=Paper=
{{Paper
|id=Vol-3097/paper38
|storemode=property
|title=Polynomial Fitting Functions for Visible Light Positioning Based on RSS With Tilted Receiver
|pdfUrl=https://ceur-ws.org/Vol-3097/paper38.pdf
|volume=Vol-3097
|authors=Cristóbal Carreño,Fabian Seguel,Patrick Charpentier,Nicolas Krommenacker
|dblpUrl=https://dblp.org/rec/conf/ipin/CarrenoSCK21
}}
==Polynomial Fitting Functions for Visible Light Positioning Based on RSS With Tilted Receiver==
https://ceur-ws.org/Vol-3097/paper38.pdf
Polynomial Fitting Functions for Visible Light
Positioning Based on RSS with Tilted Receiver
Cristóbal Carreño1 , Fabian Seguel2 , Patrick Charpentier1 and Nicolas Krommenacker1
1
Lorraine University, CRAN,CNRS UMR 7039, Nancy, France
2
Universidad de Santiago de Chile, Department of Electrical Engineering, Santiago, Chile
Abstract
In this work, we present the construction and comparison of three multivariate fitting functions to
model the relation between received signal strength (RSS) and tilted receiver angle for 2-D visible light
positioning systems. The performance of the different functions has been analysed by simulations for
a mobile node. The tilted angle has been obtained from an inertial measurement unit (IMU) sensor
considering his deviation from experimental measurements. This mismatch measurement is introduced
in the Monte-Carlo simulation to measure its impact on the position estimation. According to the results,
third order polynomial function overcomes first and second-order polynomial functions achieving a 0.16
m accuracy error.
Keywords
Visible Light Positioning, Tilted Photodiode, Polynomial Fitting, Inertial Measurement Unit, Received
Signal Strength.
1. Introduction
Positioning information has become an important issue every day. An example of this is the
Global Positioning System (GPS), and it is used for positioning and navigation services. GPS
achieves great performance in outdoor environments. Nonetheless, this technology faces several
issues when facing indoor environments. Satellites signals cannot penetrate the building’s
walls, producing distortion or shot down at communication link [1]. For solving this drawback,
several indoor positioning systems (IPS) had been developed in the last decade. To provide
IPS different technologies have been used Radiofrequency (RF), Ultrasound (US), and Visible
light. RF is widely used but it is affected by the multi-path effect indoor and his signal produces
interference with other RF signals, especially in areas as hospitals, airplanes or RF crowded.
The US is affected to multi-path, depend strongly on room temperature and the Doppler shift
effect causes several problems in the communication link. In recent years, the development
of Visible Light Communications (VLC) has encouraged the research on positioning systems
based on Optical Wireless Communications (OWC) technology. It has some advantages as No
interference in RF crowded areas, deployed lamps at buildings can be used as transmitters, low
energy consumption, and security aspects in indoor environments. These reasons have become
IPIN 2021 WiP Proceedings, November 29 – December 2, 2021, Lloret de Mar, Spain
Envelope-Open carrenom1@univ-lorraine.fr (C. Carreño); fabian.seguelg@usach.cl (F. Seguel);
patrick.charpentier@univ-lorraine.fr (P. Charpentier); nicolas.krommenacker@univ-lorraine.fr (N. Krommenacker)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
VLC an attractive technology for IPS systems. [2].
The IPS based on VLC referred to as Visible Light Positioning systems (VLP) can be classified
into two types according to the sensor used to capture visible light, namely, Image Sensor (IS)
or Photodiode (PD). There is another sensor, photoresistor (PR) can be used on IPS, an example
is in[3] but VLP advances have preferred PD over PR due to high data rates for communica-
tion proposes. Concerning IS sensors, they provide information from transmitters positioning
to visions analysis algorithm, but they have big limitations in data rate. On the other hand,
PDs provide information about optical power from lights and they have an easy hardware
implementation but they need a transimpedance amplifier (TIA), in contrast with PRs is not
needed. An example of going to work at a higher data rate of PD than IS is depicted in[4].
