sensors
Article
Impact of Transmitter Positioning and Orientation Uncertainty
on RSS-Based Visible Light Positioning Accuracy †
Neha Chaudhary 1, * , Luis Nero Alves 1
1
2
*
†
Citation: Chaudhary, N.; Alves, L.N.;
Ghassemlooy, Z. Impact of
and Zabih Ghassemlooy 2
Instituto de Telecomunicações and Departamento de Electrónica, Telecomunicações e Informática,
Universidade de Aveiro, 3810-193 Aveiro, Portugal; nero@ua.pt
Optical Communications Research Group, Faculty of Engineering and Environment, Northumbria University,
Newcastle upon Tyne NE1 8ST, UK; z.ghassemlooy@northumbria.ac.uk
Correspondence: neha.chaudhary@ua.pt
Chaudhary, N., Alves, L.N., Ghassemlooy, Z. Impact of transmitter positioning uncertainty on RSS-based
visible light positioning accuracy. In Proceedings of the 12th International Symposium on Communication
Systems, Networks and Digital Signal Processing (CSNDSP), Porto, Portugal, 20–22 July 2020.
Abstract: This paper present simulation-based results on the impact of transmitter (Tx) position
and orientation uncertainty on the accuracy of the visible light positioning (VLP) system based on
the received signal strength (RSS). There are several constraining factors for RSS-based algorithms,
particularly due to multipath channel characteristics and set-up uncertainties. The impact of Tx
uncertainties on positioning error performance is studied, assuming a statistical modelling of the
uncertainties. Simulation results show that the Tx uncertainties have a severe impact on the positioning error, which can be leveraged through the usage of more transmitters. Concerning a smaller
Tx’s position uncertainty of 5 cm, the average positioning errors are 23.3, 15.1, and 13.2 cm with the
standard deviation values of 6.4, 4.1, and 2.7 cm for 4-, 9-, and 16-Tx cases, respectively. While for a
smaller Tx’ orientation uncertainty of 5◦ , the average positioning errors are 31.9, 20.6, and 17 cm with
standard deviation values of 9.2, 6.3, and 3.9 cm for 4-, 9-, and 16-Tx cases, respectively.
Transmitter Positioning and
Orientation Uncertainty on
RSS-Based Visible Light Positioning
Keywords: localization; VLC; visible light positioning; received signal strength; localization algorithm; Tx’s uncertainty
Accuracy . Sensors 2021, 21, 3044.
https://doi.org/10.3390/s21093044
Academic Editor: Yang Yue
Received: 6 April 2021
Accepted: 24 April 2021
Published: 27 April 2021
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1. Introduction
The demand for highly precise indoor positioning (IP) systems is growing rapidly
due to its potential in the increasingly popular techniques of Internet of Thing (IoT), smart
mobile devices, and artificial intelligence. Consequently, IP becomes a promising research
domain that is getting wide attention due to its benefits in several working scenarios, such
as, industries, health sectors, indoor public locations, and autonomous navigation [1]. The
traditional positioning methods, which depend on satellites such as the global positioning
system (GPS), is common nowadays for outdoor positioning; however, it is not well-suited
for indoor environments. This is because the GPS signals from the satellites suffer from
high penetration loss and multipath fading due to building walls. Some other systems also
have been proposed for IP, for example, Bluetooth, ultrasound, ultra-wideband (UWB),
wireless-Fidelity, radio frequency identification (RFID), and radio-frequency (RF)-based
techniques [2–4]. For instance, the UWB technology transmits short RF pulses with a low
duty cycle, which provides precise localization and tracking of mobile devices in indoor
environments [4]. Despite the advantage of precise localization, the UWB technology is
still not perfect for IP systems, and it has not been embraced widely because of its cost,
complexity, and need for synchronization between transmitters (Txs) and the targets [5].
Moreover, these RF-based systems may not be suitable in RF restricted areas, such as
hospitals due to the RF-induced interference [2].
4.0/).
Sensors 2021, 21, 3044. https://doi.org/10.3390/s21093044
https://www.mdpi.com/journal/sensors
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Another developed technology that makes use of the pre-installed lighting infrastructure is visible light positioning (VLP). VLP is based on visible light communications
(VLC), which is license-free, and free from RF induced electromagnetic interference-free,
thus making it ideal in many applications including hospitals. In addition, VLC uses the
pre-existing light-emitting diodes (LEDs) infrastructure as a Tx which has the ability to provide illuminance and communication simultaneously [6]. VLP is an emerging technology
promising high accuracy, high security, low deployment cost, shorter time response, and
low relative complexity when compared with RF-based positioning [7].
Existing VLP approaches estimate the position of a receiver (Rx) based on several characteristics of light including received signal strength (RSS) [8], angle of arrival (AOA)/angle
difference of arrival (ADOA) [9], image [10], and time of arrival (TOA)/time difference
of arrival (TDOA) [11]. It has been established that, VLP based on AOA/ADOA, image,
and TOA/TDOA require auxiliary devices to capture angle/image/time. On the contrary, the RSS approach can be achieved by utilizing a single photodiode (PD) without the
need of any additional auxiliary devices, which makes the RSS-based approach the most
well-known method for VLP [12].
