1. Introduction
The development of visible light communication (VLC) in the last decade and its dependence to user location had made ways in strengthening the developments on visible light positioning (VLP), which offers higher accuracy for indoor positioning in compare to other methods such as GPS (Yan, Guan, Wen, Huang, & Song, 2021). On the current discussion VLP is classified in two, based on the light receptor component. The first class is using photodetector or PD based, and the second class is using camera as image sensor or IS based. The PD based VLP is more attuned to VLC design for user equipment, and required small resource in signal processing. The IS based VLP has the potential to give higher accuracy by having more data to process, at the same time
International Journal of Engineering Advanced Research eISSN: 2710-7167 [Vol. 4 No. 1 March 2022]
Journal website: http://myjms.mohe.gov.my/index.php/ijear
IMPROVED EXTENDED KALMAN FILTER FOR
PHOTODETECTOR BASED VISIBLE LIGHT POSITIONING
Dwi Astharini1*, Sekar W. Wasiati2 and Ahmad H. Lubis3
1 2 3 Faculty of Science and Technology, Universitas Al Azhar Indonesia, Jakarta, INDONESIA
*Corresponding author: [email protected]
Article Information:
Article history:
Received date : 15 January 2022 Revised date : 1 February 2022 Accepted date : 2 March 2022 Published date : 10 March 2022
To cite this document:
Astharini, D., Wasiati, S. W., & Lubis, A. H. (2022). IMPROVED EXTENDED KALMAN FILTER FOR
PHOTODETECTOR BASED VISIBLE LIGHT POSITIONING. International Journal of Engineering Advanced Research, 4(1), 73-84.
Abstract: Visible light positioning (VLP) depends on light reception at the detector to determine equipment’s indoor position. This paper present estimation scheme for VLP using extended Kalman filter (EKF) with improvement on the model, in which the light intensity received is incorporated as state parameter. The dynamic model is established so that the estimation process is directly in accordance with Lambertian model of visible light reception. The VLP in discussion is using photodetector as light receptor, enabling small data to process. The result is compared with reference model in which position parameters were being the central role, for variety of measurement noise. The proposed EKF model with the signal strength as state parameter showed over all smaller error than the referred model for positioning.
Keywords: estimation, extended Kalman filter, indoor light intensity, visible light positioning.
requiring more rigorous calculation and processing time. This paper discuss the PD based VLP, in conjunction to VLC system.
Empowered by progresses in VLC, this communication method is not the only use for VLP. Indoor positioning is a growing subject in itself, as a support system in the era of 5G. VLP can be used for wireless sensor network (WSN) and IoT as discussed in (Jia, 2012; Vatansever, Brandt-Pearce,
& Bezzo, 2019), with or without the use of VLC within the system. Discussion on IS based VLP mostly are for indoor robot, equipped with camera (Amsters et al., 2020; Yan et al., 2021). While camera is quite a customary feature in many indoor robots or other equipment, image processing has its basic disadvantage of heavy processing.
Kalman filter (KF) has been widely used in broad ranging type of tracking and navigation for a long time. Specifically using extended Kalman filter (EKF), there has been model proposed for PD based VLP, among others in (Amsters, Demeester, Slaets, & Stevens, 2018; Nguyen, Nguyen, Nguyen, Sripimanwat, & Suebsomran, 2015; Rahaim, Prince, & Little, 2012; Vatansever &
Brandt-Pearce, 2017). In all four publications the dynamic models contain solely the position parameters in the state vector. The light intensity or signal strength received on PD appeared in the model only as the output or estimated parameter. To reduce the error, this paper proposed an improved estimation using EKF, in which the estimated parameter of light intensity is also incorporated as state parameter.
2. Related Works
As mentioned VLP for indoor positioning has been a blooming subject of study, comprosing the PD based and IS based ones. The PD based VLP used the Lambertian VLC channel on its system model, such as in (Eroglu, Erden, & Guvenc, 2019) where position prediction with Kalman filter is used for adaptive positioning. Lamberitan model is used also (Keskin, Sezer, & Gezici, 2019) which investigated the LED power allocation effect on VLP. The IS based on the other hand derived the position from analytic projection of image processing, and in various references are applied in synch with odometry (Guan et al., 2021; Song et al., 2021; Yan et al., 2021). In (Song et al., 2021; Yan et al., 2021) position is determined using rolling camera shutter and odometry. In (Guan et al., 2021) the combination is enhanced with LiDAR to achieved multi-sensor localizer.
Without the use of angle sensor (Zhang, Zhong, Kemao, & Zhang, 2017) proposed positioning based on geometric features in image of circular LED projection.
