摘要
Cubature卡尔曼滤波(Cubature Kalman Filter, CKF)是一种性能优越的非线性滤波,具有算法实现简单、滤波精度高、收敛性好等优点,正逐渐成为当前及未来研究非线性估计问题的热点和有效方法。本文围绕CKF展开相关研究。论文的主要工作有：
对非线性滤波过程捕获的非线性函数均值进行泰勒展开,推导出不同维数非线性系统下CKF与UKF (Unscented Kalman Filter, UKF)(?)非线性滤波方法估计精度之间存在差别,指出了产生这种差别的主要原因：不同维数系统下,它们捕获的函数均值泰勒展开式高阶项与真实值的接近程度不一致且数值稳定性不同。分析研究后指出了不同维数非线性系统下两种滤波方法的选取原则。
通过分析研究滤波过程捕获传播的函数均值、方差和奇阶矩等统计信息,获得了基于Cubature变换的扩展CKF与非扩展CKF两种滤波方法之间的差别,指出了不同维数非线性系统下两者滤波精度不同以及所产生的原因：一维系统下,扩展CKF除了捕获的均值和方差更靠近真实值还额外捕获传播部分奇阶矩信息,导致其精度更高；而二维及以上系统,扩展CKF传播的统计信息反而误差更大,导致其精度比非扩展CKF差。从而指出了不同维数非线性系统下如何从两种滤波方法中选择最佳方法。
针对条件线性高斯模型中同时存在非线性与线性状态估计的问题,提出CKF-KF混合滤波算法,在滤波过程的两个不同阶段分别引入CKF及KF,该算法先用CKF估计模型中的非线性状态,再用KF估计模型中的线性状态。通过先后对非线性状态及线性状态进行Cubature采样而将CKF与KF匹配在一起,对线性状态也进行Cubature采样是它们融合的关键。由于CKF使用较少的采样点,在略降低精度下,整个CKF-KF计算量远远小于RBPF算法。CKF-KF滤波过程中,通过用非线性状态的估计误差反馈修正线性状态的估计获得了比直接使用CKF具有更高的精度。
重要性密度函数缺乏最新观测信息以及重采样破坏粒子多样性是粒子滤波产生粒子退化及滤波精度降低的重要原因,为此将CKF、高斯混合模型及EM算法引入粒子框架中,对其算法进行改进,分别提出CPF (Cubature Particle Filter)及GMCPF (Gaussian Mixture Cubature Particle Filter)算法。两算法均通过CKF设计重要性密度函数而考虑了最新观测信息,此夕(?)GMCPF通过高斯混合模型来更加真实地近似后验概率密度,同时在重要性采样结束后不执行重采样,而是用EM算法从重要性采样后的粒子集中恢复出高斯模型,以减轻粒子退化,改善滤波精度。
最后本文将CKF、CPF及GMCPF用于惯导方位大失准角初始对准中的失准角估计以及重力异常辅助惯导组合系统中,解决观测方程无法精确获知下的估计问题。仿真结果表明了上述非线性滤波的有效性。
Cubature Kalman filter (CKF) is a superior nonlinear filter that has a simple algorithm, high precision and good convergence, etc. And it is becoming a hot and efficient method to study the nonlinear estimation problems for the current and future. The thesis focuses on CKF to spread related research. The main works are as follows.
The nonlinear function mean captured in the nonlinear filter process is executed the Taylor expansion to derive the different estimation precision between CKF and UKF (Unscented Kalman Filter) under the nonlinear systems of different dimensions. And the main reasons for above difference are pointed out:the approaching degree between Taylor expansion high-order item of function mean captured by two filters and the true is not the same, besides,their numerical stabilities are different under different dimensions. The selection principle of the two filters under different dimension nonlinear systems is proposed after analysis and research.
The mean, variance and odd order moment of the function's statistical information captured in filter process is analyzed and researched to obtain the difference between augmented CKF and non-augmented CKF based on cubature transformation and point out that the filter precision is different and the reasons under the different dimension nonlinear systems:for one dimension system, the augmented CKF gains higher precision through capturing and propagating the mean,variance which are closer to true and also extra odd-order moment. But for the two dimension system and above, the function statistical information done by the augmented CKF largely deviates from true to make augmented CKF inferior to non-augmented CKF. Thus how to choose a better method from the two filters is put forward under different dimension systems.
