2.1. System Model

Fig. 1 illustrates the proposed scheme model. Here, the radiant intensity of a LED can be assumed to follow a Lambertian radiation pattern due to its large beam divergence  [12]. The line-of-sight (LoS) channel gain can be given by  [13] $$\begin{equation} H= \left\lbrace \begin{array}{ll}\frac{(m + 1)A}{2\pi d^2}{{\cos }^m(\phi)}{\cos (\psi\;)}{T_s(\psi\;)}{G(\psi\;)}, & 0 \leq \psi < \Psi _C\\ 0,& \psi \geq \Psi _C, \end{array} \right. \end{equation}$$ View Source \begin{equation} H= \left\lbrace \begin{array}{ll}\frac{(m + 1)A}{2\pi d^2}{{\cos }^m(\phi)}{\cos (\psi\;)}{T_s(\psi\;)}{G(\psi\;)}, & 0 \leq \psi < \Psi _C\\ 0,& \psi \geq \Psi _C, \end{array} \right. \end{equation} where the parameters are as follows. $m$ is the Lambertian order and defined as: $m=\frac{-ln2}{ln(cos(\Phi _{1/2}))}$, where $\Phi _{1/2}$ is the semi-angle at half illuminance of a LED. $A$ is the area of an optical detector. $d$ is the distance between a transmitter and a receiver. $T_s(\psi \;)$ is the gain of an optical filter, and $G(\psi \;)$ is the gain of an optical concentrator. $\phi$ is the irradiant angle, $\psi\;$ is the incident angle, and $\Psi _C$ is the field-of-view of the receiver.

Fig. 1.

Model of the proposed positioning scheme.

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The 3-D locations of the LED and the receiver are assume to be ${\mathbf l}_t=(x_t, y_t, z_t)^{T}$ and ${\mathbf l}_r=(x_r, y_r, z_r)^{T}$, respectively, and their normal vectors are denoted as ${\mathbf n}_t$ and ${\mathbf n}_r$, which can be defined as $$\begin{eqnarray} {\mathbf n}_t &=& (\cos (\alpha _t)\sin (\theta _t), \sin (\alpha _t)\sin (\theta _t), \cos (\theta _t))^{T}\\ {\mathbf n}_r &=& (\cos (\alpha _r)\sin (\theta _r), \sin (\alpha _r)\sin (\theta _r), \cos (\theta _r))^{T} \end{eqnarray}$$ View Source \begin{eqnarray} {\mathbf n}_t &=& (\cos (\alpha _t)\sin (\theta _t), \sin (\alpha _t)\sin (\theta _t), \cos (\theta _t))^{T}\\ {\mathbf n}_r &=& (\cos (\alpha _r)\sin (\theta _r), \sin (\alpha _r)\sin (\theta _r), \cos (\theta _r))^{T} \end{eqnarray} where $\alpha _t$ and $\alpha _r$ are the azimuth angles of the LED and the receiver respectively, which are defined as the angles between the $x$-axis and the orthogonal projections of the normal vectors of the LED and the receiver to the $xy$-plane. $\theta _t$ and $\theta _r$ present the polar angles of the LED and the receiver respectively, which are defined as the angles between the $z$-axis and the normal vectors of the LED and the receiver. Moreover, $\alpha _t$ and $\theta _t$ are determined by the location of the LED while $\alpha _r$ and $\theta _r$ are determined by the measurements of a gyroscope in the receiver.

Meanwhile, the irradiant angle and the incident angle are given by [14] $$\begin{eqnarray} \cos (\phi) &=& {{\mathbf n}_t}\cdot {\frac{{{\mathbf l}_r}-{{\mathbf l}_t}}{d}}\\ \cos (\psi \;) &=& {{\mathbf n}_r}\cdot {\frac{{{\mathbf l}_t}-{{\mathbf l}_r}}{d}}. \end{eqnarray}$$ View Source \begin{eqnarray} \cos (\phi) &=& {{\mathbf n}_t}\cdot {\frac{{{\mathbf l}_r}-{{\mathbf l}_t}}{d}}\\ \cos (\psi \;) &=& {{\mathbf n}_r}\cdot {\frac{{{\mathbf l}_t}-{{\mathbf l}_r}}{d}}. \end{eqnarray} For the case of the ceiling LED, $\alpha _t=0^\circ$ and $\theta _t=180^\circ$, and therefore, $\cos (\phi)={(z_t-z_r)}/{d}$, and for the case of the wall LED, if we assume LED's normal vector towards the positive direction of the $y$-axis of the room, $\alpha _t=90^\circ$ and $\theta _t=90^\circ$, so $\cos (\phi)={(y_t-y_r)}/{d}$.