Recent research works have addressed the impact of (i) LED power uncertainty [13];
(ii) reflections from walls and objects within the transmission paths [14]; and (iii) noise
on the positioning performance [15]. The multipath channel has a direct influence on the
model for the estimation of the received signal power, which has been addressed previously
using machine learning algorithms [16]. The Tx and the Rx design specifications, such as
the Tx beam width and its tilting angle have been investigated before. For instance, in [13],
the impact of LED output power uncertainty on the accuracy of the RSS-based VLP system
was explored, with the maximum error of 17 and 40 cm for a tolerance (possible variations)
value of 5% and 20%, respectively. The performance analysis of various VLP systems
relies on line-of-sight (LOS) transmission path, which can underestimate the achievable
error bounds, due to the fact that a real scenario will definitely include non-line of sight
(NLOS) paths [14]. Therefore, NLOS transmission paths should not be neglected. In [15],
an RSS-based VLP system using received optical power from the emitting LEDs was
investigated considering signals from both LOS and NLOS paths. The results revealed that,
the positioning accuracy of <10 cm on average can be achieved at a signal-to-noise ratio
(SNR) of >12 dB.
The location of the Rx can be estimated by using different estimation methods, such
as linear least square (LLS) and non-linear least square (NLLS) [14]. In [17], a polynomial
regression-based method was used to investigate the accuracy of an RSS-based VLP system
along with LLS and NLLS estimation algorithms. The results revealed that, the positioning
error εp was <0.6 m by using the regression approach, which is much lower than other
traditional methods. In [18], the impact of the Tx’s orientation (i.e., the tilting angle) on
the positioning accuracy of the RSS-based VLP system was studied. Another estimation
method, i.e., normalized least square estimation was utilized to estimate the positioning accuracy. In [19], an artificial neural network (ANN) based 4-LED VLP system was proposed
to reduce εp for the LOS path, which is affected by the random and unknown static Tx tilt
angle with a maximum variation of 2◦ . It was revealed that ANN achieved localization
errors below 1 cm. In general, the positions and orientations of Txs may not be symmetrical,
which (i) depends on the indoor environments such as museums, galleries, train stations,
shopping centers, etc., where lights are pointing in different directions; and (ii) can change
when replacing lights, carrying out maintenance, etc. Therefore, the random variations in
the Tx’s orientation will lead to the random errors in VLP systems, which requires further
studies. The impact of Tx’s position and its orientation uncertainties on the positioning
estimation have not yet been systematically explored, which is the objective of this paper.
In this paper, we investigate the impact of Tx’s position and its orientation uncertainties on εp of the RSS-based VLP system under multipath reflections. The uniform
distribution of light inside the illuminated place is a necessity in indoor environments. As
a result, lighting uniformity becomes a vital aspect for a well-lit environment. Moreover,
Sensors 2021, 21, 3044
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both lighting uniformity and the positioning performance are related to the Tx’s positions
and the Lambertian half-power angles (HPA). Therefore, in this work we investigate (i)
how the uniformity of light in the room changes for different HPA and the Tx positions; and
(ii) the impact of Tx position and its orientation uncertainties on the positioning accuracy
considering the optimized Tx positions from a lighting uniformity perspective. This works
focused only on the LED uncertainties caused during installation and is an extension of our
previous work [20], where the problem of Tx’s orientation uncertainty was not considered.
The rest of the paper is organized as follows, Section 2 introduces the set-up of
the system and position estimation approaches in detail. The impact of different set-up
uncertainties and uniformity are presented in Section 3. In Section 4, a discussion of the
simulation results attained is made, followed by the final concluding remarks in Section 5.
2. VLP System Modelling
2.1. Channel Model
Our proposed VLP system is composed of a PD as an Rx, which is placed on the
ground and a number of LEDs as the Txs (i.e., 4-, 9-, and 16-LED) that are installed on
the ceiling of the room as depicted in Figure 1. The field of view (FOV) and the detection
area Ar of the PD are 70◦ and 10−4 m2 , respectively. All Txs are located at the same height
h from the ground level and the coordinates of kth Tx (k = 1, . . . , K) is (xk , yk , zk ), where
K is the total number of Txs. The Rx coordinate is denoted by (xr , yr , zr ). The position
and orientation of the Txs is best illustrated by the Txk , with the coordinate of (δx, δy) and
the angles of α, β, and γ. Both δx and δy, and α, β, and γ, are assumed to be Gaussian
variables with N(0, σ2 ) and N(0, ζ2 ) probability distribution functions, respectively. The
distance between the Txs relies on the lighting uniformity considerations that are described
in a later section. An empty room is considered as a reference to study the impact of Tx’s
position and its orientation uncertainty on the positioning accuracy. In this work, both
LOS and NLOS transmission paths are assumed between the Txs and the Rx. However, for
the NLOS path, we only consider the first reflection due to the fact that the second order
reflections have much reduced intensities and therefore can be neglected [21]. Each Tx
broadcast a unique 2-bit ID information, which is encoded and modulated using on-off
keying (OOK), which allows separation at the Rx using a correlation method that can be
received at the Rx in advance of location identification.
Figure 1. Overview of the simulation set-up.
(
(
)
=
)
+ 1 cos ( )
‖ ‖
2
( )
( ) ( ),
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The total received optical power Pr at the PD comprises the power from the LOS and
NLOS paths, as given by:
Pr =
∑ Pr(LOS) + ∑ Pr(NLOS) ,
(1)
where Pr(LOS) and Pr(NLOS) represent the received power for LOS and NLOS, respectively.
