偏差未補(bǔ)償自適應(yīng)邊緣化容積卡爾曼濾波跟蹤方法

鄧洪高; 余潤華; 紀(jì)元法; 吳孫勇; 孫少帥

doi:10.11999/JEIT240469

偏差未補(bǔ)償自適應(yīng)邊緣化容積卡爾曼濾波跟蹤方法

doi: 10.11999/JEIT240469

鄧洪高^{1, 4},
余潤華^{1, 2},
紀(jì)元法^{1, 2, 4},
吳孫勇^{2, 3, ,},
孫少帥¹

1.
桂林電子科技大學(xué)信息與通信學(xué)院桂林 541004
2.
桂林電子科技大學(xué)廣西精密導(dǎo)航技術(shù)與應(yīng)用重點(diǎn)實(shí)驗(yàn)室桂林 541004
3.
桂林電子科技大學(xué)數(shù)學(xué)與計(jì)算科學(xué)學(xué)院桂林 541004
4.
衛(wèi)星導(dǎo)航定位與位置服務(wù)國家地方聯(lián)合工程研究中心桂林 541004

基金項(xiàng)目: 廣西重點(diǎn)研發(fā)項(xiàng)目(AB23026150, AB23026147)，國家自然科學(xué)基金(U23A20280)

詳細(xì)信息

作者簡介:
鄧洪高：男，研究員，研究方向?yàn)槔走_(dá)信號(hào)處理和衛(wèi)星導(dǎo)航

余潤華：男，碩士生，研究方向?yàn)槔走_(dá)多目標(biāo)跟蹤、空間誤差配準(zhǔn)和多源信息融合

紀(jì)元法：男，教授，研究方向?yàn)樾l(wèi)星導(dǎo)航和信號(hào)處理

吳孫勇：男，教授，研究方向?yàn)槔走_(dá)信號(hào)處理、多目標(biāo)檢測與跟蹤和多源信息融合

孫少帥：女，碩士，研究方向?yàn)轫?xiàng)目管理

通訊作者:
吳孫勇　wusunyong121991@163.com

中圖分類號(hào): V243.2; TN953
計(jì)量
- 文章訪問數(shù): 155
- HTML全文瀏覽量: 35
- PDF下載量: 41
- 被引次數(shù): 0
出版歷程
- 收稿日期: 2024-06-11
- 修回日期: 2024-11-30
- 網(wǎng)絡(luò)出版日期: 2024-12-09
- 刊出日期: 2025-01-31

An Adaptive Target Tracking Method Utilizing Marginalized Cubature Kalman Filter with Uncompensated Biases

DENG Honggao^{1, 4},
YU Runhua^{1, 2},
JI Yuanfa^{1, 2, 4},
WU Sunyong^{2, 3
, ,},
SUN Shaoshuai¹

1.
School of Information and Communication, Guilin University of Electronic Technology, Guilin 541004, China
2.
Guangxi Key Laboratory of Precision Navigation Technology and Application, Guilin University of Electronic Technology, Guilin 541004, China
3.
School of Mathematics and Computing Science, Guilin University of Electronic Science and Technology, Guilin, 541004, China
4.
National & Local Joint Engineering Research Center of Satellite Navigation Positioning and Location Service, Guilin 541004, China

Funds: Guangxi Key Research and Development Project (AB23026150, AB23026147), The National Natural Science Foundation of China (U23A20280)

