Aircraft Inertial Measurement Unit Fault Identification ...

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Engineering Notes

Downloaded by TECHNISCHE UNIVERSITEIT DELFT on November 3, 2015 | http://arc.aiaa.org | DOI: 10.2514/1.G001247

Aircraft Inertial Measurement Unit Fault Identification with Application to Real Flight Data P. Lu,∗ L. Van Eykeren,† E. van Kampen,‡ C. C. de Visser,§ and Q. P. Chu¶ Delft University of Technology, 2600 GB Delft, The Netherlands DOI: 10.2514/1.G001247

Nomenclature Ax , Ay , Az fAx , fAy , fAz fp , fq , fr fi p, q, r

= = = = =

ua , va , wa

=

uw , vw , ww

=

uBGS , vBGS , wBGS

=

Vt α, β ϕ, θ, ψ

= = =

linear accelerations along the body axis, m∕s2 faults in the accelerometers, m∕s2 faults in the rate gyroscopes, rad∕s faults in the input roll, pitch, and yaw rates along the body axis, rad∕s airspeed velocity components along the body axis, m∕s wind velocity components along the body axis, m∕s groundspeed velocity components along the body axis, m∕s true airspeed, m∕s angle of attack and sideslip angle, rad roll, pitch, and yaw angles along the body axis, rad

I.

Introduction

F

AULT detection and diagnosis (FDD) are important problems in aerospace engineering [1]. For flight control systems, sensor or actuator faults may cause serious problems. Therefore, quick detection and isolation of these faults is highly desirable. During the last few decades, many approaches have been proposed for FDD [2–5]. Some recent advances and trends can be found in work by Zolghadri [6], Goupil [7], and Goupil and Marcos [8]. Presently, the sensor FDD onboard the aircraft is mainly based on hardware redundancy. Despite the hardware-redundant setup, recent airliner accidents have indicated that sensor malfunctions can result in critical

Presented as Paper 2015-0859 at the AIAA Guidance, Navigation, and Control Conference, Kissimmee, FL, 5–9 January 2015; received 7 January 2015; revision received 11 April 2015; accepted for publication 13 April 2015; published online 9 June 2015. Copyright © 2015 by Peng Lu. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-3884/ 15 and $10.00 in correspondence with the CCC. *Ph.D. Student, Control and Simulation Division, P.O. Box 5058; P.Lu-1@ tudelft.nl. † Ph.D. Student, Control and Simulation Division, P.O. Box 5058; L [email protected]. ‡ Assistant Professor, Control and Simulation Division, P.O. Box 5058; e [email protected]. § Assistant Professor, Control and Simulation Division, P.O. Box 5058; c.c [email protected]. ¶ Associate Professor, Control and Simulation Division, P.O. Box 5058; q.p [email protected].

