A Survey on Indoor Visible Light Positioning Systems: Fundamentals, Applications, and Challenges

While VLC has garnered significant attention in recent years, the utilization of light for communication traces back centuries. From ancient times, light has served as a medium for transmitting information, evident in early practices like torch signals and smoke signals. Since the earliest times, the light was already used in information transmission, such as torches and smoke signals.

In the modern era, the roots of VLC extend back to the 1880s in Washington, D.C., with Alexander Graham Bell’s invention of the photophone. This groundbreaking device modulated sunlight to transmit speech, and the transmission distance can reach to several hundred meters. Fast forward over a century later, the advent of LED technology sparked a new era for VLC, presenting a fresh frontier for technological advancement.

VLC based on LED dates back to 1999, when researchers from Hong Kong University first proposed to modulate visible light signals on LED [6, 7], and the proposed system has been used for intelligent transportation. Subsequently, in 2000, the Nakagawa Laboratory in Keio University, Japan, in collaboration with Sony’s Computer Science Laboratories, introduced the concept of employing visible light LEDs for indoor data transmission [8]. They successfully established a data transmission system using a white LED bulb within an indoor environment [9], marking a pivotal moment in VLC research in the 21st century.

Following these developments, the first standardization for VLC, IEEE 802.15.7, was spearheaded by the IEEE 802.15 working group in 2011. The same year witnessed Harald Haas demonstrating Li-Fi in a TED Talk, and Li-Fi was named by Time magazine as one of the top 50 inventions. This milestone propelled VLC into an extensive realm of study encompassing modulation technology, multiple input multiple output (MIMO) technology, networking, and beyond [10]. Simultaneously, the exploration of VLC applications garnered increasing attention, extending into diverse fields including indoor access, positioning technology, and even underwater communication. The subsequent section will delve deeper into the history of VLP, outlining its evolution and early accomplishments.

As VLC emerged, it spurred the development of VLP, a highly promising indoor positioning technology. This section will delve into the evolution of VLP, highlighting key milestones and achievements during its initial phases of development.

The origins of VLP can be traced back to as early as 2001, when Pang and Liu modulation of location codes within an LED location beacon system [11]. This innovative system used a complementary metal-oxide semiconductor (CMOS) vision sensor to extract location code for calibration of a vehicle positioning system, which may consist of a Global Positioning System (GPS), inertial navigation system (INS), and other sensors. While serving as an auxiliary positioning scheme, this work marked the pioneering introduction of LED beacons into positioning methodologies. The first dedicated VLP system tailored for indoor positioning emerged through the work of Horikawa et al. from Japan [12]. This system employed optical intensity modulation for LED illumination, transmitting visible light signals received by an image sensor. To compute the receiver’s position, at least three LEDs and two image sensors were required, assuming the receiver was oriented toward the ceiling. Then, in 2008, Yoshino et al. [13] proposed to use a “collinearity condition” to calculate not only the position but also the orientation of the receiver. This condition stipulated that the projection of each spatial point and the center of the lens of the receiver must be on the same straight line.

Following the inception of VLP, researchers began to investigate innovative methods to enhance the performance of VLP. The earliest positioning techniques within VLP [11, 12, 13, 14] predominantly centered around image sensing, often coupled with trilateration or triangulation methodologies. For instance, the authors used two image sensors to establish the geometrical relationship between the distance and the position difference of the LED images and to calculate the distances between the center point of two lenses and three LEDs [14]. In a preliminary study [14], researchers employed two image sensors to establish a geometric relationship among the positional differences observed in LED images. Subsequently, distances were computed by analyzing the interplay between the center points of the lenses and three LEDs. The three distances were used to formulate the equations for location estimation according to triangulation. Subsequently, researchers identified and capitalized on geometric attributes such as circular features [15, 16] and rectangular features [17], leveraging these distinctive shapes to further refine the accuracy and efficiency of VLC-based positioning techniques.

In 2010, a pioneering step was taken with the introduction of the initial fingerprinting algorithm for VLP [18]. This innovative algorithm introduced the correlation sum ratio (CSR), a novel value derived from RSS information obtained from four LEDs. CSR functioned as distinct fingerprints corresponding to various locations. Utilizing pre-assigned information obtained offline, a receiver could determine its position by analyzing CSRs received at any given location. Subsequently, Vongkulbhisal et al. [19] formally categorized this approach as the fingerprinting-based positioning method in VLP. Their work expanded on this concept, processing multiple signals from six transmitters to generate the fingerprint map in the offline stage or estimate its location in the online stage.

Then, the technique based on RSS was introduced for VLP in 2011 [20]; however, it can only achieve a one-dimensional (1D) or two-dimensional (2D) positioning. Another RSS-based VLP scheme [21] was proposed in 2012. The proposed scheme used four LEDs and assumed Lambertian model to formulate a transmission equation group that can be solved for three-dimensional (3D) location estimation. Since then, the RSS method has been extensively used in VLP, which later gave rise to another method known as received signal strength ratio (RSSR).

Meanwhile, researchers developed a time difference of arrival (TDOA) approach for VLP in 2011. For instance, Bai et al. [22] proposed a TDOA-based VLP technique that can determine a vehicle’s position by using photodiodes LED traffic lights. This approach leveraged TDOA measurements between the traffic light signals and two photodiodes, formulating equations to estimate position. They proposed two distinct methods tailored for scenarios involving one traffic light and those with two traffic lights. In addition, Jung et al. [23] also developed a TDOA-based indoor positioning algorithm. This method synchronized the frequency address of each LED. Using the Hilbert transform, they extracted the phase difference to estimate distances, subsequently employing these distance values to determine the receiver’s position.

Furthermore, the pioneering VLP technique utilizing angle of arrival (AOA) was introduced in 2012 [24]. This method presented an AOA estimation algorithm tailored for pinpointing the location of LEDs within a VLC environment. Leveraging a circular PD array, the system aimed to accurately determine AOA, supplemented by a truncated weighting algorithm designed to bolster AOA estimation precision. In 2014, Yamazato and Haruyama used image sensors to detect AOAs of light emitted from visible light transmitters, which were further used in pose estimation [25]. The researchers also systematically introduced the pose and position estimation method by introducing the computer vision method.

As the field of VLP advanced, efforts intensified toward integrating two or more techniques in VLP to enhance the accuracy and practicability of indoor positioning [25, 26, 27]. Notably, these endeavors included amalgamating AOA with image sensing [25], blending RSS with AOA [26], and integrating RSSR with image sensing [27] for more comprehensive pose and position estimation methodologies.