When the emitted optical power $P_{ot}$ is received by the receiver, the incident optical power from LoS path $P_{or}$ can be given by $P_{or}=P_{ot}H$, and the corresponding electrical power $P_{er}$ is given by $P_{er}=R({\lambda }P_{or})^2$, where $R$ is the equivalent impedance of the receiver, and $\lambda$ is the optical detector responsivity. In our system, there are no optical filter and concentrator, and therefore, we can get $$\begin{equation} P_{er}= \frac{c}{d^4}{{\cos }^{2m}(\phi)}{{\cos }^2(\psi \;)} \end{equation}$$ View Source \begin{equation} P_{er}= \frac{c}{d^4}{{\cos }^{2m}(\phi)}{{\cos }^2(\psi \;)} \end{equation} where $c$ is a constant and defined as $$\begin{equation} c=\frac{R{{\lambda }^2}{P_{ot}}^2{A^2}(m+1)^2}{4{\pi ^2}}. \end{equation}$$ View Source \begin{equation} c=\frac{R{{\lambda }^2}{P_{ot}}^2{A^2}(m+1)^2}{4{\pi ^2}}. \end{equation}

In practical system, the total incident optical power $P_{total}$ contains the ambient light power $P_{bg}$ , the incident optical power from LoS path $P_{or}$ and NLoS path $P_{NLoS}$. In general, only $P_{or}$ is used for positioning in LPS, while $P_{NLoS}$ and $P_{bg}$ are both viewed as noise power. The total noise variance can be given as $$\begin{equation} N={\sigma ^2_{shot}}+{\sigma ^2_{thermal}}+R({\lambda }P_{NLoS})^2 \end{equation}$$ View Source \begin{equation} N={\sigma ^2_{shot}}+{\sigma ^2_{thermal}}+R({\lambda }P_{NLoS})^2 \end{equation} where $\sigma ^2_{shot}$ is the shot noise variance which is depended on $P_{total}$, and $\sigma ^2_{thermal}$ is the thermal noise variance which is depended on the receiver's parameters. $\sigma ^2_{shot}$, $\sigma ^2_{thermal}$, and $P_{NLoS}$ are all described detailedly in [13]. The signal-to-noise (SNR) due to LoS path can be defined as $$\begin{equation} SNR=10\log _{10} \left(\frac{P_{er}}{N}\right). \end{equation}$$ View Source \begin{equation} SNR=10\log _{10} \left(\frac{P_{er}}{N}\right). \end{equation}