Generally, the SNR will be high (i.e., >20 dB) in standard VLC, which could be considered
noise-free in normal cases [22]. Thus, in this study, no actual noise (e.g., shot noise or
thermal noise) is considered as a source of “measurement noise”, to unambiguously
evaluate the effect of the Tx’s position and its orientation uncertainty.
2.2. RSS-Based Positioning
Using the RSS algorithm, Pr(LOS) for the LOS path can be expressed as [22,23]:
∑ Pr(LOS)
K
m+1
= ∑ Pt Ar R
2π
k =1
where
m=−
cosm (ωk ) cos( ϕ)
k d k k2
Ts ( ϕ) g( ϕ),
ln(2)
,
ln cos Θ 1
(2)
(3)
2
where Pt is the transmit power, dk is the distance between kth Tx and the Rx, K is the total
number of Txs, and Θ1/2 is the light source irradiance half-power angle (HPA) [22]. ωk
and ϕ are the irradiance angle from the kth Tx to the Rx and the receiving incident angle,
respectively. Ar and R are the PD’s active area and responsivity, respectively. Ts ( ϕ) and
g( ϕ) are the gains of the optical filter and the concentrator at the Rx, respectively. Note,
Ts ( ϕ) and g( ϕ) are set to unity.
Considering the NLOS path, i.e., the first-order reflection, the total received power
can be expressed as [22]:
∑ Pr(NLOS)
cosm (ωk,w ) cos( ϕk,w )
m+1
= ∑ ∑ ρPt Ar R
Ts ( ϕw,r ) g( ϕw,r ) cos(ωw,r ) cos( ϕw,r ),
Aref
2π
π (kdk,w kkdw,r k)2
k =1 wall
K
(4)
where dk,w , ϕk,w , and ωk,w are the distances, receiving incident angle, and the irradiance
angle between the kth Tx and the reflective area, respectively. dw,r , ϕw,r , and ωw,r are the
distances, receiving incident angle, and the irradiance angle between the reflective area and
the Rx, respectively, see Figure 2. ρ is the reflection coefficient that relies on the reflective
surface material, and Aref is the reflection area. Pr(NLOS) for the signals from the NLOS
paths is obtained based on Matlab [22]. For each Tx, we integrate (4) over all the walls
accompanied by the assumption of a grid area with a resolution Aref of 0.1 m.
Figure 2. Overview of the visible light channel.
‖ ‖ =‖
‖ −
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2.3. Distance Estimation Using Polynomial Regression
The block diagram of the proposed VLP system is shown in Figure 3. In the case of
LOS, the horizontal distance can be computed as krk k2 = kdk k2 − h2 and therefore can be
determined from (2) as given by:
‖ ‖ =‖ ‖ −
v
u
! 2
u
u (m + 1) Ar R Pt (h)m+1 m+3
− h2 ,
rk = t
( +2πP
1)
( )
r
(
LOS
)
,k
=
− ,
2 ( ),
(5)
where h is the vertical distance between the Tx and the Rx. Pr(LOS),k is the LOS received
(
), stated in terms of the
power at the Rx from the kth Tx. In (5), the cosine terms are clearly
set-up geometry, as h/kdk k. However, in case of NLOS links, high errors are introduced
‖ presence
‖
in the channel due to ⁄the
of reflections [17,24], thus, the above method fails to
determine the distance. Another method that can be used is to employ a polynomial fitted
model for the power-distance relationship [25]. In this particular case, the relation between
dk and Pr, k for the kth Tx is represented as:
,
dk = α0 + α1 Pr,k + α2 ( Pr,k )2 + . . . + αs ( Pr,k )s
(6)
=
+
+ ⋯+
, +
,
,
where Pr, k is the total received power at the Rx from the kth Tx, and αs represent the
,
coefficients of the fitted polynomial. Note, dk is computed using (6), which is later employed
to determine rk .
LED Tx
Modulation
Total power received,
PD
Reverse process
Ideal Tx
coordinates
Optical wireless
channel (LoS
and NLoS)
at
Polynomial regression
fitting model
Compute distance,
Data encoding
Direct process
LED ID
generation
LED Luminaire
Transmitter side
Tx’s position or
orientation
uncertainty
Position estimation
using NLLS
Estimated
position of Rx
)
(
RSS
Performance
metric
Receiver side
Figure 3. Block diagram of the proposed system.
We estimated the best polynomial fitting using 3600 data points extracted from channel
simulation, see Figure 4. Note that, the 3600 data points are considered as 3600 different
locations each within the room. The best-fitted solution was exhibited by analyzing the
value of R2 (coefficient of determination) for different orders, and a fourth order polynomial
was selected with a value of R2 of 0.8942. Note, the polynomial fitting is not highly accurate.
This is because of the data points considered within the entire room for both LOS and
NLOS paths. At the center of the room, the impact of NLOS is negligible when compared
to the regions near walls and corners. As such, fitting all the data will be dominated by the
data near walls and corners. However, it shows the best fitted solution considering that
the data points corresponding to lower received power have higher contributions to the
error compared with the data points in the center of the room. The values of polynomial
coefficients are listed in Table 1.
Sensors 2021, 21, 3044
dk (m)
6 of 16
Figure 4. Estimation using the polynomial fitting method.
Table 1. The value of polynomial coefficients for the best polynomial fitting.
