改进的GNSS/SINS组合导航系统自适应滤波算法 Improved Adaptive Filtering Algorithm for GNSS/SINS Integrated Navigation System 1 first-author 林雪原，博士，教授，主要从事信息融合与组合导航工作。linxy_ytcn@126.com 林雪原，博士，教授，主要从事信息融合与组合导航工作。linxy_ytcn@126.com linxy_ytcn@126.com 1 1 1 2 烟台南山学院，山东 烟台，265713 烟台南山学院，山东 烟台，265713 Yantai Nanshan University, Yantai 265713, China Yantai Nanshan University, Yantai 265713, China 东海热电有限公司，山东 烟台，265713 东海热电有限公司，山东 烟台，265713 Donghai Thermal Power Co. Ltd, Yantai 265713, China Donghai Thermal Power Co. Ltd, Yantai 265713, China GNSS/SINS(global navigation satellite system/strapdown inertial navigation system)组合导航系统已得到广泛的应用与研究，当处于复杂环境时，GNSS输出容易出现误差均方差突变、误差均方差缓变、硬故障和软故障4种现象，进而影响组合导航系统滤波精度及载体的导航安全。为了解决上述问题，提出了一种改进的GNSS/SINS组合导航系统自适应滤波算法。首先，利用滤波过程中的观测异常检验统计量与滤波器门限值构建观测因子，然后，将变分贝叶斯原理与抗野值滤波方法结合，设计了改进的组合导航系统自适应滤波算法。仿真实验表明，相较于传统算法，当GNSS输出误差均方差发生变化时，所提算法可将位置精度及速度精度提高11.8%及13.7%；在GNSS输出发生硬故障时，所提算法可将位置精度及速度精度提高70.8%及69.6%。实验结果表明，所提算法具有较强的自适应性，可提升复杂环境下组合导航系统的精度和连续可用性。 Objectives 2 GNSS/SINS (global navigation satellite system/strapdown inertial navigation system) integrated navigation system has been widely used and researched. In complex environment, GNSS output is prone to four phenomena, including root mean square (RMS) of error sudden change, RMS of error slow change, hard fault and soft fault, which affect the filtering accuracy of integrated navigation system and the navigation safety of carrier. Methods 2 In order to solve the above problems, an improved adaptive filtering algorithm for GNSS/SINS integrated navigation system is proposed. Firstly, an observation factor was constructed by using the observation anomaly test statistics and filter threshold value in the filtering process. Then, the variable Bayesian principle and the outlier-restrained filtering method are combined to design an improved adaptive filtering algorithm for the integrated navigation system. Results 2 The simulation experiment shows that compared with the conventional algorithms, the proposed algorithm can improve the position accuracy and speed accuracy by about 11.8% and 13.7%, respectively, when the RMS of GNSS output error changes. And it can also improve the position accuracy and speed accuracy by about 70.8% and 69.6%, respectively, when a hard fault occurs on the GNSS output. Conclusions 2 The experimental results show that the proposed algorithm has strong adaptability and can improve the accuracy and continuous availability of integrated navigation systems in complex environments. 变分贝叶斯 自适应滤波 抗野值滤波 组合导航系统 观测因子 variational Bayesian adaptive filtering outlier-restrained filtering integrated navigation system observation factor 目前，GNSS/SINS（global navigation satellite system/strapdown inertial navigation system）组合导航系统的研究热点基本集中于最新滤波方法在该领域的应用以及抗野值方法的研究。当GNSS受到电磁干扰和其他人为干扰时 ［ 1 1 ］ ，GNSS的输出会发生改变，如突变、缓变，导致输出误差均方差（root mean square，RMS）发生变化，如输出发生硬故障或软故障 ［ 2 2 - 3 3 2-3 ］ 。 为了解决上述问题，抗野值自适应滤波方法得到不断的完善及应用 ［ 4 4 ］ 。为了解决GNSS输出误差RMS将会发生变化的问题，文献［ 5 5 ］利用滑窗检测卡尔曼滤波过程中新息的统计特性，进而在线估计量测噪声RMS的变化情况，然而该算法需存储窗口内的新息序列，而且算法的复杂度随着窗口尺度的增长而大幅提高。在多传感器组合导航系统中，为了解决子导航传感器输出发生硬故障与软故障的问题，文献［ 6 6 ］提出构建动态信息分配因子，进而增大有故障子导航传感器的时变量测噪声，以降低对整个系统导航精度的影响；在GNSS/SINS组合导航系统中，为了解决GNSS输出发生硬故障或软故障的问题，杨元喜院士多年来致力于自适应抗差滤波理论的研究，其主要思路为构建自适应因子，进而控制卡尔曼滤波器的增益矩阵，以降低GNSS输出故障信息对系统的影响 ［ 7 7 ］ 。 