The CKF-KF hybrid filter is proposed for conditionally linear Gaussian state model involving nonlinear state and linear state. The CKF and KF are introduced into the two different stages of the filter process; the hybrid algorithm firstly uses the CKF to estimate the nonlinear state and then uses KF to do the linear state. The CKF and KF are matched together through successively conducting cubature-sample to nonlinear and linear state, and cubature-sample to linear state is the key to their combination. On account of CKF using less sample points, the calculation amount of CKF-KF which loses slight precision is far less than the RBPF algorithm. The CKF-KF which performs a feedback correction to the linear state through the nonlinear state estimation error has higher precision than directly using CKF.
Importance density function lacking of the latest observation information and the re-sampling damaging the particle diversity are the important reasons for particle degradation and filter precision reduction. So the CKF, Gaussian mixture model and EM algorithm are introduced into particle frame to improve the algorithm, and the CPF (Cubature Particle Filter) and GMCPF (Gaussian Mixture Cubature Particle Filter) algorithm are proposed respectively. The two algorithms both use the latest observation information through CKF designing of the importance density function, and GMCPF use Gaussian mixture model to truly approach the posterior probability density. At the same time, importance sampling is not performed after re-sampling instead of using EM algorithm to regain Gaussian model from the importance sampling particle set, and thus the particle degradation and filter precision are improved.
At last, this thesis uses CKF to estimate the misalignment angle error under large azimuth misalignment angle initial alignment, CPF and GMCPF are carried out to solve the estimation problem when the observation equation can not be obtained precisely in the gravity anomaly assisted inertial navigation system. Simulation results show the effectiveness of the above nonlinear filters.
引文
[1]J.L.Crassidis and F.L.Markley. Unscented filtering for spacecraft attitude estimation. Journal of Guidance Control and Dynamics.2003,26(4):536-542P.
[2]Bin Jia, Ming Xin and Yang Cheng. Sparse Gauss-Hermite Quadrature Filter with application to Spacecraft Attitude Estimation. Journal of Guidance Control and Dynamics.2011,34(2):367-379P.
[3]Mohammad Abdelrahman and Sang-Young Park. Sigma-Point Kalam Filtering for Spacecraft Attitude and Rate Estimation using Magnetometer Measurements. IEEE Transactions on Aerospace and Electronic Systems.2011,47(2):1401-1415P.
[4]Yafeng Wang, Fuchun Sun, Youan Zhang, et al. Central Difference Particle Filter Applied to Transfer Alignment for SINS on Missiles. IEEE Transactions on Aerospace and Electronic Systems.2012,48(1):375-387P.
[5]David J. Allerton and Huamin Jia. Redundant Multi-Mode Filter for a Navigation System. IEEE Transactions on Aerospace and Electronic Systems.2007,43(1): 371-391P.
[6]Hang Liu, Sameh Nassar and Naser El-Sheimy. Two-Filter Smoothing for Accurate INS/GPS Land-Vehicle Navigation in Urban Centers. IEEE Transactions on Vehicular Technology.2010,59(9):4256-4267P.
[7]Jacques Georgy, Aboelmagd Noureldin, Michael J. Korenberg, et al. Low-Cost Three-Dimensional Navigation Solution for RISS/GPS Integration Using Mixture Particle Filter. IEEE Transactions on Vehicular Technology.2010,59(2):599-615P.
[8]陈永冰,钟斌.惯性导航原理.北京：国防工业出版社.2007
[9]秦永元.惯性导航.北京：科学出版社.2005
[10]秦永元,张洪钺,汪叔华.卡尔曼滤波与组合导航原理.西北工业大学出版社.1998.
[11]A. Farina, B. Ristic and D. Benvenuti. Tracking a ballistic target:Comparison of several nonlinear filters. IEEE Transactions on Aerospace and Electronic Systms. 2002,38(3):854-867P.
[12]B. Ristic, A. Farina, D. Benvenuti and M. S. Arulampalam. Performance bounds and comparison of nonlinear filters for tracking a ballistic object on reentry. IEE Proceedings Radar Sonar and Navigation.2003,150(2):65-70P.
[13]Ming Lei, Barend J. van WYK and Yong Qi. Online Estimation of the Approximation Posterior Cramer-Rao Lower Bound for Discrete-Time Nonlinear Filtering. IEEE Transactions on Aerospace and Electronic Systems.2011,47(1):37-57.
