目前在導航及動態定位的領域中，擴展卡曼濾波器（EKF）被公認是整合系統開發過程中核心演算法的唯一選擇。但整合式導航系統之誤差模型與觀測模型均為非線性，對模型進行線性化的擴展卡曼濾波器無可避免地會引入線性化過程中忽略高次項的誤差。當線性化假設不成立時，採用這種算法會導致濾波器性能下降甚至造成發散。

近年來，直接使用系統非線性模型的UKF（Unscented Kalman Filter）逐漸成為研究非線性估計的有效方法。UKF不需要如二次濾波方程計算Jacobian矩陣或Hessians矩陣，且運算方式和EKF相同。對於線性系統，UKF和EKF具有相同的估算能力；但在非線性系統下，UKF可以得到更好的估計。

基於上述理由，本文提出以UKF發展鬆耦合INS/GPS整合定位之演算法，同時比較分析UKF相對於EKF在整合式導航系統中精度提昇的程度。實驗結果證明了將UKF應用在鬆耦合INS/GPS整合系統之狀態估算可獲得較高精度的解。同時實驗分析結果顯示EKF必須使用特殊的誤差模型來處理較大的初始對準誤差，但是UKF不需要額外的誤差模型即可以無縫地將較大的初始姿態誤差經過最佳化的估算而得到較為精確的姿態解。

In the navigation and kinematic positioning field, the extended Kalman filter (EKF) has become the only and best way to develop the core algorithm of the integrated systems. However, the process models and measurement models of the integrated navigation systems are all nonlinear. In this case, when the local linearity assumption breaks down, i.e, the effects of the higher order terms of the Taylor series expansion become significant, this can seriously affect the accuracy or even lead to divergence of any inference system that is based on the EKF or that uses the EKF as a component part.
Recently, using the unscented Kalman filter (UKF), a systematic and nonlinear model, has become a significant method to research the nonlinear estimation. UKF requires no analytic derivation of Jacobian matrix or Hessians matrix as in the EKF. In a wide range of applications in the areas of nonlinear state estimation, UKF method is superior to standard EKF based estimation approaches.
Based on these reasons, this paper proposes the development of the algorithm of the UKF based loosely coupled INS/GPS integrated POS and also the analysis of the accuracy improvement of the UKF compared to the EKF in the integrated navigation system. The experimental results prove that the applications of UKF in loosely coupled INS/GPS integrated system would obtain higher accuracy solutions. Also, the analysis shows that the EKF needs to use specific process models to procedure the larger errors of the alignment. However, it is unnecessary for the UKF to have extra process models, and it could demonstrated the UKF’s capability of dealing with large and small attitude errors seamlessly.

