1. Introduction

Recently, rapid development of mobile communication networks and electronic information technologies has promoted numerous location-based services [1–3]. Satellite positioning systems such as global positioning system (GPS) can provide users with meter-level localization services in outdoor environments [4]. Nevertheless, in indoor environments, the positioning accuracy of GPS cannot satisfy the demands of real-time localization and navigation [5]. To realize high-accuracy positioning in indoor environments, various positioning technologies have been developed using different wireless signals, such as WiFi [6], infrared [7], ultrasonic [8], Bluetooth [9], ultra-wideband [10] and radio-frequency identification (RFID) [11]. However, these technologies still suffer low positioning accuracy or high deployment cost.

Compared with the above positioning technologies, light emitting diode (LED)-based visible light positioning (VLP) enjoys appealing advantages such as high accuracy, low implementation cost and long life time [12–15]. Existing VLP schemes estimate the terminal position from different optical measurements such as image [16], time of arrival (TOA) [17], time difference of arrival (TDOA) [18], angle of arrival (AOA) [19], angle difference of arrival (ADOA) [20] and received signal strength (RSS). Among these techniques, the RSS-based VLP is the most popular since RSS values can be easily acquired using a single photodiode (PD) with no need of auxiliary devices [21].

There have been plenty of studies on RSS-based VLP [22–28]. In [22–24], the receiver position was estimated using the received direct current (DC) signal strength, which is susceptible to ambient light and less practical in real environments. Reference [25] experimentally demonstrated a VLP system using the output alternating current (AC) signal strength of PD. As it requires specific values of all transceiver parameters, the VLP scheme in [25] is not robust to receiving coefficient. In order to achieve the robustness to receiving coefficient, Ref. [26] proposed a VLP scheme for the case where the receiver is perpendicular to the ceiling. However, the receiver may be tilted in practice, which limits the application scenarios of the VLP scheme in [26]. References [27–30] considered the RSS-based VLP in the case of tilted receiver. However, they assumed perfect knowledge of the receiver orientation angle, which is a strong assumption in practical applications.

To address the above issues, this work proposes a robust RSS-based VLP scheme with unknown receiving angle. In the proposed scheme, the receiver coordinate and receiver characteristic vector are jointly estimated from the RSS vector. Through mathematical derivations, the original estimation problem can be equivalently converted into the problem that maximizes the projection of the RSS vector on the column space spanned by the measurement matrix. The proposed scheme is robust to the receiver characteristics, as it does not require the prior knowledge of receiving coefficient and receiving angle. To further reduce the computational complexity, we also propose an importance sampling (IS) method for solving the VLP problem instead of brute-force search (BFS). Simulation and experimental results demonstrate the robustness of the proposed scheme to receiving coefficient and receiving angle. Specifically, simulation results show that the computation time of the BFS method is about 26 times that of the IS method. Experimental results show that the proposed VLP scheme achieves a stable positioning accuracy below 7 cm under different PD tilting angles in a 60 cm $\times$ 60 cm $\times$ 150 cm space, and is not sensitive to height perturbation.

The remainder of this paper is organized as follows. The signal model and benchmark VLP schemes are presented in Section 2. Section 3. elaborates the proposed VLP scheme. Simulation and experimental results are given in Section 4. and Section 5. Finally, Section 6. concludes this paper.

2. System descriptions

In this section, we describe the considered VLP system and present the corresponding signal model.

2.1 System configuration

Figure 1(a) illustrates the considered indoor VLP system with $L$ LEDs as the transmitter and one tilted PD as the receiver. Assume that all the $L$ LEDs are at the same height $z_t$ and pointing downwards, which is the common case. Each LED acts as one anchor for RSS-based VLP, and the coordinate of LED $l$ is known as $\boldsymbol {c}_l = [x_l, y_l, z_t]^\top \in {\mathbb {R}}^3$, $l=1,\ldots, L$. The tilted PD receiver is located at the position ${\boldsymbol {c}}_r=[x_r, y_r, z_r]^\top \in {\mathbb {R}}^3$. The vertical tilting angle is denoted by $\phi$, and the horizontal tilting angle is denoted by $\beta$. As in Refs. [22–24], we assume that the receiver height $z_r$ is known and the receiver moves in a horizontal plane. This assumption holds in various practical applications, e.g., where the receiver is installed on a robot or a cart. Without loss of generality, the receiver orientation vector is denoted by ${\boldsymbol {u}}_r = [\sin \beta \cos \phi, \cos \beta \cos \phi, \sin \phi ]^\top \in {\mathbb {R}}^3$ with $\| \boldsymbol {u}_r \|_2 = 1$. The distance between the LED $l$ and PD is calculated as $d_{r,l} = \|\boldsymbol {c}_l - \boldsymbol {c}_r\|_2 = \sqrt {(x_l - x_r)^2 + (y_l - y_r)^2 + z^2}$ with $z = z_t - z_r$.

