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Yang, Xilin, Warren, Michael, Arain, Bilal, Upcroft, Ben, Gonzalez, Luis Felipe, &Mejias, Luis
(2013)
A UKF-based estimation strategy for actuator fault detection of UASs. In Proceedings of the 2013 International Conference on Unmanned Aircraft Systems, ICUAS, IEEE Control Society, Atlanta, Georgia, pp. 516-525. This file was downloaded from: http://eprints.qut.edu.au/60681/
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A UKF-based Estimation Strategy for Actuator Fault
Detection of UASs
Xilin Yang, Michael Warren, Bilal Arain, Ben Upcroft,
Felipe Gonzalez and Luis Mejias
Abstract—This paper presents a recursive strategy for online detection of actuator faults on a unmanned aerial system (UAS) subjected to accidental actuator faults. The proposed detection algorithm aims to provide a UAS with the capability of identifying and determining characteristics of actuator faults, offering necessary flight information for the design of fault-tolerant mechanism to compensate for the resultant side-effect when faults occur. The proposed fault detection strategy consists of a bank of unscented Kalman filters (UKFs) with each one detecting a specific type of actuator faults and estimating correspond-ing velocity and attitude information. Performance of the proposed method is evaluated using a typical nonlinear UAS model and it is demonstrated in simulations that our method is able to detect representative faults with a sufficient accuracy and acceptable time delay, and can be applied to the design of fault-tolerant flight control systems of UASs.
I. INTRODUCTION
UASs have been increasingly employed in a number of civilian and military programs due to their low maintenance cost, easy operability and removal of humans from frontline operations [1], [2], [3]. Recently, UASs have become an indispens-able platform to perform a wide range of flight missions (power plant inspection, bush fire monitor-ing and flood loss assessment, etc.) which require a UAS to be equipped with enhanced intelligence and autonomy capabilities. However, development of available autonomy technology is still on the way. This greatly limits application of a UAS to
X. Yang, M. Warren, B. Arain, F. Gonzalez and L. Mejias are with the Australian Research Center for Aerospace Automation (ARCAA) and Queensland University of Technology (QUT), Brisbane, Australia. Email: {xilin.yang, michael.warren, bilal.arain, felipe.gonzalez,
luis.mejias}@qut.edu.au
B. Upcroft is with the School of Electrical Engineering and Computer Science, QUT, Brisbane, Australia. Email: [email protected]
flight operations which require a certain level of intelligent capabilities. Also, to make UASs suit-able for performing future missions, the ability to accommodate accidental faults/failures should be improved to ensure operational reliability during routine flight.
Fault detection of UASs has been addressed in the literature [4], [5]. In most practical scenarios, online detection techniques are preferred as they make it possible to instantly access to fault char-acteristics, thus providing opportunities to analyze faults/failures onboard and accommodate faulty ac-tuators by using either a gain tuning control algo-rithm or control structure reconfiguration strategy. This removes the need to conduct offline analysis for fault diagnosis and handling.
The multiple model adaptive estimation (MMAE) framework has been used in numerous applications [6], [7]. This framework outlines a structure of em-ploying a bank of online filters working in parallel for fault/failure detection. Maybeck [8] employed a group of linear Kalman filters (KFs) and developed a transition scheme between adjacent filters when system dynamics change. Each KF aims to capture an operating regime effectively and estimate system states (angle-of-attack, attitude rates, etc). To cover the whole flight envelope, this technique requires the setup of a series of system models for accurate fault detection. To overcome the requirement of building system models for different flight regimes, Ducard [9], [10], [11] proposed an estimation frame-work which is composed of extended Kalman filters (EKFs). The EKF is supposed to cover a wide range of possible flight envelopes by linearizing nonlinear aircraft dynamics around operational con-ditions. The authors employed a simplified model which exclusively considers angle-of-attack (AoA), sideslip and attitude rates. The validity of EKFs
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is based on the existence of Jacobian matrices of the system model under all possible flight condi-tions. In the considered application, the nonlinear 6 degree-of-freedom (DOF) UAS model makes it time-consuming to calculate the Jacobian matrices. Also, each type of actuator faults is to be mon-itored by an individual EKF. Thus, a group of EKFs are required and this indicates calculation of Jacobian matrices would be conducted in par-allel, which exacerbates difficulties in implemen-tation on our autopilot which has limited memory and computational power. Moreover, in situations where system dynamics change abruptly, Jacobian matrices might not exist or have singular issues [12], and EKF-based MMAE would fail to perform the fault detection task. Therefore, in our case where flight dynamics are subject to abrupt changes when accidental faults occur, UKF-based estimation is preferred and a group of UKFs are designed with each one monitoring a specific type of faults.