[14]Jinwhan Kim, S. S. Vaddi, P. K. Menon, et al. Comparison Between Nonlinear Filtering Techniques for Spiraling Ballistic Missile State Estimation. IEEE Transactions on Aerospace and Electronic Systems.2012,48(1):313-328.
[15]Haihua Chen, Shahram Shahbazpanahi and Alex B. Gershman. Filter-and-Forward Distributed Beamforming for Two-Way Relay Networks with Frequency Selective Channels. IEEE Transactions on Signal Processing.2012,60(4):1927-1941P.
[16]Fong W, Godsill S J, Doucet A, et al. Monte Carlo smoothing with application to audio signal enhancement. IEEE Transaction on Signal Processing.2002,50(2): 438-449P.
[17]Albin Svensson and Andreas Jakobsson. Adaptive detection of a partly known signal corrupted by strong interference. IEEE Signal Processing Letters.2011,18 (12):729-732P.
[18]Davide Anastasia and Yiannis Andreopoulos. Throughput-distortion computation of generic matrix multiplication:toward a computation channel for digital signal processing systems. IEEE Transaction on Signal Processing.2012,60(4):2024-2037P.
[19]M. OConnor, G. Elkaim, T. Bell and B. Parkinson. Automatic steering of farm vehicles using GPS. In 3rd International Conference on Precision Agriculture.1996,767-778P.
[20]N. D. Tillet. Automatic guidance for agricultural filed machines:a review. Journal of Agricultural Engineering Research.1991,50:167-187P.
[21]武元新.对偶四元数导航算法与非线性高斯滤波研究.长沙：国防科学技术大学博士论文.2005.
[22]Ienkaran Arasaratnam and Simon Haykin. Cubature Kalman Filters. IEEE Transactions on Automatic Control.2009,54(6):1254-1269P.
[23]S. J. Julier and J. K. Uhlman. A new extension of the Kalman filter to nonlinear systems. Proceedings of the Society of Photo-Optical Instrumentation Engineers.1997, 3068:182-193P.
[24]S. J. Julier, J. K. Uhlman and H. F. Durrant-Whyte. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Transactions on Automatic Control.2000,45(3):477-482P.
[25]S. J. Julier and J. K. Uhlman. Unscented filtering and nonlinear estimation. Proceedings of the IEEE.2004,92(3):401-422P.
[26]A. Doucet, N. de Freitas and N. Gordon. Sequential Monte Carlo Methods in Practice. Spring-Verlag.2001.
[27]N. J. Gordon, D. J. Salmond and A. F. M. Smith. A novel approach to nonlinear /non-Gaussian Bayesian state estimation. IEE Proceeding-F Radar and Signal Processing.1993,140(2):107-113.
[28]A. Doucet, Neil J. Gordon and Vikram Krishnamurthy. Particle Filters for State Estimation of Jump Markov Linear Systems. IEEE Transactions on Signal Processing. 2001,49(3):613-624P.
[29]A. Doucet, S. Godsill and C. Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing.2000,10:197-208P.
[30]C. Carter and R. Kohn. Markov chain Monte Carlo in conditionally Gaussian state space models. Biometrika.1996,83:589-601P.
[31]Chen R. and Liu J. S. Mixture Kalman filters. Journal of the Royal Statistical Society B.2000,62:493-508P.
[32]Mark Briers, Arnaud Doucet and Simon Maskell. Smoothing algorithms for state-space models. Annals of the institute of statistical mathematics.2010,62(1): 61-89P.
[33]M. S. Arulampalm, S. Maskell, N. Gordon, et al. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing. 2002,50(2):174-188P.
[34]Kong A, Liu J. S and Wong W. H. Sequential imputations and Bayesian missing data problems. Journal of the American Statiatical Association.1994,89(426):278-288P.
[35]万德钧,房建成.惯性导航初始对准.南京：东南大学出版社.1998.
[36]T. Kailath. A review of three decades of linear filtering theory. IEEE Transactions on Automatic Control.1974,20(2):146-181P.
[37]A. H. Jazwinski. Stochastic Processing and Filtering Theory. New York and London: Academic Press.1970.
[38]Z. Chen. Bayesian filtering:from Kalman filters to particle filters, and beyond. Hamilton:MeMaster University.2003.
[39]Fred Daum. Nonlinear Filters:Beyond the Kalman Filter. IEEE Aerospace and Electronic Systems Magazine.2005,20(8):57-69P.