 

Fig. 1. Illustration of the considered indoor VLP system. (a) System configuration. (b) Signal flow.

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Remark 1: Note that even in real scenarios where the receiver is relatively fixed on a terminal (e.g., a robot or an auto-cart), there will be height and angle perturbations due to the terminal movement and certain environmental factors (e.g., uneven ground, inaccurate installation and measurement error). Typically, the height perturbation is small while the angle perturbation can be large. Therefore, we consider the case where the receiver height is relatively fixed but the receiver angle can be updated, similarly as in Refs. [27–30].

Figure 1(b) shows the signal flow of the considered VLP system. At the transmitter, each LED transmits its own beacon signal containing both DC and AC components to the PD in different time slots to eliminate the mutual interference. In the $l$-th time slot, the PD detects the optical signal from the LED $l$ and converts it into an electrical signal. After removing the DC component via a DC-blocking circuit, the output AC signal strength of PD is given by

Note that parameters $\{ g_l \}_{l=1}^{L}$ and $\{ k_l \}_{l=1}^{L}$ are prior knowledge in a VLP system, because the LEDs are generally deployed in advance and fixed in practical applications. In addition, the values of $\{ g_l \}_{l=1}^{L}$ and $\{ k_l \}_{l=1}^{L}$ may be different for each LED even when all LEDs are driven under the same DC bias and AC signal. As for the receiver characteristic vector $\boldsymbol {\mu }_r$, it is usually unknown since the PD device or its operating states (e.g., orientation angle and detection gain) can be updated in a VLP system, where perfect knowledge of the receiver parameters is typically infeasible.

2.2 Benchmark schemes

In this work, we consider two benchmark VLP schemes, namely the typical least square (LS) scheme and the VLP scheme in Ref. [26] based on the direction of RSS vector (named “RSSD” scheme in this work).

We first present the details of the LS scheme which assumes perfect knowledge on parameters $\{ g_l \}_{l=1}^{L}$, $\{ k_l \}_{l=1}^{L}$ and $\boldsymbol {\mu }_r$ with $\phi =0^\circ$. Under this assumption, we have $\boldsymbol {u}_r=[0,0,1]^\top$ and $(\boldsymbol {c}_l - \boldsymbol {c}_r)^\top \boldsymbol {\mu }_r=\frac {BA_r}{2\pi z^2}$. Ignoring the noise in Eq. (4), the distance $d_{r,l}$ can be approximated as

The RSSD scheme in Ref. [26] jointly estimates the receiver position and receiver coefficient, which is equivalently transformed into the position estimation problem maximizing the projection of the RSS vector $\boldsymbol {p}_r$ on the signal feature vector $\boldsymbol {s}^{fe}=[s_1^{fe},\ldots,s_L^{fe}]^\top$ with $s_l^{fe}=g_l \frac {z^{k_l}}{\|{\boldsymbol {c}}_l - {\boldsymbol {c}}_r \|^{k_l}}$. Consequently, it realizes VLP based on the direction of RSS vector and only requires the knowledge of LED parameters, which is robust to the receiver coefficient. However, it only considers the case of $\phi = 0^\circ$, which limits the application scenarios. In summary, the LS scheme and RSSD scheme are proposed based on strong assumption of perfect knowledge of receiver tilting angle. In real scenarios, the receiver tilting angle can be updated, e.g., in the case of receiver perturbation. As will be shown in Sections 4. and 5, imperfect knowledge of receiver tilting angle has a great impact on the positioning accuracy of the LS scheme and RSSD scheme. To address the mentioned issue, this work considers the VLP problem of estimating the receiver horizontal coordinate $(x_r,y_r)$ from the RSS vector ${\boldsymbol {p}}_r = [p_{r,1}, \ldots, p_{r,L}]^\top \in {\mathbb {R}}^L$ with unknown receiver characteristic vector $\boldsymbol {\mu }_r$.

