Integration of GPS with Other Sensors and Network Assistance - Understanding GPS: Principles and Applications - page 517

dp dt δ = δ v (9.27)

3 3

dv dt δ =− βδ v + w (9.28)

3 3

In (9.27) and (9.28), units are consistent with those already referenced, with posi-
tion errors in meters, and velocity errors in meters per second.
In (9.28), the velocity error is modeled as a Markov process [56], which,
through appropriate choice of the variance associated with the white noise, w ,
reaches a steady-state error variance in the absence of updates. This error variance
represents the expected variation in the vertical velocity of the car. Altitude and ver-
tical velocity can be maintained through GPS measurement processing and can also
be augmented with barometric altimeter measurements. In this case, as discussed in
Section 9.3.2.5, a barometric altimeter bias state should be added to the state vec-
tor, resulting in a total of 22 states.
Implementing a Kalman filter with 21 or 22 states may pose some problems
from a computational burden standpoint, depending on the processing bandwidth
available to the filter. Some of the states can perhaps be removed. Leading candi-
dates for removal are the scale factor errors associated with the pitch and roll gyros,
since pitch and roll rates are not expected to be large for car maneuvers (except for
relatively high frequency effects, as could be induced by speed bumps, but which do
not integrate to significant attitude error). It may also be worthwhile to consider
removing the accelerometer bias states, since the initial determination of vehicle
pitch and roll will remove their effect. Their inclusion is therefore largely a function
of the bias instability and the expected pitch and roll agility of the vehicle.
Because of the potentially significant temperature sensitivities associated with
the gyro and accelerometer bias and scale factor errors, it is highly desirable that
temperature information be supplied with their high-rate outputs (i.e., the gyro
measured delta-angles and the accelerometer measured delta-velocities). The tem-
perature sensitivities can be measured in a laboratory environment (as previously
discussed, this must be done for each gyro and accelerometer), and the resulting bias
and scale factor error estimates will be comprised of a precomputed tempera-
ture-dependent component, preferably represented as a curve fit, and a correction
to that generated by the Kalman filter from processing GPS. A consistent and statis-
tically significant trend in the correction component away from the sensitivity curve
may result in a modification of the temperature sensitivity curve, as could be
determined using the following statistic for the gyro bias:

( )

∑

S = δσ

θ

b n (9.29)

t bg

b θ represents the corrections to a component of the gyro bias vector
b , preferably represented in radians per second, over the most recent set of n
Kalman filter updates. The value in the denominator of the summation,

In (9.29), δ

σ bg in (9.29)
represents the a priori uncertainty associated with each gyro bias component cor-
rection, representing the designer's best knowledge about its temporal stability. If
the process noise associated with the gyro bias state considered in (9.29) assumes
that the factory-generated temperature compensation curve is effective in removing
the gyro bias sensitivity, then the value of the normalized statistic in (9.29) can be
used to detect a departure from those conditions. Such a detection must be gated by