To offer a clear overview, we have encapsulated the historical evolution of VLC and VLP in Fig. 1. Despite having about two decades of development, VLP has already achieved centimeter-level high positioning accuracy and is one of the most promising positioning technologies in indoor positioning. In Section III-B, we will delve into the principles and latest advancements across diverse VLP techniques to provide a comprehensive understanding.

Proximity is the simplest positioning technique among the existing VLP techniques. Proximity takes the location of the closest transmitter as the location of the receiver, and thus this technique can only provide an approximate position result. In particular, each transmitter constantly broadcasts a unique ID code [55], which is associated with a specific location of the transmitter stored in a database. The receiver captures and detects the ID information that will be matched to the locations of the transmitters so that the receiver can estimate its location. When simultaneously receiving ID signals from multiple transmitters, the PD-based receiver will determine its position by the transmitter with the strongest RSS value since RSS is closely related to the distance between the transmitter and the receiver. To deal with the data transmission and identification in such multiple signals scenarios, Cherntanomwong et al. [56] used TDM in their proximity-based VLP system for identification in the light overlapping area. In addition, the camera can also be leveraged for proximity. For instance, Xie et al. [57] identified the IDs of the transmitters by processing the proposed LED-ID recognition method on the captured image based on the Fisher discriminant analysis method.

Overall, proximity is simple and highly feasible in practice. However, it can only provide limited positioning accuracy, which significantly depends on the density of the LEDs.

TOA is the absolute arrival time of signals from the transmitter to the receiver, which is a common technique for GPS [58]. In particular, TOA algorithms estimate the distances between transmitters and receivers according to the arrival time of signals, and further use the estimated distances to derive the location of the receivers. In VLC-based TOA algorithms, once the propagation delay of the signal is measured, the distance can be obtained by multiplying the delay and the speed of light. Then, TOA typically utilizes the trilateration method to obtain the position of the receiver.

Fig. 7 illustrates the principle of the trilateration method. The LEDs locate at the center of the circles, and the radius denotes the distance between the transmitter and the receiver. Suppose that $\left(x,y,z\right)$ is the coordinate of the receiver, $\left(x_{i},y_{i},z_{i}\right),i=1,2,...,n$ are the coordinates of $n$ LEDs, and $\tau_{i},i=1,2,...,n$ are the times of arrival. Then, the distance from LED $i$ to the receiver can be expressed as

$${{d}_{i}}=\sqrt{{{\left(x-{{x}_{i}}\right)}^{2}}+{{\left(y-{{y}_{i}}\right)}^{2}}+{{(z-{{z}_{i}})}^{2}}}=c\cdot{{\tau}_{i}}$$

(7)

where $c$ represents the speed of the light prorogation. On the 2D plane, each TOA can determine a circle, and thus the location of the receiver can be determined by calculating the intersection of the three circles. In the 3D space, each TOA can determine a sphere, and thus four LEDs are needed to form four spheres and intersect at a unique point, which is the location of the receiver. Wang et al. [59] proposed a TOA positioning system, in which, the LEDs and the receiver were assumed to be synchronized perfectly. The LEDs used the orthogonal frequency division multiplexing (OFDM) technique to transmit the signals, so that the receiver can separate the signals from different LEDs and estimate the location of the receiver.

However, due to the imperfect hardware, the measured distance usually deviates slightly from the truth, and thus the drawn circles or spheres may not exactly intersect at a point. Instead, the intersection forms an overlapped area. Therefore, the least square method is applied to obtain the optimal solution [60], and the cost function can be expressed as

$$F(x,y)\!=\!\sum\limits_{i=1}^{n}{\!\left(\!c{{\tau}_{i}}\!-\!\sqrt{{{\left(x\!-\!{{x}_{i}}\!\right)}^{2}}\!+\!{{\left(y\!-\!{{y}_{i}}\right)}^{2}}\!+\!{{(z\!-\!{{z}_{i}})}^{2}}}\right)}.$$

(8)

Since the arrival time is typically rather short in VLP, TOA requires that transmitters and receivers have extremely accurate time synchronization, which is difficult to implement. The TDOA algorithms can alleviate the rigorous time synchronization at the receiver. They typically need at least three receivers to measure the propagation time difference between the mobile device and transmitters, and it only requires time synchronization among transmitters[61]. Based on the time difference, the distance difference can also be derived. Letting the distances between the receiver and LED $i$ be $d_{i}$, and that between the receiver and LED $j$ be $d_{j}$, the distance difference between $d_{i}$ and $d_{j}$ can be expressed as

$$\begin{split}{{d}_{ij}}&={{d}_{i}}-{{d}_{j}}\\ &\text{ =}\sqrt{{{\left(x-{{x}_{i}}\right)}^{2}}+{{\left(y-{{y}_{i}}\right)}^{2}}+{{(z-{{z}_{i}})}^{2}}}-\\ &\text{ }\sqrt{{{\left(x-{{x}_{j}}\right)}^{2}}+{{\left(y-{{y}_{j}}\right)}^{2}}+{{(z-{{z}_{j}})}^{2}}}=c\left({{\tau}_{i}}-{{\tau}_{j}}\right).\\ \end{split}$$

(9)

According to (9), a hyperbolic positioning method can be used to determine the location of the receiver [62, 61]. The positioning principle of TDOA is shown in Fig. 7. Each distance difference can determine a hyperbola, where the two LEDs are the two focal points, and three hyperbolas corresponding to three LEDs can intersect at a point, which is the 2D location of the receiver. Similarly, the 3D location can be determined by four LEDs.

T. H. Do et al. [61] proposed a positioning system based on TDOA values. In the system, the receiver was a single PD, which received pilot signals from the LEDs. The TDOA values of the pilot signals are used to estimate the location of the receiver. The proposed system can be employed easily since the receiver can achieve positioning without embedded ID information at the LEDs. In addition, J. H. Y. Nah et al. [62] considered additive white Gaussian noise (AWGN) in the TDOA-based VLP system, and they proposed a Fuzzy logic algorithm and a Spring model to minimize the noise affect after position estimation. Then, a practical TDOA-based VLP system that used a virtual local oscillator to replace the real local one was proposed [63], which could reduce the hardware complexity. This system also applied cubic spline interpolation to the correlation function to reduce the rigorous requirement on the sampling rate and enhance the time resolution of cross correlation. Pergoloni et al. [64] proposed wavelength-based localization and color-based localization mechanisms by combining traditional RSS and TDOA approaches. They assumed that each anchor point used a unique spectral signature on the wavelength domain so that the receiver could identify it and compute its location through RSS or TDOA approaches.