failures [9]. One recent example of an inertial measurement unit (IMU) failure can be found in the accident investigation of a Qantas Airbus A330-303, flight QF72. In this case, one of the air data inertial reference units was providing erroneous data on many parameters to other aircraft systems, which led to a flight upset and 11 occupants seriously injured [9]. Therefore, fault detection and identification of the IMU faults is important to the safety of the aircraft. Aircraft IMU fault identification (FI) has been dealt with by a number of researchers [10–12]. Most research uses the aerodynamic model of the aircraft. However, the calculation of the aerodynamic forces and moments may contain uncertainties. Alwi and Edwards [11] use linear parametric varying models, which may explicitly contain information about the aerodynamic coefficient variations over the flight envelope. They use the same approach to cope with yaw rate sensor faults in the Advanced Fault Diagnosis for Sustainable Flight Guidance and Control [8] benchmark problem [12]. There are also studies using the robust FDD approach [13,14] to detect actuator faults that can cope with the model uncertainties. Alternatively, the kinematic model (KM) of the aircraft can be used instead of the aerodynamic model. It should be noted that few researchers [15–18] use the KM to cope with aircraft sensor fault detection. These studies show that using the KM can reduce the influence of model uncertainties caused by the calculation of the aerodynamic forces and moments. Van Eykeren and Chu [16] used a sliding-mode differentiator [19] to estimate the IMU faults. However, it requires the selection of a Lipschitz constant [16,19]. Lu et al. [18] used a Selective-reinitialization multiple-model adaptive estimation approach to estimate the faults in the IMU. However, the computational load is intensive when dealing with simultaneous faults. Furthermore, [16,18] assumed the absence of disturbances. More recently, Lu et al. [17] used the KM to detect the air data sensor faults of real flight-test data. In [17], the air data measurements are the output of the KM. In the present Note, the IMU measurements are the input, which leads to an input fault identification problem. In this study, an iterated optimal two-stage extended Kalman filter (IOTSEKF), which improves the performance of the optimal twostage extended Kalman filter (OTSEKF) [20–22] when dealing with nonlinear systems, is applied to estimate both the system states and the IMU faults. The simulation data are taken from the Aero-Data Model in a Research Environment (ADMIRE) benchmark model [23]. The computational complexity of the IOTSEKF is compared to an augmented extended Kalman filter (AEKF). The second contribution is that the influence of disturbances, such as turbulence, is considered. The original KM (KM1) is modified by proposing a novel KM (KM2) with different state and measurement vectors. It is shown that the KM2 is less affected by varying wind, such as turbulence. The performance of the IMU sensor FI using the KM1 and KM2 is compared. Simulation results demonstrate the effectiveness of using the KM2, even in the presence of turbulence. The third contribution of this Note lies in the fact that it uses real flight data to validate the proposed approaches for IMU FI. Few researchers [10,24] identify the IMU sensor faults using real flight data. Berdjag et al. [10] and Zolghadri et al. [25] detected oscillatory faults [26] using three inertial aircraft sensors. Their approach is validated on a normalized real flight dataset. Hu and Seiler [24] assessed the false alarm probability of a simple model-based unmanned aerial vehicle (UAV) fault detection system. Freeman et al. [27] used the same UAV for the detection of aileron faults. The present Note uses real flight-test data to validate the performance of the proposed approaches on identifying the faults that are injected into the IMU sensor data. The fault types not only include bias and drift but oscillatory faults as well. The real flight data are taken from a Cessna Citation 2 aircraft. The results show that the proposed

Article in Advance / 1

2

Article in Advance

approaches are able to identify the faults in the IMU sensors, which verifies that they can be applied in practice. The structure of this Note is as follows: Section II presents the novel KM, which includes the influence of the IMU sensor faults. In Sec. III, the IMU FI using KM2 is dealt with by the IOTSEKF. The performance is also compared to that using the KM1 in the absence and presence of turbulence. The performance of the IOTSEKF using the KM1 and KM2 is further validated using real flight-test data in Sec. IV. Finally, the conclusions are presented in Sec. V.


II.

Novel Aircraft Kinematic Model with IMU Sensor Faults

The original KM [9,28], which is called “KM1” in this Note, is used for flight-path reconstruction. Van Eykeren and Chu [15] and Lu et al. [18] used the KM1 to detect and identify aircraft sensor faults. However, the KM1 does not incorporate the influence of turbulence. To cope with that, a new KM, called the “KM2,” is proposed in this section.

/ ENGINEERING NOTES

The process model of the KM2 is given as follows: u_ BGS Axm − fx vBGS rm − fr − wBGS qm − fq − g sin θ (10)

v_BGS Aym − fy − uBGS rm − fr wBGS pm − fp g cos θ sin ψ (11)

w_ BGS Azm − fz uBGS qm − fq − vBGS pm − fp g cos θ cos ϕ (12)

ϕ_ pm − fp qm − fq sin ϕ tan θ rm − fr cos ϕ tan θ (13)

A. KM2 with IMU Sensor Faults

The general model of the aircraft using the KM2 including IMU sensor faults is described as follows: xt _ fxt; um t; fi t; t Gxtwt

θ_ qm − fq cos ϕ − rm − fr sin ϕ

(14)

sin ϕ cos ϕ rm − fr cos θ cos θ

(15)

(1) ψ_ qm − fq

yt hxt; um t; t

(2)

The measurement model of the KM2 is (3)

uGSm T eb uBGS ; vBGS ; wBGS T vuGS

(16)

where x ∈ Rn represents the system states, um ∈ Rb is the measured input, and ym ∈ Rm is the measured output. w and v are assumed to be white Gaussian noise sequences. The function fi represents the input faults. The system equation variables are defined as follows:

vGSm T eb uBGS ; vBGS ; wBGS T vvGS

(17)

wGSm T eb uBGS ; vBGS ; wBGS T vwGS

(18)

ϕm ϕ v ϕ

(19)

(6)

