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An optical wireless location (OWL) system is introduced for indoor positioning. The OWL system makes use of a mobile photoreceiver that facilitates triangulation by measuring angle-of-arrival (AOA) bearings from LEDs in an optical beacon grid. The photoreceiver has three photodiodes (PDs), arranged in a corner-cube, to facilitate differential photocurrent sensing of the incident light AOA, by way of azimuthal ϕand polar θ angles. The AOA error for indoor positioning is characterized empirically. Optical AOA positioning is shown to have a fundamental advantage over known optical received signal strength (RSS) positioning, as AOA estimation is insensitive to power and alignment imbalances of the optical beacon grid. The OWL system is built, and a performance comparison is carried out between optical AOA and RSS positioning. It is shown that optical AOA positioning can achieve a mean 3-D positioning error of only 5 cm. Experimental design and future prospects of optical AOA positioning are discussed.
© 2015 Optical Society of America
1. Introduction
An exponential growth in demand has been witnessed for wireless positioning systems over the past decade, and there are many technologies to facilitate wireless positioning—with the Global Positioning System (GPS) being the most well-known. The worldwide coverage of GPS supports its integration into many handheld devices, although there exist many challenges for GPS in indoor environments [1]. Indoor GPS signals can suffer from significant radio-frequency (RF) attenuation [2] and multipath distortion [3] to the detriment of indoor positioning accuracy. There is a long-standing challenge to augment GPS devices with an effective indoor technology. Optical wireless positioning has emerged to meet this need [4–10], with simplified architectures (not requiring imaging elements or processing [11, 12]) operating in two formats: time-based positioning and amplitude-based positioning.
Time-based positioning has been realized with optical time-difference-of-arrival (TDOA) and optical time-of-arrival (TOA) systems. Optical TDOA and TOA systems employ time-based trilateration with synchronized signals from three or more transmitters in an overhead optical grid [5, 6]. Such systems are an optical analogy to established RF positioning systems [14, 15]. Time/phase delays from transmitters at known positions are used via ranging to establish three-dimensional (3-D) position estimates. Park et al. have shown that optical TDOA trilateration, with indoor line-of-sight optical links having a 1 W power and 1 cm2 receiver size, can resolve 3-D positions with centimeter-scale errors [5]. The excellent TDOA and TOA accuracy comes at a cost, however, as such systems demand transmitter timing and clock synchronization—yielding positioning algorithm and hardware complexities [5–7].
Amplitude-based positioning has also been realized for optical wireless systems. These systems use optical received signal strength (RSS) positioning with amplitude-based trilateration off three or more transmitters in an optical beacon grid [8]. The technique is often referred to as optical proximity detection [16] and is analogous to standard RF RSS trilateration [17]. Signal amplitudes arriving at the receiver are weighted to form 3-D position estimates according to transmitter proximity, with close transmitters yielding large amplitudes and distant transmitters yielding small amplitudes. The transmitter and receiver hardware for this is minimal, as time-based processing is not required. Optical RSS positioning has been implemented with fixed visible [18] or infrared [8, 9] optical transmitter grids using mobile receivers, as well as fixed receiver grids using mobile transmitters or thermal sources [4]. Unfortunately, optical RSS positioning has a fundamental sensitivity to imbalances and fluctuations of optical beacon grid powers and optical channels [19]. Optical power imbalances, shadowing, fading or turbulence [19] can cause imbalanced signal strengths, which distort the relative amplitude weighting and degrade the positioning accuracy.
To avoid the synchronization complexity of time-based positioning and the imbalance sensitivity of amplitude-based positioning, this work introduces an Optical Wireless Location (OWL) system employing optical angle-of-arrival (AOA) positioning. The system introduced here complements the technologies emerging for indoor optical wireless communication (OWC) systems [20–22]. A multi-element photoreceiveris used to extract AOA estimates of incident optical beams from LEDs in an overhead optical beacon grid. Measured AOAs, defined with azimuthal ϕ and polar θ angles, are used for triangulation to establish accurate 3-D positions in environments with line-of-sight links. The use of triangulation with optical AOAs (i.e., signal bearing vectors) rather than trilateration with optical powers (i.e., signal strength scalars) minimizes the sensitivity to optical beacon power imbalances and fluctuations. The work here is a full optical wireless realization, with analyses of AOA precision [10] and geometrical dilution of precision (DOP) [11]. The findings can support future indoor localization and navigation systems [13, 23, 24].
