Impact of a Photodiode’s Angular Characteristics on RSS-Based VLP Accuracy

Photodiode (PD)-based Visible Light Positioning (VLP)-based localisation systems seem propitious for the low-cost tracking and route-configurable navigation of automated guided vehicles, found in warehouse settings. Delivering the required high accuracy, currently necessitates measuring and fitting the received power - distance relation. This paper shows that accurately modelling the PD receiver’s angular characteristics obsoletes this calibrating fit, while still providing accurate positioning estimates. A new responsivity model <italic>Square (SQ)</italic> is proposed, which is a function of the square of the incidence angle rather than its cosine. Both its aptitude in matching real-life propagation and its associated localisation accuracy are verified using two extensive measurement sets, each detailing the propagation of a PD moving across a 2D plane 3 m below a 4-LED plane. <italic>SQ</italic> is compared to the responsivity and calibration fit models available in the literature. In conjunction with model-based fingerprinting positioning, <italic>SQ</italic> outscores the Lambertian and generalised Lambertian model in terms of the 90<sup>th</sup> percentile root-mean-square error (rMSE) <inline-formula> <tex-math notation="LaTeX">$p_{90}$ </tex-math></inline-formula> by 45.36 cm (83.1%) and 0.84 cm (8.4%) respectively for the non-Lambertian-like receiver. <inline-formula> <tex-math notation="LaTeX">$SQ$ </tex-math></inline-formula> exhibits an equivalent performance as the generalised Lambertian model for the Lambertian-like photodiode. Accounting for the appropriate receiver model can also boost trilateration’s rMSE. A 50<sup>th</sup> percentile rMSE reduction of respectively 1.87 cm and 2.66 cm is found in the setup.


I. INTRODUCTION A. INDOOR LOCATION TRACKING
The ever-growing degree of automation, particularly in industry, prompts a rapid innovation in indoor localisation. Precise indoor (location) tracking systems should enable a more efficient, more productive and human-safer value chain [1]. A promising indoor tracking application is found in the autonomous 2D navigation of automated guided vehicles (AGVs). Envisioned use cases are e.g. an AGV transporting goods across the factory floor, and an indoor picking robot manoeuvring itself in front of ripe fruit. For highly accurate 2D navigation, most current systems employ either slotted wires embedded in the floor or lasers. Both techniques bring The associate editor coordinating the review of this manuscript and approving it for publication was Wen Chen . at least one distinct restriction with. The use of slotted wires makes it difficult to dynamically adapt the AGV's routes. Highly-accurate lasers then again add a substantial cost per vehicle and require a reflector infrastructure that is often perceived as disturbing. In addition, recently, the demand for indoor location tracking systems tuned to (miniature) unmanned aerial vehicles (UAVs) has been skyrocketing as well [2].

B. INDOOR LOCALISATION USING VISIBLE LIGHT
New indoor positioning technologies are being evaluated, in an ongoing search for localisation systems that empower low-cost AGVs and UAVs that are self-driving/self-flying along any route. Visible Light Positioning (VLP) might be a well-suited technology for these applications [3].
It employs a photodiode-(PD) or image sensor-based receiver to infer positioning estimates based on the interpretation of signals embedded in the visible light originating from a light-emitting diode (LED) infrastructure. Considering both that LEDs are regarded as the light sources of the future, and that they can simultaneously serve for positioning and illumination, VLP should manage to deliver cost-effective systems. As VLP is also able to deliver outstanding (i.e. sub-decimetre) positioning accuracies [4], the technology allows to reap the benefits of automation with relative ease.
Although not yet a mature technology, market intelligence companies, interestingly, already forecast that VLP systems might be the key solutions to unlock the highly-valued indoor location market.
The future will tell whether the dominant VLP system for 2D AGV or 3D UAV tracking will be (smartphone) camera-or photodiode-based. Fact is that both allow accurate positioning. In [5], the authors obtain a post-processed average error of respectively 5.98 cm and 8.68 cm in x and y-direction, when localising a commercial smartphone camera in 82 points underneath a high LED-density setup. Nakazawa et al. [6] equipped a 256 by 240 pixels camera with a fish-eye lens in order to achieve a reported minimal horizontal accuracy of 10 cm in a 5.4 m × 7.5 m × 3 m room.
However, in applications that are energy-or costconstrained, but still require a high positioning refresh rate and a significant robustness against stroboscopic or flicker effects (e.g. 3D drone navigation), photodiode-based VLP is favoured. Wang et al. [7] proposed to append a PD to a smartphone. Their experiments reported a 90 th percentile root-mean-square error (rMSE) p 90 of approximately 0.9 m, 1.1 m and 1.8 m in an open-plan office, a cubicle and a corridor environment, respectively. Wang et al. do first require performing a set of measurements to fit the received light intensity as a function of the LED-receiver distance d.

C. PROBLEM STATEMENT AND RELATED WORKS
Actually, many works on received signal strength (RSS)based VLP necessitate the use of an (arbitrary) calibration fit of the RSS − d relation to provide accurate positioning estimates. The fitting can be either direct via curve fitting [8]- [11] or indirect via machine learning models [12]- [14].
The 16 considered measurement points to test the orthogonal frequency-division multiplexing access (OFDMA)-based quadrature phase-shift keying (QPSK) VLP system proposed in [8], also served as reference points to fit a power law RSS − d relation. In a small-scale setup of 20 cm by 20 cm by 15 cm, a mean positioning error 1.68 cm was achieved. A centimetre order positioning accuracy was also reported by Zheng et al. in [9] when localising 36 points in a 1 m by 1.19 m by 1.29 m roll-out, also employing a (Lambertian) power RSS − d relation. Alam et al. [10]  Machine learning methods fit the RSS − d relation on a (sparse) collection of training fingerprints. References [12], [13] both ensure accurate localisation using feed-forward neural networks (FNNs). The authors of [12] obtain a factor 7.6 accuracy enhancement with respect to traditional trilateration. Du et al. put forward a hybrid approach [14]. To avoid the tedious gathering of a grid of fingerprints, a minimum of 2 reference points serve to fit the power law RSS − d relation. Both the trained 5-hidden layer FNN-based and the trilateration-based 3D localisation then yield approximately the same p 90 of 12 cm in a 1.2 m × 1.2 m × 2 m space. Combining the results of the latter 2 studies [12], [14] implicitly indicates the disparity between the measured and modelled RSS − d relation.
The requirement of directly or indirectly fitting the propagation model is a consequence of an insufficient characterisation of the VLP channel. Furthermore, the RSS − d fit generally increasingly diverges from the RSS − d measurements for larger d, limiting its general applicability.

D. PAPER CONTENT AND STRUCTURE
This paper shows that well-modelling the PD receiver's angular characteristics i.e. responsivity or acceptance obsoletes this calibrating fit, while still providing accurate trilaterationand model-based fingerprinting (MBF) [15] positioning estimates. Measurements of the receiver's angular dependence for two photodiodes will justify the proposition of a new responsivity model that is a function of the square of the incidence angle rather than its cosine. Extensive measurement sets of the Line-Of-Sight (LOS) propagation of two commercial receivers covering 151 2 points across a 2D plane 3 m below 4 LEDs, allow comparing the performance of a responsivity model with various calibration fit approaches for 2D or 3D trilateration [16] and MBF. While in trilateration, the requirement for an invertible channel model (RSS − d) restrains the complexity of the responsivity model, MBF does allow the application of arbitrary models [17]. As such, MBF permits exploring the trade-off in model complexity and positioning performance. A following simulation section treats the influence that the shape of the receiver's responsivity has on the positioning accuracy.
The main contributions of this paper can be summarised to: • An experimental study of a photodiode-based receiver's angular dependency and its impact on the VLP channel and the associated 2D/3D RSS positioning performance.
• The introduction of a new responsivity model, in which a square dependence on the incidence angle is assumed. It is shown that this model obsoletes the calibrating fit VOLUME 8, 2020 that is often required, and that in conjunction with MBF, accurate 2D localisation is ensured. This paper extends the receiver acceptance analysis of [17] by performing both additional simulations and single LED experiments in order to verify (and study the parameters of) existing acceptance models. Moreover, new models are proposed that are better suited for (trilateration-based) RSS VLP, and the impact of the receiver's responsivity on the 2D and 3D positioning accuracy is reported for the first time.
The subsequent parts of this paper are grouped in 3 sections. Section II describes the background of, and the materials and methods used throughout the work. Sections II-A to II-E describe respectively the VLP channel model, the available and newly proposed receiver responsivity/acceptance models and the calibrating fits, the positioning algorithms and the measurement setup used. Section III discusses the work's results. Section III-A discusses the result of the propagation measurements performed with both commercial receivers. It looks at how well the treated propagation models, associated with the responsivity and fit models, match with the measured channel. Section III-B builds on Section III-A by now considering the positioning performance of both receivers in LOS conditions. Section III-C discusses the short simulation study on the responsivity shape. Finally, Section IV lists the most important conclusions, and denotes the future work.

A. CHANNEL MODELLING
Presently, the most used VLP channel models in literature still remain the IR-propagation models of Kahn and Barry [18]. Supposing the channel to both have a flat frequency characteristic and be time-invariant stationary, the LOS propagation between a PD and K incoherent white chip-on-board (COB) LEDs LED i , i = 1 . . . K can be described using a DC channel gain h c relates the received radiant flux/power P R at the photodiode to the transmitted radiant flux/power P t,i of the LED, via: where P R,i are demultiplexable using appropriate (de)modulation techniques [19], [20] and where h (i) denote respectively the LED-PD distance, the PD's active area, the receiver field of view, the incidence angle from LED i , the azimuth angle of irradiance in the horizontal LED plane, the irradiance angle and LED i 's radiation pattern. As approximately-point source LEDs are generally assumed to be ideal Lambertian radiators (of order m i = 1): . If the PD receiver can be approximated to exhibit Lambertian behaviour with order m R : R P (ψ) = cos m R (ψ), with m R often equal to 1. T S (ψ) and T R (ψ) represent the receiver's optical filter and concentrator gain respectively.
In [21], Miramirkhani et al. described the discrepancies between IR and VL propagation. Cossu et al. in [22] showed that the R P (ψ) of the photodiode under consideration (Hamamatsu's C5331-11) started diverging from cos m R (ψ) for |ψ| > 55 • , when nor a filter nor a lens were used (T S (ψ) = T R (ψ) = 1). As a result, new or addendum models have been published. In [23], Kim et al. propose to generalise the R E (φ i ), R P (ψ i ), T S (ψ i ) and T R (ψ i ) contributions, so that P R,i satisfies: C opt,i is the optical power constant (absorbing the influence of A R and R E (0)), while G t (φ) and G R (ψ) denote the transmitter and receiver gain dependencies. When G t (φ) = cos m t (φ) and G R (ψ) = cos m R (ψ), (2) reduces to (1) and can be interpreted as a generalised Lambertian propagation model. Kim et al. [23] also proposed a more general form for G t (φ) and G R (ψ): G t (φ) and G R (ψ) are now a function of a slope constant (S t /S R for the LED/PD lens respectively) and a semi-angle (φ1 /2 and ψ1 /2 for the LED/PD respectively). R P (ψ) collects the ψ-dependencies and is dubbed the angular acceptance or normalised responsivity: R P (ψ) = T S (ψ) · T R (ψ) · R P (ψ) in (1) and R P (ψ) = G R (ψ) in (2).

B. ANGULAR ACCEPTANCE/RESPONSIVITY MODELS
In response to incident light, the PD generates photocurrent contributions {I PD,i }. The observed {P R,i } values are related to {I PD,i } by means of the photodiode's nominal responsivity R P P R,i : As reverse-biased PIN PDs have an extremely linear P R,i − I PD,i response, while P R,i is below saturation, R P (P R,i ) can be assumed to be independent of P R,i : R P (P R,i ) = R P (0). A gain mismatch factor M is introduced to account for the wavelength λ mismatch between transmitter (R E (0)) and receiver (R P (0)). M ensures that a wavelength λ-abstracted channel model approach remains valid. In this paper, all receiver dependencies (exempt A R ) are collected in the term responsivity R P (ψ), with R P (ψ) = R P (0) · R P (ψ). This paper focusses solely on the impact of R P (ψ) on the 2D/3D RSS VLP's positioning performance. Obviously, R P (ψ) is derived from R P (ψ) by division by the maximum responsivity R P (0).
Based on Section II-A, the following R P (ψ) models can be identified: (a) the standard Lambertian model R P (ψ) = cos(ψ) with acronym m = 1, (b) the generalised Lambertian model R P (ψ) = cos m R (ψ) denoted by m R , (c) the exponential model R P (ψ) (see (3)) dubbed Exp. In m = 1 and m R , T S (ψ) and T R (ψ) are equal to 1 as both the LEDs and the PD are not explicitly equipped with a lens. Technically, Exp, also describes a lens, but is here considered as a ψ dependence.
This paper also proposes (d) a new quadratic R P (ψ) model, denoted as Square (SQ) and presented in (5).
in which ψ 3dB represents the ψ for which R P (ψ) = 1 / √ 2 (i.e. a 3 dB electrical power decrease). |ψ C | is the minimum of the receiver field of view (FOV) and SQ displays a steeper R P (ψ) gradient when ψ approaches |ψ C | as to better cope with the cos m R (ψ) model's divergence for |ψ| > 55 • . SQ is noninvertible, which will hinder straightforward application in trilateration-based positioning (see hereinbelow in Section II-D2). Therefore, an approximated (e) SQ approx R P (ψ) model is proposed: with |ψ C | the minimum of the receiver field of view (FOV) and acos 1 − ψ 2 3dB / 2− √ 2 . Importantly, all treated R P (ψ) models are LED independent and the photodiode is operating in overfill condition as is expected in optical wireless. Each of the 5 R P (ψ) models' aptitude in matching the measured R P (ψ) behaviour will be evaluated in Section III in terms of the coefficient of determination R 2 (and later the positioning performance).

C. CALIBRATING FITS OF THE CHANNEL MODEL
Ideally, all parameters in the described channel models (see Section II-A) can be predetermined and are constant per LED and receiver type. In practice however, as a consequence of the channel models diverging from real-life measurements due to both variations on the LEDs' radiation pattern and the improper modelling of R P (ψ) [22], many works in literature calibrate certain channel parameters, often even per LED. Off-line, on-site measurements serve to fit the P R,i − d relation [7]- [10]. In [9], [10], the authors simplify (1) to a power law P R,i − d i relationship ν i : d i,0 and P R,i,0 denote the calibration distance d i and received radiant power P R,i respectively. ν i represents the generalised Lambertian modelling of the LED m and PD ν i (i.e. ν i = 2 + m+ν i ). ν i , see (7), can be generalised to P R,i = α i · d β i i , which will be referred to by Power. Some works even calibrate a seemingly more arbitrary propagation model, e.g. in [7] a second order polynomial fit is worked with. The drawback of 'fitting' lies in the recalibration effort needed when either the LED, the PD or the environment parameters change.

D. POSITIONING
The propagation models of the previous sections can be used to infer positioning, when combined with the algorithms discussed next. The receiver position is denoted by (x u , y u , z u ). A prerequisite for accurate positioning is that the LED coordinates (x S,i , y S,i , z S,i ) are meticulously known.

1) MODEL-BASED FINGERPRINTING (MBF)
Model-based fingerprinting RSS VLP employs a precomputed propagation map, accounting for the relevant channel characteristics. It holds the RSS values per LED, i.e. {P R,i }, for all locations on a predefined positioning grid. Upon a set of {P R,i } measurements { P R,i }, the grid position that has the closest, in terms of rMSE, set of modelled {P R,i } values to the measured { P R,i } is taken to be the positioning estimate [15]. The additional propagation map dimension needed for (realtime) 3D localisation (whether or not in the presence of receiver tilt) renders MBF infeasible (in terms of latency and the required storage) for e.g. drone localisation.

2) TRILATERATION a: 2D TRILATERATION
Trilateration-based RSS estimates the distances d i directly from P R,i by inverting one of the above propagation models. Afterwards, the d i are translated into a location estimate of (x u , y u , z u ) by means of the well-known least-squares matrix formalism (as used in [24]). The invertible channel model requirement restrains the complexity of the employed P R,id i relation, and thus of the R P (ψ) model.

b: 3D TRILATERATION
In 3D trilateration, z u is no longer a priori known and needs to be estimated as well. In this paper, the localisation algorithm of Plets et al. is employed. The algorithm performs an iterative evaluation of the 2D procedure of hereinabove and is detailed in [16].

