Optimal Configuration of Redundant Inertial Sensors for Navigation and FDI Performance

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.


Introduction
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 safety-critical 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 squared-error The navigation solution such as the position, velocity, and attitude, is calculated from (t) x . Let us define the estimation error of x(t) as (t) x -x(t) e(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:

Definition 1: Optimal sensor configuration for navigation performance
For redundant inertial sensor systems, the optimal configuration for the navigation performance is defined as the configuration which minimizes the figure of merit J in (4).
holds. From the singular value decomposition, the as the figure of merit for the navigation performance. However, the figure of merit p F and J in (4) give similar results.

Various Optimal Configurations for Navigation Performance
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               Table 6. Configurations which satisfy  Figure 1 shows the various Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedrons. The sensor configurations whose input axes are placed perpendicular to the surface of the Platonic solids satisfy the condition, H T H = (n/3)I. Thus, Platonic solids provide the optimal navigation performance. The tetrahedron corresponds to the 1 st configuration in Table 2, the cube to the configuration in Table 1, the octahedron to the 2nd configuration in Table 2, the dodecahedron to the 1st configuration in Table 4, and the icosahedrons to the 1st and 3rd configurations in Table 6.

Configuration Diagram Measurement Matrix
where 1  and 2  are the angles between the z-axis and the inner and outer cone surfaces, respectively.   , which is the 1 st configuration in Table 6.
where 1  and 2  are the angles between the z-axis and the inner and outer cone surfaces, respectively. , which is the 3rd configuration in Table 6.

FDI Performance due to the Number of Sensors
When a fault is included in the measurement equation (1), it can be described as follows: is the fault vector.
The parity vector p(t) is calculated from the measurement using the matrix V as follows: where the matrix V satisfies: 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 (n-3)/1. As the number of sensors increases, the dimension of v i increases and thus the FDI performance is enhanced. The FDI performance is related to many parameters such as the existence of a false alarm, miss-detection, correct isolation, and wrong isolation. The probability of correct isolation (PCI) can be used as the main index of the FDI performance. Figure 2 shows that as the fault magnitude to noise ratio increases or as the number of sensors increases, the PCI increases. Cone configurations, viz. the first one in Table 3, the second one in Table 4, and the first one in Table 5, are used in the simulation of Figure 2. Remark 3: It is well-known that the navigation performance improves as the number of sensors increases. In other words, the figure of merit for the navigation performance J in (4) decreases as the number of sensors increases. The FDI performance shows a similar trend with respect to the number of sensors. That is, as the number of sensors increases, the PCI increases.

Various Optimal Configurations for Navigation 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 Table 4), in which case the cone angle from the center axis is 54.7356°. Figure 3 shows the PCIs for the cone configurations with cone angles of 80° and 20°. The simulation result shows that the three PCIs are the same. The V matrices in (8) for the above three cases turn out to be the same, while the measurement matrices are different. Lemma 4 states more general cases of the cone configuration. Proof. For the cone configuration, the measurement matrix H can be obtained as follows:

Optimal Sensor Configuration for both Navigation and FDI Performance
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: Table 7 shows the result of Equations (12) or (13) applied to the configurations in Table 3 through Table 6. The first row in Table 7 is the result of Equations (12) or (13) when five sensors are used. The first configuration in Table 3 gives the maximum (minimum) of the inner product (angles) as 0.5393 (57.3640 0 ). Among the three configurations in Table 3, which provide the best navigation performance, the first configuration shows the best FDI performance. Table 7 shows that when 10 sensors are used, configurations 1 and 3 are the best. The reason for this is that configurations 1 and 3 in Table 6 use different sets of sensors from the same icosahedron. Symmetric configurations such as Platonic solids are known to be the best configurations for both the navigation and FDI performance. Table 7 shows that Platonic solids provide the best configuration for both the navigation and FDI performance.

Simulations
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 [8].

Monte Carlo Simulations Using PCI
In this section, we describe the Monte Carlo simulations performed for the configurations in Tables 3 through 6. For each configuration, we assume that a fault occurs and calculate the PCI for the faulty sensor using GLT method [2]. Each PCI is calculated from 3,000 simulation runs and the 3,000 PCIs are averaged to reduce the variation due to noise. The results are given in Tables 8 through 11. For each configuration, the minimum value of the PCI among all of the sensors is underlined. Among the underlined values, the configuration which gives the maximum value is the best one. The results of the best configuration for Tables 8 through 11 are exactly the same as those in Table 7.    [14] In this section, we calculate the figure of merit for the FDI performance for the configurations in Tables 3 through 6. Harrison and Gai [14] suggested a figure of merit for systematically evaluating alternative sensor configurations. To confirm the results of Section 5.1, the figure of merit in [14] is calculated and the results are shown in Table 12.

Simulation Using the Figure of Merit 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: which is the distance measure between the statistics for the parity vector with a bias fault and the parity vector without a fault. Since there are n measurements, there is an n-dimensional vector: Among the various sensor configurations, the configuration which yields the maximum 1 d J is the one which provides the best FDI performance.
The value in the cell of  (16) for each sensor configuration in Table 3 through Table 6. The results in Table 12 are the same as those in Tables 8 through 11.

Conclusions
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 [6] is used to reconfirm the performance of the suggested method, and the FOM results were identical to those of the Monte Carlo simulations.