The aim of this paper is to present a fault detection algorithm (FDI) based on signal
processing techniques developed for an inertial measurement unit (IMU) with minimal
redundancy of fiber optic gyros. In this work the recursive median filter is applied in
order to remove impulses (outliers) arising from data acquisition process and parity vector
operations, improving the fault detection and isolation performance. The FDI algorithm is
divided into two blocks: fault detection (FD) and fault isolation (FI). The FD part of the
algorithm is used to guarantee the reliability of the isolation part and is based on parity
vector analysis using
The main goal of fault detection, isolation, and recovery (FDIR) is to effectively detect faults and accurately isolate them to a failed component in the shortest time possible. This capability leads to reduction in diagnostic time or downtime in general, increasing the system availability. A good inherent diagnostic of a system also enhances the selfconfidence in operating system, the main aspect of mission success. FDIR is especially required to an onorbit system where maintenance may be difficult, very expensive, or even impossible. The spacecraft AOCS (Attitude and Orbit Control Subsystem) includes components such as sensors and actuators. Among the attitude control devices, the gyros are widely used for the sake of attitude control, mainly when highaccuracy requirements are imposed on the AOCS. This work belongs to this area and is addressed in the scenario of the orbital dynamics for spacecraft AOCS including inertial measurement unities (IMU) and FDIR software. This paper does not cover the recovery problem associated with FDIR but only with the FDI, that is, the fault detection and isolation problem. The methodology is based on signal processing techniques developed for an inertial measurement unit (IMU) with minimal redundancy of fiber optic gyros.
In the FDI problem with minimal redundancy, we are limited only to detect faults, generally with the aid of the parity vector analysis [
This paper is organized as follows. Section
The theoretical background and calibration methodology applied for FD algorithm (geometrical and parity vector analyses, recursive median filter, and
The geometrical aspect plays an important role in the inertial measurement units design. Several works [
Tetrahedral base and respective gyro axes orientation.
where
The parity vector (
where the matrix
In the absence of biases and faults, the parity vector is a white Gaussian noise vector, and in this configuration (four sensors in a skewed arrangement), it is possible to detect only the fault occurrence.
The wavelet packet (WP) decomposition is a method based on conventional wavelet transform theory whose difference is the processing of both filter outputs. In this method, the lowpass and highpass filter outputs are filtered at each level of decomposition in the form of a binary symmetric decomposition tree. This method is useful when a high spectral resolution analysis with low computational cost is needed. The WP decomposition structure is shown in Figure
Wavelet packet decomposition tree up to level
The singular value decomposition (SVD) is a mathematical operation defined in the linear algebra branch, that allows the real or complex matrix factorization. Its application is very useful in signal processing and analyses [
Let
or
where,
The SVD and the eigenvalues and eigenvectors decomposition are related as follows.
The lefthand side singular vectors of
The righthand side singular vectors of
The nonnull singular values of
Being the covariance matrix of
By using SVD the principal components from a given matrix can be obtained. Mathematically, the PCA is defined as a linear operation that projects a data set into a new coordinate systems, so that the major variance component be associated with the first coordinate, the second major variance component with the second coordinate, and so on [
Block of data set with rank
Seq.  Samples  

