Global Positioning System Reference
In-Depth Information
tude errors). The fundamental (
F
) matrix associated with the INS error dynamics
has two distinct frequencies of oscillation when the INS is at rest: the Schuler fre-
quency, with an 84-minute period, and Earth rate, with a 24-hour period. Because
the longest GPS outages in the automotive environment are expected to be no more
than several minutes long, the Earth rate dynamics can be ignored, and the Schuler
dynamics well approximated by much simpler equations. Now returning to the state
vector selection, the basic nine error states (i.e., three position errors, three velocity
errors, and three attitude errors) will be augmented by three gyro bias states, two
accelerometer biases, three gyro scale factor errors, and two accelerometer scale fac-
tors, resulting in a total of 19 states. The resulting state vector is summarized here:
[
]
T
T T T TT TT
=
x
p
v
b s b
aa
(9.21)
Of course, the 19 states must be augmented by GPS clock phase and frequency
errors if a measurement domain integration approach is chosen, resulting in a total
of 21 states. Preferably, the modeled position errors (
p
) in (9.21) are represented in
meters, velocity errors (
) in radians, gyro
biases (
b
) in radians per second, and accelerometer biases (
b
a
) in m/s
2
. Note that
scale factor errors for both the gyro (
s
) and accelerometer (
s
a
) are unitless.
Given that most GPS outages due to signal blockage are less than a few minutes
in duration, the sine or cosine of the Schuler angle, which appear in various terms in
the INS error dynamics equations, can be well approximated by (9.22) and (9.23):
v
) in meters per second, attitude errors (
()
sin
ωω
s
t
=
t
(9.22)
s
()
cos
wt
=−
1
ω
22
t
2
(9.23)
s
s
Given these substitutions, the INS error dynamics simplify significantly and become
more intuitive:
d t
p
=
v
(9.24)
d t
v
=+ +
b
g
S a
(9.25)
a
a
d t
=+
bS
(9.26)
where
S
a
and
S
are matrices with the scale factor elements on the diagonal, and
instrument input axis misalignments as off-diagonal terms, with
g
representing grav-
itational acceleration in m/s
2
. Our Kalman filter state vector per (9.21) only esti-
mates the accelerometer and gyro scale factor errors (i.e., the misalignments are set
to zero in these equations). A real-time Kalman filter would generally have a very
difficult time observing these misalignments, as controlled maneuvers are generally
required for observability, so they are generally assumed to be calibrated to negligi-
ble levels prior to the filter's operation. The altitude and vertical velocity error
behavior is noninertial, yet must be modeled by the filter, since errors in these states
drive the inertial errors. A simplified model providing acceptable performance for
many applications is
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