Actually CLoc performance is strongly affected by the number of neighbors and their ge-ometric configuration while processing and fusing all incoming information. On the other hand integrating fusion-oriented data from numerous neighbors generates high computa-tional complexity and requires significant overhead (and possibly, extra channel load) at the network level in comparison with more conventional CAM usage. Thus relevant op-erating trade-offs (e.g., in terms of required number of packets, CAM payload occupancy, refresh rates) must still be found for a better exploitation of cooperation potential, while complying with practical protocol constraints. Regarding the link selection itself, previ-ous works relying on the approximated Cram´er-Rao Lower Bound (CRLB) of cooperative position estimates as criterion (e.g., [90, 91] or more recently, a combination of this bound and a pre-validation step through innovation monitoring in the V2V context [92]). In this case, the selection is simply based on a comparison of the best positioning errors expected for given sub-sets of the available neighbors. The best sub-set leading to the presumed
3 2
Figure 3.4: Sets of selected cooperative neighbors (green) with respect to the ”ego” vehicle (red), following (a) non-Bayesian and (b) Bayesian CRLB criteria. In this example, the wrongly positioned vehicle 5 could trick the non-Bayesian selection scheme (and thus, be included in the selected fusion set), whereas the Bayesian version would account for its location uncertainty (and reject it as unreliable neighbor).
minimum error is selected. It is approximated in the sense that the theoretical bounds calculation, which would require the knowledge of all exact positions, admits erroneous positions as inputs (e.g., estimated or predicted), while considering that the latter would not fundamentally change the aspect of the relative VANET topology (i.e., in comparison with the true topology)8. However, they cannot properly account for mobile neighbors uncertainty, whereas more recent Bayesian formulations of such bounds [93], which can account for the prior uncertainty of all estimated positions. Figure 3.4 illustrates the dif-ference between them. And most of them have not yet been applied into the V2V context.
Besides, the simpler but complementary filter innovation monitoring approach in [92] is used to detect link-wise inconsistent measurements and thus, reject harmful ones.
We thus propose new link selection algorithms that aim at more efficient CLoc proce-dures under various GNSS conditions, by enabling lower footprint with respect to com-munication means and lower computational complexity. More specifically, we propose a couple of low complexity link selection criteria based on non-Bayesian and Bayesian versions of the CRLB characterizing cooperative location estimates given a subset of the available neighbors, in conjunction with a fast sub-optimal closest search instead of per-forming a computationally greedy exhaustive search (i.e., by restricting heuristically the CRLB-based comparison to a subset of the geographically nearest neighbors).
8This may be sufficient already in non-CLoc, when considering only a selection of V2I links and mea-surements with respect to known static anchors (i.e., RSUs).
3.5.1 Link Selection Criteria
Non-Bayesian Cram´er-Rao Lower Bound
The non-Bayesian CRLB or CRLB characterizes here the best achievable performance (in the minimum expected mean squared error (MSE) sense) for any non-biased (position) estimator (i.e., conditioned on a given set of reference neighbors). From the positioning point of view, this criterion reflects both the pair-wise radio link quality and the geometry of the reference vehicles relatively to the “ego” one or GDOP. The bound is determined by processing an inverse of the Fisher information matrix (FIM) [94, 95]. Consider at the
“ego” estimation instantk,xi,k, the position of the “ego” vehicleiand{xj,ki}j∈S→i,k, the positions of its selected reference vehicles, the FIM is defined as
Ji,k = X
j∈S→i,k
Esj→i,k
−∆xxi,ki,klogp(zRSSIj→i,k|xi,k,xj,ki) , (3.12)
where ∆xxf(x) denotes the Laplacian of f(x). Note that as its name suggests, the non-Bayesian CRLB treats bothxi,k and{xj,ki}j∈S→i,k as deterministic variables even though they are actually random (i.e., affected by estimation noise). Accordingly, the expectation in (3.12) is taken with respect to the measurement noise only (i.e., over the shadowing).
Under the assumption of centered Gaussian shadowing in (3.6), the expectation can be computed in closed-form solution [94]: where ˜σSh = σShlog 10/(10np). Nevertheless, neither the true position xi,k of the “ego”
vehicle nor that of its neighbors {xj,ki}j∈S→i,k are known, thus, the approximate FIM bJi,k can be computed with the predicted positions instead i.e., bxi,k|k−1, {bxj,ki}j∈S→i,k as
Thus, the bound on the location MSE can be expressed in terms of the FIM as follows:
MSE(xbi,k)≥tr
bJ−1i,k
. (3.15)
This expression shows the expected MSE conditioned on a particular subsetS→i,k ⊆ N→i,k of neighbors, as the cost function to be minimized by the link selection algorithm (i.e., with the subset as optimization variable).