3. Proposed VLP scheme

In this section, we propose a robust VLP scheme to estimate the receiver position with unknown receiving angle.

3.1 Estimation of receiver position

We consider joint maximum-likelihood (ML) estimation of the receiver horizontal coordinate $(x_r, y_r)$ and receiver characteristic vector $\boldsymbol {\mu }_r$ based on RSS vector ${\boldsymbol {p}}_r$. Mathematically, the joint ML estimation problem can be formulated as

Under the assumption of Gaussian noise in Eq. (1), the conditional probability density function $f(p_{r,l}|x_r, y_r, \boldsymbol {\mu }_r)$ is given by

It is seen from Eq. (14) that the proposed scheme requires that the measurement matrix $\boldsymbol {S}$ is full column rank. Regarding this requirement, we have the following results.

Theorem 1 With all LEDs located at the same height, matrix $\boldsymbol {S}$ is full column rank if and only if the $L$ LEDs are non-collinear.

Proof: Please see Appendix. $\hfill\blacksquare$

Theorem 1 implies that the $L$ LEDs should be non-collinear to ensure the proposed VLP scheme is feasible. Moreover, it is noteworthy that when matrix $\boldsymbol {S}$ is full rank with $L=3$, $\boldsymbol {S} \in \mathbb {R}^{3\times 3}$ has a unique inverse matrix $\boldsymbol {S}^{-1}$. Then, the objective function in problem (14) can be rewritten as

Remark 2: Note that the positioning accuracy increases with the number of LEDs, since signals from additional LEDs can provide more information about the receiver position. To achieve a good complexity-performance tradeoff, it is essential to properly choose the number of LEDs in real VLP scenarios. Based on the results in Section 4.2, we can choose 5 LEDs to realize a relatively high-accuracy positioning.

3.2 IS method for solving the VLP problem

Problem (14) is non-convex in $(x_r, y_r)$ and it is difficult to obtain its optimal solution. One straightforward solution is to adopt the BFS method in the 2-dimensional positioning area. Specifically, we divide the positioning area into multiple grid points, and choose the grid point yielding the largest value of $||\boldsymbol {S} (\boldsymbol {S}^\top \boldsymbol {S})^{-1} \boldsymbol {S}^\top \boldsymbol {p}_r||_2^2$ as the estimated receiver position. However, the computational complexity of BFS method can be very high under large number of divided grid points, which may lead to high latency for VLP in real scenarios. To reduce the computational complexity, we propose a Monte Carlo method for solving problem (14) based on the Pincus’ theorem [33] and IS method [34–36].

Pincus’ theorem: For the optimization problem $\max _{\boldsymbol {c} \in \mathbb {R}^n} F(\boldsymbol {c})$ with $\boldsymbol {c}= [c_1, \ldots, c_n]^\top$, its optimal solution is given by

Particularly, in the VLP problem (14), we have $\boldsymbol {c}_{rh} = [x_r, y_r]^\top$ and

Derivation of Sampling Distribution: The major challenge of implementing IS for calculating $\mathbb {E}_{f} \{\boldsymbol {c}_{rh}\}$ is to find a proper PDF $q(\boldsymbol {c}_{rh})$ which admits an efficient sampling. We consider the following simplifications of $F (\boldsymbol {c}_{rh})$ in (18):

Algorithm 1. The proposed VLP scheme with IS

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3.3 Estimation of LED parameters

To implement the proposed VLP scheme, we need to know some LED parameters, e.g., the values of $\{k_l\}_{l=1}^L$ and ratios of $\{g_l\}_{l=1}^L$. These LED parameters can be considered as prior information of VLP and can be estimated using the received signals of one PD (“PD-EST”) at $I$ pre-known positions. The detailed procedure of LED parameter estimation can be found in Ref. [26] and is omitted here. With the estimated LED parameters, robust VLP can be realized via Algorithm 1 using a positioning PD (“PD-POS”, not the same one as PD-EST) in the case of unknown receiving coefficient and receiving angle.