[1]Britting, K. R. (1971): Inertial Navigation systems Analysis. John Wiley & Sons, Inc.
[2]Brown, R. G. and Hwang, P. Y. C. (1992): Introduction to random signals and applied Kalman filtering. John Wiley & Sons, Inc. New York.
[3]Bucy R S, Renne, K D. (1971): Digital synthesis of nonlinear filter, Automatical, 7(3): 287-289.
[4]Chiang, K.W. (2002): INS/GPS Integration Using Neural Networks for Land Vehicular Navigation Applications. In Proceeding of ION GPS 2002 Meeting, Portland.
[5]Chiang, K.W. (2004): INS/GPS Integration Using Neural Networks for Land Vehicular Navigation Applications, Department of Geomatics Engineering, The University of Calgary, Calgary, Canada, UCGE Report 20209.
[6]Chiang, K.W., Noureldin, A., and El-Sheimy, N. (2007): Developing a Low Cost MEMS IMU/GPS Integration Scheme Using Constructive Neural Networks, IEEE Transactions on Aerospace and Electronic Systems,(In-press, Vol.44, No.1 )
[7]Farrell, J. A. and Barth, M. (1998): The Global Positioning System and Inertial Navigation. McGraw-Hill.
[8]Gelb, A. (1974): Mathematical Methods for Neural Network Analysis and Design.
[9]Grewal, M. S. and Andrews, A. P. (2001): Kalman Filtering: Theory and Practice Using MATLAB○R. John Wiley & Sons, Inc.
[10]Haykin, S. (1994): Neural Networks, A Comprehensive Foundation. Macmillan College Publishing Company, Inc, Englewood Cliffs, NJ.
[11]Julier, S. J. and Durrant-Whyte, H.F. (1995): Navigation and Parameter Estimation of High Speed Road Vehicles. In Robotics and Automation Conference, 101-105.
[12]Julier, S. J., Uhlmann, J. K., and Durrant-Whyte, H. F. (1995): A New Approach for Filtering Nonlinear System, In Proceedings of American Control Conference, pages 1628-1632.
[13]Julier, S. J., and Uhlmann, J. K. (1996): A General Method for Approximating nonlinear Transformations of Probability Distributions. Technical report, Department of Engineering Science, University of Oxford, Oxford, OX1 3PJ
[14]Julier, S. J., and Uhlmann, J. K. (1997): A New Extension of Kalman Filter to Nonlinear Systems. In Proc. SPIE – Int. Soc. Opt. Eng. (USA) (Orlando, FL, April 1997), vol. 3068, pp. 182-193.
[15]Julier, S. J. (1998): A Skewed Approach to Filtering, In SPIE Conference on Signal and Data Processing of Small Target (Orlando, Florida, April 1998), vol. 3373, SPIE, 99, 271-282.
[16]Julier, S. J. and Durrant-Whyte, H.F. (2000): A New Method for Nonlinear Transform of Means and Covariance in Filters and Estimators, IEEE Trans. Automatic Control, vol. 45, 477-482.
[17]Julier, S. J. (2002): The Scaled Unscented Transformation, In Proceeding of the American Control Conference, vol. 6, pp. 4555-4559.
[18]Julier, S. J. (2003): The Spherical Simplex Unscented Transformation. In Proceedings of the IEEE American Control Conference, pp. 2430-2434, Denver, Colorado.
[19]Kalman, R. E. (1960): A new approach to linear filtering and prediction problem. Journal of Basic Eng (ASME), 82D: 95-108.
[20]Lefebvre, T., Bruyninckx, H., and de Schutter, J. (2002): Comment on “A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimations”. IEEE Transformations on Automatic Control, 47(8):1406-1408.
[21]Maybeck, P. S. (1994): Stochastic Models, Estimation, and Control: Volume 1. Navtech Book & Software Store.
[22]Montemerlo, S. M. (2005): Stanford Racing Team’s Entry In The 2005 DARPA Grand Challenge. Stanford Artificial Intelligence Laboratory, Stanford, CA 94305-9010.
[23]Norgaard, M., Poulsen, N. K., and Ravn, O. (2000): New Development in State Estimation of Nonlinear Systems. Automatica, 36:1672-1638.
[24]Parkinson, B. W., Spilker, J.J. Jr. (1996): Global positioning System: Theory and Applications, vols.1 and 2, American Institute of Teronautics, 370 L’Enfant Promenade, SW, Washington.
[25]Rogers, R. M. (2000): Applied Mathematics in Integrated Navigation Systems. American Institute of Aeronautics and Astonautics, Inc.
[26]Rogers, R. M. (2003): Applied Mathematics in Integrated Navigation Systems, Second Edition. American Institute of Aeronautics and Astonautics, Inc.
[27]Nassar, S. (2003): Improving the Inertial Navigation System (INS) Error Model for INS and INS/DGPS Application.
[28]Savant, Jr, C. J., Howard, R. C., Solloway, C. B., and Savant, C. A. (1961): Principles of Inertial Navigation, McGraw-Hill.
[29]Schwarz, K.-P. (1999): Fundamentals of Geodesy: Lecture Notes ENGO 421. Dept. of Geomatics Eng., The University of Calgary, Calgary, Canada.
[30]Schwarz, K.-P. and Wei, M. (2000): INS/GPS Integration for Geodetic Applications: Lecture Notes ENGO 623. Dept. of Geomatics Eng., The University of Calgary, Calgary, Canada.
[31]Shin, E.-H. (2005): Estimation Techniques for low-Cost Inertial Navigation. PhD thesis, Department of Geomatics Engineering, University of Calgary, Canada.
[32]Sunahara, Y. (1969): An approximate method of state estimation for nonlinear dynamical systems, joint Automatic Control Conf, Univ. of Colorado.
[33]Titterton, D. H. and Weston, J. L. (1997): Strapdown Inertial Navigation Technology. Peter Peregrinus Ltd.
[34]Van der Merwe, R., Doucet, A., de Freitas, N., and Wan, E. A. (2000): The Unscented Particle Filter. Technical Report CUED/F-INFENG/TR 380, Cambridge University Engineering Department.
[35]Van der Merwe, R. and Wan, E. A. (2001): Efficient Derivative-Free Kalman Fliters for Online Learning. In European Symposium on Artificial Neural Networks (ESANN), Bruges, Belgium.
[36]Van der Merwe, R. (2004): Sigma-Point Kalman Filters for Probabilistic Inference in Dynamic State-Space Models. PhD thesis, OGI School of Science & Engineering at Oregon Health & Science University, Portland, OR.
[37]Wan, E. A. and Van der Merwe, R. (2000): The Unscented Kalman Filter for Nonlinear Estimation. In Proceeding of IEEE Symposium on Adaptive systems for Signal Processing Communications and Control (AS-SPCC), Lake Louise, Alberta, Canada.