Scalable Visible Light Indoor Positioning System Using RSS

Abstract

This paper proposes a visible light positioning system that utilizes commercial Light-Emitting Diode (LED) lamps as transmitters and Silicon PIN photodiodes as receivers. The light signals are transmitted and received using Intensity Modulation and Direct Detection (IMDD). The lamps are modulated using On–Off Keying (OOK) with the Manchester code, and the medium access control is achieved by Time-Division Multiplexing (TDM). The position is estimated using trilateration based on the Received Signal Strength (RSS). The system’s scalability is accomplished by replicating primary localization cells composed of seven lamps and drawing on the neighborhood synchrony, exploiting the spatial multiplexing property of the light. A basic unit in the cell comprises three lamps forming a localization triangle; then, one primary localization cell shall consist of six triangles sharing lights among basic neighbor units. The cell prototype was implemented to prove the working principle of the system. Three estimation methods were used to compute the position: a deterministic approach based on least-squares regression, an Artificial Neural Network (ANN) per lamp, and an ANN for the complete system. The best per lamp estimator was the ANN, computing positions that reached an experimental accuracy of 2.5 cm under indoor conditions.

1. Introduction

It is well known that Global Navigation Satellite Systems (GNSS) have trouble locating objects in indoor environments. This limitation involves two main factors: (1) in roofed places, the objects are out of the Field of View (FoV) of the satellites, and (2) the inherent accuracy of the GNSS (around 3.0 m) [1] is unsuitable for indoor environments. This situation has motivated research in developing Indoor Positioning Systems (IPSs), which are local positioning systems composed of their localization network and explicitly designed to work indoors. The interest in IPSs has increased considerably in the last decades due mainly to the number of emergent applications derived from the development and deployment of new technologies, such as the Internet of Things (IoT), applied robotics in industry and home activities, assisted sales, and indoor games, among others [2,3,4,5,6,7,8].

Some IPS proposals use RGB-D sensors to construct maps and the position of autonomous systems [9]. Chen et al. [10] use the fusion of an Inertial Navigation System (INS) with a Wireless Sensor Network (WSN) to improve the positioning that could not be achieved independently. Moreover, Dong et al. [11] proposed instrumenting the environment to achieve better precision in specific cases such as mines. IPS can use different technologies to create a positioning network [2,3,4]. The most popular of these technologies is the Radio-Frequency (RF) indoor positioning systems, mainly due to the wide deployment of the RF technologies (e.g., Wi-Fi and Bluetooth), which reduces the deployment cost.

Additionally, the nature of the radiofrequency signals helps reduce the deployment cost. These signals can pass through walls and obstacles, so fewer transmitters per unit area are necessary. However, this characteristic adversely impacts the privacy of the IPS because it is impossible to control who is able or unable to know the objects’ position [2,3,5]. RF IPSs are also limited in accuracy and commonly require complex signal processing. Limitations in these parameters are related to their electromagnetic wave’s nature and associated effects, such as path loss and the multiple reflections caused by walls, objects, and human bodies. Most of the IPS using radiofrequency signals and implemented over widespread technologies such as Wi-Fi and Bluetooth provide the positioning with the best accuracy results in about 1.0 m and use complex algorithms to locate objects, such as fingerprinting or Maximum Likelihood Estimation (MLE) [2,3,5]. Conversely, nowadays, technologies based on radiofrequency signals must deal with two more relevant issues: energy consumption and the saturation of its spectrum. In addition, the IoT is expected to increase the number of connected devices in subsequent years exponentially, which may saturate the frequency spectrum even more [12,13,14]. Therefore, radiofrequency IPSs could be more of a problem than a solution.

Visible Light Communications (VLC) are a different way to transmit and receive digital information through visible light signals, i.e., electromagnetic waves between 380 and 700 nm. The VLC has advantages over radiofrequency communications, such as a free spectrum with a virtually infinite bandwidth (between 430 and 770 THz). Furthermore, due to the nature of the light, these signals are harmless for humans and can be utilized in restricted areas for RF signals such as hospitals, aircraft, and others. So, they are channel secure because it is easy to delimit the influence area of the LED transmitters. Notwithstanding, this technology has some drawbacks; for instance, a direct line of sight between the transmitters and receivers is necessary, signals’ power decays quickly with the distance, natural light could interfere with data signals, and dark places are consequently a significant limitation [6,15,16].

