Evolution of Integrity Concept – From Galileo to Multisystem - Global Navigation Satellite Systems: Signal,Theory and Applications - page 94

Applying a linearization to the (38), we get the expression of the pseudo range model:

Δ ρ Δ ε = G X + (39)

ρ

The matrix G is named Design Matrix, and it consists of the linear coefficients obtained by
the partial derivatives of the observation's equations with respect to the estimated
coordinates. This matrix characterizes the user-satellite geometry. The number of the
columns of G agree with the number of unknowns to be determined (X,Y,Z and b), while the
rows equal the number of the available observations (number of satellites in view for both
navigation systems). The union of the Galileo and the GPS constellations causes a change in
the G matrix. The number of unknowns in fact become five, in order to compute the clock's
offset for both systems. In order to estimate the user position's ( Δ ) we have to apply the

weighted least mean square method to the pseudo range model, organizing the weight
matrix (W) with the information contained in the navigation message sent by EGNOS or by
the Galileo satellites (considering only SWs in view, or those with an elevation angle greater
than 10°):

( ~ T T

−

Δ XGWG GW = ⋅ ⋅ ⋅ ⋅ ⋅ Δρ (40)

where G and W are two matrices of dimension N×5 and N×N respectively, with N
representing the number of the satellites used in the positioning algorithm.

5.3.3 Outputs of the implemented algorithm

In this Section we will describe the characteristics of the implemented multisystem integrity
algorithm. We will discuss the results of a few simulation tests organized by different
typologies (with or without failure) and different durations, in order to test the validity of
the proposed algorithm and confirm the expected results.

A peculiarity of this algorithm is the allocation of the Integrity Risk, valid for the
computation of the HM P , and the F P (False Alarm Probability), required to estimate the
RAIM statistics. The false alarm probability of RAIM and the Integrity Risk of Galileo are
related to the time required for a specific flight operation. For example, in the case of safety
of life applications, this time is equal to 150 seconds. Our study refers to these applications.

The proposed algorithm elaborates the position computation, the RAIM statistics and the IR
equation in every second. It is therefore useful to refer to the probability mentioned above as
to one second. In order to perform this conversion, we use the binomial distribution,
obtaining the value of

10 −

7

P and P , both initially set 1 at 0.5 × , referred to as one

HMI FA

epoch (second).

The failures have been reproduced in two different ways:

• Introduction of a step function, at a given test epoch, on the pseudo range of a satellite
in view.
1 Equally split between the two integrity requirements from the initial value of

7
110 /150 s

−

as defined ×

by the ICAO for the avionic integrity requirements.