Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 368

r s ecef ∆ error in satellite position computation using navigation

message, in ECEF coordinates
EPH_ERD ( SV j , Site m , t k )

( SV , t ) =

j k

= satellite position error mapped into LOS to the

m th user location
CLOCK_ERD ( SV j , t k ) error in the navigation message representation of

each satellite's clock phase

=

Once ERDs are available for a given location, they can be used in a straightfor-
ward position error computation. First, compute the position solution geometry
matrix ( G ) and rotate it into local coordinates. Then, compute the inverse direction
cosine matrix ( K ) for each time t k . Several algorithms are available for computing the
inverse of a nonsquare matrix, such as the K -matrix for an overdetermined, AIV
solution. Once the K -matrix is availabl e, the SIS instantaneous positioning, naviga-
tion, and timing (PNT) error vector (

∆ x sis ) for each time t k can then be computed.

[] (

+

( ) ) ( )

allsys allsys

∆ x Site , t = G ∆ r Site , t = K ∆ r sis Site , t or

sis m k enu sis m k enu m k

( ) ( )

∆
∆
∆

e itet
n itet
u ite

,
,

ERD SV , Site , t

K ••

••
••

K

sis m k 11 1 n 1 m k

( )

•
•

K K

()

sis m k 21 2 n

= m

( )

,
,

t

K K

sis m k 31 3 n

( ) ( )

∆ t itet ERD SV , Sit et

m

,

K •• K

sis m k 41 4 n n k

where:

SIS position solution error vector in local coordinates (east,
north, up, and time) at the k th solution time for the m th site

∆ x sis ( Site , t ) =

m k

∆ ERD ( SV j , Site m , t k ) values from step 1, for all satellites used in

the k th position solution at the m th site

r sis ( Site , t ) =

m k

The resulting SIS PNT error vector can be injected with an estimate of a given
receiver's noise contribution to pseudorange measurement. This noise is mapped
through the position solution geometry and RSS into the individual PNT error com-
ponents to form an estimate of the receiver in question's total error vector. An exam-
ple result of this PNT error computation is presented in Figure 7.32 and contrasted
against the ORD-derived position errors first presented in Figure 7.29.
One of the advantages of the PNT error algorithm just discussed is that it can be
computed for any location at any desired time step. An example of initial conditions
for conducting such a core performance analysis on a global basis is provided next:

• Receiver characteristics: AIV, dual-frequency, keyed PPS, 5º mask angle,
80-cm RMS thermal noise in the dual-frequency pseudorange measurement;
• Noise environment characteristics: noise level below threshold for impacting
satellite signal acquisition or tracking and consistent with maintaining track-
ing performance within the receiver thermal noise assumption;