VLC is particularly well-suited for indoor positioning aims. The Visible Light Positioning System (VLPS), a term used to position systems based on visible light communications, has low deployment costs because they use the lighting infrastructure. VLPS can be private due to the short-range and straightness of the light signals. In addition, VLPS operate transmitters that can be used either for lighting, communications, or positioning purposes. The complexity of the VLPS is lower than radiofrequency-based IPSs since light behavior is less susceptible to reflections; it avoids no obstacles like RF signals do. Finally, the theoretical accuracy of VLPS is about a few centimeters. All these characteristics make VLPS an excellent solution for indoor positioning purposes [8,17,18].

This work is based on a preliminary study proposed by Avendano López et al. [19]. The current study proposes a visual light positioning system that utilizes commercial LED lamps as transmitters. The lamps are modulated using On–Off Keying (OOK) and the Manchester code. This scheme allows retrieving the synchronization signal on the receiver side, avoiding noticeable changes in the light intensity caused by different transmitted streaming data. The light signals are transmitted and received using Intensity Modulation and Direct Detection (IMDD). The medium access control uses Time Division Multiplexing (TDM) compared to most commercial systems, which use patterns and digital image processing to estimate the position [20]. IMDD, instead of patterns, enables a higher bandwidth for data transmission and more satisfactory reliability of the retrieved data. Contrarily, TDM allows the communication of data sequentially without noticeably affecting the intensity of the lighting system. The receivers based on Silicon PIN photodiodes offer a high-speed response and sensitivity to light intensity changes. The information captured by these sensors is used to estimate the object’s position by the trilateration principle. In addition, the system was designed to be scalable, replicating a functional unit named primary location cell and drawing on the spatial multiplexing feature of the light. The proposed framework enables a refresh rate fast enough to obtain real-time system operation. Ultimately, the system test performance delivers an accuracy of about $2.5$ cm on the estimated position. The main contribution and novelty of this work lie in formulating a complete mathematical model for the object’s location using visible light. This model comprises two submodels: the former corresponds to the power of the source of light, and the latter corresponds to the location of an object through the trilateration using a minimum of three light sources. These models were implemented and verified experimentally.

2. Materials

2.1. Transmitter

The Intensity Modulation (IM) on LED lamps can be achieved using a power NPN silicon transistor or an N-Channel Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) configured as a fast switch. The transistor commutes on/off the Direct Current (DC) power supplied to the lamp to generate an OOK modulation. Moreover, the bandwidth of the lamps provides the maximum operative frequency to control the LED lamps without affecting the communication performance. The modulation control is developed by a Microcontroller (MCU), switching the lights through a coupled power transistor. The commercial LED lamps used in this approach are limited to a few tens of kilohertz.

2.2. Receiver

The receiver uses a Silicon PIN Photodiode (PD) with high sensitivity and wide bandwidth, administered to a Direct Detection (DD) process. In our case, a high bandwidth photoreceiver is mandatory because it must capture as many harmonics of square signals as possible. Since the upper-frequency limit of the square signals is around a few tens of kilohertz, the bandwidth of the photoreceiver should be at least hundreds of kilohertz to recover the signals correctly. Oppositely, photodiodes offer high sensitivity and fast response when connected in inverse polarity, so the output current is well-known as a reversed-light current. The voltage signal corresponding to the lamps flickering is a negative pulse superposed over an offset level, proportional to the environmental light. The required operative bandwidth is also extended to the OpAmps to amplify and filter the received signals, avoiding distortion. Hence, the amplification stage must include high-speed operational amplifiers and a conditioning circuit to guarantee a proper conversion of the short-circuit current delivered by the PD to a low-noise voltage signal. Figure 1 shows the conditioning circuit and signal amplification.

$I_{sc}$ represents the light reversed current delivered by the PD. $R_L$ is the load resistor that converts the modulated light current into a voltage signal using Ohm’s Law. $+V$ and $-V$ are the voltages supplied by the OpAmp. $V_{DD}$ is the voltage source that polarizes the PD inversely, and Out-PD represents the output signal given by:

$$V_{out}=I_{sc}{R}_L\left(1+\frac{R_f}{R}\right).$$

(1)

Bearing in mind that the voltage signal corresponding to the lamps flashing is a signal of negative train pulses over a DC component, such a signal must be inverted and filtered to eliminate this DC component and reduce high-frequency noise. Therefore, OpAmps utilized at this stage also must fulfill the requirements of low noise and wide bandwidth of the previous settings. Figure 2 shows the OpAmp inverter circuit and the implemented band-pass filter.