Polynomial Coefficient
Value
α0
α1
α2
α3
3.96
×
10
−7.34
×
10
8.85 ×8.85
10 × 10−9.93
×
10
6
5
4
−9.93 × 10
−7.34 × 102
3.96 × 10
α4
7.4
2.4. Estimation Using Nonlinear Least Squares
,
,
The relation between the Tx coordinates (xk , yk ) and the Rx coordinates (xr , yr ) are
described using the following equations:
( − ) +( − ) =
2
2
(( xr−− x1)) ++((yr−− y1)) == r1 2
2
( xr − x2 ) + (⋮yr − y2 )2 = r2 2,
,
(7)
( − ) + ( .. − ) =
.
( xr − x K )2 + ( yr − y K )2 = r K 2
= ,
where K is the total number of Txs. NLLS estimation can be utilized to estimate the target
e = [ xe, ye] that minimizes a
location, in which the solution can be estimated by attaining X
cost function given by [15]:= ∑ ( − ) + ( − ) −
e=
Q
∑
i
q
( xr − x k )2 + ( yr − y k )2 − r k
2
.
(8)
e by employing the trust-region reflective
An iterative procedure is utilized to estimate X
e 0 , followed by
algorithm [26]. In this algorithm, first, an estimate is introduced as X
e
computing the corresponding cost function Q0 . Next, several points in the neighborhood
e1 is selected as X
e 0 are replaced in (8), and the one that minimizes the cost Q
e 1 . The
of X
e
Rx coordinates X will eventually be obtained following several iterative steps to ensure
e In the proposed system, the initial value for X
e 0 is estimated using a
convergence of Q.
linear least square approach.
2.5. Performance Metrics
εp is assumed to be a random variable (as it may rely on the uncertainties, i.e., the Tx’s
position or its orientation uncertainties, estimation process, or noise); thus, it is reasonable
to use the standard statistical analyses to access error performance. Here we use the
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εε
probability distribution function (PDF) and the cumulative distribution function (CDF), as
a mean to calculate the 95% quantile on εp . Hence, the PDF and CDF are composed of the
spatial distribution of εp within the entire room. The
ε PDF of εp is defined as:
ε
#ε p ε p − p ≤ ε
f ε p ( p) =
lim #ε ε − ≤
,
N
N lim
→∞
( )=
,
→
ε→
→ 0
ε
(9)
whereεε defines an error interval centered around p, and N is the number of samples. The
ε |ε -p|
ε < ε. The limiting
cardinal operator # signifies the counting of occurrences were
p
process is naturally implied by the discrete nature of the simulation. The CDF can be
expressed as:
=
FCDF ((p)) =
Zp
0
f ε p (( p))dp..
(10)
ε
Figure 5 depicts the CDF against εp for NLLS estimation with and without polynomial
ε
regression. The maximum εp values estimated by NLLS and NLLS with polynomial regression are 0.6 and 1.57 m, respectively. It is observed that, there is an evident improvement
in the positioning accuracy by using NLLS estimation with polynomial regression for
power-distance modeling. Therefore, the polynomial regression can improve the accuracy
of position estimation without the inclusion of high complexity algorithms. However,
there are some limitations of the polynomial regression; for instance, the coefficients of the
polynomial model must be provided along with the Tx’s positions in practical scenarios.
Moreover, the polynomial model can deal with the empty and non-empty rooms with the
fixed furniture and objects, but with no user’s mobility.
100
80
60
40
20
NLLS
NLLS with Polynomial Regression
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
εp (m)
Figure 5. CDF of theε εp computed by NLLS and NLLS with polynomial regression estimation.
3. Set-Up Uncertainties
The major challenge in studying the Tx’s position or orientation uncertainties occurs in
simulation, i.e., we must use random Tx positions in order to investigate the uncertainties.
This raises the necessity to re-simulate the channel for each iteration (new random TX
position). Give that the channel estimation is by itself complex and time-consuming, this
problem becomes complex. We have developed two approaches for both Tx’s position and
its orientation uncertainties to fully analyze the problem, which is termed direct and reverse
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processes. The reason for these designations is straightforward. The direct process entails
the estimation of the position given the effect of the uncertainties on the channel. It is in this
sense, complex and time-consuming. In the reverse process, the channel is assumed fixed
and the effects of the Tx uncertainties are coupled directly into the estimation equations,
this avoids the channel simulation at each set of newly generated Tx positions.
3.1. Uncertainties of the Tx’s Position
For Tx’s position uncertainties, in the direct process, we assume that the channel is
re-stimulated for each newly generated set of Tx positions. The Tx positions are composed
by the ideal position (fixed) and a random perturbation in two-dimensions. While in the
reverse process, we assume that the channel is fixed for a set of ideal Tx positions, and the
uncertainties are directly coupled into the estimation Equation (7). To illustrate the concept,
let us consider one of the terms in (7), in the direct process, the effect of the uncertainties is
on the right side of the equation, as follows:
( xr − xk )2 + (yr − yk )2 = f R ( xk + δx, yk + δy),
(11)
where (xr , yr ) and (xk , yk ) are the coordinates of the Rx and the Tx (ideal positions), respectively. (δx, δy) express the uncertainty at the Tx coordinates. Both δx and δy are assumed to
be Gaussian variables with N(0, σ2 ). The channel needs to be re-simulated at each iteration
in order to obtain fR .
In the reverse process, it is not required to re-simulate the channel at each iteration as
the uncertainties are part of the Tx position, as follows:
( xr − xk − δx )2 + (yr − yk − δy)2 = f R ( xk , yk ).