[40]H. Tanizaki and R. S. Mariano. Nonlinear filters based on Taylor series expansion. Communications in Statistics-Theory and Methods.1996,25(6):1261-1282P.
[41]F. E. Daum. Industrial strength nonlinear filters. Proceedings of the Workshop on Estimation, Tracking and Fusion:A Tribute to Yaakov Bar-Shalom, Monterey CA. 2001.
[42]D. Brigo. Filtering by projection on the manifold of exponential densities. The Netherlands:PhD thesis, Free University of Amsterdam.1996.
[43]K. Ito. Approximation of the Zakai equation for nonlinear filtering. SIAM Journal on Control and Optimization.1996,34:620-634P.
[44]S. V. Lototsky, R. Mikulevicius and B. L. Rozovskii. Nonlinear filtering revisited:A spectral approach. SIAM Journal on Control and Optimization.1997,35:435-461P.
[45]M. Norgaard, N. K. Poulsen and O. Ravn. New developments in state estimation for nonlinear systems. Automatica.2000,36(11):1627-1638P.
[46]K. Ito and K. Q. Xiong. Gaussian filters for nonlinear filtering problems. IEEE Transactions on Automatic Control.2000,45(5):910-927P.
[47]J. H. Kotecha and P. A. Djuric. Gaussian particle filtering. IEEE Transactions on Signal Processing.2003,51(10):2592-2601P.
[48]Jayesh H. Kotecha and Petar M. Djuric. Gaussian Sum Particle Filtering. IEEE Transactions on Signal Processing.2003,51(10):2602-2612P.
[49]C. Kwok, D. Fox and M. Meila. Real-time particle filters. Proceedings of the IEEE. 2004,92(3):469-484P.
[50]M. Athans, R. P. Wishner and A. Bertolini. Suboptimal state estimation for continuous time nonlinear systems from discrete noisy measurements. IEEE Transactions on Automatic Control.1968,13:504-514P.
[51]R. E. Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engineering.1960,82:34-45P.
[52]R. H. Battin. Some funny things happened on the way to the moon. Journal of Guidance Control and Dynamics.2002,25(1):1-7P.
[53]S. J. Julier. A skewed approach to filtering. The 12th International Symposium on Aerospace Defense Sensing, Simulation and Controls:SPIE.1998,271:282P.
[54]E. A. Wan and R. v. d. Merwe. The Unscented Kalman Filter. In Kalman Filtering and Neural Networks, Chapter 7, S. Haykin. Ed:Wiley Publishing.2001,221-280P.
[55]R. v. d. Merwe. Sigma-point Kalman filters for probabilistic inference in dynamic state-space models. PhD thesis. Oregon Health & Science University.2004.
[56]K. Xiong, C. W. Chan and H. Y. Zhang. Detection of Satellite Attitude Sensor Faults using the UKF. IEEE Transactions on Aerospace and Electronic Systems.2007,43(2): 480-491P.
[57]T. Lefebvre, H. Bruyninckx and J. De Schutter. Comment on "A new method for the nonlinear transformation of means and covariances in filters and estimators". IEEE Transactions on Automatic Control.2002,47(8):1406-1408P.
[58]M. Norgaard, N. K. Poulsen and O. Ravn. New developments in state estimation for nonlinear systems. Automatica.2000,36(11):1627-1638P.
[59]W. Li, H. Leung and Y. Zhou. Space-time registration of radar and ESM using unscented Kalman filter. IEEE Transactions on Aerospace and Electronic Systems. 2004,40(3):824-836P.
[60]S. J. Julier. Comprehensive process models for high-speed navigation. PhD thesis. University of Oxford.1996.
[61]S. Clark. Autonomous land vehicle navigation using millimeter wave radar. PhD thesis. University of Sydney.1999.
[62]Yunqiang Chen, Yong Rui and Thomas S. Huang. Multicue HMM-UKF for Real-Time Contour Tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2006,28(9):1525-1529P.
[63]Deok-Jin Lee. Nonlinear Estimation and Multiple Sensor Fusion Using Unscented Information Filtering. IEEE Signal Processing Letters.2008,15:861-864P.
[64]Dahir H. Dini, Danilo P. Mandic and Simon J. Julier. A Widely Linear Complex Unscented Kalman Filter. IEEE Signal Processing Letters.2011,18(11):623-626P.
