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This paper considers the best sensor configuration and fault accommodation problem for inertial navigation systems which use seven inertial sensors such as gyroscopes and accelerometers. We prove that when six inertial sensors are used, the isolation of a double fault cannot be achieved for some combinations of fault magnitudes, whereas when seven inertial sensors are used, the isolation of any double fault can be achieved. There are many configurations which provide the minimum position errors. This paper proposes four configurations which show the best navigation performance and compares their FDI performances. Considering the FDI performance and the complexity of the accommodation rule, we choose one sensor configuration and provide accommodation rules for double faults. A Monte Carlo simulation is performed to show that the accommodation rules work well.

The reliability of any system can be enhanced by fault detection, isolation, and accommodation (FDIA). FDI methods have been studied since the 1960s in various areas of engineering problems. The reviews [

To make them reliable and enhance their navigation accuracy, inertial navigation systems (INS) often use redundant sensors. Numerous studies have been performed on the use of redundant inertial sensors in FDI, and there are many FDI papers on hardware redundancy, such as those based on look-up tables and the squared error (SE) method [

Yang

The present paper proves that when six inertial sensors are used, the isolation of a double fault cannot be achieved for some combinations of fault magnitudes whereas, when seven inertial sensors are used, isolation can be achieved for any double fault. The configuration which shows the best navigation performance is not unique. Actually, there are many configurations which provide minimum position errors. This paper proposes the four best configurations from the navigational viewpoint and provides accommodation rules for double faults for one of them. For the four best sensor configurations, the probability of correct isolation (PCI) is obtained and compared to select the configuration for which the accommodation rules are obtained. Considering the FDI performance and the complexity of the accommodation rule, we choose one sensor configuration among the four suggested configurations and provide accommodation rules for double faults.

This paper is organized as follows. The sensor configuration and null space of the measurement matrix are explained in Section 2. The fact that seven inertial sensors should be used to isolate any double fault is proved, and the four best sensor configurations for navigation performance are given in Section 3. For these four sensor configurations, the PCIs are simulated and compared with each other in Section 3. The accommodation rules for a double fault for seven inertial sensors are given in Section 4. The simulation results and conclusions are given in Sections 5 and 6, respectively.

Consider a typical measurement equation for redundant inertial sensors such as their acceleration or angular rate:

m(t) = [m_{1} m_{2} … m_{n}]^{T} ∈ R^{n} : inertial sensor measurement.

H(t) = [h_{1} h_{2} … h_{n}]^{T}: n×3 measurement matrix of sensor configuration with rank(H) = 3.

x(t) ∈ R^{3} : triad-solution(acceleration or angular rate).

f(t) = [f_{1} f_{2} … f_{n}]^{T} ∈ R^{n} : fault vector.

ε(t) ∼ N(0_{n}, σI_{n}) : a measurement noise vector with normal distribution(white noise), all sensors are assumed to have the same noise characteristics.

N(x, y): Gaussian probability density function with mean x and standard deviation y.

The triad solution x(t) in (

The navigation accuracy of INS depends on the estimation error of the triad solution x(t), as shown in

Consider the measurement ^{n×3} denotes the sensor configuration. When the eigenvalues of H^{T}H are all equivalent to n/3, the sensor configuration provides the minimum estimation error of the triad solution x(t), which gives the best navigation performance.

A parity vector P(t) is obtained from the measurement using a matrix V as follows:

The following Lemma shows the well-known singular value decomposition (SVD) result.

Suppose that n>3. Every matrix H ∈ R^{n×3} with rank 3 can be transformed into the form H = UΛ = U[Σ 0]^{T}= U_{1}Σ

where U and Σ satisfy the following. UU* =U*U=I_{n}, U=[U_{1} U_{2}], U_{1} ∈ R^{n×3}, U_{2} ∈ R^{n×(n-3)}, Σ = diag{σ_{1}, σ_{2}, σ_{3}} with σ_{1} > σ_{2} > σ_{3} > 0. ( )* denotes a complex conjugate transpose.

Measurement _{2}* on the left:

If we temporarily ignore the noise, we can obtain the null space projection of the fault f.
_{2}U_{2}* is the projector into the null space of the measurement matrix H. Thus, we can estimate the fault by using f̂_{null}.

The matrix U_{2}* in (

It is well-known that two faulty sensors can be isolated among six sensors. Gilmore

Consider the measurement _{i} ≠ f_{j} for i ≠ j. Then, the isolation of the double fault cannot always be achieved for some combinations of f_{i} and f_{j}.