θ m θ vθ

(20)

(7)

ψ m ψ vψ

(21)

where uGSm , vGSm , and wGSm are the measurements for the ground velocity expressed in the Earth reference frame. T eb is the transformation matrix from the body frame to the Earth frame. If the KM1 is used, then the system variables are defined as follows [29]:

ym t yt vt;

x uBGS

vBGS

t ti ;

i 1; 2; : : :

ϕ

wBGS

θ

ψ T

(4)

um Axm Aym Azm pm qm rm Ax Ay Az p q rT w fi (5)

ym uGSm

w wx

vGSm

wy

ϕm

wGSm

wz

wp

θm

wq

ψ m T

wr T

fi fAx

fAy

fAz

fp

fq

fr T

(8)

v vuGS

vvGS

vwGS

vϕ

vθ

vψ T

(9)

where uBGS , vBGS , and wBGS are the groundspeed components in the body reference frame. Ax , Ay , and Az are the linear accelerations; ϕ, θ, and ψ are the Euler angles; and p, q, and r are the rotational rates of the aircraft. It can be noticed that um is the measurement of the IMU and fi are the faults in the IMU sensors. The faults in the accelerometers are represented by fAx, fAy , and fAz ; whereas fp , fq , and fr represent the faults in the rate gyros.

x 0 ua

va

wa

ϕ

ym0 V tm

αm

βm

ϕm

v 0 vV t

vα

vβ

vϕ

θ

θm

vθ

ψ T

ψ m T

vψ T

(22)

(23)

(24)

Article in Advance

where ua , va , and wa are the airspeed components defined in the body-fixed reference frame. V tm , αm , and βm are the measurements of the true airspeed, angle of attack, and angle of sideslip. The advantages of using the KM2 over the KM1 for IMU FI is explained in the following. Let V Bg , V Ba , and V Bw denote the groundspeed, airspeed, and windspeed components expressed in the body frame, respectively. The angular rates and body accelerations are denoted by ω and A, respectively. The gravitational acceleration is g. The derivative of V Bg is as follows: V_ Bg V_ Ba V_ Bw A T be g − ω × V Bg

Table 1 Measurement V tm uGSm , vGSm , wGSm αm , βm ϕm , θm , ψ m Axm , Aym , Azm pm , qm , rm


Update rates and standard deviations of the noises in the sensors Update rate, Hz 100 10 100 100 100 100

Standard deviation Unit 0.01 m∕s 0.01 m∕s 0.01π∕180 rad 0.01π∕180 rad 0.01 m∕s2 0.01π∕180 rad∕s

(25)

Since V Bg V Ba V Bw , the derivative of V Ba is as follows: V_ Ba A T be g − ω × V Ba − V_ Bw ω × V Bw

3

/ ENGINEERING NOTES

Table 2

(26)

Time interval 10 s < t < 30 s

If the wind is constant, i.e., dV Iw 0 dtI

40 s < t < 60 s

then dV Bw 0 dtI

Sensor Axm Aym Azm pm qm rm Axm Aym Azm pm qm rm

Fault scenario

Fault type Fault magnitude Fault unit Bias 1 m∕s2 Drift 0.1 t m∕s3 Oscillatory 2π sin0.5πt m∕s2 Bias 0.5π∕180 rad∕s Oscillatory π sin0.5πt∕180 rad∕s Bias π∕180 rad∕s Oscillatory −π sin0.5πt∕180 m∕s2 Drift −0.1t m∕s3 Bias −2 m∕s2 Bias −0.5π∕180 rad∕s Bias −0.5π∕180 rad∕s Oscillatory −π sin0.5πt∕180 rad∕s

The subscript I denotes the inertial reference frame. Since dV Bw V_ Bw ω × V Bw 0 dtI

The detailed fault scenario in this section, including the fault type and magnitude, is presented in Table 2. The fault scenario is arbitrarily chosen to test the performance of the proposed approach.

Eq. (26) reduces to

B. Iterated Optimal Two-Stage Kalman Filter

V_ Ba

A T be g − ω × V Ba

(27)

The KM1 is based on Eq. (27), whereas the KM2 is based on Eq. (25). Apparently, when the wind is not constant, Eq. (27) is no longer exact. Therefore, the KM1 is influenced by nonconstant wind such as turbulence, whereas the KM2 is not. Remarks: Another advantage of using this the KM2 is that the FI of the IMU sensors and air data sensors (ADSs) is separated. If the KM1 is used, it is assumed that there are no faults in the ADSs. However, this assumption can be removed if the KM2 is used since the ADS measurement information is not required.