1 




2 




3 












With data set centered at zero mean, the SVD on the matrix
By sorting the data in decreasing order, the major values, and respective eigenvectors, can be selected considering a properly threshold criteria. By defining
where
a threshold is established in order to define a number
After defining the
where
The basic structure of the gyros processing with FDI blocks embedded is shown in Figure
Connection between Scale Factor block and FD/FI blocks is missed.
The detection of faults in an IMU with redundant sensors can be performed by PV analysis as defined in (
The fault isolation (FI) algorithm begins processing the data information stored in the buffers when the fault occurrence is indicated by FD block. The four buffers (one for each gyro) operate in FIFO mode. The operation of the FI algorithm is based on the analysis of the principal components in the energy subbands of the data stored in the buffers. The energy subbands are obtained by using the wavelet packet decomposition up to certain level. The idea behind this approach is to obtain a
The modeling of the FI algorithm begins with WP decomposition of a large amount of data from sensors divided in to pieces with properly defined size. The WP decomposition follows the schema illustrated in Figure
where
At each decomposition level
From wavelet theory, the detailed coefficients are like noise with zero mean, and perturbations with rich spectral content will introduce the variations in the subbands pseudoenergy magnitude. So, the SVD/PCA analysis can be applied to obtain the hidden structure in the matrix
The idea behind this is to consider the noise and other noiselike variations as hidden structures of the sensors or, in other words, a sensor pattern.
By decomposing
and selecting its principal components (PCA) as stated in (
where
From PCA theory, it is known that
In the sequence, applying
The meaning of (
Performing the operation given in (
2D example of the null space projection distances of the pseudoenergy vectors.
As a consequence, the mean distance and its variance are given by
respectively.
The noise influence can be reduced by using a normalization technique as presented by de Oliveira [
where
Assuming the average pseudoenergy at each subband as the autocorrelation of
and the normalization is given by
Considering the variation of the noise energy between buffers, (
where the
Therefore, the normalized pseudoenergy matrix becomes
and can be used instead of
The analyses of algorithm performance were made upon two different data sets: One for analysis and calibration (alpha) and the other for validation (beta). Using the model developed in Section
rotation:
normalized rotation:
parity space projection:
normalized parity space projection:
whose tridimensional projections are shown in Figure
Scatter plot of 200 projections. (a) Rotation
Normalized parity space tridimensional projection of the sensor (gyro #2) noise decomposed by wavelet packet db4 at level
Normalized parity space tridimensional projection of the sensor (gyro
After extensive analyses comparing wavelet families (db2, db4, db6, haar, symlet 3, coiflet 1, and biorthogonal 3.1), decomposition level, buffer dimension, and fault level, it is summarized in Figures
Validation results using the parameters obtained in the calibration over a different data set. Parameters: buffer = 64; WP db6;
gyro  %  Run  Mean  

r1  r2  r3  r4  r5  r6  r7  r8  r9  r10  
#1  PFA  2,6316  2,6316  2,6316  2,6316  2,6316  2,6316  2,6316  2,6316  2,6316  2,6316 

PMD  0,2506  0,2506  0,2506  0,2506  0,2506  0,5013  0,1253  0  0,5013  0,3759 


CD  99,7494  99,7494  99,7494  99,7494  99,7494  99,4987  99,8747  100  99,4987  99,6241 
 
 
#2  PFA  1,8797  1,8797  1,8797  1,8797  1,8797  1,8797  1,8797  1,8797  1,8797  1,8797 

PMD  0,3760  0,5013  0,5013  0,6266  0,6266  0,3760  0,5013  0,6266  0,6266  0,8772 


CD  99,6241  99,4987  99,4987  99,3734  99,3734  99,6241  99,4987  99,3734  99,3734  99,1228 
 
 
#3  PFA  2,8822  2,8822  2,8822  2,8822  2,8822  2,8822  2,8822  2,8822  2,8822  2,8822 

PMD  0,2506  0  0  0  0,5013  0,2506  0  0  0,6266  0,1253 


CD  99,7494  100  100  100  99,4987  99,7494  100  100  99,3734  99,8747 
 
 
#4  PFA  4,01  4,01  4,01  4,01  4,01  4,01  4,01  4,01  4,01  4,01 

PMD  6,3910  6,1404  4,8872  5,3883  5,5138  4,7619  5,8897  4,8872  6,1404  4,3860 


CD  93,6090  93,8596  95,1128  94,6115  94,4862  95,2381  94,1103  95,1128  93,8596  95,6140 

PFA, PMD and CD results for seven different wavelet families applied to gyro
PFA, PMD and CD results for buffers dimensions 64, 128 and 256 applied to gyro
In this paper, it was developed a FDI algorithm based on signal processing techniques applied to IMU with minimal redundancy of fiber optic gyros. This unity was built with lowquality FOGs and a prototype of acquisition system that generates a high quantity of impulsive noise, justifying the use of recursive median filter. The first of median filters (MF1) was applied to remove the highlevel impulses, outliers with huge magnitude arising from data acquisition process. This filter is adjusted to introduce a short delay. The second filter (MF2) was used to reduce the lowlevel impulses arising from parity vector operations and,consequently, to improve the fault detection performance. However, this filter adds a new delay in the detection process, being greater than MF1. The fault detection part of the algorithm is used to guarantee the reliability of the isolation part, avoiding the actions when the fault occurrence is not confirmed by
By following the development presented in Section
For the subband average energy (
Gyro #1:
Gyro #2:
Gyro #3:
Gyro #4:
For the normalized null space projection matrix
Gyro #1:
Gyro #2:
Gyro #3:
Gyro #4:
For the
Gyro #1:
Gyro #2:
Gyro #3:
Gyro #4: