On the Sensitivity of RSS Based Localization Using the Log-Normal model: An Empirical Study Jos´e Vallet∗ , Ossi Kaltiokallio∗ , Jari Saarinen∗ , Matthieu Myrsky∗ , Maurizio Bocca† ∗ Automation

and Systems Technology Department, Aalto University, Otaniementie 17, Espoo, Finland † Electrical and Computer Engineering Department The University of Utah. Salt Lake City, UT, USA

Abstract—The accuracy of radial radio propagation models, e.g. the log-normal path loss model, is severely degraded by the effects of multipath propagation, environmental differences and hardware variability. This has a direct impact on the performance of node localization algorithms that use these models. In this paper, first we study the effect of the environment and hardware variability on the model parameters of the log-normal path loss model. We empirically show that, even in the same environment, the model parameters can vary significantly depending on the nodes used for the calibration. Second, we present node localization results obtained using a maximum likelihood algorithm and evaluate the sensitivity of the algorithm to model parameter changes. Third, we show that the localization results can be improved using an individual model for each node. Using a robot for nodes localization, we report experimental results in three different environments: an open sports hall, a semi-open lobby, and a cluttered office. In corresponding order, accuracy of 0.33 m, 1.07 m and 0.78 m is achieved using individual models.

I. I NTRODUCTION Selecting an appropriate radio propagation model is fundamental when using the received signal strength (RSS) for node localization, since distance is not the only factor affecting the RSS. In environments with multipath propagation, spatial and temporal fading can cause the RSS to deviate from the model significantly [1], [2]. In addition, differences in hardware (HW) and orientation of the nodes [3] can contribute to the RSS measurements significantly. Simplistic propagation models need little information to be used but result in limiting accuracy. On the contrary, more complex models can provide better accuracy, but typically require more knowledge e.g. about the environment to be used effectively. When deriving a simplistic RSS vs. distance model, such as the log-normal path loss model as in this work, differences in the environment and HW affect the measured RSS and consequently the model parameters. Thus, a model calibrated using measurements from different nodes in a specific environment can result in a model incapable of accounting for local variability in the environment and the differences in HW. In this paper, we refer to a model used by all the nodes as a common model. When using a common model for localization, the performance can be significantly affected by which nodes were used to calibrate the model. The log-normal path loss model [2] is a widely used RSS vs. distance model in the research field of wireless sensor networks (WSNs). In this paper, we present the results of an

extensive measurement campaign conducted in three different environments. First, we study the variability of the model parameters in the different environments. Then, we investigate the variability of the model parameters between different nodes within the same environment. The results suggest that WSN deployments in specific environments are better represented by probability distributions of the model parameters rather than by deterministic values. It is also discussed how differences in the environment and HW influence the expected value and standard deviation of the probability distributions. We present results of a localization scenario, where a robot is used as a mobile beacon to localize static nodes of a WSN. The robot location is known at each time instant and this information can be effectively used to localize the other nodes of the network. We investigate the performance of a maximum likelihood (ML) algorithm to localize the nodes in three different environments. First, we present the sensitivity of the ML algorithm to variations of the model parameters to emphasize the importance of model calibration. Second we show that models calibrated with many nodes are more robust against errors from node particularities, and give on average better position estimates. Third, we demonstrate that using an independent model for each node results in better localization accuracy compared to a common model. In this paper, we refer to such a model as individual model. Using individual models improves significantly the localization accuracy in all the environments. II. R ELATED W ORK Various radio propagation models exist in the literature such as the log-normal path loss model, breakpoint model, attenuation factor model and ray-tracing models [2]. The log-normal path loss model and the break point model are simple radial models. On the contrary, the attenuation factor and ray-tracing models require detailed information about the environment. The log-normal path loss model is the simplest of the above and predicts the received power in dB as follows P = Pd0 − 10 n log10 (d/d0 ) + χσ ,

(1)

where Pd0 is the received power at reference distance d0 , n is the path loss exponent and χσ is zero mean Gaussian noise with standard deviation σ. Typically n and σ are larger in