As the PD based VLP is closely related to VLC, quite a large part of literature we studied are of VLC, that is on the subject of channel estimation. Kalman filter (KF) is widely used in broad ranging type of tracking and navigation. Also known to be used on channel estimation, KF has made its way on VLC system. In (Rahaim et al., 2012) KF was used for a hybrid system of VLC and Wi-Fi. User position was modelled with x and y coordinate of indoor area partitioned in grid.
The estimators were extended Kalman filter (EKF) and KF with unscented transformation. Also working on multi transmitter models of VLC system is (Eroglu et al., 2019) where the Kalman filter is made adaptive for handover between the access points (AP). On (Nguyen et al., 2015), EKF is applied on dynamic model of horizontal coordinates of x, y. While KF has appeared numerously, all of the works (Eroglu et al., 2019; Nguyen et al., 2015; Rahaim et al., 2012) above are using dynamic model with x, y, z of Cartesian coordinate. On the work of (Zou et al., 2018)
unscented Kalman filter (UKF) is applied for dynamic position based on inertial measurement unit (IMU) data. IMU data is also used in (Liang & Liu, 2020) on EKF for user positioning of IS based.
The channel is then derived from estimated position. Another IS based VLP is (Guan, Liu, Wen, Xie, & Zhang, 2019) where high performance positioning is proposed by using Camshift-Kalman algorithm. The use of image requires rigorous processin but allowed more possibilities for more precise results.
2.1 Contribution
From above references it can be observed that the existing model for VLC channel estimation are using EKF are based on the dynamic models which contains solely the position parameters in the state vector. The light intensity or signal strength received on PD appeared in the model only as the output or estimated parameter.
While the estimations and predictions have succeeded in general, improvements can be made for the performance. In PD based VLP the measurement parameter is the light intensity received by the PD, and this measurement is very exposed towards noise and interference. In our observation, one critical aspect is instigated by the fact that the measured parameter is the light intensity, while the dynamic model is of position parameters, and the relation between them is highly nonlinear.
In order to lessen the error, we proposed an improvement on the dynamic model, in which the estimated parameter of light intensity is also used as state parameter. As the dynamic model is the core of estimation by EKF, this would improve the EKF performance in VLP.
3. System Model
The discussion of model in this part comprises the signal strength model, the dynamic model, and the EKF design.
3.1 Light Intensity as Signal Strength
Propagation of indoor visible light is described in Lambertian model which can be found among others in (Ghassemlooy, Popoola, & Rajbhandari, 2012; Marshoud, Sofotasios, Muhaidat, Karagiannidis, & Sharif, 2017). The channel gain is defined in (1).
ℎ =𝑚 + 1
2𝜋𝑑2 𝑐𝑜𝑠𝑚𝜃. 𝑐(𝜓). 𝑐𝑜𝑠(𝜓) (1) with d and 𝜃 the distance and angle between LED-PD, c is PD gains, ψ is PD reception angle, and
𝑚 = − 𝑙𝑛(2)
𝑙𝑛(𝑐𝑜𝑠(𝜑1 2
))
is the Lambertian order as function of LED coverage 𝜑. Figure 1 illustrate this lay out.
Using Cartesian coordinates (x, y, z), the angular parameters above are replaced by
𝑑 = √(𝑥2+ 𝑦2 + 𝑧2) (2)
cos 𝜃 = 𝑧
√(𝑥2+ 𝑦2+ 𝑧2). (3)
Figure 1: PD Reception as Function of Position
The LED is assumed to be time invariant throughout the process, thus m and 𝜑 is pre-set. The same is assumed for c of PD specifications. The PD reception angle 𝜓 is changeable, but on this work, it is assumed to be a fixed one that is 𝜓 = 𝜃. Probable cause of reception dynamic is the change in position, presented in (x, y, z).
