Pedestrian Positioning Using an Enhanced Ensemble Transform Kalman Filter
Abstract
:1. Introduction
- No indoor pedestrian positioning system based on the ETKF has been found to date, despite nearly a decade of searching. In this paper, an efficient approach for implementing the ETKF applied to the indoor pedestrian localization system that should consider the model error, which is called QETKF, is introduced. The QETKF is similar to the ETKF in that it is based on the ensemble transformation performed with the posterior error covariance in ensemble space, thereby providing more rapid positioning results than EnKF and its variants when many observations are assimilated as in the atmospheric data assimilation. However, the QETKF differs from the ETKF in that it can prevent the systematic underestimation of the error covariance that appears in the ETKF, since it can estimate the error covariances where the model error covariance is included by considering the model error in the prediction model, thereby producing more inflated ensemble spread than the ETKF. This enables the QETKF to yield higher estimation accuracy than the ETKF.
- The indoor localization system in this study is carried out using the sensors and the positioning algorithm as in the previous study [13]. In the localization system, the sensor part employs the IMU sensors (gyroscope and accelerometer) and wireless signal receivers (iBeacon and WiFi modules) on the smartphone in the same manner as the previous study [13]. The positioning algorithm part estimates the user position using the offline training and online positioning phases as in the previous study [13]. In the offline training phase, RSS data obtained from iBeacon and WiFi modules and user heading data obtained from the gyroscope and accelerometer on the smartphone are first collected at the positions predetermined for the localization indoors in the same manner as the previous study [13]. Then, the collected data (fingerprints) are transmitted to the web server and then transformed into the fingerprint database using the machine learning algorithm in the same manner as the previous study [13]. The difference between this study and the previous study is that this study utilizes the QETKF algorithm for the indoor positioning of users in the online positioning phase, whereas the previous study employs the SKPF, as shown in Figure 1. In the prediction step (i.e., PDR) of the online positioning phase, the QETKF predicts the user position with both the traveled distance of the user obtained from the adaptive step length estimation using the accelerometer and the heading information of the user obtained from the gyroscope and accelerometer on the smartphone as in the prediction step of the SKPF of the previous study [13]. In the update step of the online positioning phase, the QETKF first estimates the positional measurement using the RSS fingerprinting approach based on the machine learning algorithm without the help of the GNSS that provides considerably inaccurate localization information owing to the obstruction of the signals indoors, in the same manner as the update step of the SKPF of the previous study [13]. Then, the QETKF estimates the updated user location by fusing the predicted position by the prediction step (PDR) and the positional measurement estimated by the RSS fingerprinting scheme using the ensemble transformation, whereas the SKPF infers the updated user location by fusing them using both the unscented transformation (UT) of UKF and the weighting method of PF. The position estimation procedure of the QETKF using the ensemble transformation is addressed in more detail in Section 3.
- Similar to the previous study, pedestrian positioning experiments in this study were executed using the indoor localization system implemented on the smartphone in a campus building. However, unlike the previous study, the performance benefits of the QETKF applied to the indoor localization system were evaluated using the existing EBKFs as benchmarks, including the EnKF, EnKF-PO, EnSRF, and ETKF that have not been widely used for indoor pedestrian localization. Experimental results show that the QETKF can offer more accurate positioning results than the EnKF and ETKF and can accomplish positioning performance that is as accurate as the EnKF-PO and EnSRF. This indicates that the QETKF has great potential in carrying out better position estimation with more accurately estimated error covariances for the indoor pedestrian localization system. Furthermore, to examine the validity of the QETKF algorithm as an ensemble data assimilation method, the QEKTF was applied to the Lorenz-96 (L96) model [31], which is commonly used as a simple chaotic dynamical system in ensemble data assimilation. Data assimilation experimental results executed using the L96 model and many observation variables indicate that the QEKTF can provide higher state estimation accuracy than other EBKFs while consuming less computational time compared to the EnKF, EnKF-PO, and EnSRF.