3.4 Complexity analysis

In this subsection, we analyze the computational complexity of the VLP schemes. The complexity of the LS scheme is $\mathcal {O} (L)$ due to the matrix computation. Letting $M$ denote the number of divided grid points in the receiver plane, the complexities of the RSSD scheme in Ref. [26] and the VLP scheme (14) with BFS are both $\mathcal {O} (M \cdot L)$. For the VLP scheme (14) with IS, it requires a complexity of $\mathcal {O} (T \cdot L)$ with $T$ being the number of generated samples. Since $T$ in IS can be much lower than $M$ in BFS, the complexity of the VLP scheme (14) with IS can be significantly reduced compared to that with BFS.

4. Simulation results and discussions

In this section, the performance of the proposed VLP scheme is verified via simulation results.

4.1 Simulation setup

The simulations are carried out using MATLAB R2020b on a personal computer with 2 GHz Intel Core i5 CPU and 16 GB memory. In the simulations, we consider the scenario where $L=5$ LEDs are installed in the ceiling of a room. The coordinates of the five LEDs are $(\pm 1, \pm 1, 2)$ m and $(0, 0, 2)$ m. The LED parameters are set to be $g_1 = \cdots = g_5 = 1$ and $k_1 = \cdots = k_5 = 5$. The PD is located in the $2\, {\rm m} \times 2 \,{\rm m}$ horizontal plane with height $z_r = 0$ m. The vertical tilting angle $\phi$ of PD is chosen from set $\{0^\circ, 5^\circ, 10^\circ, 15^\circ, 20^\circ, 25^\circ \}$, and the horizontal tilting angle $\beta$ is uniformly chosen from $[0^\circ, 360^\circ )$ at each trial. The detector area of the PD is $A_r= 7.07 \, \rm {mm}^2$ and detection gain is $B=1$. The signal-to-noise ratio (SNR) is defined as ${\rm SNR} = 10\lg \frac {BA_rg_l}{2\pi z^2 n_{r,l}}[\rm {dB}]$, which adopts the receiver point directly below each LED as the reference point. Unless otherwise specified, the SNR is set to be 50 dB. The number of grid points in BFS is $M=40000$, and the sample size in IS is $T=1000$. For comparison, we also present the results of the LS scheme and RSSD scheme mentioned in Section 2.2. In this work, the mean positioning error is defined using mean absolute deviation, i.e., $MPE=\mathbb {E}\left \{\sqrt {(x_r-\hat {x}_r)^2+(y_r-\hat {y}_r)^2} \right \}$.

4.2 Simulation results

We first investigate the positioning accuracy of the proposed VLP scheme using BFS and IS methods. Figure 2 compares the CDF of positioning errors using the two methods (labeled as “Prop-BFS” and “Prop-IS”). We consider two cases of PD vertical tilting angles, namely $\phi =0^\circ$ and $\phi =15^\circ$. The CDF results are obtained using 100 PD locations which are uniformly distributed in the 2 m $\times$ 2 m horizontal plane. At each test point, 100 trials are adopted to obtain a statistical result. It can be observed that the IS method has a similar CDF of positioning errors as the BFS method. The mean positioning errors of BFS and IS methods are 3.7 cm and 4.0 cm (4.3 cm and 4.5 cm) when $\phi =0^\circ$ ($\phi =15^\circ$). It implies that the BFS and IS methods yield close positioning accuracy. In the simulations, the average running time of the BFS method is 0.79 s for each positioning point, while that of the IS method is 0.03 s for each positioning point. Compared to the BFS method, the IS method can achieve a close positioning accuracy with much lower complexity. On the whole, the IS method can achieve a good complexity-performance tradeoff and is appealing for practical applications.

 

Fig. 2. CDF of positioning errors with BFS and IS methods under different tilting angles.

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To investigate the robustness of the proposed VLP scheme to the variation of receiving angle, Fig. 3 plots the simulated positioning error versus PD vertical tilting angle $\phi$. Since actual value of $\phi$ is unknown, assume that LS and RSSD schemes adopt $\phi = 0^\circ$. It can be seen that the mean positioning error of the proposed scheme (with BFS or IS) increases slowly with $\phi$, which results from the decrease of the received SNR. On the contrary, the positioning errors of the LS and RSSD schemes increase rapidly as $\phi$ increases, due to imperfect knowledge of the receiver orientation angle. The results imply that the proposed VLP scheme is much more robust to the receiving angle than the LS and RSSD schemes. We can also see that under small $\phi$, the mean positioning error of the proposed scheme is a little higher than those of the LS and RSSD schemes. The reason is that (approximately) perfect knowledge of the PD tilting angle by the LS or RSSD scheme is beneficial for improving the positioning accuracy.

