Integration of GPS with Other Sensors and Network Assistance - Understanding GPS: Principles and Applications - page 499

accelerometers will read the bias error level associated with their current operation.
The assumptions of initial level operation, while probably good for an aircraft on a
runway or in a hangar, are generally not good for an automobile. Even if the road
on which the car is parked is level, the road crown will induce a nonzero roll angle.
In general, both the car's pitch and roll angles will be nonzero at IMU turn on. From
(9.12), this implies that each level accelerometer will sense a component of gravity.
The sum of the sensed gravity component will be nulled by assumed roll and pitch
angles as part of the process that initializes the vehicle's attitude: this so determined
pitch and roll will not, of course, in general, match the actual pitch or roll of the
vehicle. These initial attitude errors, through the actions of the inertial system, will
induce a Schuler oscillation [4] in both attitude and position and velocity error in
the level axes. The Schuler oscillation period is 84 minutes. This error oscillation, if
not disturbed by other error-inducing effects (e.g., maneuvers), will persist until the
Kalman or integration filter has had time to estimate the sensor errors. Typical
Kalman filter designs will be addressed in Section 9.3.3.
Unlike the initialization of pitch and roll, however, because low-cost gyros have
bias errors that are very large relative to Earth rate, the heading of the vehicle must
be initialized by an auxiliary sensor (e.g., a magnetic compass), use of a GPS deter-
mined heading, or use of the vehicle heading as last computed by the navigation sys-
tem. In the case of a GPS heading, care should be taken that a minimum speed has
been attained and that at least four GPS satellites are tracked to ensure adequate
accuracy.
Returning to the two-accelerometer INS, use of a vertical accelerometer in an
INS brings a potential stability problem. As is well known [4], an INS vertical chan-
nel is inherently unstable due to the dependence of gravitational acceleration upon
altitude (in general, a gravity model is needed to remove gravitational acceleration
from the accelerometer outputs to enable sensing of inertial acceleration). The fact
that modeled gravitational acceleration may decrease with altitude increase leads to
an effective positive feedback loop in the error equations for the vertical channel [4],
which produces an exponential error growth. This error growth will produce more
than a doubling of altitude error roughly every 10 minutes if not corrected. Thus, an
independent source of altitude information is needed, which could be provided by
an additional sensor (e.g., a barometric altimeter) or an altitude constraint (e.g., the
assumption that the vehicle is at mean sea level or at the known altitude for a certain
road).
Because gyro design and development is generally more complex and less reli-
able than accelerometer design and development [23], it is attractive to consider an
accelerometer-only INS, which develops angular acceleration estimates by placing
dual accelerometers at known displacements (referred to as lever arms ) from the
vehicle's center of gravity. For example, as illustrated in Figure 9.24, the two accel-
erometers illustrated could be used to sense both linear and angular acceleration.
Before discussing a recent reference [24] where such a prototype system is con-
structed, some high-level comments are worth making. First, since we have replaced
the gyro, an angular rate sensor, with an angular acceleration sensor, accelerometer
errors will have a different effect on the INS position and velocity error. Any biases
in the accelerometers will produce a time-varying rate error in angular velocity: the
accelerometer biases add, while error effects due to sensing of gravitational acceler-