3.2 Hybrid Network-wide Localization Bound
3.2.3 A Note on Hybrid Capability
As mentioned earlier, the majority of the proposed FIM formulations assume that wireless localization is performed under a single type of signaling or measurement techniques [58–61] .
In today’s context, however, due to the importance of localization accuracy, wireless localization systems increasingly tend to be heterogeneous and the broad take-home message learned from literature with regards to this is that a mechanism to improve accuracy is to rely on multiple types of information, such as distances and angle, which leads to hybrid or heterogeneous algorithms [40, 52, 53] that have been shown to have advantages over algorithms admitting only a given type of input.
Our “information-centric” construction is, to the best of our knowledge, the only generic FIM formulation that is capable to address such trending hybrid localization approaches in an easy manner.
In order to illustrate the power of our formulation and clearly highlight its distinction to those presented in [58, 59, 61], consider the hybrid localization scenario illustrated in Figure 3.2, where three location-aware references (which can be taken to be physical
“anchors” or cooperating targets whose locations have been estimated, possibly with uncertainty), namely θ1, θ2 and θ3 cooperate to localize one target (θt) in the two-dimensional space.
Since the references may not only be fully-equipped “anchors” but also devices with hardware limitations, we consider for the purpose of illustration that the three refer-ences in this scenario have the following distinct features.
Let us assume then that the first reference θ1 is a low-cost device with knowledge of location (e.g. a cooperative target) with a single antenna and therefore only capable to
measure its distance to other devices. In this case, the dissimilarity function associated with the input contributed by this device is a simple ranging measurement and therefore given by (see Table 3.1)
g(θt|θ1) =kθt− θ1k = dt1, (3.16) such that the corresponding information gradient is given by
∂g(θt|θ1)
∂θt = 1
kdt1k[(xt− x1), (yt− y1)]T = dt1
dt1 , vt1. (3.17) We may model the imperfect measurements of dt1by the variables rt1with a Nakagami probability density function, i.e.
p(rt1; g(θt|θ1)) = 2mm
Γ(m)Υmrt12m−1exp
−m Υrt12
, (3.18)
such that the corresponding information intensity is given simply by
λt1= s
m(4Υm2− 4Υm + 2m − 1)
Υ(m− 1) , (3.19)
where m and Υ parameters related to the shape and spread of the ranging distribution.
With possession of the information gradient and intensity, the total contribution of the input provided by the first reference is, in our formulation, easily, generally and flexibly modeled by the term λt1vt1vt1T.
Now consider that the second reference θ2 is a fixed base-station equipped with a narrow-band linear antenna array, thus capable of estimating the AoA of the target’s signal with respect to a global angular reference, which we shall denote φt2 with dissimilarity function (see Table 3.1)
g(θt|θ2;{a, b}) = acos
hdt2, bi kdt2− hdt2, aiak
, (3.20)
where a is vector normal to the plane on which AoA is measured, such that in this example a = [0, 0, 1], while b is the reference vector for AoA measurement (see Figure 3.1 and 3.2).
The second anchor contributes with an AoA input with corresponding information
gradient given by
∂g(θt|θ2)
∂θt = 1
kdt2k2[(yt− y2),−(xt− x2)]T =h
0 1
−1 0
idt2
d2t2 , vt2. (3.21) Now, if amongst other possibilities a Von-Mises angular error model is adopted, the AoA measurements follow the distribution
p(rt2; φt2) = exp (ωt2cos(rt2− φt2)
2πI0(ωt2) , (3.22)
where −π ≤ rt12 ≤ π, ωt2 is the shape parameter of the distribution, φt2 is its centrality parameter, and I0(·) denotes the modified Bessel function of the first kind and 0-th order.
In this case, the information intensity related to the input contributed by the second anchor is given by
λt2= ωt2
√2, (3.23)
such that the contribution of this anchor to the FIM is fully, generally and flexibly captured by the term λt2vt2vTt2.
