Read integrity monitoring techniques for vision navigation systems - 4 multiple faults in vision system

stay 3.7 Summarized in section [16-18] The study will use parity space and slope GPS Integrity monitoring technology is transformed into visual measurement , It also provides a common framework for two navigation systems . However , Yes GPS The assumptions made do not apply to the visual system . stay GPS In the system , It is unlikely that more than one fault will occur at the same time .GPS The constellation is closely monitored , And robust . In the case of visual measurement , The probability of having more than one bad measurement is much higher . Previous studies have not addressed this possibility . The first 4.1 This section summarizes the statistical tests discussed in the previous section , The statistical test quantity is designed to detect bad measurements . then , In the 4.2 This method is used as a test in section , To determine whether the data subset is good , The iterative method is used to eliminate the bad measurement in the data .

4.1 Statistical inspection quantity

stay [16-18] in ,Larson Adopted GPS Slope method in integrity monitoring , Among them, the decision variable $D=p^{T}p$ The relationship with the horizontal position error is the ratio of the square of the vector norm of the horizontal position error to the square of the odd and even vector norm ：
$$\frac{||\delta x|{|}^2}{||p|{|}^2}=\frac{\delta x^T\delta x}{p^{T}p}=\frac{b^{T}Gb}{b^{T}Sb}\text{\hspace{1em}(3.1)}$$
among $G={\bar{H}}_h^T{\bar{H}}_h$,$S=P^{T}P$,$\bar{H}=(H^{T}H{)}^{-1}{H}^T$, It's a matrix $H$ Of Moore-Penrose Pseudo inverse . Suppose we divide $b_i$ and $b_j$ Outside component , Deviation vector $b$ by 0（ Corresponds to a deviation in a set of pixel coordinates , The magnitude and direction of the error are $\theta$）, The formula （3.1） It can be written. （ For detailed derivation, see [16]）：
$$\frac{||\delta x|{|}^2}{||p|{|}^2}=[\frac{si{n}^2(\theta )(G_{ii}-G_{jj})+sin(2\theta )G_{ij}+G_{jj}}{si{n}^2(\theta )(S_{ii}-S_{jj})+sin(2\theta )S_{ij}+S_{jj}}{]}^{\frac{1}{2}}\text{\hspace{1em}(3.2)}$$

4.2 Bayesian algorithm for separating fault measurement

The algorithm of separating fault measurement is based on formula （3.3） Bayesian theorem given , And discussed in many books on probability and Statistics .
$$P(A_i|B)=\frac{P(B|A_i)P(A_i)}{{\sum }_{j=1}^{\infty }P(B|A_j)P(A_j)}\text{\hspace{1em}(3.3)}$$
When the entire measurement data set fails 4.1 When testing as described in section , It is assumed that there is at least one faulty measurement in the measurement data set , And the probability of failure of each measurement data is equal . therefore , vector $\vec{P}$ All elements of represent the error probability of each measurement , Initialize to $1/m$, among $m$ Is the number of measurements .

Create and test multiple random data subsets from the original data set . If you pass , Then use the formula （3.5） Update the values related to the measurements in the subset $\vec{P}$ The corresponding element of . If you fail , Then use the formula （3.4） to update $\vec{P}$ Corresponding elements in . After several tests on different subset combinations of measurements , If there is a high enough probability to transfer the data subset ,$\vec{P}$ Will converge .
P ⃗ { E r r o r = 1 ∣ A l a r m } ( k + 1 ) = P ˉ m d P ⃗ ( k ) ∑ ( P ˉ m d P ⃗ ( k ) ) + P f a P ˉ e (3.4) \vec{P}\{Error=1|Alarm\}(k+1)=\frac{\bar{P}_{md}\vec{P}(k)}{\sum(\bar{P}_{md}\vec{P}(k))+P_{fa}\bar{P}_e} \tag{3.4} P { Error=1∣Alarm}(k+1)=∑(Pˉmd​P (k))+Pfa​Pˉe​Pˉmd​P (k)​(3.4)
P ⃗ { E r r o r = 1 ∣ P a s s } ( k + 1 ) = P m d P ⃗ ( k ) ∑ ( P m d P ⃗ ( k ) ) + P ˉ m d P ˉ e (3.5) \vec{P}\{Error=1|Pass\}(k+1)=\frac{P_{md}\vec{P}(k)}{\sum(P_{md}\vec{P}(k))+\bar{P}_{md}\bar{P}_e} \tag{3.5} P { Error=1∣Pass}(k+1)=∑(Pmd​P (k))+Pˉmd​Pˉe​Pmd​P (k)​(3.5)
among $P_{md}$ Is the probability of missing alarm in the test ,$P_{fa}$ Is the false alarm probability of the test ,$P_e$ Is in a given subset $\vec{P}$ Probability of failure in , and ${\bar{P}}_e$ Satisfy ${\bar{P}}_e=1-P_e$.

The probability of obtaining the passing random data subset is based on the given hypergeometric distribution ,
$$P(X=x|N,M,n)=\frac{\left(\begin{array}{c}M\\ x\end{array}\right)\left(\begin{array}{c}N-M\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}\text{\hspace{1em}(3.6)}$$
among $N$ Indicates the total number of measurements ,$n$ Is the number of samples of the test subset ,$M$ Is the number of fault measurements ,$x$ Is the number of faults in a subset .

Assuming that only one fault measurement is required, the test will fail , Then the probability of passing through the subset is ：
$$P(GoodSet)=P(X=0|N,M,n)\text{\hspace{1em}(3.7)}$$

$$=\frac{\left(\begin{array}{c}N-M\\ n\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}\text{\hspace{1em}(3.8)}$$

chart 4.1 It shows the relationship between the probability of passing the test and the number of fault measurements , Let's say... At one time 5 A measurement .