2.2. AOA Estimation

By using an image sensor and a gyroscope, the AOA bearing from the LED beacon can be estimated. Here, AOA involves the incident light azimuth angle $\alpha _p$, which is defined as the angle between the $x$ -axis and the orthogonal projection of incident light to the $xy$-plane, and the incident light polar angle $\theta _p$ which presents the angle between the $z$ -axis and incident light. In addition, two frames are used as reference, which are room frame of reference $(x, y, z)^T$ and receiver frame of reference $(x^{\prime }, y^{\prime }, z^{\prime })^T$. If the room coordinates of the LED and the image sensor are presented as $(x_t, y_t, z_t)^T$ and $(x_{r1}, y_{r1}, z_{r1})^T$ respectively, and the receiver coordinates of the LED projection center on the image are $(x_I, y_I, z_I)^T$, we can get $$\begin{equation} (x_t, y_t, z_t)^T={\mathbf R}\times {k(x_I, y_I, z_I)^T}+(x_{r1}, y_{r1}, z_{r1})^T. \end{equation}$$ View Source \begin{equation} (x_t, y_t, z_t)^T={\mathbf R}\times {k(x_I, y_I, z_I)^T}+(x_{r1}, y_{r1}, z_{r1})^T. \end{equation} Subsequently $$\begin{equation} (x_t-x_{r1}, y_t-y_{r1}, z_t-z_{r1})^T={\mathbf R}\times {k(x_I, y_I, z_I)^T}, \end{equation}$$ View Source \begin{equation} (x_t-x_{r1}, y_t-y_{r1}, z_t-z_{r1})^T={\mathbf R}\times {k(x_I, y_I, z_I)^T}, \end{equation} where $k$ is a scale factor of the receiver coordinates of the LED and its projection to the image. ${\mathbf R}$ is a 3 $\times$ 3 rotation matrix which is given by $$\begin{equation} {\mathbf R}= \left[ \begin{array}{ccc}\cos (\rho) & \sin (\rho) & 0\\ -\sin (\rho) & \cos (\rho) & 0\\ 0&0&1 \end{array} \right] \left[ \begin{array}{ccc}\cos (\eta) & 0 & -\sin (\eta)\\ 0&1&0\\ \sin (\eta) & 0 & \cos (\eta) \end{array} \right] \left[ \begin{array}{ccc}1&0&0\\ 0 & \cos (\delta) & \sin (\delta)\\ 0 & -\sin (\delta) & \cos (\delta) \end{array} \right] \end{equation}$$ View Source \begin{equation} {\mathbf R}= \left[ \begin{array}{ccc}\cos (\rho) & \sin (\rho) & 0\\ -\sin (\rho) & \cos (\rho) & 0\\ 0&0&1 \end{array} \right] \left[ \begin{array}{ccc}\cos (\eta) & 0 & -\sin (\eta)\\ 0&1&0\\ \sin (\eta) & 0 & \cos (\eta) \end{array} \right] \left[ \begin{array}{ccc}1&0&0\\ 0 & \cos (\delta) & \sin (\delta)\\ 0 & -\sin (\delta) & \cos (\delta) \end{array} \right] \end{equation} where $\delta$, $\eta$, and $\rho$ are the receiver's pitch angle, roll angle, and yaw angle which are all measured by a gyroscope in the receiver. From (12) , the receiver's azimuth angle $\alpha _r$ and polar angle $\theta _r$ defined in (2) and (3) can be determined by $$\begin{equation} \left\lbrace \begin{array}{l}\alpha _r=\tan ^{-1} \left({\frac{\tan (\delta) \;+ \;\sin (\eta)\tan (\rho)}{\tan (\delta)\tan (\rho)\; -\; \sin (\eta)}}\right)\\ \theta _r=\cos ^{-1}(\cos (\eta)\cos (\rho)). \end{array}\right. \end{equation}$$ View Source \begin{equation} \left\lbrace \begin{array}{l}\alpha _r=\tan ^{-1} \left({\frac{\tan (\delta) \;+ \;\sin (\eta)\tan (\rho)}{\tan (\delta)\tan (\rho)\; -\; \sin (\eta)}}\right)\\ \theta _r=\cos ^{-1}(\cos (\eta)\cos (\rho)). \end{array}\right. \end{equation}

Therefore, from the geometric relationship of the LED beacon and the image sensor, the azimuth angle $\alpha _p$ and the polar angle $\theta _p$ of the incident light can be defined as $$\begin{equation} \left\lbrace \begin{array}{l}\alpha _p=\tan ^{-1} \left(\frac{x_{t}\,-\,x_{r1}}{y_{t}-y_{r1}}\right)\\ \theta _p=\tan ^{-1} \left(\frac{\sqrt{(x_{t}\,-\,x_{r1})^2\,+\,(y_{t}-y_{r1})^2}}{(z_{t}\,-\,z_{r1})^2}\right).\\ \end{array} \right. \end{equation}$$ View Source \begin{equation} \left\lbrace \begin{array}{l}\alpha _p=\tan ^{-1} \left(\frac{x_{t}\,-\,x_{r1}}{y_{t}-y_{r1}}\right)\\ \theta _p=\tan ^{-1} \left(\frac{\sqrt{(x_{t}\,-\,x_{r1})^2\,+\,(y_{t}-y_{r1})^2}}{(z_{t}\,-\,z_{r1})^2}\right).\\ \end{array} \right. \end{equation} Substituting (11) and (12) into (14), the angles $\alpha _p$ and $\theta _p$ can be estimated.