Another aspect of PD’s is orientation changes produce an error on positioning estimation for
VLP algorithms based on RSS because the trilateration technique depends on distance estimation
between transmitters and receiver. The distance can be estimated using regression functions
as shown in [5] that includes non-line-on-sight (NLOS) components, but without considering
the PD inclination effect. Another example is presented in [6] but under the perspective of
unknown transmitter orientation. Although the problem is the same, the angles transmitter
are fixed in contrast with the orientation of PD which is variable. Another case is presented
in [7] and his work is based on channel DC gain. Results demonstrate there exist changes at
distance estimation due to the mobile user’s dynamics. In [8] there is a mathematical analysis of
influence receiver’s orientation at a change of planar projection distances between transmitter
and receiver. The error in the estimated distance is corrected introducing a planar distances
bias from height and PD´s parameters. In [9, 10] is proposed a function to correct the received
power and then correct the estimated distance. These functions correspond to a correction
factor that depends on incident angle changes and is modeled by exponential functions that
include PD’s parameters. The problem with these functions is depending on the position of PD,
and this is normally unknown. An alternative way is to approximate the incidence angle factor
in the channel model and this is shown in [11], technique used there consists in constructs
the incidence angle factor through functions of first and second order, but they are focused
on incident angle and not of tilted angle. Another approach to overcome this drawback is to
solve an optimization problem that identifies the position of PD observing changes between
the ideal channel gain and the actual channel gain. The results are good but they still consider
the tilted angle is always fixed. A different point of view is present in literature by using a
multi-PD. In [12] four PDs are used to estimate the normal vector where the PDs are mounted.
Normal vector has been modeled by multivariate Gaussian distribution, but the problem is there
exists more information to process than in the case of one PD. Machine learning technique
is used too with multi-PD in [13], where an artificial neural network (ANN) and Weighted
K-Nearest Neighbour (WKNN) are programmed to recognize the patterns of RSS in each PD.
The receiver’s orientation is obtained through an inertial measurement unit (IMU). Then it
seems the orientation problems are solved, but the IMU sensor is very susceptible to external
forces, and magnetic fields for getting the angles, heuristic methods, complementary filtering,
and Extended Kalman filter are few techniques used to overcome those problems [14, 15].
The approach of our work besides the fact it considers an inclined receiver with tilt angle
obtained from a real inertial sensor(IMU) including its uncertainty in the measure. The behavior
of IMU has been characterized by variance and mean value of angle error. These values are used
to build a Gaussian distribution function and incorporated on Monte-Carlo simulation [16, 17],
with the proposal to train the polynomial functions for distance estimation and showing the
performance of the entire positioning system. The paper is organized as follows: Section II will
present the system model, the positioning system that includes, the fitting function definition,
the optimization problem, and the trilateration equations. Section III is dedicated to present the
parameters, IMU measurements distribution, and simulation results. Conclusions are presented
in section IV.
2. System Model
The system is composed of four Transmitters (Lamps) and one mobile receiver in a square room
with dimensions. Each lamps is separated from others to distance 𝐷𝑥 = 𝐷𝑦 and height is ℎ. The
room is depicted in figure 1.