在上述研究中，当GNSS输出误差RMS发生变化时，以文献［ 5 5 ］为代表的算法虽然能估计GNSS输出的时变量测噪声RMS，但该算法成立的前提条件是量测噪声为高斯噪声，所以当GNSS输出发生硬故障或软故障时，该算法将会产生较大的误差。当GNSS输出发生硬故障或软故障时，以文献［ 7 7 ］为代表的算法虽然能识别故障，用抗野值自适应算法对系统进行处理以提高系统的导航精度，但是当GNSS输出误差RMS发生变化时，该算法对该情况的处理将是一种非最优滤波处理。 为此，本文将变分贝叶斯理论 ［ 8 8 - 10 10 8-10 ］ 与抗野值自适应滤波理论结合，提出了一种改进的GNSS/SINS组合导航系统抗野值自适应滤波算法。该算法不仅能够利用当前信息实时估计GNSS输出误差RMS发生变化时的时变量测噪声RMS，还能够在GNSS输出发生硬故障或软故障时对系统进行抗野值自适应信息处理，进而提高系统的滤波精度。 1　GNSS/SINS组合导航滤波模型 1 GNSS/SINS系统的离散线性化状态方程 ［ 11 11 ］ 为： <math display="block" id="M1"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi 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mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115698&type= 75.01467133 31.41133499 各参数的详细含义参见文献［ 12 12 - 15 15 12-15 ］。 2　改进的自适应滤波算法 1 2.1　变分贝叶斯原理 2 当量测噪声方差未知、系统噪声方差已知时，包含量测噪声的最优贝叶斯分为预测与更新两个过程 ［ 16 16 - 17 17 16-17 ］ ，其计算式分别为： <math display="block" id="M16"><mtable columnalign="left"><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>:</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mo>=</mo><mo>∫</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mo>×</mo></mtd></mtr><mtr><mtd><mtext>          </mtext><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>:</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>⋅</mo><mi mathvariant="normal">d</mi><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115704&type= 73.57534027 12.44600010 <math display="block" id="M17"><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>:</mo><mi>k</mi></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mo>∝</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>:</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115739&type= 72.30532837 4.06400013 上述关系理论上是最优的，但难以求解。而变分贝叶斯可以利用多个已知分布来近似求解后验分布。 式（4） 式（4） 、 式（5） 式（5） 可近似表达为： <math display="block" id="M18"><mtable columnalign="left"><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>:</mo><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mo>×</mo></mtd></mtr><mtr><mtd><mtext>             </mtext><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mi>I</mi><mi>G</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115735&type= 65.02400208 16.25600052 <math display="block" id="M19"><mtable columnalign="left"><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>:</mo><mi>k</mi></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo><mo>×</mo></mtd></mtr><mtr><mtd><mtext>       </mtext><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>m</mi></mrow></munderover></mstyle><mrow><mi>I</mi><mi>G</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced><mo>,</mo><mi>i</mi></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115755&type= 54.61000061 16.25600052 式中， N 、 IG 分别表示正态分布和逆 Г 分布； <math id="M20"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115861&type= 2.87866688 2.79399991 为量测噪声方差的维数； <math id="M21"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115777&type= 4.31799984 3.97933316 为高斯分布的噪声未知方差，且有： <math display="block" id="M22"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115787&type= 33.78200150 10.32933331 式中， <math id="M23"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115799&type= 15.23999977 4.82600021 ，其他参数的含义及上述方程的推导过程详见文献［ 18 18 ］。 