(a) Basketball field

(b) Lobby

(c) Office

Fig. 1: Environments

more cluttered environments due to shadowing and multipath propagation. Considerable efforts in the late 80s and early 90s were made to characterize and model the indoor propagation channel to aid in the design and development of personal wireless communication systems [4]. Many of these works, e.g. [5], use the log-normal path loss model to derive the relationship between RSS and distance using high quality radios. These works propose general guidelines for choosing the model parameters in different environments. However, to achieve a better fit for the model, site-specific information about the environment is required [5]. Therefore, in this paper we calibrate a common model in each environment. WSN deployments comprise of many nodes usually equipped with low-cost radios with slightly different characteristics that affect the RSS [3], [6]. Therefore, we also calibrate an individual model for each node to take into consideration local differences in the environment and to account for HW variability. Furthermore, due to multipath propagation, spatial and temporal fading have been identified to cause significant fluctuations in the RSS [1]. To mitigate the undesirable effects of fast fading, we average RSS measurements over small distances [7] and exploit multichannel communication. When using a RSS-distance model for localization the performance of the algorithm depends strongly on the validity and appropriateness of the model. Many studies emphasize the importance of the model calibration prior to localization [8]. However, studying the sensitivity of localization algorithms on variation of the model parameters is not straightforward, because it depends on the geometrical configuration of the problem and on the algorithm itself. As a first approximation, the performance of a log-normal model used for localization can be analytically evaluated using the Cramer-Rao Lower Bound (CRLB). In [9] the CRLB is calculated assuming that the model parameters are known a priori. In [10], Malaney extends this idea considering the path loss exponent and the height of the node as nuisance parameters. He shows that these can affect the position accuracy significantly (a factor of two for the particular configuration presented), and underlines the importance of considering the combined effects of nuisance parameters rather than individually. In this paper, we make

an empirical study of the sensitivity of ML position estimates with respect to variations of the log-normal model parameters. We identify the parameters to which ML estimates are most sensitive to and show that the sensitivity depends on the type of environment. This sensitivity can be used as a guideline for a correct calibration campaign. III. M ETHODS A. Experimental setup We conducted three experiments in three different environments: a) an unobstructed basketball field (bfield), b) a partially-obstructed lobby, and c) an office where non lineof-sight (LoS) communication was dominant (see Fig. 1). In each experiment, a robot capable of simultaneous localization and mapping (SLAM) was used to explore the environment following a different trajectory. The maps from different tests for the same environment were matched as in [11] to obtain a global localization frame. One node was mounted on top of the robot to collect approximately ten RSS measurements per second from the other nodes of the WSN. In the experiments, 18-20 nodes were placed in fixed random positions mounted on 1 m high poles and covering an area of approximately 20x20, 20x30 and 12x20 m2 respectively for the basketball field, lobby and office. Detailed maps can be found in [12]. From the map of the robot we can also measure the ground truth position of the nodes (see also [12] for details). The nodes were equipped with omnidirectional antennas and IEEE 802.15.4 compatible TI CC2420 transceivers [13] operating in the 2.4 GHz ISM band. The communication protocol was following a token passing scheme using a predefined set of frequencies. After every round of communications the nodes synchronously switched to the next radio channel in schedule. The RSS measurements were averaged over a distance of 25 cm moved by the robot. This reduced the effect of fast fading and uniformized the density of effective measurements along the robot trajectory. B. Node Localization The position of the nodes is estimated using ML, which is the base of many other probabilistic methods. The robot’s position is known at all times, and therefore it can be used

as a mobile beacon [12]. From (1), the likelihood of one RSS measurement at time instant i, zi , with a target node at position ~x and the robot at position r~i is   (zi − z¯)2 1 exp − , (2) pZ (zi ; di , θ) = √ 2σk2 2πσ 2

bfield lobby office

0

1

2

L(z, d(~x, r), θ) = ln

N Y

pZ (zi ; di (~x, r~i ), θ),

3

n

where di = k~x − r~i k2 , z¯ = Pd0 − 10 n log10 (di /d0 ) and θ = [n, Pd0 , σ]. Assuming independence among observations, the log-likelihood of a set of measurements z = [z1 , z2 , ..., zN ] with the robot at positions r = [r~1 , r~2 , ..., r~N ] is