The work is laid out as follows. Section 2 presents the photoreceiver angular and intensity responses with both direct and reflected (multipath) contributions. Section 3 defines the photoreceiver angular accuracy and precision, and evaluates the effect of geometry on positioning uncertainty. Section 4 gives a performance comparison between the optical AOA and RSS positioning techniques. Section 5 provides some concluding remarks on the presented technology and its use in future OWL systems.
2. Photoreceiver response
To facilitate AOA estimation, a photoreceiver must have a directional dependence to the incident optical beam azimuthal ϕ and polar θ angles. Such a photoreceiver is shown in the Fig. 1 inset. The photoreceiver has three orthogonal photodiodes (PDs), defined as PD1, PD2 and PD3 in the y’-z’, x’-z’, and x’-y’ planes, respectively. With this form, PD1 is preferentially sensitive to light incident along the x’-axis, PD2 is preferentially sensitive to light incident along the y’-axis, and PD3 is preferentially sensitive to light incident along the z’-axis. This primed coordinate system is the photoreceiver body frame. An incident beam from an optical beacon is characterized by an AOA in the body frame with an azimuthal angle ϕ defined in the x'y' plane and a polar angle θ defined relative to the z' axis. The AOA azimuthal ϕ and polar θ angles are determined by measuring and comparing the three PD photocurrents. The photoreceiver uses two or more AOAs to define its 3-D position in the overall navigation frame, defined by the unprimed x, y, z coordinate system.
Fig. 1 Schematics of the optical AOA positioning system with a photoreceiver establishing position estimates by way of triangulation off an optical beacon grid.
The corner-cube PD structure of the photoreceiver has three 9.7 × 9.7 mm2orthogonal silicon PDs with a 400-1100 nm wavelength range and a 0.27 A/W effective responsivity. The effective responsivity is defined by weighting the spectral responsivity curve of the PDs with the measured spectral response of the LEDs. The PDs are Thorlabs FDS1010. The LEDs are OPTEK Technology OVS5MxBCR4, having with a maximum optical power of 57mW.The three PD output photocurrents are processed for AOA computation with the photoreceiver electronics. The three PD photocurrents pass through Butterworth bandpass filters spanning 0.5 to 3 kHz (to reject ambient noise), transimpedance amplifiers with adjustable gains (to balance the photocurrents in an initial calibration procedure having uniform illumination along the central axis of symmetry), and a differencing circuit (to form differential photocurrents). The two differential photocurrents are input to a digital signal processing (DSP) unit. The DSP unit uses a fast Fourier transform to distinguish LED frequency channels, and it uses an AOA estimation algorithm (described below) to estimate azimuthal ϕ and polar θ angles. With this configuration, the photocurrents are effectively balanced and amplified, while ambient noise from sunlight (DC) and electrical sources (60 Hz) is rejected.
The photoreceiver is tested in a small-scale OWL system, with the overhead optical beacon grid shown in Fig. 1. The scale is restricted here to control contributions from the surrounding environment, through effects such as multipath reflections, and introduce these contributions in a controlled manner. Testing is carried out with two optical beacons, LEDs A1 and A2 in the figure, or with four optical beacons, LEDs B1, B2, B3, and B4 in the figure. The optical beacon LEDs emit white light, with a 120° cone of emission. White light LEDs are selected to facilitate piggybacking with existing visible room lighting—alternatively monochromatic infrared LEDs can be dedicated to this positioning. The LEDs are modulated with distinct frequency channels, between 2 and 3 kHz, to allow the photoreceiver to separately process each channel and calculate AOAs for all observable optical beacons. It will be shown, by way of the photoreceiver angular and intensity responses in the following subsections, that the optical beacon grid can be scaled to large dimensions with an appropriate increase in optical beacon powers (to keep the photoreceiver intensity above a minimum intensity threshold).
