As the most common navigation system, SINS can provide attitude, velocity and position of a vehicle, such as a ship, a spacecraft and an aircraft [1]. To get correct navigation information, high reliability of SINS is needed. However, SINS suffers a lot due to poor working environment or improper operation. In consequence, high reliability of SINS cannot be guaranteed all the time [2].
作为最常见的导航系统，SINS可以提供车辆的姿态、速度和位置，如船舶、航天器和飞机[1]。为了获得正确的导航信息，SINS需要高可靠性。然而，由于工作环境恶劣或操作不当，SINS遭受了很大的损失。因此，无法始终保证SINS的高可靠性[2]。

Among all components of SINS, inertial sensors (including gyros and accelerometers) are the most critical and easily damaged components [3]. As a result, redundant SINS is developed to increase reliability of SINS, which is obtained by utilizing multiple inertial sensors in SINS [4]. In redundant SINS, four-gyro SINS has a wide range of application as it supplies big reliability considering cost and volume [5], [6].
在SINS的所有组件中，惯性传感器（包括陀螺仪和加速度计）是最关键和最容易损坏的组件[3]。因此，开发了冗余SINS来提高SINS的可靠性，这是通过在SINS中利用多个惯性传感器获得的[4]。在冗余SINS中，考虑到成本和体积，四陀螺SINS具有很高的可靠性，因此具有广泛的应用[5]、[6]。

In four-gyro SINS, navigation result can be calculated from eight inertial sensors’ outputs. Incorrect navigation results will be obtained if any one of the gyros or accelerometers fails.
在四陀螺捷联惯导系统中，可以根据八个惯性传感器的输出计算导航结果。如果任何一个陀螺仪或加速计发生故障，将获得不正确的导航结果。
The International Federation of Automatic Control (IFAC) Safe Process Technical Committee defines a fault as an unpermitted deviation of at least one characteristic property or parameter of the system from the acceptable/usual/standard condition [7]. There are two ways to describe a fault: hard fault and soft fault. Hard faults present a sudden change on the basis of the normal outputs of sensors. Hard faults generally result from hardware damage, and make sensors fail immediately.
国际自动控制联合会（IFAC）安全过程技术委员会将故障定义为系统的至少一个特征属性或参数与可接受/通常/标准条件的不允许偏差[7]。描述故障有两种方法：硬故障和软故障。硬故障在传感器正常输出的基础上呈现突然变化。硬故障通常是由硬件损坏引起的，并使传感器立即失效。
Soft faults present a slow change on the basis of the normal outputs of sensors. Soft faults generally result from complex reasons, and cannot be detected immediately [7], [8].
软故障在传感器正常输出的基础上呈现缓慢变化。软故障通常是由复杂的原因引起的，不能立即检测[7]，[8]。

FI is necessary when a fault occurs in four-gyro SINS. FI is to determine which sensor has a fault and judge the fault’s type in redundant SINS. Current FI approaches are mainly for single fault.
当四陀螺捷联惯导系统发生故障时，FI是必要的。FI是确定冗余SINS中哪个传感器发生故障并判断故障类型。目前的FI方法主要针对单一故障。
There are three kinds of FI approaches for redundant SINS: direct comparison approach, signal processing approach and parity space approach. Direct comparison approach isolates a fault through linear relationship of multiple sensors.
冗余捷联惯导系统的FI方法有三种：直接比较法、信号处理法和奇偶空间法。直接比较法通过多个传感器的线性关系隔离故障。
However, in four-gyro SINS, there is only one group of linear FI equation, and no FI truth table can be built up. As a result, direct comparison approach cannot isolate a fault in four-gyro SINS [9]–​[11]. In addition, direct comparison approach cannot be applied when considering errors of inertial sensors.
然而，在四陀螺捷联惯导系统中，只有一组线性FI方程，无法建立FI真值表。因此，直接比较方法无法隔离四个陀螺SINS[9]-[11]中的故障。此外，在考虑惯性传感器误差时，不能采用直接比较法。

Signal processing approach, such as linear estimation approach [12], variance identification approach [13] and wavelet transform approach [14], [15], can be applied to FI in four-gyro SINS. While they are only useful to a hard fault. Moreover, they show bad isolation performance when SINS change its motion state. Yan proposed mean test approach [16] to isolate a fault by testing mean value of an inertial sensor. However, the isolation results is not correct when considering errors of inertial sensors.
信号处理方法，如线性估计方法[12]、方差识别方法[13]和小波变换方法[14]、[15]，可以应用于四陀螺SINS中的FI。虽然它们只对严重的错误有用。此外，当SINS改变其运动状态时，它们表现出较差的隔离性能。Yan提出了平均测试方法[16]，通过测试惯性传感器的平均值来隔离故障。然而，当考虑惯性传感器的误差时，隔离结果是不正确的。
In general, signal processing approach are all especially suitable for a hard fault, and they lose effectiveness when a soft fault happens.
一般来说，信号处理方法都特别适用于硬故障，当软故障发生时，它们会失效。

Parity space approach, such as local estimation approach, optimal parity vector (OPT) approach, singular value decomposition (SVD) approach and GLT, is the most popular FI approach. They are useful when there is a hard fault or a soft fault.
奇偶空间方法，如局部估计方法、最优奇偶向量（OPT）方法、奇异值分解（SVD）方法和GLT，是最流行的FI方法。当出现硬故障或软故障时，它们很有用。
Local estimation approach can detect and isolate a small fault, but its amount of calculation is large, and it cannot diagnose a fault in time [17]. OPT has good robustness, but the rate of correct isolation is lower than that of correct detection [18], [19]. SVD cannot diagnose a fault with an opposite direction [20], [21].
局部估计方法可以检测和隔离小故障，但其计算量很大，无法及时诊断故障[17]。OPT具有良好的鲁棒性，但正确隔离率低于正确检测率[18]，[19]。SVD无法诊断相反方向的故障[20]，[21]。