After that, the combination of TDOA with other received information, such as RSS and fingerprinting, was applied in the VLP systems [65, 66]. Sheikh et al. [67] proposed a TDOA-based positioning system using VLC links in a network simulator. This system computed the cross correlation between signals arriving at the target node from all sensors to formulate the distance equations between the transmitter and the target. Under the assumption that the beacon nodes are stationary and well synchronized, the proposed TDoA-based VLC scheme outperformed the existing WiFi schemes. Cao et al. [68] proposed a phase-difference-based TDOA method, in which, a convolution neural network-based phase difference estimation was designed to relieve the influence of different error sources such as Gibbs phenomenon and time synchronization error. Then, a motion-based particle filter was proposed to improve the accuracy and robustness further. The simulation results showed that an average accuracy of 0.280 m was achieved.

There is a scarcity of applications of the TOA and TDOA methods in VLP. The limited adoption of TOA/TDOA in VLP can be primarily attributed to the stringent requirement for precise time synchronization. Visible light has a much shorter wavelength than radio waves, which may necessitate more precise and sophisticated hardware for accurate time difference measurements.

The AOA algorithms estimate the location of the receiver based on the angles between the transmitters and receivers. Fig. 8 illustrates the principle of AOA algorithm. When the AOA information from multiple VLC links is obtained, the location of the receiver can be determined as the intersection of VLC links according to geometric relationships. Suppose that the angles of incidence of the receiver from two LEDs are $\theta_{1}$ and $\theta_{2}$, respectively, and the coordinates of the receiver and the transmitters are $\left(x,y\right)$ and $\left(x_{i},y_{i}\right)$, respectively. We have

$$\tan{{\theta}_{i}}=\frac{y-{{y}_{i}}}{x-{{x}_{i}}},i=1,2,...$$

(10)

Then, triangulation can be used for AOA algorithms according to (10) [26, 69, 70]. In particular, at least two transmitters are needed for 2D AOA positioning algorithm, while at least three transmitters are needed for 3D positioning. Sun et al. [71] provided a traditional AOA algorithm, and the authors derived Cramer-Rao bound to analyze the theoretical accuracy of the algorithm. Yu et al. [72] employed an optimal optical omnidirectional angle estimator for AOA system, which can mitigate the side effect of the uncertain orientation of the receiver. When more than two LEDs were exploited, two supplemented methods that selected the LEDs with more reliable measured power data were proposed to reduce the estimation error. In addition, Soner et al. [73] proposed a quadrant photodiode-based AOA algorithm for vehicle positioning to avoid collision and platooning.

The performance of AOA algorithms significantly depends on the acquisition of AOA values. AOA values are often calculated by the VLC signals received by PDs and the image captured by the camera.

The first approach estimates AOA based on the Lambertian channel model, which is relative to radiation angle and incidence angle according to (3). Therefore, it is typically assumed that the transmitter plane should be horizontal and parallel to the receiver plane [69, 71, 72], which is inapplicable to scenarios with random receiver orientation. To overcome this limitation, a positioning framework was proposed based on angle differences of arrival in a 3D coordinates system [4], which had no receiver orientation limitations. Some researchers investigated different designs of AOA detectors [74, 75, 76] to relax the orientation limitation at the receiver. For instance, Zhu et al. [74] proposed to use pairs of PDs, namely complementary PDs, to construct AOA estimators. Stefanie et al. [75] investigated a new form of AOA detector that was a quadrant PD placed below a transparent aperture in an opaque screen. Zhang et al. [76] used two optical AOA estimators for locating LEDs. The AOA estimators had fixed relative positions, and each estimator consisted of four PDs with different orientations. In addition, MEMS sensors like accelerators and gyroscopes were also used to measure the orientation of the receiver, including azimuth, roll, and pitch angles [70].

There are also approaches that use the camera to estimate the position of the receiver. They obtain the three angles by calculating the trigonometric relationship between the coordinates of the transmitters and the receivers from the captured image. For instance, Hossei et al. [77] proposed a geometrical optics positioning algorithm (GPOA) based on AOA to locate smartphones.

The RSS algorithm is one of the most widely-used methods in indoor VLP, and it estimates the location of the receiver based on the power of the received signal, i.e., RSS values from LEDs. RSS values can be easily obtained using PDs equipped on the receiver. As illustrated in (3), the RSS increases as the distance between the transmitter and the receiver decreases. When the angles of incidence and radiation are fixed, the distance between the receiver and the corresponding LED can be calculated using the detected RSS values based on the Lambertian channel model. Then, the trilateration method can be applied for positioning.

Gu et al. [78] proposed to calculate the horizontal coordinates of the receiver given the RSS information from four LEDs, after which the height of the receiver was also estimated. It also used Kalman and particle filters to realize target tracking. An RSS-based trilateration method using the code division multiple access technique was proposed by Guan et al. [79]. Assuming that the transmitter and the receiver planes were parallel, they used RSS to estimate the distance by increasing the height of the transmitter from 0, and the distance was used to calculate the location of the receiver using the trilateration method. Zhang et al. [80] proposed a deep neural network (DNN)-based RSS positiong system. The DNN was trained by the Bayesian regularization based on the Levenberg-Marquardt algorithm, so that unknown positions across the same area can be estimated by using the trained DNN on limited training points. Then, Saboundji et al. [81] implemented the artificial neural network learning for an RSS algorithm to achieve highly accurate and efficient indoor positioning. Salman et al. [82] suggested the demonstration of a 3D VLP system, combining RSS and trilateration solution so that the speed of the computation may be increased considerably. Shen et al. [83] used hybrid maximum likelihood/maximum a posteriori principle for a multiple LEDs - multiple PDs system to achieve positioning, which took into account the presence of prior information on the orientation.

