some
noticeable
w
arp
in
the
agreemen
t
from
nonlinearities
that
this
t
w
as
unable
to
describ
e
Clearly
this
can
b
e
impro
v
ed
Nonlinear
MA
TLAB
Fit
In
order
to
impro
v
e
on
the
linear
t
a
more
complicated
nonlinear
t
is
tried
This
t
attempts
to
mo
del
some
of
the
kno
wn
sources
of
error
and
accoun
t
for
these
The
nonlinear
t
is
accomplished
b
y
the
follo
wing
equations
x
r
r
S
S
r
r
in
of
f
set
r
r
q
in
of
f
set
cos
S
x
in
of
f
set
of
f
set
y
r
r
S
S
r
r
in
of
f
set
r
r
q
in
of
f
set
sin
S
y
in
of
f
set
of
f
set
with
r
and
directly
pro
duced
b
y
the
range nder
and
all
other
v
ariables
adjusted
in
in
b
y
the
t
The
rst
thing
that
m
ust
b
e
accoun
ted
for
is
the
X
and
Y
displacemen
t
of
the
scanner
from
the
screen
This
is
easily
accoun
ted
for
b
y
solving
for
an
o set
x
y
in
eac
h
of
these
v
ariables
Once
the
scanner
is
translated
to
the
origin
of
f
set
of
f
set
w
e
can
examine
the
errors
in
r
and
Most
of
our
nonlinear
errors
will
o
ccur
in
the
r
v
alue
as
the
v
alues
are
directly
tak
en
from
an
accurate
bit
coun
ter
on
the
micro
con
troller
The
v
alue
m
ust
simply
b
e
mapp
ed
to
the
appropriate
scan
angle
e g
to
This
is
accomplished
b
y
adding
an
o set
and
scaling
the
sum
b
y
an
appropriate
n
um
b
er
suc
h
the
desired
range
of
angles
is
computed
A
similar
t
is
accomplished
with
r
except
for
the
addition
of
a
second order
term
The
hop
e
is
that
this
will
remo
v
e
some
of
the
w
arping
due
to
quadrature
errors
of
the
electronics
The
v
arious
co
e cien
ts
w
ere
solv
ed
for
using
the
MA
TLAB
f
mins
optimization
routine
and
the
results
are
sho
wn
in
Fig
The
is
somewhate
less
than
the
linear
t
and
can
still
b
e
impro
v
ed
up
on
Additional
p
erturbations
arising
from
amplitude
and
phase
errors
in
the
quadrature
demo
dulation
bias
errors
residual
crosstalk
and
unmo
delled
geometry
m
ust
b
e
tak
en
in
to
accoun
t