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A Two-LED Based Indoor Three-Dimensional Visible Light Positioning and Orienteering Scheme for a Tilted Receiver

A Two-LED Based Indoor Three-Dimensional Visible Light Positioning and Orienteering Scheme for a... hv photonics Article A Two-LED Based Indoor Three-Dimensional Visible Light Positioning and Orienteering Scheme for a Tilted Receiver 1 , 1 2 1 Xiaodi You * , Xiaobai Yang , Zile Jiang and Shuang Zhao School of Electronic and Information Engineering, Soochow University, Suzhou 215006, China; 20205228029@stu.suda.edu.cn (X.Y.); szhaoszhao@stu.suda.edu.cn (S.Z.) Kuang Yaming Honors School, Nanjing University, Nanjing 210023, China; 181240026@smail.nju.edu.cn * Correspondence: xdyou@suda.edu.cn Abstract: Conventional visible light positioning (VLP) systems usually require at least three light- emitting diodes (LEDs) to enable trilateration or triangulation, which is infeasible when the LED condition is constrained. In this paper, we propose a novel indoor three-dimensional (3D) VLP and orienteering (VLPO) scheme. By using only two LEDs and two photo-detectors (PDs), our scheme can achieve simultaneous 3D localization and receiver orientation estimation efficiently. Further, to eliminate the location uncertainty caused by receiver tilt, we propose a location selection strategy which can effectively determine the true location of the receiver. Through extensive simulations, it is found that when the receiver faces upwards, the proposed scheme can achieve a mean 3D positioning error of 7.4 cm and a mean azimuthal error of 7.0 . Moreover, when the receiver tilts with a polar angle of 10 , accurate VLPO can still be achieved with 90.3% of 3D positioning errors less than 20 cm and 92.6% of azimuthal errors less than 5 . These results indicate that our scheme is a promising solution to achieve accurate VLPO when there is only two LEDs. Results also verify the effectiveness of the VLPO scheme when locating a tilted receiver. Keywords: visible light positioning (VLP); three-dimensional (3D); visible light positioning and orienteering (VLPO); light-emitting diode (LED) Citation: You, X.; Yang, X.; Jiang, Z.; Zhao, S. A Two-LED Based Indoor Three-Dimensional Visible Light Positioning and Orienteering Scheme 1. Introduction for a Tilted Receiver. Photonics 2022, 9, The emerging location-based services (LBSs) have raised increasing interest in localiza- 159. https://doi.org/10.3390/ tion technologies [1]. Satellite-based global positioning systems are widely used in outdoor photonics9030159 environments. However, satellite signals suffer from fading when passing though solid Received: 30 January 2022 walls, thus their positioning accuracy will be severely influenced in indoor environments Accepted: 3 March 2022 such as underground parking, tunnels, and office buildings surrounded by skyscrapers. Published: 5 March 2022 With the development of visible light communication (VLC) technologies, the visible light positioning (VLP) system using light-emitting diodes (LEDs) has been considered as a Publisher’s Note: MDPI stays neutral potential candidate for the next-generation indoor positioning system, due to its advan- with regard to jurisdictional claims in tages of high accuracy, high efficiency, and low cost [2,3]. By modulating signals to existing published maps and institutional affil- lighting facilities, VLP systems can offer various indoor LBSs such as high-accurate indoor iations. navigation, asset tracking, and autonomous robot control. The simplest indoor VLP scheme is based on the principle of proximity and its position- ing accuracy is usually in meters [4]. To improve the positioning accuracy, different VLP Copyright: © 2022 by the authors. schemes based on trilateration or triangulation were reported, which can achieve centimeter- Licensee MDPI, Basel, Switzerland. level VLP [5–7]. However, to offer positioning information for trilateration or triangulation, This article is an open access article conventional VLP systems usually require at least three LEDs as transmitters to convey distributed under the terms and VLP signals according to the time domain multiplexing (TDM) or frequency domain mul- conditions of the Creative Commons tiplexing (FDM) protocols to avoid mutual interference at the receiver and make it easy Attribution (CC BY) license (https:// for the receiver to distinguish between VLP signals [8,9]. Thus, at least three time slots or creativecommons.org/licenses/by/ frequency bands are allocated for VLP. Since the bandwidth of off-the-shelf LEDs is narrow 4.0/). Photonics 2022, 9, 159. https://doi.org/10.3390/photonics9030159 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 159 2 of 15 and usually a few of tens of MHz [10], VLP will inevitably limit the available bandwidth of VLC in an integrated VLC and VLP system [11]. To ensure the ample capacity of VLC, new approaches are required to reduce the time or frequency resources allocated for VLP. This study will focus on the VLP mechanism to propose a new VLP model and its corresponding strategies to support VLP with fewer bandwidth resources. Another issue is that the LED condition of a VLP system could be constrained in some common scenarios, e.g.,: (i) when the LED number is insufficient, i.e., less than three; and (ii) when LED lamps are installed along a straight line, e.g., in a corridor or a tunnel. In these scenarios, the receiver is likely to detect the VLP signals from only one or two LEDs. Thus, conventional VLP schemes based on trilateration or triangulation cannot be directly applied. Different approaches have been proposed to perform indoor VLP with less than three LEDs, which can be categorized as two-dimensional (2D) [12–14] or three-dimensional (3D) [15–17] schemes. However, all these schemes require additional equipment such as a camera [12,13,15], a mirror [14], line lasers [16], or at least three PDs [17], which could increase the system complexity and needs to be simplified. Motivated by this, this work will particularly focus on the 3D VLP scheme, which requires fewer PDs when the LED condition is constrained. Recently, receiver orientation estimation has become an additional function of VLP. Joint VLP and orientation estimation schemes were proposed in [18,19], and they both support only 2D VLP with quite a number of LEDs (e.g., twenty LEDs in [18], and forty- eight LEDs in [19]). However, there have been no related works specifically dedicated to VLP schemes that can support simultaneous 3D localization and receiver orienteering using less than three LEDs and a PD-based receiver. Therefore, this work will also try to fill in the gap. In this paper, we propose a novel indoor 3D VLP and orienteering (VLPO) scheme using two LEDs and a pair of PDs. To the best of our knowledge, this is the first time that a two-LED-two-PD-based indoor 3D VLPO scheme is proposed, by which only two time slots or frequency bands are required to convey VLP signals. Thus, the bandwidth resources allocated for VLP can be effectively reduced as compared with conventional schemes based on trilateration or triangulation. In addition, the proposed scheme can achieve a good trade-off between the system complexity and the localization capability. As shown in Table 1, by using two LEDs and two PDs without any other types of equipment, our scheme is the only one that supports simultaneous 3D VLP and orientation estimation for a tilted receiver, thus it can be considered as a promising solution, especially when the LED condition is constrained. Table 1. Comparison between different VLP schemes. 2D or 3D Support Support Reference Transmitter Receiver VLP Orienteering Receiver Tilt [12] 2 LEDs 1 camera 2D No No [13] 1 LED 1 camera 2D No No [14] 2 LEDs 1 PD + 1 mirror 2D No No [15] 1 LED 1 camera 3D No Yes [16] 2 line lasers 2 PDs 3D Yes No [17] 1 LED array 3 PDs 3D No Yes [18] 20 LEDs 1 PD 2D Yes No [19] 48 LEDs 7 PDs 2D Yes Yes This work 2 LEDs 2 PDs 3D Yes Yes The concept of the VLPO scheme was shown in our conference paper [20], wherein we have preliminarily verified its feasibility assuming the receiver to face upwards. However, when the receiver tilts, the model in [20] cannot be directly applied and needs to be modified. Therefore, in this work, we further develop the VLPO scheme by considering the scenario of a tilted receiver. To eliminate the potential location uncertainty caused by the receiver tilt, we propose a new location selection strategy for the VLPO system to determine the Photonics 2022, 9, x FOR PEER REVIEW 3 of 16 Photonics 2022, 9, 159 3 of 15 the scenario of a tilted receiver. To eliminate the potential location uncertainty caused by the receiver tilt, we propose a new location selection strategy for the VLPO system to de- termine the true location of the receiver. We conduct extensive simulations to evaluate the tr po ue sit location ioning aof ccurac the ry eceiver and orient . We conduct eering accur extensive acy in a simulations 3 m × 5 m to × evaluate 3 m indoor s the positioning pace. The influence from the interval between PDs, the ranging error between LEDs and PDs, and accuracy and orienteering accuracy in a 3 m 5 m 3 m indoor space. The influence from the polar angle of the tiled receiver is also evaluated. Results show that the proposed the interval between PDs, the ranging error between LEDs and PDs, and the polar angle of the scheme tiledir seceiver efficient to a is also chi evaluated. eve accura Results te 3D VL show PO w that hen the the pr L oposed ED conscheme dition isis co ef ns ficient traineto d. achieve Specifically, accurate when the rec 3D VLPO eiver when face the s upward LED condition s, the proposed sch is constrained. eme can Specifically achieve a me , when the an receiver faces upwards, the proposed scheme can achieve a mean 3D positioning error of 3D positioning error of 7.4 cm and a mean azimuthal error of 7.0°. Moreover, when the 7.4 cm and a mean azimuthal error of 7.0 . Moreover, when the receiver tilts with a polar receiver tilts with a polar angle of 10°, the proposed scheme can still achieve accurate VLP angle of 10 , the proposed scheme can still achieve accurate VLP and orienteering, with and orienteering, with 90.3% of 3D positioning errors less than 20 cm and 92.6% of azi- 90.3% of 3D positioning errors less than 20 cm and 92.6% of azimuthal errors less than 5 . muthal errors less than 5°. Results also indicate that increasing the interval between PDs Results also indicate that increasing the interval between PDs and reducing the ranging and reducing the ranging error between LEDs and PDs help enhance the accuracy of error between LEDs and PDs help enhance the accuracy of VLPO, and the receiver tilt only VLPO, and the receiver tilt only slightly degrades the performance of the VLPO scheme. slightly degrades the performance of the VLPO scheme. The rest of this paper is organized as follows. Section 2 presents the principle of the The rest of this paper is organized as follows. Section 2 presents the principle of the proposed 3D VLPO system in detail. Simulations and performance analyses are con- proposed 3D VLPO system in detail. Simulations and performance analyses are conducted ducted in Section 3. Finally, we conclude the paper in Section 4. in Section 3. Finally, we conclude the paper in Section 4. 2. Proposed VLPO Scheme 2. Proposed VLPO Scheme In this study, we assume a multi-cell VLPO system, in which there are multiple LEDs In this study, we assume a multi-cell VLPO system, in which there are multiple installed along a straight line in the ceiling, and mobile receivers are distributed at differ- LEDs installed along a straight line in the ceiling, and mobile receivers are distributed at ent indoor locations. The LEDs are divided into different groups in pairs to spatially form different indoor locations. The LEDs are divided into different groups in pairs to spatially multiple VLPO cells. Based on this multi-cell configuration, we focus on VLPO within a form multiple VLPO cells. Based on this multi-cell configuration, we focus on VLPO single cell where a mobile receiver can only detect the VLP signals emitted from two LEDs within a single cell where a mobile receiver can only detect the VLP signals emitted from (LED1 and LED2). Without loss of generality, we abstract the receiver into a line segment two LEDs (LED and LED ). Without loss of generality, we abstract the receiver into a 1 2 and assume that a pair of photo-detectors (PD1 and PD2) are mounted on both ends of the line segment and assume that a pair of photo-detectors (PD and PD ) are mounted on 1 2 receiver. Figure 1 shows the architecture of the proposed VLPO system. The 3D coordi- both ends of the receiver. Figure 1 shows the architecture of the proposed VLPO system. nates of LED1 and LED2 are (xt1, yt1, zt1) and (xt2, yt2, zt2), respectively, which are a priori The 3D coordinates of LED and LED are (x , y , z ) and (x , y , z ), respectively, 1 2 t 1 t 1 t 1 t 2 t 2 t 2 knowledge at the receiver. The 3D coordinates of PD1 and PD2 are (xr1, yr1, zr1) and (xr2, yr2, which are a priori knowledge at the receiver. The 3D coordinates of PD and PD are 1 2 zr2), respectively, which are unknown at the receiver and need to be estimated. We use the (x , y , z ) and (x , y , z ), respectively, which are unknown at the receiver and need to r 1 r 1 r 1 r 2 r 2 r 2 midpoint of PD1 and PD2 to represent the location of the receiver, i.e., (xR, yR, zR) = be estimated. We use the midpoint of PD and PD to represent the location of the receiver, 1 2 ((x +x )/2, (y +y )/2, (z +z )/2). We also define the direction pointing from PD1 towards r1 r2 r1 r2 i.e., (x , y , z r1 ) = r(( 2 x + x )/2, (y + y )/2, (z + z )/2). We also define the direction R R R r1 r2 r1 r2 r1 r2 PD2 as the orientation of the receiver. Thus, the azimuthal angle of the receiver is defined pointing from PD towards PD as the orientation of the receiver. Thus, the azimuthal angle 1 2 by the angular rotation η (−180° < η ≤ 180°) from the positive X-axis to the projection of the of the receiver is defined by the angular rotation h (180 < h  180 ) from the positive receiver (from PD1 to PD2) on the XY-plane. Therefore, if we successfully estimate the 3D X-axis to the projection of the receiver (from PD to PD ) on the XY-plane. Therefore, if 1 2 coordinates of PD1 and PD2, then the location and the orientation of the receiver can be we successfully estimate the 3D coordinates of PD and PD , then the location and the 1 2 calculated. orientation of the receiver can be calculated. Figure 1. Proposed VLPO system. Figure 1. Proposed VLPO system. To estimate the coordinates of PD and PD , we assume that the LEDs and PDs 1 2 are perfectly synchronized to a common clock, and the VLP signals launched by the LEDs are used as the basis of ranging. Based on the ranging methods of time-of-arrival Photonics 2022, 9, 159 4 of 15 (TOA) or phase-difference-of-arrival (PDOA) [21,22], the distances between the two LEDs and two PDs can be measured, and their estimation accuracy is typically in the order of centimeters [21]. We define the real distance between LED and PD , LED and PD , LED 1 1 2 1 1 and PD , LED and PD to be d , d , d , and d , respectively. Therefore, we obtain the 2 2 2 1 2 3 4 following geometrical relationships: 2 2 2 d = (x ˆ x ) + (y ˆ y ) + (z ˆ z ) (1) r1 t1 r1 t1 r1 t1 2 2 2 d = (x ˆ x ) + (y ˆ y ) + (z ˆ z ) (2) r1 t2 r1 t2 r1 t2 2 2 2 d = (x ˆ x ) + (y ˆ y ) + (z ˆ z ) (3) 3 r2 t1 r2 t1 r2 t1 2 2 2 d = (x ˆ x ) + (y ˆ y ) + (z ˆ z ) (4) r2 t2 r2 t2 r2 t2 where d (i = 1, 2, 3, 4) is the estimated values of d (i = 1, 2, 3, 4) obtained by ranging, i i x ˆ , y ˆ , z ˆ is the coordinate of PD to be estimated, and x ˆ , y ˆ , z ˆ is the coordinate of ( ) ( ) r1 r1 r1 1 r2 r2 r2 PD to be estimated. To offer richer information for performing VLPO, the interval between PD and PD is known at the receiver and fixed at l. Additionally, we assume that the 1 2 polar angle q (0  q  90 ) of the tilted receiver can be acquired by extra sensors at the receiver [22]. When q = 0 , the receiver faces upwards. Based on these presumptions, we can derive additional geometrical relationships: 2 2 2 2 l = (x ˆ x ˆ ) + (y ˆ y ˆ ) + (z ˆ z ˆ ) (5) r2 r1 r2 r1 