This paper proposes an online fault detection algorithm based on the UKF technique in consider-ation of computconsider-ational burden and reliability. In the considered application, reliable and accurate filters are desired such that they can be executed over the entire operational envelopes of a nonlinear 6-DOF UAS. The UKF is based on the unscented transformation (UT) which has the advantage of better capturing the high-order moment caused by the nonlinear transform and removing the need to compute the Jacobian matrices [13]. Also, it uses the same order of calculations as the linearization, yet produces more accurate estimates compared with the EKF [12]. Therefore, in our project which is dedicated to developing a fault-tolerant flight control system with the capability to detect health status of actuators and handle accidental faults, the UKF is preferred. In most fault-detection aeronautical applications which employ the MMAE framework, linear KFs are used and very few papers address the nonlinear cases [10]. Campbell and Brunke [14] describe a recursive procedure to estimate and track parameters of a nonlinear aircraft. The UKF algo-rithm demonstrated more accurate results in simula-tions based on a well-developed F-15 like aircraft in wind frame. Cork and Walker [15] proposed a UKF-based algorithm for detecting inertial sensor faults based on the north-east-down navigation frame. In
our case, due to the fact that the AoA and side-slip angle are available, it is more convenient to proceed with the design of fault-tolerant control in the body frame, and we concentrate on the nonlinear 6-DOF dynamic model described in body frame and develop a feasible fault detection algorithm. This algorithm is designed for a typical UAS model and can be adapted to suit a variety of UASs with limited modifications.
The remainder of this paper is outlined as fol-lows: Section II derives a complete description of the 6-DOF aerodynamic model in consideration of detailed forces and moments. In Section III, we present the procedure of designing the UKF-based fault detection method. Section IV provides the simulation results obtained, and finally brief conclusions are given in Section V.
II. NONLINEARFLIGHTDYNAMICS OF AUAS
The nonlinear aerodynamic model under con-sideration in the body frame is described as follows[16], and we repeat these equations for clar-ifying notations, ˙u = rv − qw − g sin θ + 1 Ma Fx (1) ˙v = pw − ru + g sin φ cos θ + 1 Ma Fy (2) ˙ w = qu − pv + g cos φ cos θ + 1 Ma Fz (3) ˙ p = k1pq + k2qr + k3l + k4n (4) ˙ q = k5pr + k6(r2− p2) + k7m (5) ˙r = k8pq − k1qr + k4l + k9n (6) ˙
φ = p + tan θ(q sin φ + r cos φ) (7) ˙
θ = q cos φ − r sin φ (8)
where (u, v, w) are local velocities in longitudinal, lateral and vertical directions. (φ, θ, ψ) are roll, pitch and yaw angles with (p, q, r) denoting correspond-ing angular rates. Symbol Ma is mass of the UAS and g the gravitational acceleration. The constant coefficients k(·) in Eq. (4)-(6) are
Γ = IxxIzz− Ixz2 k1= (Iyy− Izz)Izz− Ixz2 Γ k2= (Ixx− Iyy+ Izz)Ixz Γ k3= Izz Γ k4= Ixz Γ k5= Izz− Ixx Iyy k6= Ixz Iyy k7= 1 Iyy k8= Ixx(Ixx− Iyy) + Ixz2 Γ k9= Ixx Γ
with Ixx, Izz, Ixz denoting moments and cross-product of inertia. In the considered application, we are concerned with a general UAS model with variable flight speeds as faults might occur during any phase of flight. Thus, the current model is different from the one which assumes constant flight speeds in Ref. [17].