4

5

bfield lobby office

(3)

i=1

where d(~x, r) = [d1 , . . . , dN ]. The vectors z and r are known experimental data, and θ is known after calibration of the model. The only independent variable is then ~x. The position estimate is the value of ~x that maximizes (3). To find the maximum likelihood we used a Broyden Fletcher Goldfarb Shanno (BFGS) method [14], with the initial guess on the nodes’ position calculated using a grid search ML [15] with a grid resolution of 0.5 m. IV. VARIABILITY OF THE M ODEL PARAMETERS As stated in section III-A, three experiments are conducted in each environment. In each experiment the robot explores the environment following a different trajectory measuring RSS from all the fixed nodes. The position of the nodes and the robot are known at all times. Thus, every RSS measurement can be associated to a transmitter-receiver distance. We then calibrate an individual model for each fixed node, trajectory of the robot and environment. The model parameters n and Pd0 are estimated using linear regression of (1) using least squares and the set of known RSS-distance pairs. Then σ is then taken as the standard deviation of the residuals. The model calibration results for one sample trajectory in each environment are shown in Fig. 2 and summarized in Table I. In the figure, the large dots represent the parameter values of the common models for the different environments. The small dots represent parameters for the different individual models. The curves depict the distributions of the individual model parameters. As shown, the model parameters differ considerably among the three environments. In addition, parameters of the individual models are remarkably different even within the same environment. A. Global Variability In this paper, we call large differences among various environments as global variability, e.g. one is unobstructed and the other one is cluttered. Global variability can be quantified through the parameters of the common models. It is well known, that the environment has a significant effect on the propagation phenomenon. As a consequence, we can see in Fig.2 different values of the parameters of common models in the different environments. The path loss exponent n describes the rate of power decay, and the larger it is, the more rapidly the signal attenuates as it

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−50

−45

Pd 0 bfield lobby office

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2

3

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σ

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Fig. 2: Common and individual model parameters in the three environments for one robot trajectory. TABLE I: Mean and standard deviation of the distributions from Fig. 2, and parameters of common models (cm). Bfield

Lobby

Office

n

mean std cm

1.71 0.14 1.68

1.70 0.25 1.74

2.87 0.48 2.85

Pd0

mean std cm

-54.19 1.54 -54.24

-55.67 1.60 -55.60

-58.66 2.18 -59.00

σ

mean std cm

2.65 0.28 3.00

2.98 0.47 3.40

4.76 0.71 5.34

propagates through the wireless medium. As shown in Fig. 2, n is higher in obstructed environments than in open areas. Pd0 is the received power at a reference distance d0 . Typically a short reference distance is chosen (e.g. one meter) for which the free space propagation is taken as the only responsible of the signal loss. In this paper we choose d0 = 5 m so that the values of Pd0 include more clearly environmental effects. Therefore, Pd0 values are lower in the obstructed environments. The standard deviation of the noise process σ describes how the signal fluctuates around the mean value. In unobstructed environments the wireless medium is homogeneous and σ is fairly low as shown in Fig. 2. In cluttered environments, the effect of multipath propagation increases as signals are scattered, reflected and diffracted more on its way from the transmitter to receiver. This causes significant deviations in the

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measured RSS with respect the average, and therefore σ also increases. The values of the common model parameters are in agreement with the literature [2].

1 kn

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(c) Office

Fig. 3: Sensitivity contours for the three environments. kn and kP are the multiplication factors for n and Pd0 respectively. The black crosses indicate points associated to the unmodified models (reference calibration point), and the red plus signs to models with which the minima from our grid are attained. The color-bars indicate normalized errors.