r2 r1 z ˆ = z ˆ + l sin q (6) r2 r1 According to the above nonlinear system of equations from Equations (1)–(6), we ex- pect to solve the coordinates of PD and PD . However, it would be difficult to directly solve 1 2 the coordinates of PDs from Equations (1)–(6) in the current XYZ-coordinate system. Thus, we divide the solving process into three steps, which includes establishing intersection circles, conducting coordinate transformations, and excluding redundant solutions. Figure 2a shows the schematic diagram of establishing intersection circles. In this step, we first subtract between Equations (1) and (2), i.e., the two spherical surfaces centered at LED and LED , to obtain a plane equation, which is described by a plane P . PD is located 1 2 1 1 in the plane P , specifically, on the intersection circle K between the spherical surfaces 1 1 Equations (1) and (2). Here, K is centered at (a , b , c ) with a radius of R . Similarly, after 1 1 1 1 1 subtracting between Equations (3) and (4), PD is located in a plane P , specifically, on the 2 2 intersection circle K centered at (a , b , c ) with a radius of R . For the intersection circles 2 2 2 2 2 K and K , the coordinates of K (i = 1, 2) are given by: 1 2 i w w w i i i K (a , b , c ) = x + (x x ) , y + (y y ) , z + (z z ) (7) t2 t2 t2 i i i i t1 t1 t1 t1 t1 t1 L L L where L is the distance between LED and LED , and w (i = 1, 2) is the distance from LED 1 2 i 1 to the Plane P (i = 1, 2), derived as: 2 2 2 2 2 ˆ ˆ (x x ) + (y y ) + (z z ) + d d t2 t1 t2 t1 t2 t1 1 2 w = q (8) 2 2 2 2 x x + y y + z z ( ) ( ) ( ) t2 t1 t2 t1 t2 t1 2 2 2 2 2 ˆ ˆ (x x ) + (y y ) + (z z ) + d d t2 t1 t2 t1 t2 t1 w = (9) 2 2 2 2 (x x ) + (y y ) + (z z ) t2 t1 t2 t1 t2 t1 Additionally, the radiuses R (i = 1, 2) of the intersection circles can be written by: ˆ 2 R = d w (10) 1 1 1 Photonics 2022, 9, x FOR PEER REVIEW 5 of 16 2 2 2 2 x − x + y − y + z − z +d − d t2 t1 t2 t1 3 4 t2 t1 w = 2 (9) 2 2 2 x − x + y − y + z − z t2 t1 t2 t1 t2 t1 Additionally, the radiuses Ri (i = 1, 2) of the intersection circles can be written by: (10) R = d − w 1 1 1 Photonics 2022, 9, 159 5 of 15 2 (11) R = d − w 2 3 2 ˆ 2 R = d w (11) 2 2 (a) (b) Figure 2. Schematic diagrams of the solving process: (a) establishing intersection circles; and (b) Figure 2. Schematic diagrams of the solving process: (a) establishing intersection circles; and conducting coordinate transformations. (b) conducting coordinate transformations. Now, based on the circles K1 and K2, the solving process for the coordinates of PDs Now, based on the circles K and K , the solving process for the coordinates of PDs 1 2 can be simplified can be by simpli constr fied ucting by co new nstruct coor ing dinate new coordin systems. ate s Thus, ystems. Th in the us, second in the step, second step, we we conduct coor conduct coordina dinate transformations, te transforma as tions, shown asin show Figur n in Figure e 2b. We first 2b. We transform first tran the sform the XYZ- 0 0 0 XYZ-coordinate system into the X Y Z -coordinate system, in which K is the origin coordinate system into the X′ Y′ Z′-coordinate system, in which K1 is the origin point, line point, line K K forms the Z -axis, and the intersection line between the plane P and the K1K2 forms the Z′-axis, and the intersection line between the plane P1 and the XY-plane 1 2 1 0 0 0 0 XY-plane forms forms t the X he-axis. X′-axis. Next, Next, w we transform e transform the the X X Y′ Y Z′ Z -coor ′-coordin dinate atsystem e system int into the o the cylindrical cylindrical coordinate system, in which the cylindrical coordinates of PD and PD are coordinate system, in which the cylindrical coordinates of PD1 and PD2 are denoted by 1 2 denoted by (R , F , 0) and (R , F , S), respectively. Here, S is the distance between K and (R 1 1, Φ 1 1, 0) and (2 R2, Φ 2 2, S), respectively. Here, S is the distance between K 1 1 and K2. After K . After coordinate transformations, Equations (5) and (6) can be written in the form of 2 coordinate transformations, Equations (5) and (6) can be written in the form of the cylin- the cylindrical coordinates: drical coordinates: F F = arccos M (12) 1 2 (12) Φ −Φ = ± arccos M 1 2 b(R sin F R sin F )= l sin q gS (13) 2 2 1 1 β R sinΦ -R sinΦ = l sinθ −γS (13) 2 r 2 1 1 2 2 2 2 where M = (R + R + S l )/(2R R ), b = a +b /S, g = c/S, a = a a , b = b 1 2 1 2 2 1 2 2 2 2 2 where M = (R1 + R2 + S − l )/(2R1R2), β = − a +b /S, γ = c/S, a = a2 − a1, b = b2 − b1, c = c2 − b , c = c c , and S = a +b +c . By solving (12) and (13), F and F can be obtained. 1 2 1 1 2 2 2 c1, and S = a +b +c . By solving (12) and (13), Φ1 and Φ2 can be obtained. Then, we per- Then, we perform coordinate inverse transformations to recover the coordinates of PD form coordinate inverse transformations to recover the coordinates of PD1 and PD2 in the and PD in the XYZ-coordinate system, which are, respectively, given by: XYZ-coordinate system, which are, respectively, given by: 0 0 0 (x ˆ , y ˆ , z ˆ ) = e , e , e (R cos F , R sin F , 0) + (a , b , c ) (14) x y z r1 r1 r1 1 1 1 1 1 1 1 x ,y ,z = e ,e ,e R cosΦ ,R sinΦ ,0 + a ,b ,c (14) r1 r1 x y z 1 1 1 1 1 1 1 r1 0 0 0 (x ˆ , y ˆ , z ˆ ) = e , e , e (R cos F , R sin F , S) + (a , b , c ) (15) x y z r2 r2 r2 2 2 2 2 1 1 1 p p 2 2 2 2 Here, the orthonormal bases are given by e = b/ a +b , a/ a +b , 0 , e = x y r r r 2 2 2 2 2 2 ac/ S a +b , bc/ S a +b , a +b /S , and e = (a/S, b/S, c/S) . Based on the solving process from Equations (12)–(15), we get a total of four solutions. In other words, due to geometrical symmetry, a total of four pairs of the estimated coor- dinates of PD and PD are distributed in the space, among which only one is true. This 1 2 means that the location of the receiver cannot be determined due to multiple solutions, and we call this situation as location uncertainty. To eliminate location uncertainty, the third step is to exclude the redundant solutions and get the true solution. In practice, the height of the VLP receiver is lower than the height of LED transmitters. Thus, we first exclude the solutions that are above the ceiling by setting the first constraint condition: Photonics 2022, 9, x FOR PEER REVIEW 6 of 16 x ,y ,z = e ,e ,e R cosΦ ,R sinΦ ,S + a ,b ,c (15) r2 r2 x y z 2 2 2 2 1 1 1 r2 2 2 2 2 Here, the orthonormal bases are given by e = b/ a +b , − a/ a +b , 0 , e = x y 2 2 2 2 2 2 ac/ S a +b , bc/ S a +b , − a +b /S , and e = a/S, b/S, c/S . Based on the solving process from Equations (12)–(15), we get a total of four solu- tions. In other words, due to geometrical symmetry, a total of four pairs of the estimated coordinates of PD1 and PD2 are distributed in the space, among which only one is true. This means that the location of the receiver cannot be determined due to multiple solu- tions, and we call this situation as location uncertainty. To eliminate location uncertainty, the third step is to exclude the redundant solutions and get the true solution. In practice, Photonics 2022, 9, 159 6 of 15 the height of the VLP receiver is lower than the height of LED transmitters. Thus, we first exclude the solutions that are above the ceiling by setting the first constraint condition: z +z )/2< min zt1, zt2 . Among the rest of solutions, we perform a further exclusion by r1 r2 z ˆ + z ˆ /2 < min z , z . Among the rest of solutions, we perform a further exclusion ( ) ( ) r1 r2 t1 t2 considering the possible moving range of the receiver. In this study, we assume that the by considering the possible moving range of the receiver. In this study, we assume that the receiver moves within the constrained space on the outside of the YZ-plane by setting the receiver moves within the constrained space on the outside of the YZ-plane by setting the second constraint condition: x +x ≥ 0, which is shown in Figure 1. r1 r2 second constraint condition: x ˆ + x ˆ  0, which is shown in Figure 1. r1 r2 During our study, we find that if the receiver faces upwards, i.e., the polar angle θ = During our study, we find that if the receiver faces upwards, i.e., the polar angle q = 0 , 0°, then only one solution remains after considering the above two constraint conditions. then only one solution remains after considering the above two constraint conditions. This This solution is taken as the estimated coordinates of PD1 and PD2, based on which the 3D solution is taken as the estimated coordinates of PD and PD , based on which the 3D 1 2 location and the orientation of the receiver can be finally calculated. However, if the re- location and the orientation of the receiver can be finally calculated. However, if the receiver ceiver tilts, i.e., θ ≠ 0°, we may obtain one or two solutions remained after considering the tilts, i.e., q 6= 0 , we may obtain one or two solutions remained after considering the above above two constraint conditions, depending on the location of the receiver, the orientation two constraint conditions, depending on the location of the receiver, the orientation of the of the receiver, and the ranging error between LEDs and PDs. Therefore, to eliminate the receiver, and the ranging error between LEDs and PDs. Therefore, to eliminate the location location uncertainty caused by the receiver tilt, we propose a location selection strategy to uncertainty caused by the receiver tilt, we propose a location selection strategy to select the select the true location of PDs. true location of PDs. Figure 3 shows a flow chart of the location selection strategy. For a tilted receiver, if Figure 3 shows a flow chart of the location selection strategy. For a tilted receiver, if there remains only one pair of the estimated coordinates of PD1 and PD2, i.e., the case of there remains only one pair of the estimated coordinates of PD and PD , i.e., the case of 1 2 one solution, then they can be directly used to perform VLPO for the receiver. However, one solution, then they can be directly used to perform VLPO for the receiver. However, if if there remain two solutions, then we need to exclude the redundant solution by making there remain two solutions, then we need to exclude the redundant solution by making comparisons between the received signal power at the PDs and the expected received sig- comparisons between the received signal power at the PDs and the expected received signal nal power at the specific locations. Without loss of generality, we use the Solution A and power at the specific locations. Without loss of generality, we use the Solution A and the the Solution A’ to represent the two possible solutions remained, among which only one Solution A to represent the two possible solutions remained, among which only one is true. iFor s true. For the Sol the Solution A,u the tion estimated A, the esti coor ma dinates ted coordi of PD nate and s ofPD PD1ar an e,d PD respectively 2 are, re , sp defined ectively by , 1 2 es es es es es es es es es es es es 0 defin (x , y ed by ( , z )xand , y(x, z, y) an , zd ( ), x and , y for , zthe ), and fo Solution r th A e S , the olutes ion timated A’, the estimated c coordinates of oordi- PD r1 r1 r2 r2 1 r1 r1 r1 r2 r2 r2 r1 r2 0 0 0 0 0 0 es’ es’ es’ es’ es’ es’ es es es es es es nates o and PD f PD are, 1 and PD respectively 2 are,, re defined spectively by (, de x fined , y , by z ) (x and , y (x , ,zy ) and , z ( ). x , y , z ). 2 r1 r1 r2 r2 r2 r2 r2 r1 r1 r1 r1 r2 Figure 3. Flow chart of location selection strategy. We first measure the received signal power from LED (i = 1, 2) to PD (j = 1, 2), which i j is expressed by: r total P = s(t) h (t)dt/T (16) i,j i,j total where s(t) is the transmitted VLP signal, T is the duration of s(t), and h (t) is the total i,j channel impulse response (CIR) from LED (i = 1, 2) to PD (j = 1, 2), which can be further i j written by [23,24]: +¥ total LOS NLOS LOS LOS NLOS NLOS NLOS NLOS h (t)= h (t)+h (t)= H (0)d t t + A t d t t dt (17) i,j i,j i,j i,j i,j i,j i,j i,j i,j NLOS t =0 i,j Photonics 2022, 9, 159 7 of 15 LOS Here, d t is Dirac function. From LED (i = 1, 2) to PD (j = 1, 2), h t is the ( ) ( ) i j i,j NLOS LOS NLOS line-of-sight (LOS) CIR, h (t) is the non-LOS (NLOS) CIR, t and t represent i,j i,j i,j LOS the signal time delays of the LOS and NLOS channels, respectively, H (0) is the DC i,j NLOS NLOS gain of the LOS channel, and A t is the gain of the NLOS channel with a time i,j i,j NLOS delay t . i,j We assume all the LEDs have a Lambertian radiation pattern. Then, by supposing that the Solution A is true, we can estimate the expected received signal power from LED (i = 1, 2) to PD (j = 1, 2) based on the coordinates in Solution A, which is derived by [7,23]: m+1 es (m + 1)A z z T gP r ti s rj A LOS t P = H (0) P = (18) i,j i,j i (m+3)/2 2 2 2 es es es 2p x x + y y + z z ti ti ti rj rj rj where P is the transmitted power of LED (i = 1, 2), m is the Lambertian emission order of LEDs, A is the effective area of PDs, T is the gain of an optical filter, g is the gain of r s an optical concentrator. Similarly, by supposing that the Solution A is true, we can also estimate the expected received signal power from LED (i = 1, 2) to PD (j = 1, 2) based on i j the coordinates in Solution A , which is derived by [7,23]: m+1 es t (m + 1)A z z T gP r s ti 0 rj i P = (19) i,j (m+3)/2 2 2 2 0 0 0 es es es 2p x x + y y + z z ti ti ti rj rj rj r A A Next, by comparing the power differences between P , P , and P , we construct a i,j i,j i,j metric F to determine the true solution, which is written as: 2 2 2 2 r A r A F = P P P P (20) å å i,j i,j i,j i,j j = 1 i = 1 If F > 0, then the Solution A is selected as the true solution for the subsequent location and orientation estimation of the receiver; otherwise, the Solution A is selected. Finally, by using the single solution remained, the 3D coordinate of the receiver is estimated by (x ˆ , y ˆ , z ˆ ) = ((x ˆ + x ˆ )/2, (y ˆ + y ˆ )/2, (z ˆ + z ˆ )/2), and its azimuthal R R R r1 r2 r1 r2 r1 r2 angle h is estimated by: 2 2 h ˆ = sign(y ˆ y ˆ )arccos (x ˆ x ˆ )/ (x ˆ x ˆ ) + (y ˆ y ˆ ) (21) r2 r1 r2 r1 r1 r2 r1 r2 where sign () represents sign function. 