The forces (Fx, Fy, Fz) and moments (l, m, n) result from external aerodynamic and propulsive contributions. The forces (Fx, Fy, Fz) can be de-composed as
Fx = FAx+ FT x (9)
Fy = FAy+ FT y (10)
Fz = FAz + FT z (11)
where (FAx, FAy, FAz) are aerodynamic forces (in body frame) and (FT x, FT y, FT z) are thrust forces. The external forces (Fx, Fy, Fz) can be obtained based on the knowledge of lift force L, sideways force Fy and drag force D.
Fx Fy Fz = cos α 0 − sin α 0 1 0 sin α 0 cos α −D Fy −L + FT x FT y FT z (12) The aerodynamic forces (D, Fy, L) (in wind frame) and moments (l, m, n) are computed in the following forms:
The lift force L is L = ¯qS(CL0+ CLαα + ¯ c 2Va CLα˙α +˙ ¯ c 2Va CLqq + CLδeδe) (13) Here, S is total wing area and ¯c is the chord length. Dynamic pressure is ¯q = 0.5ρV2
a where ρ is air density, Va the aircraft velocity through the surrounding air mass and δe the elevator.
The drag force D is
D = ¯qS(CD0 + CDδaδa+ CDδeδe) (14)
with the aileron deflection denoted by δa.
The side-force Fy is Fy = ¯qS(CYββ + b 2Va CYpp + b 2Va CYrr + CYδaδa) (15) where b is the wing span.
The roll moment l is l = ¯qS(Clββ + b 2Va Clpp + b 2Va Clrr + Clδaδa) (16) The pitch moment m is
m = ¯qS(Cm0+ Cmαα + ¯ c 2Va Cmα˙α +˙ ¯ c 2Va Cmqq + Cmδeδe) (17) The yaw moment n is
n = ¯qS(Cnββ + b 2Va Cnpp + b 2Va Cnrr + Cnδaδa) (18) The aerodynamic coefficients C(·)(·) in Eq. (13)-(18)
can be found in [18].
Remark 1 Due to the fact that only aileron (for lat-eral motion control) and elevator (for longitudinal motion control) actuators are available onboard the UAS under consideration, the UKFs are designed to detect occurrence of faults on these actuators. Practically, since these actuators are installed in pairs, control means are introduced to indicate the resultant aileron δa provided by deflecting left and right actuator surfaces differentially, i.e.,
δa= (δalef t− δ
right
a )/2
Also, the mean elevator control δe is caused by deflecting the left and right actuator parts symmet-rically,
δe = (δelef t+ δrighte )/2
Remark 2 There are no flaps or rudders on our UAS, and corresponding aerodynamic coefficients are neglected when deriving analytic expressions for forces and moments. This applies to quite a few UASs with a similar aerodynamic configuration. Remark 3 Control of yaw motion is not considered in the force and moment equations. The stabilization of roll, pitch, pitch rate and yaw rate will ensure the
˙u = rv − qw − g sin θ + ρV 2 ∞S 2m (CW 0+ CLαα sin α + ¯ c 2Va CLα˙α sin α +˙ ¯ c 2Va CLqq sin α
− CDδa cos α · δa + CWδe δa ) +
FT m (19) ˙v = pw − ru + g sin φ cos θ + ρV 2 ∞S 2m (CYββ + b 2Va CYpp + b 2Va CYrr + CYδa δa ) (20) ˙ w = qu − pv + g cos φ cos θ + ρV 2 ∞S 2m (CZ0 − CLαα cos α − ¯ c 2Va CLα˙α cos α −˙ ¯ c 2Va CLqq cos α
− CDδa sin α · δa + CZδe δe ) (21)
˙ p = k1pq + k2qr + ρ2 ∞S 2 (k3Clβ + k4Cnβ)β + b 2Va (k3Clp+ k4Cnp)p + b 2Va (k3Clr + k4Cnr)r + (k3Clδa + k4Cnδa) δa (22) ˙ q = k5pr + k6(r2− p2) + k7ρV∞2S 2 (Cm0+ Cmαα + ¯ c 2Va CMα˙α +˙ ¯ c 2Va CMqq + CMδe δe ) + k7FTZT P (23) ˙r = k8pq − k1qr + ρV∞2S 2 (k4Clβ + k9Cnβ)β + b 2Va (k4Clp + k9Cnp)p + b 2Va (k4Clr + k9Cnr)r + (k4Clδa + k9Cnδa) δa (24) ˙
φ = p + q sin φ tan θ + r cos φ tan θ (25)
˙
θ = q cos φ − r sin φ (26)
where
CW 0 = −CD0cos α + CL0sin α, CWδe = −CDδecos α + CLδesin α
CZ0= −CD0sin α − CL0cos α, CZδe = −CDδesin α − CLδe cos α
stability of yaw motion as the yaw update equation is only related to these states.