Global variation is not the only factor affecting the model parameters. The nodes are also impacted by local effects of the environment. Furthermore, HW differences, radiation patterns of the antennas and orientation of the nodes can significantly influence the RSS. Quantifying the specific causes of local variability is more complicated than for global variability, since it is a sum of multiple causes interacting in a complex way. However, its effects can be clearly seen in the dispersion of the individual model parameters. As Fig.2 shows, the distributions tend to spread as the inhomogeneity of the environment grows. In environments with obstacles, the robot can have LoS communication with a node at some instants, whereas at other times the communication is carried out through walls and reflections. Different LoS conditions result in different values of the model parameters, specially n and σ, and therefore their distribution widen. In the office some nodes are located in rooms, whereas others are located in corridors or more open spaces with more dominant LoS communication. Consequently, the model parameters vary considerably among nodes. The dispersion of Pd0 is affected mostly by the HW variability. From Table I we see that the relative variation of the standard deviations when changing environments is larger for n and σ when compared to Pd0 . Therefore, the dispersion of Pd0 is less affected by the change of environment. The variability of the parameters seen in Fig. 2 suggests that WSN deployments are better characterized by distribution of the model parameters rather than deterministic values, whose mean values and dispersion are affected by the environment and the HW variability. V. L OCALIZATION R ESULTS In this section we study empirically the accuracy of ML position estimation. First, we calculate the localization error using common models calibrated with data from all the nodes and study the sensitivity of the algorithm to changes on the model parameters. Second, we study the effects of calibrating the model with different number of nodes. Third, we compare the use of individual models vs. a common model. Throughout this section, we consider errors associated to using common models calibrated with data collected from different sets of nodes. These errors are calculated as follows. Consider a set S containing L selected nodes out of M available (L ≤ M ). Three models are calibrated for the set using only the data from the selected nodes, one for each robot trajectory. The resulting models are then crossvalidated calculating the position of all the M nodes as explained in section III-B using independently the data from the other two remaining robot trajectories. The localization error associated to the set S is then the average distance between the true and all the estimated node positions, i.e.,

1.4

Normalized mean error


 P3 P2 PM 1 xi − x ˆSijk k2 , where x ˆijk e¯S = 3·2·M k=1 j=1 i=1 k~ is the position estimate of node i using the trajectory j for validation and the data from the set of nodes S and trajectory k for model calibration. We will also compare the localization error associated to different sets of nodes with respect a reference set whose models are calibrated using data from all the available nodes (L = M ). The errors associated to the reference set for different environments can be seen in Table II. In the following we refer to these errors as reference calibration errors.

bfield lobby office 1.2

1

A. Sensitivity of ML localization Using a Common Model From Table II, the best localization accuracy of 0.94 m is achieved in the basketball field, whereas the worst in the lobby, i.e., 1.44 m. These results can be affected by variations of the model parameters. To study this effect we modify the parameters of the common models multiplying them by factors ranging from 0.5 to 2. Then we calculate the new localization errors and compare them with the errors when using the unmodified models (i.e., the reference calibration errors). Fig. 3 shows a contour plot of the sensitivity of ML position estimates in the different environments. The horizontal and vertical axis show the multiplication factors kn and kP used to modify n and Pd0 . The figure covers a detailed area of interest close to the point corresponding to the unmodified model (kn = 1 and kP = 1). We will refer to this point as the calibration reference point. The resolutions of kn and kP are 0.03 and 0.01 respectively. The color bars present the localization errors normalized with the reference calibration errors. Variations of σ have a negligible impact when compared to variations of n and Pd0 , and therefore σ is not included in the figure. Fig.3 shows that the normalized errors can change significantly with modifications of the parameters. Relative variations of Pd0 in general have notably larger impact than relative variations of n. The curvature of the error is higher in the basketball field and lobby, suggesting that the sensitivity is higher in open environments. Interestingly, we can see that sometimes artificial variations of the calibration parameters can be beneficial. The minima (red plus signs) are not attained with the unmodified models (black crosses). We find them with parameter variations corresponding to kn = 1.01 and kP = 1.03 for the basketball field, kn = 0.86 and kP = 1.02 for the lobby and kn = 0.74 and kP = 1.00 for the office. These correspond to error reductions of 33%, 10% and 21% respectively (errors of 0.63 m, 1.29 m and 0.76 m respectively). The impact of parameter variations depends on the direction of change. Considering variations near the calibration reference point, we can see that relative reductions of the parameters (in absolute value) can increase the error rapidly specially in the basketball field and lobby. The HW variability is one factor that affects directly the variability of Pd0 . The large sensitivity to variations of Pd0 then suggests that the HW variability can have a relatively large impact on the localization accuracy. This is specially relevant in open environments