2.1 Angular response
When a beam illuminates the photoreceiver, the three respective photocurrents i1(ϕ,θ), i2(ϕ,θ), and i3(ϕ,θ) are generated from PD1, PD2, and PD3 respectively. For processing, the photocurrents are normalized with respect to the maximum photocurrent, and differential photocurrents are formed by using PD3 as a reference and subtracting i3(ϕ,θ) from the remaining two photocurrents. The resulting two ϕ- and θ-dependent differential photocurrents are normalized (unitless) quantities defined according to
The trivial result Δi3(ϕ,θ) = i3(ϕ,θ) - i3(ϕ,θ) = 0 is omitted. The Δi1(ϕ,θ)and Δi2(ϕ,θ) functions are piecewise [24], and thus difficult to solve algebraically for the desired inverse functions ϕ(Δi1,Δi2) and θ(Δi1,Δi2), but the expressions for Δi1(ϕ,θ)and Δi2(ϕ,θ) can be approximated for the photoreceiver as the respective polynomials in Eqs. (1) and (2). Least-angle regression is used for this [25]. A root-mean-square error of 0.2% is obtained for numerical fitting parameters of C0 = −9.08 × 10−1, C1 = 1.58 × 10−2 (°)−1, C2 = −2.44 × 10−4 (°)−1, C3 = 1.51 × 10−1 (°)−2, C4 = −1.06 × 10−4 (°)−2, C5 = 8.40 × 10−6 (°)−2, C6 = −1.17 × 10−6(°)−3, C7 = 2.32 × 10−6 (°)−3, C8 = −2.51 × 10−6 (°)−3, D0 = −8.63 × 10−1, D1 = - 1.40 × 10−2 (°)−1, D2 = −1.22 × 10−3 (°)−1, D3 = 3.60 × 10−4 (°)−2, D4 = 5.58 × 10−4 (°)−2, D5 = 7.98 × 10−6 (°)−2, D6 = −1.17 × 10−6 (°)−3, D7 = −2.31 × 10−6 (°)−3, and D8 = −2.52 × 10−6 (°)−3. Given the polynomial functions for Δi1(ϕ,θ)and Δi2(ϕ,θ) in Eqs. (1) and (2), a simple numerical algorithm can be used to solve for the inverse functions and output the desired ϕ(Δi1,Δi2) and θ(Δi1,Δi2) angles. The process is illustrated in Fig. 2 for symmetric illumination of the photoreceiver, with an incident AOA of ϕ = 45° and θ = arccos(1/√3) ≈54.7°. The Δi1(ϕ,θ)and Δi2(ϕ,θ) surfaces from Eqs. (1) and (2) are shown intersecting at the precise condition for which Δi1 = Δi2 = 0 and ϕ = 45° and θ ≈54.7°.Fig. 2 Analytical results are shown for normalized differential photocurrents Δi1(ϕ,θ) and Δi2(ϕ,θ) versus azimuthal ϕ and polar θ angles.
2.2 Intensity response
The AOA estimation capability of the photoreceiver allows it to triangulate using bearings from multiple optical beacons. If two or more optical beacons are used for AOA estimation, the photoreceiver 3-D position can be determined. Inherent to this optical AOA positioning, and in stark contrast to amplitude-based positioning, is the fact that AOA estimation can be made to be largely intensity independent and less sensitive to optical transmitter and channel imbalances. The conditions for such intensity independence are tested here.
A minimum intensity is expected for successful AOA estimation—being brought about by a sufficiently large optical beacon power or sufficiently short distance between the photoreceiver and optical beacon. To experimentally determine the minimum intensity for mitigation of systematic AOA errors, the optical beacon power is varied while incident light ϕ and θ angles are scanned with the photoreceiver in a gyroscopic mount. Incident intensities from 0.01 to 2.0 µW/cm2 are tested for each AOA alignment, and the ϕ and θ angles are estimated from the photoreceiver differential photocurrents using Eqs. (1) and (2).