After the amplification and conditioning stages, the received signal is transferred to a pair of complementary but independent processing circuits, the digitalization and peak detector stages. The digitalization module enables the MCU to decode data sent by the lamps. The whole circuit includes an IC comparator, an MCU’s digital input, and a trimpot, as shown in Figure 3.

Therefore, the IC comparator provides the digital signal sent to the MCU. The peak detector circuit is used as an envelope detector for the data flow acquired from each lamp, as shown in Figure 4.

In addition, this circuit helps estimate how far the receiver is from a lamp. The resistor, OpAmp on the left, and two diodes form the precision rectifier. In addition, a capacitor is used to hold the voltage signal when it is charged with the pulses amplitude of the received light data stream. A resistor serves to couple the precision rectifier output and the capacitor. A Single Pole Single Throw (SPST) analog switch charges and discharges the capacitor, all driven by the MCU. Finally, the last OpAmp was configured as a signal follower, coupling the peak detector output into the Analog to Digital Converter (ADC) in the MCU.

The peak detection co-occurs with the decoding process. Once the MCU starts receiving data from a lamp, it sends a digital signal to open the SPST analog IC switch. From this, the capacitor is charged to hold the voltage amplitude of the stream data during decoding. Once the capacitor holds the signal, the MCU can periodically read this voltage signal using its built-in ADC module (the Sampling and Hold (S/H) process). Moreover, after the MCU completes decoding the data packet, it changes the digital output connected to the peak detector to discharge the capacitor. In this way, the peak detector is ready to receive a new data stream from another lamp. Lastly, the data obtained from the decoding (digitalization) and S/H process (peak detector) are processed by the MCU or a computer to estimate the object’s positioning. Figure 5 shows the interaction of all components of the proposed system.

3. Methods

3.1. System Overview

The proposed system is scalable, replicating a positioning unit along with deploying particular units. The replicable positioning unit is called the primary localization cell, and the particular units, which are partial primary localization cells, are called special localization cells. Generally, an MCU works in every cell location as a hub of transmitters and controls every cell lamp, as shown in Figure 6.

Additionally, Figure 7 shows a study case requiringthe deployment of one particular localization cell (down) along with a primary cell (up).

Figure 8 shows the deployed system using three primary localization cells.

Each cell is identified by a specific color, comprising seven lamps. A small circle represents each lamp, and a hexagon symbolizes the halo of such a lamp; this is only to obtain a more straightforwardvisual representation of the system modularity.

Additionally, every system lamp has its unique identification code, noted with the number preceded by the tag “ID-”, and every lamp in each cell has a specific time to send its ID. This specific time is represented by the number preceded by the tag “Dif-” and followed by a letter identifying the cell. The lamp sends this code periodically to be decoded by the receiver identifying this lamp.

3.2. Medium Access Control

The Medium Access Control (MAC) is performed using TDM to avoid all the lamps sending their identification codes simultaneously. The TDM implementation is achieved by assigning a specific transmission time to each lamp in a localization cell and making the lamps’ transmission sequence cyclic. The transmission lamp shift is represented in Figure 8 by a number preceded by the tag “Dif-” and followed by a letter identifying the cell. It is worth mentioning that the lamps are switched on when the system is operating, even when they are not transmitting data. TDM was chosen instead of Frequency-Division Multiplexing (FDM) to take advantage of light physical properties, keeping a low complexity in the modulation and demodulation processes. The main characteristic of light allowing this transmission model is the quick decay in the power transmission with the distance, easing spatial multiplexing. Spatial multiplexing permits the spatial separation between signals guaranteeing that two simultaneous signals can be demodulated and decoded satisfactorily. Figure 8 shows TDM advantages, explicitly noting that the transmission turn assigned to each lamp is repeated in the different cells. Accordingly, this functionality is possible thanks to spatial multiplexing. It is noteworthy that lamps with the same assigned moment of transmission must be separated at a safe distance for this purpose. Furthermore, the use of TDM warrants that the light power emitted by the lamps is not significantly reduced because the light is only affected when the lamp transmits data, unlike most of the modulation schemes based on FDM, which remarkably affect the light power emitted by the lamps.

3.3. Synchronization

A problem that arises while using TDM is the inter-cell synchrony. The synchronization of all cells may turn into a considerable challenge because the propagation delay becomes higher with distance. Consequently, the proposed transmission array includes individual cells using a noncentralized control. Naturally, the propagation delay must be corrected continuously in each cell, but spatial multiplexing also solves this problem. Since light power decays quickly with distance, it offers a short-range characteristic to transmitted signals. Therefore, synchronization is mandatory in neighbor cells, whose signals are perceived by the receiver at a specific point. The synchronization between two neighbor cells is achieved by taking a parallel derivation of a lamp control output of the MCU from the first cell and connecting it to an MCU input in the second cell.