In addition, the coordinates of the LED projection center in the image can be determined by the following steps. Firstly, the image captured by the image sensor is grayed. Secondly, the image is filtered by a binary OTSU filter [15]. Lastly, the image is blurred and the LED projection center is determined by means of averaging all points in the projection of the LED beacon. Moreover, the gyroscope accuracy is also discussed. In general, the accuracy of the raw gyroscope outputs is quite low on account of its serious zero drift effect. However, after compensation and calibration, this effect can be suppressed effectively and the accuracy can be improved greatly. According to our experiment results, the maximum angle error measured by the gyroscope in iPhone 6 is about 2$^\circ$. Besides, the distance between the lens and imager in the image sensor $z_I$ is an important parameter, which is used to estimate the AOA bearing, and it would change with the distance between the object and lens, but fortunately, it deviates only $0.7\%$ when the distance from the object to lens changes from 1 m to infinity  [10]. Moreover, the distance between LED and receiver is only several meters, so it's reasonable to assume $z_I$ to be a constant for positioning. Based on known locations of the LED beacon and the image sensor (the front facing camera of iPhone 6 is used in this paper), $z_I$ is set with 1230 pixels.

2.3. RSS/AOA-Based Hybrid 3-D Positioning Algorithm

In addition, by using the RSS/AOA-based hybrid positioning algorithm, 3-D positioning can be achieved. The schematic of the positioning algorithm is illustrated in Fig. 2. Here, the coordinates of the LED and the receiver are still assumed to be ${\mathbf l}_t=(x_t, y_t, z_t)^T$ and ${\mathbf l}_{r}=(x_r, y_r, z_r)^T$, respectively which indicate their center coordinates. In the receiver, an image sensor and a PD are assumed to be deployed on the positions of ${\mathbf l}_{r1}=(x_{r1}, y_{r1}, z_{r1})^T$ and ${\mathbf l}_{r2}=(x_{r2}, y_{r2}, z_{r2})^T$. Although the image sensor and the PD can be deployed on any positions of the receiver, the relationships of their positions and the receiver center should be known in advance. As shown in Fig. 2 , $d_1$ and $d_2$ are defined as the distances from the image sensor and the PD to the receiver center, $\omega _1$ is the angle between the receiver's axis and the line through the receiver center as well the image sensor, and $\omega _2$ is the angle between the receiver's axis and the line through the receiver center as well the PD, the relationships of ${\mathbf l}_{r1}$, ${\mathbf l}_{r2}$, and ${\mathbf l}_{t}$ can be given as $$\begin{equation} \left\lbrace \begin{array}{l}{\mathbf l}_{r1}={\mathbf l}_{r}+{d_1(\cos ({\alpha _r}+{\omega _1})\sin (\theta _r),\sin ({\alpha _r}+{\omega _1})\sin (\theta _r),\cos (\theta _r))^T} \\ {\mathbf l}_{r2}={\mathbf l}_{r}+{d_2(\cos ({\alpha _r}+{\omega _2})\sin (\theta _r),\sin ({\alpha _r}+{\omega _2})\sin (\theta _r),\cos (\theta _r))^T}.\\ \end{array} \right. \end{equation}$$ View Source \begin{equation} \left\lbrace \begin{array}{l}{\mathbf l}_{r1}={\mathbf l}_{r}+{d_1(\cos ({\alpha _r}+{\omega _1})\sin (\theta _r),\sin ({\alpha _r}+{\omega _1})\sin (\theta _r),\cos (\theta _r))^T} \\ {\mathbf l}_{r2}={\mathbf l}_{r}+{d_2(\cos ({\alpha _r}+{\omega _2})\sin (\theta _r),\sin ({\alpha _r}+{\omega _2})\sin (\theta _r),\cos (\theta _r))^T}.\\ \end{array} \right. \end{equation}

Fig. 2.

Schematic of the RSS/AOA-based hybrid 3-D positioning algorithm.