E. MEASUREMENT SETUP
To both characterise R P (ψ) for two commercial Thorlabs PDs, namely Thorlabs' PDA100A2 1 or PDA36A2, 2 and study the effect of R P (ψ) on the positioning performance, it is required to use two different measurement setups. The former setup is detailed in Section II-E1, the latter in Section II-E2. Common to both experiments, the considered PD's photocurrents are amplified with a transimpedance gain G PD to photovoltages, which are then N times sampled at a rate f S using National Instrument's USB-6212 DAQ Device. A MATLAB R backend then processes the measurement. The characteristics of both photodiodes are listed in Table 1. The COB LEDs utilised in the experiments are of the Bridgelux BXRE-50C3001-D-24 3 type.  Fig. 1 shows the R P (ψ) measuring setup, consisting of a single LED (highlighted in purple) that looms directly over the centre point of a receiving platform (highlighted in green). This in-house-made platform consists of a linear actuator, moving the tilting structure on top accurately up and down (in the z-direction). The schematic representation of the tilter is visualised in Fig. 2. By placing either of the PDs on top of a Thorlabs GNL20/M 4 goniometer, itself fixed on the tilter, ψ can be accurately set. The GNL20/M fine-grainedly tunes ψ between [−10 • , 10 • ] around a ψ set point determined by the tilter. As the tilter not only tilts the PD's normal, but also displaces the PD at each tilter setting, the receiving platform moves a predefined (according to the laws of trigonometry) distance both vertically and horizontally to keep the PD at the same 3D location. The tilter set points are

1) SINGLE LED EXPERIMENT FOR R P (ψ)
The LED-PD distance amounts to 1.25 m. The centre point of the receiving platform coincides with the gravity centre of the PD's active area. G PD here is a tenth of the values of Table 1. The BXRE LED is transmitting DC light in response to a DC driving current magnitude of 750 mA. f S = 256 kHz, N = 51.2 kS and 50 photocurrent waveforms are averaged. The obtained R P (ψ) dataset is referred to as Point, as R P (ψ) is measured for 1 receiver location, underneath the LED.

2) LOCALISATION MEASUREMENT SETUP
The extent of the impact of the normalised responsivity R P (ψ) models on the VLP channel model quality and the corresponding positioning performance is studied in WAVES' VLP lab [17]. One of either PDs is installed on top of a 2D slider (Velmex' BiSlides) that covers 1 m 2 with a uniform grid of 41 2 points. By sequentially displacing the slider 1 m, 3 https://www.bridgelux.com/sites/default/files/resource_media/DS43% 20V15%20Array%20Data%20Sheet%20Rev%20E%2020160606.pdf 4 https://www.thorlabs.com/thorproduct.cfm?partnumber=GNL20/M   as specified in [19]. Per measurement location, 10 I PD,i values are averaged to reduce the noise impact. Fig. 4 depicts the schematic representation of this receiver chain. At a height of 3 m above the receiver plane, a nearly square-shape LED constellation of (K = 4) BXRE COBs is installed. These LEDs are intensity-modulated as to transmit pulse trains (with current magnitudes ≈ [600, 750, 750, 750] (mA)) with frequencies f c,i and with duty cycle δ i = 0.5 [25]. To demultiplex the contributions of the different LEDs at the receiver side, f c,i are chosen to satisfy: f c,i = 2 i−1 f 0 [19]. f 0 is set to 500 Hz to exceed the flicker threshold. The LED coordinates are depicted in Fig. 3 Rearranging the expression of (1), also allows deriving R P (ψ) from the I PD,i measurements. The LEDs' measurement data I PD,i is overlaid, compensated by R E (φ i , γ i ) and d 2 i , and then per LED normalised. The resulting R P (ψ) dataset is named Plane, as R P (ψ) is obtained by measuring spatially distributed receiver locations i.e. receiver locations on the ground plane.

A. CHANNEL CHARACTERISATION
In order to both model the receivers' (angular) characteristics and study their impact on I PD,i − d i and thus on the positioning, the propagation is measured across the 2D receiver plane. The receivers' normal n R is assumed to remain equal to [0 0 1]. The (demodulated) received photocurrents I PD,i are shown in Fig. 5, for LED 3 and respectively the (a) PDA100A2 and (c) PDA36A2.

1) CALIBRATION OF MEASUREMENT SETUP IMPERFECTIONS
To limit the influence of all non-responsivity channel contributions in deriving R P (ψ) and in the positioning, additional calibration steps are required.

a: RADIATION PATTERN
In trilateration, R E (φ i , γ i ) is assumed to be Lambertian (generally with order m = 1). Fig. 6 shows by means of the C0, C45 and C90 slices of BXRE-50C3001-D-24's C0/C90 photometric diagram that R E (φ i , γ i ) slightly deviates from the best fitting Lambertian radiator with m = 1.138, which is depicted in black. During MBF localisation and the derivation of R P (ψ), the tabulated R E (φ i , γ i ) is used.