Moreover, synchronization also functions by setting on a first digital MCU output when the lamps’ sequence starts or ends and connecting such output to the second MCU input. In the first case, parallel derivation of the connection in all cells must be taken from the lamp control output with the same assigned transmission turn. For instance, if the derivation in a cell is taken from a lamp control output transmitting in turn “Dif-1”, the derivation of the following cells must also be taken from the lamp transmitting in the same turn. On another note, the digital input arriving at the second cell MCU works as a synchronization signal in both cases.

Figure 9 shows the system deployment in a scene with a specific geometric form, highlighting the synchronization signals between the different cells.

In this figure, the clusters of lamps with a particular color represent every cell. The blue numbers correspond to the absolute delay of the synchronization signals, and the blue arrows describe the synchronization connection between the MCUs, indicating the origin and the destination.

Note that the synchronization signal of one cell can be linked to several cells. Additionally, the propagation delay between cells increases as the distance between them increases. The propagation delay at any inter-cell point can be neglected because the maximum signal’s retardation is caused by the distance between two neighbor cells, usually a short-distance. Nonetheless, it is possible to achieve a local synchronization at any point in the room where the receiver is placed by using only one electrical connection between every cell and following some straightforward spreading rules.

3.4. Deployment and Refresh Time

Figure 9 shows the system deployment that uses 8 primary location cells, 11 special location cells, and 96 lamps. Special location cells are incomplete primary location cells used only to fit the system according to the geometric distribution of the experimental area. Therefore, the actual system deployment may require many lamps. Hence, the maximum number of supported lights is an essential feature. However, in the proposed modular design method, the maximum number of lamps is only limited by the number of bits allocated to the id-code of the lights.

Conversely, the number of receivers supported by the system is only limited by the room space. It can be considered virtually unlimited because every receiver processes and computes its position. Figure 10 shows the system modularity from another perspective, where each localization cell may be a primary or a special cell. These cells can be added or removed to cover a larger or smaller surface or replaced in case of malfunction. When a cell fails, it only affects the system performance in the area below and around it. Even if a lamp in a cell fails, the system can keep working, affecting only the system functioning nearby the lamp while this is repaired or replaced.

Moreover, the system exhibits a short refresh time due to the module-based transmission schema. The refresh time of our design depends on the transmission sequence length in a primary cell $t_c$:

$$t_c=7.0(t_{xL}+t_t),$$

(2)

where $t_{xL}$ and $t_t$ are the transmission time of each lamp and between lamps, respectively. The $t_t$ represents the system time required to decode and process the data sent by a lamp before the next lamp starts its transmission. Meanwhile, $t_{xL}$ depends on the number of bits $N_{Lb}$ sent by each lamp and the modulation frequency bits $f_b$ in the transmitter:

$$t_{xL}={f_b}^{-1}{N}_{Lb}={f_b}^{-1}(N_{db}+N_{cb}),$$

(3)

where $N_{Lb}$ bits include the data bits $N_{db}$ to address the lamps and the control bits $N_{cb}$ to identify the start and end of the transmission.

3.5. Localization Triangle

The trilateration method is used to calculate the positioning in the system. This method requires identifying the position of at least three reference points forming a triangle and the target’s distance to these three reference points. Suppose we want to obtain the exact position $(x,y)$ of point T in Figure 11.

For this purpose, we know the position of points $A=(x_1,y_1)$, $B=(x_2,y_2)$, and $C=(x_3,y_3)$. Likewise, we know the distances from A to T ($R_1$), from B to T ($R_2$), and, finally, from C to T ($R_3$). Lastly, by using the Euclidean distance expression, the Equations (4)–(6) are obtained:

$$\begin{array}{ccc}\hfill {R_1}^2& =& \hfill {(x-x_1)}^2+{(y-y_1)}^2;\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill {R_2}^2& =& \hfill {(x-x_2)}^2+{(y-y_2)}^2;\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {R_3}^2& =& \hfill {(x-x_3)}^2+{(y-y_3)}^2.\hfill \end{array}$$

(6)

Moreover, correlating the Equations (4)–(6), we obtain the next useful relations:

$$\begin{array}{ccc}\hfill 2(x_2-x_1)x+2(y_2-y_1)y& =& \hfill ({R_1}^2-{R_2}^2)-({x_1}^2-{x_2}^2)-({y_1}^2-{y_2}^2);\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill 2(x_3-x_1)x+2(y_3-y_1)y& =& \hfill ({R_1}^2-{R_3}^2)-({x_1}^2-{x_3}^2)-({y_1}^2-{y_3}^2).\hfill \end{array}$$