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From the measured RSS on the position ${\mathbf l}_{r2}$, we can get $$\begin{equation} \left\lbrace \begin{array}{l}P_{er}={\frac{c}{d^4}}{{\cos ^{2m}(\phi)}{\cos ^2(\psi \;)}} \\ d=\sqrt{{(x_{r2}-x_{t})^2+(y_{r2}-y_{t})^2+(z_{r2}-z_{t})^2}}.\\ \end{array} \right. \end{equation}$$ View Source \begin{equation} \left\lbrace \begin{array}{l}P_{er}={\frac{c}{d^4}}{{\cos ^{2m}(\phi)}{\cos ^2(\psi \;)}} \\ d=\sqrt{{(x_{r2}-x_{t})^2+(y_{r2}-y_{t})^2+(z_{r2}-z_{t})^2}}.\\ \end{array} \right. \end{equation}

Based on the estimated AOA bearing (the azimuth angle $\alpha _{p}$ and the polar angle $\theta _{p}$, as shown in Fig. 2), we can get $$\begin{equation} \left\lbrace \begin{array}{l}\tan (\alpha _{p})=\frac{x_{t}\,-\,x_{r1}}{y_{t}\,-\,y_{r1}}\\ \tan (\theta _{p})=\frac{\sqrt{(x_{t}\,-\,x_{r1})^2\;+\;(y_{t}\,-\,y_{r1})^2}}{(z_{t}\,-\,z_{r1})^2}.\\ \end{array} \right. \end{equation}$$ View Source \begin{equation} \left\lbrace \begin{array}{l}\tan (\alpha _{p})=\frac{x_{t}\,-\,x_{r1}}{y_{t}\,-\,y_{r1}}\\ \tan (\theta _{p})=\frac{\sqrt{(x_{t}\,-\,x_{r1})^2\;+\;(y_{t}\,-\,y_{r1})^2}}{(z_{t}\,-\,z_{r1})^2}.\\ \end{array} \right. \end{equation} Moreover, the locations of the LED beacon and the image sensor follow as $$\begin{equation} \left\lbrace \begin{array}{l}x_{r1} > x_t, y_{r1}>y_t, 0 < \alpha _p\leq {\frac{\pi }{2}}\\ x_{r1}<x_t, y_{r1}>y_t, {\frac{\pi }{2}}<\alpha _p\leq \pi \\ x_{r1}<x_t, y_{r1}<y_t, \pi <\alpha _p\leq {\frac{3\pi }{2}}\\ x_{r1}>x_t, y_{r1}<y_t, {\frac{3\pi }{2}}<\alpha _p\leq {2\pi }.\\ \end{array} \right. \end{equation}$$ View Source \begin{equation} \left\lbrace \begin{array}{l}x_{r1} > x_t, y_{r1}>y_t, 0 < \alpha _p\leq {\frac{\pi }{2}}\\ x_{r1}<x_t, y_{r1}>y_t, {\frac{\pi }{2}}<\alpha _p\leq \pi \\ x_{r1}<x_t, y_{r1}<y_t, \pi <\alpha _p\leq {\frac{3\pi }{2}}\\ x_{r1}>x_t, y_{r1}<y_t, {\frac{3\pi }{2}}<\alpha _p\leq {2\pi }.\\ \end{array} \right. \end{equation}

Based on (15)–​(18), the receiver's 3-D position coordinates can be estimated. Note that our scheme is not limited in the application of one LED beacon, and it still works in multiple LED beacons case. By using the RF carrier allocation technology [16], different LED projections in the image can be distinguished by means of the image sensor's rolling shutter effect  [10] and the mixed beacon signals from different LED beacons can be extracted, respectively, by using a PD, a fast fourier transform (FFT) module, and the corresponding filters in the receiver. Moreover, every LED beacon can determine one 3-D location of the receiver, thus the positioning can be achieved by an optimization process trying to minimize the linear mean square (LMS) error [8] .

