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 diferent 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.
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 communication 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 efect. 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 diferent 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.
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 = and height is ℎ. The room is depicted in figure 1.
VLC channel is modelled considering the line of sight (LOS) components from each LED in
some mobile node´s point . 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
efect, blockages scenarios or LED installation errors). The noise is additive Gaussian (0,
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 (
is the DC gain of LOS channel and is the power transmitted for LED . In (
Ψ . depends of Lambertian order transmission , the area of PD
and factors ( ) and ( ) that represents the optical filter gain and the optical concentrator
2
)
is
in (
4 4 ∑ = ∑ =1 =1
+ 0 = { 2 cos ( )cos( ) 0 ≤ ≤ Ψ
elsewhere
= ( 2+ 1) ( ) ( ) (
The factor cos( ) is the same that cos( ) 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( ) from tilted angle. This is possible from the expression in
(
cos( ) = ⃗ ⋅ ̂ ⃗ ‖ ‖2‖ ̂ ‖2 ̂ =
[sin() cos(), sin() sin(), cos()] ‖[sin() cos(), sin() sin(), cos()]‖ 2
The transmitter send power square waves (without zero values) to the receiver with frequencies
. This waves are modulated by sine functions with carrier frequencies . The displacement
in frequency domain serves to mitigate the inter channel interference (ISI) and allowing to
(
x x x x + + + + Transmitter's side
Distances calculation depend of the tilted angle of PD, which is delivered from IMU sensor
and RSS. We define the vector ⃗= ( 1, 2, 3, 4, 5) = ( ̂1 , ̂2 , ̂3 , ̂4 , ) that contains all
the information necessary for next step. To use all this data, we propose multidimensional
polynomial functions ̂ with argument vector ⃗. 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 (
are the third order terms. First and second order function are construct by deleting the second and third order terms respectively.
̂ (⃗) = 0 + ∑ + ∑ 5 =1 5 =1
∑ =1 + ∑ 5 =1
∑ =1
=1 ∑
It is necessary to find the constants in (
This matrix changes depends on the considered function. Each element in matrix represents the data obtain from vector ⃗, if we consider vectors ⃗() , the Notation 1() represents nth fist order terms, 1 ⋅ 1() … 5 ⋅ 5() are the second order terms and 1 ⋅ 1 ⋅ 1() … 5 ⋅ 5 ⋅ 5() the third order terms in . contains the values of and they represent the perfect distance from PD to Tx . With vectors ⃗, and matrix , it is possible to solve the optimisation problem in (14) by Non linear least Square (NLLS) algorithm.
minimize
⃗ subject to ̂ ⪰ 0 =1 ∑ ( ̂ − ) 2
The simulation procedure consists of estimation for 2D position of a mobile node in a room of
3x3x3 . The parameters used for the channel model, transmitters, and receiver are exposed in
to consider a FOV value diferent 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.
=
⎡
⎢
⎢
⋮
1 1(
⋅ ( ) = ⃗= [ 0 1 ⋯ 5 11 ⋯ 55 111
First step in simulation process consisted on finding the coeficients of polynomial functions. As we showed before, this coeficients were calculated from NLLS algorithm. For forming the matrix a mobile node is positioned in 361 diferent points over the grid, due to grid step is 0.1 . This value is considered at the most cases of VLP simulations on literature). For each grid point, the used tilted angles were = [−15 ∘, −10∘, −5∘, 0∘, 5∘, 10∘, 15∘]. This means, we used a total of 2527 point cases and represent all possibilities of vector ⃗. For each angle case of vector ⃗, its available the true distance and all the true distances conform the vector . With Matrix and vector ⃗, 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 diference between the estimated distance and correct distance. Third order function presents a mean and deviation error less than 0, 1 . Second order polynomial present a best performance than first order, but all the proposer polynomial have error less than 5 in distance estimation.
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 efect. The results are summarised in table 3. Again the best performance corresponds to third order polynomial with errors less than 2 .
In figures 4a and 4b are plotted the cumulative distribution of positioning. The behaviour in ifrst 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.
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 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.
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 diference between the other polynomials has become less with an mean error up to 2 . Figures 7a and 7b shows the CDF positioning error for non perfect tilted angle. Plots deliver a change on third order CDF when = 15 ∘, instead of = 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 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.
With all of previous results, in table 5 shows the percent diference 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 changes.
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 afected performance 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 coeficients 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 afect the training phase of iftting 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.
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. estimator with complementary photodiodes, Journal of Lightwave Technology 37 (2019) 2932–2945. doi:1 0 . 1 1 0 9 / J L T . 2 0 1 9 . 2 9 0 7 9 6 9 . [13] A. H. A. Bakar, T. Glass, H. Y. Tee, F. Alam, M. Legg, Accurate visible light positioning using multiple-photodiode receiver and machine learning, IEEE Transactions on Instrumentation and Measurement 70 (2021) 1–12. doi:1 0 . 1 1 0 9 / T I M . 2 0 2 0 . 3 0 2 4 5 2 6 . [14] M. Kok, T. B. Schön, A fast and robust algorithm for orientation estimation using inertial sensors, IEEE Signal Processing Letters 26 (2019) 1673–1677. doi:1 0 . 1 1 0 9 / L S P . 2 0 1 9 . 2 9 4 3 9 9 5 . [15] E. M. Diaz, D. B. Ahmed, S. Kaiser, A review of indoor localization methods based on inertial sensors, Geographical and Fingerprinting Data to Create Systems for Indoor Positioning and Indoor/Outdoor Navigation (2019) 311–333. [16] N. Stevens, D. Plets, L. De Strycker, Monte carlo algorithm for the evaluation of the distance estimation variance in rss-based visible light positioning, in: 2017 20th International Symposium on Wireless Personal Multimedia Communications (WPMC), IEEE, 2017, pp. 212–216. [17] D. Plets, S. Bastiaens, N. Stevens, L. Martens, W. Joseph, Monte-carlo simulation of the impact of led power uncertainty on visible light positioning accuracy, in: 2018 11th International Symposium on Communication Systems, Networks & Digital Signal Processing (CSNDSP), IEEE, 2018, pp. 1–6. [18] Z. Ghassemlooy, W. Popoola, S. Rajbhandari, Optical wireless communications: system and channel modelling with Matlab®, CRC press, 2019. [19] M. Kavehrad, R. Aminikashani, Visible Light Communication Based Indoor Localization,
CRC Press, 2019. [20] A. Cecen, Multivariate polynomial regression, 2021. URL: https://fr.mathworks.com/ matlabcentral/fileexchange/34918-multivariate-polynomial-regression, program code ver.1.4.0.0. [21] H. Tai, Arduino library for mpu9250 nine-axis (gyro + accelerometer + compass) mems motion tracking device, 2021. URL: https://www.arduino.cc/reference/en/libraries/mpu9250/, library code ver.0.4.2.