Define a unit vector e_{i} for which only the i^{th} component is 1 and the other components are zero. Then we obtain double faults as follows:

The difference between f̂_{jk} and f̂_{lm}
_{j}, f_{k}, f_{l} and f_{m} since U_{2}U_{2}* has a maximum of three independent columns and, thus, there exist some combinations of f_{j}, f_{k}, f_{l} and f_{m} which make (_{i} and f_{j}.

The simulation result for Theorem 1 can be seen in [_{i} and f_{j}, seven sensors should be used.

Consider the measurement _{i} and f_{j}.

Consider the double fault in (_{2} ∈ R^{7×4}. The difference value:
_{j}, f_{k}, f_{l} and f_{m} since U_{2}U_{2}* has a maximum of four independent columns. Thus, a double fault can be isolated for any combination of f_{i} and f_{j}.

Considering the result of Theorem 2, we need to use seven sensors to isolate double faults in any situation.

Lemma 1 gives the condition for the sensor configuration to provide the least estimation error of x in (^{T}H = 7/3 I_{3}. The configurations in ^{T}H = 7/3 I_{3}.

In this section, we suggest an appropriate sensor configuration in the case where seven inertial sensors are used, which takes into consideration simultaneously the navigation performance, FDI performance, and the complexity of the accommodation rule. This section considers the FDI performances for the four sensor configurations described in Section 3.2. Even though these four configurations all show the best navigation performance, their FDI performances are different from each other. For each configuration, we obtain the probability of correct isolation (PCI) with respective to each sensor. The PCI is obtained from 3,000 simulation runs and the PCI value in

The FDI performances of the four sensor configurations do not differ from each other very much. However, we can recognize that configuration 1 shows the worst FDI performance among the four configurations, while configurations 2 and 4 are better than configuration 3. Even though the magnitude of the fault varies, the PCI values of the four configurations show similar trends to those in _{7}C_{2} = 21.

In this section, the results of [

When a single fault satisfies the following inequality
_{i} is the i^{th} column of matrix V in (

When double faults satisfy the following three inequalities:

| f_{j} | < | f_{i} |

where
^{T}H)^{−1}h_{i}, (H^{T}H)^{−1}h_{i} > < v_{i}, v_{j} >, D_{ij} = ‖V_{i}‖^{2}‖V_{j}‖^{2} − <V_{i}, V_{j}>^{2}.

the two faulty sensors should not be excluded.

When double faults satisfy the following three inequalities:

| f_{j} | < | f_{i} |

^{th}sensor should be excluded, but not the j

^{th}sensor.

When double faults satisfy the following three inequalities:

| f_{j} | < | f_{i} |

When double faults satisfy the following three inequalities:

| f_{j} | < | f_{i} |

For categories I through IV above, we consider only half of the first quadrant in two dimensional space. i.e., 0 ≤ θ ≤ π/4.

In Section 3.3, we choose sensor configuration 2, as shown in

In this case, the measurement matrix H is given in (8) and has the following relations:

Configuration 2 in _{i}, i=1, …, 6) : adjacent double faults (f_{i} and f_{i+1}), double faults skipping a sensor (f_{i} and f_{i+2}), and double faults skipping two sensors (f_{i} and f_{i+3}). The other double fault combination takes place between the Z-axis (f_{7}) and one sensor on the cone (f_{i}, i=1, …, 6). For simplicity, we call these faults the 1^{st} and 2^{nd}, 1^{st} and 3^{rd}, 1^{st} and 4^{th}, and 1^{st} and 7^{th} faults, respectively. The regions of Category 0 through Category IV in Section 4.1 are shown in ^{th} sensor should be excluded and “+j” means that the j^{th} sensor should be included.

The accommodation rule is implemented by updating matrix W as in (9)–(12), where W_{-i-j} denotes the identity matrix with the i^{th} and j^{th} diagonal components having zero values, and W_{-i} denotes the identity matrix with the i^{th} diagonal component having zero value.

In this section, Monte Carlo simulations are performed 10,000 times for each double fault combination to confirm that the accommodation rules are correct. Seven identical sensors are used with configuration 2 as shown in ^{T} = I can be obtained by using the SVD method as follows:
_{i}‖_{2} = 0.7559 (i = 1,2, …, 7).