III.

IMU Sensor FI Using KM2

This section uses the proposed KM2 to identify the faults in the IMU sensors fi . First, the fault scenario of the IMU sensors is shown. Then, the IOTSEKF, which is an improved version of the OTSEKF [20], is introduced to estimate fi . The performance of the IOTSEKF using the KM2 is compared to that using the KM1 to demonstrate the advantages of using KM2. A. Fault Scenario Using the Simulated Data

The simulated measurement data are taken from the ADMIRE aircraft benchmark model [23]. These simulation data are used in this section to validate the performance of the proposed approaches, whereas the real flight-test data are used in Sec. IV. The noise variances and the update rates of the sensors are given in Table 1. The update rate of the Global Positioning Systems (GPS) receivers was slow a decade ago. However, recently, rates of 10, 20, 50, or even 100 Hz have been used for various purposes [30]. In addition, the GPS receiver is only used to measure the speed in this research. In this section, the update rate of the GPS measurements is 10 Hz. In this Note, all the IMU sensors fail simultaneously during certain a period. Furthermore, different types of faults are considered, including bias faults and drift faults, as well as oscillatory faults.

This section first presents the equations of the IOTSEKF for the estimation of fi . Then, the computational complexity comparison between the IOTSEKF and an AEKF is presented. 1. IOTSEKF

The OTSEKF can estimate the bias of the system in an optimal minimum variance error sense [20]. It is composed of a modified bias-free filter and a bias filter [20]. However, its performance may be degraded when dealing with nonlinear systems [29]. To cope with that, an IOTSEKF is proposed. The state estimate and its covariance matrix using the IOTSEKF are given as follows: x^ kjk x kjk V k fikjk

i

V k Pfkjk V k0 Pxkjk Pxkjk

(28)

(29)

The bias-free filter of the IOTSEKF is as follows: x kjk−1 Φk−1 x k−1jk−1 u k−1

(30)

0 Pxkjk−1 Φk−1 Pxk−1jk−1 Φk−1 Q k−1

(31)

where Φk−1 is calculated as follows:

Φk−1 eFk−1 Δt

∞ X Fnk−1 Δtn n! n

(32)

4

Article in Advance

∂fxt; um t; fi t; t ; ∂x xx^k−1jk−1 ∂hxt; um t; t Hk ∂x xη1

/ ENGINEERING NOTES

fikjk−1 fik−1jk−1

Fk−1

(33)

(41)

i

fikjk fikjk−1 Kfk yk − H k xkjk−1 − Sk fikjk−1

(42)

and Q~ k−1

Z

tk tk−1

Φk−1 Gxk−1jk−1 QGT x k−1jk−1 ΦTk−1 dτ

Q EfwtT wtg;

R EfvtT vtg

i

i

i

i

H Tk R R~ k H k Pxkjk−1

(36)

Kxk Pxkjk−1 H Tk R~ −1 k

u k U k1 − Uk1 fikjk

(46)

i xfi fi 0 0 Q k Qk − Qxf k U k1 − Uk1 Qk − Uk1 Qk

(47)

U k Ak−1 V k−1 Ek−1

(48)

Sk H k U k

(49)

(38)

η2 − η1 η2

and ϵ0 the desired parameter to stop the iteration; if jϵj > jϵ0 j, repeat step 1 and step 2. After each iteration, η1 ≔ η2 . 3) The new time update is as follows: After the iteration, the time update is obtained as follows: x kjk η2

fp (rad/s)

40

50

40

50

30 time (s)

40

50

a) True and estimated fAx , fAy , and fAz using KM2

Fig. 1

(50)

60

(51)

true estimation

10

20

30

40

50

60

10

20

30

40

50

60

10

20

30 time (s)

40

50

60

0 −0.02 0 0.02

60 fr (rad/s)

30

0 20

−1

0 −0.02 0 0.02

60 fq (rad/s)

30

0

10

i

It can be noticed that Eqs. (35–40) are different from those of the OTSEKF [20]. The computational load of the IOTSEKF is more intensive than the OTSEKF due to the iteration of Eqs. (35–38). However, application of this iteration is only used in the first 10 time