In addition, RSSR algorithms are also applied to VLP. Different from RSS algorithms that directly use the received power to calculate the distance between the transmitter and the receiver, RSSR algorithms calculate the ratio of distances according to the ratio of the received power from multiple transmitters and estimate the location based on the distance ratios. RSSR has the advantage that the ratio of received power can avoid the error caused by the non-zero irradiance [30], and thus it has less error than the distance directly obtained from the power. However, RSSR algorithms typically require the receiver plane to be parallel to the transmitter plane [84, 85], so that the incidence angle can be equal to the irradiance angle. Hence, the received power can be derived from (3) as

$${{P}_{r}}={{P}_{t}}\frac{(m+1)A}{2\pi{{d}^{2}}}g(\psi){{\cos}^{m}}(\phi){\cos}(\psi)=\frac{U}{{{d}^{m+3}}}$$

(11)

where $U=\frac{(m+1){P_{t}}Ag(\psi)h^{m+1}}{2\pi}$ is a constant that can be calculated. Then, the distance ratio can be expressed as

$$\frac{{{d}_{1}}}{{{d}_{2}}}=\sqrt[{}^{{}^{m+3}}]{\frac{{{P_{r_{2}}}}}{{{P_{r1}}}}}.$$

(12)

By substituting the coordinates of two LEDs, $({{x}_{1}},{{y}_{1}},{{z}_{1}})$ and $({{x}_{2}},{{y}_{2}},{{z}_{2}})$ into (12), we have

$$\frac{\sqrt{{{\left(x-{{x}_{1}}\right)}^{2}}+{{\left(y-{{y}_{1}}\right)}^{2}}+{{(z-{{z}_{1}})}^{2}}}}{\sqrt{{{\left(x-{{x}_{2}}\right)}^{2}}+{{\left(y-{{y}_{2}}\right)}^{2}}+{{(z-{{z}_{2}})}^{2}}}}=\frac{{{d}_{1}}}{{{d}_{2}}}.$$

(13)

In this way, two simultaneous equations can achieve 2D positioning when three LEDs are detected. Three simultaneous equations with four LEDs used can achieve 3D positioning.

Since RSS or RSSR algorithms rely on a perfect channel model, they are susceptible to the incidence and irradiance angles. Several RSS and RSSR algorithms have been proposed to tackle the transmitter and receiver orientation limitations [86, 87, 88]. Wang et al. proposed two designs for RSSR-based VLP algorithms, including multiple directional LED array [86] and multiple direction PD array [87], to reduce the error caused by the orientation of transmitters and receivers, respectively. In addition, Li et al. [88] also considered the orientation uncertainty of the receiver by coping with the non-linear relationship between the RSS and the orientation uncertainty. They utilized the first and second-order Taylor series expansion of RSS to find an accurate approximation for the RSS when the orientation of the receiver was uncertain.

In fingerprinting algorithms, one or multiple features related to the receiver’s position are selected as fingerprints. The position of the receiver is estimated by matching the measured data with the prestored location-related data. In particular, there are two phases for the fingerprinting algorithm, including the offline phase and the online phase. In the offline phase, the data related to the location are collected and stored in the database. In the online phase, the location of the target is estimated by matching the currently measured data to the pre-stored database.

There are various features of the signal that can be selected as fingerprints. Most of the existing literature adopted RSS as the fingerprints, and they typically used RSS vector to further enhance the positioning accuracy[89, 90, 91, 92]. RSS vector consists of several RSS values from multiple transmitters [93, 94, 95]. Received signal strength indicator (RSSI), as another representation of RSS, was used as a fingerprint [96, 97]. In addition, the channel impulse responses (CIR) were also selected as a fingerprint [98]. Yang et al. [99] chose the extinction ratio that represented the ratio of received powers when bit 1 and bit 0 are transmitted as the fingerprint. In another work [100], light power distribution calculated from grayscale images was used as fingerprints for indoor positioning and tracking. Moreover, Zhao et al. [101] proposed a LightPrint as the fingerprint, which is a vector of multiple light intensity values obtained from existing lighting infrastructure with any unmodified light source during the user’s walks.

To match the measured data to the database accurately, a number of match methods have been studied. In particular, probabilistic methods and k-nearest neighbors (kNN) algorithms are extensively used.

The probabilistic method utilizes the probability distribution of RSS to estimate the location of the target with the Bayesian method. Supposing that the transmitters are independent of each other, the probability of a location candidate $L_{i}$ can be calculated as the probability of the RSS from all transmitters by

$$P\left(\bm{R}|{{L}_{i}}\right)=\prod\limits_{j=1}^{n}{P\left({{R}_{j}}|{{L}_{i}}\right)}$$

(14)

where $\bm{R}=\left({{R}_{1}},{{R}_{2}},...,{{R}_{n}}\right)$ is an RSS vector composed of RSS values from $n$ transmitters, and $P\left({{R}_{j}}|{{L}_{i}}\right)$ represents the probability of the RSS value $R_{j}$ from the $j$th transmitter when the receiver locates at location $L_{i}$. The probability distribution of the RSS is calculated and stored in the offline stage. Then, the positioning coordinate can be estimated by a weighted average of the coordinates as

$$\left(\widehat{x},\widehat{y}\right)=\sum\limits_{i=1}^{n}{P\left({{L}_{i}}|\bm{R}\right)}\left({{x}_{{{L}_{i}}}},{{y}_{{{L}_{i}}}}\right)$$

(15)

in the online stage. Kail et al. [102] proposed a probabilistic positioning system that addressed the problem of unpredictable obstructions and synchronization error based on a Bayesian model. Another research [103] proposed a light-signal decomposition method to extract fingerprints, and then a Bayesian localization framework was applied to improve the precision. Ou et al. [104] used kernel functions to model the spatial correlation nature for the reflected lights. Based on that, target location estimates are given by the Bayesian inference both in a discrete setup and a continuous framework, which are the probabilistic fingerprinting and the maximum likelihood estimate.

The kNN method is another typical method adopted in fingerprinting algorithms based on Euclidean distance [89, 90, 91, 92, 93]. The Euclidean distance $D_{j}$ between the online measured RSS vector $\bm{R}=\left({{R}_{1}},{{R}_{2}},...,{{R}_{n}}\right)$ and the offline RSS vector ${{\bm{S}}_{j}}=\left({{S}_{j,1}},{{S}_{j,2}},...,{{S}_{j,n}}\right)$ can be calculated by

$${{D}_{j}}=\sqrt{\sum\limits_{i=1}^{n}{{{\left({{R}_{i}}-{{S}_{j,i}}\right)}^{2}}}}$$

(16)

where ${S}_{j,i}$ denotes the prestored RSS value from the $i$th transmitter in RSS vector, ${{\bm{S}}_{j}}$, which is related to the $j$th location candidate. According to the Euler distance, there are usually several location candidates can be selected. When $k$ location candidates are selected, the final location of the target is determined by averaging $k$ location candidates. For simplicity, $k=1$ was typically selected [19, 98].