The explicit description of aerodynamic equation of the UAS can be obtained by substituting Eq. (13)-(18) into Eq. (1)-(8), which is given in Eq. (19)-(26). The continuous-time system model is discretized using the Euler integral method, and can be described by
X(k + 1) = f (X(k), u(k), k) + (k) (27) where state vector X refers to 8 state variables
X = [u, v, w, p, q, r, φ, θ]T (28) and actuator inputs are u = [δa, δe]T. Process noise = (1, · · · , 8)T is mutually independent Gaussian
distributions and satisfies
E[(k)] = 0, E[(k)T(i)] = δ(k − i)Q(k) (29) where δ(·) is the Kronecker function.
The measurement equation is
Y (k) = C · X(k) + ξ(k) (30) The constant matrix C = I8×8 and measurement noise with Gaussian distribution ξ = [ξ1, ..., ξ8]T satisfies
III. UKF-BASEDFAULTDETECTION METHOD A. The Unscented Kalman Filtering Technique
Given the system and measurement model, a UKF can be developed to fulfill the fault detec-tion task by following the procedure in [19]. The UKF begins with defining an augmented state vec-tor Xka = [XT
k, Tk, ξkT]T. Then the UT applies to the new state to generate a sigma matrix χak = [(χxk)T, (χk)T, (χξk)T]T. The UKF procedure can be summarized as follows [20]: 1) Initialization: ˆ X0= E[X0], P0= E[(X0− ˆX0)(X0− ˆX0)T] ˆ X0a = [ ˆXk−1a , 0, 0]T, P0a = diagP0, R, Rξ (32)
2) Compute sigma points: χak−1 = ˆ Xk−1a , Xk−1a + γqPa k−1, ˆ Xk−1a − γqPa k−1 (33) 3) Instantiate each point through process model
χxk|k−1 = f (χxk−1, uk−1, χk−1) (34)
4) Compute predicted mean and covariance ˆ Xk−= 2L X i=0 Wi(m)χxi,k|k−1 (35) Pk−1− = 2L X i=0 Wi(c)[χxi,k|k−1− ˆXk−][χxi,k|k−1− ˆXk−]T (36) 5) Compute predicted measurement and
instan-tiate each predicted point
Yk|k−1= h[χxk|k−1, χ ξ k−1] (37) ˆ Yk− = 2L X i=0 Wi(m)Yi,k|k−1 (38)
6) Compute innovation and cross covariance ma-trices PY˜ kY˜k= 2L X i=0
Wi(c)[Yi,k|k−1− ˆYk−][Yi,k|k−1− ˆYk−] T (39) PXkYk= 2L X i=0
Wi(c)[χxi,k|k−1− ˆXk−][Yi,k|k−1− ˆYk−] T
(40)
7) Update state and error covariance Kk = PXkYkP −1 ˜ YkY˜k (41) ˆ Xk = ˆXk−+ Kk(Yk− ˆYk−) (42) Pk = Pk−− KkPY˜kY˜kKkT (43) where γ = √L + λ, λ=composite scaling factor, L=dimension of augmented state, R and Rξ=process and measurement noise, Wi(·) are weights calculated by the UT.