[65]Wenling Li and Yingmin Jia. Consensus-Based Distributed Multiple Model UKF for Jump Markov Nonlinear Systems. IEEE Transactions on Automatic Control.2012, 57(1):230-236P.
[66]Yuanxin Wu, Dewen Hu, Meiping Wu, et al. A numerical-integration perspective on Gaussian filters. IEEE Transactions on Signal Processing.2006,54(8):2910-2921P.
[67]Hammersley J. M and Morton K. W. Poor Man's Monte Carlo. Royal Statistical Society B.1954,16:23-38P.
[68]Rosenbluth M. N and Rosenbluth A. W. Monte Carlo calculation of the average extension of molecular chains. Chemical Physics.1955,23:356-359P.
[69]H. Akashi and H. Kumamoto. Construction of discrete-time nonlinear filter by Monte Carlo methods with variance-reducing techniques. Systems and Control.1975,19:211-221P.
[70]V. S. Zaritskii, V. B. Svetnik and L. I. Shimelevich. Monte Carlo technique in problems of optimal data processing. Automation and Remote Control.1975,12:95-103P.
[71]J. E. Handschin. Monte Carlo techniques for prediction and filtering of non-linear stochastic processes. Automatica.1970,6:555-563P.
[72]J. E. Handschin and D. Q. Mayne. Monte Carlo techniques to estimate the conditional expectation in multi-stage non-linear filtering. International Journal of Control.1969, 9:547-559P.
[73]王婷婷,郭圣权.粒子滤波算法综述.仪表技术.2009,6(3)：64-66P.
[74]Herman S. M. A particle filtering approach to joint passive radar tracking and target classification. PhD thesis. University of Illinois at Urbana-Champaign.2002.
[75]Simon Godsill and Tim Clapp. Improvement strategies for Monte Carlo particle filter. Signal Processing Group, University of Cambridge.1998,2(1):17-23P.
[76]梁军.粒子滤波算法及其应用研究.哈尔滨：哈尔滨工业大学博士论文.2009.
[77]T. Higuchi. Monte Carlo filtering using the genetic algorithm operators. Journal of Statistical Computation and Simulation.1997,59:1-23P.
[78]A. Doucet. On sequential simulation-based methods for Bayesian filtering. Technical Report. University of Cambridge, CUED-F-ENG-TR310.1998.
[79]B. D. Ripley. Stochastic Simulation. New York:Wiley.1987.
[80]J. Carpenter, P. Clifford and P. Fearnhead. An improved particle filter for nonlinear problems. Technical Report. University of Oxford.1997.
[81]M. K. Pitt and N. Shephard. Filtering via simulation:Auxiliary particle filters. Journal of the American Statistical Association.1999,94:590-599P.
[82]G. Kitagawa and G. Gersch. Smoothness Priors Analysis of Time Series.In Lecture Notes in Statistics, Springer.1996.
[83]G. Kitagawa. Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics.1996,1:1-35P.
[84]C. Musso, N. Oudjane and F. Legland. Improving Regularized Particle Filters. In: Sequential Monte Carlo Methods in Practice. Springer-Verlag,2001.
[85]莫以为,萧德云.进化粒子滤波算法及其应用.控制理论与应用.2005,22(2)：269-272P.
[86]杜正聪,唐斌,李可.混合退火粒子滤波器.物理学报.2006,55(3)：999-1003P.
[87]方正,徐国峰,徐心和.基于粒子群优化的粒子滤波定位方法.控制理论与应用.2008,23(3)：533-537P.
[88]邹国辉,敬忠良,胡洪涛.基于优化组合重采样的粒子滤波算法.上海交通大学学报.2006,40(7)：12-16P.
[89]张琪,王鑫,胡昌华等.人工免疫粒子滤波算法的研究.控制与决策.2008,23(3)：293-301P.
[90]Vandermerwe R, Doucet A, Defreitas N, et al. The unscented particle filter. Technical Report CUED/F-INFEG/TR 380. Cambridge University Engineering Department. 2000.
[91]G. Casella and C. P. Robert. Rao-Blackwellisation of sampling schemes. Biometrika. 1996,83(1):81-94P.
[92]C. Andrieu and A. Doucet. Particle filtering for partially observed Gaussian state space models. Journal of the Royal Statistical Society B.2002,64(4):827-836P.