2.2　改进的自适应滤波算法 2 由 式（3） 式（3） 可以定义新息如下： <math id="M24"><msub><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115804&type= 29.63333130 5.75733376 ，其方差为 <math id="M25"><msub><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115817&type= 35.64466476 5.07999992 ，则定义归一化新息为 <math id="M26"><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115822&type= 20.99733353 3.97933316 。 假设 H 0 代表系统无故障发生， H 1 代表系统有故障发生，由上述关系构成假设检验问题： <math display="block" id="M27"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>:</mo><msubsup><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msubsup><mrow><mi mathvariant="bold-italic">v</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mo>∇</mo><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">S</mi></mrow><mrow><mi>k</mi></mrow><mrow/></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115836&type= 29.12533188 9.99066734 <math id="M28"><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115846&type= 3.64066648 3.72533321 服从自由度为 <math id="M29"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115861&type= 2.87866688 2.79399991 的 <math id="M30"><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115955&type= 3.21733332 3.89466691 分布，并且： <math display="block" id="M31"><mi>P</mi><mo stretchy="false">[</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>></mo><msub><mrow><mi>T</mi></mrow><mrow><mi>D</mi></mrow></msub><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">]</mo><mo>=</mo><mi>α</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115877&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115865&type= 31.66533470 4.57200003 式中， <math id="M32"><msub><mrow><mi>T</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115880&type= 4.23333359 3.72533321 代表检测门限； <math id="M33"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115887&type= 1.94733346 2.79399991 代表显著性水平。 构建观测因子为： <math display="block" id="M34"><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>D</mi></mrow></msub></mtd><mtd/></mtr><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>/</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>></mo><msub><mrow><mi>T</mi></mrow><mrow><mi>D</mi></mrow></msub></mtd><mtd/></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115895&type= 38.69266510 9.90600014 假设SINS已采用余度配置技术，不会发生故障；GNSS输出发生误差RMS突变、误差RMS缓变、硬故障和软故障这4种情况是相互独立的。则本文所提出的改进自适应滤波算法如下 ［ 19 19 - 20 20 19-20 ］ ： 1）如果 <math id="M35"><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115898&type= 9.73666668 4.57200003 ，利用贝叶斯自适应滤波器对系统进行滤波处理，预测过程为： <math display="block" id="M36"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">ρ</mi><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">ρ</mi><mo>⋅</mo><msub><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115919&type= 57.99666977 23.79133224 更新过程为： <math display="block" id="M37"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">R</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow><mrow/></msubsup><mo>/</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">R</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Ι</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow><mrow/></msubsup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo></mtd></mtr><mtr><mtd><mtext>                      </mtext><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115923&type= 76.03067017 43.51866913 2）如果 <math id="M38"><mn mathvariant="normal">1</mn><mo><</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115929&type= 18.20333290 4.82600021 ，对系统进行滤波处理。 <math id="M39"><msub><mrow><mi>γ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115949&type= 3.64066648 4.57200003 是一个大于1的整数，其取值可根据经验选取，主要是因为基于残差 <math id="M40"><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115955&type= 3.21733332 3.89466691 的故障检测方法对硬故障检测较灵敏，而对软故障效果较差 ［ 21 21 ］ 。对于硬故障， <math id="M41"><msub><mrow><mi>γ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115960&type= 3.64066648 4.57200003 的选值可大于并接近于1，而对于软故障， <math id="M42"><msub><mrow><mi>γ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115960&type= 3.64066648 4.57200003 的选值可稍微大一些。预测过程和更新过程分别为： <math display="block" id="M43"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115944&type= 57.99666977 11.68400002 <math display="block" id="M44"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">R</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>+</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">R</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Ι</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow><mrow/></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115967&type= 75.01467133 37.84600067 3）如果 <math id="M45"><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>></mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115972&type= 11.51466751 4.57200003 ，对系统进行时间更新，可以认为GNSS的输出信息没有利用价值。预测过程同 式（13） 式（13） ，更新过程为： <math display="block" id="M46"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">X</mi><mo>^</mo></mover></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow><mrow/></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mfenced open="|" close="" separators="|"><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>k</mi><mfenced open="|" close="" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115977&type= 28.27866554 22.69066620 3　仿真及分析 1 在仿真实验中，各传感器的误差参数设置见 表1 表1 。GNSS的采样周期设定为1 s，而SINS的采样周期设定为0.02 s，仿真时间为3 600 s。本文将与文献［ 3 3 ］对应的传统抗野值自适应算法定义为算法1，将与文献［ 2 2 ］对应的变分贝叶斯滤波算法定义为算法2，为了便于对本文算法与算法1、算法2的滤波精度进行比较，对于相同的SINS、GNSS仿真数据，本文进行了上述算法的滤波对比分析。仿真实验中的系统飞行轨迹如 图1 图1 所示。 误差项 数值 陀螺噪声RMS/（°·h -1 ） 1.8 加速度计噪声RMS/g 1×10 -4 GNSS定位误差（每个方向）/m 10 (1 σ ) GNSS的测速误差（每个方向）/(m·s -1 ) 0.2 (1 σ ) 在仿真实验中， 式（9） 式（9） 中取 <math id="M47"><mi>α</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116005&type= 21.33600044 3.04800010 ，故 <math id="M48"><msub><mrow><mi>T</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">27.8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115989&type= 15.74800014 3.72533321 ； 式（11） 式（11） 中 <math id="M49"><mi mathvariant="bold-italic">ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116010&type= 2.03200006 3.64066648 、 <math id="M50"><mi mathvariant="bold-italic">α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45115992&type= 2.11666679 2.79399991 、 <math id="M51"><mi mathvariant="bold-italic">β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116016&type= 2.28600001 3.64066648 的初始值分别为 <math id="M52"><mi mathvariant="bold-italic">ρ</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0.99</mn></mtd><mtd><mn mathvariant="normal">0.99</mn></mtd><mtd><mn mathvariant="normal">0.99</mn></mtd><mtd><mn mathvariant="normal">0.99</mn></mtd><mtd><mn