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Impact of Transmitter Positioning and Orientation Uncertainty on RSS-Based Visible Light Positioning Accuracy^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

_{p}was <0.6 m by using the regression approach, which is much lower than other traditional methods. In [18], the impact of the Tx’s orientation (i.e., the tilting angle) on the positioning accuracy of the RSS-based VLP system was studied. Another estimation method, i.e., normalized least square estimation was utilized to estimate the positioning accuracy. In [19], an artificial neural network (ANN) based 4-LED VLP system was proposed to reduce ε

_{p}for the LOS path, which is affected by the random and unknown static Tx tilt angle with a maximum variation of 2°. It was revealed that ANN achieved localization errors below 1 cm. In general, the positions and orientations of Txs may not be symmetrical, which (i) depends on the indoor environments such as museums, galleries, train stations, shopping centers, etc., where lights are pointing in different directions; and (ii) can change when replacing lights, carrying out maintenance, etc. Therefore, the random variations in the Tx’s orientation will lead to the random errors in VLP systems, which requires further studies. The impact of Tx’s position and its orientation uncertainties on the positioning estimation have not yet been systematically explored, which is the objective of this paper.

_{p}of the RSS-based VLP system under multipath reflections. The uniform distribution of light inside the illuminated place is a necessity in indoor environments. As a result, lighting uniformity becomes a vital aspect for a well-lit environment. Moreover, both lighting uniformity and the positioning performance are related to the Tx’s positions and the Lambertian half-power angles (HPA). Therefore, in this work we investigate (i) how the uniformity of light in the room changes for different HPA and the Tx positions; and (ii) the impact of Tx position and its orientation uncertainties on the positioning accuracy considering the optimized Tx positions from a lighting uniformity perspective. This works focused only on the LED uncertainties caused during installation and is an extension of our previous work [20], where the problem of Tx’s orientation uncertainty was not considered.

## 2. VLP System Modelling

#### 2.1. Channel Model

_{r}of the PD are 70° and 10

^{−4}m

^{2}, respectively. All Txs are located at the same height h from the ground level and the coordinates of kth Tx (k = 1, …, K) is (${x}_{k},{y}_{k},{z}_{k}$), where K is the total number of Txs. The Rx coordinate is denoted by (${x}_{r},{y}_{r},{z}_{r}$). The position and orientation of the Txs is best illustrated by the ${\mathrm{Tx}}_{k}$, with the coordinate of ($\delta x,\delta y$) and the angles of $\alpha $, $\beta $, and $\gamma $. Both $\delta x$ and $\delta y$, and $\alpha $, $\beta $, and $\gamma $, are assumed to be Gaussian variables with N(0, ${\sigma}^{2}$) and N(0, ζ

^{2}) probability distribution functions, respectively. The distance between the Txs relies on the lighting uniformity considerations that are described in a later section. An empty room is considered as a reference to study the impact of Tx’s position and its orientation uncertainty on the positioning accuracy. In this work, both LOS and NLOS transmission paths are assumed between the Txs and the Rx. However, for the NLOS path, we only consider the first reflection due to the fact that the second order reflections have much reduced intensities and therefore can be neglected [21]. Each Tx broadcast a unique 2-bit ID information, which is encoded and modulated using on-off keying (OOK), which allows separation at the Rx using a correlation method that can be received at the Rx in advance of location identification.

#### 2.2. RSS-Based Positioning

_{r}and $\mathcal{R}$ are the PD’s active area and responsivity, respectively. ${T}_{s}\left(\phi \right)$ and $g\left(\phi \right)$ are the gains of the optical filter and the concentrator at the Rx, respectively. Note, ${T}_{s}\left(\phi \right)$ and $g\left(\phi \right)$ are set to unity.

#### 2.3. Distance Estimation Using Polynomial Regression

^{2}(coefficient of determination) for different orders, and a fourth order polynomial was selected with a value of R

^{2}of 0.8942. Note, the polynomial fitting is not highly accurate. This is because of the data points considered within the entire room for both LOS and NLOS paths. At the center of the room, the impact of NLOS is negligible when compared to the regions near walls and corners. As such, fitting all the data will be dominated by the data near walls and corners. However, it shows the best fitted solution considering that the data points corresponding to lower received power have higher contributions to the error compared with the data points in the center of the room. The values of polynomial coefficients are listed in Table 1.

