Global Positioning System Reference
In-Depth Information
ation from pitch or roll error is largely cancelled. The quality of the angular acceler-
ation sensing improves as the separation between the accelerometers increases. To
understand this, consider the treatment in (9.15), valid for two accelerometers
placed along the longitudinal axis of the vehicle:
(
)
m
a
=−
1
f
f
2
(9.15a)
2
(
)
ddt
ω
m
=+
1
f f
2
L
(9.15b)
2
(
)
f
=+
1
s
a
+
Ld
ω
dt
+ −
b
g
sin
ϕ
(9.15c)
1
1
1
(
)
f
=−
1
+
s
a
+
Ld
ω
dt
+
b
+
g
sin
ϕ
(9.15d)
2
2
2
[
]
(
)
(
)
δω
ddt
=
s sab b
−
+
+
2
L
(9.15e)
1
2
1
2
In (9.15), (9.15a) and (9.15b) represent the equations that would be used to
measure linear and angular acceleration, labeled
a
m
and
d
/
dt
m
, respectively; (9.15c)
and (9.15d) represent the error equations associated with the measured quantities in
(9.15a) and (9.15b). Thus,
a
represents the true acceleration of the vehicle along the
sensitive (lateral) axis,
b
1
and
b
2
are the accelerometer biases, all preferably repre-
sented in units of m/s
2
. As used previously,
ω
is the roll angle of the vehicle in radians,
and
g
represents gravitational acceleration in m/s
2
. The accelerometer scale factor
errors (unitless quantities) are denoted
s
1
and
s
2
, respectively. The lever arm is repre-
sented by the variable
L
, expressed in meters to maintain consistent units. Finally,
note that (9.15c) is an equation for the rate of change of the error in sensing angular
rate (i.e., yaw rate, which is roughly heading rate), which would typically be mod-
eled in a Kalman filter that attempted to reduce this error by processing GPS mea-
surement data.
Thus, the error contributors to angular acceleration—the individual accelerom-
eter bias and scale factor errors,
b
1
and
b
2
and
s
1
and
s
2
—are reduced by increasing
the lever arm,
L
, between each sensor and the center of gravity of the vehicle. In the
specific case illustrated in Figure 9.24, best performance would be achieved by plac-
ing one accelerometer near the front of the car and the second near the rear of the
car. Of course, the lever arm does not affect the quality of the determined linear
acceleration. Because the accelerometer bias contributes to an angular rate bias in
this formulation, it produces different position and velocity error behavior than its
gyro bias counterpart. As is well known [4], level axis gyro bias errors produce
biased velocity errors superimposed on a Schuler oscillation in the level axes. The
bias component of the velocity error can dominate the INS drift for periods that are
less than the Schuler period, leading to the familiar “nmi/h” rating often associated
with inertial systems [2]. A bias angular acceleration error can therefore be expected
to produce a ramping velocity error over a similar time period.
The concept of using accelerometers to sense angular acceleration is not new
[25]. Only fairly recently, however, has this concept received new attention, driven
largely by the presence of very-low-cost microelectromechanical sensors (MEMS)
accelerometer technology for cars and the ability to fabricate accelerometers for a
fraction of the cost of gyros [26, 27]. Recent work has also focused on the placement
of accelerometers within the vehicle for best performance [28]. The most recent
ϕ
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