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In this paper we propose a global positioning algorithm of multiple assets based on Received Signal Strength (RSS) measurements that takes into account the gain uncertainties of each hardware transceiver involved in the system, as well as the uncertainties on the Log-Distance Path Loss (LDPL) parameters. Such a statistical model is established and its Maximum Likelihood Estimator (MLE) is given with the analytic expression of the Cramér-Rao Lower Bound (CRLB). Typical values of those uncertainties are given considering whether calibration is done in production,

Assets positioning raised a great interest in the last decade, especially with the Internet of Things (IoT) business. In this context, we expect indoor and outdoor positioning to be achieved with the same hardware, a low energy constraint for an autonomy of a few years. Economical aspect can even constrain each reference node, named Access Point (AP), used to locate the target assets, to run on battery.

Outdoor positioning is mainly achieved using Global Navigation Satellite System (GNNS) with an accuracy of a few meters that is enough for this kind of applications. Even if GNNS receiver energy consumption has been greatly improved in the past decade [

Indoor positioning using GNNS requires additional infrastructure because of the poor signal level; the monolithic solutions [

This paper focuses on indoor solutions that meet the constraints of low energy (for assets and APs) with minimal infrastructure and setup to reach a precision of a few meters. We particularly focus on the static assets positioning use-case, which implies that some techniques such as pedestrian dead reckoning are not applicable for the solutions we are studying. This area of indoor positioning using LPWANs, WIFI, or Bluetooth Low Energy (BLE) raised a great interest in the last decade and many solutions have been proposed, based on Received Signal Strength Indication (RSSI), Time of Arrival (ToA), and Angle of Arrival (AoA). Reviews on recent advances and capacities can be found [

The common framework widely used proceeds in two steps instead of direct estimation to reduce complexity even if this is suboptimal in general [

A given number of measurements related to distance are collected in a short time slot; outliers and noise are filtered and an approximate distance measurement is inferred from those, which are not coherent one with the other due to multipath and measurement noise.

Those distances are then aggregated using algorithms, which can belong to the following domains:

Geometry (finding the intersection of distance circles).

Machine learning (neural networks, Smallest M-vertex Polygon (SMP), or Support Vector Machine (SVM)).

Cellular algorithms (closest neighbour, weighted neighbors, etc. …).

Statistical estimation algorithms (mainly maximum likelihood).

ToA based solutions can be very accurate (within a meter or centimeter), such as Ultra Wide Band (UWB) or collaborative positioning, but they require expensive hardware and moreover synchronization signals are involved which increases the power consumption. On the other hand such a precision is not needed for asset positioning.

This paper then considers less precise but lower cost solutions like Received Signal Strength (RSS) positioning using low energy protocols like BLE. In this case the accuracy is strongly limited by the fast and slow fading effect arising in dense multipath environments. Fast fading can be mitigated using several measurements in time and their first step preprocessing (averaging and removing outliers from the average, i.e., values that are far from the mean value, or using median value) before running the second step positioning algorithm (which is known as time diversity [

Contrariwise, slow fading effects need spatial diversity or frequency diversity to be reduced. For a given multipath configuration, frequency diversity changes fading effects on the RSS allowing simple algorithms like averaging or maximum selection to improve the measurement at the preprocessing stage of the positioning [

Fingerprinting methods include the multipath effect for each specific room in the propagation model to compensate the static part of slow fading effects. Depending on the AP placement, this method can drastically improve the accuracy of positioning compared to physical models based solutions. The main drawback is the learning process energy consumption and hardware cost that should be deployed at the setup for static environments, and moreover the continuous learning process needed to correct slow changes in the environment.

An energy efficient solution to reduce slow fading is to use a sufficient number of APs in sparse places of the room to give redundant measurements, with different multipath biases. This spatial diversity may increase the accuracy of the positioning algorithm when it is able to take advantage of redundant measurements. There is then a trade-off between the hardware cost of a great number of APs and the desired accuracy of positioning. Part of the hardware cost resides in its installation process which is greatly reduced when those receivers are battery powered (so they do not need to be connected to a power source), when no calibration of the hardware gain is needed, and moreover when precise coordinates of AP placement in the room are not needed.

Calibration of APs gains can be done in the fabrication process or at setup time which takes into account the antenna coupling with the environment, or calibration can be run jointly to positioning as an unsupervised learning process. Many works deal with the calibration of radio map with pure machine learning [

To help the design of a positioning system, the authors of the previous work proposed theoretical bounds of positioning error with uncalibrated propagation model that are reachable in an analytic way [

We first consider in this paper the whole estimation of multiple assets positions, as it may improve performances compared to the independent estimation of each single target in presence of common APs miss-calibrated parameters. Then full equations are given and compared to previous paper single asset positioning [

In the next section, we first give the stochastic propagation model and state the positioning problem as an uncertain static parameter estimation. Then analytic CRLB and the MLE reaching this bound are derived. Section

We consider the positioning problem of I target assets which sends N signals to J APs placed in a given room. The positioning algorithm takes part of a propagation model which contains uncertain parameters model for each asset and APs.