mathvariant="normal">0.99</mn></mtd><mtd><mn mathvariant="normal">0.99</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116024&type= 68.24134064 4.06400013 ， <math id="M53"><mi mathvariant="bold-italic">α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116032&type= 2.11666679 2.79399991 = ［1 1 1 1 1 1］， <math id="M54"><mi mathvariant="bold-italic">β</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">2</mn></mtd><mtd><mn mathvariant="normal">0.1</mn></mtd><mtd><mn mathvariant="normal">0.1</mn></mtd><mtd><mn mathvariant="normal">0.1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116034&type= 48.51399994 3.97933316 。 3.1　GNSS输出误差RMS发生突变 2 假定在时间为600~1 400 s、2 201~3 000 s期间，GNSS输出的三维位置及三维速度的误差RMS突变为 表1 表1 中相关数值的5倍；在时间为1 401~2 200 s期间，各RMS突变为 表1 表1 中相关数值的9倍。采用本文算法与算法1对系统进行仿真，由于篇幅限制，本文选取部分导航参数绘制误差曲线，如 图2 图2 所示；本文算法对GNSS输出误差RMS突变的跟踪如 图3 图3 所示。 由 图3 图3 可以看出，本文算法对GNSS输出误差RMS突变的跟踪能力较强。为了综合对比本文算法与算法1的性能，将GNSS输出分为RMS突变和RMS不变两个时段，并分别对各导航参数的RMS进行统计，结果见 表2 表2 。 2 时段 2 滤波算法 3 位置误差/m 3 速度误差/（m·s -1 ） 3 姿态误差/(°) 经度 纬度 高度 东向速度 北向速度 天向速度 横滚角 俯仰角 航向角 2 RMS突变 本文算法 9.42 8.82 9.13 0.393 0.368 0.189 0.084 0.099 1.273 算法1 10.56 9.85 10.87 0.433 0.420 0.262 0.124 0.138 1.468 11 2 RMS不变 本文算法 2.22 1.49 1.98 0.082 0.084 0.056 0.068 0.066 0.801 算法1 2.20 1.47 1.99 0.083 0.085 0.057 0.071 0.068 0.803 由 图2 图2 和 表2 表2 可以看出，当GNSS输出误差RMS发生突变时，相对于算法1，基于本文算法的位置精度及速度精度分别提高12.7%及14.9%，并能较实时、准确地预测RMS变化，具有自适应的特性；当GNSS输出误差RMS未发生变化时，两个算法性能等同。 3.2　GNSS输出误差RMS发生缓变 2 假定在时间为600~2 600 s期间，GNSS输出的三维位置及三维速度的误差RMS缓变规律为 <math id="M55"><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">10</mn><mo>×</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>×</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">[</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">(</mo><mi>t</mi><mo>-</mo><mn mathvariant="normal">600</mn><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">400</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116051&type= 60.79066849 4.48733330 。当 <math id="M56"><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116076&type= 3.38666677 3.72533321 代表位置误差RMS时， <math id="M57"><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116081&type= 3.38666677 3.72533321 =10 m；当 <math id="M58"><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116076&type= 3.38666677 3.72533321 代表速度误差RMS时， <math id="M59"><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116081&type= 3.38666677 3.72533321 =0.2 m/s。 采用本文算法与算法1对系统进行仿真，选取部分导航参数绘制误差曲线，如 图4 图4 所示。经统计，相对于算法1，基于本文算法的位置精度及速度精度分别提高了11.8%及13.7%；本文算法对GNSS输出误差RMS缓变的跟踪如 图5 图5 所示。由 图5 图5 可以看出，本文算法对GNSS输出误差RMS缓变的跟踪能力较强。 3.3　GNSS输出发生硬故障 2 假定在时间为600~700 s期间，GNSS输出的三维位置及三维速度发生硬故障，其中GNSS输出的三维位置各有100 m的恒定误差，而GNSS输出的三维速度各有2 m/s的恒定误差。采用本文算法与算法2对系统进行仿真，选取部分导航参数绘制误差曲线，如 图6 图6 所示。经统计，相对于算法2，基于本文算法的位置精度及速度精度分别提高了70.8%及69.6%。 3.4　GNSS输出发生软故障 2 假定在时间600~700 s期间，GNSS输出的三维位置及三维速度发生软故障，其中GNSS输出的三维位置、三维速度发生软故障的规律分别为 <math id="M60"><mn mathvariant="normal">40</mn><mo>×</mo><mo stretchy="false">(</mo><mi>t</mi><mo>-</mo><mn mathvariant="normal">600</mn><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">100</mn><mtext> </mtext><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116118&type= 34.03599930 3.55599999 、 <math id="M61"><mn mathvariant="normal">1</mn><mo>×</mo><mo stretchy="false">(</mo><mi>t</mi><mo>-</mo><mn mathvariant="normal">600</mn><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">100</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116121&type= 35.47533035 3.55599999 。