(12)
Subtracting (11) and (12) and extracting the mathematical expectation of both sides gives:
h
i
Ur (σr ) = E δx2 + δy2 − 2δx ( xr − xk ) − 2δy(yr − yk ) ,
(13)
where σr is the Tx’s uncertainty of the reverse process. The reverse process has an expectation as stated in (13), which is given by (after solving (13)):
Ur (σr ) = 2σr 2 .
(14)
Performing the same analysis for the direct process we have:
Ud (σd ) = E[ f R ( xk , yk ) − f R ( xk + δx, yr + δy)].
(15)
Here, the value of Ud is estimated using numerical simulation, where σd is the Tx’s
uncertainty of the direct process. Assuming that, the reverse and direct processes are
equivalent, the expectation values Ud and Ur must be equal. In this case, it is possible to
infer the relation between σd and σr . This relation can be estimated assuming that the Rx is
placed at the center of the room and a single Tx is placed in a uniform grid of positions for
the whole room. Using mathematical simulation, we have computed the values of Ud (σd )
for different values of σd , see Figure 6. Note, the two processes generate uncertainties, i.e.,
Ud and Ur and these processes would be equivalent if these uncertainties are the same for
certain values of σd and σr . This indicates that, the uncertainties from the reverse process
can be generated with the knowledge of the direct process as given by:
σr2 =
Ud (σd )
.
2
(16)
Simulating the uncertainty for the reverse process using the correction in (16) provides
the same values for the expectation as depicted in Figure 6 (curve labelled Ur ).
Sensors 2021, 21, 3044
(
=
Ud (
40
Ur(
9 of 16
)
2
)
d
2
)
r
30
20
10
0
0
5
10
15
20
25
(cm)
d
Figure 6. Study on Tx’s position uncertainty based on the direct and reverse processes.
3.2. Uncertainties of the Tx’s Orientation
The Tx’s orientation uncertainties can be treated in a similar fashion as before. There
are, however, some aspects that need addressing. In this paper, the Tx’ orientation uncertainties are modeled via three random axis-rotations of the Tx’s heading vector. Let us
consider the kth Tx’s heading vector, nˆk , which is given by:
nˆk = R x (α)· Ry ( β)· Rz (γ)·nˆ0k = R(α, β, γ)·nˆ0k
=
( )∙
( )∙
( )∙
(17)
= ( , , )∙
where R x (α), Ry ( β) and Rz (γ) represent the three rotation matrix relative to the x-axis,
( )
( )
)
α, β, and γ are three random angles with probability
y-axis,( and
z-axis,
respectively,
distribution function N(0, ζ2 ), finally, nˆ0k represents the unperturbed heading vector of the
kth Tx.( R, (α,
, ) β, γ) is a composed rotation matrix. The effect of (17) on the Tx’s heading is
due to the cosine terms due to the Txs in (2) and (4), showing that a rotation will impact the
received signal. The next step resorts to the evaluation of the expectations due to the direct
and reverse process. In the direct process, the channel is re-simulated for each random
iteration, generating an expectation value, for the kth Tx, which is given by:
( )=
(
,
)−
,
, ( , , )
Ud (ζ d ) = E[ f R ( xk , yk ) − f R ( xk , yr , R(α, β, γ))].
(18)
E[X] (cm)
Ud cannot be cast in a closed-form solution. Instead, we resorted to simulation using
the approach described in the previous section, results are depicted in Figure 7.
Figure 7. Study on Tx’s orientation uncertainty based on the direct and reverse processes.
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The expectation of the reverse process reveals some difficulties. The quadratic forms
in the left side of (7) are invariant to rotations, which would imply that orientation uncertainties do not affect error performance. Since this is not the case, we approach the problem
in a similar manner as in the case of Tx’s position uncertainties, that is, assuming that the
effect of random headings can be reproduced by affecting the Tx’s positions with a random
perturbation. Following this approach, the expectation of the reverse process can be cast as:
h
i
Ur (ζ r ) = E δx2 + δy2 − 2( xr − xk )δx − 2(yr − yk )δy = 2ζ r 2
(19)
where δx and δy are the equivalent position perturbations, which are functions of the
random orientation angles α, β, and γ. Equating Ud and Ur , allows to retrieve the correction
values for the angle uncertainty, ζ r as a function of ζ d . To show that the processes are
indeed equivalent, we performed the simulation for Ur , the results are depicted in Figure 7.
As it can be seen the two expectations match well and reveal that the effect of orientation
uncertainty is worse than the position uncertainty.
3.3. Lighting Uniformity
In the indoor environments, it is essential to uniformly distribute Pr inside the illuminated zone [27]. This constraint is generally assumed in well-lit spaces, where the
uniformity of light is linked to the best perception of the objects. The uniform distribution
of light in the room, Ul , is represented as the ratio of the minimum to maximum power
intensity at the receiving plane, which is given by:
Ul =
min( Pr )
,
max( Pr )
(20)
Ul (%)
Lighting uniformity relies on the three factors related to the Txs, i.e., the distance
between the Txs, HPA, and the number of Txs. Figure 8 illustrates the lighting uniformity
as a function of the number of Txs (i.e., 4 and 9) and for different values of HPA.
(a)
Figure 8. Cont.
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Ul (%)
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(b)
Figure 8. Comparison of uniformity for different HPA and distance between the Txs in case of (a)
4-Tx, and (b) 9-Tx.
It is noticed from the figure that, a higher number of Txs provide improved uniformity.