It must be noted that this estimation problem is not a joint estimation of the model parameters and the position at the same time (like it is often done when dealing with fingerprinting) but rather a position estimator which takes into account the residual uncertainties of model parameters from the calibration process to estimate a position.

The propagation model and estimation problem are stated in the next section; the likelihood and maximum likelihood estimator is derived in Section

We consider the position estimation problem to be a static parameter estimation: knowing the RSSI channel model, the uncertainties on the parameters model, we want to find

The use of probabilistic algorithms requires a propagation model; the Log-Distance Path Loss (LDPL) model is a common choice for indoor (NLOS) propagation [

where

Fast fading phenomenon is removed by the preprocessing step, usually by removing outliers using median or Kalman filtering. NLOS situation is taken into account by the log-normal probability of shadowing

Path Loss parameters are known with relative precision depending on the calibration setup when it exists; its value differs from the theoretical value of 2 (valid in the case of isotropic propagation in vacuum). Density of obstacles and antenna directivity change this value which can be different from an AP to another. However, it is possible to find average values for those parameters working fine in most case and improving them with the collected RSSI over time [

If we do not calibrate

Typical uncertain parameters models when production calibration is run, or when calibration is done at the setup process

Parameters | Production calibration | | Not calibrated |
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| | Non Applicable | |

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| Nonrelevant | | |

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We propose studying the influence of those variances over the accuracy and precision of the likelihood estimate using CRLB, for instance, to know which accuracy we could expect by skipping or not the calibration of the receivers or the APs.

Let us consider that

The path loss model (

It is shown in the Appendix that the vector

Thus the measurements are affected linearly by a multivariate covariant random uncertain vector, so the likelihood and MLE can be obtained.

The positioning problem is to find all the assets coordinates components

Then the maximum likelihood to be solved is

Analytic solution to this optimization problem seems complicated as long as uncertainties on

In this paper we consider that all path-loss exponents are certain, or sufficiently calibrated, with known values

The positioning problem becomes the following nonlinear least square formulation:

which is efficiently solved in an iterative way [

To compute the theoretical bound, the FIM matrix should be derived using

Once again, the log-likelihood derivative with respect to

In this case the FIM can only be obtained in a numerical way. Taking the assumption of a well-calibrated path-loss exponents (

Then the CRLB can be used to obtain the inequality

Then as

As the measurements sensibility is independent from an asset to another and the expectation of measurements is independent from a measure to another, the derivative

The inverse FIM shows that each asset positioning accuracy is independent one from the other, which shows that estimating positions of

This formula shows that the increase of assets to be positioned does not improve accuracy (as the positioning algorithm does not calibrate the gain mismatch, this result seems fine). The number of measurements

In this section we compare analytic expression with simulations. The first subsection validates the analytic form of the FIM with simulations, showing analytic versus simulated covariance matrices. Then Section

For the numerical verification of previous equations, we used a simulator generating RSSIs with an additional pseudorandom noise. From those values we computed a position estimate using the MLE, and we measured the covariance matrix of those estimates. We then plotted the estimates coordinates, the 2-sigma ellipsis of the numerical verification, and the CRLB covariance matrix. An example of the results is shown in Figure

Numerical verification of the CRLB equation. The left figure shows 2-sigma covariance of simulated position estimates using MLE estimator vs. the covariance from our CRLB expression. Right figure shows the RMSE from simulation data and the CRLB trace.

To make sure that the CRLB matches real case scenarios, we conducted experiments in different room configurations, with several APs configurations at various known positions and computed the Cumulated Density Function (CDF) of the distance error between estimated and real positions. This subsection depicts the setup and the results.

After a numerical verification of the equations, we confronted the CRLB results with real measurements. Experiments took place in 11 by 6 meters office with two devices from firm FFLY4U: FFLYdot and Myria, respectively, for APs and tracked devices (see Figure

Measurement setup in one of the rooms. The grid shows all the measurements points.

To compare those measurements, we also computed a CDF using our RSSI simulator: for all 135 positions of measurements, we simulated 140 measurements, from which we computed the estimated position. For the CRLB, we generated 1000 positions estimates from the covariance matrix at each measuring point. It must be noted that the number of estimates is significantly higher than the CRLB because we want to generate a theoretical smooth curve and the random generated position is just an easier way to compute the theoretical curve from the analytic expression, taking into account all the different values of the CRLB at the different positions of the room. We made measurements with 7 and 4 APs placed at easiest mount points in the room (close to existing pillars, e.g., not on windows); the resulting CDF curves of those simulations and measurements are shown in Figure

Experimental values matched against CRLB.