[93]Z. Khan, T. Balch and F. Dellaert. A Rao-Blackwellized particle filter for Eigen Tracking. Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.2004,2:980-986P.
[94]A. Doucet, N. Gordon and V. Krishnamurthy. Stochastic sampling algorithms for state estimation of jump Markov linear systems. IEEE Transactions on Automtic Control. 2000,45:188-202P.
[95]K. Murphy and S. Russell. Rao-Blackwellised particle for dynamic Bayesian networks. In Sequential Monte Carlo Methods in Practice, Chapter 24, A. Doucet, N. Freitas and N. Gordon. Ed:New York Springer-Verlag.2001.
[96]F. Gustafsson, F. Gunnarsson, N. Bergman, et al. Particle filters for positioning, navigation, and tracking. IEEE Transactions on Signal Processing.2002,50(2): 425-437P.
[97]Fox D. Adapting the sample size in particle filters through KLD-sampling. The International Journal of Robotics Research.2003,22:985-1003P.
[98]P. J. Davis and P. Rabinowitz. Methods of Numerical Integration. New York, Academic Press.1975.
[99]A. H. Stroud. Approximate calculation of multiple integrals. NJ:Prentice Hall.1971.
[100]A. Genz and B. D. Keister. Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. Journal of Computational and Applied Mathematics.1996,71(2):299-309P.
[101]P. Rabinowitz and N. Richter. Perfectly symmetric two-dimensional integration formulas with minimal numbers of points. Math of Computation.1969,23(108): 765-779P.
[102]H. Pesonen and R. Piche. Cubature-based Kalman filters for positioning. Workshop on Positioning Navigation and Communication.2010,45-49P.
[103]Ienkaran Arasaratnam. Cubature Kalman Filtering:Theory and application. PhD thesis. McMaster University.2009.
[104]Ienkaran Arasaratnam and Simon Haykin. Cubature Kalman Filtering:A Powerful New Tool for Aerospace Applications. http://grads.ece.mcmaster.ca/-aienkaran/ Article_IRC09.pdf.
[105]Kumar Pakki. Bharani Chandra, Da-Wei Gu and Ian Postlethwaite. Cubature Information Filter and its Applications.2011 American Control Conference.2011, 3609-3614P.
[106]Anup Dhital, Pau Closas and Carles Fernandez-Prades. Bayesian filtering for indoor localization and tracking in wireless sensor networks. EURASIP Journal on Wireless Communications and Networking 2012.
[107]Liu Jiang, Tang Tao, Gai BaiGen, et al. Integrity assurance of GNSS-based train integrated positioning system. Science China (Technological Sciences).2011,54(7): 1779-1792P.
[108]Ienkaran Arasaratnam and Simon Haykin. Cognitive Tracking Radar Systems and Cubature Kalman Filtering. US Provisional Patent No:61/150830.2009.
[109]Mingjie Wan, Pengfei Li and Tao Li. Tracking Maneuvering Target with Angle-only Measurements Using IMM Algorithm Based on CKF. International on Communications and Mobile Computing.2010,92-96P.
[110]Jing Mu and Yuan-li Cai. Iterated Cubature Kalman Filter and its Application. Proceeding of the 2011 IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems.2011,33-37P.
[111]Quan-bo GE, Wen-bin Li and Cheng-lin Wen. SCKF-STF-CN:a universal nonlinear filter for maneuver target tracking. Journal of Zhejiang University-SCIENCE C (Computers & Electronics).2011,12(8):678-686P.
[112]Xudong Wang, Baoyu Xu and Shaobo Zhong. Passive Location of Angle Sensors Using Extend Square-Root Cubature Kalman Filter. Applied Mechanics and Materials (Electrical Information and Mechatronics and Applications).2011,143-144:577-581P.
[113]Chunyang Sheng, Jun Zhao, Ying Liu, et al. Prediction for noisy nonlinear time series by echo state network based on dual estimation. Neurocomputing.2012,82:186-195 P.
[114]Chao Li and Quan-Bo Ge. SCKF for Mav Attitude Estimation. Proceedings of the 2011 International Conference on Machine Learning and Cybernetics.2011,1313 1318P.
[115]Wenling Li and Yingmin Jia. Location of Mobile Station with Maneuvers Using an IMM-Based Cubature Kalman Filter. IEEE Transactions on Industrial Electronics. 2011, unpublished.
