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This paper considers the optimal sensor configuration for inertial navigation systems which have redundant inertial sensors such as gyroscopes and accelerometers. We suggest a method to determine the optimal sensor configuration which considers both the navigation and FDI performance. Monte Carlo simulations are performed to show the performance of the suggested optimal sensor configuration method.
Inertial navigation systems (INS) require at least three accelerometers and three gyroscopes to calculate the navigation information such as the position, velocity and attitude. However, the use of redundant sensors is preferable to ensure their reliability and enhance their navigation accuracy and, thus, the problem of the proper placement of the redundant inertial sensors has been studied since the 1970s. For over four decades reliability has been a subject of interest in various complex systems, such as industrial process systems and power systems, as well as in safetycritical systems such as nuclear power systems and the control of military and space aircraft. Hardware redundancy has been studied from the early stages of the introduction and development of FDI (fault detection and isolation). The various FDI approaches to hardware redundancy include the following methods: the squarederror (SE) [
The optimal configuration problem of redundant inertial sensors was studied in [
In the early 1970s, nine inertial sensors were employed in aircraft, with three sensors in each axis, since it was not known how to optimally configure the sensors. One of the earliest references to redundancy in inertial units uses two sets of orthogonal triads skewed against one another [
Harrison
In this paper, we focus on hardware redundancy in INS and especially on the optimal configuration and suggest a figure of merit for a sensor configuration considering both the navigation and FDI performance. The proposed figure of merit can be used to compare the alternative sensor configurations and, thus, it is possible to obtain the optimal configuration of the redundant sensors considering both the best navigation and FDI performance. Section 2 discusses the condition of the optimal sensor configuration for the navigation performance and gives some sensor configurations providing the best navigation performance, and Section 3 discusses the FDI performance of the sensor configurations with respect to the number of sensors and the angles between them. Section 4 discusses the main results of this paper and suggests a figure of merit for a sensor configuration considering both the navigation and FDI performance. Section 5 shows some simulation results to confirm the validity of the suggested method and in Section 6 we give our conclusions.
Consider an inertial sensor system which uses more than three gyroscopes and three accelerometers. Then, a typical measurement equation for the redundant inertial sensors can be described as follows:
m = [m_{1} m_{2} … m_{n}]^{T} ∈ R
H = [h_{1} … h_{n}]^{T}:
x(t) ∈ R^{3}: the triadsolution (acceleration or angular rate)
ε(t)= [
The triad solution x̂ =[x̂_{x} x̂_{y} x̂_{z}]^{T} for x(t) in (1) can be obtained by the least squares method, as follows:
The navigation solution such as the position, velocity, and attitude, is calculated from x̂(t). Let us define the estimation error of x(t) as e(t) = x(t) − x̂(t). Then, the navigation accuracy of the INS depends on the error covariance:
The figure of merit for the navigation performance can be described as follows:
For redundant inertial sensor systems, the optimal configuration for the navigation performance is defined as the configuration which minimizes the figure of merit
This section shows that there exist many configurations which provide the best navigation performance. The necessary and sufficient condition for the best navigation performance is
There are an infinite number of configurations which satisfy the condition
When a fault is included in the measurement
The parity vector p(t) is calculated from the measurement using the matrix V as follows:
The parity vector p(t) is used for fault detection and isolation(FDI) and the matrix V in (9) is used for various algorithms of FDI. The column vector v_{i} has a dimension of (
The FDI performance is related to many parameters such as the existence of a false alarm, missdetection, correct isolation, and wrong isolation. The probability of correct isolation (PCI) can be used as the main index of the FDI performance.
Generally speaking, the wider the orientation vector corresponding to the spread of the inertial sensors, the better the navigation performance. However, this trend does not apply to the FDI performance. For example, consider the cone configuration with six sensors (the second one in
The row vectors of the matrix V satisfying VH = 0 forms the null space of H. The range space of H is given as follows:
In this chapter, we suggest a method to provide the optimal sensor configuration from the viewpoint of both the navigation and FDI performance. Chapter II shows that there are many optimal configurations to obtain the best navigation performance for each value of n, the number of sensors. Among the optimal configurations providing the best navigation performance, we need to pick the one that gives the best FDI performance.
Considering both the navigation and FDI performance, we suggest a figure of merit for a sensor configuration H as follows:
Among the configurations providing the best navigation performance, the optimal configuration is the one which makes the angle between the nearest two sensors the largest, which suggests a method to provide the best sensor configuration for both the navigation and FDI performance as follows:
The inner product between
In this chapter, we describe some simulations that were performed to show that the method suggested in (13) works well to obtain the optimal sensor configuration for both the navigation and FDI performance. In Section 5.1, we describe Monte Carlo simulations that were performed to calculate the PCI for the FDI performance, while Section 5.2 describes the simulations conducted using the figure of merit suggested in [
In this section, we describe the Monte Carlo simulations performed for the configurations in
In this section, we calculate the figure of merit for the FDI performance for the configurations in
A distance measure (14) is used to compare the detectability (and hence the potential FDI performance) inherent in the different configurations of the sensors:
The figure of merit is defined as in (16):
The value in the cell of
This paper considers the optimal sensor configuration for inertial navigation systems which have redundant inertial sensors. We show that the condition which affords the optimal sensor configuration for the best navigation performance is a necessary and sufficient condition, and enumerate some of the best sensor configurations for navigation performance. We suggest a figure of merit to determine the optimal sensor configuration which considers both the navigation and FDI performance. The main criterion is that among the configurations providing the best navigation performance, the optimal configuration is the one which makes the angle between the nearest two sensors the largest
Monte Carlo simulations are performed to demonstrate the performance of the suggested optimal sensor configuration method. For the FDI performance, the probability of correct isolation is used. To obtain one PCI value in the table, 3,000 Monte Carlo simulation runs are performed and the resulting 3,000 values are averaged. The results of the Monte Carlo simulations were found to be the same as those of the suggested method. The figure of merit (FOM) for the FDI performance suggested in [
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 20100014697).
Platonic solids (Regular Polyhedron).
PCI with respect to the number of sensors and fault size.
PCI for various cone angles from the center axis with n = 6.
Configurations which satisfy