3.1. Performance of the AOA Estimation

The performance of the image sensor-based AOA estimation is investigated by experiment in the cases of the ceiling-mounted and the wall-mounted LED, respectively. In the following part of this paper, the projection center of the LED beacon on the $xy$-plane is assumed to be the origin of room frame. Moreover, the vertical distance between the LED beacon and the receiver is set to 2 m. The error distribution of the azimuth angle $\alpha _p$ is firstly depicted in Fig. 3. As shown in Fig. 3(a) (the case of the ceiling-mounted LED), a great error is found on the origin of coordinates. The reason is that, as the smartphone is just deployed on the position below the LED, due to the mismatch between the real and assumed LED positions, a small center deviation of the LED projection in the image maybe induce a great estimation error of $\alpha _p$. If the error on the origin is neglected, the mean error is only 3.1$^\circ$. Moreover, as shown in Fig. 3(b) (the case of the wall-mounted LED), the greater errors are achieved on the positions near the $x$-axis (here, we still assume the LED's normal vector towards the positive direction of the $y$-axis). This is because as the smartphone gets close to the wall, the irradiant angle gets close to 90 $^\circ$, and therefore, the LoS light becomes weaker. Meanwhile, the NLoS light reflected by the ceiling gets stronger, which would affect the estimation of the LED projection center in the image dramatically so that the greater errors are found near the wall. This time, the mean error is 19.5$^\circ$. The estimation error of the polar angle $\theta _p$ is also investigated and the results are shown in Tables I and II. As observed, when the LED beacon is mounted on the ceiling, the relative small estimation errors can be achieved, and only 0.9$^\circ$ mean error is achieved. However, when the LED beacon is mounted on the wall, the estimation errors become greater due to the influence of the NLoS light reflected by the ceiling. At this time, the mean error becomes 3.8$^\circ$.

Fig. 3.

Error distribution of the azimuth angle $\alpha _p$ in the cases of (a) the ceiling-mounted LED and (b) the wall-mounted LED.

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TABLE 1 Polar Angle Errors in the Case of Ceiling-Mounted LED

TABLE 2 Polar Angle Errors in the Case of Wall-Mounted LED

3.2. Performance of the Proposed Positioning Scheme

In this section, the performance in terms of positioning error is evaluated with the ceiling-mounted and the wall-mounted LED. Here, the positioning error is defined as the Euclidean distance between the estimated position and the real position. In our scheme, positioning error is mostly influenced by the AOA ($\alpha _p$ and $\theta _p$) estimation error, the RSS measurement error, and the errors of the receiver's azimuth angle $\alpha _r$ and polar angle $\theta _r$. Because the errors of $\alpha _r$ and $\theta _r$ are irrelevant to the measurement positions, the impacts of the AOA and RSS on positioning error at different positions are investigated in the following. First, according to the real positions of the LED beacon and the receiver (the PD or the image sensor), the theoretical values of RSS and AOA can be easily obtained by using the estimation methods of RSS and AOA which are introduced in Section 2. Then, the theoretical RSS and the estimated AOA are substituted into the RSS/AOA-based hybrid positioning algorithm. The positioning error distribution is depicted in Fig. 4. From Fig. 4(a) (the case of the ceiling-mounted LED), 12 cm maximum positioning error and 2.1 cm minimum one are achieved on the positions of (0.8 m, 0.8 m) and (0 m, 0.2 m), respectively. The average positioning error is 5.8 cm. Moreover, compared to the results in Fig. 3(a), although a great estimations error of $\alpha _p$ is achieved on the origin of coordinates, the positioning error on this position is insensitive to the error of $\alpha _p$. Contrastively, as observed in Fig. 4(b) (the case of the wall-mounted LED), 2.15 m maximum positioning error and 1.4 cm minimum one are achieved on the positions of (0.8 m, 0.1 m) and (0.2 m, 0.5 m) respectively. The average positioning error is 57.6 cm.

Fig. 4.

Distribution of the positioning error induced by AOA in the cases of (a) the ceiling-mounted LED and (b) the wall-mounted LED.