采用本文算法与算法2对系统进行仿真，选取部分导航参数绘制误差曲线，如 图7 图7 所示。 从以上仿真分析可得出如下结论： 1）从 图2 图2 ~ 图7 图7 可以看出，在系统没有发生硬故障或软故障时，算法1对GNSS输出误差RMS发生突变或缓变的处理能力较弱，而本文算法可以实时估计GNSS输出误差RMS发生变化的规律，并进一步提高系统的滤波精度。 2）从 图6 图6 可以看出，在GNSS发生硬故障时，本文算法相较于算法2优势明显。 3）在 图2 图2 、 图4 图4 的0~100 s期间，本文算法相对于算法1滤波精度较差，主要是因为实验开始时 <math id="M62"><mi mathvariant="bold-italic">ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116142&type= 2.03200006 3.64066648 、 <math id="M63"><mi mathvariant="bold-italic">α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116149&type= 2.11666679 2.79399991 、 <math id="M64"><mi mathvariant="bold-italic">β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116154&type= 2.28600001 3.64066648 的初始值依据 式（12） 式（12） 确定的方差与实际方差有较大差距；在100~600 s期间，两种滤波算法确定的误差曲线重合，进而说明了本文算法对GNSS输出误差RMS的实时跟踪能力较强。 4）从 图7 图7 可以看出，在GNSS发生软故障时，本文算法与算法2的处理能力基本一致，效果较差。上文已提到，目前公开的各种自适应滤波算法均对硬故障较为灵敏，而对软故障的检测不灵敏，且鲁棒性较差。 4　结　语 1 根据滤波过程中的新息及故障检测门限构建观测因子，结合变分贝叶斯估计理论与常规抗野值自适应滤波算法，本文提出了一种改进的自适应滤波算法。实验结果表明，所提算法在处理GNSS输出误差RMS发生突变及缓变、GNSS输出发生硬故障的问题时优势明显。随着新理论、新方法在处理GNSS输出发生软故障上获得突破，所提思路将会获得更加完美的效果。 参考文献 Zhao Ang ， Yang Yuanxi ， Xu Yangyin ， et al . Integrity Analysis of GNSS Single System and Multi-System Combination ［J］. Geomatics and Information Science of Wuhan University ， 2020 ， 45 （ 1 ）： 72 - 80 . Integrity Analysis of GNSS Single System and Multi-System Combination 赵昂 ， 杨元喜 ， 许扬胤 ， 等 . GNSS单系统及多系统组合完好性分析 ［J］. 武汉大学学报（信息科学版） ， 2020 ， 45 （ 1 ）： 72 - 80 . GNSS单系统及多系统组合完好性分析 Li Bolin ， Xu Bin ， Wang Fengju ， et al . VB-CKF for SINS/GPS Integrated Navigation System Based on Variational Bayesian Theory ［J］. Science Technology and Engineering ， 2019 ， 19 （ 28 ）： 396 - 400 . VB-CKF for SINS/GPS Integrated Navigation System Based on Variational Bayesian Theory 李博琳 ， 徐彬 ， 王凤聚 ， 等 . 基于变分贝叶斯原理的SINS/GPS组合导航VB-CKF滤波算法 ［J］. 科学技术与工程 ， 2019 ， 19 （ 28 ）： 396 - 400 . 基于变分贝叶斯原理的SINS/GPS组合导航VB-CKF滤波算法 Wang Dingjie ， Meng Deli ， Li Zhaoyang ， et al . Adaptively Outlier-Restrained GNSS/MEMS-INS Integrated Navigation Method ［J］. Chinese Journal of Scientific Instrument ， 2017 ， 38 （ 12 ）： 2952 - 2958 . Adaptively Outlier-Restrained GNSS/MEMS-INS Integrated Navigation Method 王鼎杰 ， 孟德利 ， 李朝阳 ， 等 . 抗野值自适应卫星/微惯性组合导航方法 ［J］. 仪器仪表学报 ， 2017 ， 38 （ 12 ）： 2952 - 2958 . 抗野值自适应卫星/微惯性组合导航方法 Miao Yuewang ， Zhou Wei ， Tian Liang ， et al . Extended Robust Kalman Filter Based on Innovation Chi-Square Test Algorithm and Its Application ［J］. Geomatics and Information Science of Wuhan University ， 2016 ， 41 （ 2 ）： 269 - 273 . Extended Robust Kalman Filter Based on Innovation Chi-Square Test Algorithm and Its Application 苗岳旺 ， 周巍 ， 田亮 ， 等 . 基于新息 <math id="M65"><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=45116166&type= 2.79399991 3.30200005 检测的扩展抗差卡尔曼滤波及其应用 ［J］. 武汉大学学报（信息科学版） ， 2016 ， 41 （ 2 ）： 269 - 273 . 