摘要

摘要: 針對(duì)存在突變測量偏差和未知時(shí)變量測噪聲場景下的目標(biāo)跟蹤問題，該文提出一種偏差未補(bǔ)償自適應(yīng)邊緣化容積卡爾曼濾波跟蹤方法。首先通過建立差分量測方程來消除恒定的測量偏差，同時(shí)構(gòu)建滿足beta-Bernoulli分布的指示變量識(shí)別突變測量偏差，將相鄰時(shí)刻目標(biāo)狀態(tài)擴(kuò)維以滿足實(shí)時(shí)濾波需求，利用逆Wishart分布建模未知量測噪聲協(xié)方差矩陣，從而建立目標(biāo)狀態(tài)、指示變量、噪聲協(xié)方差矩陣的聯(lián)合分布，并通過變分貝葉斯推斷來求解各個(gè)參數(shù)的近似后驗(yàn)。為減小濾波負(fù)擔(dān)，對(duì)擴(kuò)維后的狀態(tài)向量進(jìn)行邊緣化處理，結(jié)合容積卡爾曼濾波方法實(shí)現(xiàn)邊緣化容積卡爾曼濾波跟蹤。仿真實(shí)驗(yàn)結(jié)果表明，所提方法能夠同時(shí)處理突變測量偏差和未知時(shí)變量測噪聲，從而對(duì)目標(biāo)進(jìn)行有效跟蹤。
- 突變測量偏差 /
- Beta-Bernoulli分布 /
- 逆Wishart分布 /
- 變分貝葉斯推斷 /
- 邊緣化容積卡爾曼濾波
Abstract: Objective In radar target tracking, tracking accuracy is often influenced by sensor measurement biases and measurement noise. This is particularly true when measurement biases change abruptly and measurement noise is unknown and time-varying. Ensuring effective target tracking under these conditions poses a significant challenge. An adaptive target tracking method is proposed, utilizing a marginalized cubature Kalman filter to address this issue. Methods (1) Initially, measurements taken at adjacent time points are differentiated to formulate the differential measurement equation, thereby effectively mitigating the influence of measurement biases that are either constant or change gradually between adjacent observations. Concurrently, the target states at these moments are expanded to create an extended state vector facilitating real-time filtering. (2) Following the differentiation of measurements, sudden changes in measurement biases may cause the differential measurement at the current moment to be classified as outliers. To identify the occurrence of these abrupt bias changes, a Beta-Bernoulli indicator variable is established. If such a change is detected, the differential measurement for that moment is disregarded, and the predicted state is adopted as the updated state. In the absence of any abrupt changes, standard filtering procedures are conducted. The Gaussian measurement noise, despite having unknown covariance, continues to follow a Gaussian distribution after differentiation, allowing its covariance matrix to be modeled using the inverse Wishart distribution. (3) A joint distribution is formulated for the target state, indicator variables, and the covariance matrix of the measurement noise. The approximate posteriors of each parameter are derived using variational Bayesian inference. (4) To mitigate the increased filtering burden arising from the high-dimensional extended state vector, the extended target state is marginalized, and a marginalized cubature Kalman filter for target tracking is implemented in conjunction with the cubature Kalman filtering method. Results and Discussions The target tracking performance is clearly illustrated, indicating that the proposed method accurately identifies abrupt measurement biases while effectively managing unknown time-varying measurement noise. This leads to a tracking performance that significantly exceeds that of the comparative methods. The findings further support the conclusions by examining the Root Mean Square Error (RMSE). Additionally, the stability of the proposed method is demonstrated. The results reveal that the computational load associated with the proposed method is greatly reduced through marginalization processing. This reduction occurs because, during the variational Bayesian iteration process, cubature sampling and integration are performed multiple times. Once the target state is marginalized, the dimensionality of the cubature sampling is halved, and the number of sampling points for each variational iteration is also reduced by half. As a result, the computational load during the nonlinear propagation of the sampling points decreases, with the amount of computation reduction increasing with the number of variational iterations. Furthermore, the results demonstrate that marginalization does not compromise tracking accuracy, thereby further validating the effectiveness of marginalization processing. This finding also confirms that marginalization processing can be extended to other nonlinear variational Bayesian filters based on deterministic sampling, providing a means to reduce computational complexity. Conclusions This paper proposes an adaptive marginalized cubature Kalman filter to improve target tracking in scenarios with measurement biases and unknown time-varying measurement noise. The approach incorporates measurement differencing to eliminate constant biases, constructs indicator variables to detect abrupt biases, and models the unknown measurement noise covariance matrix using the inverse Wishart distribution. A joint posterior distribution of the parameters is established, and the approximate posteriors are solved through variational Bayesian inference. Additionally, marginalization of the target state is performed before implementing tracking within the CKF framework, reducing the filtering burden. The results of our simulation experiments yield the following conclusions: (1) The proposed method demonstrates superior target tracking performance compared to existing techniques in scenarios involving abrupt measurement biases and unknown measurement noise; (2) The marginalization processing strategy significantly alleviates the filtering burden of the proposed filter, making it applicable to more complex nonlinear variational Bayesian filters, such as robust nonlinear random finite set filters, to reduce filtering complexity; (3) This filtering methodology can be extended to target tracking scenarios in higher dimensions.
- Abrupt measurement biases /
- Beta-Bernoulli distribution /
- Inverse Wishart distribution /
- Variational Bayesian inference /
- Marginalized cubature Kalman filter