Although the kNN method is simple and highly feasible, it is limited by the size of the cell, i.e., the density of the grid. To address this issue, weighted K-nearest neighbors (WkNN) was adopted for fingerprinting algorithms [89, 90, 91, 97, 105]. The WkNN method assigns weights to the distances so that the neighbor with a smaller distance has a greater weight compared with the neighbor with a greater distance. Based on the WkNN method, Cui et al. [97] proposed a clustering algorithm and used Spearman distance to improve the performance of the fingerprint algorithm. Oh et al. [105] constructed a fingerprinting database based on the channel characteristics of VLC and obtained the approximate location of the user by applying WkNN. In addition to kNN, extreme learning machine (ELM) and random forest were also applied to fingerprinting [92].

With the popularity of the camera equipment, the image sensing-based VLP technique has also attracted massive attention. This technique estimates the location of the target by analyzing the geometric relation between the LEDs’ coordinates and their corresponding projection on the image. The image sensing in VLP is similar to the multi-view geometry in computer vision. The key difference is that the camera can receive the VLC signals to obtain the information of transmitters in VLP.

To identify the transmitters and obtain the information in image sensing algorithms, the VLC signals should be received and decoded. Thus the under-sampled phase shift on-off keying [106] based modulation and camera rolling shutter effect [107, 108] based modulation methods are adopted for communications.

In image sensing, there are four coordinate systems established for calculating the geometric relation between the corresponding points. They are a 3D world coordinate system, a 3D camera coordinate system, a 2D image coordinate system, and a 2D pixel coordinate system. The image coordinate $\left(x^{\rm{i}},y^{\rm{i}}\right)^{\rm{T}}$ can be easily obtained through image processing [109], while the world coordinate of the LED $\left(x^{\rm{w}},y^{\rm{w}},z^{\rm{w}}\right)^{\rm{T}}$ can be obtained by receiving the modulated VLC signal from the LED. The camera coordinate $\left(x^{\rm{c}},y^{\rm{c}},z^{\rm{c}}\right)^{\rm{T}}$ can be calculated by

$${{\left({{x}^{\text{i}}},{{y}^{\text{i}}},1\right)}^{\text{T}}}=\mathbf{A}{{\left({{x}^{\text{c}}},{{y}^{\text{c}}},{{z}^{\text{c}}},1\right)}^{\text{T}}}$$

(17)

according to pinhole camera model as illustrated in Fig. 9, where

$$\mathbf{A}=\begin{bmatrix}f&0&0&0\\ 0&f&0&0\\ 0&0&1&0\\ \end{bmatrix}$$

(18)

is the calibration matrix of the camera, which can be calibrated and known in advance, and $f$ is the focal length of the lens. Then the geometric relation between the world coordinate system and the image coordinate system can be described as

$${{\left({{x}^{\text{i}}},{{y}^{\text{i}}},1\right)}^{\text{T}}}=\mathbf{A}\cdot\left(\begin{matrix}\mathbf{R}&\mathbf{T}\\ \mathbf{0}&\mathbf{1}\\ \end{matrix}\right){{\left({{x}^{\text{w}}},{{y}^{\text{w}}},{{z}^{\text{w}}},1\right)}^{\text{T}}}$$

(19)

where $\mathbf{R}$ is a $3\times 3$ rotation matrix representing the pose of the receiver, and $\mathbf{T}$ is a $3\times 1$ translation vector representing the location of the receiver. The goal of image sensing algorithms is to find the pose and location of the receiver, i.e., the camera.

Single view geometry is widely used in image sensing based VLP algorithms [108, 109, 110, 111, 15, 112, 113]. Guan et al. [108] proposed a double-LED positioning (DLP) system using CMOS image sensor. The authors utilized a rolling shutter mechanism and machine learning algorithm to identify the IDs of LEDs. DLP required the camera to parallel to the LEDs and solved the symmetry problem in this circumstance. Lain et al. [111] proposed a $K$-pairwise LED image-sensor-based VLP (IS-VLP) algorithm for indoor positioning. $K$-parwise LED IS-VLP identified the mapping between the received LED IDs and their images to determine the coordinates of the LEDs, which was then used to estimate the location of the receiver. Moreover, Huang et al. [114] proposed a 3D NLOS VLP system using a single LED and an image sensor (IS) to address the problem of obstructed LOS paths. Two virtual LEDs reflected from the ground together with the real LED are captured for positioning.

In addition, the researchers also gradually focus on the geometric features captured by the image sensing. For instance, Zhang et al. [15] extracted the feature of a single circular LED to locate the target by assuming a weak projection model with the assistance of another built-in sensor. Zhu et al. [16] proposed a VLC-assisted perspective circle and arc algorithm (V-PCA), which exploited the geometric features between two LEDs and their circle images for positioning. To reduce the required luminaires to one, they further proposed a visual odometry-assisted VLP algorithm [115]. Meanwhile, Bai et al. [17] considered the rectangular features of a single luminaire in the positioning process.

To attain practical and highly accurate positioning, recent research has focused on merging multiple VLP algorithms, termed hybrid VLP algorithms in this context.

These approaches often amalgamate two classical VLP algorithms to achieve superior positioning performance. We categorize these hybrid algorithms into several distinct types for clarity and classification.

When combined with RF, a heterogeneous indoor positioning system using both LiFi and WiFi was envisaged to improve the accuracy of indoor positioning [141]. Zigbee wireless network was constructed in a VLP system to transmit VLC data to reduce the position estimation error caused by nearby visible light channels [142]. There was also a two-stage positioning system developed [143]. In the first stage, RF was used to detect the room where the device located. In the second stage, LiFi was employed to detect the specific position of the device. The estimation error was reported to be only 5.8 cm. In addition, Shi et al. [144] proposed a 5G indoor positioning scheme based on VLC and indoor broadband communication for the museum. The system utilized unlicensed visible light of the electromagnetic spectrum to provide visitors with high-accuracy positioning on a mobile device, realizing a mean positioning error of 0.18 m.

When combined with Bluetooth, Luo et al. [145] proposed a Bluetooth signal based spring model to hybrid VLP and Bluetooth positioning. The intensity of visible light signals was detected through the Bluetooth beacon set in advance to match the fingerprint database. Simulation results showed that the system can achieve an average positioning accuracy of 6 cm. Hussain et al. [146] used a VLC-based indoor mapping application to facilitate Bluetooth MAC address mapping. In this way, the advantages of VLC and Bluetooth can be combined to achieve superior positioning performance. Albraheem et al. [149] utilized VLC proximity for initial location determination and Bluetooth RSS trilateration for refinement, achieving an accuracy of up to 0.03 meters. The VLC system estimated the receiver’s location based on modulated information from the light source, which was demodulated at the receiver into a distinct identifier code. Subsequently, the Bluetooth system used an RSSI-based trilateration technique to further refine the receiver’s location.