基于新息χ2检测的扩展抗差卡尔曼滤波及其应用 Soken H E ， Hajiyev C . Adaptive Unscented Kalman Filter with Multiple Fading Factors for Pico Satellite Attitude Estimation ［C］// The 4th International Conference on Recent Advances in Space Technologies ， Istanbul， Turkey ， 2009 . Adaptive Unscented Kalman Filter with Multiple Fading Factors for Pico Satellite Attitude Estimation Xiong Zhi ， Shao Hui ， Hua Bing ， et al . An Improved Fault Tolerant Federated Filter with Fault Isolation ［J］. Acta Aeronautica et Astronautica Sinica ， 2015 ， 36 （ 3 ）： 929 - 938 . An Improved Fault Tolerant Federated Filter with Fault Isolation 熊智 ， 邵慧 ， 华冰 ， 等 . 改进故障隔离的容错联邦滤波 ［J］. 航空学报 ， 2015 ， 36 （ 3 ）： 929 - 938 . 改进故障隔离的容错联邦滤波 Yang Yuanxi ， Ren Xia ， Xu Yan . Main Progress of Adaptively Robust Filter with Applications in Navigation ［J］. Journal of Navigation and Positioning ， 2013 ， 1 （ 1 ）： 9 - 15 . Main Progress of Adaptively Robust Filter with Applications in Navigation 杨元喜 ， 任夏 ， 许艳 . 自适应抗差滤波理论及应用的主要进展 ［J］. 导航定位学报 ， 2013 ， 1 （ 1 ）： 9 - 15 . 自适应抗差滤波理论及应用的主要进展 Xu Dingjie ， Shen Chen ， Shen Feng . Adaptive Kalman Filtering with Time-Varying Colored Measurement Noise by Variational Bayesian Learning ［J］. Journal of Electronics ＆ Information Technology ， 2013 ， 35 （ 7 ）： 1593 - 1598 . Adaptive Kalman Filtering with Time-Varying Colored Measurement Noise by Variational Bayesian Learning 徐定杰 ， 沈忱 ， 沈锋 . 时变有色观测噪声下基于变分贝叶斯学习的自适应卡尔曼滤波 ［J］. 电子与信息学报 ， 2013 ， 35 （ 7 ）： 1593 - 1598 . 时变有色观测噪声下基于变分贝叶斯学习的自适应卡尔曼滤波 Lim K L ， Wang H . MAP Approximation to the Variational Bayes Gaussian Mixture Model and Application ［J］. Soft Computing ， 2018 ， 22 （ 10 ）： 3287 - 3299 . MAP Approximation to the Variational Bayes Gaussian Mixture Model and Application Vrettas M D ， Cornford D ， Opper M . Estimating Parameters in Stochastic Systems： A Variational Bayesian Approach ［J］. Physica D： Nonlinear Phenomena ， 2011 ， 240 （ 23 ）： 1877 - 1900 . Estimating Parameters in Stochastic Systems： A Variational Bayesian Approach Lin Xueyuan ， Li Rongbing ， Gao Qingwei . Integrated Navigation and Its Information Fusion Method ［M］. Beijing ： National Defense Industry Press ， 2017 . Integrated Navigation and Its Information Fusion Method 林雪原 ， 李荣冰 ， 高青伟 . 组合导航及其信息融合方法 ［M］. 北京 ： 国防工业出版社 ， 2017 . 组合导航及其信息融合方法 Lin Xueyuan ， Sun Yumei ， Dong Yunyun ， et al . Improved Multi-Scale Filtering Algorithm of Multi-Sensor Integrated Navigation System ［J］. Chinese Space Science and Technology ， 2020 ， 40 （ 4 ）： 61 - 68 . Improved Multi-Scale Filtering Algorithm of Multi-Sensor Integrated Navigation System 林雪原 ， 孙玉梅 ， 董云云 ， 等 . 多传感器组合导航系统的改进多尺度滤波算法 ［J］. 中国空间科学技术 ， 2020 ， 40 （ 4 ）： 61 - 68 . 多传感器组合导航系统的改进多尺度滤波算法 Dai Qing ， Sui Lifen ， Tian Yuan ， et al . Gaussian Mixture Filter Based on Variational Bayesian Learning Optimization and Its Application to Integrated Navigation ［J］. Geomatics and Information Science of Wuhan University ， 2019 ， 44 （ 5 ）： 699 - 705 . Gaussian Mixture Filter Based on Variational Bayesian Learning Optimization and Its Application to Integrated Navigation 戴卿 ， 隋立芬 ， 田源 ， 等 . 变分优化的高斯混合滤波及其在导航中的应用 ［J］. 武汉大学学报（信息科学版） ， 2019 ， 44 （ 5 ）： 699 - 705 . 变分优化的高斯混合滤波及其在导航中的应用 Dai Q ， Sui L F ， Wang L X ， et al . An Efficiency Algorithm on Gaussian Mixture UKF for BDS/INS Navigation System ［J］. Geodesy and Geodynamics ， 2018 ， 9 （ 2 ）： 169 - 174 . An Efficiency Algorithm on Gaussian Mixture UKF for BDS/INS Navigation System Wei Erhu ， Sun Danshi ， Wan Wei ， et al . A Processing Method for Strapdown Inertial Navigation System Initial Alignment Under Large Misalignment Angles ［J］. Journal of Geomatics ， 2020 ， 45 （ 6 ）： 7 - 11 . A Processing Method for Strapdown Inertial Navigation System Initial Alignment Under Large Misalignment Angles 魏二虎 ， 孙聃石 ， 万伟 ， 等 . 