(8)

Finally, applying the Cramer’s rule to (7) and (8), the trilateration is formally computed by:

$$x=\frac{\left|\begin{array}{ccc}({R_1}^2-{R_2}^2)-({x_1}^2-{x_2}^2)-({y_1}^2-{y_2}^2)& & (y_2-y_1)\\ ({R_1}^2-{R_3}^2)-({x_1}^2-{x_3}^2)-({y_1}^2-{y_3}^2)& & (y_3-y_1)\end{array}\right|}{2\left|\begin{array}{cc}(x_2-x_1)& (y_2-y_1)\\ (x_3-x_1)& (y_3-y_1)\end{array}\right|},$$

(9)

$$y=\frac{\left|\begin{array}{ccc}(x_2-x_1)& & ({R_1}^2-{R_2}^2)-({x_1}^2-{x_2}^2)-({y_1}^2-{y_2}^2)\\ (x_3-x_1)& & ({R_1}^2-{R_3}^2)-({x_1}^2-{x_3}^2)-({y_1}^2-{y_3}^2)\end{array}\right|}{2\left|\begin{array}{cc}(x_2-x_1)& (y_2-y_1)\\ (x_3-x_1)& (y_3-y_1)\end{array}\right|}.$$

(10)

The reference points are represented by the lamps forming the localization triangle in the proposed system. In addition, the distance from the receiver to the lights is estimated through the RSS module. Hence, the primary location cells are formed by six localization triangles, as shown in Figure 12.

A localization red triangle in the cell is highlighted to improve precision. Figure 8 and Figure 9 should be looked at closely. So, several localization triangles (represented by the dotted red line) form and join the different primary and special localization cells across the arrangements. The localization triangle is the functional unit of the system; therefore, its requirements and limitations are extended to the entire system. Thereby, the system requires at least three lamps to operate, and its accuracy will depend on the experimental accuracy reached by the localization triangle. In the proposed system experiments, the receiver will capture signals from more than three lamps. For those cases, the signals acquired by the receiver should be sorted by their signal strength, choosing the highest three to compute the trilateration.

3.6. Distance Estimation

The accuracy of the localization triangle is mainly affected by the precision in distance estimations based on the RSS. Hence, the performance and robustness of the system will depend on the adaptability of the estimator under different operation conditions. Theoretically, the optical signal in the receiver $P_r$ with a Direct Line of Sight (LOS) through the transmitter [21] is modeled by:

$$P_r=\frac{P_t}{d^2}{R}_0\left(\varphi \right)A_{eff}\left(\psi \right),$$

(11)

where $P_t$ is the transmitted optical power, d is the distance between the transmitter and receiver, $R_0$ and $\varphi$ are the radiant intensity and radiation angle of the transmitter, and $A_{eff}$ and $\psi$ are the effective collection area and the incidence angle of the receiver. $R_0\left(\varphi \right)$ and $A_{eff}\left(\psi \right)$ modify the optical power transferred from the transmitter to the receiver depending on the angles $\varphi$ and $\psi$ and other parameters related to the transmitter and receiver internal structures. Experimental measurements of the light behavior in a controlled environment have enlightened how the parameters $R_0\left(\varphi \right)$ and $A_{eff}\left(\psi \right)$ affect the proposed system. Light power was measured using a lux-meter placed at different radii around the lamp. Figure 13 shows the experimental acquisition process of these measurements.

Figure 14 shows the results of such light behavior; different color lines are used to represent every experiment and tested radii, and these colors match with the measurement sequences illustrated in Figure 13.

Next, (11) is used to conduct a comparison with the obtained experimental results. Two regression models serving as ground truth were obtained using the power light measures. The power light $P_r$ estimated just beneath the lamp (radius = 0 m, angles $\varphi$ and $\psi$ are unchanged) is given by:

$$P_r=\frac{116.68}{d^{1.848}},\forall R^2=0.9932.$$

(12)

This power regression approximates (11) when $R_0\left(\varphi \right)$ and $A_{eff}\left(\psi \right)$ are constants. The parameter $R^2$ expresses that this regression closely fits the experimental data.