Before we move to the next anchor, let us emphasize that although φt2 is the centrality parameter of the AoA input, as can be readily seen from equation (3.26), the information direction associated with an AoA input is actually orthogonal to the direction indicated φt2. Albeit counter-intuitive at first glance, this observation quickly becomes obvious, since the AoA is a quantity whose variation (at a small scale) occurs on directions orthogonal to the line between the anchor and the target (see Figure 3.3).
Note that intuition such as this cannot be discussed nor captured by many of the state of the art formulations such as [58, 59, 61].
Finally, consider that the third reference is a mobile station equipped with a passive antenna array, and therefore incapable of performing distance estimation to target.
Assume the unit is furthermore equipped with a GPS, but no gyroscope or similar equipment, such that it is unaware of its orientation and thus incapable of estimating an absolute AoA due to the lack of an angular reference. However, the device is still capable of measuring the ADoA between the signals of the target and the first reference.
Succinctly, the dissimilarity function of the ADoA input contributed by this reference
can then be expressed as follows
g(θt|θ3, θ1) = ∆φt3= acos d2t3+ d213− d2t1
2dt3d13
= acos
h(θt|θ1) + 1/4− h2(θt|θ3) h(θt|θ1)
, (3.24) where we have implicitly defined the auxiliary function
h(θt|θi) , kθt− θik 2d13
. (3.25)
Given the above model and referring to Table 3.1, the corresponding information gradient becomes
∂g(θt|θ1, θ3))
∂θt = dt1
kdt3× dt1k(ρ1dt3+ ρ2dt1) , vt3, (3.26) where
ρ1 , d213− (d2t3+ d2t1) 2d13dt3
and ρ2 , 1 d13
. (3.27)
Notice that from a sheer inspection of the latter equation, one can immediately observe that the information direction associated with an ADoA input is a vector within the convex cone defined by the vectors dt1 and dt3.
For conciseness, we assume that ADoAs are measured also under the influence of Von-Mises errors (again, with the remark that any other suitable model could be adopted as easily), such that the ADoA measurements follows the distribution
p(rt3; ∆φt3) = exp (ωt3cos(rt3− ∆φt3)
2πI0(ωt3) , (3.28)
which in turn implies that the associated information intensity is given by λt3= ωt3
√2, (3.29)
such that the information obtained from the third anchor contributes to the FIM with the term λt3vt3vTt3.
t3 t2
d13
dt3
✓3
= [x2, y2]
✓t
✓1
✓2
= [x1, y1]
= [x3, y3] dt1
b
Figure 3.2: A planar hybrid localization scenario in which a combination of distance, AoA and ADoA measurements are used for localization of a single target.
✓t
t2
✓
2= [x
2, y
2] b
Figure 3.3: Highlight of the information direction associated with AoA measurements.
In light of all the above, we are ready to construct a concise, elegant, and highly tractable FIM based on our contributed formulation for the hybrid example used in this illustration, which simply yields
Fθt= The above example is a rather simple scenario for the readers to be able to better un-derstand the hybrid capabilities and informative structure of the new FIM formulation introduced in this work.
However, our generic hybrid bound can be applied to other scenarios and algorithms of interest. In order to further validate this, we compare the bound and performance of one such hybrid localization algorithm, namely super MDS (SMDS) which was proposed in [29, 30] and it generalizes the isometric embedding technique known as MDS [54, 55]
so as to enable the joint processing of distance and angle measurements to deliver localization accuracies far superior to its classic predecessor. Due to its relevance, the detailed description of this algorithm is given in subsection 4.2.1.
The simulation setup consists of a 10m-by-10m room equipped with 4 anchor nodes, one at each corner, and populated with 8 target nodes located randomly in its interior with x and y coordinates following a uniform distribution.
It is assumed that all nodes can cooperate with each other in terms of collecting their pairwise distances however AoA measurement is only possible between a target node and an anchor. The reason for such assumption is that obtaining AoA information over the air in an actual system requires at least one of the two involved nodes to have an absolute reference of orientation (which could be provided by a magnetic compass).