一种针对大失准角捷联惯导系统初始对准的处理方法 ［J］. 测绘地理信息 ， 2020 ， 45 （ 6 ）： 7 - 11 . 一种针对大失准角捷联惯导系统初始对准的处理方法 Pan Quan ， Hu Yumei ， Lan Hua ， et al . Information Fusion Progress： Joint Optimization Based on Variational Bayesian Theory ［J］. Acta Automatica Sinica ， 2019 ， 45 （ 7 ）： 1207 - 1223 . Information Fusion Progress： Joint Optimization Based on Variational Bayesian Theory 潘泉 ， 胡玉梅 ， 兰华 ， 等 . 信息融合理论研究进展： 基于变分贝叶斯的联合优化 ［J］. 自动化学报 ， 2019 ， 45 （ 7 ）： 1207 - 1223 . 信息融合理论研究进展： 基于变分贝叶斯的联合优化 Sarkka S ， Nummenmaa A . Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations ［J］. IEEE Transactions on Automatic Control ， 2009 ， 54 （ 3 ）： 596 - 600 . Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations Song Huijie ， Dong Shaowu ， Wang Yanping ， et al . An Improved Kalman Filter Time Scale Algorithm Based on Forgetting Factor ［J］. Geomatics and Information Science of Wuhan University ， 2019 ， 44 （ 8 ）： 1205 - 1211 . An Improved Kalman Filter Time Scale Algorithm Based on Forgetting Factor 宋会杰 ， 董绍武 ， 王燕平 ， 等 . 基于渐消因子的改进Kalman滤波时间尺度估计算法 ［J］. 武汉大学学报（信息科学版） ， 2019 ， 44 （ 8 ）： 1205 - 1211 . 基于渐消因子的改进Kalman滤波时间尺度估计算法 Tang Luliang ， Yang Xue ， Jin Chen ， et al . Traffic Lane Number Extraction Based on the Constrained Gaussian Mixture Model ［J］. Geomatics and Information Science of Wuhan University ， 2017 ， 42 （ 3 ）： 341 - 347 . Traffic Lane Number Extraction Based on the Constrained Gaussian Mixture Model 唐炉亮 ， 杨雪 ， 靳晨 ， 等 . 基于约束高斯混合模型的车道信息获取 ［J］. 武汉大学学报（信息科学版） ， 2017 ， 42 （ 3 ）： 341 - 347 . 基于约束高斯混合模型的车道信息获取 Stepanov O A ， Chelpanov I B ， Motorin A V . Accuracy of Sensor Bias Estimation and Its Relationship with Allan Variance ［J］. Gyroscopy and Navigation ， 2017 ， 8 （ 1 ）： 51 - 57 . Accuracy of Sensor Bias Estimation and Its Relationship with Allan Variance Zhang Hao ， Xiao Yong ， Yang Chaoxu ， et al . Integrated Navigation System Based on Fault Detection Using Double State Chi-Square Test ［J］. Acta Aeronautica et Astronautica Sinica ， 2020 ， 41 （ S2 ）： 724271 . Integrated Navigation System Based on Fault Detection Using Double State Chi-Square Test 张浩 ， 肖勇 ， 杨朝旭 ， 等 . 基于双状态卡方故障检测的组合导航系统 ［J］. 航空学报 ， 2020 ， 41 （ S2 ）： 724271 . 基于双状态卡方故障检测的组合导航系统 10.13203/j.whugis20200436 P228 A 1671-8860（2023）01-0127-08 林雪原,刘丽丽,董云云，等.改进的GNSS/SINS组合导航系统自适应滤波算法[J].武汉大学学报（信息科学版）,2023,48(1):127-134.DOI: 10.13203/j.whugis20200436 http://dx.doi.org/10.13203/j.whugis20200436 林雪原,刘丽丽,董云云，等.改进的GNSS/SINS组合导航系统自适应滤波算法[J].武汉大学学报（信息科学版）,2023,48(1):127-134.DOI:10.13203/j.whugis20200436 LIN Xueyuan,LIU Lili,DONG Yunyun,et al.Improved Adaptive Filtering Algorithm for GNSS/SINS Integrated Navigation System[J].Geomatics and Information Science of Wuhan University,2023,48(1):127-134.DOI: 10.13203/j.whugis20200436 http://dx.doi.org/10.13203/j.whugis20200436 LIN Xueyuan,LIU Lili,DONG Yunyun,et al.Improved Adaptive Filtering Algorithm for GNSS/SINS Integrated Navigation System[J].Geomatics and Information Science of Wuhan University,2023,48(1):127-134.DOI:10.13203/j.whugis20200436 国家自然科学基金（61673208， 60874112）；山东省自然科学基金（2016ZRA06068）；烟台市“双百计划”人才项目（YT201803）。 2022-04-09 2023-01-05 2023-06-29 33226001 武汉大学学报（信息科学版） 1 48