3. Simulation Results and Discussions We evaluate the performance of the proposed VLPO scheme in a 3 m  5 m  3 m (length  width  height) indoor space based on the XYZ-coordinate system, as shown in Figure 1. In a considered VLPO cell, there are only two LED transmitters (LED and LED ) 1 2 and a mobile receiver. LED and LED are located at (0, 1.5, 3) and (0, 3.5, 3), respectively, 1 2 both of which have a Lambertian radiation pattern. Two PDs (PD and PD ) are mounted at 1 2 both ends of the receiver to jointly estimate the location and the orientation of the receiver. Around the VLPO cell, there stands three walls, which are represented by three plane equations: (i) x = 3, 0  y  5, 0  z  3; (ii) y = 0, 0  x  3, 0  z  3; and (iii) y = 5, 0  x  3, 0  z  3. Thus, the channel between the LEDs and PDs consists of both LOS and NLOS channels, wherein the first indoor reflection is considered. We assume that the TOA ranging method is used to estimate the distances between the LEDs and PDs. Photonics 2022, 9, x FOR PEER REVIEW 9 of 16 Photonics 2022, 9, 159 8 of 15 However, due to many factors including the geometry of the room, the frequency and transmitted power of the VLP signal, and the physical characteristics of LEDs and PDs, the distance estimation accuracy of the TOA method will be influenced [21]. Therefore, random estimation errors occur with respect to the real distances d , d , d , and d between 1 2 3 4 LEDs and PDs. We assume these ranging errors are zero-mean, independent and identically Gaussian distributed, and they have the same standard deviation denoted as Dd. In our simulations, we assume that the receiver is located at certain test locations and the interval between adjacent test locations at the same height is 0.5 m. The receiver faces upwards (q = 0 ) in Figures 4–7, and can be tilted (q > 0 ) in Figures 8–11. The azimuthal angle of the receiver is randomly distributed in the range of 180 < h  180 . To calculate the Figure 4. CDFs of 3D positioning errors at (0.5, 1, 1). (θ = 0°). cumulative distribution functions (CDFs) of the positioning error and the azimuthal error, we conduct Monte Carlo simulations at least 50 times at each test location. Table 2 lists the Photonics 2022, 9, x FOR PEER REVIEW 9 of 16 key parameters of the indoor VLPO system. Other parameters of the receiver are the same To evaluate the performance of the VLPO scheme at different locations, Figure 5 as those in [23]. shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. Here, we assume that the interval between PDs is 0.2 m and the standard deviation of the ranging error is 0.025 m. We test 77 locations in each receiving plane, thus a total of 231 locations are tested for performance evaluation in the considered indoor space. We use “+” to represent the real locations of the receiver and “○” to repre- sent the coordinates estimated from the proposed VLPO scheme. On the left of Figure 5, it can be observed that most of the estimated 3D locations are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different locations. By comparison, we find that the 3D positioning accuracy is related to the height of the receiver. For example, at the height of 0.5 m, 1 m, and 1.5 m, the mean 3D positioning error is 5.9 cm, 8.3 cm, and 14.2 cm, respectively. Thus, when the receiver is at a lower location, it tends to get a more accurate 3D positioning result. The reason is that lower PDs have larger distances to LEDs, thus enhancing their tolerance against the ranging error. On the right of Figure 5, we show the bird-eye view of the 3D positioning results, i.e., 2D positioning results, for the test locations at the height of 1 m, wherein the mean positionin Figure g e 4. CDFs rror is of 3D 6.9 cm, which c positioning errors at an m (0.5,eet 1, 1).the needs (q = 0 ). of most location-based services. Figure 4. CDFs of 3D positioning errors at (0.5, 1, 1). (θ = 0°). To evaluate the performance of the VLPO scheme at different locations, Figure 5 shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. Here, we assume that the interval between PDs is 0.2 m and the standard deviation of the ranging error is 0.025 m. We test 77 locations in each receiving plane, thus a total of 231 locations are tested for performance evaluation in the considered indoor space. We use “+” to represent the real locations of the receiver and “○” to repre- sent the coordinates estimated from the proposed VLPO scheme. On the left of Figure 5, it can be observed that most of the estimated 3D locations are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different locations. By comparison, we find that the 3D positioning accuracy is related to the height of the receiver. For example, at the height of 0.5 m, 1 m, and 1.5 m, the mean 3D positioning error is 5.9 cm, 8.3 cm, and 14.2 cm, respectively. Thus, when the receiver is at a lower location, it tends to get a more accurate 3D positioning result. The reason is that lower PDs have larger distances to LEDs, thus enhancing their tolerance against the ranging error. On the right of Figure 5, we show the bird-eye view of the 3D positioning results, Figure 5. Figure An ex 5. An am example ple of 3D of 3D positioning positioning r resu esults.lt(s. ( q =θ 0 = ). 0°). i.e., 2D positioning results, for the test locations at the height of 1 m, wherein the mean positioning error is 6.9 cm, which can meet the needs of most location-based services. By considering the test locations in the receiving plane at the height of 1 m in Figure 5, Figure 6a shows the simulated CDFs of the 3D positioning errors at different locations. We assume Δd = 0.025 m and compare the CDFs obtained by adopting different intervals l between PDs. As can be seen, due to the same reason as in Figure 4, increasing l can Figure 5. An example of 3D positioning results. (θ = 0°). By considering the test locations in the receiving plane at the height of 1 m in Figure 5, Figure 6a shows the simulated CDFs of the 3D positioning errors at different locations. We assume Δd = 0.025 m and compare the CDFs obtained by adopting different intervals l between PDs. As can be seen, due to the same reason as in Figure 4, increasing l can Photonics 2022, 9, x FOR PEER REVIEW 10 of 16 effectively improve the overall positioning accuracy in the considered receiving plane. Specifically, when l is 0.1 m, 0.2 m, 0.5 m, and 1 m, around 90% of 3D positioning errors are less than 44.9 cm, 28.4 cm, 13.2 cm, and 7.9 cm, respectively. Therefore, to achieve a targeted positioning accuracy for the proposed VLPO scheme, we may appropriately in- crease the interval between PDs while considering the size of a receiver terminal. Take the shopping scenario as an example, with the proposed VLPO scheme, by installing a pair of PDs with their interval to be 0.5 m on a shopping cart, we may achieve accurate 3D posi- Photonics 2022, 9, 159 9 of 15 tioning for a customer with 92.7% of 3D positioning errors less than 20 cm using only two LEDs. Photonics 2022, 9, x FOR PEER REVIEW 11 of 16 (a) (b) Photonics 2022, 9, x FOR PEER REVIEW 11 of 16 Figure 6. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 0°). Figure 6. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. (q = 0 ). Corresponding to Figure 6a, Figure 6b plots the simulated CDFs of the azimuthal errors by considering the test locations in the receiving plane at the height of 1 m in Figure 5. We assume Δd = 0.025 m and compare the CDFs obtained by using different l. It is found that increasing l can effectively improve the overall orienteering accuracy in the consid- Figure 7. Mean positioning errors and azimuthal errors at the height of 1 m. (θ = 0°). ered receiving plane. The reason is the same as in Figure 6a. Specifically, around 77.7%, 87.4%, 91.6%, and 92.8% of azimuthal errors are less than 3° when l is 0.1 m, 0.2 m, 0.5 m, Next, we evaluate the performance of the VLPO scheme when the receiver is tilted and 1 m, respectively, thereby verifying the high orienteering accuracy of the proposed (θ > 0°). Figure 8a shows the simulated CDFs of the 3D positioning errors when the re- VLPO scheme at different locations. ceiver is located at (1.5, 2, 1) with its polar angle θ fixed at 30°. We assume Δd = 0.025 m Based on the test locations in Figure 6, Figure 7 compares the mean positioning errors and use different curves to represent the CDFs obtained by adopting different intervals l and azimuthal errors versus different standard derivations Δd of the distance estimation between PDs. It can be seen that, when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of errors caused by ranging. We set Δd = 0.0125 m, 0.025 m, 0.05 m, and 1 m, and plot the achieving 3D positioning errors less than 15 cm is 70.5%, 78.1%, 83.8%, and 85.7%, respec- mean positioning errors and azimuthal errors obtained by using different intervals l be- tively, thereby verifying the effectiveness of the proposed VLPO scheme for a tilted re- tween PDs. The three curves above are the CDFs of mean azimuthal errors, and the three ceiver. Besides, a total of around 80% of 3D positioning errors are less than 28.5 cm when curves below are the CDFs of the mean azimuthal errors. We note that both the position- l is 0.1 m, and by further increasing l to 0.5 m, 80% of 3D positioning errors are less than ing accuracy and the orienteering accuracy degrade with an increased Δd. Specifically, 8.9 cm. Therefore, for the scenario of receiver tilt, adopting a larger l can effectively en- when l is 0.5 m and Δd increases from 0.0125 m to 0.05 m, the mean positioning error hance the Figure 7. 3D Mean positionin positioning g err accurac ors and yazimuthal of the proposed errors at the VLPO scheme height of 1 m. (. q = 0 ). Figure 7. Mean positioning errors and azimuthal errors at the height of 1 m. (θ = 0°). increases from 3.5 cm to 15.2 cm, and the mean azimuthal error increases from 6.2° to 8.1°. Further, when Δd reaches 0.1 m, the mean positioning error is as large as 30.5 cm, which Next, we evaluate the performance of the VLPO scheme when the receiver is tilted is inapplicable for high-accurate location-based services, whereas the mean azimuthal er- (θ > 0°). Figure 8a shows the simulated CDFs of the 3D positioning errors when the re- ror is 11.2°. Thus, compared with the orienteering accuracy, the positioning accuracy is more sensitiv ceiver is locat e to the distance estim ed at (1.5, 2, 1) with it atio s po n error betw lar angle θeen LEDs an fixed at 30°. W d PDs. Ad e assumdit e Δ ionally d = 0.02 , 5 m when and use d Δd is ist ffe ab ril ent curves to ized at 0.025 represent the m, the mean p CDFs osition obta ing er inror ed by adopting dif is 7.4 cm and the fem rent interva ean azi- ls l muthal error is 7.0° when l is 0.5 m. Therefore, high ranging accuracy is crucial to achieve between PDs. It can be seen that, when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of the high-accurate performance of the proposed VLPO scheme. achieving 3D positioning errors less than 15 cm is 70.5%, 78.1%, 83.8%, and 85.7%, respec- tively, thereby verifying the effectiveness of the proposed VLPO scheme for a tilted re- ceiver. Besides, a total of around 80% of 3D positioning errors are less than 28.5 cm when l is 0.1 m, and by further increasing l to 0.5 m, 80% of 3D positioning errors are less than 8.9 cm. Therefore, for the scenario of receiver tilt, adopting a larger l can effectively en- hance the 3D positioning accuracy of the proposed VLPO scheme. (a) (b) Figure 8. CDFs at (1.5, 2, 1) for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 30°). Figure 8. CDFs at (1.5, 2, 1) for: (a) 3D positioning errors; and (b) azimuthal errors. (q = 30 ). By using the same the simulated conditions as in Figure 8a, Figure 8b plots the sim- ulated CDFs of the azimuthal errors. We see that when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of achieving azimuthal errors less than 5° is 89.5%, 93.1%, 95.2%, and 95.3%, respectively, thus proving the high orienteering accuracy of the proposed VLPO scheme for a tilted receiver. Furthermore, increasing the interval between PDs can enhance the orienteering accuracy effectively. For example, by increasing l from 0.1 m to 0.5 m, the probability of achieving azimuthal errors less than 3° can be improved from 79.4% to 93.4%, when θ = 30°. (a) (b) Figure 8. CDFs at (1.5, 2, 1) for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 30°). By using the same the simulated conditions as in Figure 8a, Figure 8b plots the sim- ulated CDFs of the azimuthal errors. We see that when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of achieving azimuthal errors less than 5° is 89.5%, 93.1%, 95.2%, and 95.3%, respectively, thus proving the high orienteering accuracy of the proposed VLPO scheme for a tilted receiver. Furthermore, increasing the interval between PDs can enhance the orienteering accuracy effectively. For example, by increasing l from 0.1 m to 0.5 m, the probability of achieving azimuthal errors less than 3° can be improved from 79.4% to 93.4%, when θ = 30°. Photonics 2022, 9, x FOR PEER REVIEW 12 of 16 To evaluate the performance of the VLPO scheme at different locations for a tilted receiver, Figure 9 shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. We fix the receiver polar angle θ at 20° to ensure that the LOS channels between LEDs and PDs will not be blocked due to the receiver tilt. We assume the interval between PDs is 0.2 m and the standard derivation Δd of the rang- ing error is 0.025 m. The test locations are the same as in Figure 5. On the left of Figure 9, we observe that most of the estimated 3D coordinates are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different loca- tions for a tilted receiver. Specifically, the mean 3D positioning error is 9.2 cm, 14.2 cm, and 20.9 cm at the height of 0.5 m, 1 m, and 1.5 m, respectively, from which the positioning accuracy is related to the height of the receiver. The reason is the same as in Figure 5. On the right of Figure 9, we show the corresponding 2D positioning results with a tilted re- ceiver at the height of 1 m, wherein the mean positioning error is 10.7 cm. Besides, it can be observed that small positioning errors have a higher possibility to occur at the test lo- cations below LEDs, e.g., where xR = 0. This is because, when the receiver is tilted and located at the boundary of the considered indoor space with xR = 0, multiple solutions of the estimated coordinates of PD1 and PD2 may stay close due to geometrical symmetry, thus their midpoints, i.e., the estimated locations of the receiver, have a higher chance to Photonics 2022, 9, 159 10 of 15 overlap, which potentially enhances the positioning accuracy at the test locations with xR = 0. Photonics 2022, 9, x FOR PEER REVIEW 13 of 16 Photonics 2022, 9, x FOR PEER REVIEW 13 of 16 Figure 9. An example of 3D positioning results. (θ = 20°). Figure 9. An example of 3D positioning results. (q = 20 ). (a) (b) Figure 10. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 0°, By considering the test locations in the receiving plane at the height of 1 m in Figure 10°, and 20°). 9, Figure 10a shows the simulated CDFs of the 3D positioning errors at different locations. We assume the standard derivation Δd of the ranging error is 0.025 m and the interval l Corresponding to Figure 10a, Figure 10b plots the simulated CDFs of the azimuthal between PDs is 0.5 m. We use different curves to represent the CDFs obtained by consid- errors by considering the test locations in the receiving plane at the height of 1 m in Figure ering different polar angles with θ to be 0°, 10°, and 20°. For a performance evaluation, 9. We assume the standard derivation Δd of the ranging error is 0.025 m and the interval we also plot the simulated CDF curves of the 3D positioning errors when excluding the l between PDs is 0.2 m, and compare the CDFs obtained by considering different polar test locations below LEDs with xR = 0. We see that increasing the polar angle of a tilted angles θ of the tilted receiver. It is found that increasing θ slightly degrades the overall receiver slightly degrades the overall positioning accuracy in the considered receiving orienteering accuracy in the considered receiving plane. Specifically, around 92.6%, 92.6%, plane. Specifically, when θ is 0°, 10°, and 20°, the probability of achieving 3D positioning and 90.9% of azimuthal errors are less than 5° when θ is 0°, 10°, and 20°, respectively, errors thereby verifying the h less than 20 cm isi 92 gh .7%, 90 orienteerin .3%, g an accurac d 87.9%, res y of thp e V ecti LP vel O s y c . In contra heme at dis ft f, when excl erent locatio u n d si ng the test l for a tilted ocat r ions wi eceiver. Mo th xR r = eover, wh 0, the prob en ab exclud ility o ing f achiev the test ing 3D loca p tioonsit s w iointh ing er xR = rors 0, around less than 99.5%, 99.2%, and 97.7% of azimuthal errors are less than 5° when θ is 0°, 10°, and 20°, 20 cm decreases to 91.5%, 88.7%, and 86.0% when θ is 0°, 10°, and 20°, respectively. By respectively. Obviously, large azimuthal errors tend to occur at the test locations with xR comparison, smaller positioning errors tend to occur at the test locations below LEDs with = 0. The reason lies in that, when the receiver tilts, although multiple solutions of the esti- xR = 0, which confirms the conclusion in Figure 9. mated coordinates of PD1 and PD2 may stay close due to geometrical symmetry as dis- (a) (b) cussed in Figure 9, the estimated azimuthal angles of the receiver are completely different Figure 10. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 0°, Figure 10. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. and their sum is 180°, thus degrading the orienteering accuracy at the test locations with 10°, and 20°). (q = 0 , 10 , and 20 ). xR = 0. Corresponding to Figure 10a, Figure 10b plots the simulated CDFs of the azimuthal errors by considering the test locations in the receiving plane at the height of 1 m in Figure 9. We assume the standard derivation Δd of the ranging error is 0.025 m and the interval l between PDs is 0.2 m, and compare the CDFs obtained by considering different polar angles θ of the tilted receiver. It is found that increasing θ slightly degrades the overall orienteering accuracy in the considered receiving plane. Specifically, around 92.6%, 92.6%, and 90.9% of azimuthal errors are less than 5° when θ is 0°, 10°, and 20°, respectively, thereby verifying the high orienteering accuracy of the VLPO scheme at different locations for a tilted receiver. Moreover, when excluding the test locations with xR = 0, around 99.5%, 99.2%, and 97.7% of azimuthal errors are less than 5° when θ is 0°, 10°, and 20°, respectively. Obviously, large azimuthal errors tend to occur at the test locations with xR = 0. The reason lies in that, when the receiver tilts, although multiple solutions of the esti- mated coordinates of PD1 and PD2 may stay close due to geometrical symmetry as dis- cussed in Figure 9, the estimated azimuthal angles of the receiver are completely different and their sum is 180°, thus degrading the orienteering accuracy at the test locations with xR = 0. Figure 11. Mean positioning errors and azimuthal errors at the height of 1 m. (θ = 0°, 10°, and 20°). Figure 11. Mean positioning errors and azimuthal errors at the height of 1 m. (q = 0 , 10 , and 20 ). Finally, based on the test locations in Figure 10, Figure 11 compares the mean posi- tioning errors and azimuthal errors versus different polar angles θ of the tilted receiver. Figure 11. Mean positioning errors and azimuthal errors at the height of 1 m. (θ = 0°, 10°, and 20°). Finally, based on the test locations in Figure 10, Figure 11 compares the mean posi- tioning errors and azimuthal errors versus different polar angles θ of the tilted receiver. Photonics 2022, 9, 159 11 of 15 Table 2. Key parameters of the VLPO system. Name of Parameters Values Indoor space (length  width  height) 3 m  5 m  3 m Height of receiver 0.5 m/1 m/1.5 m Launch power of each LED 5 Watt Modulation index 0.1 Lambertian emission order of LEDs 1 LED semi-angle at half power 60 Field-of-view (FOV) at the receiver 170 4 2 Effective area of photo-detector 10 m Photo-detector responsivity 0.35 A/W Reflection coefficient of wall 0.83 Gain of the optical filter 1 Refractive index of the optical concentrator 1.5 Baud rate of the VLP signal 10 Msymbol/s We first evaluate the performance of the VLPO scheme when the receiver faces up- wards (q = 0 ). Figure 4 shows the simulated CDFs of the 3D positioning errors when the receiver is fixed at a specific location (0.5, 1, 1). Different curves represent the CDFs obtained by adopting different intervals l between PDs. We assume the standard deviation Dd of the ranging error is 0.025 m. We note that, when l is 0.1 m, 0.2 m, 0.5 m, and 1 m, the probability of achieving 3D positioning errors less than 10 cm is 83.2%, 90.2%, 93.7%, and 94.1%, respectively. This is because when Dd is fixed, adopting a larger l helps enhance the tolerance against the ranging error, thereby improving the positioning accuracy. Specifically, around 90% of 3D positioning errors are less than 9.8 cm when l is 0.2 m, and by further increasing l to 0.5 m, 90% of 3D positioning errors are less than 4.5 cm, thus verifying the high positioning accuracy of the proposed VLPO scheme when q = 0 . To evaluate the performance of the VLPO scheme at different locations, Figure 5 shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. Here, we assume that the interval between PDs is 0.2 m and the standard deviation of the ranging error is 0.025 m. We test 77 locations in each receiving plane, thus a total of 231 locations are tested for performance evaluation in the considered indoor space. We use “+” to represent the real locations of the receiver and “#” to represent the coordinates estimated from the proposed VLPO scheme. On the left of Figure 5, it can be observed that most of the estimated 3D locations are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different locations. By comparison, we find that the 3D positioning accuracy is related to the height of the receiver. For example, at the height of 0.5 m, 1 m, and 1.5 m, the mean 3D positioning error is 5.9 cm, 8.3 cm, and 14.2 cm, respectively. Thus, when the receiver is at a lower location, it tends to get a more accurate 3D positioning result. The reason is that lower PDs have larger distances to LEDs, thus enhancing their tolerance against the ranging error. On the right of Figure 5, we show the bird-eye view of the 3D positioning results, i.e., 2D positioning results, for the test locations at the height of 1 m, wherein the mean positioning error is 6.9 cm, which can meet the needs of most location-based services. By considering the test locations in the receiving plane at the height of 1 m in Figure 5, Figure 6a shows the simulated CDFs of the 3D positioning errors at different locations. We assume Dd = 0.025 m and compare the CDFs obtained by adopting different intervals l between PDs. As can be seen, due to the same reason as in Figure 4, increasing l can effectively improve the overall positioning accuracy in the considered receiving plane. Specifically, when l is 0.1 m, 0.2 m, 0.5 m, and 1 m, around 90% of 3D positioning errors are less than 44.9 cm, 28.4 cm, 13.2 cm, and 7.9 cm, respectively. Therefore, to achieve a targeted positioning accuracy for the proposed VLPO scheme, we may appropriately increase the interval between PDs while considering the size of a receiver terminal. Take the shopping scenario as an example, with the proposed VLPO scheme, by installing a pair of PDs with Photonics 2022, 9, 159 12 of 15 their interval to be 0.5 m on a shopping cart, we may achieve accurate 3D positioning for a customer with 92.7% of 3D positioning errors less than 20 cm using only two LEDs. Corresponding to Figure 6a, Figure 6b plots the simulated CDFs of the azimuthal errors by considering the test locations in the receiving plane at the height of 1 m in Figure 5. We assume Dd = 0.025 m and compare the CDFs obtained by using different l. It is found that increasing l can effectively improve the overall orienteering accuracy in the considered receiving plane. The reason is the same as in Figure 6a. Specifically, around 77.7%, 87.4%, 91.6%, and 92.8% of azimuthal errors are less than 3 when l is 0.1 m, 0.2 m, 0.5 m, and 1 m, respectively, thereby verifying the high orienteering accuracy of the proposed VLPO scheme at different locations. Based on the test locations in Figure 6, Figure 7 compares the mean positioning errors and azimuthal errors versus different standard derivations Dd of the distance estimation errors caused by ranging. We set Dd = 0.0125 m, 0.025 m, 0.05 m, and 1 m, and plot the mean positioning errors and azimuthal errors obtained by using different intervals l between PDs. The three curves above are the CDFs of mean azimuthal errors, and the three curves below are the CDFs of the mean azimuthal errors. We note that both the positioning accuracy and the orienteering accuracy degrade with an increased Dd. Specifically, when l is 0.5 m and Dd increases from 0.0125 m to 0.05 m, the mean positioning error increases from 3.5 cm to 15.2 cm, and the mean azimuthal error increases from 6.2 to 8.1 . Further, when Dd reaches 0.1 m, the mean positioning error is as large as 30.5 cm, which is inapplicable for high-accurate location-based services, whereas the mean azimuthal error is 11.2 . Thus, compared with the orienteering accuracy, the positioning accuracy is more sensitive to the distance estimation error between LEDs and PDs. Additionally, when Dd is stabilized at 0.025 m, the mean positioning error is 7.4 cm and the mean azimuthal error is 7.0 when l is 0.5 m. Therefore, high ranging accuracy is crucial to achieve the high-accurate performance of the proposed VLPO scheme. Next, we evaluate the performance of the VLPO scheme when the receiver is tilted (q > 0 ). Figure 8a shows the simulated CDFs of the 3D positioning errors when the receiver is located at (1.5, 2, 1) with its polar angle q fixed at 30 . We assume Dd = 0.025 m and use different curves to represent the CDFs obtained by adopting different intervals l between PDs. It can be seen that, when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of achieving 3D positioning errors less than 15 cm is 70.5%, 78.1%, 83.8%, and 85.7%, respectively, thereby verifying the effectiveness of the proposed VLPO scheme for a tilted receiver. Besides, a total of around 80% of 3D positioning errors are less than 28.5 cm when l is 0.1 m, and by further increasing l to 0.5 m, 80% of 3D positioning errors are less than 8.9 cm. Therefore, for the scenario of receiver tilt, adopting a larger l can effectively enhance the 3D positioning accuracy of the proposed VLPO scheme. By using the same the simulated conditions as in Figure 8a, Figure 8b plots the simulated CDFs of the azimuthal errors. We see that when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of achieving azimuthal errors less than 5 is 89.5%, 93.1%, 95.2%, and 95.3%, respectively, thus proving the high orienteering accuracy of the proposed VLPO scheme for a tilted receiver. Furthermore, increasing the interval between PDs can enhance the orienteering accuracy effectively. For example, by increasing l from 0.1 m to 0.5 m, the probability of achieving azimuthal errors less than 3 can be improved from 79.4% to 93.4%, when q = 30 . To evaluate the performance of the VLPO scheme at different locations for a tilted receiver, Figure 9 shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. We fix the receiver polar angle q at 20 to ensure that the LOS channels between LEDs and PDs will not be blocked due to the receiver tilt. We assume the interval between PDs is 0.2 m and the standard derivation Dd of the ranging error is 0.025 m. The test locations are the same as in Figure 5. On the left of Figure 9, we observe that most of the estimated 3D coordinates are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different locations for a tilted receiver. Specifically, the mean 3D positioning error is 9.2 cm, 14.2 cm, Photonics 2022, 9, 159 13 of 15 and 20.9 cm at the height of 0.5 m, 1 m, and 1.5 m, respectively, from which the positioning accuracy is related to the height of the receiver. The reason is the same as in Figure 5. On the right of Figure 9, we show the corresponding 2D positioning results with a tilted receiver at the height of 1 m, wherein the mean positioning error is 10.7 cm. Besides, it can be observed that small positioning errors have a higher possibility to occur at the test locations below LEDs, e.g., where x = 0. This is because, when the receiver is tilted and located at the boundary of the considered indoor space with x = 0, multiple solutions of the estimated coordinates of PD and PD may stay close due to geometrical symmetry, 1 2 thus their midpoints, i.e., the estimated locations of the receiver, have a higher chance to overlap, which potentially enhances the positioning accuracy at the test locations with x = 0. By considering the test locations in the receiving plane at the height of 1 m in Figure 9, Figure 10a shows the simulated CDFs of the 3D positioning errors at different locations. We assume the standard derivation Dd of the ranging error is 0.025 m and the interval l between PDs is 0.5 m. We use different curves to represent the CDFs obtained by considering different polar angles with q to be 0 , 10 , and 20 . For a performance evaluation, we also plot the simulated CDF curves of the 3D positioning errors when excluding the test locations below LEDs with x = 0. We see that increasing the polar angle of a tilted receiver slightly degrades the overall positioning accuracy in the considered receiving plane. Specifically, when q is 0 , 10 , and 20 , the probability of achieving 3D positioning errors less than 20 cm is 92.7%, 90.3%, and 87.9%, respectively. In contrast, when excluding the test locations with x = 0, the probability of achieving 3D positioning errors less than 20 cm decreases to 91.5%, 88.7%, and 86.0% when q is 0 , 10 , and 20 , respectively. By comparison, smaller positioning errors tend to occur at the test locations below LEDs with x = 0, which confirms the conclusion in Figure 9. Corresponding to Figure 10a, Figure 10b plots the simulated CDFs of the azimuthal errors by considering the test locations in the receiving plane at the height of 1 m in Figure 9. We assume the standard derivation Dd of the ranging error is 0.025 m and the interval l between PDs is 0.2 m, and compare the CDFs obtained by considering different polar angles q of the tilted receiver. It is found that increasing q slightly degrades the overall orienteering accuracy in the considered receiving plane. Specifically, around 92.6%, 92.6%, and 90.9% of azimuthal errors are less than 5 when q is 0 , 10 , and 20 , respectively, thereby verifying the high orienteering accuracy of the VLPO scheme at different locations for a tilted receiver. Moreover, when excluding the test locations with x = 0, around 99.5%, 99.2%, and 97.7% of azimuthal errors are less than 5 when q is 0 , 10 , and 20 , respectively. Obviously, large azimuthal errors tend to occur at the test locations with x = 0. The reason lies in that, when the receiver tilts, although multiple solutions of the estimated coordinates of PD and PD may stay close due to geometrical symmetry as discussed in Figure 9, the 1 2 estimated azimuthal angles of the receiver are completely different and their sum is 180 , thus degrading the orienteering accuracy at the test locations with x = 0. Finally, based on the test locations in Figure 10, Figure 11 compares the mean posi- tioning errors and azimuthal errors versus different polar angles q of the tilted receiver. We assume Dd = 0.025 m. We set q = 0 , 10 , and 20 , and plot the mean positioning errors and azimuthal errors obtained by using different intervals l between PDs. As can be seen, the change in the polar angle of the tilted receiver has only a small influence on both the positioning accuracy and the orienteering accuracy. Specifically, when l is 0.2 m and q changes from 0 to 20 , the mean positioning error slightly increases from 12.5 cm to 13.3 cm, and the mean azimuthal error slightly increases from 7.4 to 7.7 . Therefore, the proposed VLPO scheme is robust to the variation of the receiver polar angle, and can achieve stable 3D positioning and orienteering accuracy with a tilted receiver. 4. Conclusions To enable indoor VLP when the LED condition is constrained, we proposed a novel 3D VLP and orienteering (VLPO) scheme. By using only two LEDs and a pair of PDs, the Photonics 2022, 9, 159 14 of 15 scheme can simultaneously estimate the 3D coordinate and the orientation of the receiver. To support VLPO for a tilted receiver, we further proposed a location selection strategy to overcome the location uncertainty caused by receiver tilt. Simulation studies showed that, when the receiver faces upwards, the proposed VLPO scheme can achieve a mean 3D positioning error of 7.4 cm and a mean azimuthal error of 7.0 . Moreover, when the receiver tilts with a polar angle of 10 , the proposed scheme can still achieve accurate VLP with 90.3% of 3D positioning errors less than 20 cm, and accurate receiver orienteering with 92.6% of azimuthal errors less than 5 . The evaluation also indicated that appropriately increasing the interval between PDs can help enhance the tolerance against the ranging error, thus improving the VLPO accuracy effectively. In the future work, we will study and improve the VLPO scheme under scenarios when the LOS channels between LEDs and PDs are partially blocked. Author Contributions: Conceptualization, X.Y. (Xiaodi You); methodology, X.Y. (Xiaodi You) and Z.J.; validation, X.Y. (Xiaodi You) and X.Y. (Xiaobai Yang); investigation, X.Y. (Xiaobai Yang) and S.Z.; resources, X.Y. (Xiaodi You); data curation, X.Y. (Xiaobai Yang); writing—original draft prepara- tion, X.Y. (Xiaodi You); writing—review and editing, X.Y. (Xiaodi You); visualization, Z.J. and S.Z.; supervision, X.Y. (Xiaodi You); project administration, X.Y. (Xiaodi You); funding acquisition, X.Y. (Xiaodi You). All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China (62001319); Suzhou Science and Technology Bureau-Technical Innovation Project in Key Industries (SYG202112); and Open Fund of IPOC (BUPT) (IPOC2020A009). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Liu, H.; Darabi, H.; Banerjee, P.; Liu, J. Survey of wireless indoor positioning techniques and systems. IEEE Trans. Syst. Man Cybern. Part C-Appl. 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You, X.; Chen, J.; Yu, C. Performance of location-based equalization for OFDM indoor visible light communications. IEEE Trans. Cogn. Commun. Netw. 2019, 5, 1229–1243. [CrossRef] 24. Jiang, Z.; You, X.; Shen, G.; Mukherjee, B. Location-aware spectrum sensing for cognitive visible light communications over multipath channels. Opt. Express 2021, 29, 43700–43719. [CrossRef] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Photonics Multidisciplinary Digital Publishing Institute