GLT become most widely used in FI as it achieves high sensitivity, small calculation and easy implementation [22]–​[24]. Lots of researchers apply KF algorithms to GLT, making GLT useful when considering errors of inertial sensors [25], [26]. However, GLT isolation approach based on KF cannot be applied to isolate a hard fault or a soft fault in four-gyro SINS.
GLT在FI中的应用最为广泛，因为它实现了高灵敏度、小计算和易于实现[22]-[24]。许多研究人员将KF算法应用于GLT，使GLT在考虑惯性传感器误差时非常有用[25]，[26]。然而，基于KF的GLT隔离方法不能应用于隔离四陀螺SINS中的硬故障或软故障。

A few researchers come up with an idea for FI—using other kinds of sensors to provide angular velocity baseline information. Fault can be isolated by checking difference between angular velocity baseline information and gyro outputs.
一些研究人员提出了FI的想法——使用其他类型的传感器提供角速度基线信息。通过检查角速度基线信息和陀螺仪输出之间的差异，可以隔离故障。
Since 1990s, lots of domestic and foreign scholars do research on star sensors to get accurate angular velocity of a vehicle in non-gyro system, and many research results have been obtained [27]–​[30]. In recent years, with the development of star sensors and the maturity of angular velocity’s algorithm based on a star sensor, the accuracy of angular velocity can reach 10−6rad/s [31], [32]. These make star sensors provide accurate angular velocity baseline information for FI. A few researchers utilize star sensors to isolate a faulty gyro. Faulty gyro can be detected and isolated through comparison between gyro outputs and angular velocity from star sensor [33], [34].This FI approach is applied in SINS, however, it cannot be used in redundant SINS when considering errors of sensors.
自20世纪90年代以来，国内外许多学者对星敏感器进行了研究，以在非陀螺系统中获得车辆的精确角速度，并取得了许多研究成果[27]-[30]。近年来，随着星敏感器的发展和基于星敏感器的角速度算法的成熟，角速度的精度可以达到 10−6rad/s [31]，[32]。这使得星敏感器为FI提供了准确的角速度基线信息。一些研究人员利用星敏感器来隔离有故障的陀螺仪。通过比较陀螺仪输出和星敏感器的角速度，可以检测和隔离有故障的陀螺仪[33]，[34]。这种FI方法应用于SINS，但在考虑传感器误差的情况下，它不能用于冗余SINS。

In this paper, with the aid of star sensors, an improved residual chi-square test fault isolation approach improves the thought of residual chi-square test, which has been widely applied to detect a fault for integrated navigation system (INS) [35], [36]. This approach first modifies the model of GLT considering the errors of inertial sensors, and redesigns a residual vector generated by gyros and star sensors for four-gyro SINS. Second, a separate residual chi-square test approach is designed to isolate a fault for each gyro.
本文借助星敏感器，提出了一种改进的残差卡方检验故障隔离方法，改进了残差卡方试验的思想，该方法已被广泛应用于组合导航系统（INS）的故障检测[35]，[36]。该方法首先修改了GLT模型，考虑了惯性传感器的误差，并为四陀螺SINS重新设计了由陀螺仪和星敏感器生成的残差矢量。其次，设计了一种单独的残差卡方检验方法来隔离每个陀螺仪的故障。
Third, to reduce soft fault isolation time, an average separate residual vector and new isolation threshold are designed. Fourth, star sensors provide accurate angular velocity baseline information to generate residual vector for FI in four-gyro SINS.
第三，为了减少软故障隔离时间，设计了一个平均分离残差向量和新的隔离阈值。第四，星敏感器提供精确的角速度基线信息，为四陀螺SINS中的FI生成残差矢量。

The contribution of our research are as follows. To recognize a fault in four-gyro SINS, first, a new observation model for isolation is designed. Second, separate residual chi-square test fault isolation approach is proposed to isolation a fault in four-gyro SINS.
我们的研究贡献如下。为了识别四陀螺捷联惯导系统中的故障，首先，设计了一种新的隔离观测模型。其次，提出了一种分离残差卡方检验故障隔离方法，用于隔离四陀螺捷联惯导系统中的故障。
Third, average separate residual chi-square test fault isolation approach improves separate residual chi-square test approach, and reduce the isolation time of a soft fault.
第三，平均分离残差卡方检验故障隔离方法改进了分离残差卡平方检验方法，减少了软故障的隔离时间。

The rest of this paper is organized as follows. In part II, the limitation of GLT algorithm based on KF is proved. In part III, an improved residual chi-square test fault isolation approach for four-gyro SINS is proposed.
本文的其余部分组织如下。第二部分证明了基于KF的GLT算法的局限性。在第三部分中，提出了一种改进的四陀螺捷联惯导系统残差卡方检验故障隔离方法。
Part IV describes a redundant model, tetrahedral structure. Experimental results and discussions are presented in part V. Finally, conclusions are summarized in Part VI.
第四部分描述了一个冗余模型，四面体结构。第五部分给出了实验结果和讨论。最后，第六部分总结了结论。

