IEEE - Aerospace and Electronic Systems - July 2022 - 8

I Am Not Afraid of the GPS Jammer: Resilient Navigation Via Signals of Opportunity in GPS-Denied Environments
Table 1.
Figure 2.
Clock error states dynamics model.
where xpv , rT
w~x; ~wy; ~wz
T
r ; _rT
r
of the navigator-mounted receiver, and
T
, rr , xr;yr;zr½T is the 3-D position
w~pv ¼
is the process noise vector, whose elements of
are modeled as zero-mean, mutually independent white
noise processes with power spectral densities ~qx ~qy,and ~qz,
respectively. Note that here, the superscript g is dropped and
the subscript r is used to denote the navigator-mounted
receiver's position instead of b, since the navigator in this
case is not relying on an INS and the orientation of its body
is not estimated.
Discretizing the navigator's dynamics (5) at a constant
sampling period T yields the discrete-time model
xpv k þ 1ðÞ¼ Fpv xrðkÞþ wpvðkÞ;k ¼ 0; 1; 2; ...
Fpv ¼
I33 TI33
033
I33
where wpv
Qpv¼
2
6
6
6
6
6
6
6
6
6
is a discrete-time zero-mean white noise
sequence with covariance Qpv given by
x
T3
3
q~x
T2
2
4 0
00 q~x
q~y
q~y
T3
3
00 q~z
T2
2
00 q~z
T3
3
T2
2
00 q~y
00
T2
2
00 q~z
T2
2
T2
2
3
7
7
7
7
7
7
7
7
7
xclk ¼
dt
d_t
;
w~ clk ¼
w~dt
w~ _dt
; Aclk ¼
01
00
(6)
:
00 q~xT 00
00 q~yT 0 5
00 q~zT
where the elements of ~wclk are modeled as zero-mean,
mutually independent white noise processes, and the
power spectral density of
diag S~wdt
be related to the power-law coefficients hafg2
hi
;S~w _dt .The power spectra S~wdt
w~clk is given by Q~ clk ¼
and S~w _dt
can
a¼2,
CLOCK ERROR DYNAMICS MODEL
GNSS satellites are equipped with atomic clocks, are
synchronized, and their clock errors are transmitted in
the navigation message along with the satellites' positions.
Therefore, in GNSS-based navigation, only the
receiver's clock error is estimated. In contrast, SOPs
are equipped with less stable oscillators than GNSS satellites,
are typically roughly synchronized to GNSS,
and their clock error states (bias and drift) are mostly
unknown. As such, the SOP clock errors must be simultaneously
estimated with the receiver's clock error. To
facilitate this estimation in the radio SLAM framework,
the clock error state dynamics must be specified. To
this end, a typical model for the dynamics of the clock
errorstatesisthe so-called two-state model, composed

IEEE - Aerospace and Electronic Systems - July 2022

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