HTML全文

圖 1 跟蹤效果

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圖 2 E[r]值變化圖

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圖 3 不同方法的位置RMSE

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圖 4 不同方法的速度RMSE

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圖 5 不同方法的位置ARMSE

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圖 6 不同方法的速度ARMSE

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圖 7 不同$\varepsilon $值的位置ARMSE

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圖 8 不同$\varepsilon $值的速度ARMSE

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圖 9 邊緣化處理前后的RMSE差值

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圖 10 邊緣化處理前后的ARMSE差值

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1 偏差未補(bǔ)償自適應(yīng)邊緣化容積卡爾曼濾波跟蹤算法

輸入：狀態(tài)估計(jì)值${{\boldsymbol{\tilde x}}_{0\|0}}$，誤差協(xié)方差${{\boldsymbol{P}}_{0\|0}}$，逆Wishart分布參數(shù) 　${u_{0\|0}}$, ${{\boldsymbol{U}}_{0\|0}}$，貝塔分布參數(shù)${\alpha _0}$, ${\beta _0}$，初始時(shí)刻伯努利變量的期望　值${\text{E}}[{r_0}]$，遺忘因子$ \rho $，迭代次數(shù)N。
輸出：${{\boldsymbol{\tilde x}}_{k\|k}}$,${{\boldsymbol{P}}_{k\|k}}$, ${u_{k\|k}}$, ${{\boldsymbol{U}}_{k\|k}}$。
(1) for k = 1:K
(2) 　通過式(23)計(jì)算${{\boldsymbol{\tilde x}}_{k\|k - 1}}$, ${{\boldsymbol{P}}_{k\|k - 1}}$, $ {{\boldsymbol{C}}_{k\|k - 1}} $；
(3) 　計(jì)算：$ {u_{k\|k - 1}} = \rho {u_{k - 1\|k - 1}} $, ${{\boldsymbol{U}}_{k\|k - 1}} = \rho {{\boldsymbol{U}}_{k - 1\|k - 1}}$；
(4) 　初始化：$ {\boldsymbol{\tilde x}}_{k\|k}^{{\text{(0)}}} = {{\boldsymbol{\tilde x}}_{k\|k - 1}} $, ${\boldsymbol{P}}_{k\|k}^{{\text{(0)}}} = {{\boldsymbol{P}}_{k\|k - 1}}$, 　　　　$ u_{k\|k}^{(0)} = {u_{k\|k - 1}} $, $ {\boldsymbol{U}}_{k\|k}^{(0)} = {{\boldsymbol{U}}_{k\|k - 1}} $；
(5) 　for i = 0:N
(6) 　　通過式(14)計(jì)算${{\stackrel \frown{{\boldsymbol{R}}} }}_k^{(i + 1)}$；
(7) 　　通過式(35)計(jì)算${\boldsymbol{\tilde x}}_{k\|k}^{{\text{(}}i{\text{ + 1)}}}$, ${\boldsymbol{P}}_{k\|k}^{{\text{(}}i{\text{ + 1)}}}$；
(8) 　　通過式(18)計(jì)算${({\text{E}}[{r_k}])^{(i + 1)}}$；