𝑅𝑜𝑜𝑚 𝑥
𝐷𝑦 𝑅𝑜𝑜𝑚 𝑦
𝐷𝑥
Figure 1: Room scenario
2.1. Channel Model
VLC channel is modelled considering the line of sight (LOS) components from each LED 𝑗 in
some mobile node´s point 𝑖. For simulation proposes make it easier, we consider the received
power of PD only depends from the power of lamps and the noise present in receiver device
(leaving out frequency responses of PD and LEDs, PD construction materials, ambient light
effect, blockages scenarios or LED installation errors). The noise 𝑁 is additive Gaussian 𝒩 (0, 𝜎 2 )
including shot and thermal noises. Noise expressions are available in [18]. Total power received
corresponds to sum of power from every LED, this is showed in (1). 𝑅𝑝 is the responsivity of PD,
𝐻𝑖𝑗𝐿𝑂𝑆 is the DC gain of LOS channel and 𝑃𝑗 is the power transmitted for LED 𝑗. In (2) 𝐻𝑖𝑗𝐿𝑂𝑆 is
obtained from the incidence angle 𝜓𝑖𝑗 and transmitter angle 𝜑𝑖𝑗 , distance between transmitter and
receiver 𝑑𝑖𝑗 and constant 𝐶𝑖𝑗 . DC gain is valid just when the incidence angles is less than PD’s
field of view (FOV) angle Ψ𝑙 . 𝐶𝑖𝑗 depends of Lambertian order transmission 𝑚𝑙 , the area of PD 𝐴
and factors 𝐺(𝜓𝑖𝑗 ) and 𝑇𝑠 (𝜓𝑖𝑗 ) that represents the optical filter gain and the optical concentrator
in (3).
4 4
∑ 𝑃𝑖𝑗 = ∑ 𝑅𝑝 𝑃𝑗 𝐻𝑖𝑗𝐿𝑂𝑆 + 𝑁 (1)
𝑗=1 𝑗=1
𝐶𝑖𝑗 𝑚𝑙
𝐿𝑂𝑆 2 cos (𝜑𝑖𝑗 ) cos(𝜓𝑖𝑗 ) 0 ≤ 𝜓𝑖𝑗 ≤ Ψ𝑙
𝐻𝑖𝑗 = { 𝑑𝑖𝑗 (2)
0 elsewhere
(𝑚𝑙 + 1)𝐴
𝐶𝑖𝑗 = 𝐺(𝜓𝑖𝑗 )𝑇𝑠 (𝜓𝑖𝑗 ) (3)
2𝜋
The factor cos(𝜓𝑗𝑖 ) is the same that cos(𝜑𝑖𝑗 ) when the normal vector of PD is perpendicular to
ground plane. This is not the general case. We consider in this work a tilted PD, in consequence,
it is necessary to find the cos(𝜓𝑗𝑖 ) from tilted angle. This is possible from the expression in
(4).Let be define cos(𝜓𝑗𝑖 ) as the dot product between distance vector 𝑑⃗𝑗𝑖 and tilted PD normal
𝑁̂ 𝑖 , divided by product of their norms. In spherical coordinates, the PD normal vector can be
calculated from (5), where 𝛽 is the tilt angle and 𝛼 is the rotation angle. This work consider
𝛼 = 0∘ . The tilt angle 𝛽 is delivered by an inertial sensor, because the PD can not knows this
angle by itself. Figure 2 are showed the mentioned angles.
𝑑⃗𝑗𝑖 ⋅ 𝑁̂ 𝑖
cos(𝜓𝑗𝑖 ) = (4)
‖𝑑⃗𝑗𝑖 ‖2 ‖𝑁̂ 𝑖 ‖2
[sin(𝛽) cos(𝛼), sin(𝛽) sin(𝛼), cos(𝛽)]
𝑁̂ 𝑖 = (5)
‖[sin(𝛽) cos(𝛼), sin(𝛽) sin(𝛼), cos(𝛽)]‖2
Figure 2: Tilted Photodiode
2.2. Positioning System
The transmitter send power square waves (without zero values) to the receiver with frequencies
𝑓𝑚𝑗 . This waves are modulated by sine functions with carrier frequencies 𝑓𝑐𝑗 . The displacement
in frequency domain serves to mitigate the inter channel interference (ISI) and allowing to
separate the received signal in PD´s side from each LED. Once RSS is added with PD noise,
Receiver apply four bandpass filter centred at frequencies 𝑓𝑐𝑗 and bandwidth 𝐵𝑊 = 2𝑓𝑚𝑗 . A fast
Fourier transform is used to obtain the RSS from each transmitter, using the Parseval’s power
theorem. With the transmitter identification by frequency of separated signal, the position of
transmitter Tx𝑗 is obtained. After that, IMU sensor delivers the tilted angle of PD. A fit function
𝑓𝑗 (𝑣)̂ of five variables (four powers and one angle) represented by vector 𝑣⃗ are used to estimate
the distances to the LEDs from the PD 𝑑̂𝑗 , as showed in figure 3. Each 𝑑̂𝑗 is employed to calculate
the ratios 𝑟𝑗 = √𝑑̂𝑗2 − ℎ2 . The matrices 𝐴 in (6) and 𝐵 in (7) correspond to trilateration technique
and 𝑥𝑗 , 𝑗 = 1, ..., 4 𝑦𝑗 , 𝑗 = 1, ..., 4 are the LEDs coordinates. Solution is obtained through Least
Square Algorithm in (8).