2. Related Work
2.1. EBKF
2.2. ETKF
2.2.1. Prediction (Ensemble Forecast)
2.2.2. Update (Observation Assimilation)
2.3. Model Error Issues with ETKF
3. Methodology
3.1. QETKF
3.2. Indoor Positioning System
3.3. QETKF-Based Localization Algorithm
Algorithm 1 QETKF-based indoor positioning approach |
Calculate ensemble members, ensemble mean, and its error covariance • Prediction (ensemble forecast) Determine the prior ensemble by transforming the posterior ensemble at time to the next time k using (6) in [13] for do end for Compute the prior ensemble mean and its error covariance using (4) and (23) • Update (observation assimilation) Determine the prior ensemble in observation space using (7) in [13] for do end for Compute the posterior error covariance in ensemble space using (12) Determine the Kalman gain , posterior ensemble mean , error covariance , and (24), respectively Determine the posterior ensemble by the ensemble sampling using (25) |
4. Results and Discussion
4.1. Indoor Pedestrian Positioning Experiments
4.1.1. Experimental Settings
4.1.2. Experiment with Ensemble Size and Positioning Scheme
4.1.3. Positioning Accuracy and Computational Time
4.1.4. Rank Histogram
4.2. Cycling Data Assimilation Experiments
4.2.1. L96 Model
4.2.2. Experimental Settings
4.2.3. Estimation Accuracy and Computational Time
5. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Posterior Error Covariance for the QETKF
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Notation | Description |
---|---|
P | Position prediction (i.e., PDR) through user direction and acceleration values obtained by accelerometer and gyroscope |
PU1 | Position prediction by P and position update (correction) through the positional observation obtained with user direction and WiFi RSS values |
PU2 | As in PU1, but with user direction and iBeacon RSS values |
PU3 | As in PU1, but with user direction, WiFi RSS, and iBeacon RSS values |
Bayes Filter | PU1 | PU2 | PU3 | |||||
---|---|---|---|---|---|---|---|---|
Positioning Error (cm) | Computational Time (s) | Positioning Error (cm) | Computational Time (s) | Positioning Error (cm) | Computational Time (s) | |||
KF | 32.32 | 2.41 × 10 | 31.01 | 2.42 × 10 | 32.76 | 2.41 × 10 | ||
EnKF (with ) | 26.83 | 2.52 × 10 | 27.69 | 2.52 × 10 | 28.73 | 2.51 × 10 | ||
EnKF-PO (with ) | 20.24 | 2.55 × 10 | 21.34 | 2.54 × 10 | 23.70 | 2.53 × 10 | ||
EnSRF (with ) | 21.29 | 2.54 × 10 | 22.92 | 2.53 × 10 | 23.22 | 2.54 × 10 | ||
ETKF (with ) | 24.59 | 2.52 × 10 | 24.32 | 2.52 × 10 | 27.38 | 2.53 × 10 | ||
QETKF (with ) | 14.57 | 2.53 × 10 | 16.31 | 2.52 × 10 | 17.02 | 2.52 × 10 |
Bayes Filter | PU1 | PU2 | PU3 | |||||
---|---|---|---|---|---|---|---|---|
Positioning Error (cm) | Computational Time (s) | Positioning Error (cm) | Computational Time (s) | Positioning Error (cm) | Computational Time (s) | |||
KF | 199.19 | 2.41 × 10 | 184.12 | 2.42 × 10 | 192.78 | 2.41 × 10 | ||
EnKF (with ) | 188.20 | 2.60 × 10 | 177.48 | 2.59 × 10 | 182.68 | 2.59 × 10 | ||
EnKF-PO (with ) | 165.59 | 2.59 × 10 | 164.56 | 2.58 × 10 | 169.76 | 2.59 × 10 | ||
EnSRF (with ) | 175.65 | 2.59 × 10 | 147.85 | 2.62 × 10 | 172.40 | 2.58 × 10 | ||
ETKF (with ) | 180.41 | 2.59 × 10 | 162.66 | 2.59 × 10 | 177.94 | 2.59 × 10 | ||
QETKF (with ) | 157.08 | 2.58 × 10 | 128.43 | 2.59 × 10 | 145.83 | 2.59 × 10 |
EBKF Algorithm | RMSE (Actual Uncertainty) | Ensemble Spread (Estimated Uncertainty) | Computational Time (s) |
---|---|---|---|
EnKF () | 0.97 | 0.45 | 16.84 |
EnKF-PO () | 0.65 | 0.56 | 17.72 |
EnSRF () | 0.64 | 0.57 | 17.03 |
ETKF () | 0.83 | 0.51 | 14.37 |
QETKF () | 0.55 | 0.53 | 14.95 |
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Sung, K. Pedestrian Positioning Using an Enhanced Ensemble Transform Kalman Filter. Sensors 2023, 23, 6870. https://doi.org/10.3390/s23156870
Sung K. Pedestrian Positioning Using an Enhanced Ensemble Transform Kalman Filter. Sensors. 2023; 23(15):6870. https://doi.org/10.3390/s23156870
Chicago/Turabian StyleSung, Kwangjae. 2023. "Pedestrian Positioning Using an Enhanced Ensemble Transform Kalman Filter" Sensors 23, no. 15: 6870. https://doi.org/10.3390/s23156870
APA StyleSung, K. (2023). Pedestrian Positioning Using an Enhanced Ensemble Transform Kalman Filter. Sensors, 23(15), 6870. https://doi.org/10.3390/s23156870