In this part, GLT algorithm is first presented. Second, GLT algorithm based on KF is described. Third, limitation of GLT isolation algorithm is proved.
本部分首先介绍了GLT算法。其次，描述了基于KF的GLT算法。第三，证明了GLT隔离算法的局限性。

A. GLT Algorithm A.GLT算法

m measurements are generated by a number of redundant inertial sensors:
m 测量值由多个冗余惯性传感器生成：

Z=HX+f+ε,(1)

View Source 查看源代码 \begin{equation*} \boldsymbol {Z}=\boldsymbol {HX}+\boldsymbol {f}+\boldsymbol {\varepsilon },\tag{1}\end{equation*} where Z∈Rm denotes inertial sensors’ measurements. X∈Rn is the angular velocity. H∈Rm×n is the inertial sensors’ measurement matrix. f is a fault vector. ε is a measurement noise vector satisfies:
其中 Z∈Rm 表示惯性传感器的测量值 X∈Rn 是角速度 H∈Rm×n 是惯性传感器的测量矩阵。3#是故障向量 ε 是满足以下条件的测量噪声向量：

E(ε)=0,E(εεT)=R.(2)

View Source 查看源代码 \begin{equation*} E(\boldsymbol {\varepsilon })=0,\quad E(\boldsymbol {\varepsilon \varepsilon }^{T})=\boldsymbol {R}.\tag{2}\end{equation*}

A (m−3) -dimensional parity vector P is determined to detect and isolate a fault.
确定 (m−3) 维奇偶校验向量 P 以检测和隔离故障。

P=VZ=VHX+Vf+Vε,(3)

View Source 查看源代码 \begin{equation*} \boldsymbol {P}=\boldsymbol {VZ}=\boldsymbol {VHX}+\boldsymbol {Vf}+\boldsymbol {V}\boldsymbol {\varepsilon },\tag{3}\end{equation*} where V is a (m−3)×m -dimensional parity matrix that satisfies
其中 V 是满足以下条件的 (m−3)×m 维奇偶校验矩阵

VVT=Im−3VH=0.(4)

View Source 查看源代码 \begin{equation*} \boldsymbol {VV}^{T}=\boldsymbol {I}_{m-3} \boldsymbol {VH}=\boldsymbol {0}.\tag{4}\end{equation*}

(3) can be rewritten as
（3） 可以改写为

P=Vf+Vε.(5)

View Source 查看源代码 \begin{equation*} \boldsymbol {P}=\boldsymbol {Vf}+\boldsymbol {V}\boldsymbol {\varepsilon }.\tag{5}\end{equation*}

The parity vector P is a vector only related to fault and noise, and P has no relationship with angular velocity.
奇偶校验向量 P 是仅与故障和噪声相关的向量，而 P 与角速度无关。

GLT fault detection decision function is [22]:
GLT故障检测决策函数为[22]：

FDGLT=If FDGLT≥if FDGLT<1σ2(PTP).TD, a fault occurs;TD, no fault occur.(6)

View Source 查看源代码 \begin{align*} FD_{GLT}=&\frac {1}{\sigma ^{2}}\left ({{\boldsymbol P^{\boldsymbol T}\boldsymbol P} }\right). \\ \text {If}~FD_{\text {GLT}}\ge&T_{D},~\text {a fault occurs}; \\ \text {if}~FD_{\text {GLT}} < &T_{D},~\text {no fault occur}.\tag{6}\end{align*} where FDGLT∼χ2(m−n) . TD denotes a fault detection threshold derived from Chi square distribution table according to the given false alarm rate and degree of freedom (m−n) .
其中 FDGLT∼χ2(m−n) TD 表示根据给定的误报率和自由度从卡方分布表导出的故障检测阈值 (m−n) 。

GLT fault isolation function is shown:
GLT故障隔离功能如图所示：

FIGLT(i)=(PTVi)2σ2VTiVi,(7)

View Source 查看源代码 \begin{equation*} FI_{\text {GLT}} (i)=\frac {(\boldsymbol P^{\text {T}}\boldsymbol V_{i})^{2}}{\sigma ^{2}\boldsymbol V_{i}^{\text {T}} \boldsymbol V_{i}},\tag{7}\end{equation*} faulty sensor has the maximum of FIGLT(i) . Vi is the i th column of V .
故障传感器的最大值为 FIGLT(i) Vi 是 V 的第2#列。

B. GLT Algorithm Based on KF
B.基于KF的GLT算法

In traditional GLT algorithm, the errors of sensors-input misalignment, scale factor and bias are not taken into consideration. As a result, traditional GLT algorithm cannot detect and isolate a fault. KF has been applied to compensate the errors in GLT, and makes GLT work.
在传统的GLT算法中，没有考虑传感器输入失准、比例因子和偏差的误差。因此，传统的GLT算法无法检测和隔离故障。KF已被应用于补偿GLT中的误差，并使GLT工作。

When faults are not taken into consideration, (1) will be rewritten as:
当不考虑故障时，（1）将改写为：

Z=(I+Hse)[(H+Hme)X+b1+ε1],(8)

View Source 查看源代码 \begin{equation*} \boldsymbol {Z}=(\boldsymbol {I}+\boldsymbol {H}_{se})[(\boldsymbol {H}+\boldsymbol {H}_{me})\boldsymbol {X}+\boldsymbol {b}_{1} +\boldsymbol {\varepsilon }_{1}],\tag{8}\end{equation*} where Hse∈Rm×m is a scale factor error matrix. Hme∈Rm×n is an input misalignment error matrix. b1 is a bias vector. ε1 is Gaussian white noise with zero mean.
其中 Hse∈Rm×m 是比例因子误差矩阵。 Hme∈Rm×n 是输入未对准误差矩阵。 b1 是偏置向量 ε1 是均值为零的高斯白噪声。

b=ε=Hm=(I+Hse)b1(I+Hse)ε1H+Hme+HseH.(9)