x +
x +
+
x +
x +
Transmitter's side Receiver's side
Figure 3: Positioning system description
𝑥2 − 𝑥1 𝑦2 − 𝑦1
𝐴 = [ 𝑥3 − 𝑥1 𝑦3 − 𝑦1 ] (6)
𝑥4 − 𝑥1 𝑦4 − 𝑦1
(𝑟12 − 𝑟22 ) + (𝑥22 + 𝑦22 ) − (𝑥12 + 𝑦12 )
𝐵 = [ (𝑟22 − 𝑟32 ) + (𝑥32 + 𝑦32 ) − (𝑥12 + 𝑦12 ) ] (7)
(𝑟12 − 𝑟42 ) + (𝑥42 + 𝑦42 ) − (𝑥12 + 𝑦12 )
−1
𝑥⃗̂ = (𝐴⊤ 𝐴) 𝐴⊤ 𝐵 (8)
2.3. Distance Estimation
Distances calculation depend of the tilted angle of PD, which is delivered from IMU sensor
̂ , 𝑃𝑟2
and RSS. We define the vector 𝑣⃗ = (𝑣1 , 𝑣2 , 𝑣3 , 𝑣4 , 𝑣5 ) = (𝑃𝑟1 ̂ , 𝑃𝑟3
̂ , 𝑃𝑟4
̂ , 𝛽𝑖𝑚𝑢 ) that contains all
the information necessary for next step. To use all this data, we propose multidimensional
polynomial functions 𝑑̂𝑗 with argument vector 𝑣⃗. This functions can be constructed, with fist
order terms, second order terms and third order terms. Second and third order terms come from
the product combination among two or three variables. In (9) shows the third order polynomial
function and that contains all other low order terms. Every term is multiplied by constants.
There are first order constants 𝑎𝑖 , second order constants 𝑏𝑖𝑗 and third order constants 𝑐𝑖𝑗𝑘
.Respect the terms ∑ 𝑎𝑖 𝑣𝑖 corresponds to first order terms, ∑ 𝑏𝑖𝑗 𝑣𝑗 are the second order terms
and ∑ 𝑣𝑖 ∑ 𝑣𝑗 ∑ 𝑐𝑖𝑗𝑘 𝑣𝑘 are the third order terms. First and second order function are construct by
deleting the second and third order terms respectively.
5 5 𝑖 5 𝑖 𝑗
𝑑̂𝑗 (⃗𝑣) = 𝑎0 + ∑ 𝑎𝑖 𝑣𝑖 + ∑ 𝑣𝑖 ∑ 𝑏𝑖𝑗 𝑣𝑗 + ∑ 𝑣𝑖 ∑ 𝑣𝑗 ∑ 𝑐𝑖𝑗𝑘 𝑣𝑘 (9)
𝑖=1 𝑖=1 𝑗=1 𝑖=1 𝑗=1 𝑘=1
It is necessary to find the constants in (9) to obtain the fitting functions, this step is called
“Training Phase”. To do this, we reorganise the terms in matrix form (10). Vector 𝑔⃗ in (11) repre-
sents all polynomial coefficients and they are the targets. Matrix 𝐷 in (13) regroup all the terms.