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Number of nodes in calibration Fig. 4: Mean error of ML position estimates when using a common model depending on the number of nodes used in the calibration

where the sensitivity is the highest. As more obstacles are added in the lobby and office the relative contribution of the HW variability to the model uncertainty diminishes when compared to the effects from the environment. B. Calibration of a Common Model Next we study how using different number of nodes in the calibration affects the localization errors as a consequence of variations of the model parameters. In order to make this study we calibrate common models using data from between one to all the available nodes (L = {1, .., M }). For each number of nodes used in the calibration L we form (up to) ten different sets of L randomly selected nodes. For each of the sets we calculate its corresponding localization error as explained in the beginning of this section. Finally we calculate the average of all the errors over the ten sets and normalize it with the calibration reference error. Fig. 4 shows the normalized average error as a function of the number of nodes used in the calibration for the three environments. Due to the fact that the combination of nodes are selected randomly, different runs of the procedure to create Fig. 4 yield different results. But the trend is always the same: an initial tendency of error reduction and a stabilization as the number of nodes used in the calibration increases. When using few nodes for calibration, the model parameters can reach areas far of the calibration reference point (see Fig.3). If, by chance, we select the right set of nodes, the model might reach points near the minima. But, without a priori information, we have a high risk of selecting nodes whose models fall in regions with high normalized error. Using more nodes, on the other hand, attracts the parameters towards the calibration reference point, making them fall closer and diminishing the risk of reaching areas of large errors. Therefore, using more nodes for calibration of common models results in more robust models against errors from HW variability and local environmental inhomogeneity.

TABLE II: Average errors of ML position estimates in different environments when using a common model (CM) and individual models (IM). Errors for CM are the reference calibration errors. Distance in meters

Basketball field Lobby Office

CM

IM

Ratio(CM/IM)

0.94 1.44 0.97

0.33 1.07 0.78

2.83 1.34 1.25

C. Individual Models The variability of the parameters shown in Fig. 2 together with the sensitivity displayed in Fig. 3 suggest that it would be beneficial to use individual models instead of a common one calibrated with all the nodes. This was already demonstrated in [12] for several types of polynomial models in the context of simultaneous localization and model calibration. Here, Table II shows a comparison using the cross validation scheme described in the beginning of this section. The improvements range from 25% in the office to 183% in the basketball field. Individual models incorporate more effects of the particular HW and close obstacles. A common model, in contrast, looses the information about the variability of parameters among different nodes, i.e, it ignores the dispersion of the curves from Fig. 2 for n and Pd0 . To accommodate the differences among nodes, a common model needs to increase its standard deviation. We can see this in the lower plot of Fig. 2, where the value of σ for the common models (thick dots) are greater than the average of σ for individual ones in all environments. In terms of information, this essentially means that a collection of individual models contains more information than a single common model calibrated using all the data from all the nodes. And because of this, the localization algorithm will give better position estimates when using individual models with respect to using a common one (Table II). It is worth noting that the improvement due to using individual models is the largest in the basketball field. To understand the reasons behind this we shall go back to Fig. 2. We can see that σ for common models are always over the mean of the distribution for individual models in all the environments. However, for the case of the basketball field the thick dots are further in the right tails of the distributions. Thus, the increase relative to the dispersion is the largest. That is: (σcm − σ ¯ )/std(σ) is largest, where σcm is the standard deviation of the common model, and σ ¯ and std(σ) are the mean and standard deviation of σ for the individual models. This means, in essence, that the gain of information when switching from common to individual models is relatively larger in the basketball field. The HW variability has the largest contribution to the dispersion of the model parameters in the basketball field with respect to the other environments. This suggests, again, that the use of individual models is specially effective for the HW variability problem in open spaces.