[116]Ienkaran Arasaratnam, Simon Haykin and Thomas R. Hurd. Cubature Kalman Filtering for Continuous-Discrete Systms:Theory and Simulations. IEEE Transactions on Signal Processing.2010,58(10):4977-4993P.
[117]Kumar Pakki. Bharani Chandra, Da-Wei Gu and Ian Postlethwaite. Cubature Kalman Filter based Localization and Mapping. The 18th IFAC World Congress.2011, 2121-2125P.
[118]Martin Havlicek, Karl J. Friston, Jiri Jan, et al. Dynamic modeling of neuronal responses in fMRI using cubature Kalman filtering. Neurolmage.2011,56(4): 2109-2128P.
[119]Huang Jianjun, Zhong Jiali and Jiang Feng. A CKF Based Spatial Alignment of Radar and Infrared Sensors. IEEE 10th International Conference on Signal Processing.2010, 2386-2390P.
[120]熊剑,刘建业,赖际舟等.一种基于粒子滤波的惯性导航系统初始对准方法.中国专利.2010.
[121]P. Dmitriyev, O. A. Stepanov and S. V. Shepel. Nonlinear Filtering Methods Application in INS Alignment. IEEE Transactions on Aerospace and Electronic Systems.1997,33(1):260-271P.
[122]M. J. Yu, J. G. Lee and H. W. Park. Comparison of SDINS in flight Alignment using Equivalent Error Models, IEEE Transactions on Aerospace and Electronic Systems. 1999,35(3):1046-1054P.
[123]王丹力,张洪钺.惯导系统初始对准的非线性滤波算法.中国惯性技术学报.1999,73(3)：17-21P.
[124]陈兵舫,张育林,杨乐平.H∞鲁棒滤波在惯导初始对准中的应用.中国惯性技术学报.2000,8(4)：58-62P.
[125]崔培玲,张会娟,全伟.一种基于MRUPF的捷联惯导系统大方位失准角初始对准方法.中国专利.2011.
[126]张召友,郝燕玲,张鑫Sigma点Kalman滤波在惯性导航初始对准中的性能评估.中国惯性技术学报.2010,18(6)：639-644P.
[127]郝燕玲,杨峻巍,陈亮等.基于平方根中心差分卡尔曼滤波的大方位失准角初始对准.中国惯性技术学报.2011,19(2)：180-189P.
[128]李涛.非线性滤波方法在导航中的应用研究.长沙：国防科学技术大学博士论文.2003.
[129]丁杨斌,王新龙,申功勋等.Unscented卡尔曼滤波在SINS静基座大方位失准角 初始对准中的应用研究.宇航学报.2006,27(6)：1201-1204P.
[130]周卫东,乔相伟,任蕾等.QUKF算法及其在SINS初始对准中的应用.仪器仪表学报.2010,31(2)：264-269P.
[131]周战鑫,高亚楠,陈家斌.基于无轨迹卡尔曼滤波的大失准角INS初始对准.系统仿真学报.2006,18(1)：173-175P.
[132]王志刚,陈良友,边少锋.EPF算法在惯导非线性初始对准中的应用.中国惯性技术学报.2007,15(2)：164-167P.
[133]丁杨斌,申功勋Unscented粒子滤波在静基座捷联惯导系统大方位失准角初始对准中的应用研究.宇航学报.2007,28(2)：397-4014P.
[134]R. v. d. Merwe, E. A. Wan and S. Julier. Sigma-point kalman filters for nonlinear estimation and sensor-fusion:Applications to integrated navigation. AIAA Guidance, Navigation and Control Conference, Providence, RI,2004.
[135]J. L. Crassidis. Sigma-Point Kalman Filtering for Integrated GPS and Inertial Navigation. IEEE Transactions on Aerospace and Electronic Systems.2006,42(2): 750-756P.
[136]H. Carvalho and P. D. Moral. Optimal Nonlinear Filtering in GPS/INS Integration. IEEE Transactions on Aerospace and Electronic Systems.1997,33(3):835-849P.
[137]C. Musso and N. Oudjane. Particle Methods for Multimodal Filtering Application to Terrain Navigation. IEE Colloquium on Target Tracking:Algorithms and Applications. 1999,6:1-5P.
[138]钱东.重力梯度辅助定位与地形反演关键问题研究.哈尔滨：哈尔滨工程大学博士论文.2011.