0.02

0

20

i

V k Uk − Kxk Sk

true estimation

2

2

fAx (m/s )

fAy (m/s2) 2 fAz (m/s )

i

f f Uk U k Qxf k−1 − U k Qk−1 Pkjk−1

(40)

The bias filter of the IOTSEKF, which is used to estimate the faults, is as follows:

−5 0

where Uk and V k are the two-stage blending matrices that are given by

(39)

Pxkjk I − K xk H k Pxkjk−1 I − Kxk H k T Kxk RK xk T

10

i

(45)

Define

−2 0 5

i

Pfkjk I − Kfk Sk Pfkjk−1

(37)

η2 xkjk−1 K xk zk − hη1 − Hk x kjk−1 − η1

20

(44)

and

2) The measurement update is

10

−1

(35) i

−2 0 2

(43)

H k0 R Sk Pfkjk−1 Sk0 Kfk Pfkjk−1 Sk0 H k Pxkjk−1

η1 x kjk−1

ϵ≔

i

(34)

The iteration part of the bias-free filter is as follows: 1) The Kalman gain calculation is Downloaded by TECHNISCHE UNIVERSITEIT DELFT on November 3, 2015 | http://arc.aiaa.org | DOI: 10.2514/1.G001247

i

Pfkjk−1 Pfk−1jk−1 Qfk−1

0 −0.02 0

b) True and estimated fp , fq , and fr using KM2

IMU sensor FI of simulated aircraft model using the KM2 in the absence of turbulence.

Article in Advance

u (m/s) w

10

vw (m/s)

10

20

30

40

50

60

10

20

30

40

50

60

0 −10 0 10 0

C. IMU Sensor FI Using KM2 and IOTSEKF in the Absence of Turbulence

w

w (m/s)

The dimension of the state vector using the AEKF is (n b). The dimension of the bias-free filter is n, and that of the bias filter is b using the IOTSEKF. Since the computational load of the Kalman filter is approximately related with the cube of the state dimension [22], the most intensive computationally complexity of the AEKF is On b3 , whereas that of the IOTSEKF is On3 b3 . It is straightforward to see that the IOTSEKF is less computational complex than the AEKF. For the case in this Note, n b. The most intensive complexity of the AEKF is O8n3 , whereas that of the IOTSEKF is O2n3 .

0 −10 0 10

−10 0

10

20

30 time (s)

40

50

60

Speed of the turbulence in the north, east, and down directions.

steps [9]. Therefore, the increased computational load is limited. The advantage of the IOTSEKF and the performance comparison of the IOTSEKF and the OTSEKF can be found in work by Lu and van Kampen [29]. 2. Computational Complexity Comparison

The AEKF [31] can also estimate fi by augmenting fi as states. The state vector of the AEKF is

xa uBGS vBGS wBGS ϕ θ ψ fAx fAy fAz fp fq fr T

20

30

40

50

10

20

30

40

50

fr (rad/s)

Az

f

10

20

30 time (s)

40

50

50

60

Ay

f

30

40

50

Az

0 −5 0

10

20

30 time (s)

40

50

c) True and estimated fAx , fAy , and fAz using KM2

Fig. 3

10

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60

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60

true estimation

10

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10

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0 −0.02 0 0.02

60 fr (rad/s)

20

60

0 −0.02 0 0.02

fq (rad/s)

40

0 10

50

0.02 fp (rad/s)

0 30

40

b) True and estimated fp , fq , and fr using KM1

true estimation

20

30

time (s)

2

10

20

0 −0.02 0

60

10

0 −0.02 0 0.02

60

a) True and estimated fAx , fAy , and fAz using KM1

−2 0 5

0 −0.02 0 0.02

60

2

(m/s )

10

0

−2 0 2

true estimation

0.02

p

−5 0 5

−10 0

2 fAx (m/s )

In this subsection, the performance using the KM2 is demonstrated in the presence of turbulence. A Dryden wind model is used to generate the turbulence, for which the speed is shown in Fig. 2. The results using the KM1 and KM2 are shown in Fig. 3. The estimation of fAx , fAy , and fAz using the KM1, shown in Fig. 3a, is corrupted with turbulence, whereas that of fp , fq , and fr using the KM1 (shown in Fig. 3b) is satisfactory. The reason is that, in the presence of turbulence, Eq. (27) is not equal to Eq. (26) and is no longer correct. Consequently, the estimated fAx , fAy , and fAz using Eq. (27) are also biased.