The maximum uniformity is achieved at distances of 3.4 and 2.6 m for HPA of 60◦ for 4- and
9-Tx, respectively. It is interesting to notice that, the same factors influence both lighting
uniformity and positioning accuracy. It is usually believed that, positioning accuracy can
be enhanced with higher uniformity. However, our study reveals that, this is not a general
rule, moreover, HPA seems to have a more prominent impact on the positioning accuracy
than the uniformity.
4. Simulation Results
In this section, the performance of the proposed VLP system is evaluated by simulation
results. We consider the scenario where the LED-based Txs are located on the ceiling of
a room of dimension 6 × 6 × 3 m3 , which are assumed to be the same and modelled as
pointwise Lambertian sources with order m depending ζ on the value of HPA, see (2).
In practical environments also, the square LED placement layout is common. We have
considered three different cases of 4-, 9-, and 16-LEDs, which are arranged in a square grid
on the ceiling plane. The Rx is located on the ground plane. For all simulation purposes,
the resolution of the grid is fixed at 1 cm, which implies that the PD can be placed at 3600
different locations. All the key parameters for the simulation are detailed in Table 2.
Table 2. The key parameters for the indoor VLP system.
Parameter
Value
Room size
Number of LED Txs
Transmit power of each Tx
Rx’s field of view
Reflection coefficient
Area of PD
Responsivity of PD
Tx elevation
Tx azimuth
Rx elevation
Rx azimuth
6 × 6 × 3 m3
4/9/16
1W
75◦
0.7
10−4 m2
0.5 A/W
−90◦
0◦
90◦
0◦
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4.1. Positioning Error Dependence on Lighting Uniformity
This section analyses the influence of lighting uniformity and HPA on εp . Here, we
vary the uniformity between 0.5 and 0.8 in steps of 0.05 and for HPA of 40◦ and 60◦ . For this,
we first select the values of distance between the Txs according to the results of Figure 8.
Following this, we estimate εp for each set of conditions using NLLS and the polynomial
regression power-distance model. εp for different values of uniformity for 4-, 9-, and 16-Tx
are illustrated in Figure 9. It is observed that the minimum εp is attained for the case of
4-Tx with HPA and uniformity of 60◦ and 0.65, respectively, whereas, in the case of 9-Tx,
the minimum εp is achieved with HPA of 60◦ and uniformity equal to 0.55. For the 16-Tx
case, the minimum εp is accomplished with an HPA of 40◦ and with the uniformity of 0.65.
It is clear from Figure 9 that, (i) low to a moderate value of lighting uniformity can support
low εp ; and (ii) an optimal value for the number of Txs and the associated HPA, which
do not match the optimal value of lighting uniformity conditions. Under the simulated
conditions, the optimal values for the HPA, uniformity, and distance between the Tx based
on the lowest εp are shown in Table 3.
40
4-Tx
9-Tx
16-Tx
30
HPA = 40°
20
10
HPA = 60°
0
50
55
60
65
Ul (%)
70
75
80
ε
Figure 9. Comparison of εp for different Txs and uniformity.
Table 3. The optimized values for all cases of Txs.
σ
Number of Txs
Uniformity
σ
4
9
16
ε
Distance between
Txs (m)
HPA (◦ )
2.4
1.3
1.34
60
60
40
0.65
0.55,
( )
Q0.65
4.2. Impact of Tx’s Position and Orientation Uncertainty on Error Performance
Q ( )=
( )
In this section, we investigate the impact of Tx’s position uncertainties σ and the Tx’s
orientation uncertainties ζ on the error performance. We follow the same approach that
ε reverse method
has been described in detail in Section 3.1. The simulation follows the
σ
with 1000 random iterations for each σ value using the adjusted variance values given
σ
by (16), where a random displacement vector (δx, δy) is added in the ideal Tx positions
metric
as explained in (11). The quantile function QF ( p) is employed as a performance
σ
considering both, its average and standard deviation,
over the 1000 random iterations to
σ
obtain the confidence interval of εp , which is given by:
QF ( p) = FCDF −1 ( p),
where p is the percentage of the confidence interval.
(21)
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QF (95%) (cm)
Figure 10 illustrates the average and standard deviation of the 95% quantile of εp as a
function of σ for 4-, 9-, and 16-Tx for the Tx’s positioning uncertainty. It is observed that
the effect of Tx’s position uncertainty, σ, traduces in an increasing error dependence, which
is more prominent for set-ups with a lower number of Txs. The standard deviation of the
error quantile, i.e., the increasing error also confirms an increasing trend with σ, is more
evident for the 4-Tx case. It is noticed that, for σ of 5 cm the average positioning errors are
23.3, 15.1, and 13.2 cm, with the standard deviation values of 6.4, 4.1, and 2.7 cm for 4-, 9-,
and 16-Tx cases, respectively. Figure 10 suggests that the error dependence on the Tx’s
position uncertainty can be lowered by increasing the number of Txs.
σ
σ Txs and σ.
Figure 10. Comparison of positioning error for different
Figure 11 depicts the average and standard deviation of the 95%εεquantile of εp or as a
function of
ߞߞ ζ for 4-, 9-, and 16-Tx for the Tx’s orientation uncertainty. It is clear from the
ߞߞ effect ζ has more significant error
figure that, alike the case of Tx’s position uncertainty, the
ߞߞ that for ζ of 5◦ , the
dependence for set-ups with a lower number of Txs. It is observed
average positioning errors are 31.9, 20.6, and 17 cm, with the standard deviation values of
9.2, 6.3, and 3.9 cm for 4-, 9-,
and 16-Tx cases, respectively. The increasing error also proves
ߞߞ
an increasing trend with ζ, therefore, the placement of Tx with the accurate Tx’s position
and its orientation should be taken into consideration, as the error may rise even with low
values of set-up uncertainties.