A representative result for this direct intensity characterization of the photoreceiver is shown in Fig. 3(a) for ϕ = 45° and θ = arccos(1/√3) ≈54.7°. One hundred samples are used. It is seen that AOA angles are accurately determined for incident intensities above a 0.2 µW/cm2minimum intensity threshold. For the implemented (small) optical beacon grid, the corresponding minimum LED optical power is6.3mW. Larger link distances with higher optical beacon powers, such as those in [8], can be implemented with appropriate scaling. Optical beacon grids with increasing distances between beacons require increasing optical beacon powers, scaling in proportion to the area of the optical beacon grid, to keep the photoreceiver intensity above the minimum intensity threshold.
Fig. 3 Experimental results are shown for the photoreceiver AOA estimation by way of (a) a direct intensity characterization of measured azimuthal ϕ and polar θ angles versus incident optical intensity and (b) a reflected intensity characterization of measured azimuthal ϕ and polar θ angles vs. reflective surface distance for plywood, stainless steel, and drywall surfaces.
Operation well above the minimum intensity threshold is advisable, as it allows greater numbers of observable optical beacons to be used in optical AOA positioning. With only one observable optical beacon, the photoreceiver registers one AOA, and a corresponding line-of-position (LOP) between the photoreceiver and optical beacon. However, this single LOP vector cannot define a 3-D estimate of the photoreceiver position. With two observable optical beacons, the photoreceiver registers two AOAs, and two corresponding LOPs between the photoreceiver and optical beacons, and these two LOP vectors can define a 3-D estimate of the photoreceiver position if the photoreceiver orientation is known. With three observable optical beacons, the photoreceiver registers three AOAs, and three corresponding LOPs between the photoreceiver and optical beacons, and these three LOP vectors can define a 3-D estimate of the photoreceiver position, even if the photoreceiver orientation is unknown. With four or more observable optical beacons, the photoreceiver can estimate 3-D positions, and the accuracy of these estimates improves as the number of observable optical beacons increases.
The remaining consideration that must be addressed for optical AOA positioning relates to multipath reflections. Obstacles in the environment can form reflected intensities on the photoreceiver at the distinct LED modulation frequencies, and this can yield erroneous AOA estimates. To investigate this, a reflected intensity characterization is carried out for the photoreceiver with a variety of reflective materials (plywood, stainless steel, drywall) in the proximity of the photoreceiver. The results are shown in Fig. 3(b) as a function of the reflective surface distance, with the photoreceiver central axis of symmetry parallel to the reflective surface and the incident optical beam along ϕ = 45° and θ = arccos (1/√3) ≈54.7°.
Multipath effects are immediately apparent in the Fig. 3(b) when the distance between the photoreceiver and the surface is less than approximately 0.5 m. Beyond this distance, ϕ converges to the true value of 45° and θ converges to the true value of θ = arccos (1/√3) ≈54.7°. Optical AOA positioning has a 7% error between measured and true angles for the minimum distance of 0.5 m (which can be decreased to1% error if a minimum distance of 0.7 m is used). These conclusions apply to surfaces with diffuse (scattered) reflections, as surfaces with specular (mirror-like) reflections can degrade the AOA estimates at larger distances.
3. Photoreceiver accuracy and precision
Positioning accuracy and precision have three contributing factors when defined for this work.
The first factor relates to systematic errors, i.e., biases, that can cause deviations between measured and true AOA angles. Systematic errors are assumed to be negligible for our AOA measurements, when the photoreceiver is operated with the above-threshold and reflection-free conditions of the prior section. Any remaining systematic errors, from differing PD responsivities, are mitigated by the initial calibration procedure (where transimpedance gains for the photocurrents are balanced while the structure is illuminated along the central axis of symmetry). The negligible systematic errors means that accuracy and precision are equivalent.
The second factor relates to angular precision which quantifies the standard deviation of the error in the measured AOA angles (i.e., measured uncertainty). In Section 3.1 measured angle errors are presented. Due to the lack of systematic errors, for equivalence of accuracy and precision, the mean of measured angle errors is assumed to be equivalent to the standard deviation of measured angle errors. Thus, in Section 3.1 angular precision will be quantified by measuring the mean angular error and the value so obtained will be used in Section 3.2.