In addition, the combination of acoustic positioning and VLP has also been studied [147] and [148]. Akiyama et al. [147] measured the propagation time of acoustic signal through an acoustic sensor equipped in the phone. With the obtained propagation time, TOA was then used in VLP to achieve precise localization. Experiments showed that the positioning accuracy can be achieved up to 100 to 200 mm. Png et al. [148] integrated two acoustic sensors on the Arduino hardware board. The sensors can detect the moving distance of pedestrians on the X and Y axes in a 2D plane respectively, which improved the positioning readings to centimeters and helped the system obtain higher accuracy.

IMU can measure the acceleration and angular velocity of the object jointly in real-time, and thus it can be combined with VLP when visible light signals are not available. Liang et al. [136] proposed an EKF-based visual and inertial fusion method for visible light positioning with an IMU and a rolling shutter camera. The results showed that the method can keep the RMSE errors of positioning to 5 cm. Zhang et al. [156] developed an indoor positioning system based on VLP and sensor fusion techniques. The system enhanced positioning accuracy by fusing data collected by the image sensor and motion sensors of a smartphone. The experiment results showed that the proposed algorithm improves the position accuracy by 44% when compared with the algorithm employing a single image sensor. Then, Qin et al. [157] fused the IMU and VLC information in a tightly coupled scheme so that the VLP system could continually provide location service under the aforementioned intermittent outage condition. Experimental results showed that the proposed method had better stability than the PnP methods. At the same time, fusion positioning schemes using a single circular LED were proposed [137, 158]. In particular, the geometric features of LED projection and plane intersection lines were used for deriving relative position relation between the LED and the receiver, and IMU was used to estimate the orientation of the receiver [137].

PDR can achieve positioning without GPS by analyzing the data obtained by inertial sensors. However, due to the accumulation of external noise and internal drift error, PDR is arduous to be adopted as an independent navigation or positioning scheme without proper correction [159]. If PDR can be integrated with VLP, its accuracy can be significantly improved with the assistance of a VLP system.

There have been studies on the combination of VLP and PDR[160, 161, 162, 138, 140, 139, 164, 165]. Li et al. [160] proposed to simultaneously process the data from PDR and VLP by particle filter. In particular, after obtaining positioning data of PDR in some specific positions, the system put it into a particle filter together with VLP data. Filter processed two kinds of data and obtained the final coordinates of pedestrians. The scheme solved the performance degradation caused by the multipath effect and light transmission-blocking and alleviated the problem of cumulative error of inertial navigation. The experimental results showed that positioning error could be as low as 0.14 m. After that, they have also proposed to use the Kalman filter to simultaneously process VLP and PDR data [161]. The inertial navigation without requiring external sources improved the positioning performance of the VLP system in the region with weak visible light intensity or occlusion, while the accumulated error of inertial navigation was reduced with the assistance of VLP data. The experimental results showed that the hybrid system was better compared to the single VLP or inertial navigation.

Different from works using PD as the receiver [160, 161], some works [162, 138] employed the image sensor. In particular, Huang et al. [162] completed cell recognition and 3D positioning by capturing the image of a single luminaire, realizing continuous and stable indoor positioning by using sparse light source beacons in actual scenes. Wang et al. [138] proposed to use mode average to process heading angle data for pedestrian positioning, which improved positioning accuracy and shortened reliable running time. In the positioning process, the LED with modulation code was used as the absolute position beacon, and then the LED image was captured and decoded by the camera to obtain the absolute position. The information would be used to periodically correct the position of the PDR, thereby avoiding the accumulation of PDR technology and obtaining high precision. Wen et al. [139] designed a VLP-integrated PDR system for real-time localization using a common smartphone, achieving 98.5% decoding accuracy for 20-bit IDs at 2.1 m height, even under sparse LED distribution and variable shutter speed. In the system, the VLP provided global information and periodic calibration to the PDR, while the PDR addresses the discontinuity problem caused by the limited FoV of the camera sensor. Alcázar-Fernández et al. [140] presented a mobile application-based indoor positioning system for museums that combines smartphone IMU sensors with PDR and VLP AoA algorithms. The system estimates the user’s position using the device’s inertial sensors when no light source is detected. Simulation results demonstrated that this approach reduces the average error to 0.85 meters and provides accurate navigation.

VLP algorithms such as RSS typically require simultaneously receiving multiple visible light signals for positioning. Therefore, appropriate multiplexing schemes are needed to ensure that signals from different LEDs can be differentiated. Here, we summarize the state-of-the-art multiplexing as follows:

Time Division Multiplexing (TDM): In TDM, each lamp is assigned to a time slot and transmits its unique code sequence in its time slot. In the remaining time slots, the LEDs keep illuminating without transmitting any information. Therefore, TDM requires strict time synchronization among LEDs. Zhou et al. [21] developed a TDM method, in which each time was encoded into three 16-bit location codes representing the $x$, $y$, and $z$ coordinates of the LEDs. To keep the transmission power stable, the TDM slot structure was also specially designed so as to provide stable illuminations. However, since each lamp must take up a time slot, the positioning delay can be large when there are a large number of LED lamps, which can limit the application of VLP in mobile scenarios. To tackle this challenge, Hou et al. [175] proposed a novel block encoding TDM scheme, in which nine LEDs composed one block to reduce the system delay. Meanwhile, in order to reduce inter-cell interference, an extended binary coded decimal code was used to encode signals transmitted by the LEDs. In addition, two TDM schemes [176] were proposed for VLP systems with/without communication capability, and the required total time slots for estimating channel gains are studied to facilitate positioning.

Frequency Division Multiplexing (FDM): In FDM, each lamp is assigned to a carrier frequency and the transmitted information will be modulated on this frequency. With the assigned carrier frequencies, the LEDs can send information continuously. At the receiver side, light signals from different LEDs can be distinguished by filters. Kim et al. [177] proposed a carrier allocation VLC system to mitigate inter-cell interference. At the transmitter side, the QPSK signals with 2 MHz, 2.5 MHz and 3 MHz carriers are generated. The receiver position can be calculated by the trilateration method. For instance, a VLP system based on FDM method was proposed [178], which utilized the properties of square waves in the frequency domain. Neighboring LEDs used multiples of the ground frequency of the first LED, and the receiver performed an FFT on the received signals to obtain the RSS values. The system can utilize unsynchronized low-bandwidth transmitters, thus achieving an easy implementation with current high-efficiency LED drivers.