This matrix changes depends on the considered function. Each element in matrix 𝐷 represents
the data obtain from vector 𝑣⃗, if we consider 𝑛 vectors 𝑣⃗(𝑛) , the Notation 𝑣1 (𝑛) represents nth fist
order terms, 𝑣1 ⋅ 𝑣1 (𝑛) … 𝑣5 ⋅ 𝑣5 (𝑛) are the second order terms and 𝑣1 ⋅ 𝑣1 ⋅ 𝑣1 (𝑛) … 𝑣5 ⋅ 𝑣5 ⋅ 𝑣5 (𝑛) the
third order terms in 𝐷. 𝐹 contains the values of 𝑑𝑗 and they represent the perfect distance from
PD to Tx𝑗 . With vectors 𝑔⃗, 𝐹 and matrix 𝐷, it is possible to solve the optimisation problem in
(14) by Non linear least Square (NLLS) algorithm.
𝐷 ⋅ (𝐶)𝑇 = 𝐹 (10)
𝑔⃗ = [𝑎0 𝑎1 ⋯ 𝑎5 𝑏11 ⋯ 𝑏55 𝑐111 ⋯ 𝑐555 ] (11)
𝑇
𝐹 = [𝑑𝑗 (⃗𝑣)(1) 𝑑𝑗 (⃗𝑣)(2) ⋯ 𝑑𝑗 (⃗𝑣)(𝑛) ] (12)
1 𝑣1 (1) ⋯ 𝑣5 (1) 𝑣1 ⋅ 𝑣1 (1) ⋯ 𝑣5 ⋅ 𝑣5 (1) 𝑣1 ⋅ 𝑣1 ⋅ 𝑣1 (1) ⋯ 𝑣5 ⋅ 𝑣5 ⋅ 𝑣5 (1)
⎡ ⎤
⎢1 𝑣1 (2) ⋯ 𝑣5 (2) 𝑣1 ⋅ 𝑣1 (2) ⋯ 𝑣5 ⋅ 𝑣5 (2) 𝑣1 ⋅ 𝑣1 ⋅ 𝑣1 (2) ⋯ 𝑣5 ⋅ 𝑣5 ⋅ 𝑣5 (2) ⎥
𝐷=⎢ ⎥ (13)
⎢⋮ ⋮ ⋮ ⋮ ⋮ ⎥
(𝑛) ⋯ 𝑣 (𝑛) 𝑣 ⋅ 𝑣 (𝑛) ⋯ 𝑣 ⋅ 𝑣 (𝑛) 𝑣 ⋅ 𝑣 ⋅ 𝑣 (𝑛) ⋯ 𝑣 ⋅ 𝑣 ⋅ 𝑣 (𝑛)
⎣1 𝑣 1 5 1 1 5 5 1 1 1 5 5 5 ⎦
𝑛
2
minimize ∑ (𝑑̂𝑗 − 𝑑𝑗 )
𝑔⃗ 𝑖=1 (14)
subject to 𝑑̂𝑗 ⪰ 0
3. Simulation and Results
The simulation procedure consists of estimation for 2D position of a mobile node in a room of
3x3x3 𝑚. The parameters used for the channel model, transmitters, and receiver are exposed in
table 1. Other parameters as noise, PD responsivity and so on, are available in [19]. The reason
to consider a FOV value different from those exposed in most cases in literature is to regard
a harder situation for fitting functions with a lower FOV value. This section is divided into
three-part. Each part represents one step done inside the simulation process.
Table 1
System parameters
Parameter Value
Semi-angle Half power 70∘
Lambertian Order 1
Source power 10 (𝑊)
Area of PD 10−4 (𝑚2 )
𝑇𝑠 (𝜓𝑖𝑗 ) 1
Half angle FOV receiver 60∘
Room dimensions 3 × 3 (𝑚)
LED height 3 (𝑚)
𝑔(𝜓𝑖𝑗 ) 1
Number of LEDs 4
PD height 1 (𝑚)
x-y Grid step 0.1 (𝑚)
3.1. Fitting Functions Performance
First step in simulation process consisted on finding the coefficients of polynomial functions.