View Source 查看源代码 \begin{align*} \boldsymbol {b}=&(\boldsymbol {I}+\boldsymbol {H}_{se})\boldsymbol {b}_{1} \\ \boldsymbol {\varepsilon }=&(\boldsymbol {I}+\boldsymbol {H}_{se})\boldsymbol {\varepsilon }_{1} \\ \boldsymbol {H}_{m}=&\boldsymbol {H}+\boldsymbol {H}_{me} +\boldsymbol {H}_{se} \boldsymbol {H}.\tag{9}\end{align*}

From (3) and (4), parity vector is expressed:
根据（3）和（4），奇偶向量表示为：

P=VZ=VHmX+Vb+Vε+Vf,(10)

View Source 查看源代码 \begin{equation*} \boldsymbol {P}=\boldsymbol {VZ}=\boldsymbol {VH}_{m} \boldsymbol {X}+\boldsymbol {Vb}+\boldsymbol {V}\boldsymbol {\varepsilon }+\boldsymbol {Vf},\tag{10}\end{equation*} where estimation X^ is:
其中估计 X^ 是：

X^=(HTH)−1HTZ.(11)

View Source 查看源代码 \begin{equation*} \hat {\boldsymbol {X}}=(\boldsymbol {H}^{\text {T}}\boldsymbol {H})^{-1}\boldsymbol {H}^{\text {T}}\boldsymbol {Z}.\tag{11}\end{equation*}

The state vector e related to VHm and Vb satisfies the Discrete Markov process [25]:
与 VHm 和 Vb 相关的状态向量 e 满足离散马尔可夫过程[25]：

ek=ek−1+wk−1.(12)

View Source 查看源代码 \begin{equation*} \boldsymbol {e}_{k} =\boldsymbol {e}_{k-1} +\boldsymbol {w}_{k-1}.\tag{12}\end{equation*}

From KF, the estimation e^ can be obtained.
从KF可以获得估计 e^ 。

After compensation, compensated parity vector, GLT fault detection function and isolation function based on KF are
补偿后，补偿奇偶向量、GLT故障检测功能和基于KF的隔离功能

P∗=FDGLT=FIGLT(i)=P−VHmX−Vb=Vε+Vf.1σ2(P∗TP∗).(P∗TVi)2σ2VTiVi.(13)(14)(15)

View Source 查看源代码 \begin{align*} \boldsymbol {P}^{\ast }=&\boldsymbol {P}-\boldsymbol {VH}_{m} \boldsymbol {X}-\boldsymbol {Vb}=\boldsymbol {V}\boldsymbol {\varepsilon }+\boldsymbol {Vf}.\tag{13}\\ FD_{\text {GLT}}=&\frac {1}{\sigma ^{2}}\left ({{\boldsymbol P^{\ast T}\boldsymbol P^{\ast }} }\right).\tag{14}\\ FI_{\text {GLT}} (i)=&\frac {(\boldsymbol P^{\ast \text {T}}\boldsymbol V_{i})^{2}}{\sigma ^{2}\boldsymbol V_{i}^{\text {T}} \boldsymbol V_{i} }.\tag{15}\end{align*}

C. Limitation of GLT Isolation Based on KF
C.基于KF的GLT分离的局限性

GLT algorithm can only detect a fault, but fails to isolate a fault in four-gyro SINS. Here is a proof of the conclusion.
GLT算法只能检测故障，但无法隔离四陀螺SINS中的故障。这是结论的证明。

From (4) and (13), the fault vector f ’s projection onto the column space of V is described:
根据（4）和（13），描述了故障向量 f 在 V 列空间上的投影：

f^null=Vf.(16)

View Source 查看源代码 \begin{equation*} \hat {\boldsymbol {f}}_{null} =\boldsymbol {Vf}.\tag{16}\end{equation*}

Assuming that gyro i has a fault. The fault’s projection onto the column space of V is shown as f^i :
假设陀螺仪 i 有故障。断层在 V 柱距上的投影如 f^i 所示：

f^i=Vfi=V(fei)=fVi,(17)

View Source 查看源代码 \begin{equation*} \hat {\boldsymbol {f}}_{i} =\boldsymbol {Vf}_{i} =\boldsymbol {V}(f{\boldsymbol {e}}_{i})=f\boldsymbol {V}_{i},\tag{17}\end{equation*} where f is the inertial sensor’s fault magnitude. e is a m -dimensional column vector. The i th element of ei is 1, and the others are 0.
其中 f 是惯性传感器的故障幅度 e 是一个2#维列向量。 ei 的第3#个元素为1，其他元素为0。

According to (15), (17) can be rewritten as:
根据（15），（17）可以改写为：

FIGLT(i)=(P∗TVi)2σ2VTiVi=f2(VTiVi)2σ2VTiVi.(18)

View Source 查看源代码 \begin{equation*} FI_{\text {GLT}} (i)=\frac {(\boldsymbol P^{\ast \text {T}}\boldsymbol V_{i})^{2}}{\sigma ^{2}\boldsymbol V_{i}^{\text {T}} \boldsymbol V_{i} }=\frac {f^{2}(\boldsymbol V_{i}^{\text {T}}\boldsymbol V_{i})^{2}}{\sigma ^{2}\boldsymbol V_{i}^{\text {T}} \boldsymbol V_{i} }.\tag{18}\end{equation*}