(9) 　　根據(jù)式(34)判斷傳感器測量偏差是否突變
(10) 　　若傳感器測量偏差突變：
(11) 　　 ${{\boldsymbol{\tilde x}}_{k\|k}} = {\boldsymbol{\tilde x}}_{k\|k}^{{\text{(}}0{\text{)}}}$, ${{\boldsymbol{P}}_{k\|k}} = {\boldsymbol{P}}_{k\|k}^{(0)}$；
(12) 　　 ${u_{k\|k}} = u_{k\|k}^{(0)}$, ${{\boldsymbol{U}}_{k\|k}} = {\boldsymbol{U}}_{k\|k}^{(0)}$；
(13) 　　跳出循環(huán)；
(14) 　若傳感器測量偏差不突變：
(15) 　　 ${{\boldsymbol{\tilde x}}_{k\|k}} = {\boldsymbol{\tilde x}}_{k\|k}^{{\text{(}}i{\text{ + 1)}}}$,${{\boldsymbol{P}}_{k\|k}} = {\boldsymbol{P}}_{k\|k}^{(i + 1)}$；
(16) 　　通過式(20)計(jì)算$\alpha _k^{(i + 1)}$和$\beta _k^{(i + 1)}$；
(17) 　　通過式(22)計(jì)算$u_{k\|k}^{(i + 1)}$和$ {\boldsymbol{U}}_{k\|k}^{(i + 1)} $；
(18) 　　 ${u_{k\|k}} = u_{k\|k}^{{\text{(}}i{\text{ + 1)}}}$, $ {{\boldsymbol{U}}_{k\|k}} = {\boldsymbol{U}}_{k\|k}^{(i + 1)} $；
(19)　　計(jì)算迭代停止閾值$\kappa $：　　　　　　$\kappa = \|\|{\boldsymbol{\tilde x}}_{k\|k}^{{\text{(}}i + 1{\text{)}}} - {\boldsymbol{\tilde x}}_{k\|k}^{{\text{(}}i{\text{)}}}\|\|/\|\|{\boldsymbol{\tilde x}}_{k\|k}^{{\text{(}}i{\text{)}}}\|\|$；
(20) 　　當(dāng)$\kappa \le {10^{ - 6}}$時(shí)：
(21) 　　　跳出循環(huán)。
(22)　　 end for
(23) end for

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表 1 運(yùn)行時(shí)間對(duì)比(s)

方法	時(shí)間
傳統(tǒng)CKF	1.038 1
增量CKF	1.210 0
NRCKF	11.343 8
提出的方法(非邊緣化)	12.866 4
提出的方法(邊緣化)	7.333 3

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表 2 僅測量偏差${{\boldsymbol}_k}$變化時(shí)各方法ARMSE對(duì)比

	傳統(tǒng)CKF	增量CKF	NRCKF	提出的邊緣化CKF
${\text{ARMS}}{{\text{E}}_{{\text{pos}}}}$(m)	155.359 1	78.411 8	47.686 8	15.359 0
${\text{ARMS}}{{\text{E}}_{{\text{vel}}}}$(m/s)	6.533 4	3.533 9	2.062 2	1.549 0

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表 3 僅量測協(xié)方差矩陣${{\boldsymbol{R}}_k}$變化時(shí)各方法ARMSE對(duì)比

	傳統(tǒng)CKF	增量CKF	NRCKF	提出的邊緣化CKF
${\text{ARMS}}{{\text{E}}_{{\text{pos}}}}$(m)	49.822 0	29.690 6	22.204 5	9.597 0
${\text{ARMS}}{{\text{E}}_{{\text{vel}}}}$(m/s)	1.481 2	1.655 1	1.696 7	1.518 7

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參考文獻(xiàn)(21)

[1]	LIN X and BAR-SHALOM Y. Multisensor target tracking performance with bias compensation[J]. IEEE Transactions on Aerospace and Electronic Systems, 2006, 42(3): 1139–1149. doi: 10.1109/TAES.2006.248212.
[2]	董云龍, 張焱. 雷達(dá)系統(tǒng)偏差精確配準(zhǔn)技術(shù)研究綜述[J]. 現(xiàn)代雷達(dá), 2024, 46(3): 1–8. doi: 10.16592/j.cnki.1004-7859.2024.03.001. DONG Yunlong and ZHANG Yan. A review of radar system deviation accurate registration technology[J]. Modern Radar, 2024, 46(3): 1–8. doi: 10.16592/j.cnki.1004-7859.2024.03.001.
[3]	崔亞奇, 熊偉, 何友. 基于MLR的機(jī)動(dòng)平臺(tái)傳感器誤差配準(zhǔn)算法[J]. 航空學(xué)報(bào), 2012, 33(1): 118–128. doi: 11-1929/V.20111213.1132.001. CUI Yaqi, XIONG Wei, and HE You. Mobile platform sensor registration algorithm based on MLR[J]. Acta Aeronautica et Astronautica Sinica, 2012, 33(1): 118–128. doi: 11-1929/V.20111213.1132.001.
[4]	XIU Jianjuan, DONG Kai, and HE You. Systematic error real-time registration based on modified input estimation[J]. Journal of Systems Engineering and Electronics, 2016, 27(5): 986–992. doi: 10.21629/JSEE.2016.05.06.
[5]	GENG Han, LIANG Yan, LIU Yurong, et al. Bias estimation for asynchronous multi-rate multi-sensor fusion with unknown inputs[J]. Information Fusion, 2018, 39: 139–153. doi: 10.1016/j.inffus.2017.03.002.
[6]	CHUGHTAI A H, MAJAL A, TAHIR M, et al. Variational-based nonlinear Bayesian filtering with biased observations[J]. IEEE Transactions on Signal Processing, 2022, 70: 5295–5307. doi: 10.1109/TSP.2022.3217921.
[7]	HUANG Yulong, JIA Guangle, CHEN Badong, et al. A new robust Kalman filter with adaptive estimate of time-varying measurement bias[J]. IEEE Signal Processing Letters, 2020, 27: 700–704. doi: 10.1109/LSP.2020.2983552.
[8]	傅惠民, 吳云章, 婁泰山. 欠觀測條件下的增量Kalman濾波方法[J]. 機(jī)械強(qiáng)度, 2012, 34(1): 43–47. doi: 10.16579/j.issn.1001.9669.2012.01.014. FU Huimin, WU Yunzhang, and LOU Taishan. Incremental Kalman filter method under poor observation condition[J]. Journal of Mechanical Strength, 2012, 34(1): 43–47. doi: 10.16579/j.issn.1001.9669.2012.01.014.
[9]	傅惠民, 婁泰山, 吳云章. 欠觀測條件下的擴(kuò)展增量Kalman濾波方法[J]. 航空動(dòng)力學(xué)報(bào), 2012, 27(4): 777–781. doi: 10.13224/j.cnki.jasp.2012.04.004. FU Huimin, LOU Taishan, and WU Yunzhang. Extended incremental Kalman filter method under poor observation condition[J]. Journal of Aerospace Power, 2012, 27(4): 777–781. doi: 10.13224/j.cnki.jasp.2012.04.004.
[10]	馬麗麗, 趙甜甜, 陳金廣. 欠觀測條件下的增量容積卡爾曼濾波[J]. 計(jì)算機(jī)工程, 2014, 40(10): 228–231,238. doi: 10.3969/j.issn.1000-3428.2014.10.043. MA Lili, ZHAO Tiantian, and CHEN Jinguang. Incremental cubature Kalman filtering under poor observation condition[J]. Computer Engineering, 2014, 40(10): 228–231,238. doi: 10.3969/j.issn.1000-3428.2014.10.043.