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Furthermore, the positioning error induced by RSS is also investigated. Similarly, the theoretical AOA and the measured RSS are substituted into the hybrid positioning algorithm. By reason of RSS easily influenced by the NLoS light, we assume two applications in the case of the ceiling-mounted LED, which are the LED mounted on the center of ceiling (the room size is 4 m $\times$ 6 m $\times$ 3 m) and the LED mounted on the edge of ceiling (the LED is 1 m away from the wall). The positioning error results of the ceiling center-mounted LED is presented in Fig. 5(a). A 22 cm maximum positioning error and 0.1 cm minimum one are achieved on the positions of (0.8 m, 0.8 m) and (0 m, 0.6 m), respectively. The average positioning error is 6.8 cm. Besides, it is easily got that, the positioning errors on the $x$-axis are greater than the ones on the $y$-axis. These can be attributed to that, the position on the $x$-axis is nearer to the wall than the symmetrical position on the $y$-axis, which makes the receiver suffer from more influences of the NLoS light along the positive direction of the $x$-axis, hence achieving the greater positioning errors. Meanwhile, the positioning error results of the ceiling edge-mounted LED is presented in Fig. 5(c). Comparatively, 40 cm maximum positioning error and 7.9 cm minimum one are achieved on the positions of (0.8 m, 0.8 m) and (0 m, 0.4 m), respectively, while 19.5 cm average positioning error is achieved. In addition, the positioning error in the case of the wall-mounted LED is shown in Fig. 5(c). The LED's normal vector is still towards the positive direction of the $y$-axis, and the greater errors are found on the positions near the $x$ -axis. These errors are not only contributed by the influence of the NLoS light but by the weak incident light strength as well.

Fig. 5.

Distribution of the positioning error induced by RSS in the cases of (a) the ceiling center-mounted LED, (b) the ceiling edge-mounted LED, and (c) the wall-mounted LED.

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The overall positioning error results are shown in Fig. 6. It is easily got that, when the LED is deployed in the center of ceiling, 16.2 cm maximum positioning error and 2.1 cm minimum one are achieved on the positions of (0.8 m, 0.8 m) and (0 m, 0.2 m) respectively, and 7.3 cm average positioning error is obtained. However, when the LED is mounted on the edge of ceiling, the maximum and minimum positioning error become 33.4 cm and 8.5 cm respectively, and the average error is 19.1 cm. That indicates the NLoS light would influence the scheme performance greatly, especially, when the receiver gets close to the wall. In addition, when the LED is mounted on the wall, the worst performance is achieved. As observed in Fig. 6(c), the average positioning error increases to 67.2 cm. Therefore, from the experimental results, we can find the performance with the ceiling-mounted LED is better than the one with the wall-mounted case. Moreover, due to the influence of the NLoS light, the performance at the edge of room is worse than the one in the center of room.

Fig. 6.

Distribution of the overall positioning error in the cases of (a) the ceiling center-mounted LED, (b) the ceiling edge-mounted LED, and (c) the wall-mounted LED.

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In the following, the impact of the receiver tilting on our scheme is discussed. This impact can be divided into two aspects. One is the impact on RSS estimation, and the other is the one on AOA estimation. First, the former is discussed. Due to the receiver (the PD) tilting, the incident angle $\psi\;$ in (1) is changed. That makes the channel model inaccurate, so great positioning errors are achieved. In order to improve the channel accuracy, the incident angle can be compensated based on the gyroscope measurements. Moreover, the high channel accuracy in the case of the PD tilting has been demonstrated [11], [12], [16]. Therefore, if the influence of the NLoS light is neglected, the receiver tilting would have little influence on RSS estimation. In addition, as for the impact on AOA estimation, the receiver tilting has little influence on the determination of the projection center of the LED beacon to the image. Meanwhile, the gyroscope accuracy has no selectivity to the receiver tilting angle. Therefore, the receiver tilting has also little influence on AOA estimation. Therefore, if the influence of the NLoS light is neglected, the performance in the case of the receiver tilting would be similar with the one in the case of the receiver horizontal.

At last, the complexity of our scheme is discussed. Due to the use of a PD, an image sensor, and a gyroscope in our receiver, the hardware complexity would be higher than other LPSs. However, our scheme is designed to be applied for smartphones and these devices have been integrated in most of commercial available smartphones, so the hardware complexity can be neglected. With respect to the software complexity, only one image captured by the image sensor in our scheme is necessary in one positioning process no matter how many the beacon signal is, and therefore, the image processing would be more simple than the image sensor-based LPS [10]. In addition, the power consumption can be reduced by combining with the inertial navigation system (INS). That is, our scheme only provide the target's original coordinates and calibration for INS.