 

Fig. 3. Simulated positioning error versus PD vertical tilting angle $\phi$.

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To investigate the effect of receiver coefficient mismatch on the positioning accuracy, we consider the scalar error model of the receiver characteristic vector: ${\boldsymbol {\mu }}_r = \alpha {\boldsymbol {\mu }}_{r0}$, where ${\boldsymbol {\mu }}_{r0}$ is a known vector, $\alpha$ is an unknown scalar, and ${\boldsymbol {\mu }}_{r}$ is the unknown true vector. In the positioning, ${\boldsymbol {\mu }}_{r0}$ is used instead of ${\boldsymbol {\mu }}_{r}$. Figure 4 shows the simulated positioning error versus the unknown scalar $\alpha$ under $\phi = 0^\circ$. It can be observed that the positioning error of the proposed scheme (with BFS or IS) decreases with $\alpha$ due to the increase of received SNR. Similar results are also observed for RSSD scheme. In contrast, the positioning accuracy of the LS scheme deteriorates sharply under severe mismatch of scalar $\alpha$. The results in Fig. 3 and Fig. 4 demonstrate the robustness of the proposed scheme to the receiver characteristics including the receiving angle and receiving coefficient.

 

Fig. 4. Simulated positioning error versus the unknown scalar $\alpha$ under $\phi = 0^\circ$.

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Figure 5 shows the estimated positions of 100 test points under $\phi =0^\circ$ and $L = 4, 5$. When $L =4$, the coordinates of the LEDs are $(\pm 1, \pm 1, 2)\, {\rm m}$; When $L =5$, the coordinates of the LEDs are $(\pm 1, \pm 1, 2)\, {\rm m}$ and $(0, 0, 2)\, {\rm m}$. In Fig. 5, 100 trials are adopted at each test point to obtain a statistical result. Note that ML estimator is an asymptotically unbiased estimator (e.g., as the number of samples approaches infinity) rather than an exactly unbiased one [38]. Therefore, there exists estimation biases in VLP for small number of LEDs. Similar results can also be observed in Refs. [39–41]. It is seen that the positioning accuracy using 4 LEDs is much worse than that using 5 LEDs. Similar results can be obtained for other tilting angles. The results provide guidance for us to adopt 5 LEDs in real positioning scenarios.

 

Fig. 5. Estimated positions of 100 test points under $\phi =0^\circ$. (a) BFS. (b) IS.

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5. Experimental results and discussions

In this section, experimental results are presented to further evaluate the performance of the proposed VLP scheme.

5.1 Experimental setup

The experimental environment is shown in Fig. 6. In the experiments, we adopt 5 LEDs (Kecent SL-COB) with coordinates $(\pm 0.3, \pm 0.3, 1.5)$ m and $(0, 0, 1.5)$ m. All LEDs are driven by the same DC bias and sinusoid signal using a bias-tee circuit. An AC-coupled avalanche photodiode (APD, C12702) receives the optical signals and outputs the electrical AC signals. A real-time oscilloscope (Agilent Technologies MSO-X 6004A) is used to collect the output AC signals of APD for further processing performed with MATLAB. We first estimate the LED parameters $\{g_l\}_{l=1}^L$ and $\{k_l\}_{l=1}^L$ using one APD (“PD-EST”) with $\phi = 0^\circ$. Then, we locate the position of one test APD (“PD-POS”) based on the estimated LED parameters. The VLP experiments are conducted under two vertical tilting angles of PD-POS, namely $\phi = 0^\circ$ (Case 1) and $\phi = 13^\circ$ (Case 2), and the horizontal tilting angle $\beta$ is not fixed at each test point as it is difficult to accurately measure $\beta$ in real scenarios. The SNR in the experiments is about 35 dB.

 

Fig. 6. The experimental environment.

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We measure the output AC signals at 49 receiver points of PD-POS. Figure 7 depicts the horizontal positions of LEDs and test receiver points. The relevant experimental parameters are listed in Table 1.