[11]	傅惠民, 吳云章, 婁泰山. 自適應(yīng)無跡增量濾波方法[J]. 航空動(dòng)力學(xué)報(bào), 2013, 28(2): 259–263. doi: 10.13224/j.cnki.jasp.2013.02.008. FU Huimin, WU Yunzhang, and LOU Taishan. Adaptive unscented incremental filter method[J]. Journal of Aerospace Power, 2013, 28(2): 259–263. doi: 10.13224/j.cnki.jasp.2013.02.008.
[12]	孫小君, 周晗, 閆廣明. 基于新息的自適應(yīng)增量Kalman濾波器[J]. 電子與信息學(xué)報(bào), 2020, 42(9): 2223–2230. doi: 10.11999/JEIT190493. SUN Xiaojun, ZHOU Han, and YAN Guangming. Adaptive incremental Kalman filter based on innovation[J]. Journal of Electronics & Information Technology, 2020, 42(9): 2223–2230. doi: 10.11999/JEIT190493.
[13]	LOU Taishan, YANG Ning, WANG Yan, et al. Target tracking based on incremental center differential Kalman filter with uncompensated biases[J]. IEEE Access, 2018, 6: 66285–66292. doi: 10.1109/ACCESS.2018.2879118.
[14]	黃玉龍, 張勇剛, 武哲民, 等. 帶有色厚尾量測噪聲的魯棒高斯近似濾波器和平滑器[J]. 自動(dòng)化學(xué)報(bào), 2017, 43(1): 114–131. doi: 10.16383/j.aas.2017.c150810. HUANG Yulong, ZHANG Yonggang, WU Zhemin, et al. Robust Gaussian approximate filter and smoother with colored heavy tailed measurement noise[J]. Acta Automatica Sinica, 2017, 43(1): 114–131. doi: 10.16383/j.aas.2017.c150810.
[15]	WANG Guoqing, ZHAO Jiaxiang, YANG Chunyu, et al. Robust Kalman filter for systems with colored heavy-tailed process and measurement noises[J]. IEEE Transactions on Circuits and Systems II: Express Briefs, 2023, 70(11): 4256–4260. doi: 10.1109/TCSII.2023.3283547.
[16]	WANG Hongwei, LI Hongbin, FANG Jun, et al. Robust Gaussian Kalman filter with outlier detection[J]. IEEE Signal Processing Letters, 2018, 25(8): 1236–1240. doi: 10.1109/LSP.2018.2851156.
[17]	HOSTETTLER R and S?RKK? S. Rao-blackwellized Gaussian smoothing[J]. IEEE Transactions on Automatic Control, 2019, 64(1): 305–312. doi: 10.1109/TAC.2018.2828087.
[18]	JIA Guangle, HUANG Yulong, ZHANG Yonggang, et al. A novel adaptive Kalman filter with unknown probability of measurement loss[J]. IEEE Signal Processing Letters, 2019, 26(12): 1862–1866. doi: 10.1109/LSP.2019.2951464.
[19]	GRANSTR?M K and ORGUNER U. On the reduction of Gaussian inverse Wishart mixtures[C]. 2012 15th International Conference on Information Fusion, Singapore, 2012: 2162–2169.
[20]	GU Peng, JING Zhongliang, and WU Liangbin. Robust adaptive multi-target tracking with unknown measurement and process noise covariance matrices[J]. IET Radar, Sonar & Navigation, 2022, 16(4): 735–747. doi: 10.1049/rsn2.12216.
[21]	ZHU Jiangbo, XIE Weixin, and LIU Zongxiang. Student’s t-based robust Poisson multi-Bernoulli mixture filter under heavy-tailed process and measurement noises[J]. Remote Sensing, 2023, 15(17): 4232. doi: 10.3390/rs15174232.