From the property of V , the rank of V is 1 in four-gyro SINS. Thus, Vi in (18) is only a scalar but not a vector. At this time, (18) is rewritten as:
从 V 的性质来看，在四个陀螺捷联惯导系统中， V 的秩为1。因此，（18）中的 Vi 只是标量，而不是向量。此时，（18）被改写为：

FIGLT(i)=f2(VTiVi)2σ2VTiVi=f2(VTiVi)σ2=P∗TP∗σ2.(19)

View Source 查看源代码 \begin{equation*} FI_{\text {GLT}} (i)=\frac {f^{2}(\boldsymbol V_{i}^{\text {T}}\boldsymbol V_{i})^{2}}{\sigma ^{2}\boldsymbol V_{i}^{\text {T}} \boldsymbol V_{i} }=\frac {f^{2}(\boldsymbol V_{i}^{\text {T}}\boldsymbol V_{i})}{\sigma ^{2}}=\frac {\boldsymbol P^{\ast \text {T}}\boldsymbol P^{\ast }}{\sigma ^{2}}.\tag{19}\end{equation*}

From (19), no matter which inertial sensor fails in four-gyro SINS, GLT fault isolation function value will remain the same. There is no maximum value in the isolation functions. Thus, GLT isolation function based on KF cannot isolate a fault in four-gyro SINS.
从（19）可以看出，无论四个陀螺捷联惯导系统中哪个惯性传感器发生故障，GLT故障隔离函数值都将保持不变。隔离函数中没有最大值。因此，基于KF的GLT隔离功能无法隔离四陀螺SINS中的故障。

From (15), GLT isolation works through judging which column of V has the same direction with the parity vector P∗ . Since P∗ is a scalar in four-gyro SINS system, it cannot have a direction. Thus, parity vector can’t be applied to isolate a fault in four-gyro SINS system either.
根据（15），GLT隔离是通过判断 V 的哪一列与奇偶校验向量 P∗ 的方向相同来实现的。由于#2是四陀螺捷联惯导系统中的标量，因此它不能有方向。因此，奇偶向量也不能用于隔离四陀螺SINS系统中的故障。

Simulations are conducted to verify the limitation of GLT isolation based on KF. Assuming that gyro 4 has a hard fault in the thirtieth second. The results are shown below. The gyro providing the biggest isolation value is the faulty one. Fig. 1 shows that gyro 4’s isolation function value is the same as others. As a result, we can’t tell which gyro is faulty. The results demonstrate that fault isolation are wrong. Same situation happens when there is a soft fault in four-gyro SINS.
通过仿真验证了基于KF的GLT隔离的局限性。假设陀螺仪4在第三十秒内出现硬故障。结果如下所示。提供最大隔离值的陀螺仪是有故障的陀螺仪。图1显示，陀螺仪4的隔离函数值与其他陀螺仪相同。因此，我们无法判断哪个陀螺仪有故障。结果表明，故障隔离是错误的。当四个陀螺捷联惯导系统出现软故障时，也会出现同样的情况。

FIGURE 1.  图1。

GLT fault isolation based on KF for a hard fault.
基于KF的GLT故障隔离用于硬故障。

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FIGURE 2.  图2。

Improved residual chi-square test fault isolation approach brief flowchart.
改进的残差卡方检验故障隔离方法简要流程图。

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FIGURE 3.  图3。

Improved residual chi-square test fault isolation approach.
改进的残差卡方检验故障隔离方法。

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From these above, we can see that GLT fault isolation based on KF cannot be applied to FI in four-gyro SINS. In order to isolate a fault in four-gyro SINS, a new fault isolation approach must be designed.
综上所述，基于KF的GLT故障隔离不能应用于四陀螺SINS中的FI。为了隔离四陀螺捷联惯导系统中的故障，必须设计一种新的故障隔离方法。

Improved residual chi-square test fault isolation approach, first, designs a residual vector based on KF, which is for isolating faulty gyro. Second, separate residual vector chi-square test fault isolation algorithm is proposed to isolate a fault for four-gyro SINS system.
改进的残差卡方检验故障隔离方法，首先设计了一种基于KF的残差向量，用于隔离故障陀螺仪。其次，提出了一种分离残差向量卡方检验故障隔离算法，用于隔离四陀螺SINS系统的故障。
Third, average residual vector chi-square test fault isolation algorithm is designed to improve isolation performance of a soft fault.
第三，设计了一种平均残差向量卡方检验故障隔离算法，以提高软故障的隔离性能。
At last, an approach to obtain angular velocity of a vehicle with the aid of star sensor is presented, which can provide important observation baseline information for isolation approach.
最后，提出了一种借助星敏感器获取车辆角速度的方法，该方法可以为隔离方法提供重要的观测基线信息。

A. Residual Vector Based on KF
A.基于KF的残差向量

In this part, first, a new isolation observation model and the state vector are designed. Second, on the basis of KF, a residual vector can be obtained through KF algorithm.
在这一部分中，首先，设计了一个新的隔离观测模型和状态向量。其次，在KF的基础上，通过KF算法可以获得残差向量。