#### 2.4. Estimation Using Nonlinear Least Squares

#### 2.5. Performance Metrics

_{p}is assumed to be a random variable (as it may rely on the uncertainties, i.e., the Tx’s position or its orientation uncertainties, estimation process, or noise); thus, it is reasonable to use the standard statistical analyses to access error performance. Here we use the probability distribution function (PDF) and the cumulative distribution function (CDF), as a mean to calculate the 95% quantile on ε

_{p}. Hence, the PDF and CDF are composed of the spatial distribution of ε

_{p}within the entire room. The PDF of ε

_{p}is defined as:

_{p}for NLLS estimation with and without polynomial regression. The maximum ε

_{p}values estimated by NLLS and NLLS with polynomial regression are 0.6 and 1.57 m, respectively. It is observed that, there is an evident improvement in the positioning accuracy by using NLLS estimation with polynomial regression for power-distance modeling. Therefore, the polynomial regression can improve the accuracy of position estimation without the inclusion of high complexity algorithms. However, there are some limitations of the polynomial regression; for instance, the coefficients of the polynomial model must be provided along with the Tx’s positions in practical scenarios. Moreover, the polynomial model can deal with the empty and non-empty rooms with the fixed furniture and objects, but with no user’s mobility.

## 3. Set-Up Uncertainties

#### 3.1. Uncertainties of the Tx’s Position

_{R}.

#### 3.2. Uncertainties of the Tx’s Orientation

^{2}), finally, $\widehat{{n}_{k}^{0}}$ represents the unperturbed heading vector of the kth Tx. $R\left(\alpha ,\beta ,\gamma \right)$ is a composed rotation matrix. The effect of (17) on the Tx’s heading is due to the cosine terms due to the Txs in (2) and (4), showing that a rotation will impact the received signal. The next step resorts to the evaluation of the expectations due to the direct and reverse process. In the direct process, the channel is re-simulated for each random iteration, generating an expectation value, for the kth Tx, which is given by:

#### 3.3. Lighting Uniformity

## 4. Simulation Results

^{3}, which are assumed to be the same and modelled as pointwise Lambertian sources with order m depending ζ on the value of HPA, see (2). In practical environments also, the square LED placement layout is common. We have considered three different cases of 4-, 9-, and 16-LEDs, which are arranged in a square grid on the ceiling plane. The Rx is located on the ground plane. For all simulation purposes, the resolution of the grid is fixed at 1 cm, which implies that the PD can be placed at 3600 different locations. All the key parameters for the simulation are detailed in Table 2.

#### 4.1. Positioning Error Dependence on Lighting Uniformity

_{p}. Here, we vary the uniformity between 0.5 and 0.8 in steps of 0.05 and for HPA of 40° and 60°. For this, we first select the values of distance between the Txs according to the results of Figure 8. Following this, we estimate ε

_{p}for each set of conditions using NLLS and the polynomial regression power-distance model. ε

_{p}for different values of uniformity for 4-, 9-, and 16-Tx are illustrated in Figure 9. It is observed that the minimum ε

_{p}is attained for the case of 4-Tx with HPA and uniformity of 60° and 0.65, respectively, whereas, in the case of 9-Tx, the minimum ε

_{p}is achieved with HPA of 60° and uniformity equal to 0.55. For the 16-Tx case, the minimum ε

_{p}is accomplished with an HPA of 40° and with the uniformity of 0.65. It is clear from Figure 9 that, (i) low to a moderate value of lighting uniformity can support low ε

_{p}; and (ii) an optimal value for the number of Txs and the associated HPA, which do not match the optimal value of lighting uniformity conditions. Under the simulated conditions, the optimal values for the HPA, uniformity, and distance between the Tx based on the lowest ε

_{p}are shown in Table 3.