Nevertheless, while the radius grows, the power regression estimator deviates from (11), obtaining a weak estimator to model the experimental data and reflected on the $R^2$ coefficient. For example, the power regression model for measurements taken at a radius of 1.0 m is given by:

$$P_r=\frac{39.125}{d^{0.809}},\forall R^2=0.8842,$$

(13)

where the angles $\varphi$ and $\psi$ change at each measurement. Therefore, the terms $R_0\left(\varphi \right)$ and $A_{eff}\left(\psi \right)$ start to play a significant role in the model analysis. In addition, a polynomial regression model at a radius of 1.0 m was obtained for comparison purposes as follows:

$$P_r=-7.2999d^4+60.359d^3-179.51d^2+210.86d-49.354,\forall R^2=0.9886.$$

(14)

The results show that the coefficient $R^2$ for the polynomial regression (14) is higher than such of the power regression (13). Therefore, polynomial regression is a better estimator than power regression when the radius increases, as shown in Figure 15.

The Taylor series of functions could help explain these regression dissimilarities by considering that any function can be approximated by:

$$\begin{array}{c}\hfill f\left(x\right)=f\left(0\right)+\frac{f^{\prime }\left(0\right)}{1!}x+\frac{f^{″}\left(0\right)}{2!}{x}^2+\dots +\frac{f^{\left(n\right)}\left(0\right)}{n!}{x}^n.\end{array}$$

(15)

The regression (14) can be seen as the truncated Taylor series of (11) when the received light power is known and the angles $\varphi$ and $\psi$ change.

Moreover, it is possible to obtain the behavior of the light power when a receiver is moving in a horizontal plane using the same experimental data but changing the approach in which the data is analyzed. The new analysis approach to the experimental data is shown in Figure 16.

Figure 17 shows the measurements acquired when the lux-meter moves to different heights of the lamp’s plane.

In addition, the data belonging to the receiver’s movement over each horizontal plane are given in different color lines and their corresponding distances. This last analysis is more relevant than the vertical movement analysis because we aim to localize targets moving horizontally by trilateration.

Equation (11) models the behavior of the light in an optical channel. However, we have proven that the same behavior can be modeled as a polynomial regression when the receiver moves around the lamp over a changing radius. Moreover, polynomial regressions model the behavior of the light power when the receiver is moving in a horizontal plane. Experiments noted that the polynomial order with the best fitting decreases when the vertical distance between the plane and the lamp increases. This behavior is a drawback to obtaining a general heuristics model of the light power behavior due to a different model being required each time the plane in which the receiver is moving changes. Consequently, the moving plane is fixed at a certain height to obtain a unique polynomial regression as follows:

$$P_r=a_n{d}^n+a_{n-1}{d}^{n-1}+\dots +a_2{d}^2+a_{1}d+a_0.$$

(16)

The above analysis allows estimating expressions from which it is possible to compute the distance between a lamp and the receiver based on the captured signal strength.

The distance estimation from (16) results straightforwardly when a second-order polynomial is obtained, challenging for high-order polynomials. The experimental tests showed that if the horizontal plane in which the object moves is superior to 1.5 m of the lamp’s flat, a second-order polynomial regression fits satisfactorily, as shown in Figure 18.

The captured data and their corresponding regressions are shown when the object moves in the planes at 1.5 m, 1.8 m, 1.95 m, 2.3 m, and 2.8 m of the lamp plane, respectively. The obtained polynomial regressions and their coefficients are given by:

$$\begin{array}{ccc}\hfill P{r}_{1.5}& =& \hfill -10.157d^3-29.908d^2-3.3482d+54.54,\forall R_{1.5}^2=0.9989;\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill P{r}_{1.8}& =& \hfill -0.7167d^2-15.067d+40.9;\forall R_{1.8}^2=0.9889,\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill P{r}_{1.95}& =& \hfill -1.6408d^2-9.4633d+34.223;\forall R_{1.95}^2=0.9897,\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill P{r}_{2.3}& =& \hfill -1.106d^2-6.3901d+25.528;\forall R_{2.3}^2=0.9938,\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill P{r}_{2.8}& =& \hfill +0.7737d^2-3.2078d+18.235\forall R_{2.8}^2=0.9903.\hfill \end{array}$$

(21)

Therefore, to keep a low complexity processing in the system, the object should move in planes at such distances to the lamp, guaranteeing a second-order polynomial is a good estimator for light power behavior, i.e., distances greater than 1.5 m. However, in an application requiring that the object moves closer to the lamp, estimating the distance between the object and the lamp is also possible, increasing the processing and model complexity.