A Two-LED Based Indoor Three-Dimensional Visible Light Positioning and Orienteering Scheme for a Tilted Receiver

Photonics , Volume 9 (3) – Mar 5, 2022

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hv photonics Article A Two-LED Based Indoor Three-Dimensional Visible Light Positioning and Orienteering Scheme for a Tilted Receiver 1 , 1 2 1 Xiaodi You * , Xiaobai Yang , Zile Jiang and Shuang Zhao School of Electronic and Information Engineering, Soochow University, Suzhou 215006, China; 20205228029@stu.suda.edu.cn (X.Y.); szhaoszhao@stu.suda.edu.cn (S.Z.) Kuang Yaming Honors School, Nanjing University, Nanjing 210023, China; 181240026@smail.nju.edu.cn * Correspondence: xdyou@suda.edu.cn Abstract: Conventional visible light positioning (VLP) systems usually require at least three light- emitting diodes (LEDs) to enable trilateration or triangulation, which is infeasible when the LED condition is constrained. In this paper, we propose a novel indoor three-dimensional (3D) VLP and orienteering (VLPO) scheme. By using only two LEDs and two photo-detectors (PDs), our scheme can achieve simultaneous 3D localization and receiver orientation estimation efficiently. Further, to eliminate the location uncertainty caused by receiver tilt, we propose a location selection strategy which can effectively determine the true location of the receiver. Through extensive simulations, it is found that when the receiver faces upwards, the proposed scheme can achieve a mean 3D positioning error of 7.4 cm and a mean azimuthal error of 7.0 . Moreover, when the receiver tilts with a polar angle of 10 , accurate VLPO can still be achieved with 90.3% of 3D positioning errors less than 20 cm and 92.6% of azimuthal errors less than 5 . These results indicate that our scheme is a promising solution to achieve accurate VLPO when there is only two LEDs. Results also verify the effectiveness of the VLPO scheme when locating a tilted receiver. Keywords: visible light positioning (VLP); three-dimensional (3D); visible light positioning and orienteering (VLPO); light-emitting diode (LED) Citation: You, X.; Yang, X.; Jiang, Z.; Zhao, S. A Two-LED Based Indoor Three-Dimensional Visible Light Positioning and Orienteering Scheme 1. Introduction for a Tilted Receiver. Photonics 2022, 9, The emerging location-based services (LBSs) have raised increasing interest in localiza- 159. https://doi.org/10.3390/ tion technologies [1]. Satellite-based global positioning systems are widely used in outdoor photonics9030159 environments. However, satellite signals suffer from fading when passing though solid Received: 30 January 2022 walls, thus their positioning accuracy will be severely influenced in indoor environments Accepted: 3 March 2022 such as underground parking, tunnels, and office buildings surrounded by skyscrapers. Published: 5 March 2022 With the development of visible light communication (VLC) technologies, the visible light positioning (VLP) system using light-emitting diodes (LEDs) has been considered as a Publisher’s Note: MDPI stays neutral potential candidate for the next-generation indoor positioning system, due to its advan- with regard to jurisdictional claims in tages of high accuracy, high efficiency, and low cost [2,3]. By modulating signals to existing published maps and institutional affil- lighting facilities, VLP systems can offer various indoor LBSs such as high-accurate indoor iations. navigation, asset tracking, and autonomous robot control. The simplest indoor VLP scheme is based on the principle of proximity and its position- ing accuracy is usually in meters [4]. To improve the positioning accuracy, different VLP Copyright: © 2022 by the authors. schemes based on trilateration or triangulation were reported, which can achieve centimeter- Licensee MDPI, Basel, Switzerland. level VLP [5–7]. However, to offer positioning information for trilateration or triangulation, This article is an open access article conventional VLP systems usually require at least three LEDs as transmitters to convey distributed under the terms and VLP signals according to the time domain multiplexing (TDM) or frequency domain mul- conditions of the Creative Commons tiplexing (FDM) protocols to avoid mutual interference at the receiver and make it easy Attribution (CC BY) license (https:// for the receiver to distinguish between VLP signals [8,9]. Thus, at least three time slots or creativecommons.org/licenses/by/ frequency bands are allocated for VLP. Since the bandwidth of off-the-shelf LEDs is narrow 4.0/). Photonics 2022, 9, 159. https://doi.org/10.3390/photonics9030159 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 159 2 of 15 and usually a few of tens of MHz [10], VLP will inevitably limit the available bandwidth of VLC in an integrated VLC and VLP system [11]. To ensure the ample capacity of VLC, new approaches are required to reduce the time or frequency resources allocated for VLP. This study will focus on the VLP mechanism to propose a new VLP model and its corresponding strategies to support VLP with fewer bandwidth resources. Another issue is that the LED condition of a VLP system could be constrained in some common scenarios, e.g.,: (i) when the LED number is insufficient, i.e., less than three; and (ii) when LED lamps are installed along a straight line, e.g., in a corridor or a tunnel. In these scenarios, the receiver is likely to detect the VLP signals from only one or two LEDs. Thus, conventional VLP schemes based on trilateration or triangulation cannot be directly applied. Different approaches have been proposed to perform indoor VLP with less than three LEDs, which can be categorized as two-dimensional (2D) [12–14] or three-dimensional (3D) [15–17] schemes. However, all these schemes require additional equipment such as a camera [12,13,15], a mirror [14], line lasers [16], or at least three PDs [17], which could increase the system complexity and needs to be simplified. Motivated by this, this work will particularly focus on the 3D VLP scheme, which requires fewer PDs when the LED condition is constrained. Recently, receiver orientation estimation has become an additional function of VLP. Joint VLP and orientation estimation schemes were proposed in [18,19], and they both support only 2D VLP with quite a number of LEDs (e.g., twenty LEDs in [18], and forty- eight LEDs in [19]). However, there have been no related works specifically dedicated to VLP schemes that can support simultaneous 3D localization and receiver orienteering using less than three LEDs and a PD-based receiver. Therefore, this work will also try to fill in the gap. In this paper, we propose a novel indoor 3D VLP and orienteering (VLPO) scheme using two LEDs and a pair of PDs. To the best of our knowledge, this is the first time that a two-LED-two-PD-based indoor 3D VLPO scheme is proposed, by which only two time slots or frequency bands are required to convey VLP signals. Thus, the bandwidth resources allocated for VLP can be effectively reduced as compared with conventional schemes based on trilateration or triangulation. In addition, the proposed scheme can achieve a good trade-off between the system complexity and the localization capability. As shown in Table 1, by using two LEDs and two PDs without any other types of equipment, our scheme is the only one that supports simultaneous 3D VLP and orientation estimation for a tilted receiver, thus it can be considered as a promising solution, especially when the LED condition is constrained. Table 1. Comparison between different VLP schemes. 2D or 3D Support Support Reference Transmitter Receiver VLP Orienteering Receiver Tilt [12] 2 LEDs 1 camera 2D No No [13] 1 LED 1 camera 2D No No [14] 2 LEDs 1 PD + 1 mirror 2D No No [15] 1 LED 1 camera 3D No Yes [16] 2 line lasers 2 PDs 3D Yes No [17] 1 LED array 3 PDs 3D No Yes [18] 20 LEDs 1 PD 2D Yes No [19] 48 LEDs 7 PDs 2D Yes Yes This work 2 LEDs 2 PDs 3D Yes Yes The concept of the VLPO scheme was shown in our conference paper [20], wherein we have preliminarily verified its feasibility assuming the receiver to face upwards. However, when the receiver tilts, the model in [20] cannot be directly applied and needs to be modified. Therefore, in this work, we further develop the VLPO scheme by considering the scenario of a tilted receiver. To eliminate the potential location uncertainty caused by the receiver tilt, we propose a new location selection strategy for the VLPO system to determine the Photonics 2022, 9, x FOR PEER REVIEW 3 of 16 Photonics 2022, 9, 159 3 of 15 the scenario of a tilted receiver. To eliminate the potential location uncertainty caused by the receiver tilt, we propose a new location selection strategy for the VLPO system to de- termine the true location of the receiver. We conduct extensive simulations to evaluate the tr po ue sit location ioning aof ccurac the ry eceiver and orient . We conduct eering accur extensive acy in a simulations 3 m × 5 m to × evaluate 3 m indoor s the positioning pace. The influence from the interval between PDs, the ranging error between LEDs and PDs, and accuracy and orienteering accuracy in a 3 m 5 m 3 m indoor space. The influence from the polar angle of the tiled receiver is also evaluated. Results show that the proposed the interval between PDs, the ranging error between LEDs and PDs, and the polar angle of the scheme tiledir seceiver efficient to a is also chi evaluated. eve accura Results te 3D VL show PO w that hen the the pr L oposed ED conscheme dition isis co ef ns ficient traineto d. achieve Specifically, accurate when the rec 3D VLPO eiver when face the s upward LED condition s, the proposed sch is constrained. eme can Specifically achieve a me , when the an receiver faces upwards, the proposed scheme can achieve a mean 3D positioning error of 3D positioning error of 7.4 cm and a mean azimuthal error of 7.0°. Moreover, when the 7.4 cm and a mean azimuthal error of 7.0 . Moreover, when the receiver tilts with a polar receiver tilts with a polar angle of 10°, the proposed scheme can still achieve accurate VLP angle of 10 , the proposed scheme can still achieve accurate VLP and orienteering, with and orienteering, with 90.3% of 3D positioning errors less than 20 cm and 92.6% of azi- 90.3% of 3D positioning errors less than 20 cm and 92.6% of azimuthal errors less than 5 . muthal errors less than 5°. Results also indicate that increasing the interval between PDs Results also indicate that increasing the interval between PDs and reducing the ranging and reducing the ranging error between LEDs and PDs help enhance the accuracy of error between LEDs and PDs help enhance the accuracy of VLPO, and the receiver tilt only VLPO, and the receiver tilt only slightly degrades the performance of the VLPO scheme. slightly degrades the performance of the VLPO scheme. The rest of this paper is organized as follows. Section 2 presents the principle of the The rest of this paper is organized as follows. Section 2 presents the principle of the proposed 3D VLPO system in detail. Simulations and performance analyses are con- proposed 3D VLPO system in detail. Simulations and performance analyses are conducted ducted in Section 3. Finally, we conclude the paper in Section 4. in Section 3. Finally, we conclude the paper in Section 4. 2. Proposed VLPO Scheme 2. Proposed VLPO Scheme In this study, we assume a multi-cell VLPO system, in which there are multiple LEDs In this study, we assume a multi-cell VLPO system, in which there are multiple installed along a straight line in the ceiling, and mobile receivers are distributed at differ- LEDs installed along a straight line in the ceiling, and mobile receivers are distributed at ent indoor locations. The LEDs are divided into different groups in pairs to spatially form different indoor locations. The LEDs are divided into different groups in pairs to spatially multiple VLPO cells. Based on this multi-cell configuration, we focus on VLPO within a form multiple VLPO cells. Based on this multi-cell configuration, we focus on VLPO single cell where a mobile receiver can only detect the VLP signals emitted from two LEDs within a single cell where a mobile receiver can only detect the VLP signals emitted from (LED1 and LED2). Without loss of generality, we abstract the receiver into a line segment two LEDs (LED and LED ). Without loss of generality, we abstract the receiver into a 1 2 and assume that a pair of photo-detectors (PD1 and PD2) are mounted on both ends of the line segment and assume that a pair of photo-detectors (PD and PD ) are mounted on 1 2 receiver. Figure 1 shows the architecture of the proposed VLPO system. The 3D coordi- both ends of the receiver. Figure 1 shows the architecture of the proposed VLPO system. nates of LED1 and LED2 are (xt1, yt1, zt1) and (xt2, yt2, zt2), respectively, which are a priori The 3D coordinates of LED and LED are (x , y , z ) and (x , y , z ), respectively, 1 2 t 1 t 1 t 1 t 2 t 2 t 2 knowledge at the receiver. The 3D coordinates of PD1 and PD2 are (xr1, yr1, zr1) and (xr2, yr2, which are a priori knowledge at the receiver. The 3D coordinates of PD and PD are 1 2 zr2), respectively, which are unknown at the receiver and need to be estimated. We use the (x , y , z ) and (x , y , z ), respectively, which are unknown at the receiver and need to r 1 r 1 r 1 r 2 r 2 r 2 midpoint of PD1 and PD2 to represent the location of the receiver, i.e., (xR, yR, zR) = be estimated. We use the midpoint of PD and PD to represent the location of the receiver, 1 2 ((x +x )/2, (y +y )/2, (z +z )/2). We also define the direction pointing from PD1 towards r1 r2 r1 r2 i.e., (x , y , z r1 ) = r(( 2 x + x )/2, (y + y )/2, (z + z )/2). We also define the direction R R R r1 r2 r1 r2 r1 r2 PD2 as the orientation of the receiver. Thus, the azimuthal angle of the receiver is defined pointing from PD towards PD as the orientation of the receiver. Thus, the azimuthal angle 1 2 by the angular rotation η (−180° < η ≤ 180°) from the positive X-axis to the projection of the of the receiver is defined by the angular rotation h (180 < h  180 ) from the positive receiver (from PD1 to PD2) on the XY-plane. Therefore, if we successfully estimate the 3D X-axis to the projection of the receiver (from PD to PD ) on the XY-plane. Therefore, if 1 2 coordinates of PD1 and PD2, then the location and the orientation of the receiver can be we successfully estimate the 3D coordinates of PD and PD , then the location and the 1 2 calculated. orientation of the receiver can be calculated. Figure 1. Proposed VLPO system. Figure 1. Proposed VLPO system. To estimate the coordinates of PD and PD , we assume that the LEDs and PDs 1 2 are perfectly synchronized to a common clock, and the VLP signals launched by the LEDs are used as the basis of ranging. Based on the ranging methods of time-of-arrival Photonics 2022, 9, 159 4 of 15 (TOA) or phase-difference-of-arrival (PDOA) [21,22], the distances between the two LEDs and two PDs can be measured, and their estimation accuracy is typically in the order of centimeters [21]. We define the real distance between LED and PD , LED and PD , LED 1 1 2 1 1 and PD , LED and PD to be d , d , d , and d , respectively. Therefore, we obtain the 2 2 2 1 2 3 4 following geometrical relationships: 2 2 2 d = (x ˆ x ) + (y ˆ y ) + (z ˆ z ) (1) r1 t1 r1 t1 r1 t1 2 2 2 d = (x ˆ x ) + (y ˆ y ) + (z ˆ z ) (2) r1 t2 r1 t2 r1 t2 2 2 2 d = (x ˆ x ) + (y ˆ y ) + (z ˆ z ) (3) 3 r2 t1 r2 t1 r2 t1 2 2 2 d = (x ˆ x ) + (y ˆ y ) + (z ˆ z ) (4) r2 t2 r2 t2 r2 t2 where d (i = 1, 2, 3, 4) is the estimated values of d (i = 1, 2, 3, 4) obtained by ranging, i i x ˆ , y ˆ , z ˆ is the coordinate of PD to be estimated, and x ˆ , y ˆ , z ˆ is the coordinate of ( ) ( ) r1 r1 r1 1 r2 r2 r2 PD to be estimated. To offer richer information for performing VLPO, the interval between PD and PD is known at the