60
4-Tx
9-Tx
16-Tx
50
40
Average error
30
20
10
Standard deviation
0
0
2
4
(°)
6
8
10
ߞߞ Txs and ζ.
Figure 11. Comparison of positioning error for different
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5. Conclusions
In this paper, we demonstrated the influence of Tx’s position and orientation uncertainty on the performance of VLP systems based on RSS. From the results, we can conclude
that light uniformity and Tx’s HPA are crucial design parameters for developing an efficient
VLP system. The selection of the Tx’s HPA as well as the optimum distance between Txs
have to be carefully implemented. Moreover, we showed that the best uniformity and
optimum error performance were not met for the same conditions, inferring necessary
design trade-offs. Furthermore, the effect of error dependence on Tx’s position and orientation uncertainty reduced with increasing the number of Txs. In case of square grid
Tx placement for a VLP system, the number of Txs can be further explored as an added
variable to optimize both light uniformity and error performance, as it reduces the HPA for
smaller distances between Txs.
Author Contributions: The contributions of the authors in this paper are the following: conceptualization: N.C. and L.N.A.; investigation: N.C.; methodology: N.C., L.N.A. and Z.G.; project
administration: L.N.A. and Z.G.; software: N.C.; validation: L.N.A. and Z.G. All authors have read
and agreed to the published version of the manuscript.
Funding: This research was funded by H2020/MSCA-ITN funding program under the framework
of the European Training Network on Visible Light Based Interoperability and Networking, project
(VisIoN) grant agreement no 764461.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: This paper was raised under the frameworks of the COST Action NEWFOCUS
(CA19111), supported by COST (European Cooperation in Science and Technology).
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature
ADOA
ANN
AOA
CDF
FOV
GPS
HPA
IoT
IP
LEDs
LLS
LOS
NLLS
NLOS
OOK
PD
PDF
PR
RF
RFID
RSS
Rx
SNR
Angle difference of arrival
Artificial neural network
Angle of arrival
Cumulative distribution function
Field of view
Global positioning system
Half power angle
Internet of things
Indoor positioning
Light-emitting diodes
Linear least square
Line of sight
Nonlinear least square
Non-line of sight
On-off keying
Photodiode
Probability distribution function
Polynomial regression
Radio frequency
Radio frequency identification
Received signal strength
Receiver
Signal-to-noise ratio
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TDOA
TOA
Tx
UWB
VLC
VLP
Time difference of arrival
Time of arrival
Transmitter
Ultra-wide band
Visible light communication
Visible light positioning
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Chaudhary, N.; Alves, L.N.; Ghassemblooy, Z. Current Trends on Visible Light Positioning Techniques. In Proceedings of the
2019 2nd West Asian Colloquium on Optical Wireless Communications (WACOWC), Tehran, Iran, 27–28 April 2019; pp. 100–105.
Do, T.-H.; Yoo, M. An In-Depth Survey of Visible Light Communication Based Positioning Systems. Sensors 2016, 16, 678.
[CrossRef] [PubMed]
Yassin, A.; Nasser, Y.; Awad, M.; Al-Dubai, A.; Liu, R.; Yuen, C.; Raulefs, R.; Aboutanios, E. Recent Advances in Indoor
Localization: A Survey on Theoretical Approaches and Applications. IEEE Commun. Surv. Tutor. 2017, 19, 1327–1346. [CrossRef]
Gezici, S.; Tian, Z.; Giannakis, G.B.; Kobayashi, H.; Molisch, A.F.; Poor, H.V.; Sahinoglu, Z. Localization via Ultra-Wideband
Radios: A Look at Positioning Aspects for Future Sensor Networks. IEEE Signal Process. Mag. 2005, 22, 70–84. [CrossRef]
Tariq, Z.B.; Cheema, D.M.; Kamran, M.Z.; Naqvi, I.H. Non-GPS Positioning Systems: A Survey. ACM Comput. Surv. 2017, 50,
1–34. [CrossRef]
Luo, J.; Fan, L.; Li, H. Indoor Positioning Systems Based on Visible Light Communication: State of the Art. IEEE Commun. Surv.
Tutor. 2017, 19, 2871–2893. [CrossRef]
Armstrong, J.; Sekercioglu, Y.; Neild, A. Visible Light Positioning: A Roadmap for International Standardization. IEEE Commun.
Mag. 2013, 51, 68–73. [CrossRef]
Zhuang, Y.; Wang, Q.; Shi, M.; Cao, P.; Qi, L.; Yang, J. Low-Power Centimeter-Level Localization for Indoor Mobile Robots Based
on Ensemble Kalman Smoother Using Received Signal Strength. IEEE Internet Things J. 2019, 6, 6513–6522. [CrossRef]
Zhu, B.; Cheng, J.; Wang, Y.; Yan, J.; Wang, J. Three-Dimensional VLC Positioning Based on Angle Difference of Arrival With
Arbitrary Tilting Angle of Receiver. IEEE J. Sel. Areas Commun. 2018, 36, 8–22. [CrossRef]
Xu, J.; Gong, C.; Xu, Z. Experimental Indoor Visible Light Positioning Systems With Centimeter Accuracy Based on a Commercial
Smartphone Camera. IEEE Photonics J. 2018, 10, 1–17. [CrossRef]
Naeem, A.; Hassan, N.U.; Pasha, M.A.; Yuen, C.; Sikora, A. Performance Analysis of TDOA-Based Indoor Positioning Systems
Using Visible LED Lights. In Proceedings of the 2018 IEEE 4th International Symposium on Wireless Systems within the
International Conferences on Intelligent Data Acquisition and Advanced Computing Systems (IDAACS-SWS), Lviv, Ukraine,
20–21 September 2018; pp. 103–107.