f (rad/s)

0

0

2 (m/s )

D. IMU Sensor FI Using KM2 and IOTSEKF in the Presence of Turbulence

true estimation

5

−5 0 10

2 (m/s )

In this subsection, the performance of using KM2 in the absence of turbulence is tested. The results of using the KM2 are shown in Fig. 1. It can be seen from the figure that the estimation of fi is satisfactory. The fault estimation can follow the dynamics of the faults including the oscillatory faults. The results using the KM1 are similar to those using the KM2 in this case and are omitted here.

fq (rad/s)

2 fAy (m/s )

2 fAx (m/s )

(52)

f


Fig. 2

5

/ ENGINEERING NOTES

0 −0.02 0

time (s)

d) True and estimated fp , fq , and fr using KM2

IMU sensor FI of simulated aircraft model using the KM1 and KM2 in the presence of turbulence.

6

Article in Advance

Table 3 Variable V tm Axm , Aym , Azm αm , βm ϕm , θm ψm pm , qm , rm uGSm , vGSm , wGSm

Specifications of the sensors using the real flight dataa

Standard deviations N/A 2 · 10−2 3.5 · 10−3 8.7 · 10−3 N/A 2 · 10−3 N/A

Update frequency, Hz 100 100 100 100 10 100 1

The estimation of fAx , fAy , and fAz using the KM2 is shown in Fig. 3c. It can be seen that the performance is not influenced by the turbulence. The estimation of fp , fq , and fr using the KM2 is shown in Fig. 3d, which is also comparable to the performance in the absence of turbulence, shown in Fig. 1b. This demonstrates the superiority of using the KM2 over the KM1 for IMU sensor FI.

IV.

IMU Sensor FI with Application to Real Flight Data

In the previous section, the performance of the IOTSEKF using the KM1 and the KM2 is tested and compared using simulated aircraft data. In this section, they are validated by making use of the recorded real flight data of the Cessna Citation II aircraft. It is a fixed-wing business jet aircraft, which is owned by Delft University of Technology and the Dutch Aerospace Laboratory. The real flight data are taken from flight tests, which were designed for the objective of aerodynamic model identification [32]. As stated in work by de Visser [32], sensor measurements contain biases. These biases can influence the FI performance of the IOTSEKF. It should be noted that the IMU FI using the KM1 requires the assumption of constant wind or no wind, which may not be satisfied

in real flight; its performance may be degraded in the presence of changing wind. The IMU FI using the KM2 does not assume this. A. Measurements of the Real Flight Data

In reality, different sensors may have different update rates. Our GPS receiver for the groundspeed measurements was manufactured before 2000, and its update rate was 1 Hz. It should be noted that GPS receivers with higher rates have recently become available [30]. Also note that the GPS receiver is only required to measure the speed. The specific update rates of the onboard sensors are given in Table 3 [32]. Simultaneous faults are injected into the real flight data, which are the same as in Table 2. B. Real-Life Measurement Model

The sensors of the aircraft may contain biases and drifts. If the KM1 is used, the measurements of α and β are required. In the flight test, α and β are measured by multiple vanes mounted on the nose boom of the aircraft. However, α and β measured by the vanes are not the true angle of attack and sideslip [17,28,32]. Since the vanes are 0.02 φ (rad)

u (m/s)

108

a

106 20

30

40

50

60

10

20

30

40

50

60

10

20

30 time (s)

40

50

60

−2

4 2 0

20

30

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60

0.05 0 4.7

10

20

30

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60

4.65 0

10

20

30 time (s)

40

50

60

true estimation

2

2

(m/s )

10

b) Estimation of φ , θ , and ψ

a) Estimation of u a , va , and wa

true estimation

f (rad/s)

0.02

−2 0 5

10

20

30

40

50

60

−0.02 0 0.02 f (rad/s)

10

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30 time (s)

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0

q

0 10

20

30

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−0.02 0 0.02

60 f (rad/s)

2

−5 0 5

0

p

f

Ax

0

2

fAy (m/s )

−0.02 0 0.1

ψ (rad)

wa (m/s)

−4 0 6

0

θ (rad)

10

a

v (m/s)

104 0 0

(m/s )

Sensor Pitot-static probe Q-Flex 3100 accelerometer (IMU) Vanes Sperry VG-14H vertical gyro Gyrosyn compass LITEF μ FORS rate gyro (IMU) TANS III

N/A denotes not available.

