Visible Light Positioning (VLP) is considered as one of the most promising technologies for achieving low-cost and massive coverage indoor location-based service. However, traditional trilateration-based VLP methods su er the Received Signal Strength (RSS) uctuation problem which would signi cantly limit the positioning performance. This paper proposes an interval analysis-based set-membership approach to improve the positioning accuracy and stability of the VLP system in noisy environments. The proposed method utilizes a statistics method to construct con dence intervals from the uctuated RSS measurements and casts the positioning process into a set-inversion problem which is then solved via an interval analysis-based algorithm in the framework of set-membership. Simulation results have been compared with the traditional least-square based positioning method, showing that the proposed method can provide more accurate and stable positioning results in different noisy interference environments.
In the upcoming 5G Internet of Things (IoT) era, most of the mobile services will be generated in indoor environments and there would be an explosive growth of Location-Based Service (LBS). As the core technology of LBS, indoor positioning technology lays a technical foundation for housekeeping services, emergency security, smart warehousing, crowd monitoring, precision marketing, mobile health, cultural entertainment, etc. To meet the diversi ed and massive LBS demands in indoor environment, di erent technologies, namely UWB, WLAN, RFID, BLE and VLP, have been proposed and developed to tackle the positioning issues, committed to achieve accurate, reliable, and full coverage solution. ? The authors gratefully acknowledge the nancial support of the EU Horizon 2020 program towards the Internet of Radio-Light project H2020-ICT 761992.
Among the aforementioned technologies, VLP is gaining more and more attention nowadays due to numbers of inherent advantages: modest infrastructure cost, adequate coverage, free of magnetic interference and potential centimeter accuracy. Received Signal Strength (RSS) is perhaps the mostly used metric in VLP system due to its simplicity and low hardware requirement [6]. However, the main challenge of such method is the continuous signal uctuations [3]. The measured RSS values usually have a high variability over time due to the uctuating nature of wireless signals. Besides the thermal and shot noise, they could be signi cantly a ected by shadowing, fading, and multipath propagation in indoor scenarios [10]. Such high variability will a ect the ranging results and degrade the performance of the VLP system in terms of accuracy and reliability.
To deal with the RSS signal uctuation problem while still maintaining the simplicity of VLP system, this paper proposes an interval analysis-based setmembership approach to improve the accuracy and stability of the RSS-based trilateration method. Interval analysis based methods have achieved promising results in parameter and state estimation tasks [4], as well as the mobile robotic localization and mapping area [5, 8, 9]. Our proposed method constructs con dence intervals from uctuated RSS measurements by utilizing a statistics method, and then characterizes the con dence region of the receiver's position with the Set Inversion Via Interval Analysis (SIVIA) algorithm. Afterwards, the nominal position is characterized by a weight-coe cients method.
This paper is organized as follows: Section 2 details the framework of our proposed method and implementation. Section 3 presents the simulation results with a comparison to least-square method. Section 4 concludes the paper and proposes the perspective of future work. 2
Proposed Set-membership method for VLP system 2.1
Denoting the RSS original data obtained during the ranging phase by Pr = (RSS1; RSS2; ; RSSKr )T . Traditional methods usually utilize a Gaussian lter to rstly remove the irregular values and then use the averaged RSS value is used for distance estimation. Our proposed method adopts the Bootstrap method to construct con dence intervals from the raw RSS measurement data. Firstly, we randomly sample with replacement from origin data Pr, which leads to a new series of measurement data
Pr1 = (RSS1(
Secondly, for each Bootstrap sample Pri, we can compute the Bootstrap statistics Pri by
Pri =
Kr j=1 1 XKr RSSj(i) where i = 1; 2; ; Kb. Then the distance between the VLC receiver and transmitter can be estimated by using the Optical Wireless Channel (OWC) model described in [2]. At the receiver side, the RSS value can be expressed as
Pr = (HLOS + HNLOS) Pt + wn where Pt and Pr are the transmitted and received signal power. HLOS and HNLOS represent the VLC channel gain of light-of-sight (LOS) and non-light-of-sight (NLOS) channel respectively. wn denotes the noise power at the receiver, i.e. the shot noise and thermal noise power. Since only the LOS channel signals are useful for RSS-based positioning algorithm, Eq. 3 can be rewritten as:
Pr = HLOS Pt + Pn
where Pn = HNLOS Pt + wn represents the total noise power that a ects the
RSS value of the LOS channel. According to Lambertian radiation model, the
typical VLC channel gain HLOS can be expressed as:
(
HLOS =
0 ( (m+1)Ar cosm(') cos( ) 0 2 d2 >
FOV
FOV where ' and d are respectively the radiation angle and distance between the receiver and transmitter. Ar is the e ective area of the receiver, and is the angle of light incident to the receiving surface of the detector. m represents the order of Lambertian emission and FOV is the eld-of-view of the receiver. The distance between the transmitter and receiver is thus given by
s d = m+3 (m + 1)ArHm+1 Pt
2 Pr d1 d2
dKb
[d] = [du1 ; du2 ]
For Kb Bootstrap statistics, we can obtain Kb estimated distance results which
can be sorted from small to large as follows:
(d1; d2; ; dKb ) is called the Bootstrap distribution. From this stage, we can
utilize the Bootstrap percentile formula to construct the con dence interval of
the distance estimation with a (
To deal with the random RSS uctuation problem, we propose to de ne the trilateration problem as a set inversion problem and utilize the SIVIA algorithm to compute the con dence region where the receiver is assumed to be located.