Xie, C.; Guan, W.; Wu, Y.; Fang, L.; Cai, Y. The LED-ID Detection and Recognition Method Based on Visible Light Positioning
Using Proximity Method. IEEE Photonics J. 2018, 10, 1–16. [CrossRef]
Plets, D.; Bastiaens, S.; Stevens, N.; Martens, L.; Joseph, W. Monte-Carlo Simulation of the Impact of LED Power Uncertainty
on Visible Light Positioning Accuracy. In Proceedings of the 2018 11th International Symposium on Communication Systems,
Networks & Digital Signal Processing (CSNDSP), Budapest, Hungary, 18–20 July 2018; pp. 1–6.
Gu, W.; Aminikashani, M.; Deng, P.; Kavehrad, M. Impact of Multipath Reflections on the Performance of Indoor Visible Light
Positioning Systems. J. Light. Technol. 2016, 34, 2578–2587. [CrossRef]
Mousa, F.I.K.; Almaadeed, N.; Busawon, K.; Bouridane, A.; Binns, R.; Elliot, I. Indoor Visible Light Communication Localization
System Utilizing Received Signal Strength Indication Technique and Trilateration Method. Opt. Eng. 2018, 57, 1. [CrossRef]
Wang, X.; Shen, J. Machine Learning and Its Applications in Visible Light Communication Based Indoor Positioning. In
Proceedings of the 2019 International Conference on High Performance Big Data and Intelligent Systems (HPBD&IS), Shenzhen,
China, 9–11 May 2019; pp. 274–277.
Shawky, S.; El-Shimy, M.A.; El-Sahn, Z.A.; Rizk, M.R.M.; Aly, M.H. Improved VLC-Based Indoor Positioning System Using a
Regression Approach with Conventional RSS Techniques. In Proceedings of the 2017 13th International Wireless Communications
and Mobile Computing Conference (IWCMC), Valencia, Spain, 26–30 June 2017; pp. 904–909.
Plets, D.; Bastiaens, S.; Martens, L.; Joseph, W. An Analysis of the Impact of LED Tilt on Visible Light Positioning Accuracy.
Electronics 2019, 8, 389. [CrossRef]
Plets, D.; Bastiaens, S.; Martens, L.; Joseph, W. Performance Assessment of Artificial Neural Networks on the RSS-Based Visible
Light Positioning Accuracy with Random Transmitter Tilt. In Proceedings of the CSNDSP, Porto, Portugal, 20–22 July 2020;
Volume 8, p. 389.
Chaudhary, N.; Alves, L.N.; Ghassemlooy, Z. Impact of Transmitter Positioning Uncertainty on RSS-Based Visible Light Positioning Accuracy. In Proceedings of the 2020 12th International Symposium on Communication Systems, Networks and Digital
Signal Processing (CSNDSP), Porto, Portugal, 20 July 2020; pp. 1–6.
Shi, G.; Li, Y.; Cheng, W. Accuracy Analysis of Indoor Visible Light Communication Localization System Based on Received
Signal Strength in Non-Line-of-Sight Environments by Using Least Squares Method. Opt. Eng. 2019, 58, 1. [CrossRef]
Ghassemblooy, Z.; Popoola, W.; Rajbhandari, S. Optical Wireless Communications: System and Channel Modelling with Matlab® ; CRC
Press: Boca Raton, FL, USA, 2019.
Sensors 2021, 21, 3044
23.
24.
25.
26.
27.
16 of 16
Chaudhary, N.; Alves, L.N.; Ghassemblooy, Z. Feasibility Study of Reverse Trilateration Strategy with a Single Tx for VLP. In
Proceedings of the 2019 2nd West Asian Colloquium on Optical Wireless Communications (WACOWC), Tehran, Iran, 27–28 April
2019; pp. 121–126.
Sun, X.; Duan, J.; Zou, Y.; Shi, A. Impact of Multipath Effects on Theoretical Accuracy of TOA-Based Indoor VLC Positioning
System. OSA Publishing: Washington, DC, USA, 2015; Volume 4.
Chaudhary, N.; Younus, O.I.; Alves, L.N.; Ghassemlooy, Z.; Zvanovec, S.; Le-Minh, H. An Indoor Visible Light Positioning System
Using Tilted LEDs with High Accuracy. Sensors 2021, 21, 920. [CrossRef] [PubMed]
Shiquan, W. Computation of a Trust Region Step. ACTA Math. Appl. Sin. 1991, 7, 354–362.
Hu, Y.; Luo, M.R.; Yang, Y. A Study on Lighting Uniformity for LED Smart Lighting System. In Proceedings of the 2015 12th
China International Forum on Solid State Lighting (SSLCHINA), Shenzhen, China, 2–4 November 2015; pp. 127–130.