Therefore, given the high level of sophistication in cooperation and hardware capability, we assume AoA is not available between target nodes.
On the other hand, in order to take full advantage of the SMDS framework, angle and
distance information are both required. However, in case no or partial angle information is available, rough estimates of the missing angles can be constructed using a “quick-and-dirty” prior range-based solution of the positioning problem, using for example the centroid method proposed in [73].
In this comparison, we consider two cases, namely range-based SMDS in which ranging is the only available type of information, and hybrid SMDS in which both distance and angle information is available and therefore used for localization. In the range-based, we use centroid method to obtain the missing AoA between two target nodes.
As for the measurement statistics, distance measurements are modeled as gamma-distributed random variables [94] with the mean given by the true distance and a standard deviation σd. In other words, the probability density function (PDF) of measured distances ˜d associated with a true distance d is
p( ˜d; d) = 1
υκΓ(κ) · d(κ−1)· eυd˜, (3.31) with shape and scale parameters
κ = d2/σd2 and υ = σd2/d. (3.32) In turn, angle measurement errors are assumed to be Tikhonov-distributed [95–97], with concentration parameter ω ≥ 0 inversely proportional to the angular error variance.
The PDF of measured AoA φta between a generic target t and an anchor a is given by
p( ˜φta; φta) = exp
ω cos( ˜φta− φta
2πI0(ω) , (3.33)
Due to the non-linear relationship between ω and angular error variances, we capture the influence of angular errors by the quantity εφ, defined as the bounding angle of the 90th centered percentile, i.e.
εφ= φB
Z φB
−φB
pΘ(φ; 0, ω) dφ = 0.9, (3.34) where pΘ(φ; µ, ω) denotes the Tikhonov distribution with centered at µ and with shape parameter ω [97].
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
Super MDS localization algorithm
(CRLB vs performance comparison for angular error± 5◦)
AverageCRLB/MSE(m)
Ranging Error σd (m) Hybrid SMDS
Distance-based SMDS Hybrid CRLB
Distance-based CRLB
Figure 3.4: Performance comparison of range-based and hybrid SMDS localization algorithm against corresponding CRLB. As for the hybrid case, it is assumed that full set of ranging information subject to Gamma noise with variance ranging from 0 to 1 as well as partial (target-to-anchor) AoA information subject to Tikhonov noise with angular error ± 5◦ is available. The missing target-to-target AoAs are estimated via centroid algorithm [73].
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
Super MDS localization algorithm
(CRLB vs performance comparison for angular error± 15◦)
AverageCRLB/MSE(m)
Ranging Error σd (m) Hybrid SMDS
Distance-based SMDS Hybrid CRLB
Distance-based CRLB
Figure 3.5: Performance comparison of range-based and hybrid SMDS localization algorithm against corresponding CRLB. As for the hybrid case, it is assumed that full set of ranging information subject to Gamma noise with variance ranging from 0 to 1 and partial (target-to-anchor) AoA information subject to Tikhonov noise with angular error± 15◦ is available. The missing target-to-target AoAs are estimated via centroid algorithm [73].
Finally we compared the average CRLB against mean squared error both obtained similar to that of equation (2.72) and (2.73). The results are shown in Figure 3.4 and 3.5. It can be seen that although the performance of the two variations of SMDS are not too far from the calculated bounds, the algorithm can still be improved to achieve results closer to that of the bounds.
Note that unlike range-based SMDS, there exists a gap between the hybrid SMDS and the corresponding bound even for relatively small ranging errors. The reason for this behaviour is that, the calculated hybrid bound accounts for the fact that in the presence of large angular error, and in case ranging error is below a certain value, the optimum solution is to compute the angles from distance information and use the estimated angles instead of the measured ones. However, in practice, due to lack of knowledge on the quality of the measured data, the algorithm cannot make such decisions, therefore, it processes the input information as it is.