This paper proposes a geometric dilution-of-precision approach to quantize the privacy-aware location errors in a cooperative wearable network with opportunistic positioning. The main hypothesis is that, a wearable inside a multi-floor building could localize itself based on cooperative pseudoranges measurements from nearby wearables, as long as the nearby wearables are heard above the sensitivity limit and as long as nearby wearables choose to disclose their own positions. A certain percentage of wearables, denoted by , is assumed to not want to disclose their positions in order to preserve their privacy. Our paper investigates the accuracy limits under the privacy constraints with variable and according to various building maps and received signal strength measurements extracted from real buildings. The data (wearable positions and corresponding power maps) are synthetically generated using a floor-and-wall path-loss model with statistical parameters extracted from real-field measurements. It is found that the network is tolerant to about 30% of the wearables not disclosing their position (i.e., opting for a full location-privacy mode).
According to the survey by Grand View Research, Inc. in [
Given the widespread adoption of wearable devices and smartphones (serving as a gateway for their data processing) [3], one of the most on-demand functionalities nowadays is the ability to locate oneself within defined indoor or outdoor space [ 4]. When localization engines are used on power-constrained devices such as wearables, it is becoming more and more important to be able to use the signals coming from opportunistic networks to perform localization in the absence of a Global Navigation Satellite System (GNSS) chipset or in order to decrease the energy consumption at the wearable side. Indeed, opportunistic ad-hoc networks have been highly studied in order to provide seamless connectivity in situations where an infrastructure mode is not continuously available [5, 6, 7]. When various wearables found inside a certain area, such as an indoor mall or a commuting hall, are interacting with each other and do various measurements based on the received wireless signals, the exchange of information can be done in a faster and lower energy-consuming way than in an infrastructure mode. One of the convenient aspects of an opportunistic network is the possibility of a low-energy cooperative localization through basic information exchanges between wearables, such as pseudorange computations based on Received Signal Strengths (RSS), timing measurements, angle, or acoustic measurements [8, 9, 10]. Hence, not only high-end wearables but also low-cost wearables could perform cooperative self-positioning through these basic information exchanges. One of the drawbacks of such opportunistic positioning with wearables is the inherent risk in terms of user privacy, e.g., when the data exchange contains accurate location information of one’s wearables. For example, for an opportunistic positioning scenario, wearable devices equipped with GNSS modules and/or Inertial Measurement Units (IMUs) acting as Anchor Nodes (ANs) for their nearby wearables with lower computational resources will have to disclose their locations to nearby nodes in order to also enable the nodes in the ANs’ vicinity to self-locate (e.g., when such nearby nodes are not equipped with GNSS/IMUs). Another scenario is when all wearables in the system have only WiFi/BLE chipset, but no GNSS or IMUs, and thus the process could run iteratively, where each node takes turns to act as a mobile AN for other nodes in its vicinity, based on its previously computed position. Localization can be performed by relying on distance measurements to neighboring wearables, acting as ANs and transmitting their estimated location to the devices within range [11]. Such distance measurements, at their turn, can be obtained from time, angle, or power measurements.
RSS-based measurements are susceptible to noise, signal fluctuations, and line-of-sight (LOS) vs. non-line-of-sight (NLOS) detection dificulties and other factors, as reported, for example, in [12, 13, 14, 15]. Therefore, the study in [16] investigated the RSS-based cooperative localization challenge and used Cramer–Rao lower bound (CRLB) to compare performance of the proposed method of mitigating NLOS-related errors with the conventional approaches, showing increased performance.
Another metric, acknowledged by the research community is the geometric dilution of
precision (GDOP), which was originally developed to assess the precision of location estimates
in GNSS [
However, indoor localization systems may use various technologies, such as WiFi, Bluetooth,
ultrasonic, or infrared, for determining the position of a user or object within a building [
Based on the literature review, one can infer that applying diferent metrics allows to determine circumstances when the geometry may have a detrimental impact on the localization accuracy, leading to improvements in system performance as well as its design.