3.2 System Dynamic Model
On the ground of the model of light reception above, a dynamic model of PD position is established. The variables of interest are the coordinates (x, y, z). Changes are caused by movement input described in each axis, and process noise. The coordinates on any time k can be stated as:
𝑥(𝑘 + 1) = 𝑥(𝑘) + 𝑢𝑥(𝑘) + 𝑤𝑥(𝑘) (4) 𝑦(𝑘 + 1) = 𝑦(𝑘) + 𝑢𝑦(𝑘) + 𝑤𝑦(𝑘) (5) 𝑧(𝑘 + 1) = 𝑧(𝑘) + 𝑢𝑧(𝑘) + 𝑤𝑧(𝑘) (6)
The estimation by EKF in references (Nguyen et al., 2015; Rahaim et al., 2012; Vatansever &
Brandt-Pearce, 2017) are basically build by (4-6). In our proposed model, the signal strength h of light intensity is added into the dynamic model as the fourth state. As shown in (1-3), signal strength h is a nonlinear function of position coordinate (x, y, z), so that the dynamic model of it can be presented as
ℎ(𝑘 + 1) = 𝑐 (𝑚 + 1)𝑧𝑚+1(𝑘)
2𝜋. √(𝑥2(𝑘) + 𝑦2(𝑘) + 𝑧2(𝑘))𝑚+3 (7) Wrapping (4-7) into a state equation with the state vector 𝑞 = [𝑥 𝑦 𝑧 ℎ]𝑇, the dynamic model of the system is
𝑞(𝑘 + 1) = 𝑓(𝑞(𝑘), 𝑢(𝑘)) + 𝑤(𝑘) (8) with 𝑓1(𝑞(𝑘), 𝑢(𝑘)) to 𝑓4(𝑞(𝑘), 𝑢(𝑘)) are deducted directly from (4-7) respectively.
Output of the VLP system is the position coordinate (x, y, z). But considering estimation process by Kalman filter, the system output is by default the measured parameters. Since the only measured parameter here is the signal strength h, then it is the sole output of the model, with additive measurement noise v.
𝑝(𝑘) = ℎ(𝑘) + 𝑣(𝑘). (9)
3.3 Positioning by Extended Kalman Filter
As the system model involves a nonlinear (7), extended Kalman filter is used for estimation.
Kalman filter is a cyclic process of prediction and estimation throughout the process run time, as shown in Figure 2. Complete process of each cycle is detailed in Algorithm 1 of EKF, which in general can be acquired references of KF such as (Brown & Hwang, 2012; Reid & Term, 2001).
The EKF cope with the nonlinearity by using Jacobian matrices 𝐽𝑓 =𝜕𝑓
𝜕𝑥, 𝐽ℎ =𝜕ℎ
𝜕𝑥.
Figure 2: EKF Estimation Cycle Algorithm 1 Extended Kalman filter
Initialization: f(q,u), h(q), Q(k) = E[w(k)w(k)’], R(k) = E[v(k)v(k)T], P_init, q_init Input: u
Output: he Iterative step:
Measurement update:
Jh = 𝝏𝒉(𝒒)
𝝏𝒒
G(k) = P(k|k-1) Jh' [JhP(k|k-1) Jh'+R]-1 h(k|k-1) = h(q)
q(k|k) = q(k|k-1) + G(k)[hv(k) – h(k|k-1)]
P(k|k) = [I - G(k)Jh] P(k|k-1) Time update:
q(k+1|k) = f(k|k) + w(k) Jf = 𝝏𝒇(𝒒)
𝝏𝒒
P(k+1|k) = Jf P(k|k) Jf'’ + B*Q*B'
% Jacobian matrix of output function
% Kalman gain
% Estimated signal strength
% measurement update
% noise covariance
% next state prediction
% Jacobian matrix of state function Prediction
𝑞𝑘+1|𝑘 = 𝑓(𝑞𝑘, 𝑢𝑘) Measurement update –
estimation 𝑞𝑘 = 𝑔(𝑞𝑘|𝑘−1, 𝐺𝐾)
Output:
he = h(k|k)
The Jacobian of state and output functions for the dynamic model proposed are
𝐽𝑓 =𝜕𝑓
𝜕𝑞= [
1 0 0 0 0 1 0 0 0 0 1 0
𝑚 + 1
2𝜋 𝐽ℎ ]
, (10)
𝐽ℎ = [ −(𝑚 + 3)𝑧𝑚+1𝑥 2𝜋√(𝑥2 + 𝑦2+ 𝑧2)𝑚+5
−(𝑚 + 3)𝑧𝑚+1𝑦 2𝜋√(𝑥2+ 𝑦2+ 𝑧2)𝑚+5
𝑧𝑚(1 − (𝑚 + 3)𝑧2 𝑥2+ 𝑦2+ 𝑧2) 2𝜋√(𝑥2+ 𝑦2+ 𝑧2)𝑚+3 1]
(11)
Figure 3: Simulation Framework of EKF for VLP
4. Simulation and Discussion
Implementation of the designed EKF for VLP in simulation is as shown in Figure 3. The estimation performed by Kalman filter is in parallel to the system. The EKF performed based on the dynamic model, directed by the same input as the system, and updated to system condition by measurement.
The signal measured is disturbed by noises, making the measurement inaccurate. The EKF performance is observed in the estimation errors, by comparing the system output and the estimated parameters. Table 1 shows parameter set used on the simulation.
Figure 4 shown an example result set of the simulation. The left column shows the system output and the estimated parameter. The right column shows the error derived from the comparison. The first row of plot is for the signal strength h. The second row is of x, the third row is of y, and the fourth row is of z coordinate.