From part II, both parity vector and isolation function of GLT fail to isolate a fault in four-gyro SINS. As a result, the observation model in (10) cannot be applied to isolation anymore. A new observation model should be considered.
从第二部分可以看出，GLT的奇偶向量和隔离功能都无法隔离四陀螺SINS中的故障。因此，（10）中的观察模型不能再应用于隔离。应考虑新的观测模型。

From (8), when there is no fault, (8) can be presented as:
从（8）中，当没有故障时，（8）可以表示为：

Z=HX+HmX+b+ε.(20)

View Source 查看源代码 \begin{equation*} \boldsymbol {Z}=\boldsymbol {HX}+\boldsymbol {H}_{m} \boldsymbol {X}+\boldsymbol {b}+\boldsymbol {\varepsilon }.\tag{20}\end{equation*}

Different from (12), X is from star sensors, and details are presented in D, part III.
与（12）不同， X 来自恒星传感器，详细信息见D，第三部分。

From (20), there exists
从（20）开始，存在

Z−HX=HmX+b+ε.(21)

View Source 查看源代码 \begin{equation*} \boldsymbol {Z}-\boldsymbol {HX}=\boldsymbol {H}_{m} \boldsymbol {X}+\boldsymbol {b}+\boldsymbol {\varepsilon }.\tag{21}\end{equation*}

Supposing there is a vector q , and its form is presented as below:
假设有一个向量 q ，其形式如下：

q=[q11q12q13q21…qm3]T,(22)

View Source 查看源代码 \begin{equation*} \boldsymbol {q}=[{q_{11}} \quad {q_{12}} \quad {q_{13}} \quad {q_{21}} \quad {\ldots } \quad {q_{m3}}]^{\text {T}},\tag{22}\end{equation*} where qij denotes Hm ’s element. The first part of (21) is rewritten as:
其中 qij 表示 Hm 的元素。（21）的第一部分改写为：

HmX=Wq,(23)

View Source 查看源代码 \begin{equation*} \boldsymbol {H}_{m} \boldsymbol {X}=\boldsymbol {Wq},\tag{23}\end{equation*} where W∈Rm×3m can be denoted as:
其中 W∈Rm×3m 可以表示为：

W=⎡⎣⎢⎢⎢⎢XT000…000000XT…000…………000000…XT⎤⎦⎥⎥⎥⎥.(24)

View Source 查看源代码 \begin{equation*} \boldsymbol {W}=\left [{ {{\begin{array}{cccccccccccccccccccc} {\boldsymbol {X}^{\text {T}}} &\quad {000} &\quad {{\ldots{ }}} &\quad {000} \\ {000} &\quad {\boldsymbol {X}^{\text {T}}} &\quad {\ldots } &\quad {000} \\ {\ldots } &\quad {\ldots } &\quad {\ldots } &\quad {\ldots } \\ {000} &\quad {000} &\quad {\ldots } &\quad {\boldsymbol {X}^{\text {T}}} \\ \end{array}}} }\right].\tag{24}\end{equation*}

(21) can be expressed as:
（21）可以表示为：

Z−HX^=Wq^+b+ε.(25)

View Source 查看源代码 \begin{equation*} \boldsymbol {Z}-\boldsymbol H\hat {{\boldsymbol {X}}}=\hat {\boldsymbol {W}\boldsymbol q}+\boldsymbol {b}+\boldsymbol {\varepsilon }.\tag{25}\end{equation*}

A new state vector c can be extended from q and b ,
新的状态向量 c 可以从 q 和 b 扩展，

c=[qb].(26)

View Source 查看源代码 \begin{equation*} \boldsymbol {c}=\left [{ {{\begin{array}{cccccccccccccccccccc} {\boldsymbol {q}} \\ {\boldsymbol {b}} \\ \end{array}}} }\right].\tag{26}\end{equation*}

Substituting (26) into (25), we can derive an observation model:
将（26）代入（25），我们可以得到一个观测模型：

Z−HX^=M^c+ε,(27)

View Source 查看源代码 \begin{equation*} \boldsymbol {Z}-\boldsymbol H\hat {\boldsymbol {X}}=\hat {\boldsymbol M}\boldsymbol c+\boldsymbol {\varepsilon },\tag{27}\end{equation*} where M=^[W^I] . 其中 M=^[W^I] 。

A dynamic model is needed to get the state vector c . Supposing that c can be modeled as a discrete-time Markov process [25]:
需要一个动态模型来获得状态向量 c 。假设#1可以建模为离散时间马尔可夫过程[25]：

ck=Φk|k−1ck−1+wk−1.(28)

View Source 查看源代码 \begin{equation*} \boldsymbol {c}_{k} =\boldsymbol {\Phi }_{k\vert k-1} \boldsymbol {c}_{k-1} +\boldsymbol {w}_{k-1}.\tag{28}\end{equation*} where wk denotes a white noise, and its noise covariance matrix is Qk . Φk|k−1 is a state transition matrix.
其中 wk 表示白噪声，其噪声协方差矩阵为 Qk Φk|k−1 是一个状态转移矩阵。

The estimation c^k can be derived from KF:
估计值#0可以从KF中得出：

c^k|k−1=Ek|k−1=Kk=c^k=Ek=Φk|k−1c^k−1Φk|k−1Ek−1ΦTk|k−1+Qk−1Ek|k−1M^Tk[M^kEk|k−1M^Tk+Rk]−1c^k|k−1+Kk[Zk−HX^k−M^kc^k|k−1][I−KkM^k]Ek|k−1.(29)(30)(31)(32)(33)