The goal of this paper is to investigate the tradeof between location accuracy on one hand (i.e., how well a wearable can locate itself based on cooperative pseudoranges measurements from nearby wearables) and the location privacy on another hand (i.e., the percentage of wearables deciding to disclose their location, with or without a perturbation, as well as the amount of intentional positioning errors or perturbations with which the nearby wearables are disclosing their location). We propose a model which includes the inherent measurement errors in a cooperative and opportunistic location-estimation algorithm, the intentional positioning errors derived for example from various obfuscation or perturbation mechanisms with which the nearby wearables in a multi-floor building are disclosing their position, as well as the percentage of the wearables fully hiding their position. Our methodology is based on a GDOP approach to quantize the privacy-constraint location errors.
Some related work has been addressed by the authors in [
The user location privacy typically depends on three main factors: i) measurement errors statistics, or how accurately such a location is estimated based on prior knowledge and/or information collected from the nearby nodes, ii) intentional error statistics, i.e., if and how accurately the nearby wearables disclose their location – we assume that the users have full control of how and at which level of accuracy they can share own location information with other nodes (this is computed as the percentage of users disclosing their locations), and iii) number of wearables, namely how many wearables with similar locations are there in the area; this afects the probability to mistake a wearable for another one in the area; this is also related to the probability of several wearables to belong to the same user.
Other related work to location privacy protocols and metrics [
The rest of the paper is organized as follows: Section 2 focuses on the system mathematical model that takes into account the privacy constraints and on the proposed GDOP-based metric. Section 3 presents the simulation environment and the simulation-based results. Section 4 summarizes the findings and discusses some future research topics in the field.
To illustrate the considered environment, we provide a graphical example of a potential indoor scenario for the opportunistic exchange of positioning information in Fig. 1, and in the latter part of this section, we explain the modeling parameters used in the simulations.
Fig. 1 shows a schematic floor plan of an indoor setting, such as a shopping mall. To ofer the context, the users are assumed to be distributed across three floors, and this setting could be used for various location-based services, such as finding a friend, finding the nearest shop with certain items of interests, finding the nearest exit, etc. The notion of proximity detection may be utilized, for example, for counting passive encounters [35], to enhance the tourism experience [36], for sociometric applications [37], etc. It can also be used to assist individuals in navigating their surroundings. Another application of indoor positioning is the use of digital contact tracing in public places. In such a context, proximity detection could be employed to alert persons when they are too close to one another, assisting in maintaining of safe distances, and in the prevention of infection spread [38]. Indoor proximity detection can be used to track the whereabouts of valuable assets within a warehouse or factory, such as equipment, cars, or goods. This could help in the optimization of processes, theft prevention, and maximizing the efectiveness of equipment [39].
Let us assume an indoor system with wearables. We also assume that each wearable can perform power, time, or angle-based measurements (or a combination of those) to identify their location in the defined indoor space. We assume also that each wearable = 1, . . . , can hear a certain amount of neighbourhood wearables, denoted with ℎ . A heard wearable is a wearable from which the received power is higher than the receiver sensitivity level of the -th wearable. An additional LOS condition can be imposed if the target measurements are
In a cooperative dynamic system, each wearable can update iteratively its position estimate based on the positions of the heard neighbours, assuming that a fraction 1 − of the neighbours choose to disclose their position p˜ , but with a random intentional error as below: where p˜ is the position disclosed by the j-th wearable with a certain intentional error . The intentional error is a privacy measure. In our simulator, we assumed to be Gaussian distributed with zero mean and equal to standard deviation of error for all variables, i.e., time-based or angle-based. NLOS wearables are also assumed to be ‘heard’ for power-based measurements (but not for time or angle-based measurements). A pseudorange measurement , is then obtained based on the power, time, or angle measurements available at each -th wearable from all its heard neighbours with = 1, . . . , ℎ .