To show the navigation performance, the error covariance of the triad solution x is used. The covariance matrices are defined as follows:

Matrix C_{+i+j} denotes the error covariance of x̂ including the i^{th} and j^{th} sensor outputs and C_{-i-j} the error covariance of x̂ excluding the i^{th} and j^{th} sensors, and so on. It is known that the minimization of the trace of the error covariance matrix provides the best navigation performance. The traces of the error covariance matrices (13)–(15) will be calculated and compared with each other for the four accommodation rules, which results are shown in

Suppose that the first and second sensors have a fault such that f(t) = [f_{1} f_{2} 0 0 0 0]^{T} with f_{1} and f_{2} being constants. Simulations are performed for each point on the linear line of f_{2} = 0.5 f_{1} as shown in

_{1} and f_{2} belong to the region of Category I, the trace of C_{+1+2} is the minimum among the three traces. This means that when faults f_{1} and f_{2} belong to the region of Category I, the two faulty sensors should be used to obtain the minimum estimation error, in other words, the best navigation accuracy. When faults f_{1} and f_{2} belong to the region of Categories II or III, the trace of C_{-1+2} is the minimum, and when they belong to the region of Category IV, the trace of C_{-1-2} is the minimum. In _{1} = 1.1107 is the boundary between Category I and II, and the point at f_{1} = 2.2968 is the boundary between Category III and IV. These boundary points correspond to the crossover points between trace(C_{+1+2}) and trace(C_{-1+2}) and between trace(C_{-1+2}) and trace(C_{-1-2}), respectively, in

For inertial navigation systems which use seven sensors, this paper proves that a double fault can be isolated for any combination of fault magnitudes. This paper suggests the four sensor configurations which provide the best navigation performance when seven sensors are used. The four sensor configurations are as follows: (1) cone configuration, (2) six sensor inputs on the cone surface and one sensor input on the center axis through the cone, (3) two sensors on the x and y axes, respectively, and the other five sensors on the cone surface with the z axis as center axis of the cone, (4) three sensors on the x, y, and z axes, respectively, and the other four sensors on the cone surface with the z axis as the center axis of the cone.

For these four configurations, the PCI is obtained for each sensor in order to compare their FDI performance. The Monte Carlo simulations indicate that configuration 4 shows the best PCI, but which is only slightly better than that of configuration (4). As explained in detail in Sections 3.3 and 4.2, configuration (2) has four sets of accommodation rules, while configuration (4) has 13. Thus, we chose configuration (2) after considering the complexity of the accommodation rule. For sensor configuration (2), four accommodation rules are obtained and a Monte Carlo simulation is performed. The Monte Carlo simulation shows that the suggested accommodation rules are correct and work well.

This Research was supported by the Chung-Ang University Research Grants in 2009.

INS with redundant inertial sensor configuration and FDIA.

Configuration 1 of seven inertial sensors.

Configuration 2 of seven inertial sensors.

Configuration 3 of seven inertial sensors.

Configuration 4 of seven inertial sensors.

Configuration 4 redrawn.

Configuration 2 with seven inertial sensors.

Region of categories I through IV for 1^{st} and 2^{nd} faulty sensors for configuration 2.

Region of categories I through IV for 1^{st} and 3^{rd} faulty sensors for configuration 2.

Region of categories I through IV for 1^{st} and 4^{th} faulty sensors for configuration 2.

Region of categories I through IV for 1^{st} and 7^{th} faulty sensors for configuration 2.

Accommodation rule for 1^{st} and 2^{nd} faulty sensors for configuration 2.

Accommodation rule for 1^{st} and 3^{rd} faulty sensors for configuration 2.

Accommodation rule for 1^{st} and 4^{th} faulty sensors for configuration 2.

Accommodation rule for 1^{st} and 7^{th} faulty sensors for configuration 2.

Accommodation rule for 1^{st} and 2^{nd} faulty sensors in

Trace(C_{+1+2}), trace(C_{-1+2})and trace(C_{-1-2}) with respect to fault magnitude in

Accommodation rule for 1^{st} and 3^{rd} faulty sensors in

Trace(C_{+1+3}), trace(C_{-1+3})and trace(C_{-1-3}) with respect to fault magnitude in

Accommodation rule for 1^{st} and 4^{th} faulty sensors in

Trace(C_{+1+4}), trace(C_{-1+4})and trace(C_{-1-4}) with respect to fault magnitude in

Accommodation rule for 1^{st} and 7^{th} faulty sensors in

Trace(C_{+1+7}), trace(C_{-1+7})and trace(C_{-1-7}) with respect to fault magnitude in

PCI value for single fault (f = 7σ, 3000 simulations, 30 averaged).

0.964 | 0.963 | 0.965 | 0.965 | 0.965 | 0.964 | 0.963 | |

0.969 | 0.968 | 0.968 | 0.968 | 0.968 | 0.967 | 0.970 | |

0.966 | 0.967 | 0.969 | 0.966 | 0.966 | 0.966 | 0.967 | |

Combinations of accommodation rules for Configuration 4 related to

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
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