Simple Distributed Control Strategy for Hover Control

FEASIBILITY OF A
DISTRIBUTED FLIGHT ARRAY
By: Raymond Oung, Alireza Ramezani, and Raffaello D’Andrea
Presented at the 48th IEEE Conference on Decision and Control, Shanghai 2009
The Distributed Flight Array (DFA) is an abstract testbed that features rich dynamics and challenging design
problems. It will be used as a research and pedalogical tool for distributed estimation and control.
The Array is Modeled as a single rigid body. Forces and torques are generated
locally by each module around the hovering equilbrium.
τ
τ
Start with a Linearized Dynamics Model
(except for yaw angle of an N-module array:
τ
4
&& = β
α +γ
α
∑
=
5
5
τ
9
4
&& = β
α +γ
α
∑
=
5
The normalized force to torque conversion constant, . Experiments have shown that
the torque can be accurately modeled as a linear function of thrust, see figures above.
o
5
9
4
&& = ∑
=
5
5
where
is the total mass
and
are the principal
mass moments of inertia of
the array.
−
A
C
D
C
+
9
4
B
E
C
Euler angles
acting along the z-, y-, x-axis in this order
describe the orientation of the DFA’s body coordinate frame
B with respect to the inertial coordinate frame.
F
*
+
:
γ&& = ∑
8
=
5
5
=ε
5
@
9
l



=
4
∑
β&& = −
7
5
=
5
l



5
=ε
9
?
α&& = ∑
5
=
5
q
p
r
g
p
+
-
Simulation Results prove feasibility of the DFA for
a small and large set of modules.
4-Module Configuration
Characteristic length of the module.
<




=
Q
The Normalized
and
Linearized Dynamics
Model about hover equilibrium:
Q
l
5
:
l



9
capture the mass
distribution of the array and
are expected to be close to
p
,
:
4
6




+
>
s
=ε
;
g




=
Q
Q
G
∑
&& =
l
H
=
L
H
where
G
=
Q
This term is comparable to the radius of the
array, assuming a circular array configuration.
∑
γ&& =
Q
K
l
H
H
=
P
L
H
=
ε l
P
Parameters used in the simulation:
V
W
X
Y
W
Z
Y
W
[
Y
W
\
]
^
V
_
`
a
Y
_
b
Y
_
b
Y
_
Y
l
and
]
V
c
X
Y
c
Z
Y
c
[
Y
c
\
]
d
e
f
g
f
g
f
g
f
.
h
G
=
Q
∑
β&& = −
Q
J
H
H
O
=
L
H
=
ε l
N
20-Module Configuration
G
Equations are Normalized in
The normalized thrust,
,
is the normalized thrust
where
required to establish equilibrium
about hover and is the normalized
control input.
.
order to gain some intuition on
how the size of the array affects
flight dynamics.
1
/
0
1
/
2
3
∑
α&& =
I
H
H
M
=
εl
M
=
L
H
/
/
3
/
Simple Distributed Control Strategy for Hover Control
based on physical parameters. This strategy is generalized and assumes full state feedback of the system.
Re-write the Normalized Dynamics
Model in Compact Form:
where
where
=
γ
=
β
&
$
W
X
Y
W
Z
Y
W
&
=
γ γ&
γ
β β&
β
β
%
β
=
&
γ
(
α α&
α
γ
=
α
'
W
\
]
^
V
i
`
j
k
Y
l
`
m
_
`
m
_
Y
i
`
n
l
]
and
V
c
X
Y
c
Z
Y
c
[
Y
c
\
]
^
V
_
Y
_
Y
_
Y
=
K
γ
=
K
=
α
)
Simulation
Experiment
Consider Each Degree of Freedom
Separately. For example, the following functions can
Now
be chosen:
!
"
β
=−
K
α
=
"
K
!
γ
& = − ω ζ & −ω
γ γ& = −
β
β β& = −
α
α α& = −
!
!
−
"
"
"
ωγ ζ γ γ& + ωγ γ
ω β ζ β β& + ω β β
ωα ζ α α& + ωα α
where each degree of freedom is a second-order system
with natural frequencies
and damping
ratios
as tuning parameters.
R
R
S
T
]
.
A plot of the DFA’s 3D position relative to its take-off
origin, obtained from experimental results.
_
#
α
γ
β
&
Y
α
For a large and equal number of CW and CCW modules,
one could use the following elements of to Decouple
the Degrees of Freedom:
The matrix P contains information
pertaining to the configuration of the
array,
=
γ
β
α
where
[
with simulation results.
& γ γ& β β& α α&
$
V
Experimental Results are shown to be consistent
Consider the following Control Strategy:
=
Parameters used in the simulation:
S
T
U
U
www.idsc.ethz.ch/Research_DAndrea/DFA
Simulated results for a 4-module configuration
without any feedback on altitude, where
) = (13,13) and (
) = (1,1).
(
The same controller was also used in an
experimental flight test; the measurements
obtained from this test are shown here.