VI. C ONCLUSIONS Local inhomogeneity in the environment and HW variability cause significant fluctuations of the parameters of RSSdistance log-normal models. Therefore, WSN deployments are better represented by probability functions of the model parameters rather than deterministic values. The mean value and dispersion of the parameter densities depend on the environment and HW characteristics of the nodes. Variations of the model parameters in turn affect the performance of localization algorithms. ML localization errors using a common model are notably more sensitive to relative variations of Pd0 than n, and quite insensitive to variations of σ. The sensitivity is higher in open environments. The high sensitivity to changes in Pd0 emphasizes the potential impact of the HW variability on the localization accuracy, specially in open environments. Calibrating site-specific common models using many nodes produces more robust models against errors from HW variability and local environmental inhomogeneity. Even better is to use individual models, which in our experiments reduced the average localization error between 25% to 183%. R EFERENCES [1] H. Hashemi, “A study of temporal and spatial variations of the indoor radio propagation channel,” in IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, vol. 1, Sep. 1994, pp. 127 – 134. [2] T. S. Rappaport, Wireless Communications: Principles and Practice. Prentice Hall, July 1999. [3] D. Lymberopoulos, Q. Lindsey, and A. Savvides, “An empirical characterization of radio signal strength variability in 3-d ieee 802.15.4 networks using monopole antennas,” in Wireless Sensor Networks, ser. Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2006, vol. 3868, pp. 326–341. [4] H. Hashemi, “The indoor radio propagation channel,” Proc. IEEE, vol. 81, no. 7, pp. 943 –968, jul 1993. [5] S. Seidel and T. Rappaport, “914 mhz path loss prediction models for indoor wireless communications in multifloored buildings,” IEEE Trans. Antennas and Propagation, vol. 40, no. 2, pp. 207–217, Feb. 1992. [6] T. Stoyanova, F. Kerasiotis, A. Prayati, and G. Papadopoulos, “Evaluation of impact factors on rss accuracy for localization and tracking applications,” in ACM Proc. international workshop on Mobility management and wireless access, ser. MobiWac ’07. ACM, 2007, pp. 9–16. [7] A. F. Molisch, Wireless Communications. John Wiley & Sons, 2010. [8] G. P. Fotis Kerasiotis, Tsenka Stoyanova, “Evaluation of outdoor rssbased tracking for wsns aiming at topology parameter ranges selection,” International Journal On Advances in Networks and Services, vol. 3, no. 1 and 2, pp. 140–157, September 2010. [9] N. Patwari, I. Hero, A.O., M. Perkins, N. Correal, and R. O’Dea, “Relative location estimation in wireless sensor networks,” IEEE Trans. Signal Processing, vol. 51, no. 8, pp. 2137–2148, Aug. 2003. [10] R. Malaney, “Nuisance parameters and location accuracy in log-normal fading models,” IEEE Trans. Wireless Communications, vol. 6, no. 3, pp. 937 –947, march 2007. [11] J. Saarinen, J. Paanaj¨arvi, and P. Forsman, “Best-first branch and bound search method for map based localization,” in IEEE/RJS Proc. International Conference on Intelligent Robots and Systems, 2011, pp. 59–64. [12] J. Vallet, O. Kaltiokallio, M. Myrsky, J. Saarinen, and M. Bocca, “Simultaneous rss-based localization and model calibration in wireless networks with a mobile robot,” in Procedia Computer Science. Elsevier, 2012, pp. 1106–1113. [13] CC2420. 2.4 ghz ieee 802.15.4 / zigbee-ready rf transceiver. [Online]. Available: http://focus.ti.com/lit/ds/symlink/cc2420.pdf [14] J. Nocedal and S. J. Wright, Numerical Optimization. Springer, Aug. 2000. [15] S. M. Kay, Fundamentals of Statistical Signal Processing. Prentice Hall, 1993.