IEEE - Aerospace and Electronic Systems - October 2023 - 29

Nguyen and Kassas
ESTIMATION FRAMEWORK
Generally, one needs to estimate the state vector x, which
includes the aerial vehicle's position rr and velocity _rr as
well as relative clock error states fxclk;i
gM
r ; r_T
r ; DxT
vehicle-mounted receiver and each SOP, namely
x , rT
clk;1; ... ; DxT
clk;M
Dxclk;i , cðdtr dtsi Þ;cð_dtr _dtsi
hiT
T
Þ :
Formulating the transmitter selection problem with x 2
R6þ2 M results in a large-scale optimization problem. To scale
down the problem, two simplifications are made. First, it is
noted that terrestrial SOPs suffer from poor geometric diversity
in the vertical direction (particular as seen by high-altitude aircraft).
Therefore, relying exclusively on SOPs for 3D navigation
leads to a large vertical dilution of precision [42], [43].
Hence, it is assumed that the aerial vehicle is equipped with an
altimeter to determine its altitude. As such, in what follows, the
problem is formulated to only consider the 2D (planar) aerial
vehicle states. Second, only the position states of the aerial
vehicle will be considered, leading to the redefined state vector
x0 2 R2. It will be demonstrated in the " Effect of Timing on
the Optimal Transmitter Selection " section that this simplification,
which ignores the timing states, results in a negligible
increase in position uncertainty (on the order ofsubmeter).
A static, weighted nonlinear least-squares (WNLS)
estimator is employed on the redefined state vector x0.
The resulting Jacobian matrix Hr is given by
rT
Hrr
¼
2
6
6
6
4
rrT
.
s1
krrrs1 k2
..
rT
rrT
sM
krrrsMk2
3
7
7
7
5
TheWNLS estimation error covariancematrix is given by
Prr
, P1 þHT
0;rrr
hi1
rrR1Hrr
I1
s1 ; ... ; s2
sM.
(4)
where, a prior of x0 may be given, denoted by ^x0, with an
associated initial estimation error covariance ðP0;rrr
0;rrr Þ and R ¼ diag½s2
OPTIMAL TRANSMITTER SELECTION PROBLEM
The optimal transmitter selection problem can be cast as
the optimization problem
minimize
w
JðwÞ
subject to 1T
Mw ¼ K
wi 2f0; 1g;i ¼ 1; ...;M
where, JðwÞ denotes a desired cost function [e.g., A-, D-, and
E-optimality criterion or dilution of precision (DOP) [44],
[45]], wi is a binary decision variable which determines
whether to accept or reject the ith measurement, w ¼
½w1; ... ;wMT is a vector of the binary decision variables,
OCTOBER 2023
¼
:
(3)
i¼1 between the
1M 2 RM is a vector ofones, and K is the selection subset's
cardinality. This optimization problem is computationally
involved to solve in real time due to the integer constraints.
Instead ofsolving the abovementioned optimization problem,
two efficient transmitter selection strategies are proposed in
the next section.
TRANSMITTER SELECTION FRAMEWORK
The proposed transmitter selection framework selects the
most informative SOP subset to minimize the aerial vehicle's
position error uncertainty. According to the simplification discussed
in the " Estimation Framework " section, only the information
contribution from the ith SOP to the position states,
denoted Irr;i, is used to evaluate the cost function JðwÞ.
Ergo, the cost function is defined as the A-optimality criterion:
trace of the posterior position estimation error covariance
(equivalently, trace ofthe inverse ofFIM)
J wðÞ, tr I0;rrr þH0T
¼ tr I0;rrr
where, H0
rr , ðR1
þ
hi1
rr diagðwÞH0
XM
i¼1
rr
" #1
wiIrr;i
factorized measurement covariance (i.e., R ¼ RT
(5)
a ÞTHr, Ra is the upper triangular Cholesky
aRa), and
I0;rrr is the prior FIM corresponding to the receiver's position
states (see Figure 3).
Algorithm 1 summarizes each of the proposed transmitter
selection strategy's steps.
Algorithm 1. Transmitter Selection Strategies
Input: Prior FIM, FIM associated with each measurement,
map of all SOPs, and number ofSOPs to be selected
Output: SOP selection subset and FIM for the selected SOPs
1: Define an empty set for SOP selection
2: Perform an exhaustive search to select the two SOPs
with the largest information content
3: Update the prior FIM and SOP selection subset
One-Shot Selection (OSS)
4: Compute the posterior FIM for all SOPs, excluding
those already selected
5: Choose the K 2 SOPs which minimize the receiver's
average position error uncertainty
6: Compute the FIM for the selected SOPs (i.e., prior FIMplus
all selected SOP's FIM) and update the SOP selection subset
7: Return SOP selection subset and FIMfor the selected SOPs
Opportunistic Greedy Selection (OGS)
for K 2 iterations
8: Compute the posterior FIM for all SOPs, excluding
those already selected
9: Choose one SOP which minimizes the receiver's average
position error uncertainty
10: Redefine the prior FIM [i.e., (current) prior FIM plus
selected SOP's FIM] and update the SOP selection subset
end for
IEEE A&E SYSTEMS MAGAZINE
29
IEEE - Aerospace and Electronic Systems - October 2023

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