From this analysis, two methods are proposed to estimate the distance from the RSS: (1) a deterministic model based on the second-order least-squares regression and (2) a computational model based on Artificial Neural Networks (ANN). The former was widely discussed earlier. The latter, a Multi-Layer Perceptron (MLP) with two neurons in the hidden layer, was designed. This represents a 1-2-1 architecture with sigmoid activation functions in the input and hidden layers and a linear activation function in the output layer. The training algorithm used was gradient descent by backpropagation; MLP was trained with a dataset of light intensities measured at different distance points around the lamps. After training, the neural network provides distances to the lamps from light measurements.

4. Results

4.1. Transmitter

The experimental setup includes 10 W slim downlight LED lamps used as transmitters. The modulation system was deployed using a Texas Instruments MSP430G2553 MCU per cell and an NPN Epitaxial Silicon Transistor TIP31C driver per lamp. The lamps were modulated at a base frequency of 20 kHz using OOK modulation and Manchester code. Thus, the modulation frequency is between 10 and 20 kHz, according to the digital code sent by each lamp. The base frequency of 20 kHz was chosen as an upper-limit frequency to achieve a linear response of the lamp circuit. Figure 19 shows the signals generated by the MCU to control three lamps for transmitting the codes 1.0, 48, and 170, respectively, using TMD and $t_t=1.0$ ms.

Every transmitted frame is defined by 16 bits, which in turn are divided into four-start bits, eight data bits, and four-stop bits. The transmission time of every bit is 50 $\mathsf{\mu }$s. Therefore, $t_{xL}$ is set to 800 $\mathsf{\mu }$s, and according to (2), the transmission sequence in a primary cell is 12.8 ms, and the system’s refresh rate is about 80 Hz.

4.2. Receiver

The Quadrant Si PIN photodiode s4349 from Hamamatsu was used in the receiver circuit. The IC used to amplify the signals captured was the current feedback amplifier AD8001 from Analog Devices. In addition, the values for the resistors Rf, R, and RL in the amplification module were 10, 1.0, and 100 k$\Omega$, respectively. The ultraprecision OpAmp AD8676 from Analog Devices was used in the conditioning stage. The first part of this circuit is an inverting buffer implemented using one of the OpAmps in the AD8676 and two resistors of 10 k$\Omega$. The second part of the conditioning stage is a pass-band filter implemented utilizing passive elements. One resistor of 1.0 k$\Omega$ and one capacitor of 1.0 nF are used to obtain a low-pass filter (cutoff frequency of 160 kHz), and a resistor of 1.0 M$\Omega$ and a capacitor of 10 nF define the high-pass filter (cutoff frequency of 16 Hz). Figure 20 shows the signal captured by the PD after applying the conditioning stage.

The digitalization stage was achieved using one of the two low-power comparators in the IC TLV2352 from Texas Instruments and a trimpot of 10 k$\Omega$, which sets the reference voltage. The output of this stage is connected to a digital input of an MCU MSP430G2553 of Texas Instruments. Finally, the peak detector was implemented using a Texas Instruments SPST CMOS analog switch TS12A4515 and a dual OpAmp integrated circuit OPA2350.

Figure 21 shows that the signal obtained from this stage overlapped with the signal received from the digitalization stage.

The 10-bit ADC of the MCU samples the peak detector signal while decoding the address information from the lamp. Despite the output signal of the peak detector being noisy, the averaged signal provides the magnitude of the light’s power received from each lamp. Consequently, the MCU takes several samples from the peak detector corresponding to each lamp transmission, and an average filter provides the signal’s power. The transmitted address and the received signal power are processed and grouped by the MCU. Once a transmission sequence is completed in the nearest cell to the receiver, the data stream may contain the address of each lamp. At the same time, the power is estimated, and the address and power are used to calculate the object’s position by trilateration.

4.3. Test Scenario

Three downlights’ LED lamps were fixed on the laboratory’s ceiling, and several landmarks under these lamps were also placed. The three lamps and landmarks create a triangle of 0.83 m${}^2$ used for localization on the floor. The receiver was situated inside this triangle at 1.8 m from the ceiling. The receiver was shifted inside the localization triangle through several trajectories on this plane. At different points of these trajectories, the lighting power of the signals serves to build the dataset. Figure 22 shows the test scenario with the receiver capturing and decoding signals from the lamps.

The bar plot on the right shows the signal’s power according to the distance between the lamps and the receiver. Figure 23 shows the different performed trajectories exploited to generate the proposed dataset.

