Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 349

where |[ P 13 P 23 ] T | = PP

13

2

+

2

.

23

In general,

(range bias b on i th satellite) = (norm of parity bias vector)/(norm of i th column of P )

= $ The position error vector e is defined as: e xx

LS

−

( )

− 1

T T

e HHHyx = −

( )

− 1

( )

= HH H Hx+ x

T T

−

( )

− 1

T T

= HH H (Vector position error)

which, for a bias b in the i th satellite, can be written as

0

⋅

b

( )

−
HH H

1

T T

(position error vector) = ⋅

⋅
⋅

0

These equations provide a means of getting back and forth from a bias in parity
space to the corresponding bias in range space, and finally to the corresponding
position error. The norm of the first two components of the position error vector
provides the horizontal radial position error.
The objective is to protect against excessive horizontal position error. The
RAIM algorithm must detect whether the horizontal error goes beyond a certain
threshold within a specified level of confidence. Since the position error cannot be
observed directly, something must be inferred from the quantity that can be
observed, which in this case is the parity vector.
The magnitude of the parity vector is used as the test statistic (mathematical
indicator) for detection of a satellite failure. The inputs to the parity space algorithm
are the standard deviation of the measurement noise, the measurement geometry,
and the maximum allowable probabilities for a false alert and a missed detection.
The output of the algorithm is the HPL, which defines the smallest horizontal radial
position error that can be detected for the specified false alert and missed detection
probabilities.
A false alert is an indication of a positioning failure to the pilot when a position-
ing failure has not occurred, as the result of a false detection. The detection thresh-
old for the RAIM and FDE algorithms is determined by integrating the pdf from the
detection threshold to infinity so that the area under the curve is equal to the proba-
bility of a false alert, P FA .
The parity space method is based on modeling the test statistic using a chi distri-
bution with n 4 degrees of freedom for six or more visible satellites. The sum of

the squared measurement residuals has a chi-square distribution. A Gaussian distri-

−