The third factor is the effect of optical beacon and photoreceiver geometry on positioning error standard deviation and is quantified by Dilution of Precision (DOP). In Section 3.2 the effect of DOP on positioning error standard deviation is analyzed.
3.1 Angular precision
To characterize angular precision, the photoreceiver is mounted in a gyroscope and illuminated by a single optical beacon with an above-threshold intensity of 1.8 µW/cm2. Measured AOA ϕ and θ angles are compared to true angles. Each estimated AOA (ϕ and θ) pair is the result of one hundred samples at a modulated optical beacon frequency of 2.5 kHz.
Absolute differences between measured and true ϕ and θ angles are defined here as the measured angle errors Δϕ and Δθ, respectively, and are shown in Figs. 4(a) and 4(b) as a function of ϕ and θ. Error trends are apparent. The measured angle errors Δϕ and Δθ are at their lowest level in close proximity to the ϕ = 45° and θ = arccos(1/√3) ≈54.7° central axis of symmetry. In moving away from this central axis, the errors increase in a way that reflects the structural symmetry. In Fig. 4(a), the measured angle error Δϕ is roughly symmetric about the ϕ = 45° line, as one would expect by the structure’s mirror symmetry about a ϕ = 45° bisecting plane. At the same time, the measured angle error Δϕ is disproportionately large for small θ angles, compared to those for large θ angles. This distinction is seen from illumination asymmetry for small or large θ. When the structure is illuminated near θ ≈0°, both PD1 and PD2 yield negligible photocurrents and only PD3 yields a high photocurrent. This gives way to large measured angle errors in Δϕ. When the structure is illuminated near θ ≈90°, both PD1 and PD2 yield high photocurrents and only PD3 yields a negligible photocurrent. This gives way to low measured angle errors in Δϕ.
In Fig. 4(b), the measured angle error distribution for Δθ is roughly symmetric about ϕ = 45° when θ approaches 90°. This can be seen by looking at the photoreceiver in Fig. 1, with its symmetry between PD1 and PD2. As θ is reduced, however, symmetry in the measured angle error Δθ diminishes, and the response becomes dominated by random error.
To deploy the photoreceiver in OWL systems, the measured angle errors Δϕ and Δθ must be kept below an acceptable level. For this investigation, a mean error of 2° is deemed to be acceptable, and this is achieved by defining an operational cone of ϕ × θ = 40° × 40° through the central axis of symmetry. Optical beacons illuminating the photoreceiver within this operational cone give measured mean angle errors Δϕ and Δθ of 2°. The deployed optical beacon grid spacing and height can thus be designed to facilitate AOA reception and 3-D position estimation with at least two optical beacons being in this operational cone. More optical beacons provide greater redundancy if improved accuracy is needed.
3.2 Effect of geometry on position uncertainty
Angular measurement uncertainty (angular error standard deviation) is linked to position uncertainty (position error standard deviation) by DOP. In certain beacon/photoreceiver geometries angle errors Δϕ and Δθ have minimal effects on position uncertainty, while in other geometries angle errors have dramatic effects on position uncertainty. The geometrical precision characteristics are quantified by way of PDOP and are visualized with LOP vectors between the photoreceiver and observable optical beacons. The photoreceiver must acquire a minimum of two AOA estimates from two observable optical beacons, as the resulting two LOP vectors define the position as the intersection of these LOP vectors. In general, near-orthogonal LOPs are desirable as they establish minimal sensitivity to the measured angle errors Δϕ and Δθ, while near-parallel LOPs are undesirable as they establish significant sensitivity to the measured angle errors Δϕ and Δθ.
