CHAPTER 9:  GPS AND KALMAN FILTERING

9.1  INTRODUCTION

A GPS receiver measures pseudoranges and pseudorange rates to the satellites.

Knowing the position of the satellites from the decoded navigation messages, the

user position and GPS system time can be calculated from four or more satellites.

A GPS receiver, however, can never measure exact range to each satellite.  The

measurement process is corrupted by noise which introduces errors into the

calculation. This noise includes errors in the ionospheric corrections and system

dynamics not considered during the measurement process (e.g., user clock drift).  A

Kalman filter characterizes the noise sources in order to minimize their effect on the

desired receiver outputs.

When the GPS receiver is aided or integrated with other navigation sensors (e.g.,

INS, clock, altimeter or AHRS), then the Kalman filter can be extended to include

the measurements added by these sensors.  In fact, a typical implementation for

integrated systems would be to have a central Kalman filter incorporating

measurements from all available sources.

9.2  KALMAN FILTER PRINCIPLE

The Kalman filter is a linear, recursive estimator that produces the minimum

variance estimate in a least squares sense under the assumption of white,

Gaussian noise processes. Because the filter is a linear estimator by definition, for

navigation systems it generally estimates errors in the total navigation state.  The

Kalman filter also produces a measure of the accuracy of its error state vector

estimate.  This level of accuracy is the matrix of second central moments of the

errors in the estimate and is defined as the covariance matrix.

There are two basic processes that are modeled by a Kalman filter.  The first process is a

model describing how the error state vector changes in time.  This model is the system

dynamics model. The second model defines the relationship between the error state vector

and any measurements processed by the filter and is the measurement model.

Intuitively, the Kalman filter sorts out information and weights the relative contributions

of the measurements and of the dynamic behavior of the state vector.  The

measurements and state vector are weighted by their respective covariance matrices.

If the measurements are inaccurate (large variances) when compared to the state

vector estimate, then the filter will deweight the measurements.  On the other hand, if

the measurements are very accurate (small  variances) when compared to the state

estimate, then the filter will tend to weight the  measurements heavily with the

consequence that its previously computed state estimate will  contribute little to the

latest state estimate.

9 1