The pseudorange measurement , is equal to the true LOS distance between -th and -th wearables in the system, plus an additional measurement error, denoted here by , : , = ||p − p || + , , = 1, . . . , ; = 1, . . . , ℎ (1)
In eq. (1), || · || is the Euclidean norm and p = [, , ] is the position vector of the -th wearable, having the World Geodetic System 1984 (WGS84) 3D coordinates , , and .
In our simulator, we used the assumption that all measurement errors , are Gaussian distributed with zero mean and standard deviation of error, i.e., , ∼ (0, 2).
p˜ = p +
∼ (0, 2).
(2)
(3)
(4)
This assumption follows the model introduced in [
If we assume a maximum likelihood cooperative position estimation, e.g. similarly with [40], the estimated updated position p^ is p^ = max ℎ ∏︁ =1
√︀(2 ) , ︂( − ||p− p˜− ,− ,||
2 , = 1, . . . , ; = 1, . . . , ℎ = min
=1 = 1, . . . , ; = 1, . . . , ℎ
2 , ℎ (︂ ||p − p˜ − , − , || ︂) 2 ∑︁ where is a 0/1 flag associated to the privacy status of each wearable (i.e., = 1 if the nearby wearable choose to disclose its position and = 0 if the nearby wearable does not disclose its position) and ℎ ≤
ℎ is the number of wearables in the vicinity of -th wearable which are heard (in terms of received power being higher than the sensitivity threshold) and choosing to disclose their position (namely, the heard wearables flagged with = 1, = 1, . . . , ℎ ).
Eq. (5) is a non-linear optimization problem which can be solved iteratively after Taylor linearization, following, for example, the procedure in [40]:
︂( p^+1 = p^ +
Σ− 1 Σ− 1 ︂( [ ,1 . . . ,ℎ ] − || p^ − 1, ℎ from [40] equal to where ∈ Rℎ × 3 ≜ [,1, . . . , ,ℎ ] is the Taylor linearized matrix with rows , , = , = ︂[ − ˜ − ˜
− ˜ ]︂ ||p −
p˜j|| ||p − p˜j|| ||p − p˜j|| and Σ = ( ,1, . . . , ,ℎ ) ∈ Rℎ × ℎ is the error covariance matrix taking into account the measurement errors , coming from the cooperative pseudorange measurements. Above, is the iteration index in the location estimation process, = 1, 2, . . . . The initial point of the estimation p^
1 can be assumed, for example, to be the true position of the -th wearable (e.g., the wearable is placed just at the entrance of the building and had access to accurate GNSS-based position estimates) or it can be computed as the weighted centroid of heard wearables in range which discloses their position: p^1 = =1 ℎ ∑︁ , p˜1 ℎ ∑︁ , =1 with , = 10(, /10), and , is the RSS (in dBm) by the -th wearable from the -th wearable. ︂) 2
, , (5) (6) (7) (8) √︃(︂ Σ
If we assume that we have uncorrelated measurement errors , of zero means and equal variances , ≜ , then the accuracy of the location solution from eq. (6), measured as the overall variance of the location error, is directly proportional to the trace of the square root geometry matrix , ≜ [40]. This means that 2 = E (,) , ︂( ︂) (9) where E(· ) is the expectation operator taken with respect to all wearables in the building.
The matrix , is also known as the GDOP matrix. The overall location error accuracy becomes thus a function of both the measurement error standard deviation as well as of the intentional error standard deviation (for clarity we assumed that all wearables have the same standard deviation of the measurement and intentional errors), as matrix is a function of . Moreover, the , can be computed either based only on LOS links from wearables which choose to disclose their position (e.g., when the pseudorange measurements in eq. (1) are based on timing or angle measurements which require LOS), or based on all heard wearables which choose to disclose their position, both LOS and NLOS (e.g., this is relevant when the pseudorange estimates are based on RSS measurements, which do not necessarily require LOS).