For this study, the photoreceiver is positioned at (x,y,z). Measurements are made for AOA ϕi and θi angles for LED optical beacons at (xi,yi,zi), wherei = 1, 2. The AOA angles are related to the photoreceiver and respective optical beacon positions by
The AOA angles ϕi and θi are defined for directions toward the i = 1 and 2 optical beacons.Expressions Eqs. (5) and (6)are fundamental relationships for linking the errors in the existing measured angles to the errors in the desired position. The measured angle errors are recorded by the measured angle error vector
where Δϕi and Δθi are the respective measured angle errors for optical beacons i = 1, 2, and the T superscript denotes the transpose operation. Similarly, the position error vector iswhere Δx, Δy and Δz are the respective position errors in the x, y, and z directions. The measured angle errors Δϕ and Δθ can be linked to the absolute position errors by taking partial derivatives of the observation ϕ1, θ1 and ϕ2, θ2 with respect to the unknown photoreceiver position x, y, z. The resulting partial derivative matrix, H, is defined byIn general, one wishes to invert the matrix representation of Eq. (7) to solve for the three position errors, Δx, Δy and Δz, given the four measured angle errors, Δϕ1, Δϕ2, Δθ1, and Δθ2. This overdetermined system, with four Eqs. and three unknowns, is solved by applying a least-squares minimization procedure to the linear system and solving for the unique generalized inverse matrix [27, 28]. The resulting solution for the position error vector is
For this AOA-based system, the measured angle errors are assumed to be independent, zero-mean, Gaussian-distributed random variables with equal variance, σM2 [28, 29]. The covariance of Eq. (9) can then be used to express the position error variance as
where E(∙) and tr(∙) denote respective expectation and trace operations. Three-dimensional position DOP (PDOP) is then defined as the ratio of positioning error standard deviation σPto the standard deviation of the measured angle errors, σM:Here, position standard deviations in x, y and z directions are denoted by σx, σy and σz, respectively. Note that the PDOP in Eq. (10) is an AOA-based quantity with units of meters per radian, unlike the unitless PDOP for range-based systems such as GPS. PDOP acts as a weighting factor on the measured angle standard deviation for the calculation of the position standard deviation. The effect of the angle standard deviation on the position standard deviation will depend on the photoreceiver position with respect to observable optical beacons. Given two observable optical beacons in close proximity, for example, the photoreceiver registers two AOAs with the corresponding LOPs being nearly parallel, which in turn yields a large PDOP and large position standard deviation. Given two observable optical beacons that are well separated, in contrast, the photoreceiver registers two AOAs with the corresponding LOPs being nearly orthogonal, which in turn yields a small PDOP and small position standard deviation. For the present analysis, the 3-D position error standard deviation, σP, results from the measured angular error of σM = 2° in the operational cone and can be found by calculating the PDOP from Eq. (10) for the OWL system.A positioning system with two optical beacons is first tested. The optical beacons are positioned with LED A1 at (x1 = 15 cm, y1 = 0 cm, z1 = 50 cm) and with LED A2 at (x2 = −15 cm, y2 = 0 cm, z2 = 50 cm), with the photoreceiver scanned across the z = 0 plane. Results are shown as positioning error standard deviation in Fig. 5(a). Note that the largest position error standard deviation occurs in the plane y = 0. This is because LOPs from LEDs A1 and A2 are parallel, giving rise to large positioning error standard deviations.
Fig. 5 The 3-D positioning error standard deviation, σp, is shown for optical AOA positioning with (a) two optical beacons, LEDs A1, A2, and (b) four optical beacons, LEDs B1, B2, B3, B4.
A positioning system with more optical beacons can improve the positioning capabilities. Such improvements are demonstrated by the results in Fig. 5(b) for a positioning system with four optical beacons. The optical beacons are positioned with LED B1 at (x1 = 15 cm, y1 = 15 cm, z1 = 50 cm), LED B2 at (x2 = −15 cm, y2 = 15 cm, z2 = 50 cm), LED B3 at (x3 = −15 cm, y3 = −15 cm, z3 = 50 cm) and LED B4 at (x4 = 15 cm, y4 = −15 cm, z4 = 50 cm). Improved position error standard deviations are apparent. For the four optical beacon grid a mean 3-D position error standard deviation of σP = 2.8 cm is witnessed. This is a factor of two improvement over the mean 3-D position error standard deviation of σP = 4.7 cm for the two optical beacon grid. These geometrical precision effects can be factored into the OWL system design for meeting the desired positioning accuracy. In general, a greater number-density of optical beacons per unit area can yield a reduced 3-D standard deviation, but it should be noted that the above geometrical precision analysis, when carried out for a hexagonal-close-packed grid of the same area, yields only a minor improvement in the 3-D positioning standard deviation.