receiver and fixed at l. Additionally, we assume that the 1 2 polar angle q (0  q  90 ) of the tilted receiver can be acquired by extra sensors at the receiver [22]. When q = 0 , the receiver faces upwards. Based on these presumptions, we can derive additional geometrical relationships: 2 2 2 2 l = (x ˆ x ˆ ) + (y ˆ y ˆ ) + (z ˆ z ˆ ) (5) r2 r1 r2 r1 r2 r1 z ˆ = z ˆ + l sin q (6) r2 r1 According to the above nonlinear system of equations from Equations (1)–(6), we ex- pect to solve the coordinates of PD and PD . However, it would be difficult to directly solve 1 2 the coordinates of PDs from Equations (1)–(6) in the current XYZ-coordinate system. Thus, we divide the solving process into three steps, which includes establishing intersection circles, conducting coordinate transformations, and excluding redundant solutions. Figure 2a shows the schematic diagram of establishing intersection circles. In this step, we first subtract between Equations (1) and (2), i.e., the two spherical surfaces centered at LED and LED , to obtain a plane equation, which is described by a plane P . PD is located 1 2 1 1 in the plane P , specifically, on the intersection circle K between the spherical surfaces 1 1 Equations (1) and (2). Here, K is centered at (a , b , c ) with a radius of R . Similarly, after 1 1 1 1 1 subtracting between Equations (3) and (4), PD is located in a plane P , specifically, on the 2 2 intersection circle K centered at (a , b , c ) with a radius of R . For the intersection circles 2 2 2 2 2 K and K , the coordinates of K (i = 1, 2) are given by: 1 2 i w w w i i i K (a , b , c ) = x + (x x ) , y + (y y ) , z + (z z ) (7) t2 t2 t2 i i i i t1 t1 t1 t1 t1 t1 L L L where L is the distance between LED and LED , and w (i = 1, 2) is the distance from LED 1 2 i 1 to the Plane P (i = 1, 2), derived as: 2 2 2 2 2 ˆ ˆ (x x ) + (y y ) + (z z ) + d d t2 t1 t2 t1 t2 t1 1 2 w = q (8) 2 2 2 2 x x + y y + z z ( ) ( ) ( ) t2 t1 t2 t1 t2 t1 2 2 2 2 2 ˆ ˆ (x x ) + (y y ) + (z z ) + d d t2 t1 t2 t1 t2 t1 w = (9) 2 2 2 2 (x x ) + (y y ) + (z z ) t2 t1 t2 t1 t2 t1 Additionally, the radiuses R (i = 1, 2) of the intersection circles can be written by: ˆ 2 R = d w (10) 1 1 1 Photonics 2022, 9, x FOR PEER REVIEW 5 of 16 2 2 2 2 x − x + y − y + z − z +d − d t2 t1 t2 t1 3 4 t2 t1 w = 2 (9) 2 2 2 x − x + y − y + z − z t2 t1 t2 t1 t2 t1 Additionally, the radiuses Ri (i = 1, 2) of the intersection circles can be written by: (10) R = d − w 1 1 1 Photonics 2022, 9, 159 5 of 15 2 (11) R = d − w 2 3 2 ˆ 2 R = d w (11) 2 2 (a) (b) Figure 2. Schematic diagrams of the solving process: (a) establishing intersection circles; and (b) Figure 2. Schematic diagrams of the solving process: (a) establishing intersection circles; and conducting coordinate transformations. (b) conducting coordinate transformations. Now, based on the circles K1 and K2, the solving process for the coordinates of PDs Now, based on the circles K and K , the solving process for the coordinates of PDs 1 2 can be simplified can be by simpli constr fied ucting by co new nstruct coor ing dinate new coordin systems. ate s Thus, ystems. Th in the us, second in the step, second step, we we conduct coor conduct coordina dinate transformations, te transforma as tions, shown asin show Figur n in Figure e 2b. We first 2b. We transform first tran the sform the XYZ- 0 0 0 XYZ-coordinate system into the X Y Z -coordinate system, in which K is the origin coordinate system into the X′ Y′ Z′-coordinate system, in which K1 is the origin point, line point, line K K forms the Z -axis, and the intersection line between the plane P and the K1K2 forms the Z′-axis, and the intersection line between the plane P1 and the XY-plane 1 2 1 0 0 0 0 XY-plane forms forms t the X he-axis. X′-axis. Next, Next, w we transform e transform the the X X Y′ Y Z′ Z -coor ′-coordin dinate atsystem e system int into the o the cylindrical cylindrical coordinate system, in which the cylindrical coordinates of PD and PD are coordinate system, in which the cylindrical coordinates of PD1 and PD2 are denoted by 1 2 denoted by (R , F , 0) and (R , F , S), respectively. Here, S is the distance between K and (R 1 1, Φ 1 1, 0) and (2 R2, Φ 2 2, S), respectively. Here, S is the distance between K 1 1 and K2. After K . After coordinate transformations, Equations (5) and (6) can be written in the form of 2 coordinate transformations, Equations (5) and (6) can be written in the form of the cylin- the cylindrical coordinates: drical coordinates: F F = arccos M (12) 1 2 (12) Φ −Φ = ± arccos M 1 2 b(R sin F R sin F )= l sin q gS (13) 2 2 1 1 β R sinΦ -R sinΦ = l sinθ −γS (13) 2 r 2 1 1 2 2 2 2 where M = (R + R + S l )/(2R R ), b = a +b /S, g = c/S, a = a a , b = b 1 2 1 2 2 1 2 2 2 2 2 where M = (R1 + R2 + S − l )/(2R1R2), β = − a +b /S, γ = c/S, a = a2 − a1, b = b2 − b1, c = c2 − b , c = c c , and S = a +b +c . By solving (12) and (13), F and F can be obtained. 1 2 1 1 2 2 2 c1, and S = a +b +c . By solving (12) and (13), Φ1 and Φ2 can be obtained. Then, we per- Then, we perform coordinate inverse transformations to recover the coordinates of PD form coordinate inverse transformations to recover the coordinates of PD1 and PD2 in the and PD in the XYZ-coordinate system, which are, respectively, given by: XYZ-coordinate system, which are, respectively, given by: 0 0 0 (x ˆ , y ˆ , z ˆ ) = e , e , e (R cos F , R sin F , 0) + (a , b , c ) (14) x y z r1 r1 r1 1 1 1 1 1 1 1 x ,y ,z = e ,e ,e R cosΦ ,R sinΦ ,0 + a ,b ,c (14) r1 r1 x y z 1 1 1 1 1 1 1 r1 0 0 0 (x ˆ , y ˆ , z ˆ ) = e , e , e (R cos F , R sin F , S) + (a , b , c ) (15) x y z r2 r2 r2 2 2 2 2 1 1 1 p p 2 2 2 2 Here, the orthonormal bases are given by e = b/ a +b , a/ a +b , 0 , e = x y r r r 2 2 2 2 2 2 ac/ S a +b , bc/ S a +b , a +b /S , and e = (a/S, b/S, c/S) . Based on the solving process from Equations (12)–(15), we get a total of four solutions. In other words, due to geometrical symmetry, a total of four pairs of the estimated coor- dinates of PD and PD are distributed in the space, among which only one is true. This 1 2 means that the location of the receiver cannot be determined due to multiple solutions, and we call this situation as location uncertainty. To eliminate location uncertainty, the third step is to exclude the redundant solutions and get the true solution. In practice, the height of the VLP receiver is lower than the height of LED transmitters. Thus, we first exclude the solutions that are above the ceiling by setting the first constraint condition: Photonics 2022, 9, x FOR PEER REVIEW 6 of 16 x ,y ,z = e ,e ,e R cosΦ ,R sinΦ ,S + a ,b ,c (15) r2 r2 x y z 2 2 2 2 1 1 1 r2 2 2 2 2 Here, the orthonormal bases are given by e = b/ a +b , − a/ a +b , 0 , e = x y 2 2 2 2 2 2 ac/ S a +b , bc/ S a +b , − a +b /S , and e = a/S, b/S, c/S . Based on the solving process from Equations (12)–(15), we get a total of four solu- tions. In other words, due to geometrical symmetry, a total of four pairs of the estimated coordinates of PD1 and PD2 are distributed in the space, among which only one is true. This means that the location of the receiver cannot be determined due to multiple solu- tions, and we call this situation as location uncertainty. To eliminate location uncertainty, the third step is to exclude the redundant solutions and get the true solution. In practice, Photonics 2022, 9, 159 6 of 15 the height of the VLP receiver is lower than the height of LED transmitters. Thus, we first exclude the solutions that are above the ceiling by setting the first constraint condition: z +z )/2< min zt1, zt2 . Among the rest of solutions, we perform a further exclusion by r1 r2 z ˆ + z ˆ /2 < min z , z . Among the rest of solutions, we perform a further exclusion ( ) ( ) r1 r2 t1 t2 considering the possible moving range of the receiver. In this study, we assume that the by considering the possible moving range of the receiver. In this study, we assume that the receiver moves within the constrained space on the outside of the YZ-plane by setting the receiver moves within the constrained space on the outside of the YZ-plane by setting the second constraint condition: x +x ≥ 0, which is shown in Figure 1. r1 r2 second constraint condition: x ˆ + x ˆ  0, which is shown in Figure 1. r1 r2 During our study, we find that if the receiver faces upwards, i.e., the polar angle θ = During our study, we find that if the receiver faces upwards, i.e., the polar angle q = 0 , 0°, then only one solution remains after considering the above two constraint conditions. then only one solution remains after considering the above two constraint conditions. This This solution is taken as the estimated coordinates of PD1 and PD2, based on which the 3D solution is taken as the estimated coordinates of PD and PD , based on which the 3D 1 2 location and the orientation of the receiver can be finally calculated. However, if the re- location and the orientation of the receiver can be finally calculated. However, if the receiver ceiver tilts, i.e., θ ≠ 0°, we may obtain one or two solutions remained after considering the tilts, i.e., q 6= 0 , we may obtain one or two solutions remained after considering the above above two constraint conditions, depending on the location of the receiver, the orientation two constraint conditions, depending on the location of the receiver, the orientation of the of the receiver, and the ranging error between LEDs and PDs. Therefore, to eliminate the receiver, and the ranging error between LEDs and PDs. Therefore, to eliminate the location location uncertainty caused by the receiver tilt, we propose a location selection strategy to uncertainty caused by the receiver tilt, we propose a location selection strategy to select the select the true location of PDs. true location of PDs. Figure 3 shows a flow chart of the location selection strategy. For a tilted receiver, if Figure 3 shows a flow chart of the location selection strategy. For a tilted receiver, if there remains only one pair of the estimated coordinates of PD1 and PD2, i.e., the case of there remains only one pair of the estimated coordinates of PD and PD , i.e., the case of 1 2 one solution, then they can be directly used to perform VLPO for the receiver. However, one solution, then they can be directly used to perform VLPO for the receiver. However, if if there remain two solutions, then we need to exclude the redundant solution by making there remain two solutions, then we need to exclude the redundant solution by making comparisons between the received signal power at the PDs and the expected received sig- comparisons between the received signal power at the PDs and the expected received signal nal power at the specific locations. Without loss of generality, we use the Solution A and power at the specific locations. Without loss of generality, we use the Solution A and the the Solution A’ to represent the two possible solutions remained, among which only one Solution A to represent the two possible solutions remained, among which only one is true. iFor s true. For the Sol the Solution A,u the tion estimated A, the esti coor ma dinates ted coordi of PD nate and s ofPD PD1ar an e,d PD respectively 2 are, re , sp defined ectively by , 1 2 es es es es es es es es es es es es 0 defin (x , y ed by ( , z )xand , y(x, z, y) an , zd ( ), x and , y for , zthe ), and fo Solution r th A e S , the olutes ion timated A’, the estimated c coordinates of oordi- PD r1 r1 r2 r2 1 r1 r1 r1 r2 r2 r2 r1 r2 0 0 0 0 0 0 es’ es’ es’ es’ es’ es’ es es es es es es nates o and PD f PD are, 1 and PD respectively 2 are,, re defined spectively by (, de x fined , y , by z ) (x and , y (x , ,zy ) and , z ( ). x , y , z ). 2 r1 r1 r2 r2 r2 r2 r2 r1 r1 r1 r1 r2 Figure 3. Flow chart of location selection strategy. We first measure the received signal power from LED (i = 1, 2) to PD (j = 1, 2), which i j is expressed by: r total P = s(t) h (t)dt/T (16) i,j i,j total where s(t) is the transmitted VLP signal, T is the duration of s(t), and h (t) is the total i,j channel impulse response (CIR) from LED (i = 1, 2) to PD (j = 1, 2), which can be further i j written by [23,24]: +¥ total LOS NLOS LOS LOS NLOS NLOS NLOS NLOS h (t)= h (t)+h (t)= H (0)d t t + A t d t t dt (17) i,j i,j i,j i,j i,j i,j i,j i,j i,j NLOS t =0 i,j Photonics 2022, 9, 159 7 of 15 LOS Here, d t is Dirac function. From LED (i = 1, 2) to PD (j = 1, 2), h t is the ( ) ( ) i j i,j NLOS LOS NLOS line-of-sight (LOS) CIR, h (t) is the non-LOS (NLOS) CIR, t and t represent i,j i,j i,j LOS the signal time delays of the LOS and NLOS channels, respectively, H (0) is the DC i,j NLOS NLOS gain of the LOS channel, and A t is the gain of the NLOS channel with a time i,j i,j NLOS delay t . i,j We assume all the LEDs have a Lambertian radiation pattern. Then, by supposing that the Solution A is true, we can estimate the expected received signal power from LED (i = 1, 2) to PD (j = 1, 2) based on the coordinates in Solution A, which is derived by [7,23]: m+1 es (m + 1)A z z T gP r ti s rj A LOS t P = H (0) P = (18) i,j i,j i (m+3)/2 2 2 2 es es es 2p x x + y y + z z ti ti ti rj rj rj where P is the transmitted power of LED (i = 1, 2), m is the Lambertian emission order of LEDs, A is the effective area of PDs, T is the gain of an optical filter, g is the gain of r s an optical concentrator. Similarly, by supposing that the Solution A is true, we can also estimate the expected received signal power from LED (i = 1, 2) to PD (j = 1, 2) based on i j the coordinates in Solution A , which is derived by [7,23]: m+1 es t (m + 1)A z z T gP r s ti 0 rj i P = (19) i,j (m+3)/2 2 2 2 0 0 0 es es es 2p x x + y y + z z ti ti ti rj rj rj r A A Next, by comparing the power differences between P , P , and P , we construct a i,j i,j i,j metric F to determine the true solution, which is written as: 2 2 2 2 r A r A F = P P P P (20) å å i,j i,j i,j i,j j = 1 i = 1 If F > 0, then the Solution A is selected as the true solution for the subsequent location and orientation estimation of the receiver; otherwise, the Solution A is selected. Finally, by using the single solution remained, the 3D coordinate of the receiver is estimated by (x ˆ , y ˆ , z ˆ ) = ((x ˆ + x ˆ )/2, (y ˆ + y ˆ )/2, (z ˆ + z ˆ )/2), and its azimuthal R R R r1 r2 r1 r2 r1 r2 angle h is estimated by: 2 2 h ˆ = sign(y ˆ y ˆ )arccos (x ˆ x ˆ )/ (x ˆ x ˆ ) + (y ˆ y ˆ ) (21) r2 r1 r2 r1 r1 r2 r1 r2 where sign () represents sign function. 