Bayesian Cram´er-Rao Lower Bound
The Bayesian CRLB (BCRLB) considers the positions as realizations of random vari-ables [90, 95]. Therefore, besides the radio link quality and the geometry of the reference neighbors relatively to the “ego” vehicle, this criterion also captures the uncertainties of the
“ego” and neighbors’ estimated positions. Assume that at “ego” estimation time epoch k, xi,k ∼p(xi,k|zi,1:k−1), the position of the “ego” i and {xj,ki ∼p(xj,ki|zj,1:k)}j∈S→i,k, the positions of its selected reference vehicles, the Bayesian FIM (BFIM) is now expressed as [93]
i are the a priori FIMs of the positions of the “ego” i and its reference neighbors j ∈ S→i,k respectively, while JMj→i,k denotes the FIM obtained from the link measurement (j →i). In particular, the prior FIMs are defined as
JPi,k =Exi,k related to the measurements is now calculated as follows:
JMj→i,k =Esj→i,k,xi,k,xj,ki
Note that the expectation over the measurement noise is performed analytically in (3.19) still considering the Gaussian shadowing (in dB). Besides, as the expectation with respect toxi,k and xj,ki is tedious to derive analytically, we propose to use numerical integration instead, following a Monte Carlo approach. Accordingly, we draw P samples {x(p)i,k}Pp=1
and {x(p)j,k
Note that this Monte Carlo integration is still in compliance with the claimed low com-plexity link selection for two reasons: (i) we only calculate (3.20) for a smaller subset of potential neighbors, which will be presented in the next section and (ii) part of this calculation can be reused later on when updating the weights of the PF (e.g., particle-based V2V distance in the denominator). Finally, similarly to the non-Bayesian CRLB, the final bound on the MSE can be calculated by replacing the FIM bJi,k in (3.15) with the BFIM JBj→i,k
MSE(xi,k)≥trn
(bJBi,k)−1o
. (3.21)
The goal is again to identify the best subset S→i,k ⊆ N→i,k that minimizes the best conditional positioning MSE.
3.5.2 Link Selection Algorithm
Previously, we have derived the cost functions to be minimized (in the MSE sense) for the link selection problem. Particularly, considering the “ego” vehicleiat timekand given the set N→i,k of perceived neighboring vehicles, we are now interested in solutions to search for the minimum MSE conditioned on all possible subsets of lengthS ofN→i,k denoted by PS(N→i,k) to find S→i,k∗ yielding the best contribution to the CLoc problem resolution.
The optimal link selection would result from an exhaustive search, which is by far too complex in case of high V2V connectivity and thus, not really intended for implementation in a real system. This exhaustive search simply evaluates the cost functions for CRLB or BCRLB, for all the possible combinations listed by PS(N→i,k). For instance, choosing 4 links out of 10 leads to 210 combinations, what seems still reasonable but evaluating 4845 combinations in case of 20 neighbors appears much more challenging. Therefore, in order to reduce the computational burden, one straightforward approach is to develop a search algorithm that hopefully yields the same solution as that of the exhaustive approach (or at least an equivalent solution). A closer look at the (B)FIMs in both criteria (e.g., in (3.13)
Algorithm 2 Sup-optimal closest search of S most informative links among C most potential ones (iteration k, “ego” cari)
1: if |N→i,k|> C then
4: getCnearest neighbors fromN→i,k to buildC→i,k
5: else 12: determine the bound on the MSE
MMSE (bxi,k) [s] =
and (3.19)) reveals that its link-dependent sub-components are inversely proportional to the squared distances between the nodes. Intuitively, this means that performing CLoc with more distant neighbors leads to suffer from larger MSE or in other heuristic words, the optimal subset of neighbors is expected to be formed among the nearest ones (say, the 8–10 closest neighbors are expected sufficient on most common European highways having 3 lanes). Of course, this intuitive interpretation could be applied with other kinds of V2V metrics but it is all the more noticeable within CLoc based on RSSI measurements due to the considered log-normal path loss model. Thus, we search the best combination among a subset of the physically closest neighbors only, as shown on Algorithm 2 (lines 2–4).