Code Division Multiplexing (CDM): In CDM, each lamp is assigned to a unique code and these codes are orthogonal to each other. The LEDs use digital modulation to send information, and the receiver utilizes the same codes to obtain different signals. The work [179] used the CDM scheme for the VLC system and evaluated uni- and bi- polar codes with the appropriated receiver. The results showed that bipolar codes can reduce the distance error caused by surrounding light. Then, Szabo et al. [180] designed and built an experimental platform for the VLC CDM system, and verified the functionality of the system. Ho et al. [181] formulated a nonconvex optimization problem to find the optimal code and used majorization theory analytically to solve this optimization problem. It has been proved that a combinational code is an optimal code that simultaneously attains the minimum values of both total noise variance and maximum noise variance. Through the optimal code, a bound on the fundamental trade-off between the total noise variance and codeword length is derived. Saed et al. [182] used combinational code as a coding scheme for channel estimation, and the experimental results showed that combinational code significantly outperforms the other two schemes based on TDM in terms of noise variance. Another work [183] proposed an encoding scheme based on 1023-bit Kasami sequences for every transmission, which provided multiple access capability and robustness against low signal-to-noise ratios and harsh conditions, such as multipath and near-far effect.

Orthogonal Frequency Division Multiplexing (OFDM): In OFDM, the transmission bandwidth is divided into a series of orthogonal subcarriers, and different lamps are assigned to different subcarriers. There is no need for bandwidth protection between subcarriers, and the spectrums can overlap with each other. Therefore, compared to FDM, OFDM can fully utilize the bandwidth resources. An indoor VLP and VLP system using the OPDM scheme was proposed [184, 49] to overcome intercell interference and provide indoor positioning and data communications. The transmitted signals from LEDs were encoded with an allocated subcarrier, and the receiver recovered all signals by a DFT operation. Three subcarriers with the maximum RSS are selected for positioning based on the trilateration method. The experiment results showed that the proposed system can achieve a mean positioning error of 1.68 cm and an error vector magnitude of more than 15 dB.

Wavelength Division Multiplexing (WDM): In WDM, different lamps send optical carrier signals with different wavelengths to transmit information. At the receiver side, optical carriers with different wavelengths are separated to restore the original signals. A localization method named color-based localization [185] was proposed, which utilized the properties of metamerism. The red-green-blue LEDs provided a white light sensation to the human eye, and the CSK scheme allowed data transmission by modulating the intensity of LEDs using a color coding format. The receiver position can be estimated using the RSS or TDOA method.

VLP is based on VLC since VLP needs to know the location information of each LED, which is transmitted by VLC. Conversely, the VLC channel model is a function of the receiver’s location. They interact to achieve an efficient integrated communication and positioning system. In practical indoor scenarios, both communication and positioning services should be required simultaneously. Therefore, reasonable resource allocation provides an important guarantee for high-speed communication and high-accuracy positioning.

There have been some works focused on this issue. For instance, Ma et al. [186] revealed the intrinsic relationship between VLP and VLC in a single-LED-based VLCP system, and it demonstrated that the positioning information can be used to estimate the channel state information. A robust power allocation scheme was proposed under the practical optical constraints and QoS requirements. The result showed the efficiency in both localization and communications. Then, Yang et al.[51] presented an integrated VLCP network for indoor IoT devices and proposed jointly optimizing the access point selection, subcarrier group allocation, adaptive modulation, and power allocation approach to satisfy different QoS requirements of devices while maximizing the network data rate. They further presented an integrated visible light VLCP system and proposed a jointly coordinated subcarrier and power allocation approach with the purpose of maximizing the system sum rate while guaranteeing the minimum data rate and positioning accuracy requirements of devices [52].

In VLCP systems, the LED has a double function of illumination and communication. The LED layout can directly affect illumination uniformity, communication performance, and positioning performance. Therefore, the layout of the LEDs must be properly designed to make the illumination uniform while ensuring the communication quality for positioning. To this end, the work [187] proposed a power-efficient LED placement optimization algorithm for the VLC system, which considered the illumination level and communication requirements simultaneously. The simulation results show that the proposed LED placement algorithm can harvest a 14% power consumption gain. Li et al. [188] also studied the optimal deployment of LEDs indoors under the constraints of illumination and signal-to-noise ratio (SNR) by using simplicial complexes. It achieved seamless coverage with the minimal number of LEDs required.

Based on the VLC systems, when designing the LED placement for VLP systems, the positioning performance such as accuracy and coverage ratio, should be considered in addition to communication performance metrics. To evaluate the positioning performance, a 3D normalized positioning error metric (NPEM) [189] was characterized to explore the relationship between the NPEM and the LED cell layout. Then, the authors proposed a layout optimization algorithm for a structured square LED transmitter cell to minimize the NPEM. Limited attention has been directed towards the LED placement problem in VLP systems. The primary challenge lies in formulating a theoretical model that correlates positioning performance metrics, like accuracy, with the arrangement of LEDs. Consequently, it is meaningful to create a theoretical model for positioning accuracy that holds significant value. Such research can serve as a foundational framework for enhancing performance.

The communication quality of the VLP system is the basis to ensure the positioning function. Different from traditional radio frequency systems, the common Shannon channel capacity formula cannot be applied to VLC. The illumination demands and the LED dynamic range constraints must be considered when deriving channel capacity, and the VLC channel capacity can be expressed as [190]

$$C\geq\frac{1}{2}{\log_{2}}\left({1+\frac{2}{{2\pi}}{{\left({\frac{{\xi Ph}}{\sigma}}\right)}^{2}}}\right)$$

(21)

where $C$ is the channel capacity, and $\xi$ represents the illumination target. $P$ is the optical power of the LED, and $\sigma$ is the standard deviation of the additive white Gaussian noise. The expression of (21) has been adopted in optimizing the network in some VLC systems [187, 191]. In addition to the above channel capacity, several studies also used SNR to quantify communication quality [188, 192].

VLP systems reuse the luminaires in the room, and thus, the illumination level should be taken into consideration. The brightness is typically defined by the luminous intensity. The illumination level constraints can be determined by the minimum illumination or average illumination in the room. The illumination in lux, ${\eta_{j}}$, at the receiver point $j$ can be given by

$${\eta_{j}}=I_{0}{\cos^{m}}({\phi_{j}})\cos({\psi_{j}})/d_{j}^{2}$$

(22)

where ${I_{0}}=I\left({\Phi=0}\right)=\left({m+1}\right){{\phi_{j}}}/\left({{2\pi}}\right)$ is the maximal luminous intensity. $\Phi=683\int_{0}^{\infty}{V\left(\lambda\right)}P\left(\lambda\right)d\lambda$ is the luminous flux, where $V\left(\lambda\right)$ is the luminosity function and $V\left(\lambda\right)$ is the spectral power distribution [193].