3. Simulation Results and Discussions We evaluate the performance of the proposed VLPO scheme in a 3 m  5 m  3 m (length  width  height) indoor space based on the XYZ-coordinate system, as shown in Figure 1. In a considered VLPO cell, there are only two LED transmitters (LED and LED ) 1 2 and a mobile receiver. LED and LED are located at (0, 1.5, 3) and (0, 3.5, 3), respectively, 1 2 both of which have a Lambertian radiation pattern. Two PDs (PD and PD ) are mounted at 1 2 both ends of the receiver to jointly estimate the location and the orientation of the receiver. Around the VLPO cell, there stands three walls, which are represented by three plane equations: (i) x = 3, 0  y  5, 0  z  3; (ii) y = 0, 0  x  3, 0  z  3; and (iii) y = 5, 0  x  3, 0  z  3. Thus, the channel between the LEDs and PDs consists of both LOS and NLOS channels, wherein the first indoor reflection is considered. We assume that the TOA ranging method is used to estimate the distances between the LEDs and PDs. Photonics 2022, 9, x FOR PEER REVIEW 9 of 16 Photonics 2022, 9, 159 8 of 15 However, due to many factors including the geometry of the room, the frequency and transmitted power of the VLP signal, and the physical characteristics of LEDs and PDs, the distance estimation accuracy of the TOA method will be influenced [21]. Therefore, random estimation errors occur with respect to the real distances d , d , d , and d between 1 2 3 4 LEDs and PDs. We assume these ranging errors are zero-mean, independent and identically Gaussian distributed, and they have the same standard deviation denoted as Dd. In our simulations, we assume that the receiver is located at certain test locations and the interval between adjacent test locations at the same height is 0.5 m. The receiver faces upwards (q = 0 ) in Figures 4–7, and can be tilted (q > 0 ) in Figures 8–11. The azimuthal angle of the receiver is randomly distributed in the range of 180 < h  180 . To calculate the Figure 4. CDFs of 3D positioning errors at (0.5, 1, 1). (θ = 0°). cumulative distribution functions (CDFs) of the positioning error and the azimuthal error, we conduct Monte Carlo simulations at least 50 times at each test location. Table 2 lists the Photonics 2022, 9, x FOR PEER REVIEW 9 of 16 key parameters of the indoor VLPO system. Other parameters of the receiver are the same To evaluate the performance of the VLPO scheme at different locations, Figure 5 as those in [23]. shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. Here, we assume that the interval between PDs is 0.2 m and the standard deviation of the ranging error is 0.025 m. We test 77 locations in each receiving plane, thus a total of 231 locations are tested for performance evaluation in the considered indoor space. We use “+” to represent the real locations of the receiver and “○” to repre- sent the coordinates estimated from the proposed VLPO scheme. On the left of Figure 5, it can be observed that most of the estimated 3D locations are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different locations. By comparison, we find that the 3D positioning accuracy is related to the height of the receiver. For example, at the height of 0.5 m, 1 m, and 1.5 m, the mean 3D positioning error is 5.9 cm, 8.3 cm, and 14.2 cm, respectively. Thus, when the receiver is at a lower location, it tends to get a more accurate 3D positioning result. The reason is that lower PDs have larger distances to LEDs, thus enhancing their tolerance against the ranging error. On the right of Figure 5, we show the bird-eye view of the 3D positioning results, i.e., 2D positioning results, for the test locations at the height of 1 m, wherein the mean positionin Figure g e 4. CDFs rror is of 3D 6.9 cm, which c positioning errors at an m (0.5,eet 1, 1).the needs (q = 0 ). of most location-based services. Figure 4. CDFs of 3D positioning errors at (0.5, 1, 1). (θ = 0°). To evaluate the performance of the VLPO scheme at different locations, Figure 5 shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. Here, we assume that the interval between PDs is 0.2 m and the standard deviation of the ranging error is 0.025 m. We test 77 locations in each receiving plane, thus a total of 231 locations are tested for performance evaluation in the considered indoor space. We use “+” to represent the real locations of the receiver and “○” to repre- sent the coordinates estimated from the proposed VLPO scheme. On the left of Figure 5, it can be observed that most of the estimated 3D locations are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different locations. By comparison, we find that the 3D positioning accuracy is related to the height of the receiver. For example, at the height of 0.5 m, 1 m, and 1.5 m, the mean 3D positioning error is 5.9 cm, 8.3 cm, and 14.2 cm, respectively. Thus, when the receiver is at a lower location, it tends to get a more accurate 3D positioning result. The reason is that lower PDs have larger distances to LEDs, thus enhancing their tolerance against the ranging error. On the right of Figure 5, we show the bird-eye view of the 3D positioning results, Figure 5. Figure An ex 5. An am example ple of 3D of 3D positioning positioning r resu esults.lt(s. ( q =θ 0 = ). 0°). i.e., 2D positioning results, for the test locations at the height of 1 m, wherein the mean positioning error is 6.9 cm, which can meet the needs of most location-based services. By considering the test locations in the receiving plane at the height of 1 m in Figure 5, Figure 6a shows the simulated CDFs of the 3D positioning errors at different locations. We assume Δd = 0.025 m and compare the CDFs obtained by adopting different intervals l between PDs. As can be seen, due to the same reason as in Figure 4, increasing l can Figure 5. An example of 3D positioning results. (θ = 0°). By considering the test locations in the receiving plane at the height of 1 m in Figure 5, Figure 6a shows the simulated CDFs of the 3D positioning errors at different locations. We assume Δd = 0.025 m and compare the CDFs obtained by adopting different intervals l between PDs. As can be seen, due to the same reason as in Figure 4, increasing l can Photonics 2022, 9, x FOR PEER REVIEW 10 of 16 effectively improve the overall positioning accuracy in the considered receiving plane. Specifically, when l is 0.1 m, 0.2 m, 0.5 m, and 1 m, around 90% of 3D positioning errors are less than 44.9 cm, 28.4 cm, 13.2 cm, and 7.9 cm, respectively. Therefore, to achieve a targeted positioning accuracy for the proposed VLPO scheme, we may appropriately in- crease the interval between PDs while considering the size of a receiver terminal. Take the shopping scenario as an example, with the proposed VLPO scheme, by installing a pair of PDs with their interval to be 0.5 m on a shopping cart, we may achieve accurate 3D posi- Photonics 2022, 9, 159 9 of 15 tioning for a customer with 92.7% of 3D positioning errors less than 20 cm using only two LEDs. Photonics 2022, 9, x FOR PEER REVIEW 11 of 16 (a) (b) Photonics 2022, 9, x FOR PEER REVIEW 11 of 16 Figure 6. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 0°). Figure 6. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. (q = 0 ). Corresponding to Figure 6a, Figure 6b plots the simulated CDFs of the azimuthal errors by considering the test locations in the receiving plane at the height of 1 m in Figure 5. We assume Δd = 0.025 m and compare the CDFs obtained by using different l. It is found that increasing l can effectively improve the overall orienteering accuracy in the consid- Figure 7. Mean positioning errors and azimuthal errors at the height of 1 m. (θ = 0°). ered receiving plane. The reason is the same as in Figure 6a. Specifically, around 77.7%, 87.4%, 91.6%, and 92.8% of azimuthal errors are less than 3° when l is 0.1 m, 0.2 m, 0.5 m, Next, we evaluate the performance of the VLPO scheme when the receiver is tilted and 1 m, respectively, thereby verifying the high orienteering accuracy of the proposed (θ > 0°). Figure 8a shows the simulated CDFs of the 3D positioning errors when the re- VLPO scheme at different locations. ceiver is located at (1.5, 2, 1) with its polar angle θ fixed at 30°. We assume Δd = 0.025 m Based on the test locations in Figure 6, Figure 7 compares the mean positioning errors and use different curves to represent the CDFs obtained by adopting different intervals l and azimuthal errors versus different standard derivations Δd of the distance estimation between PDs. It can be seen that, when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of errors caused by ranging. We set Δd = 0.0125 m, 0.025 m, 0.05 m, and 1 m, and plot the achieving 3D positioning errors less than 15 cm is 70.5%, 78.1%, 83.8%, and 85.7%, respec- mean positioning errors and azimuthal errors obtained by using different intervals l be- tively, thereby verifying the effectiveness of the proposed VLPO scheme for a tilted re- tween PDs. The three curves above are the CDFs of mean azimuthal errors, and the three ceiver. Besides, a total of around 80% of 3D positioning errors are less than 28.5 cm when curves below are the CDFs of the mean azimuthal errors. We note that both the position- l is 0.1 m, and by further increasing l to 0.5 m, 80% of 3D positioning errors are less than ing accuracy and the orienteering accuracy degrade with an increased Δd. Specifically, 8.9 cm. Therefore, for the scenario of receiver tilt, adopting a larger l can effectively en- when l is 0.5 m and Δd increases from 0.0125 m to 0.05 m, the mean positioning error hance the Figure 7. 3D Mean positionin positioning g err accurac ors and yazimuthal of the proposed errors at the VLPO scheme height of 1 m. (. q = 0 ). Figure 7. Mean positioning errors and azimuthal errors at the height of 1 m. (θ = 0°). increases from 3.5 cm to 15.2 cm, and the mean azimuthal error increases from 6.2° to 8.1°. Further, when Δd reaches 0.1 m, the mean positioning error is as large as 30.5 cm, which Next, we evaluate the performance of the VLPO scheme when the receiver is tilted is inapplicable for high-accurate location-based services, whereas the mean azimuthal er- (θ > 0°). Figure 8a shows the simulated CDFs of the 3D positioning errors when the re- ror is 11.2°. Thus, compared with the orienteering accuracy, the positioning accuracy is more sensitiv ceiver is locat e to the distance estim ed at (1.5, 2, 1) with it atio s po n error betw lar angle θeen LEDs an fixed at 30°. W d PDs. Ad e assumdit e Δ ionally d = 0.02 , 5 m when and use d Δd is ist ffe ab ril ent curves to ized at 0.025 represent the m, the mean p CDFs osition obta ing er inror ed by adopting dif is 7.4 cm and the fem rent interva ean azi- ls l muthal error is 7.0° when l is 0.5 m. Therefore, high ranging accuracy is crucial to achieve between PDs. It can be seen that, when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of the high-accurate performance of the proposed VLPO scheme. achieving 3D positioning errors less than 15 cm is 70.5%, 78.1%, 83.8%, and 85.7%, respec- tively, thereby verifying the effectiveness of the proposed VLPO scheme for a tilted re- ceiver. Besides, a total of around 80% of 3D positioning errors are less than 28.5 cm when l is 0.1 m, and by further increasing l to 0.5 m, 80% of 3D positioning errors are less than 8.9 cm. Therefore, for the scenario of receiver tilt, adopting a larger l can effectively en- hance the 3D positioning accuracy of the proposed VLPO scheme. (a) (b) Figure 8. CDFs at (1.5, 2, 1) for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 30°). Figure 8. CDFs at (1.5, 2, 1) for: (a) 3D positioning errors; and (b) azimuthal errors. (q = 30 ). By using the same the simulated conditions as in Figure 8a, Figure 8b plots the sim- ulated CDFs of the azimuthal errors. We see that when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of achieving azimuthal errors less than 5° is 89.5%, 93.1%, 95.2%, and 95.3%, respectively, thus proving the high orienteering accuracy of the proposed VLPO scheme for a tilted receiver. Furthermore, increasing the interval between PDs can enhance the orienteering accuracy effectively. For example, by increasing l from 0.1 m to 0.5 m, the probability of achieving azimuthal errors less than 3° can be improved from 79.4% to 93.4%, when θ = 30°. (a) (b) Figure 8. CDFs at (1.5, 2, 1) for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 30°). By using the same the simulated conditions as in Figure 8a, Figure 8b plots the sim- ulated CDFs of the azimuthal errors. We see that when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of achieving azimuthal errors less than 5° is 89.5%, 93.1%, 95.2%, and 95.3%, respectively, thus proving the high orienteering accuracy of the proposed VLPO scheme for a tilted receiver. Furthermore, increasing the interval between PDs can enhance the orienteering accuracy effectively. For example, by increasing l from 0.1 m to 0.5 m, the probability of achieving azimuthal errors less than 3° can be improved from 79.4% to 93.4%, when θ = 30°. Photonics 2022, 9, x FOR PEER REVIEW 12 of 16 To evaluate the performance of the VLPO scheme at different locations for a tilted receiver, Figure 9 shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. We fix the receiver polar angle θ at 20° to ensure that the LOS channels between LEDs and PDs will not be blocked due to the receiver tilt. We assume the interval between PDs is 0.2 m and the standard derivation Δd of the rang- ing error is 0.025 m. The test locations are the same as in Figure 5. On the left of Figure 9, we observe that most of the estimated 3D coordinates are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different loca- tions for a tilted receiver. Specifically, the mean 3D positioning error is 9.2 cm, 14.2 cm, and 20.9 cm at the height of 0.5 m, 1 m, and 1.5 m, respectively, from which the positioning accuracy is related to the height of the receiver. The reason is the same as in Figure 5. On the right of Figure 9, we show the corresponding 2D positioning results with a tilted re- ceiver at the height of 1 m, wherein the mean positioning error is 10.7 cm. Besides, it can be observed that small positioning errors have a higher possibility to occur at the test lo- cations below LEDs, e.g., where xR = 0. This is because, when the receiver is tilted and located at the boundary of the considered indoor space with xR = 0, multiple solutions of the estimated coordinates of PD1 and PD2 may stay close due to geometrical symmetry, thus their midpoints, i.e., the estimated locations of the receiver, have a higher chance to Photonics 2022, 9, 159 10 of 15 overlap, which potentially enhances the positioning accuracy at the test locations with xR = 0. Photonics 2022, 9, x FOR PEER REVIEW 13 of 16 Photonics 2022, 9, x FOR PEER REVIEW 13 of 16 Figure 9. An example of 3D positioning results. (θ = 20°). Figure 9. An example of 3D positioning results. (q = 20 ). (a) (b) Figure 10. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 0°, By considering the test locations in the receiving plane at the height of 1 m in Figure 10°, and 20°). 9, Figure 10a shows the simulated CDFs of the 3D positioning errors at different locations. We assume the standard derivation Δd of the ranging error is 0.025 m and the interval l Corresponding to Figure 10a, Figure 10b plots the simulated CDFs of the azimuthal between PDs is 0.5 m. We use different curves to represent the CDFs obtained by consid- errors by considering the test locations in the receiving plane at the height of 1 m in Figure ering different polar angles with θ to be 0°, 10°, and 20°. For a performance evaluation, 9. We assume the standard derivation Δd of the ranging error is 0.025 m and the interval we also plot the simulated CDF curves of the 3D positioning errors when excluding the l between PDs is 0.2 m, and compare the CDFs obtained by considering different polar test locations below LEDs with xR = 0. We see that increasing the polar angle of a tilted angles θ of the tilted receiver. It is found that increasing θ slightly degrades the overall receiver slightly degrades the overall positioning accuracy in the considered receiving orienteering accuracy in the considered receiving plane. Specifically, around 92.6%, 92.6%, plane. Specifically, when θ is 0°, 10°, and 20°, the probability of achieving 3D positioning and 90.9% of azimuthal errors are less than 5° when θ is 0°, 10°, and 20°, respectively, errors thereby verifying the h less than 20 cm isi 92 gh .7%, 90 orienteerin .3%, g an accurac d 87.9%, res y of thp e V ecti LP vel O s y c . In contra heme at dis ft f, when excl erent locatio u n d si ng the test l for a tilted ocat r ions wi eceiver. Mo th xR r = eover, wh 0, the prob en ab exclud ility o ing f achiev the test ing 3D loca p tioonsit s w iointh ing er xR = rors 0, around less than 99.5%, 99.2%, and 97.7% of azimuthal errors are less than 5° when θ is 0°, 10°, and 20°, 20 cm decreases to 91.5%, 88.7%, and 86.0% when θ is 0°, 10°, and 20°, respectively. By respectively. Obviously, large azimuthal errors tend to occur at the test locations with xR comparison, smaller positioning errors tend to occur at the test locations below LEDs with = 0. The reason lies in that, when the receiver tilts, although multiple solutions of the esti- xR = 0, which confirms the conclusion in Figure 9. mated coordinates of PD1 and PD2 may stay close due to geometrical symmetry as dis- (a) (b) cussed in Figure 9, the estimated azimuthal angles of the receiver are completely different Figure 10. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. (θ = 0°, Figure 10. CDFs at the height of 1 m for: (a) 3D positioning errors; and (b) azimuthal errors. and their sum is 180°, thus degrading the orienteering accuracy at the test locations with 10°, and 20°). (q = 0 , 10 , and 20 ). xR = 0. Corresponding to Figure 10a, Figure 10b plots the simulated CDFs of the azimuthal errors by considering the test locations in the receiving plane at the height of 1 m in Figure 9. We assume the standard derivation Δd of the ranging error is 0.025 m and the interval l between PDs is 0.2 m, and compare the CDFs obtained by considering different polar angles θ of the tilted receiver. It is found that increasing θ slightly degrades the overall orienteering accuracy in the considered receiving plane. Specifically, around 92.6%, 92.6%, and 90.9% of azimuthal errors are less than 5° when θ is 0°, 10°, and 20°, respectively, thereby verifying the high orienteering accuracy of the VLPO scheme at different locations for a tilted receiver. Moreover, when excluding the test locations with xR = 0, around 99.5%, 99.2%, and 97.7% of azimuthal errors are less than 5° when θ is 0°, 10°, and 20°, respectively. Obviously, large azimuthal errors tend to occur at the test locations with xR = 0. The reason lies in that, when the receiver tilts, although multiple solutions of the esti- mated coordinates of PD1 and PD2 may stay close due to geometrical symmetry as dis- cussed in Figure 9, the estimated azimuthal angles of the receiver are completely different and their sum is 180°, thus degrading the orienteering accuracy at the test locations with xR = 0. Figure 11. Mean positioning errors and azimuthal errors at the height of 1 m. (θ = 0°, 10°, and 20°). Figure 11. Mean positioning errors and azimuthal errors at the height of 1 m. (q = 0 , 10 , and 20 ). Finally, based on the test locations in Figure 10, Figure 11 compares the mean posi- tioning errors and azimuthal errors versus different polar angles θ of the tilted receiver. Figure 11. Mean positioning errors and azimuthal errors at the height of 1 m. (θ = 0°, 10°, and 20°). Finally, based on the test locations in Figure 10, Figure 11 compares the mean posi- tioning errors and azimuthal errors versus different polar angles θ of the tilted receiver. Photonics 2022, 9, 159 11 of 15 Table 2. Key parameters of the VLPO system. Name of Parameters Values Indoor space (length  width  height) 3 m  5 m  3 m Height of receiver 0.5 m/1 m/1.5 m Launch power of each LED 5 Watt Modulation index 0.1 Lambertian emission order of LEDs 1 LED semi-angle at half power 60 Field-of-view (FOV) at the receiver 170 4 2 Effective area of photo-detector 10 m Photo-detector responsivity 0.35 A/W Reflection coefficient of wall 0.83 Gain of the optical filter 1 Refractive index of the optical concentrator 1.5 Baud rate of the VLP signal 10 Msymbol/s We first evaluate the performance of the VLPO scheme when the receiver faces up- wards (q = 0 ). Figure 4 shows the simulated CDFs of the 3D positioning errors when the receiver is fixed at a specific location (0.5, 1, 1). Different curves represent the CDFs obtained by adopting different intervals l between PDs. We assume the standard deviation Dd of the ranging error is 0.025 m. We note that, when l is 0.1 m, 0.2 m, 0.5 m, and 1 m, the probability of achieving 3D positioning errors less than 10 cm is 83.2%, 90.2%, 93.7%, and 94.1%, respectively. This is because when Dd is fixed, adopting a larger l helps enhance the tolerance against the ranging error, thereby improving the positioning accuracy. Specifically, around 90% of 3D positioning errors are less than 9.8 cm when l is 0.2 m, and by further increasing l to 0.5 m, 90% of 3D positioning errors are less than 4.5 cm, thus verifying the high positioning accuracy of the proposed VLPO scheme when q = 0 . To evaluate the performance of the VLPO scheme at different locations, Figure 5 shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. Here, we assume that the interval between PDs is 0.2 m and the standard deviation of the ranging error is 0.025 m. We test 77 locations in each receiving plane, thus a total of 231 locations are tested for performance evaluation in the considered indoor space. We use “+” to represent the real locations of the receiver and “#” to represent the coordinates estimated from the proposed VLPO scheme. On the left of Figure 5, it can be observed that most of the estimated 3D locations are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different locations. By comparison, we find that the 3D positioning accuracy is related to the height of the receiver. For example, at the height of 0.5 m, 1 m, and 1.5 m, the mean 3D positioning error is 5.9 cm, 8.3 cm, and 14.2 cm, respectively. Thus, when the receiver is at a lower location, it tends to get a more accurate 3D positioning result. The reason is that lower PDs have larger distances to LEDs, thus enhancing their tolerance against the ranging error. On the right of Figure 5, we show the bird-eye view of the 3D positioning results, i.e., 2D positioning results, for the test locations at the height of 1 m, wherein the mean positioning error is 6.9 cm, which can meet the needs of most location-based services. By considering the test locations in the receiving plane at the height of 1 m in Figure 5, Figure 6a shows the simulated CDFs of the 3D positioning errors at different locations. We assume Dd = 0.025 m and compare the CDFs obtained by adopting different intervals l between PDs. As can be seen, due to the same reason as in Figure 4, increasing l can effectively improve the overall positioning accuracy in the considered receiving plane. Specifically, when l is 0.1 m, 0.2 m, 0.5 m, and 1 m, around 90% of 3D positioning errors are less than 44.9 cm, 28.4 cm, 13.2 cm, and 7.9 cm, respectively. Therefore, to achieve a targeted positioning accuracy for the proposed VLPO scheme, we may appropriately increase the interval between PDs while considering the size of a receiver terminal. Take the shopping scenario as an example, with the proposed VLPO scheme, by installing a pair of PDs with Photonics 2022, 9, 159 12 of 15 their interval to be 0.5 m on a shopping cart, we may achieve accurate 3D positioning for a customer with 92.7% of 3D positioning errors less than 20 cm using only two LEDs. Corresponding to Figure 6a, Figure 6b plots the simulated CDFs of the azimuthal errors by considering the test locations in the receiving plane at the height of 1 m in Figure 5. We assume Dd = 0.025 m and compare the CDFs obtained by using different l. It is found that increasing l can effectively improve the overall orienteering accuracy in the considered receiving plane. The reason is the same as in Figure 6a. Specifically, around 77.7%, 87.4%, 91.6%, and 92.8% of azimuthal errors are less than 3 when l is 0.1 m, 0.2 m, 0.5 m, and 1 m, respectively, thereby verifying the high orienteering accuracy of the proposed VLPO scheme at different locations. Based on the test locations in Figure 6, Figure 7 compares the mean positioning errors and azimuthal errors versus different standard derivations Dd of the distance estimation errors caused by ranging. We set Dd = 0.0125 m, 0.025 m, 0.05 m, and 1 m, and plot the mean positioning errors and azimuthal errors obtained by using different intervals l between PDs. The three curves above are the CDFs of mean azimuthal errors, and the three curves below are the CDFs of the mean azimuthal errors. We note that both the positioning accuracy and the orienteering accuracy degrade with an increased Dd. Specifically, when l is 0.5 m and Dd increases from 0.0125 m to 0.05 m, the mean positioning error increases from 3.5 cm to 15.2 cm, and the mean azimuthal error increases from 6.2 to 8.1 . Further, when Dd reaches 0.1 m, the mean positioning error is as large as 30.5 cm, which is inapplicable for high-accurate location-based services, whereas the mean azimuthal error is 11.2 . Thus, compared with the orienteering accuracy, the positioning accuracy is more sensitive to the distance estimation error between LEDs and PDs. Additionally, when Dd is stabilized at 0.025 m, the mean positioning error is 7.4 cm and the mean azimuthal error is 7.0 when l is 0.5 m. Therefore, high ranging accuracy is crucial to achieve the high-accurate performance of the proposed VLPO scheme. Next, we evaluate the performance of the VLPO scheme when the receiver is tilted (q > 0 ). Figure 8a shows the simulated CDFs of the 3D positioning errors when the receiver is located at (1.5, 2, 1) with its polar angle q fixed at 30 . We assume Dd = 0.025 m and use different curves to represent the CDFs obtained by adopting different intervals l between PDs. It can be seen that, when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of achieving 3D positioning errors less than 15 cm is 70.5%, 78.1%, 83.8%, and 85.7%, respectively, thereby verifying the effectiveness of the proposed VLPO scheme for a tilted receiver. Besides, a total of around 80% of 3D positioning errors are less than 28.5 cm when l is 0.1 m, and by further increasing l to 0.5 m, 80% of 3D positioning errors are less than 8.9 cm. Therefore, for the scenario of receiver tilt, adopting a larger l can effectively enhance the 3D positioning accuracy of the proposed VLPO scheme. By using the same the simulated conditions as in Figure 8a, Figure 8b plots the simulated CDFs of the azimuthal errors. We see that when l is 0.1 m, 0.2 m, 0.5 m and 1 m, the probability of achieving azimuthal errors less than 5 is 89.5%, 93.1%, 95.2%, and 95.3%, respectively, thus proving the high orienteering accuracy of the proposed VLPO scheme for a tilted receiver. Furthermore, increasing the interval between PDs can enhance the orienteering accuracy effectively. For example, by increasing l from 0.1 m to 0.5 m, the probability of achieving azimuthal errors less than 3 can be improved from 79.4% to 93.4%, when q = 30 . To evaluate the performance of the VLPO scheme at different locations for a tilted receiver, Figure 9 shows an example of 3D positioning results in different receiving planes at the height of 0.5 m, 1 m, and 1.5 m. We fix the receiver polar angle q at 20 to ensure that the LOS channels between LEDs and PDs will not be blocked due to the receiver tilt. We assume the interval between PDs is 0.2 m and the standard derivation Dd of the ranging error is 0.025 m. The test locations are the same as in Figure 5. On the left of Figure 9, we observe that most of the estimated 3D coordinates are close to their corresponding real locations, thus verifying the effectiveness of the proposed VLPO scheme at different locations for a tilted receiver. Specifically, the mean 3D positioning error is 9.2 cm, 14.2 cm, Photonics 2022, 9, 159 13 of 15 and 20.9 cm at the height of 0.5 m, 1 m, and 1.5 m, respectively, from which the positioning accuracy is related to the height of the receiver. The reason is the same as in Figure 5. On the right of Figure 9, we show the corresponding 2D positioning results with a tilted receiver at the height of 1 m, wherein the mean positioning error is 10.7 cm. Besides, it can be observed that small positioning errors have a higher possibility to occur at the test locations below LEDs, e.g., where x = 0. This is because, when the receiver is tilted and located at the boundary of the considered indoor space with x = 0, multiple solutions of the estimated coordinates of PD and PD may stay close due to geometrical symmetry, 1 2 thus their midpoints, i.e., the estimated locations of the receiver, have a higher chance to overlap, which potentially enhances the positioning accuracy at the test locations with x = 0. By considering the test locations in the receiving plane at the height of 1 m in Figure 9, Figure 10a shows the simulated CDFs of the 3D positioning errors at different locations. We assume the standard derivation Dd of the ranging error is 0.025 m and the interval l between PDs is 0.5 m. We use different curves to represent the CDFs obtained by considering different polar angles with q to be 0 , 10 , and 20 . For a performance evaluation, we also plot the simulated CDF curves of the 3D positioning errors when excluding the test locations below LEDs with x = 0. We see that increasing the polar angle of a tilted receiver slightly degrades the overall positioning accuracy in the considered receiving plane. Specifically, when q is 0 , 10 , and 20 , the probability of achieving 3D positioning errors less than 20 cm is 92.7%, 90.3%, and 87.9%, respectively. In contrast, when excluding the test locations with x = 0, the probability of achieving 3D positioning errors less than 20 cm decreases to 91.5%, 88.7%, and 86.0% when q is 0 , 10 , and 20 , respectively. By comparison, smaller positioning errors tend to occur at the test locations below LEDs with x = 0, which confirms the conclusion in Figure 9. Corresponding to Figure 10a, Figure 10b plots the simulated CDFs of the azimuthal errors by considering the test locations in the receiving plane at the height of 1 m in Figure 9. We assume the standard derivation Dd of the ranging error is 0.025 m and the interval l between PDs is 0.2 m, and compare the CDFs obtained by considering different polar angles q of the tilted receiver. It is found that increasing q slightly degrades the overall orienteering accuracy in the considered receiving plane. Specifically, around 92.6%, 92.6%, and 90.9% of azimuthal errors are less than 5 when q is 0 , 10 , and 20 , respectively, thereby verifying the high orienteering accuracy of the VLPO scheme at different locations for a tilted receiver. Moreover, when excluding the test locations with x = 0, around 99.5%, 99.2%, and 97.7% of azimuthal errors are less than 5 when q is 0 , 10 , and 20 , respectively. Obviously, large azimuthal errors tend to occur at the test locations with x = 0. The reason lies in that, when the receiver tilts, although multiple solutions of the estimated coordinates of PD and PD may stay close due to geometrical symmetry as discussed in Figure 9, the 1 2 estimated azimuthal angles of the receiver are completely different and their sum is 180 , thus degrading the orienteering accuracy at the test locations with x = 0. Finally, based on the test locations in Figure 10, Figure 11 compares the mean posi- tioning errors and azimuthal errors versus different polar angles q of the tilted receiver. We assume Dd = 0.025 m. We set q = 0 , 10 , and 20 , and plot the mean positioning errors and azimuthal errors obtained by using different intervals l between PDs. As can be seen, the change in the polar angle of the tilted receiver has only a small influence on both the positioning accuracy and the orienteering accuracy. Specifically, when l is 0.2 m and q changes from 0 to 20 , the mean positioning error slightly increases from 12.5 cm to 13.3 cm, and the mean azimuthal error slightly increases from 7.4 to 7.7 . Therefore, the proposed VLPO scheme is robust to the variation of the receiver polar angle, and can achieve stable 3D positioning and orienteering accuracy with a tilted receiver. 4. Conclusions To enable indoor VLP when the LED condition is constrained, we proposed a novel 3D VLP and orienteering (VLPO) scheme. By using only two LEDs and a pair of PDs, the Photonics 2022, 9, 159 14 of 15 scheme can simultaneously estimate the 3D coordinate and the orientation of the receiver. To support VLPO for a tilted receiver, we further proposed a location selection strategy to overcome the location uncertainty caused by receiver tilt. Simulation studies showed that, when the receiver faces upwards, the proposed VLPO scheme can achieve a mean 3D positioning error of 7.4 cm and a mean azimuthal error of 7.0 . Moreover, when the receiver tilts with a polar angle of 10 , the proposed scheme can still achieve accurate VLP with 90.3% of 3D positioning errors less than 20 cm, and accurate receiver orienteering with 92.6% of azimuthal errors less than 5 . The evaluation also indicated that appropriately increasing the interval between PDs can help enhance the tolerance against the ranging error, thus improving the VLPO accuracy effectively. In the future work, we will study and improve the VLPO scheme under scenarios when the LOS channels between LEDs and PDs are partially blocked. Author Contributions: Conceptualization, X.Y. (Xiaodi You); methodology, X.Y. (Xiaodi You) and Z.J.; validation, X.Y. (Xiaodi You) and X.Y. (Xiaobai Yang); investigation, X.Y. (Xiaobai Yang) and S.Z.; resources, X.Y. 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Journal

PhotonicsMultidisciplinary Digital Publishing Institute

Published: Mar 5, 2022

Keywords: visible light positioning (VLP); three-dimensional (3D); visible light positioning and orienteering (VLPO); light-emitting diode (LED)

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