b: TRANSMITTER TILT
Nonnegligble LED tilts present in the lab setup, arising from manually installing them on rails, need compensation. The most likely LED normal is found where I PD,i · d 3 i is maximal in the receiver plane [26]. A more robust method, accounting for outliers, entails taking the I PD,i · d 3 i -weighted centroid of all locations with an I PD,i · d 3 i larger than the 95 th percentile of all I PD,i · d 3 i . The centroids are highlighted in red in Fig. 5 (b) and indicate the intersections of the LED normals and the receiver plane. The tilt is then compensated for by sequentially dividing the measurement data I PD,i on each grid location (x, y) by cos(φ m (x, y)) and then multiplying it by cos(φ id (x, y)) = (z S,i −h) /d i (x,y). φ m (x, y) and h represent the measured irradiance angle (with respect to the tilted LED normals {n S,i }) and the z-coordinate of the PD [17]. c: RADIANT FLUX P t ,i P t,i also requires a calibration, as it is never exactly known without measuring [27]. Given M and R P (0), {P t,i } are calibrated directly beneath each LED after accounting for the transmitter tilt. The calibrated P t,i satisfy approximately P t,1 · [1 1.245 1.225 1.2]. P t,i | i>1 < 1.25 P t,1 can (partly) be attributed to the LED driving current -P t,i relation starting to saturate (see Section II-B). In addition, the variation in P t,i will impede propagation models that assume a LED independent P t,i = P t , i = 2..K .
2) NORMALISED RESPONSIVITY R P (ψ) Fig. 7 depicts the normalised receiver responsivity R P (ψ) measurements of both Thorlabs receivers, over the incidence angle ψ. A clear disparity in R P (ψ) between both receivers is VOLUME 8, 2020   noticed. PDA36A2's R P (ψ) is depicted in the green shades, PDA100A2's in the purple ones. The lighter and darker shades represent the data from the 2D Plane measurement setup (see Section II-E2) and the single LED tilting Point setup (Section II-E1) respectively.

a: EXISTING R P (ψ) MODELS
The R P (ψ) acceptance models of Section II-B are fit on the Plane R P (ψ) measurement data and are subsequently depicted in Fig. 8. Plane is deemed more reliable (than Point) due to the manual operation of the tilter. It furthermore allows comparing the R P (ψ) models with the calibrating fits of Section II-C.
(b) The generalised Lambertian model dubbed m R , with fitted m R = 0.98/1.9 visualised in eucalyptus (dark) green, improves R 2 to 0.991/0.986. It particularly fits PDA36A2's R P (ψ) well. (a) and (b) allow concluding that the PDA36A2 exhibits a rather Lambertian-like responsivity behaviour (i.e. m R ∼ 1), while, in contrast, the PDA100A2 does not. PDA100A2' divergence from the ideal Lambertian response can be partly attributed to inherent photodiode physics [28], [29]. However, it is mainly due to additional internal reflections or blockage instilled by the packaging. At large ψ, both m = 1 and m R gradually saturate towards ψ = π /2. The measured R P (ψ) noticeably differs by dropping to zero (or equivalently, the noise floor).
(c) Exp, the exponential form of G R (ψ) of (3) shown in orange, identifies ψ1 /2 to be 1.06/0.80 rad and then fits S R = 2.3/2.39. Exp's consistent fitted parameters and R 2 scores (R 2 = 0.988/0.987) entails it being a more robust, universally applicable model. For ψ exceeding 1.3 rad, Exp starts diverging more and more from the Point measured data. When R P (ψ) ≈ 1 /2 is inadequately known (see Fig. 8), Exp may encounter difficulties. In [17] therefore, a slightly modified model was introduced, which utilises the ''three-quarter'' angle to deliver comparable R 2 scores.
(d) Rated at R 2 = 0.991/0.993, SQ exceedingly outperforms the m R model, certainly at ψ approaching |ψ C |. SQ is more able to account for the PDA100A2-receiver's behaviour, due to it having a larger R P (ψ) gradient when ψ → |ψ C |. It is displayed in curious (darkest) blue. ψ 3dB amounts to 0.79 and 0.61 rad respectively. Moreover, no fit is required. (e) The associated SQ approx model, shown in light steel (middle) blue, is seemingly unsuccessful in delivering the high R 2 scores (R 2 = 0.983/0.982) required.

b: OTHER R P (ψ) MODELS
The angular pseudospectrum of the MUltiple SIgnal Classification (MUSIC) algorithm [30] with R P (ψ) as input, shows a single and distinct peak indicating that R P (ψ) could be modelled by a cosine function R P (ψ) = cos(a · ψ). This model is named cos(ax). a is a scaling factor. It is returned by identifying the pseudospectrum peak. cos(ax) is plotted in red and shows that SQ fits R P (ψ) at least as well as an arbitrary cosine function. It can be remarked that all R P (ψ) models increasingly diverge from the measured data, for ψ ∼ |ψ C |. For SQ though, this divergence is mainly restrained to the neighbourhood of |ψ C | i.e. where R P (ψ) saturates. At these higher ψ, Fresnel losses manifest itself. Averaging the Fresnel s and p-polarisation equations, allows adding a Fresnel loss contribution to the R P (ψ) models. The reflection coefficient of silicon (= 4.22) is obtained by weighing the refractive index data of Schinke et al. [31] with the BXRE LEDs' wavelength spectrum. Fig. 8 shows the additional Fresnel loss impact (at high ψ) in conjunction with the m = 1 model in riptide (middle) green (i.e. the curve denoted with m = 1, FR). Adding the Fresnel losses to the SQ shifts its infliction point (i.e. |ψ C |) even closer to the actual measured R P (ψ)'s. All models' resulting coefficients of determination are collected in Table 2.
The analysis performed in this section suggests that the SQ acceptance model best approximates the measured R P (ψ). In Section III-B1, the model's goodness of fit is translated into a positioning accuracy.

3) CALIBRATING FITS
The R P (ψ) models of the previous section are LED independent. As stated in Section II-C, in VLP literature, the propagation models are frequently calibrated per LED. This subsection effectuates the fitting of the models of Section II-C on the measured received photocurrent I PD,idistance d i relation as to ready them for application for positioning. In addition, it is investigated how the SQ model fares in comparison with the calibration fits. Fig. 9 (a) and (b) depict the measured (and tiltcompensated) I PD,1d 1 (i = 1) relation in grey diamonds for respectively the PDA36A2 and the PDA100A2. Noise, interference and small measurement/setup errors (e.g. imperfectly compensated tilt) cause the curves to widen with increasing inter-PD-LED distance d. 5 calibrated models are considered (see Section II-C) and depicted in Fig. 9: (i) ν i = 1, (ii) ν i tilt, (iii) ν i , (iv) ν i = ν and (v) Power. (i) ν i = 1 is the baseline model, in which the LED and the PD receiver are assumed to be Lambertian with order 1.138 (see Section III-A1) and order 1 respectively. Its mean R 2 over the 4 LEDs equals for the PDA36A2 and PDA100A2 respectively 0.996/0.965. ν i = 1 corresponds to the (a) m = 1 R P (ψ) responsivity model. The (ii) ν i tilt and (iii) ν i are generalised models in the sense that they fit a joint exponent containing both receiver and LED ψ dependence (both fit the ν i exponent in (7)). (ii) ν i tilt is the only model that fits the uncompensated data. (ii) ν i tilt and (iii) ν i both push the R 2 score towards 0.998 for both PDs. The mean ν i is 4.01/4.87 for the PDA36A2/PDA100A2. In [17], it was shown that propagation models whichever hypothesise P t,i = P t lead to an inferior I PD,i − d i fit. Optimising a single ν i jointly over all LEDs leads to the (iv) ν i = ν, which scores comparatively at 0.9978/0.9976 in terms of R 2 . The last model Power achieves the highest R 2 = 0.9981/0.9981. It should deliver a model well-suited for trilateration as d i is easily gathered from I PD,i . Power does require to fit 2 degrees of freedom instead of 1. The parameters are obtained by employing the curve fitting tool cftool of MATLAB R (i.e. nonlinear least squares minimisation). The mean exponents β i are -3.93 and -4.9 and emphasise the difference between both PDs. All these calibration model parameters are fit by minimising the rMSE between the modelled and measured data and hence correspond to fitting a practical propagation model in situ [10]. Each model's mean coefficient of determination R 2 with respect to the measurement data is listed in Table 3 for overview. Importantly though, the fitted I PD,id i relations diverge more and more from the measured ones with increasing distance d i . This behaviour (also treated in the Section III-A2) will, depending on the VLP roll-out situation, significantly impact the positioning accuracy. To investigate whether adequately modelling the nominal receiver responsivity will spare the effort of calibration-based fitting, the SQ model's associated I PD,1 -d 1 is plotted in blue in Fig. 9 as well. Fig. 9 demonstrates that SQ better approximates the measured behaviour at higher d, due to SQ being well-suited for ψ → |ψ C |. The R 2 scores slightly diminish however to 0.9972 and 0.9978. As these scores are comparable to the ν i model's SQ should deliver accurate localisation estimates.

B. LOCALISATION PERFORMANCE
This section evaluates the aptitude of the different propagation models of the previous section in terms of their associated positioning accuracy found in the VLP roll-out configuration (Section II-E2). Section III-B1 treats model-based fingerprinting (MBF) localisation, whilst Section III-B2 details the trilateration-based positioning results.

1) MODEL-BASED FINGERPRINTING (MBF)
MBF's propagation map has a 1 cm grid resolution and accounts for the BXRE-50C3001-D-24s' tabulated R E (φ, γ ) and the measured tilt normal. The 3 strongest LEDs, in terms of I PD,i/P t,i , are used in the positioning process (as this performs better), inadvertently limiting the maximal ψ i found at the receiver. Part of MBF's benefit is that it can account for arbitrary and/or irregular propagation models, arising from arbitrary LED radiation and PD normalised responsivity patterns.
a: MBF AND R P (ψ) Fig. 10 shows the cumulative distribution (CDF) of the rMSE found during MBF localisation associated with the R P (ψ) models of Section III-A2. In addition, Table 4 shows the 50 th percentile p 50 , the 75 th percentile p 75 and the 90 th percentile p 90 rMSE.
The black curve CDF substantiates the hypothesis of Section III-A2 in that the baseline m = 1 model is unable to supply sufficient localisation accuracy for both Lambertian-like (with p 50 = 5.85 cm) and non-Lambertianlike photodiode receivers (with p 50 = 33.97 cm). m R significantly improves upon m = 1 and shifts m = 1 over approximately 30 mm for the PDA36A2 and 82% for the PDA100A2 (see the eucalyptus (dark) green curve). Although [23] showed the rationale for Exp (displayed in orange), this work finds m R to outscore Exp. The remark has to be made though that the respective photodiodes are not equipped with lenses.
SQ displays a positioning error function in curious (dark) blue in Fig. 10 that confirms the data of Table 2. Across both receivers, SQ leads to the optimal rMSE. For the PDA36A2, SQ exhibits an equivalent performance to the m R model: p 50 = 5.52 cm (versus 5.59 cm), p 75 = 7.5 cm (versus 7.5 cm) and p 90 = 9.39 cm (versus 9.18 cm). For the PDA100A2-based receiver, SQ has a pronounced performance gain e.g. in terms of p 90 : a 45.36 cm (83.1%) gain  Approximating SQ to SQ approx curtails some of the positioning rMSE gain, particularly for higher percentiles (> 75%). SQ approx 's CDF resembles the scaled cosine cos(ax) model's. It dips below the m R 's up to approximately the 60 th percentile. It even beats SQ, but solely for the PDA36A2 in the sub-60% range. Beyond 60%, SQ approx 's rMSE rises more rapidly than the rMSE of m R and SQ. The dashed line represents the positioning performance, when MBF uses the measured Point R P (ψ) data directly (i.e. a lookup table). It serves as a reference. Finally, Fig. 10 also shows that the ν i fit not necessarily performs better than SQ. In fact, well-modelling R P (ψ) should obsolete the need for calibration fitting (see hereinbelow).
As a general conclusion, highly-accurate localisation really requires incorporating a well-designed R P (ψ) model i.e. the m = 1 model simply does not perform well enough. Of the models considered, SQ scores best in both its aptitude to model the measured R P (ψ) and its associated positioning performance.

b: MBF AND THE CALIBRATING FITS
Section III-B1 finishes by revisiting the calibrating fits of Section III-A3. Fig. 11 displays the MBF rMSE CDF for the 5 fitted propagation models, as well as the CDF of SQ and ν i = 1, Lamb. ν i = 1 represents the model with ν i = 1 and R E (φ, γ ) the tabulated radiation pattern, while ν i = 1, Lamb denotes the propagation model where ν i = 1 and R E (φ, γ ) is approximated by the best-fitting Lambertian (i.e. with order m = 1.138). Comparing with the baseline ν i = 1 shown in black, the dotted line of ν i = 1, Lamb demonstrates the for the PDA36A2 substantial added error cost when the Lambertian radiation pattern is employed. The comparison hints at one of the strengths of MBF, which is that it can easily account for non-Lambertian R E (φ, γ ).
The subsequent models all rely on a Lambertian radiator. ν i and ν i tilt have a comparable positioning error, exceeding the Power and ν i = ν. Interestingly, the latter models equalled or exceeded their mean R 2 values. ν i improves upon ν i = 1, Lamb at the cost of additional rMSE outliers (from the 93 th percentile on). ν i 's rMSE is in the same ball park as SQ's in 50 th of the grid points. ν i also ensures a relatively accurate localisation for the PDA100A2. As a general conclusion, SQ outperforms the calibrating fits. The fits even exhibit a performance decrease compared to the baseline ν i due to the Lambertian approximation and overfitting on I PD,i -d i .

2) 2D AND 3D TRILATERATION a: 2D TRILATERATION
In contrast to the fits of the previous paragraph, not all R P (ψ) models are directly invertible for use in trilateration. The Lambertian transmitter-based propagation models associated with m = 1, m R , ν i and ν i = ν are readily inverted. Their 2D trilateration CDF (on tilt compensated data) is plotted in black, eucalyptus (dark) green, red and orange respectively in Fig. 12. The m R and SQ MBF plots are repeated in green grey and white lilac for comparison. For completeness,  'm R tilt' shows the CDF curve that would result from effectuating trilateration with the m R R P (ψ)-based on the nontilt-compensated dataset as well. Comparing 'm R tilt' to m R allows to conclude on the importance of the tilt correction.
ν i obviously outperforms ν i = ν due to it fitting an exponent per LED. As both fit all available measurement data, ν i = ν is deemed impractical. ν i outscores m R in approximately 75% of the grid points, though at the cost of more exuberant outliers.
As a rule, trilateration-based VLP performs inferior to its MBF variant. Some use-cases e.g. those requiring low latency and/or a high positioning estimate update rate, do necessitate the use of trilateration. 3D-drone positioning is a prime example. Improving trilateration-based VLP is hence of paramount importance. This paper considers two potential approaches: Tril SQ Corr and Tril SQ approx . The 2D positioning results of which are illustrated in the two darker blue shades.
Tril SQ Corr entails running m R -based trilateration. The obtained position estimate is then translated into {ψ}, the set of incidence angles at the estimated position. New incidence angles {ψ } are searched for which (5) equals cos m R (ψ) i.e. by solving a quadratic equation. The associated SQ R P (ψ ) is then used to recompute the PD-LED distance d i . This approach should enable trilateration-based VLP with an arbitrary R P (ψ) model (which could even be given by means of a Lookup table). Tril SQ Corr improves m R 's rMSE but a little bit, and is (in its current form) not required the additional effort. An iterative extension is part of the future work.
Tril SQ approx on the other hand starts from the SQ approx acceptance model. Solving for d i delivers: with h the perpendicular distance between the LED and receiver plane. (8) can be iteratively solved. For most LEDs, the best fitting Lambertian has an order close to 1. By rounding, the d 3+m  best-working trilateration approach (for the LEDs under consideration). Fig. 13 extends the previous analysis to 3D positioning. The investigated height range h is limited to [2 m, 4 m] (in steps of 2.5 cm) as to account for the height ambiguity of 3D trilateration in the presence of a square-like LED constellation [16]. Fig. 13 shows the rMSE for the 2D and 3D trilateration-based algorithms. In addition, in 3D trilateration, a distinction is made between the 2D rMSE (only computing the rMSE over the x-and y-estimates) and the 3D rMSE (also accounting for the error on the receiver height estimation).

