attitude error function defined in (8)
has the following critical points: the identity matrix, and
rotation matrices that can be written as exp(πˆv) for any
v ∈ S
2
. These non-identity critical points of the attitude
error function lie outside of the region of attraction. As it is
a two-dimensional subspace of the three-dimensional SO(3),
we claim that the presented controller exhibits almost global
properties in SO(3). It is impossible to construct a smooth
controller on SO(3) that has global asymptotic stability. The
two-dimensional family of non-identity critical points can be
reduced to four points by modifying the error function to be
1
2
tr[G(I −R
T
d
R)] for a matrix G 6= I ∈ R
3×3
. The presented
controller can be modified accordingly.
IV. NUMERICAL EXAMPLE
The parameters of the quadrotor UAV are chosen accord-
ing to a quadrotor UAV developed in [2].
J = [0.0820, 0.0845, 0.1377] kgm
2
, m = 4.34 kg
d = 0.315 m, c
τ f
= 8.004 × 10
−4
m.
The controller parameters are chosen as follows:
k
x
= 16m, k
v
= 5.6m, k
R
= 8.81, k
Ω
= 2.54.
We consider the following two cases.
(I) This maneuver follows an elliptical helix while rotating
the heading direction at a fixed rate. Initial conditions
are chosen as
x(0) = [0, 0, 0], v(0) = [0, 0, 0],
R(0) = I, Ω(0) = [0, 0, 0].
The desired trajectory is as follows.
x
d
(t) = [0.4t, 0.4 sin πt, 0.6 cos πt],
~
b
1
d
(t) = [cos πt, sin πt, 0].
(II) This maneuver recovers from being initially upside
down. Initial conditions are chosen as
x(0) = [0, 0, 0], v(0) = [0, 0, 0],
R(0) =
1 0 0
0 −0.9995 −0.0314
0 0.0314 −0.9995
, Ω(0) = [0, 0, 0].
The desired trajectory is as follows.
x
d
(t) = [0, 0, 0],
~
b
1
d
(t) = [1, 0, 0].
Simulation results are presented in Figures 4 and 5. For
Case (I), the initial value of the attitude error function Ψ(0) is
less than 0.15. This satisfies the conditions for Proposition 2,
and exponential asymptotic stability is guaranteed. As shown
in Figure 4, the tracking errors exponentially converge to
zero. This example illustrates that the proposed controlled
quadrotor UAV can follow a complex trajectory that involve
large angle rotations and nontrivial translations accurately.
In Case (II), the initial attitude error is 178
◦
, which yields
the initial attitude error function Ψ(0) = 1.995 > 1. This
corresponds to Proposition 3, which implies almost global
exponential attractiveness. In Figure 5(b), the attitude error
function Ψ decreases, and it becomes less than 1 at t =
0.88 seconds. After that instant, the position tracking error
and the angular velocity error converge to zero as shown
in Figures 5(c) and 5(d). The region of attraction of the
proposed control system almost covers SO(3), so that the
corresponding controlled quadrotor UAV can recover from
being initially upside down.
V. CONCLUSION
We presented a global dynamic model for a quadrotor
UAV, and we developed a geometric tracking controller
directly on the special Euclidean group that is intrinsic and
coordinate-free, thereby avoiding the singularities of Euler
angles and the ambiguities of quaternions in representing
attitude. It exhibits exponential stability when the initial
attitude error is less than 90
◦
, and it yields almost global
exponentially attractiveness when the initial attitude error is
less than 180
◦
. These are illustrated by numerical examples.
This controller can be extended as follows. In this paper,
four input degrees of freedom are used to track a three-
dimensional position, and a one-dimensional heading direc-
tion. But, without changing the controller structure, they
can be used to follow arbitrary three-dimensional attitude
commands. The remaining one input degree of freedom can
be used to maintain the altitude as much as possible. By
constructing a hybrid controller based on these two tracking
modes, we can generate complicated acrobatic maneuvers of
a quadrotor UAV.
REFERENCES
[1] M. Valenti, B. Bethke, G. Fiore, and J. How, “Indoor multi-vehicle
flight testbed for fault detection, indoor multi-vehicle flight testbed for
fault detection, isolation, and recovery,” in Proceedings of the AIAA
Guidance, Navigation and Control Conference, 2006.
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