 

Fig. 7. The horizontal positions of LEDs and test receiver points.

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Table 1. The Experimental Parameters for Indoor VLP System

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5.2 Experimental results

Recall that the parameter $I$ represents the number of pre-known positions of PD-EST used for LED parameter estimation. Table 2 lists the estimated values of LED parameters under different $I$. To study the impact of number $I$ on the positioning accuracy, Fig. 8 shows the mean positioning errors of the proposed scheme (with BFS or IS) under different $I$. It can be seen that the mean positioning error decreases with $I$. Specifically, when $I=13$, the mean positioning errors of the proposed scheme are both below 7 cm in Case 1 and Case 2. To guarantee a high positioning accuracy, we adopt $I=13$ by default in the following results.

 

Fig. 8. Mean positioning errors of the proposed scheme under different $I$.

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Table 2. Estimated Values of LED Parameters under different $I$

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Table 3 illustrates the difference between the normalized RSS in Case 1 and fitting values using the estimated LED parameters with $I = 13$. We can see that the normalized RSS is very close to the fitting values with a maximum difference below 0.03. It verifies the accuracy of the signal model given by Eq. (4).

Table 3. Difference between the normalized RSS in Case 1 and fitting values using the estimated LED parameters with $I = 13$

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In the following, we compare the positioning accuracy of different VLP schemes. The LS and RSSD schemes simply assume $\phi = 0^\circ$ in Cases 1 and 2. Figure 9 shows the CDF of positioning errors in Cases 1 and 2. The CDF results are obtained using 49 PD locations shown in Fig. 7. At each test point, 10 trials are adopted to obtain a statistical result. We can see that the LS scheme and RSSD scheme are both infeasible in Case 2 with imperfect knowledge of the receiver tilting angle. Specifically, the maximum positioning errors of the LS scheme and RSSD scheme are both about 45 cm. In contrast, the positioning accuracy of the proposed scheme (with BFS or IS) in Case 2 is close to that in Case 1 with a maximum positioning error of 12 cm, demonstrating its robustness to the receiving angle. Note that there exists a performance gap between the experimental results and simulation results, which can be explained as follows. Firstly, the received SNR in the simulations is higher than that in the experiments, yielding higher positioning accuracy. Secondly, there can be measurement error and parameter estimation error in the experiments, resulting in a positioning accuracy degradation. Therefore, it is reasonable that the positioning accuracy in the experiments is worse than that in the simulations. Similar results can be observed in Refs. [26,42,43].

 

Fig. 9. CDF of positioning errors in Cases 1 and 2.

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To better compare the simulation results and experimental results, Table 4 lists the mean positioning errors obtained in the simulation and experiment under experimental parameters. The coordinates of LEDs are ($\pm$0.3, $\pm$0.3, 1.5) m and (0, 0, 1.5) m. The SNR is about 35 dB. It is seen from Table 4 that the positioning errors in the experiment are a little larger than those in the simulation due to the measurement error and parameter estimation error in the experiment.

Table 4. Mean positioning errors in the simulation and experiment under experimental parameters

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We further investigate the effects of receiver height and angle perturbations on the positioning accuracy. With actual transceiver height difference $z = 150$ cm, Fig. 10 shows the mean positioning errors under different transceiver height assumptions in Cases 1 and 2. It can be observed that the positioning errors of all VLP schemes vary little under small mismatch of transceiver height difference, implying that the receiver height disturbance has small influence on the positioning accuracy. Also note that, the proposed scheme (with BFS or IS) retains a stable positioning accuracy below 7 cm in Cases 1 and 2. In contrast, the mean positioning error of the LS or RSSD scheme in Case 2 is much higher than that in Case 1. It implies that the LS and RSSD schemes are very sensitive to the variation of receiver tilting angle. The results demonstrate the robustness of the proposed VLP scheme under receiver height and angle disturbances.

 

Fig. 10. Mean positioning errors under different transceiver height assumptions in Cases 1 and 2.