#### 4.2. Impact of Tx’s Position and Orientation Uncertainty on Error Performance

_{p}, which is given by:

_{p}as a function of σ for 4-, 9-, and 16-Tx for the Tx’s positioning uncertainty. It is observed that the effect of Tx’s position uncertainty, σ, traduces in an increasing error dependence, which is more prominent for set-ups with a lower number of Txs. The standard deviation of the error quantile, i.e., the increasing error also confirms an increasing trend with σ, is more evident for the 4-Tx case. It is noticed that, for σ of 5 cm the average positioning errors are 23.3, 15.1, and 13.2 cm, with the standard deviation values of 6.4, 4.1, and 2.7 cm for 4-, 9-, and 16-Tx cases, respectively. Figure 10 suggests that the error dependence on the Tx’s position uncertainty can be lowered by increasing the number of Txs.

_{p}or as a function of $\zeta $ for 4-, 9-, and 16-Tx for the Tx’s orientation uncertainty. It is clear from the figure that, alike the case of Tx’s position uncertainty, the effect $\zeta $ has more significant error dependence for set-ups with a lower number of Txs. It is observed that for $\zeta $ of 5°, the average positioning errors are 31.9, 20.6, and 17 cm, with the standard deviation values of 9.2, 6.3, and 3.9 cm for 4-, 9-, and 16-Tx cases, respectively. The increasing error also proves an increasing trend with $\zeta $, therefore, the placement of Tx with the accurate Tx’s position and its orientation should be taken into consideration, as the error may rise even with low values of set-up uncertainties.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

ADOA | Angle difference of arrival |

ANN | Artificial neural network |

AOA | Angle of arrival |

CDF | Cumulative distribution function |

FOV | Field of view |

GPS | Global positioning system |

HPA | Half power angle |

IoT | Internet of things |

IP | Indoor positioning |

LEDs | Light-emitting diodes |

LLS | Linear least square |

LOS | Line of sight |

NLLS | Nonlinear least square |

NLOS | Non-line of sight |

OOK | On-off keying |

PD | Photodiode |

Probability distribution function | |

PR | Polynomial regression |

RF | Radio frequency |

RFID | Radio frequency identification |

RSS | Received signal strength |

Rx | Receiver |

SNR | Signal-to-noise ratio |

TDOA | Time difference of arrival |

TOA | Time of arrival |

Tx | Transmitter |

UWB | Ultra-wide band |

VLC | Visible light communication |

VLP | Visible light positioning |

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**Figure 8.**Comparison of uniformity for different HPA and distance between the Txs in case of (

**a**) 4-Tx, and (

**b**) 9-Tx.

Polynomial Coefficient | ${\mathit{\alpha}}_{0}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\alpha}}_{4}$ |
---|---|---|---|---|---|

Value | $8.85\times {10}^{6}$ | $-9.93\times {10}^{5}$ | $3.96\times {10}^{4}$ | $-7.34\times {10}^{2}$ | 7.4 |

Parameter | Value |
---|---|

Room size | 6 × 6 × 3 m^{3} |

Number of LED Txs | 4/9/16 |

Transmit power of each Tx | 1 W |

Rx’s field of view | 75° |

Reflection coefficient | 0.7 |

Area of PD | 10^{−4} m^{2} |

Responsivity of PD | 0.5 A/W |

Tx elevation | −90° |

Tx azimuth | 0° |

Rx elevation | 90° |

Rx azimuth | 0° |

Number of Txs | Uniformity | Distance between Txs (m) | HPA (°) |
---|---|---|---|

4 | 0.65 | 2.4 | 60 |

9 | 0.55 | 1.3 | 60 |

16 | 0.65 | 1.34 | 40 |

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**MDPI and ACS Style**

Chaudhary, N.; Alves, L.N.; Ghassemlooy, Z.
Impact of Transmitter Positioning and Orientation Uncertainty on RSS-Based Visible Light Positioning Accuracy. *Sensors* **2021**, *21*, 3044.
https://doi.org/10.3390/s21093044

**AMA Style**

Chaudhary N, Alves LN, Ghassemlooy Z.
Impact of Transmitter Positioning and Orientation Uncertainty on RSS-Based Visible Light Positioning Accuracy. *Sensors*. 2021; 21(9):3044.
https://doi.org/10.3390/s21093044

**Chicago/Turabian Style**

Chaudhary, Neha, Luis Nero Alves, and Zabih Ghassemlooy.
2021. "Impact of Transmitter Positioning and Orientation Uncertainty on RSS-Based Visible Light Positioning Accuracy" *Sensors* 21, no. 9: 3044.
https://doi.org/10.3390/s21093044