As we showed before, this coefficients were calculated from NLLS algorithm. For forming the
matrix 𝐷 a mobile node is positioned in 361 different points over the grid, due to grid step is
0.1 𝑚. This value is considered at the most cases of VLP simulations on literature). For each grid
point, the used tilted angles were 𝛽 = [−15∘ , −10∘ , −5∘ , 0∘ , 5∘ , 10∘ , 15∘ ]. This means, we used a
total of 2527 point cases and represent all possibilities of vector 𝑣⃗. For each angle case of vector
𝑣⃗, its available the true distance 𝑑𝑗 and all the true distances conform the vector 𝐹. With Matrix
𝐷 and vector 𝐹, ⃗ are introduced to polynomial fitting matlab function available in [20]. We took
the 60% of available data in a randomly form. The training step was made with perfect tilted
angles. After the training step, we validated three fitted function (linear, quadratic and cubic).
With 40% of remaining data a validation step was done. In table 2 is possible to observe the
performance from three fitted functions. The error corresponds to absolute difference between
the estimated distance and correct distance. Third order function presents a mean and deviation
error less than 0, 1 𝑐𝑚. Second order polynomial present a best performance than first order, but
all the proposer polynomial have error less than 5 𝑐𝑚 in distance estimation.
Table 2
Training Step: Error of distance estimation by fitting functions
Index Error(m) 𝑑̂1 Error(m) 𝑑̂2 Error(m) 𝑑̂3 Error(m) 𝑑̂4
Mean 1st 0, 0433 0, 0425 0, 0434 0, 0426
Std 1st 0, 0308 0, 0310 0, 0313 0, 0323
Mean 2nd 0, 0105 0, 0100 0, 0098 0, 0100
Std 2nd 0, 0107 0, 0103 0, 0105 0, 0103
Mean 3th 7, 406 ⋅ 10−4 7, 559 ⋅ 10−4 9, 5 ⋅ 10−4 8.88 ⋅ 10−4
Std 3th 0, 0021 0, 0012 0, 0019 0, 0020
Once the fitted functions are ready, the next step is test them in VLP system. To do this, the
same 361 grid points are used, but now, PD noise is included. For each grid point simulation is
repeated 10 times include the noise effect. The results are summarised in table 3. Again the
best performance corresponds to third order polynomial with errors less than 2 𝑐𝑚.
Table 3
Positioning Errors with perfect angle
Index Mean(m),15∘ Std(m),15∘ Mean(m), 10∘ Std(m),10∘
1st 0, 0324 0, 0337 0, 0249 0, 0258
2nd 0, 0273 0, 0326 0, 0187 0, 0196
3th 0, 0117 0, 0094 0, 094 0, 0068
In figures 4a and 4b are plotted the cumulative distribution of positioning. The behaviour in
first and second order functions are similar, but third order presents less CDF. In figures 5a and
5b are plotted the positing distribution of estimated coordinates for the best case: third order
function.
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25
(a) 𝛽 = 15∘ (b) 𝛽 = 10∘
Figure 4: CDF for fit functions with exact 𝛽
3.2. Effect on mismatch tilt measurement
In contrast with previous scenario, now we consider a deviation in tilted angle to discover
his influence on polynomial functions and VLP system. To characterise the behaviour of non
perfect tilted angle, we used an IMU sensor model MPU9250. An Arduino Nano platform was
used to recorder the IMU data, process it and calculate the orientations angles. For calculations
a code library was employed available in [21]. Samples recorder consisted in hold the sensor
with a hand (as a cellphone) and tilted the sensor up to get 15∘ . This angle was measured with a
standard inclinometer OOTDTY-8YY00146. Figure 6 shows the distribution of non perfect angle
2.5
2.4
2.2
2 2
1.8
1.6
1.5
1.4
1.2
1 1
0.8
0.6
0.5
0.4
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.5 1 1.5 2 2.5
(a) 𝛽 = 15∘ (b) 𝛽 = 10∘
Figure 5: Estimated position with 3th order function 𝛽.
