29
FLIGHT
CONTROL
OF
A
HOVERCRAFT
98
29
Flight
Control
of
a
Hovercraft
You
are
tasked
with
developing
simple
control
systems
for
two
types
of
hovercraft
moving
in
the
horizontal
plane.
As
you
know,
a
hovercraft
rests
on
a
cushion
of
air,
with
very
little
ground
resistance
to
motion
in
the
surge
(bodyreferenced
forward),
sway
(bodyreferenced
port),
and
yaw
(taken
positive
counterclockwise
viewed
from
above)
degrees
of
freedom.
The
simpliﬁed
dynamic
equations
of
motion
are:
u
˙
=
−
u
+
F
u
v
˙
=
F
v
r
˙
=
M,
where
the
surge
and
sway
velocities
are
u
and
v
and
the
control
forces
in
the
u
and
v

directions
are
F
u
and
F
v
,
respectively.
The
yaw
rate
is
r
and
the
control
moment
is
M
.
There
are
two
types
of
craft
we
will
consider:
1)
one
where
F
u
,
F
v
,and
M
are
all
commanded
independently,
e.g.,
using
independent
thrusters,
and
2)
one
where
F
v
=
p
v
δF
u
and
M
=
p
φ
δF
u
.
This
second
case
is
encountered
if
there
is
a
surge
thruster
(or
set),
whose
exhaust
pushes
on
a
rudder
with
(bodyreferenced)
angle
δ
.
To
create
a
lateral
force
or
moment,
you
have
to
have
some
F
u
and
some
rudder
action.
We
will
use
the
values
p
v
=0
.
2and
p
φ
=
−
0
.
2
in
this
problem.
Notice
negative
sign
in
p
φ
;
it
means
that
a
positive
rudder
action,
counterclockwise
if
viewed
from
above,
leads
to
a
negative
body
torque,
that
is,
clockwise.
This
is
typical
when
the
rudder
is
behind
the
thruster,
and
near
the
back
of
the
craft.
The
dynamic
equations
are
augmented
with
some
kinematic
relations
to
evolve
the
location
of
the
craft
in
a
ﬁxed
frame:
X
˙
=
u
cos
φ
−
v
sin
φ
Y
˙
=
u
sin
φ
+
v
cos
φ
φ
˙
=
r,
where
X
and
Y
denote
the
location
in
Cartesian
coordinates,
and
φ
is
the
yaw
angle.
See
the
ﬁgure
below.
Create
a
sixstate
model
for
each
vehicle
type
from
the
above
equations,
and
perform
the
following
tasks:
1.
By
putting
in
diﬀerent
settings
and
combinations
of
F
u
,
F
v
,
M
,
δ
,
convince
yourself
that
for
each
of
the
two
cases,
the
behavior
is
like
a
hovercraft.
In
the
ﬁrst
case,
turns
occur
completely
independently
of
translational
motion
in
the
X,
Y
plane,
whereas
for
the
second,
lateral
force
and
turning
torque
occur
together,
and
they
scale
with
both
δ
and
F
u
.
2.
For
Case
1:
From
a
fullzero
starting
condition
±s
=[
u,
v,
r,
X,
Y,
φ
]=
[0
,
0
,
0
,
0
,
0
,
0],
the
ob
jec
t
ivei
s
to
moveto
to
±s
desired
=[
u,
v,
r,
X,
Y,
φ
]
desired
=[0
,
0
,
0
,
1
m,