As explained above, the measurement is done only on signal strength h, therefor only parameter h in first row have the system error in green colour. The system error is the different between the system output and the measured signal (h-hmeasured). As the estimation error (h-he) in red line is smaller than the green line, it shows that the proposed design of EKF has performed effectively in estimating the parameter of signal strength h.
Table 1: Parameter Setting for Simulation
Variables Value
Q Process noise 10-4 – 5.10-4 R Measurement noise 0.001 – 0.2
𝒙𝟎 Initial x coordinate 0
𝒚𝟎 Initial y coordinate 0
𝒛𝟎 Initial z coordinate 2 m h0 Initial signal strength 0.5 𝝋𝟏
𝟐 Half angle of LED coverage 85°
c Total component power 8
Figure 4: Estimation Performance of the Proposed EKF x-y-z-h
The EKF performance for VLP, which is of position estimation, is observed solely by the estimation error. In this observation, comparison is made with other EKF of reference (Nguyen et al., 2015; Rahaim et al., 2012; Vatansever & Brandt-Pearce, 2017), which employed the state vector 𝑞 = [
𝑥 𝑦 𝑧
] consisting only the Cartesian coordinates. The comparison is depicted in Figure 5 and 6. For simplification the standard deviation of position error presented in figure 5 is the average of axis’ error.
𝜎𝑝𝑜𝑠= 𝜎𝑥+ 𝜎𝑦 + 𝜎𝑧
3 (12)
Figure 5: Comparison of Performance in Standard Deviation of Position Error
In Figure 5, average error of position estimation of the proposed EKF x-y-z-h has shown lower values than those of the referred EKF x-y-z, throughout a variety of measurement noise. The measurement noise is varied from the low point of R = 0.001 to a high noise of R = 0.2. In condition with high measurement noise, the error is getting high, but remains lower than the reference EKF.
The proposed EKF has succeeded in reducing the positioning error. So that it can be said that the proposed EKF has an overall better performance as method of VLP.
Comparison is also done on the performance in signal strength estimation. For the purpose of VLP the signal strength is merely the hopping stone to determine position. Nevertheless, low performance in signal strength estimation very probable would lead to high error in positioning part. In VLC system however, the signal strength is the desired parameter, crucial in signal detection at the receiver, and determining power allocation at the transmitter.
Figure 6 shows the standard deviation of signal strength estimation error, of the two EKFs. Just as before, also in variety of measurement noise from low to high of R = 0.001 to 0.2. The graphic shows that the proposed EKF model has lower error only on high noise environment. From these we can deduce that for signal strength estimation, the proposed EKF has better performance than the reference on high noise environment.
0 0.05 0.1 0.15 0.2 0.25
0.001 0.01 0.1 0.2
Measurement noise covariance R Position Estimation Error
EKF xyzh EKF xyz
Figure 6: Comparison of Performance in Standard Deviation of Signal Estimation Error
At a glance, it seemed that the proposed EKF is not entirely a success, since lower error is achieved on in certain condition. But recalling the outlaying problem, as also pictured in Figure 4 of framework, the background difficulty is the high probable presence of noise and interference, likely in high ratio toward the signal. So, in this viewpoint, the proposed EKF has succeeded in offering alternate model with lower error in high noise environment. It can be said that the proposed EKF is less susceptible to additive noise.
Naturally the proposed EKF must pay the price for the better performance. As the state dynamic contains nonlinearity in the presence of h, the cyclic process of EKF x-y-z-h involves two Jacobian matrices where both has nonlinearity, and with a certain degree of complexity, as shown in (10- 11). The referred EKF x-y-z by comparison only contains nonlinearity in its output model, giving way to a linear Jf and nonlinear Jh. Thus, the proposed EKF required more resource than the reference one.
5. Conclusion
To conclude the presentation, the objective of this work has been achieved, that is to improve error performance of EKF for PD based VLP. This was done by designing an EKF model, in which the signal strength h is included as the state vector of dynamic model. The proposed EKF has better VLP performance throughout variety of low to high measurement noise. The proposed EKF performed better high noise condition for signal estimation or VLC channel estimation. The proposed EKF is less vulnerable towards noise and interference in light.
6. Acknowledgement
The work and publication of this paper was supported by Universitas Al Azhar Indonesia PRG Grant no. 011/SPK/A-01/UAI/IV/2021.
0 0.01 0.02 0.03 0.04 0.05 0.06
0.001 0.01 0.1 0.2
Measurement noise covariance R Standard Deviation of
Intensity Estimation Error
EKF xyzh EKF xyz
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