容积卡尔曼滤波算法研究及其在导航中的应用
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摘要
随着现代科技的发展，人们对导航精度的要求越来越高，而非线性滤波算法能为提高导航精度提供有力的基础，因此得到了广泛的关注和研究。容积卡尔曼滤波（CubatureKalman Filter，CKF）是近年来新兴起来的一种具有优越性能的非线性滤波算法，该算法结构简单，估计精度高，数值稳定性好，克服了其它非线性滤波算法存在的一些问题，因此正成为非线性滤波算法中的研究热点，本文对CKF算法进行了相应研究，所做的主要工作有：
（1）针对多源信息融合等技术中经常出现的噪声相关非线性系统，基于最小方差估计准则提出了两种噪声相关非线性高斯滤波公式：基于模型变换的噪声相关非线性高斯滤波公式和基于一步预测递推的噪声相关非线性高斯滤波公式，并通过更新新息定理证明了二者的等价性，再利用三阶球面-相径容积规则近似其中高斯积分，进而提出两种等价的噪声相关CKF算法。
（2）常规确定采样型滤波算法随着处理非线性系统维数增高，采样点个数增多，滤波计算量也相应增大，因此针对一类非线性高斯系统，证明了其状态的后验均值和协方差为导致该系统非线性的部分向量的高斯积分，并求得了积分表达式，再用三阶球面-相径容积规则近似其中高斯积分，提出降维CKF算法，从而在不损失滤波精度的情况下，减少滤波过程中采样点个数，降低计算量，并对降维滤波算法做了进一步探讨，指出将该思想拓展到GHF算法更加具有现实意义。
（3）分析了强跟踪滤波算法（STF）的运行机理，指出由于对量测一步预测协方差阵近似不够准确，使得STF算法以较大概率产生渐消因子，导致对滤波增益过调节，最终产生对状态估计不够平滑，需要凭经验加入弱化因子来解决这一问题，为此提出了改进的强跟踪滤波算法，避免了靠经验选取弱化因子的麻烦。同时推导并推导了统一的非线性系统强跟踪滤波算法递推公式，只需要用不同的策略近似其中的高斯积分，便可得到不同的非线性强跟踪滤波算法。再利用三阶球面-相径容积规则近似高斯积分，进而提出了强跟踪CKF算法，并针对一类特殊非线性系统，提出了估计效果更佳的多渐消因子CKF算法，该算法能产生多个渐消因子，以不同的速率对各数据通道渐消，从而达到更好的估计效果。
（4）提出将降维CKF算法应用于SINS大方位失准角初始对准，以降低滤波过程中计算量；提出将多渐消因子CKF算法应用于噪声统计不准确的SINS大方位失准角初始对准，以提高滤波估计精度；提出将多渐消因子CKF算法应用于具有惯性器件突变的SINS/GPS组合导航，以提高导航精度。
With the development of modern science and technology, the requirement of navigationaccuracy is forward to higher, and nonlinear filtering algorithm can provide a strongfoundation to improve the navigation accuracy, which gets a lot of attention and research.Cubature Kalman Filter(CKF) is a kind of superior nonlinear filtering algorithm rising inrecent years, which has simple structure, high estimation accuracy, good numerical stability,and overcomes some problems existed in other nonlinear filtering algorithms, thus it isbecoming popular research in nonlinear filtering algorithm. In this paper, CKF algorithm isstudied, the main work is as follows.
1) According to the nonlinear system with correlative noises appeared in multiple sourceinformation confusion technology, two nonlinear Gaussian filterings with correlative noisesare proposed: nonlinear Gaussian filtering with correlative noises based on modeltransformation and nonlinear Gaussian filtering with correlative noises based on one-stepprediction recursion, then the equivalence between them is proved by updating innovationtheorem. At last the third Spherical-Radial Cubature rule is used to approximate Gaussianintegral, and two equivalent CKF algorithm with correlative noises is obtained.
2) When conventional deterministic sampling filtering algorithms process highdimension nonlinear system, the sampling points increase, and the calculation amoutincreases correspondingly. According to this problem, it is proved that the posterior mean andvariance of the state is multiple Gaussian integral of its component vector, and the integralformula is obtained. Then the third Spherical-Radial Cubature rule is used to approximatemultiple Gaussian integral, the reduced dimension CKF algorithm is proposed, which reducesthe samping points, so as to the calculation amount. At last, further discussion is carried out,and it is pointed that it will be more meaningful if this idea is extended to GHF.
3) The running mechanism of Strong Tracking Filtering (STF) is discussed, it is pointedthat owing to inaccurate approximation of the covariance of measurement one-step prediction,STF produces fading factor with high probability, leads to excessive regulation for filteringgain, and eventually makes the state estimation lack smoothness, while sofenting factor isintroduced by experience to realease this problem. According to this disadvantage, animproved STF is proposed, which can advoid this problem. Then a unified nonlinear systemstrong tracking filtering recursive formula is proposed, and different nonlinear system STFalgorithem can be derived by different approximation strategy to the Gaussian integral, then the third Spherical-Radial Cubature rule is used to approximate the Gaussian integral, andStrong Tracking CKF algorithm is derived, meanwhile, according to a class of specialnonlinear system, multiple fading factors CKF algorithm with better performance is proposed,which can generate multiple fading factors, and fades the data channel with different rates, soas to achieve better estimation result.4) Reduced dimension CKF is ultilized to SINS alignment with large azimuth misalignmentangle to reduced the calculation amout, and multiple fading factors CKF is ultilized to SINSalignment with large azimuth misalignment angle with inaccuraten oise statistics to improveestimation accuracy, at last, multiple fading factors CKF is ultilized to SINS/GPS ingegratednavigation with inertial instrument sudden change to improve navigation accuracy.