Calibration is a step costly in time, which involves measuring in postproduction the static gain of all devices (APs and tracked assets), which might also vary in time for devices on battery. However, it could be skipped by measuring the average and variance of gains and model parameters and using those values in the model. As it could save time and money for industrial deployments, we want to study its impact on the expected accuracy. Now, we showed that our CRLB is consistent with real measurements; we will use it as a criterion to see if calibration is required depending on the expected accuracy. We simulated the CRLB in the case of a squared room with an AP at each corner in function of the size of the room (

CRLB RMSE. Coordinates are

| ( | I | Not calibrated | Calibrated APs | Calibrated APs & targets |
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5 | (0.5, 0.5) | 1 | 2.13 | 0.48 | 0.48 |

5 | (0.5, 0.5) | 20 | 2.08 | 0.11 | 0.11 |

5 | (0.1, 0.5) | 1 | 3.14 | 1.95 | 1.16 |

5 | (0.1, 0.5) | 20 | 3.09 | 1.87 | 1.01 |

10 | (0.5, 0.5) | 1 | 4.26 | 0.96 | 0.96 |

10 | (0.5, 0.5) | 20 | 4.15 | 0.21 | 0.21 |

10 | (0.1, 0.5) | 1 | 6.27 | 3.89 | 2.31 |

10 | (0.1, 0.5) | 20 | 6.17 | 3.73 | 2.03 |

20 | (0.5, 0.5) | 1 | 8.51 | 1.92 | 1.92 |

20 | (0.5, 0.5) | 20 | 8.30 | 0.43 | 0.43 |

20 | (0.1, 0.5) | 1 | 12.54 | 7.79 | 4.62 |

20 | (0.1, 0.5) | 20 | 12.35 | 7.47 | 4.05 |

What we can see on those results is that if we do not calibrate the APs gains, the lower error achievable is equivalent to

The positioning problem of multiple assets with uncertain receivers gain and uncertain propagation model has been addressed. Typical values of uncertainties have been given from the observation of multiple setups and calibration realized in different industrial environments. Based on this model, the global MLE algorithm is given and formulated, having an iterative nonlinear least square solver. The CRLB is expressed in the general form and analytically for the specific case of well-calibrated path-loss exponents. Simulations show that the bound is reached by the MLE and moreover real measurements and estimations show that this result is not too much conservative; that is, the bound is a good estimation of the expected accuracy. Using this analytic CRLB, we discuss the impact on the accuracy of access points gain calibration, assets gain calibration, precision of measurements, number of measurements repetitions, and number of joint assets to be positioned.

It first shows that in contrary to joint calibration and positioning algorithms the number of assets to be estimated has no effect on the accuracy of each estimation. One can then reduce the solver complexity by running independent estimations without loss of optimality.

Secondly, it shows that the positioning of RMSE is split into two terms involving APs calibration, measurement precision and repetition, weighted by a parameter connected to the geometric position of the APs inside the room on the one hand. And on the other hand the asset calibration gain and LDPL reference gain are weighted by a different geometrical factor to impact the accuracy. Then it shows that depending on different calibration quality or efforts different geometrical terms can be involved and then different optimal APs configurations can arise. Using a numerical application, we also showed that with typical hardware the error was close to those of cellular positioning if we do not calibrate APs.

More generally, this can be used for dimension and to optimize a setup to reach a desired accuracy. It showed an example of how to infer the number of APs to reach accuracy in a given room. Future work could infer the number of measurements required for a given accuracy in function of the calibration error of the APs or even to find the optimal AP disposition in a given room for a given calibration.

Equation (

We first stack in the vector

For stacking all APs on the vector we need to define the mismatch gain vector

Finally, we note

For the full log-square-distance vector defined as

where

The path-loss exponents

Then (

Then the independent random Gaussian vector

The independent measurement noise vector

Finally measurements are modeled by

Angle of Arrival

Access Point

Bluetooth Low Energy

Cumulated Density Function

Cramér-Rao Lower Bound

Difference Time of Arrival

Fisher Information Matrix

Geometrical Dilution of Precision

Global Navigation Satellite System

Internet of Things

Log-Distance Path Loss

Low Power Wide Area Network

Maximum Likelihood Estimator

Moore-Penrose Pseudoinverse

Nonline of Sight

Probability Density Function

Root Minimum Square

Root Minimum Square Error

Received Signal Strength.

Received Signal Strength Indication

Smallest M-vertex Polygon

Support Vector Machine

Time of Arrival

Ultra Wide Band.

Position estimation algorithm and Cramér-Rao bound ellipses can be computed following the instructions and using the code provided in

The authors declare that there are no conflicts of interest regarding the publication of this paper.