Each lamp L is represented by white circles labeled with a TAG number, and the black point labeled as $CP$ represents the circumcenter. Twelve trajectories labeled with a letter T and an identifier number were used to measure data every 2.5 cm. Trajectories T1 and T4 were considered to diversify the number of scenarios and cover the same distance but in a contra sense. This experiment was repeated for several days at different hours for varying natural light conditions to verify system reliability.

4.4. Distance Estimator

The power of the received light emitted from each lamp is inversely proportional to the distance between the receiver and lamp, according to (11). Such an equation lets us estimate the distance from the receiver to each lamp based on the received power. Contrarily, the received light power based on the distance to the lamps can be modeled by a second-order polynomial. This deterministic model based on the quadratic regression of the experimental data is also helpful in estimating the distance to the lamp through the received signal power. The dataset obtained from our test scenario is then used to compute the three coefficients ($a_2$, $a_1$, and $a_0$) of the second-order estimator described by:

$$P_r=a_2{d}^2+a_{1}d+a_0.$$

(22)

Experimentally, even though the three lamps employed are of the same type, the respective emitted light power is slightly different. Therefore, each lamp has a particular model described by three different coefficients $a_2$, $a_1$, and $a_0$. The same dissimilarity effect is noticed for the light power model of the lamps. Hence, a set of coefficients was selected for providing a particular deterministic model to each lamp, choosing the lower MSE in (22). Finally, the expression to estimate the distance to the lamps according to RSS is obtained by solving (22) as follows:

$$d=\frac{-a_1\pm \sqrt{a_1^2-4a_2(a_0-P_r)}}{2a_2}.$$

(23)

The two solutions for d represent the light behavior when the receiver moves close and away from the lamp.

Conversely, the non-lineal nature of light may limit a deterministic model to obtain the distance adequately. Therefore, one MLP per lamp was trained using the acquired measurements to achieve a computational model that compensates for these non-linearities. Each MLP receives the signal power of the lamp as input and provides one output that represents the distance between the receiver and the lamp, using two internal hidden layers. The dataset was divided into two parts: 80% of the data was used for training and 20% for validation.

Moreover, a second ANN architecture was trained to obtain a general estimator to model the three lamps’ behavior. So, this model is developed to compare the use of an individual MLP per lamp and a second general model exploiting the three lamps’ measurements.

4.5. Experimental Results

The results show an average positioning error of about 2.0 cm using the deterministic estimator and the ANN model per lamp but about 40 cm using the general ANN model.

Table 1. Absolute error between the calculated position and the real position from different estimators.

Model	Mean Absolute Error	Standard Deviation
Quadratic	2.40 cm	1.90 cm
Individual ANN	2.50 cm	1.60 cm
General ANN	39.8 cm	30.1 cm

presents the best results obtained from the different estimators according to the absolute error between the calculated position and the real position of the receiver.

Likewise,

Table 2. The mean squared error and the root mean squared error between the calculated position and the real position using different estimators.

Model	MSE	RMSE
Quadratic	0.0976 mm${}^2$	3.10 cm
Individual ANN	0.0897 mm${}^2$	3.00 cm
General ANN	0.0868 mm${}^2$	9.30 cm

presents the MSE and RMSE of the calculated position using the different estimators.

Table 1 and Table 2 show the worst estimation cases for coordinates x and y computed by the trilateration method. So, Table 1 shows that three methods are good estimators. However, the numerical comparison shows that the individual ANN model is the best estimator and the general ANN is the worst.

5. Conclusions

This article presents a new indoor positioning system using visible light communications to prove the VLC positioning principle. The system comprises commercial LED lamps modulated using OOK and Manchester code. The LED lamps are generic to facilitate access to materials in an actual context. In addition, the use of OOK and Manchester code enables the recovery of the synchronization signal on the receiver side, providing better communication channels and reduced interference in VLC. In addition, OOK and Manchester code avoid noticeable changes in the light intensity or flickering caused by different stream transmissions. The medium access control uses Time Division Multiplexing to avoid interference between the lamps to preserve the lamp’s original light intensity.

The system was designed to be scalable; therefore, the primary location cells can be replicated using the spatial multiplexing feature of the light. These primary cells are composed of seven lamps, but it is possible to use incomplete primary cells, called special location cells, to cover surfaces with different geometric shapes. The position determination is estimated in the receiver, which gives privacy and security to the system and enables it to operate a virtually infinite number of receivers. The system utilizes trilateration to compute the object’s position. This method requires knowing the distance to at least three lamps to operate fast and accurately. OOK and trilateration give the system a reduced complexity feature and high reproducibility.