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6. Conclusion

In this paper, we have proposed a novel RSS-based VLP scheme with unknown receiving angles. By jointly estimating the receiver coordinate and receiver characteristic vector, the proposed scheme is robust to the receiving coefficient and receiving angle. To further reduce the computational complexity, we have also proposed an IS method for solving the VLP problem. Simulation and experimental results demonstrate the robustness of the proposed scheme. Overall, the proposed scheme can provide a good performance-complexity tradeoff and is appealing for practical applications. Note that this work assumes linear response of PD. In practical scenarios, the PD may suffer nonlinear behavior under dynamic input signals. Robust design of RSS-based VLP under PD nonlinearity remains for our future work.

7. Appendix: proof of theorem 1

According to the definition of $\boldsymbol {S}$ in Section 3, we have

(25)$$\boldsymbol{S} = \begin{bmatrix} g_1 \frac{z^{k_1}}{d_{r,1}^{k_1}} (x_1 - x_r) & g_1 \frac{z^{k_1}}{d_{r,1}^{k_1}} (y_1 - y_r) & g_1 \frac{z^{k_1}}{d_{r,1}^{k_1}} \cdot z\\ \vdots & \vdots & \vdots \\ g_L \frac{z^{k_L}}{d_{r,L}^{k_L}} (x_L - x_r) & g_L \frac{z^{k_L}}{d_{r,L}^{k_L}} (y_L - y_r) & g_L \frac{z^{k_L}}{d_{r,L}^{k_L}} \cdot z \end{bmatrix} = \begin{bmatrix} g_1 \frac{z^{k_1}}{d_{r,1}^{k_1}} & & \\ & \ddots & \\ & & g_L \frac{z^{k_L}}{d_{r,L}^{k_L}} \end{bmatrix} \cdot \widetilde{\boldsymbol{S}},$$

where $\widetilde {\boldsymbol {S}}$ is given by Eq. (21). Using (25), we obtain ${\rm rank}(\boldsymbol {S}) = {\rm rank}(\widetilde {\boldsymbol {S}})$.

Subtracting the first row of $\widetilde {\boldsymbol {S}}$ from the last $L-1$ ones, it yields

(26)$$ \widetilde{\boldsymbol{S}}=\left[\begin{array}{ccc} x_1-x_r & y_1-y_r & z \\ x_2-x_r & y_2-y_r & z \\ \vdots & \vdots & \vdots \\ x_L-x_r & y_L-y_r & z \end{array}\right] \rightarrow\left[\begin{array}{cc|c} x_1-x_r & y_1-y_r & z \\ \hline x_2-x_1 & y_2-y_1 & 0 \\ \vdots & \vdots & \vdots \\ x_L-x_1 & y_L-y_1 & 0 \end{array}\right] \triangleq \widetilde{\boldsymbol{S}^{\prime}}=\left[\begin{array}{c|c} \widetilde{\boldsymbol{S}_1^{\prime}} & z \\ \hline \widetilde{\widetilde{S}_2^{\prime}} & \mathbf{0} \end{array}\right] . $$

Using (26), we obtain ${\rm rank}(\widetilde {\boldsymbol {S}}) = {\rm rank}(\widetilde {\boldsymbol {S}}^\prime ) = {\rm rank}(\widetilde {\boldsymbol {S}}^\prime _2) + 1$. When the $L$ LEDs are collinear, we have ${\rm rank}(\widetilde {\boldsymbol {S}}^\prime _2) = 1$ and ${\rm rank}(\boldsymbol {S}) ={\rm rank}(\widetilde {\boldsymbol {S}}) = 2$; When the $L$ LEDs are non-collinear, we have ${\rm rank}(\widetilde {\boldsymbol {S}}^\prime _2) = 2$ and ${\rm rank}(\boldsymbol {S}) ={\rm rank}(\widetilde {\boldsymbol {S}}) = 3$. The two cases of collinear and non-collinear LEDs are illustrated in Fig. 11(a) and Fig. 11(b), respectively. Therefore, ${\boldsymbol {S}}$ is full column rank if and only if LEDs are non-collinear.

 

Fig. 11. Illustration of the LED layout. (a) All LEDs are collinear. (b) LEDs are non-collinear.

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Funding

National Natural Science Foundation of China (62101526, 62171428); Key Program of National Natural Science Foundation of China (61631018); Key Research Program of Frontier Sciences of CAS (QYZDY-SSW-JSC003); Fundamental Research Funds for the Central Universities (KY2100000118); open research fund of National Mobile Communications Research Laboratory Southeast University (2019D14).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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