Therefore, the overall location accuracy per wearable can be modeled via the aboveintroduced GDOP-based statistics ≜ , and it will be a function of the measurement error , the number of wearables ∑︁ , which choose to disclose their position ℎ with some intentional error, their intentional error standard deviation and, possibly, of the number of LOS links.
A Matlab-based simulator has been implemented for our studies. One building map from a three-floor shopping center was used in our models, as shown in Fig. 2. The wearables are assumed uniformly distributed across the floors of each building and a random walk model is assumed for each wearable. The wearables can have diferent heights and placements, ranging from 5 cm above the floor (e.g., foot/shoe mounted wearable) to 1.8 m above floor (e.g., head-mounted wearable). The following single-slope floor-and-wall path-loss model was assumed: , = − 10 * , * 10||p − p || − − , − , + , , (10) where , is the received signal power (in dBm) of the -th wearable from the -th wearable, is the transmit power (in dBm) of the -th wearable, , is the path-loss slope coeficient for the path connecting wearable to wearable (channel reciprocity condition , = , was assumed to be respected), , and , are the numbers of inner and outer walls,respectively, between wearable and wearable (computed based on the building map), , is the number of floors between wearable and wearable , and are a loss factor (in dB) per inner or outer wall, respectively, and is a loss factor (in dB) per floor (all floors were assumed to introduce equal wall losses). The term , models the random shadowing efects and was assumed to follow a Gaussian distribution of 0 mean and ℎ standard deviation (in our simulations, ℎ = 4 dB, based on the measurement results reported in [41].
A wearable can be heard by another wearable in the building if and only if , ≥ . In our simulations, the sensitivity threshold was set to − 100dBm. Additionally, two wearables and are in assumed to be in LOS condition to each other if and only if , + , = 0 (and they are in a NLOS condition otherwise).
Two arbitrary examples of the synthetically generated power map from two wearables in the building is shown in Fig. 3.
A random-walk mobility model [42] with a randomly distributed velocity between 0.1 and 1 m/s was used. The wearable movements were assumed to remain at the floor of their initial position (i.e., vertical/across-floor mobility is not yet included in our model). As the building maps are proprietary, the Matlab-based simulator is not currently provided in open access.
The simulation results are based on the in-house built simulator described in Section 3.1. A total of users carrying one wearable each were assumed to be distributed uniformly across the three floors of a simulated building. The buildings were based on real maps, collected from a university and from a shopping mall. The information about the building walls and floors was used in the path-loss model (see eq. (10)). All wearables are assumed to be able to hear all other wearables in the building as long as the received power is higher than the sensitivity threshold (in our simulations, it was set to − 100 dBm). The received power values depend, of course, on the distance between any two wearables as well as on the number of walls and lfoors between any pair of wearables (see eq. (10)). The path-loss coeficient, inner and outer walls attenuation factors, floor attenuation factors and transmit powers of each wearable were modeled according to models extracted from real-world measurements in the two buildings based on [41], as follows • A log-normal distribution for the transmit power of each wearable, = 1, . . . , • A normal distribution for the path loss coeficient of each wearable, = 1, . . . , • A Weibull distribution for the inner-wall and floor losses of each wearable, = 1, . . . , • A Gamma distribution for the outer-wall losses of each wearable, = 1, . . . ,
Fig. 4 illustrates the histograms of the path-loss parameters used in our simulations and based on field RSS measurements. Further details on choosing these parameters based on measurement data can be found in [41].
Under the assumption that a fraction of the wearables within a multi-floor building do not disclose their location estimates (in order to preserve their location privacy) and that all the other wearables form an opportunistic network for self-positioning, we have looked at the number of heard wearables as well as at the GDOP-based positioning accuracy in diferent buildings and for diferent levels. The number of heard wearables is shown in Fig. 5; the left-hand plot compares the two buildings (university building and shopping mall) assuming a small measurement noise standard deviation = 0.2 m and an intentional position error for the wearables that disclose their position of a moderate standard deviation = 2 m. The right-hand plot in Fig. 5 also shows the number of heard wearables (taken into account the path-loss propagation and ) for three diferent combinations of measurement and intentional errors standard deviations ( , ).