View Source 查看源代码 \begin{align*} \hat {\boldsymbol {c}}_{k\vert k-1}=&\boldsymbol {\Phi }_{k\vert k-1} \hat {\boldsymbol {c}}_{k-1}\tag{29}\\ \boldsymbol {E}_{k\vert k-1}=&\boldsymbol \Phi _{k\vert k-1} \boldsymbol {E}_{k-1} \boldsymbol \Phi _{k\vert k-1} ^{\boldsymbol {T}}+\boldsymbol {Q}_{k-1}\tag{30}\\ \boldsymbol {K}_{k}=&\boldsymbol {E}_{k\vert k-1} \hat {\boldsymbol {M}}^{T}_{k} [\hat {\boldsymbol {M}}_{k} \boldsymbol {E}_{k\vert k-1} \hat {\boldsymbol {M}}^{T}_{k} +\boldsymbol {R}_{k}]^{-1}\tag{31}\\ \hat {\boldsymbol {c}}_{k}=&\hat {\boldsymbol {c}}_{k\vert k-1} +\boldsymbol {K}_{k} [\boldsymbol {Z}_{k} -\boldsymbol {H}\hat {\boldsymbol X}_{k} -\hat {\boldsymbol {M}}_{k} \hat {\boldsymbol {c}}_{k\vert k-1}]\tag{32}\\ \boldsymbol {E}_{k}=&[\boldsymbol {I}-\boldsymbol {K}_{k} \hat {\boldsymbol {M}}_{k}]\boldsymbol {E}_{k\vert k-1}.\tag{33}\end{align*}

The practical observation is
实际观察是

Zpk=Zk−HX^k,(34)

View Source 查看源代码 \begin{equation*} \boldsymbol {Zp}_{k} =\boldsymbol {Z}_{k} -\boldsymbol {H}\hat {\boldsymbol X}_{k},\tag{34}\end{equation*} and the predicted observation is
预测的观测结果为

Zpk|k−1=M^kc^k|k−1,(35)

View Source 查看源代码 \begin{equation*} \boldsymbol {Zp}_{k\vert k-1} =\hat {\boldsymbol {M}}_{k} \hat {\boldsymbol {c}}_{k\vert k-1},\tag{35}\end{equation*}

The residual vector can be contained from the difference between the practical observation outputs and the predicted observation [35]
残差向量可以包含在实际观测输出和预测观测之间的差异中[35]

r=Zpk−Zpk|k−1=Zk−HX^k−M^kc^k|k−1,(36)

View Source 查看源代码 \begin{equation*} \boldsymbol {r}=\boldsymbol {Zp}_{k} \boldsymbol {-Zp}_{k\vert k-1} =\boldsymbol {Z}_{k} -\boldsymbol {H}\hat {\boldsymbol X}_{k} -\hat {\boldsymbol {M}}_{k} \hat {\boldsymbol {c}}_{k\vert k-1},\tag{36}\end{equation*} where r refers to a residual vector generated by gyros and star sensors.
其中 r 是指由陀螺仪和恒星传感器生成的残差向量。

B. Separate Residual Chi-Square Test Fault Isolation Approach
B.单独的残差卡方检验故障隔离方法

In traditional residual chi-square test, the function can only be used to detect a fault in INS, but it is never used to isolate a fault. In traditional residual chi-square test approach, the fault detection function [35] is
在传统的残差卡方检验中，该函数只能用于检测惯性导航系统中的故障，但从不用于隔离故障。在传统的残差卡方检验方法中，故障检测函数[35]为

λ=rTA−1kr,(37)

View Source 查看源代码 \begin{equation*} \lambda =\boldsymbol {r}^{\text {T}}\boldsymbol {A}_{k}^{-1}\boldsymbol {r},\tag{37}\end{equation*} where the residual vector r satisfies Gaussian noise with zero mean when there is no fault, and its covariance matrix is
其中，当没有故障时，残差向量 r 满足均值为零的高斯噪声，其协方差矩阵为

Ak=If λ≥if λ<M^kEk|k−1M^Tk+Rk.TD1,there is a fault;TD1,there is no fault.(38)

View Source 查看源代码 \begin{align*} \boldsymbol {A}_{k}=&\hat {\boldsymbol {M}}_{k} \boldsymbol {E}_{k\vert k-1} \hat {\boldsymbol {M}}_{k}^{\text {T}}+\boldsymbol {R}_{k}. \\ \text {If}~\lambda\ge&T_{D1},\quad \text {there is a fault}; \\ \text {if}~\lambda < &T_{D1},\quad \text {there is no fault}.\tag{38}\end{align*} where λ∼χ2(m) , as the dimension of r is m . TD1 denotes a fault detection threshold. It can be obtained through Chi square distribution table.
其中 λ∼χ2(m) ，因为 r 的维度是 m TD1 表示故障检测阈值。它可以通过卡方分布表获得。

From (37), the detection function λ contains the residual information of redundant SINS. Traditional residual chi-square test approach designs a residual vector for redundant SINS. The function cannot determine which gyro has a fault.
从（37）中，检测函数 λ 包含冗余SINS的残差信息。传统的残差卡方检验方法为冗余捷联惯导系统设计残差向量。该功能无法确定哪个陀螺仪有故障。
Thus, a new fault isolation approach is required to isolate a fault effectively in four-gyro SINS.
因此，需要一种新的故障隔离方法来有效隔离四陀螺捷联惯导系统中的故障。

