Integration of GPS with Other Sensors and Network Assistance - Understanding GPS: Principles and Applications - page 498

While this is true to excellent approximation even for low-cost gyros, the gyro, in
theory, senses inertial angular velocity along its sensitive axis, which will include a
component of the Earth's rotation rate. It is, in fact, this property that has been
exploited in initializing the heading of inertial systems, using a process generally
referred to as gyrocompassing [4]. Because the sources of error associated with
low-cost gyros are orders of magnitude greater than Earth rate (e.g., drift rates
approaching 1º/second, as contrasted with 15º/hour), an alternate means of
initializing heading is necessary until low-cost gyro technology dramatically
improves.
Let's return now to the issue of gyro and accelerometer initial alignment. Any
misalignment of either sensor, due either to imperfect mounting of the sensitive ele-
ment(s) within the sensor's housing or imperfect alignment of the sensor housing
within the vehicle upon installation, will lead to a cross-axis sensitivity, which can
be significant. From (9.12), a misalignment about the vertical axis of the host will
cause the accelerometer to sense a component of longitudinal acceleration, and a
misalignment about the roll axis will cause the accelerometer to sense gravitational
acceleration, even when the vehicle is level (i.e., at zero roll angle). In each case, the
magnitude of the error, for small misalignment angles, is the product of the angle (in
radians) and the off-axis acceleration. For example, a 5º misalignment about the
vehicle's roll axis will produce an error in the lateral accelerometer of roughly 0.1 g ,
or about 1 m/s 2 . A gyro mounted with its sensitive axis in the vertical direction,
intended to sense the turns of the vehicle, will produce an output that may be
modeled (in units of radians/second) as:

( )

m

ω =+ 1 s ω + + b m ω + m ω (9.13)

H HH H ϕ θ θ ϕ

where s H is the gyro's scale factor error, b H is the gyro bias, m ϕ and m θ are the small
angle misalignments (in radians) of the gyro sensitive axis about the roll and pitch
axes, respectively, and ω ϕ are pitch and roll rate (in radians/second), respec-

tively. In addition, any misalignment of the gyro with respect to the local vertical
will appear as a component of gyro scale factor error, since it will contribute an
error that is proportional to the angular rate about its sensitive axis. The scale factor
error term is expressed in (9.14), where

ω θ and

α (in radians), is the misalignment value:

δ s H = cos α − 1 = − α

2

2 (9.14)

So, for a gyro that is misaligned by 5º relative to the vertical axis of the car, the effec-
tive scale factor error is changed by 0.5%, which is generally not significant for
low-cost gyros (the nominal scale factor error can be 10 times this level).
Now, given this very basic review of inertial sensing technology, we can return
to the issues associated with the options for inertial sensor augmentation of GPS in
automotive vehicles. The first two options differ only in that the second abandons
the vertical accelerometer, as the vertical motion of an automobile is not expected to
be significant, and GPS aided by an altitude constraint may suffice. Referring to
(9.12) and Figure 9.23, initialization of the pitch and roll angles for both systems
begins (upon turn-on of the system) by assuming that the car is stationary and level,
which implies that the accelerometers (after gravity compensation for the vertical
axis for the first option) should read zero. Of course, under zero acceleration, the