To satisfy the illumination requirements, some studies [188, 194, 191] set a threshold for the illumination function (22) of each sample point at the receiver plane to formulate a network optimization problem. In addition, some studies [187, 195, 196, 192] ensured the illumination level from a perspective of illumination uniformity. For instance, the coefficient of variation of root mean square error (CV(RMSE)) was typically adopted [187, 195, 196], and CV(RMSE) can be expressed as

$${\rm{CV}}\left({{\rm{RMSE}}}\right)=\left(\sqrt{\frac{1}{N}\sum\limits_{j=1}^{N}{{{\left({{\eta_{j}}-{\eta_{{\rm{avg}}}}}\right)}^{2}}}}\right)/{\eta_{{\rm{avg}}}},$$

(23)

where ${\eta_{{\rm{avg}}}}={\frac{1}{N}}{\sum\limits_{j=1}^{N}{{{{{\eta_{j}}}}}}}$ is the average illumination of all sample points. There is also the definition of illumination uniformity that is given by the ratio between the minimum and the average illumination intensity among all sample points, i.e., $\vartheta=\min\left({{\eta_{j}}}\right)/{\eta_{{\rm{avg}}}}$ [197], which has been adopted to achieve acceptable illumination uniformity in the room [192].

In contrast to a singular VLC network, the precision of positioning is intricately linked to VLP networks. As highlighted in Section III, the efficacy of positioning algorithms is markedly impacted by the count of received LEDs, a factor confined by the arrangement of LEDs. Few studies delve into network optimization strategies that account for VLP accuracy. The Cramer-Rao lower bound (CRLB) is a common tool employed to assess the theoretical positioning accuracy of VLP. The location and orientation of the user are bounded from below as follows [198]

$$\left\{\begin{array}[]{l}{\mathbb{E}}\left\{{\left\|{{\bf{\hat{x}}}-{\bf{x}}}\right\|_{2}^{2}}\right\}\geq{\rm{tr}}\left({{{\rm\mathcal{B}}_{\bf{x}}}\left({{\bf{x}},{\bf{u}}}\right)}\right)\\ {\mathbb{E}}\left\{{\left\|{{\bf{\hat{u}}}-{\bf{u}}}\right\|_{2}^{2}}\right\}\geq{\rm{tr}}\left({{{\rm\mathcal{B}}_{\bf{u}}}\left({{\bf{x}},{\bf{u}}}\right)}\right)\end{array}\right.$$

(24)

where $\mathbb{E}\left\{\cdot\right\}$ is the expectation over noise. ${\bf{\hat{x}}}$ and ${\bf{\hat{u}}}$ are the estimation of the location and orientation, respectively, while $\rm{tr}\left\{\cdot\right\}$ represents the matrix trace. ${{\mathcal{B}_{\bf{u}}}\left({{\bf{x}},{\bf{u}}}\right)}$ and ${{{\mathcal{B}}_{\bf{x}}}\left({{\bf{x}},{\bf{u}}}\right)}$ denote the CRLBs on the location estimation error and orientation error, respectively, and they are given by

$$\left\{\begin{array}[]{l}{{\mathcal{B}}_{\bf{x}}}\left({{\bf{x}},{\bf{u}}}\right)=\left({{\mathcal{H}}_{\bf{x}}^{{\rm{obs}}}+{\bf{X}_{{\rm{prior}}}}}\right)^{-1}\\ {{\mathcal{B}}_{\bf{u}}}\left({{\bf{x}},{\bf{u}}}\right)=\left({{\mathcal{H}}_{\bf{u}}^{{\rm{obs}}}+{\bf{U}_{{\rm{prior}}}}}\right)^{-1}\end{array}\right.$$

(25)

where ${\mathcal{H}}_{\bf{x}}^{\rm{obs}}$ and ${\mathcal{H}}_{\bf{u}}^{\rm{obs}}$ denote the observation information of the location and orientation of the user, respectively, and ${\bf{U}_{{\rm{prior}}}}$ and ${\bf{X}_{{\rm{prior}}}}$ denote the prior information. The obtained CRLBs were then used to comprehensively study the performance limits of RSS-based VLP algorithms. The authors also pointed out that the CRLB could be used to provide guidelines for optimizing the performance of practical VLP systems. Then, Shi et al. [199] optimized the energy efficiency performance for a radio frequency-visible light communication and positioning system by guaranteeing the CRLB of the positioning error and the minimum data rate of the communication. Ma et al. [200] derived CRLB to quantify the performance of the integrated VLP and VLP system, and aimed to minimize the CRLB of VLP, i.e., the positioning errors, while satisfying the outage probability constraint of the VLC rate, and total power constraint.

In addition to CRLB, Zhang et al. [76] derived a novel closed-form error expression that related the noise statistics to the LED localization error at a certain point and provided insights into the error control of the LED coordinates databases.

However, it is still challenging to construct a theoretical accuracy model for camera-based VLP systems. Recently, Xu et al. [189] introduced a novel positioning error metric, NPEM, derived from the partial derivative of the positioning function. Their analysis revealed an inversely proportional relationship between NPEM and the number of captured LEDs. Intriguingly, NPEM exhibited independence from the rotation angle around the z-axis when the transmitter and receiver planes were parallel. This finding serves as a foundational concept for devising a quantitative model for positioning errors by amalgamating positioning functions from distinct camera-based VLP systems.

When considering the performance of VLP, the coverage ratio of the positioning service should also be considered in the VLP network. As shown in (20), the coverage ratio is the proportion of the effective area that the positioning service can be provided. In the VLP network design, eq. (20) can be formulated as an optimization objective, or as a constraint to guarantee the positioning service area. For instance, let $A_{\rm{affective}}$ be the area where the user can achieve a positioning accuracy of 5 cm, and then, we can have

$${R_{{\rm{cov}}}}=\frac{{{A_{{\rm{effective}}}}}}{{{A_{{\rm{total}}}}}}\times 100\%\geq 95\%$$

(26)

with (20) to guarantee the accuracy and coverage of the VLP network system simultaneously.

In addition, there are other objectives that can be conducted in the VLP network system design, such as maximizing the system efficiency and minimizing the deployed LEDs. In general, it is necessary to make the VLP system more efficient, accurate, and reliable.