[139]徐遵义,晏磊,宁书年等.海洋重力辅助导航的研究现状与发展.地球物理学进展.2007,22(1)：104-111 P.
[140]F. L. Lewis. Optimal Estimation. New York:Wiley.1986.
[141]A.H.Stroud.Gaussian quadrature formulas. NJ:Prentice Hall.1966.
[142]S. Capstick and B. D. Keister. Multidimensional quadrature algorithms at higher degree and/or dimension. Journal of Computational Physics.1996,123(2):267-273P.
[143]Y. Bar Shalom, X. R. Li and T. Kirubarajan. Estimation with applications to tracking and navigation. NY:Wiley & Sons.2001.
[144]Jindrich Dunik, Ondrej Straka and Miroslav Simandl. The Development of a Randomised Unscented Kalman Filter. Preprints of the 18th IFAC World Congress. 2011.
[145]E. R. Beadle and P. M. Djuric. A fast weighted Bayesian bootstrap filter for nonlinear model state estimation. IEEE Transactions on Aerospace and Electronic Systems. 1997,33(1):338-343P.
[146]M. Simandl, J. Kralovec and T. Soderstrom. Anticipative Grid Design in Point-Mass Approach to Nonlinear State Estimation. IEEE Transactions on Automatic Control. 2002,47(4):699-702P.
[147]Pengfei Li, Jianping Yu, Mingjie Wan, et al. The augmented form of cubature kalman filter and quadrature kalman filters for additive noise. IEEE Youth Conference on Information Computing and Telecommunication.2009,295-298P.
[148]S. Julier and J. K. Uhlmann. A general method for approximation nonlinear transformations of probability distributions. Technical Report. University of Oxford. 1996.
[149]Syed Ahmed Pasha, Hoang Duong Tuan and Ba-Ngu Vo. Nonlinear Bayesian Filtering Using the Unscented Linear Fractional transformation Model. IEEE Transactions on Signal Processing.2010,58(2):477-489P.
[150]Bar-Shalom Y and Li X R. Multitarget-Multisensor Tracking Principles and Techniques. New Orleans:University of New Orleans.1995.
[151]Mustiere F, Bolic M and Bouchard M. Rao-Blackwellised Particle Filters:Examples of Applications. Canadian Conference on Electrical and Computer Engineering.2006, 1196-1200P.
[152]Matti Vihola. Rao-Blackwellised Particle Filtering in Random Set Multitarget Tracking. IEEE Transactions on aerospace and electroic systems.2007,43(2): 689-705P.
[153]Nando de Freitas. Rao-Blackwellised Particle Filtering for Fault diagnosis. IEEE Aerospace Conference Proceedings.2002,4:1767-1772P.
[154]Arnaud Doucet, Neil J.Gordon and Vikram Krishnamurthy. Particle Filters for State Estimation of Jump Markov Linear systems. IEEE Transactions on Signal Processing. 2001,49(3):613-624P.
[155]Mustiere F, Bolic M and Bouchard M. A Modified Rao-Blackwellised Particle Filter. IEEE International Conference on Acoustics, Speech and Signal Processing.2006, III21-24P.
[156]A. Gil Aparicio, O. Reinoso, L. Paya, et al. Assessing the influence in the parameters of a Rao-Blackwellised particle filter to solve the SLAM problem.2008,6(1):18-27P.
[157]尹建君.线性/非线性系统的混合动态滤波理论及应用.上海：复旦大学博士论文.2008.
[158]尹建君,张建秋,林青.Unscented卡尔曼滤波-卡尔曼滤波算法.系统工程与电子技术.2008,30(4)：617-620.
[159]G. J. McLachlan and T. Krishnan. The EM Algorithm and Extensions. New Jersey, American:John Wiley & Sons.1997.
[160]吕娜,冯祖仁.非线性交互粒子滤波算法.控制与决策.2007,22(4)：378-383P.
[161]Optimal Filtering. Prentice-Hall.1979.
[162]Kong Xiaoying, Nebot E. M and Durrant-Whyte H. Development of a nonlinear psi-angle model for large misalignment errors and its application in INS alignment and calibration. IEEE International Conference on Robotics and Automation.1999,2: 1430-1435P.
[163]严恭敏,严卫生,徐德民.简化UKF滤波在SINS大失准角初始对准中的应用.中国惯性技术学报.2008,16(3)：253-264P.