Configurations which satisfy
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2 


Configurations which satisfy
1 


2 


3 


Configurations which satisfy
1 


2 


3 


Configurations which satisfy
1 


2 


3 


4 


Configurations which satisfy
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2 


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4 


Best configuration for both navigation and FDI performance in
Sensor Configurations

1  2  3  4  Best Configuration 

Number of Sensors  
 
5  0.5393(57.3640°)  0.6667(48.1871°)  0.6640(48.3943°) 

1 
6  0.4472(63.4358°)  0.6667(48.1871°)  0.5774(54.7321°)  1  
7  0.7491(41.4875°)  0.6111(52.3309°)  0.7213(43.8381°)  0.5774(54.7321°)  4 
10  0.7454(41.8065°)  0.9342(20.9007°)  0.7454(41.8065°)  0.7823(38.5284°)  1, 3 
PCI for each faulty sensor with
Faulty Sensor

1st  2nd  3rd  4th  5th  Best Configuration 

Sensor Configurations  
 
1  0.2443  0.2442  0.2442  0.2445  best  
2  0.3084  0.2050  0.1517  0.1800  
3  0.1994  0.2842  0.2219  0.2213 
PCI for each faulty sensor with
Faulty Sensor

1st  2nd  3rd  4th  5th  6th  Best Configuration 

Sensor Configurations  
 
1  0.3528  0.3527  0.3524  0.3527  0.3527  best  
2  0.2117  0.2118  0.2123  0.2123  0.2120  
3  0.3492  0.3491  0.3490  0.3485  0.3486 
PCI for each faulty sensor with
Faulty Sensor

1st  2nd  3rd  4 th  5 th  6 th  7 th  Best Configuration 

Sensor Configurations  
 
1  0.3793  0.3798  0.3796  0.3798  0.3800  0.3793  
2  0.3852  0.3845  0.3846  0.3846  0.3848  0.3869  
3  0.3825  0.3819  0.3838  0.3830  0.3812  0.3828  
4  0.3849  0.3850  0.3858  0.3858  0.3855  0.3854  best 
PCI for each faulty sensor with
Faulty Sensor

1st  2nd  3rd  4 th  5 th  6 th  7 th  8 th  9 th  10 th  Best Configuration 

Sensor Configurations  
 
1  0.4101  0.4102  0.4103  0.4106  0.4103  0.4104  0.4103  0.4105  0.4104  best  
2  0.4095  0.4103  0.4096  0.4099  0.4102  0.4107  0.4098  0.4109  0.4100  
3  0.4109  0.4103  0.4105  0.4103  0.4102  0.4106  0.4097  0.4102  0.4107  best  
4  0.4106  0.4104  0.4104  0.4103  0.4106  0.4105  0.4103  0.4110  0.4102 
FDI figures of merit for configurations in
Sensor Configurations  1  2  3  4  Best Configuration 

Number of Sensors  
5  1.5277  1.0000  1.0718 

1 
6  5.0000  2.2498  3.0000 

1 
7  3.1687  4.7606  3.4165  5.3343  4 
10  9.8000  6.2388  9.8000  8.8969  1,3 