b: 3D TRILATERATION
For the PDA36A2, SQ Corr scores equivalently to m R , both exceeding m = 1's performance for at least 90 th percent of the grid points. SQ approx works superior for around 50% of the grid locations, before the gradient of its CDF stagnates due to the distortion incurred by rounding the exponent (underestimating the LED-PD distance). After all, SQ approx is designed to operate with Lambertian radiators with a unitary Lambertian order. Interestingly, when combining m = 1 and m R with 3D trilateration, the p 50 2D rMSE values decrease by 3 mm and 2 mm respectively, compared to when 2D trilateration would be employed i.e. 3D trilateration corrects (part) of the R P (ψ) bias by incurring a height estimation error. For SQ Corr and SQ approx , this is not the case. In the PDA100A2's case, SQ Corr exceedingly outperforms the other R P (ψ). m = 1 scores disastrously. 3D trilateration's p 50 2D rMSE shaves more than 2 cm off, from the p 50 of m R and SQ Corr during 2D trilateration.
As a conclusion, adequately modelling R P (ψ) is of uttermost importance in RSS-based VLP, even for trilateration that precludes non-invertible R P (ψ) models. SQ Corr outperforms the other R P (ψ) models for an arbitrary receiver.
The 8 R P (ψ)'s associated positioning CDFs are simulated for the PDA36A2 in the idealised, virtualised VLP lab i.e. with the Lambertian LEDs (with m = 1.138) found at the coordinates displayed in Fig. 3 and with no LED tilt present. |ψ C | is set to either 65 • or 90 • , with R P (ψ) = 0, ψ ≥ |ψ C |. R P (ψ) is normalised to 1, regardless of |ψ C |. 2 receiver plane heights are considered: the ground plane (with receiver height h rec = 0 m) and h rec = 1.5 m above the ground plane. Furthermore, in the simulation the following applies: M · P t,i · R P (0) = 4 A, the Gaussian root mean square input-referred current magnitude σ amounts to 50 nA and per 2.5 cm-spread grid point 10 I PD,i measurements are averaged. Fig. 15 depicts the associated cumulative distribution of the MBF rMSE. Identifying the optimal R P (ψ) shape is far from straightforward due to the dependence on h rec (thus the typical ψ found) and |ψ C |. Let us consider (5) the ψflat rectangular shape for instance. While at h rec = 0 m, the performance is immensely underwhelming (a p 90 of 61.06 cm and 1.15 m for |ψ C | = 65 • and |ψ C | = 90 • respectively), at h rec = 1.5 m it is able to ensure accurate localisation (highlighted by a p 90 of 28.77 cm and 45.43 cm for |ψ C | = 65 • and |ψ C | = 90 • respectively). There is no need for designing complicated lens arrangements to effectuate e.g. the Laplacian behaviour, an m R shape should generally suffice. An increasing h rec demands a lower m R for optimal performance. In conclusion, per use case, an optimal R P (ψ) can be designed/chosen depending on the gradient of R P (ψ) in the found ψ range, but this choice will not be generalizable to other environments, let alone for 3D positioning. Fig. 16 also shows the influence of |ψ C | on m R = 1. Above a critical |ψ C |, which is determined by the LED type and arrangement, limiting (and normalising) the |ψ C |, increases the positioning performance. For example, when |ψ C | reduces from 60 • to 50 • , the p 90 rMSE drops from 37.96 cm to 29.66 cm. When |ψ C | is limited, not rescaled, a lower |ψ C | does not necessarily translate in performance benefits i.e. unless limiting |ψ C | decreases the interference and noise contribution of other (ambient) light sources.

IV. CONCLUSION AND FUTURE WORK
This work investigates the influence of a photodiode (PD) receiver's angular dependence R P (ψ) on the relation between the photocurrent contribution I PD,i and the receiver-transmitter distance d i . Whereas in literature RSS-based VLP systems are only able to ensure accurate localisation through (arbitrarily) fitting I PD,i − d i per LED, this paper shows that adequately modelling R P (ψ) via SQ, a proposed function of the square of the incidence angle ψ rather than its cosine, obsoletes this calibrating fit. Incorporating SQ in model-based fingerprinting, improves the p 90 by 45.36 cm (83.1%) and 0.84 cm (8.4%) with respect to the ideal m = 1 and generalised m R Lambertian acceptance models for the non-Lambertian-like PDA100A2 while rivalling the m R model's score for the Lambertian-like PDA36A2. This paper also focussed on both reconciling a noninvertertible R P (ψ) with trilateration and on the (simulated) positioning performance associated with non-standard R P (ψ) characteristics.
This work provides a first step towards VLP proprietary propagation models maximising VLP's positioning potential.
The future work concentrates itself around testing more VLP receivers on the short term, and on 3D drone localisation on the longer term. While this paper focusses on photoconductive (PIN) PDs, the applicability of the R P (ψ) models and the validity of the conclusions regarding the models' associated positioning accuracy need to be verified for other PD types used in VLP e.g. for photoelectric avalanche photodiodes. During drone flight, the induced tilt necessitates a detailed R P (ψ) modelling even more. Existing algorithms will be exerted for accurate positioning in the presence of significant receiver tilt. An experimental comparison can be made with tilt-agnostic machine learning models as well.

ACKNOWLEDGMENT
This work was executed within LEDsTrack, a research project bringing together academic researchers and industry partners. The LEDsTrack project was co-financed by imec (iMinds) and received project support from Flanders Innovation & Entrepreneurship. This paper is an extended version of our paper, titled New Photodiode Responsivity Model for RSS-based VLP, published in Proceedings of Global Lifi Congress 2019. The authors would like to thank Matthias Van den Bossche for the realisation of the tilt/displacement/rotation platform and the VLP lab.