As we can see from the left plot in Fig. 5, when all wearables disclose their position, the number of average heard number of wearables by their neighbours is a bit higher in the university building than in the shopping mall building; this can be explained if we refer to Fig. 4, where we can see that the path-loss slopes are a bit sharper and the inner wall losses are a bit higher for the shopping mall than for the university building, which means that signal attenuates faster and can go faster below the − 100 dBm sensitivity limit. Surprisingly enough, when only 1 − of the wearables disclose their position, the number of heard wearables in the two buildings are closer to each other. As expected, this number severely decreases when increases. The right plot in Fig. 5 focuses only on the shopping mall building and checks diferent combinations of ( , ). The impact of varying ( , ) on the number of heard wearables is very small, as expected.
The average positioning accuracy per wearable is shown in Fig. 6. Again, the left-hand plot compares the results for diferent buildings and the right-hand plot compares the results inside the shopping mall building, for diferent ( , ) pairs. The blue lines are independent of and show the GDOP-based accuracy when all wearables disclose their position. In theory, these curves should be completely flat, but since at every Monte Carlo run we have random placement of wearables within the building and random path-loss parameters, there are small variations with 7500 Monte Carlo runs; these blue curves would converge to completely flat curves for a suficiently high number of Monte-Carlo runs. The red lines show the deterioration in the positioning accuracy when increases. If we set a target of maximum 1 m accuracy deterioration, then the network would be tolerant to maximum 30% of wearables not disclosing their position (i.e., = 0.3 as a fraction or = 30% as a percentage; for clarity reasons, is given in percents in our figures). Despite the fact the the number of heard wearables was rather independent on the ( , ), the positioning accuracy, as expected, is highly influenced by the measurement and intentional errors, as seen in the right-hand plot of Fig. 6. Again, up to = 30% ofers very little degradation in the overall positioning accuracy, but for > 30%, the performance starts to deteriorate fast. We would also like to emphasize the diferences between the two situations: a limited number of devices all disclosing their position (let’s say = 70 devices, = 0%) and the presence of some devices not willing to share their position (let’s say = 30% out of = 100 devices not sharing the position); while the overall performance will be the same in both cases (as only 70 cooperative devices that share their positions would be available in both cases), our research question pertained to finding out how much the performance is deteriorating with respect to the maximum achievable performance (i.e., = 100 devices in our second example) when some of them are not disclosing the position. Our findings show that such performance deterioration is not high as long as the ≤ 30%, no matter on the value of .
Last but not least, the efect of the number of the wearables in the building is shown in Fig. 7, where we compare a situation with a low number of wearables = 30 with a situation with a moderate number of wearables = 80. The left-hand plots show the average number of heard wearables, which, of course, decreases when decreases. The right-hand plot of Fig. 7 show again that the positioning accuracy starts to deteriorate significantly for > 30% for both = 30 and = 80.
To sum up, our findings show that the opportunistic network for positioning tolerates up to 30% of wearables not disclosing their position, without a significant loss in the positioning accuracy and for > 30%, the accuracy starts to deteriorate significantly.
Privacy-aware opportunistic and collaborative positioning could rely on the hypothesis that only a percentage of wearables within an indoor space are willing to disclose their positions. After proposing a GDOP-based positioning accuracy metric, we have investigated the robustness to such a collaborative opportunistic setup under various scenarios (building maps, number of wearables, assumptions regarding the measurement errors in estimating the position, etc.). It was shown that up to around 30% of the wearables can choose to keep their location undisclosed without a significant impact on the overall system performance.
Diferent random distributions of the wearables within a building as well as diferent mobility
models of the wearables, including across-floor mobility, are to be investigated next.
Furthermore, the relationship between the percentage of wearables hiding their position and classical
location privacy metrics, such as entropy-based privacy of [
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