（1）针对多源信息融合等技术中经常出现的噪声相关非线性系统，基于最小方差估计准则提出了两种噪声相关非线性高斯滤波公式：基于模型变换的噪声相关非线性高斯滤波公式和基于一步预测递推的噪声相关非线性高斯滤波公式，并通过更新新息定理证明了二者的等价性，再利用三阶球面-相径容积规则近似其中高斯积分，进而提出两种等价的噪声相关CKF算法。
（2）常规确定采样型滤波算法随着处理非线性系统维数增高，采样点个数增多，滤波计算量也相应增大，因此针对一类非线性高斯系统，证明了其状态的后验均值和协方差为导致该系统非线性的部分向量的高斯积分，并求得了积分表达式，再用三阶球面-相径容积规则近似其中高斯积分，提出降维CKF算法，从而在不损失滤波精度的情况下，减少滤波过程中采样点个数，降低计算量，并对降维滤波算法做了进一步探讨，指出将该思想拓展到GHF算法更加具有现实意义。
（3）分析了强跟踪滤波算法（STF）的运行机理，指出由于对量测一步预测协方差阵近似不够准确，使得STF算法以较大概率产生渐消因子，导致对滤波增益过调节，最终产生对状态估计不够平滑，需要凭经验加入弱化因子来解决这一问题，为此提出了改进的强跟踪滤波算法，避免了靠经验选取弱化因子的麻烦。同时推导并推导了统一的非线性系统强跟踪滤波算法递推公式，只需要用不同的策略近似其中的高斯积分，便可得到不同的非线性强跟踪滤波算法。再利用三阶球面-相径容积规则近似高斯积分，进而提出了强跟踪CKF算法，并针对一类特殊非线性系统，提出了估计效果更佳的多渐消因子CKF算法，该算法能产生多个渐消因子，以不同的速率对各数据通道渐消，从而达到更好的估计效果。
（4）提出将降维CKF算法应用于SINS大方位失准角初始对准，以降低滤波过程中计算量；提出将多渐消因子CKF算法应用于噪声统计不准确的SINS大方位失准角初始对准，以提高滤波估计精度；提出将多渐消因子CKF算法应用于具有惯性器件突变的SINS/GPS组合导航，以提高导航精度。
With the development of modern science and technology, the requirement of navigationaccuracy is forward to higher, and nonlinear filtering algorithm can provide a strongfoundation to improve the navigation accuracy, which gets a lot of attention and research.Cubature Kalman Filter(CKF) is a kind of superior nonlinear filtering algorithm rising inrecent years, which has simple structure, high estimation accuracy, good numerical stability,and overcomes some problems existed in other nonlinear filtering algorithms, thus it isbecoming popular research in nonlinear filtering algorithm. In this paper, CKF algorithm isstudied, the main work is as follows.
1) According to the nonlinear system with correlative noises appeared in multiple sourceinformation confusion technology, two nonlinear Gaussian filterings with correlative noisesare proposed: nonlinear Gaussian filtering with correlative noises based on modeltransformation and nonlinear Gaussian filtering with correlative noises based on one-stepprediction recursion, then the equivalence between them is proved by updating innovationtheorem. At last the third Spherical-Radial Cubature rule is used to approximate Gaussianintegral, and two equivalent CKF algorithm with correlative noises is obtained.
2) When conventional deterministic sampling filtering algorithms process highdimension nonlinear system, the sampling points increase, and the calculation amoutincreases correspondingly. According to this problem, it is proved that the posterior mean andvariance of the state is multiple Gaussian integral of its component vector, and the integralformula is obtained. Then the third Spherical-Radial Cubature rule is used to approximatemultiple Gaussian integral, the reduced dimension CKF algorithm is proposed, which reducesthe samping points, so as to the calculation amount. At last, further discussion is carried out,and it is pointed that it will be more meaningful if this idea is extended to GHF.
3) The running mechanism of Strong Tracking Filtering (STF) is discussed, it is pointedthat owing to inaccurate approximation of the covariance of measurement one-step prediction,STF produces fading factor with high probability, leads to excessive regulation for filteringgain, and eventually makes the state estimation lack smoothness, while sofenting factor isintroduced by experience to realease this problem. According to this disadvantage, animproved STF is proposed, which can advoid this problem. Then a unified nonlinear systemstrong tracking filtering recursive formula is proposed, and different nonlinear system STFalgorithem can be derived by different approximation strategy to the Gaussian integral, then the third Spherical-Radial Cubature rule is used to approximate the Gaussian integral, andStrong Tracking CKF algorithm is derived, meanwhile, according to a class of specialnonlinear system, multiple fading factors CKF algorithm with better performance is proposed,which can generate multiple fading factors, and fades the data channel with different rates, soas to achieve better estimation result.4) Reduced dimension CKF is ultilized to SINS alignment with large azimuth misalignmentangle to reduced the calculation amout, and multiple fading factors CKF is ultilized to SINSalignment with large azimuth misalignment angle with inaccuraten oise statistics to improveestimation accuracy, at last, multiple fading factors CKF is ultilized to SINS/GPS ingegratednavigation with inertial instrument sudden change to improve navigation accuracy.
引文
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