0

r

Az

0

f


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Unit m∕s m∕s2 rad rad rad rad∕s m∕s

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fr , shown in Fig. 5d, is satisfactory, which is also comparable to the result using the KM1. However, the estimation of fAx , fAy , and fAz , shown in Fig. 5c, is degraded, even though the estimation follows the dynamics of the faults. It is caused by the low sampling rate of the groundspeed measurements (uGSm , vGSm , and wGSm ).

not located in the center of gravity of the aircraft, the measurement model should incorporate the angular velocity induced flow at the vane location. Besides, the fuselage induced upwash and sidewash also have to be taken into account. The specific measurement model for αm and βm can be found in work by Lu et al. [17]. However, if the KM2 is used, the corresponding measurement model [Eqs. (16–21)] remains, since it does not use the air data information.

E. Comparison and Discussion

The fault estimation performance using the KM1 and the KM2 is compared using the root-mean-square errors (RMSEs) of the fault estimation. The results are shown in Fig. 6. The RMSEs of the estimation of fp , fq , and fr using the two approaches, shown in Fig. 6b, are comparable. This is reasonable, since the measurements used for estimation of these faults using the two approaches are the same. The RMSEs of the estimation of fAx , fAy , and fAz using the KM2, shown in Fig. 6a, are slightly larger than those using the KM1. However, recall that the sampling rate of the air data sensors used by the KM1 is 100 Hz, whereas that of the groundspeed sensors used by the KM2 is only 1 Hz. As can be seen from Fig. 5a, the vertical speed wBGS varies continually. Consequently, the estimation of wBGS will be influenced more by the low update rate. This is the reason why the RMSEs of the estimation of fAz using the KM2 are approximately twice as much as those using the KM1. On the contrary, vBGS does not change significantly, making the RMSE of the fAy using the KM2 comparable to that using KM1. The windspeed can be roughly estimated using the airspeed and the groundspeed in the body axis, which are shown in Figs. 4a and 5a, respectively. The estimate of the windspeed in the body axis is shown in Fig. 7. As can be seen, the windspeed is time varying. However, the variation of the windspeed is not significant. The windspeeds are approximately −16, 10, and 1 m∕s, respectively. That explains why the performance of the IMU FI using the KM1 is still

In this subsection, the KM1 is used to identify fi that are injected to the real flight data. The results are given in Fig. 4. The estimation of the airspeed components (ua , va , and wa ) and the Euler angles (ϕ, θ, and ψ) are given in Figs. 4a and 4b, respectively. As can be seen, the state estimation is influenced by the faults. During 40 s < t < 60 s, there are some oscillations in the estimation of ua , va , and ψ. However, the influence is small compared to the magnitude of the corresponding states. The estimation of fAx , fAy , and fAz is shown in Fig. 4c. All the faults are estimated in the optimal minimum variance sense. The estimation can also follow the dynamics of the faults including the oscillatory faults. The estimation of fp , fq , and fr , shown in Fig. 4d, maintains satisfactory performance. D. IMU Sensor FI of Real Flight Data Using KM2

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satisfactory. If the variation of the windspeeds is significant (e.g., when the aircraft is subjected to turbulence), the fault estimation performance using the KM1 will be degraded. Although the update rate of the air data sensors is 100 times that of the groundspeed sensors, the performance of using the KM1 and the KM2 is comparable. Compared to the results from Sec. III, it highlights that, for real applications, GPS receivers with a higher update rate should be selected to guarantee the performance using the KM2.

V.

Conclusions

This Note mainly addresses three problems. First, the KM2 is proposed for the fault identification of the inertial measurement unit sensor faults. It takes the wind into account, which makes it less sensitive to turbulence compared to the KM1. Second, the FI using the KM2 is tackled by proposing a novel iterated optimal two-stage extended Kalman filter, which improves the performance of the optimal two-stage extended Kalman filter when dealing with nonlinear systems. Finally, the IMU sensor FI performance of the IOTSEKF using the KM1 and the KM2 is further validated by using the recorded real flight-test data. Results demonstrate the effectiveness of the approaches. The proposed approach using the KM1 and the KM2 can be further extended to fault tolerant control systems, which can provide more accurate information to enhance the safety of the aircraft when there are malfunctions in the IMU sensors.

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