Let's consider three deployed VLC transmitters with xed coordinates, denoted by (txi ; tyi ) (i = 1; 2; 3). The distances between the transmitters and receiver are respectively d1; d2; d3. According to the trilateration positioning formulation, the feasible values of the receiver's position (rx; ry) can be con gured via the equations: 8>(rx <
(rx >:(rx tx1 )2 + (ry tx2 )2 + (ry tx3 )2 + (ry ty1 )2 + h2 = d2
1 ty2 )2 + h2 = d2
2 ty3 )2 + h2 = d2 3 where h is a constant, denoting the vertical distance between the receiver and VLC transmitters. By using the Bootstrap method, the estimated con dence interval of the distances [d1]; [d2] and [d3] between the receiver and each VLC transmitter can be calculated. The con dence region X is thus de ned as a set of all the feasible values which satisfy the constraints (18) and can be characterized by solving the set inversion problem:
X = f(x; y) 2 R2 j g(x; y) 2 [D]g = g 1([D]) where [D] is a three dimensional interval box [D] = [d1] R2 ! R3 is a vector function de ned as: [d2]
[d3] and g( ) :
8p(x
>
g(x; y) = <p(x
>:p(x
tx1 )2 + (y
tx2 )2 + (y
tx3 )2 + (y
ty1 )2 + h2
ty2 )2 + h2
ty3 )2 + h2
(
Confidence region 0.7 0.6 The SIVIA algorithm performs the inclusion test and bisection process recursively to verify that all the boxes in the solution list belong to X. As a result, it yields a list of non-overlapping boxes described in Fig. 1b. The green boxes are those which are validated and the yellows are the undetermined ones. The union of these non-overlapping boxes thus denotes the con dence region where the receiver is deemed to be located. The con dence region obtained via the SIVIA algorithm is a list of interval boxes, from which we can calculate the receiver's nal position estimation (we call it the nominal position). In our work, we propose to use the weighted arithmetic average method based on interval box dimensions to calculate the nominal position. Denote the list of solution boxes in the con dence region by L = f[x1]; [x2]; ; [xp]g, where p is the number of boxes in the con dence region. Then the nominal position is determined through
p X k=1 (tx; ty) k mid([xk]) (12) where i is the weight-coe cient, calculated by k = pvol([xk]) , vol([xi]) and P vol([xi]) i=1 mid([xi]) represent respectively the size and the center point of the ith interval box. 3 3.1
To test our proposed method, we consider a typical VLP scenario in simulation, i.e., a 4 m 4 m 3 m area. Three LEDs are deployed on the ceiling downward
Set-membership position Set-membership method
Least-square method reb m u N reb m u N (a) Illustration of positioning results
Positioning error (m) Positioning error (m)
(b) Histograms of the positioning errors
vertically. with coordinates (
1) + N (0; n2) Pn(t) = n(t) HLOS Pt (13) where is an arbitrary number between 0 1, N (0; n2) is Gaussian white noise, and n(t) denote the LOS channel noise factor. The noise vibration level depends on the n value: the bigger the n is, the larger noise uctuation will be. 3.2
Firstly, 100 unknown points are evenly distributed on the plane, their positions are estimated through the two positioning methods with the noise level n = 0:3. The results are described in Fig. 2a, the red stars are the reference positions, the blue circles are the nominal positions estimated by our proposed method and the yellow circles are the results obtained by least-square method. We utilize the Euclidean distance between estimated position and reference position to calculate the positioning error. Fig. 2b gives the statistics of the positioning error for the two methods. Our proposed method could achieve more stable positioning results when dealing with uctuated RSS measurements, as it can be seen from Fig. 2b, the position errors of our method are all below 0.3 m, while for the least-square method, the largest error reaches 0.54 m. Calculating an average value gives 0.11 m accuracy for our method and 0.16 m for the leastsquare method, showing that our proposed method expresses better performance in terms of accuracy.
A robust positioning scheme should accommodate interference in di erent noisy environments. In order to get a quantitative evaluation of our proposed 0.40 0.35 method, we perform the experiments with di erent level of noise uctuation by changing the standard deviation of white noise in Eq. 13. The average positioning error and the variance of the positioning accuracy are computed for di erent values of n ranging from 0.01 to 0.5. Fig. 3 presents the results obtained over 1000 randomly positioned points. The black error bars denote the standard deviations of the positioning errors. As we can see from the gure, when the noise uctuation is very small ( n = 0:01), both methods obtain almost the same results, the positioning error is about 0.5 cm. When the noise uctuation increases, the positioning errors of both methods increase as well. But the positioning error of least-square method increases more rapidly which means it is more vulnerable to the noise uctuation than ours. The standard deviation of positioning error (the black error bar on the gure) of our proposed method is also smaller than the least-square method at all noise uctuation levels, which demonstrates the positioning results obtained through our method are more stable than the least-square method. 4
This paper presents an interval analysis-based trilateration positioning approach in the scheme of set-membership. The proposed approach takes advantage and combines Bootstrap and SIVIA algorithm to compute a con dence region of the feasible positions from uctuated RSS measurements and gives a nominal position estimation with a weighting method. Simulation results demonstrate that our method expresses better performance in terms of accuracy and stability in comparison with least-square method, which indicates that our method is more tolerable to RSS signal uctuation. Future work will focus on validating the performance of the proposed method in real indoor environment.