In order to achieve isolation, separate residual chi-square test fault isolation approach designs a separate residual vector for each gyro. The problem of isolation can be realized by checking whether one gyro is faulty or not.
为了实现隔离，单独的残差卡方检验故障隔离方法为每个陀螺仪设计了一个单独的残差向量。隔离问题可以通过检查一个陀螺仪是否有故障来实现。
Consequently, the proposed approach can not only isolate a fault in four-gyro SINS, but also isolate a fault in any other redundant SINS.
因此，所提出的方法不仅可以隔离四个陀螺捷联惯导系统中的故障，还可以隔离任何其他冗余SINS中的故障。

Obviously, (36) can be rewritten as
显然，（36）可以改写为

r=Zk−HX^k−M^kc^k|k−1=[r1r2…ri]T,(39)

View Source 查看源代码 \begin{equation*} \boldsymbol {r}=\boldsymbol {Z}_{k} -\boldsymbol H\hat {\boldsymbol X}_{k} -\hat {\boldsymbol {M}}_{k} \hat {\boldsymbol {c}}_{k\vert k-1} =\left [{ {r_{1}} \quad {r_{2}} \quad {\ldots } \quad {r_{i}} }\right]^{\text {T}},\tag{39}\end{equation*} where ri(i=1,2…m) refers to a separate residual vector of gyro i .
其中 ri(i=1,2…m) 表示陀螺仪 i 的单独残差向量。

In (39), the separate residual vector ri satisfies Gaussian with zero mean when there is no fault, and its covariance matrix is
在（39）中，当没有故障时，分离残差向量 ri 满足均值为零的高斯分布，其协方差矩阵为

Ai=Ak(i,i).(40)

View Source 查看源代码 \begin{equation*} A_{i} =\boldsymbol {A}_{k} (i,i).\tag{40}\end{equation*}

From (39), when there is no fault, ri is a linear function of ε . ri is subject to 1-dimensional normal distribution. When there is a fault, the mean value of ri is not zero. As a result, fault isolation function can be determined according to the property of ri .
根据（39），当没有故障时， ri 是 ε 的线性函数 ri 服从一维正态分布。当发生故障时，#3的平均值不为零。因此，可以根据 ri 的属性确定故障隔离功能。

The statistical property of ri presents differently under two situation. The normal situation is denoted as H0 , and the failure situation is denoted as H1 :
在两种情况下， ri 的统计特性表现不同。正常情况表示为 H0 ，故障情况表示为 H1 ：

H0:E(ri)=H1:E(ri)=0,E(rirTi)=Aif,E[(ri−f)(ri−f)T]=Ai.(41)(42)

View Source 查看源代码 \begin{align*} H_{0}: E(r_{i})=&0,\quad E(r_{i} r_{i}^{\text {T}})=A_{i} \tag{41}\\ H_{1}: E(r_{i})=&f,\quad E[(r_{i} -f)(r_{i} -f)^{\text {T}}]=A_{i}.\tag{42}\end{align*} where f is faulty magnitude of gyro i .
其中 f 是陀螺仪 i 的错误幅度。

A log-likelihood ratio function related to ri is defined
定义了与 ri 相关的对数似然比函数

Λ(ri)=lnPr(ri/H1)Pr(ri/H0),(43)

View Source 查看源代码 \begin{equation*} \Lambda (r_{i})=\ln \frac {P_{r} (r_{i} /H_{1})}{P_{r} (r_{i} /H_{0})},\tag{43}\end{equation*} where Pr(⋅/⋅) means normal conditional probability density function. From (43) and (44),
其中 Pr(⋅/⋅) 表示正态条件概率密度函数。根据（43）和（44），

Λ(ri)=12[rTiri−(ri−f)T(ri−f)].(44)

View Source 查看源代码 \begin{equation*} \Lambda (r_{i})=\frac {1}{2}[r_{i}^{\text {T}} r_{i} -(r_{i} -f)^{\text {T}}(r_{i} -f)].\tag{44}\end{equation*}

From (44), the maximum likelihood estimation of f can be obtained
根据（44），可以获得 f 的最大似然估计

f^=ri.(45)

View Source 查看源代码 \begin{equation*} \hat {f}=r_{i}.\tag{45}\end{equation*}

Substituting (45) into (44), we obtain (46) as follows:
将（45）代入（44），我们得到（46）如下：

Λ(ri)=12rTiri(46)

View Source 查看源代码 \begin{equation*} \Lambda (r_{i})=\frac {1}{2}r_{i}^{\text {T}} r_{i}\tag{46}\end{equation*}

Thus, separate residual chi-square test fault isolation function can be derived:
因此，可以推导出单独的残差卡方检验故障隔离函数：

FRi=If FRi≥if FRi<rTiA−1iri.TIi,gyro i has a fault;TIi,gyro i has no fault.(47)

View Source 查看源代码 \begin{align*} FR_{i}=&r_{i}^{\text {T}} A_{i}^{-1}r_{i}. \\ \text {If}~FR_{i}\ge&T_{Ii},\quad \text {gyro}~i~\text {has a fault}; \\ \text {if}~FR_{i} < &T_{Ii},\quad \text {gyro}~i~\text {has no fault}.\tag{47}\end{align*} where FRi∼χ2(1) , as the dimension of ri is 1. TIi denotes gyro i ’s fault isolation threshold. It can be obtained through Chi square distribution table when probability of false alarm is given.
其中 FRi∼χ2(1) 表示陀螺仪 i 的故障隔离阈值，因为 ri 的维度为 1. TIi 。当给出误报概率时，可以通过卡方分布表获得。