12
RANGING
MEASUREMENTS
IN
THREESPACE
24
12
Ranging
Measurements
in
ThreeSpace
The
global
positioning
system
(GPS)
and
some
acoustic
instruments
provide
longbaseline
navigation
 wherein
a
number
of
very
long
range
measurements
can
be
used
to
triangulate.
We
call
the
item
that
we
want
to
track
the
Target
,
and
the
nodes
in
the
navigation
system
the
Satellites
.
The
locations
of
the
satellites
are
assumed
to
be
well
known,
and
what
we
measure
during
tracking
are
ranges
from
the
satellites
to
the
target.
In
a
plane,
you
can
appreciate
that
two
satellites
would
provide
two
range
measurements,
and
the
target
could
then
be
located
on
one
of
two
points
that
form
the
intersection
of
two
circles.
1.
For
a
planar
setting,
how
many
satellites
are
required
to
uniquely
locate
a
target
at
an
arbitrary
location,
and
how
these
should
be
laid
out?
Include
sketches
as
needed
to
explain
your
reasoning.
Three
satellites
give
three
ranges,
corresponding
with
three
circles.
The
intersection
of
circles
is
 in
the
best
case
 a
single
point,
the
unique
localization
result.
There
are
several
notable
conditions
where
you’ll
get
bad
results
with
three
satellites.
If
they
are
colinear,
you
get
no
information
about
location
in
the
direction
perpendicular
to
the
line.
When
noise
is
included
(as
below),
we
don’t
want
to
be
even
close
to
colinear:
we
want
low
aspectratio
(”not
too
long
and
thin”)
triangles.
More
fundamentally,
if
the
target
is
located
on
or
near
the
line
connecting
any
two
satellites,
then
it
is
as
if
we
have
lost
one
satellite
 the
second
range
measurement
does
us
very
little
good.
2.
Consider
a
problem
now
in
threespace.
Two
satellites
are
given,
with
locations
[X,Y,Z]
of
[500
,
500
,
1000]
m
and
[
−
500
,
500
,
1000]
m
.
The
target
ranges
are
measured
at
724
m
and
768
.
2
m
respectively,
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 Spring '05
 GeorgeKocur
 Global Positioning System, ThreeSpace, target lo cation

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