4. Positioning performance
A performance analysis is given here for the presented optical AOA positioning technique. The methodology is similar to that of an earlier experiment [11]. The complete performance analysis quantifies positioning error, which is defined as the Euclidean distance between the measured and true 3-D positions of the photoreceiver. The performance of optical AOA positioning is compared to that of the known optical RSS positioning. The OWL system is tested with the aforementioned four optical beacons, LEDs B1, B2, B3 and B4.
Optical RSS positioning is tested first with a single flat 9.7 × 9.7 mm2 PD. The PD is rastered across the xy plane at 25 different positions of (0, 0), (0, ± 7.5), (0, ± 15), ( ± 7.5,0), ( ± 7.5, ± 7.5), ( ± 7.5, ± 15), ( ± 15, 0), ( ± 15, ± 7.5), and ( ± 15, ± 15). The incident intensities are used to quantify the respective ranges to the four LEDs. At PD position (0, 0), the received optical power is calculated from the measured photocurrent and effective responsivity. The optical power is computed for each LED according to free-space propagation [29]. A nonlinear least-squares positioning algorithm is applied to estimate the PD position. The resulting positioning error for the optical RSS positioning technique is shown in Fig. 6(a). At the centre of the grid at (x = 0, y = 0), the optical intensities from the LED beacons have beencalibrated such that they are all equal, since the PD is equidistant from all four LEDs, so the centre position gives the lowest positioning error. Motion away from the centre increases error. A mean RSS positioning error of 20 cm is found for positions across the xy plane.
Optical AOA positioning is tested next with the photoreceiver. The photoreceiver body frame shown in Fig. 1 is oriented with the PD1 normal along (x, y, z) = (1/√6, 1/√6, √2/√3), the PD2 normal along (x, y, z) = (−1/√6, 1/√6, √2/√3), and the PD3 normal along (x, y, z) = (0, -√2/√3, 1/√3). The photoreceiver is rastered across the xy plane at the same 25 test points as the RSS experiment. At each of these positions a total of one thousand AOA measurements are registered for each optical beacon and the mean AOA values from LEDs B1, B2, B3 and B4 are used in a Least Squares algorithm to estimate the position of the photoreceiver. The standard deviation of the one thousand AOA error measurements is less than 1% of the mean AOA error indicating that random AOA errors are negligible. The resulting positioning error is shown in Fig. 6(b) for this optical AOA positioning. The distribution of the positioning error highlights the symmetry of the photoreceiver and supports the conclusions of Section 3.1 with respect to Fig. 4. Orientations with significant illumination of two PDs (at large and negative y values) have improved accuracy over orientations with significant illumination of only one PD (at large and positive y values). This is because more light flux is captured by the photoreceiver. Overall, the mean positioning error of optical AOA positioning is 5 cm, a result which is four times lower than that of optical RSS positioning. Clearly, optical AOA positioning can be an effective solution for enhanced localization in OWL systems.
5. Conclusion
An OWL system was demonstrated with optical AOA positioning. The mobile photoreceiver was implemented as a corner-cube PD architecture giving AOA measurements by differential photocurrents. A direct intensity characterization was carried out for azimuthal ϕand polar θ angles, and it was found that incident intensities should remain above a 0.2 µW/cm2 threshold. Above this threshold, the photoreceiver operation is intensity-independent and therefore insensitive to fluctuations from optical beacon power imbalances, shadowing, fading, etc. A reflected intensity characterization was carried out to quantify multipath effects, and it was found that the photoreceiver should be operated at a distance of at least 0.5 m from reflective surfaces. The characterization of the AOA angular precision showed lowest AOA error close to the photoreceiver's central axis of symmetry—leading to the use of a restricted operational cone of ϕ × θ = 40° × 40°. The characterization for the effect of system geometry showed that a greater number-density of optical beacons leads to improved precision. A complete OWL was built and tested for optical RSS and optical AOA positioning. The AOA technique’s 5 cm positioning error was four times lower than that of the RSS technique. Future indoor positioning systems seeking improved accuracies can benefit from this new OWL system.
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