Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements
Abstract
:1. Introduction
- The RCKF was developed by integrating Huber’s M-estimation theory with the standard CKF to effectively handle nonlinear systems, with missing measurements characterized using random variables following the Bernoulli distribution.
- The RCKF exhibited superior performance compared to the EKF, EnKF, and CKF in terms of accuracy and reliability on two moving-target tracking models (UNGM and BOT) with missing measurements, indicating that the RCKF is the most effective approach for nonlinear systems with missing measurements.
2. Nonlinear System with Missing Measurements
- The initial state follows a Gaussian distribution, i.e., .
- The noise sequences and are independent Gaussian sequences with zero means, and the covariance matrix of is denoted as , while the covariance matrix of is denoted as .
3. Cubature Kalman Filter with Missing Measurements
- The time prediction is as follows:
- I
- The posterior probability distribution of a given time isBy Cholesky decomposition, The expression denoting the error covariance at time , denoted as , is given by
- II
- Calculating the cubature points.
- III
- Predicting state.The t-th cubature point’s predicted state from time to time k is defined as
- The measurement update is as follows, including the error covariance at time k:
- I
- Factorizing the CM of the error .
- II
- Calculating the cubature points.
- III
- Updating observation.The estimated observation of the t-th cubature point between epochs and k is denoted byFrom (24), we can obtain the predicted observation of the t-th cubature point from time to :
- IV
- Calculating the Kalman gain.
- V
- State update.
- VI
- CM of the estimate error update.
4. Robust Cubature Kalman Filter with Missing Measurements
5. Metrics of Performance
- Root mean square error (RMSE).The RMSE for the state estimate generated utilizing M Monte Carlo simulations at time instant k is as follows:
- Non-credibility index.In order to calculate the NCI, we compared the estimator’s normalized squared estimation error, which is defined asThe NCI can measure the estimator’s credibility. That is, the estimator’s CM is close to the MSE . The lower the NCI score, the more reliable the estimator; therefore, an NCI score of zero indicates an entirely credible estimator.
6. Numerical Experiments
6.1. Model Specifications
- The UNGM.This model is described as follows:
- BOT.There are two states inside the bearing-only tracking (BOT) paradigm, with the state displaying a tracked target’s positioning in Cartesian coordinates. Its nonlinear model is as follows:
6.2. Experiment and Analysis
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
Abbreviations | ||
RCKF | Robust cubature Kalman filter | |
CKF | Cubature Kalman filter | |
EKF | Extended Kalman filter | |
UKF | Unscented Kalman filter | |
EnKF | Ensemble Kalman filter | |
CM | Covariance matrix | |
MSE | Mean square error | |
RMSE | Root mean square error | |
NCI | Non-credibility index | |
UNGM | Univariate Non-stationary Growth Model | |
BOT | Bearing-only tracking | |
Symbols | ||
The state vector | ||
The measurement vector | ||
Nonlinear function of the state | ||
Nonlinear function of the measurement | ||
The process noise | ||
The measurement noise | ||
The covariance matrix of | ||
The covariance matrix of | ||
Factor of missing measurement | ||
The predicted state estimation | ||
The predicted measurement estimation | ||
Predicted error covariance estimation | ||
Estimated matrix of innovation covariance | ||
Estimated cross-covariance matrix | ||
Kalman gain | ||
Estimated update state | ||
The cubature point | ||
Estimated matrix of innovation covariance using an absence of difference M-estimation approach | ||
Residue vectors t-th | ||
Residue vectors t-th | ||
Mean variance of |
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Authors | Method | Results |
---|---|---|
Tiancheng Li et al. [17] | Huber’s M-estimation-based robust CKF and robust square root CKF adapted to anomalous measurement noise using innovation covariance comparison. | Simulations demonstrated the superior performance in terms of accuracy, robustness, and reliability compared to standard methods for target tracking. |
Zhao et al. [16] | Robust adaptive CKF to reduce kinematic model errors through covariance adjustment and dynamic disturbance processing. | The experiment confirmed the proposed strategy’s effectiveness in dynamic systems with high dynamics and weak signals. |
Cui Bingbo et al. [18] | RCKF enhanced GNSS/INS accuracy in GNSS-denied environments by considering noise using missing observations. | Numerical experiments and field tests demonstrated the RCKF’s superior robustness compared to the CKF and EKF. |
Xiangzhou Ye et al. [19] | Adaptive robust CKF (ARCKF) based on the H-infinity CKF by incorporating two adaptable algorithm components to address erroneous system models and noise statistics. | Simulations favored the recommended approach over the HCKF for handling model errors and aberrant observations. |
Method | Average RMSE When | Average RMSE When |
---|---|---|
EKF | 5.41 | 4.63 |
EnKF | 4.82 | 5.32 |
CKF | 3.60 | 4.16 |
RCKF | 3.27 | 1.60 |
Method | Average RMSE When | Average RMSE When | Average RMSE | Average RMSE When |
---|---|---|---|---|
EKF | 17.50 | 17.19 | 0.64 | 0.71 |
EnKF | 74.38 | 63.78 | 22.79 | 20.20 |
CKF | 0.067 | 0.07 | 0.55 | 0.54 |
RCKF | 0.057 | 0.07 | 0.53 | 0.40 |
Method | Average NCI When | Average NCI When |
---|---|---|
EKF | 9.08 | 7.58 |
EnKF | 9.44 | 11.07 |
CKF | 5.23 | 7.46 |
RCKF | 4.83 | 7.10 |
Method | Average NCI When | Average NCI When | Average NCI When | Average NCI When |
---|---|---|---|---|
EKF | 3.91 | 0.69 | 6.91 | 6.76 |
EnKF | 3.89 | 0.47 | 3.72 | 0.88 |
CKF | 4.71 | 0.40 | 3.53 | 1.52 |
RCKF | 3.75 | 0.18 | 2.65 | 0.29 |
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Sahl, S.; Song, E.; Niu, D. Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements. Sensors 2024, 24, 392. https://doi.org/10.3390/s24020392
Sahl S, Song E, Niu D. Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements. Sensors. 2024; 24(2):392. https://doi.org/10.3390/s24020392
Chicago/Turabian StyleSahl, Samer, Enbin Song, and Dunbiao Niu. 2024. "Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements" Sensors 24, no. 2: 392. https://doi.org/10.3390/s24020392
APA StyleSahl, S., Song, E., & Niu, D. (2024). Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements. Sensors, 24(2), 392. https://doi.org/10.3390/s24020392