“+”:Ideal Position “•”:Estimated Position
around 15∘ . The natural hand move delivers a standard deviation of 1.5∘ and a mean value of
0.5∘ . Because the samples form a normal distribution, this model is used to provided the non
perfect angle to simulation step.
Figure 6: Tilted PD angle deviation distribution 𝜎 = 1.5∘ ,𝜇 = 0.5∘
In table 4 there are the error of positioning system consider a non perfect tilted angle. The third
order once more has the best performance, but the difference between the other polynomials has
become less with an mean error up to 2 𝑐𝑚. Figures 7a and 7b shows the CDF positioning error
for non perfect tilted angle. Plots deliver a change on third order CDF when 𝛽 = 15∘ , instead of
𝛽 = 10∘ three polynomials have a similar behaviour. The most evident change can be appreciate
from figure 4a and 4b against figures 7a and 7b.With non exact 𝛽 appears a displacement on red
dots. If we focus on right corners in figure 8a appears a dispersion behaviour and this is due to
the loss Line-on-sight link from the transmitter positioned at the right side.
Table 4
Positioning Errors with non perfect angle
Index Mean(m),15∘ Std(m),15∘ Mean(m), 10∘ Std(m),10∘
1st 0, 0384 0, 0374 0, 0323 0, 0301
2ns 0, 0388 0, 0381 0, 0289 0, 0273
3th 0, 0279 0, 0244 0, 0253 0, 0217
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25
(a) 𝛽 = 15∘ (b) 𝛽 = 10∘
Figure 7: CDF for fit functions with non exact 𝛽
3.3. Comparing Results
With all of previous results, in table 5 shows the percent difference between perfect tilted angle
and non perfect case. Negative percent means there was increment respect to the ideal case.
We can infer that all polynomial are not robust to non perfect tilted angle, but the most sensible
functions is third order polynomial. In fact, this function is the most accurate to estimate
distance and producing less positioning error but is the less robust too 𝛽 changes.
Table 5
Mean and Standard deviation error according fitting functions
Index Mean(%),15∘ Std(%),15∘ Mean(%), 10∘ Std(%),10∘
Δ1st −18.518 −10.979 −29.718 −16.667
Δ2nd −42.124 −16.871 −54.545 −39.285
Δ3th −138.461 −159.574 73.085 −219.117
2.5 2.5
2 2
1.5 1.5
1 1
0.5 0.5
0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5
(a) 𝛽 = 15∘ (b) 𝛽 = 10∘
Figure 8: Estimated position with 3th order function non exact 𝛽.
“+”:Ideal Position “•”:Estimated Position
4. Conclusions and Future Works
In this work we proposed three polynomial fitting functions to estimate the distance from RSS
in tilted PD device. The functions works under tilted angles with certain deviations. Movements
could be produced by the natural move of human body, irregular plane, etc. The deviation angle
has been characterised through samples recorder from real IMU sensor. The performance of
these functions in VLP system has been tested by Monte-Carlo simulations. Results shows that
it is possible to construct multi variable polynomial functions to approximate power versus
distance with tilted receiver. Also, there are non robust behaviour from polynomial functions
performance against the deviation of non perfect tilted angle, and therefore an affected perfor-
mance on entire VLP system.
This approach could be tested in a real environment, using a PD located over the human’s
body, for example in a hospital, but with drones is possible to assure the 361 points of the grid
due to great stability inside an industrial factory room. The coefficients of fitting functions will
depend on the environment, existing blockages, multi-path, error in angle measurements (tilt
and rotation), and Transmitter’s coverage. All these factors will affect the training phase of
fitting functions. In future works we pretend to extend the polynomial functions considering
rotations angle (It is very important in real environments as we mentioned before) and test
other kinds of fitting functions as exponential, piece-wise, and machine learning, all of this to
improve the robustness against rotation changes over real experiments with embedded systems.
5. Acknowledgments
The authors acknowledge the financial support of Beca Doctorado en el Extranjero (Becas Chile)
2020 ANID (PFCHA) 72210523 and the project FONDECYT REGULAR 1211132 at the University
of Santiago of Chile.
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