B. UKF-based Fault Detection
In this section, fault detection algorithms for aileron and elevator are derived with one UKF monitoring the health status of one actuator. System model (Eq. (27)) and measurement model (Eq. (30)) can only be used for the state estimate in non-fault scenarios, and an augmented system model is desired with adequate modifications to detect a specific fault. We follow the strategy in Ref. [10] and augment the state vector as
˜
Xi = [XT, δi]T (44) where δi, i = a, e refers to the aileron or elevator. This definition considers the monitored actuator as an additional state variable. Thus, control action from the flight controller to the δi actuator is ne-glected and the status of the δi actuator is estimated through conducting the UKF procedure.
For the sake of simplicity, we only illustrate the scheme to detect the elevator status. The augmented state vector for the UKF can be defined as
˜
Xe = [u, v, w, p, q, r, φ, θ, δe]T (45) and the augmented system update equations are
˜ Xe = fδe[ ˜Xe(k), δa(k), k] + (k) (46) Ye(k) = CeX˜e(k) + ξ(k) (47) where fδe[ ˜Xe(k), δa(k), k] = f [X(k), δa(k), k] δe(k)
The detailed system model is described by Eq. (48)-(56). It is noticed that an additional update equation for δe is added and it is assumed con-stant within the sampling interval. Here, Ce = [I8×8, O8×1] and the only actuator input is δa. An
˙u = rv − qw − g sin θ + ρV 2 ∞S 2m (CW 0+ CLαα sin α + ¯ c 2Va CLα˙α sin α +˙ ¯ c 2Va CLqq sin α − CDδacos α · δa+ CWδe δa ) +
FT m (48) ˙v = pw − ru + g sin φ cos θ + ρV 2 ∞S 2m (CYββ + b 2Va CYpp + b 2Va CYrr + CYδa δa ) (49) ˙ w = qu − pv + g cos φ cos θ + ρV 2 ∞S 2m (CZ0 − CLαα cos α − ¯ c 2Va CLα˙α cos α −˙ ¯ c 2Va CLqq cos α
− CDδasin α · δa + CZδeδe) (50)
˙ p = k1pq + k2qr + ρ2 ∞S 2 (k3Clβ + k4Cnβ)β + b 2Va (k3Clp+ k4Cnp)p + b 2Va (k3Clr + k4Cnr)r + (k3Clδa + k4Cnδa) δa (51) ˙ q = k5pr + k6(r2− p2) + k7ρV∞2S 2 (Cm0+ Cmαα + ¯ c 2Va CMα˙α +˙ ¯ c 2Va CMqq + CMδeδe) + k7FTZT P (52) ˙r = k8pq − k1qr + ρV∞2S 2 (k4Clβ + k9Cnβ)β + b 2Va (k4Clp+ k9Cnp)p + b 2Va (k4Clr + k9Cnr)r + (k4Clδa + k9Cnδa) δa (53) ˙
φ = p + q sin φ tan θ + r cos φ tan θ (54)
˙
θ = q cos φ − r sin φ (55)
˙δe = 0 (56)
Fig. 1. Fault detection structure
augmented system for aileron failure detection can be set up in a similar way.
The fault detection process is conducted by sepa-rately designing UKFs for both aileron and elevator as shown in Fig. 1. Inputs to each EKF are measured
system states together with other available infor-mation (Airspeed, AoA, sideslip, thrust, etc.). The UKF can detect actuator faults without knowledge of health status of any actuator. As is seen in the structure, when detecting aileron faults, the esti-mated elevator signal from UKF2 is considered as a replica of actual elevator command. Thus, UKF1 is able to estimate status of aileron which is included in the augmented state vector of UKF1. Similarly, UKF2 takes estimated aileron from UKF1 as actual aileron and outputs estimate of elevator status. Thus, detection structure can be considered as a black-box which takes measured states as inputs and outputs status of actuators and estimated states. Rudders and flaps can also be contained in a more complicated system model and faults in these actuators can be detected by following the proposed detection scheme.
IV. SIMULATIONRESULTS
In this section, performance of the UKF-based fault detection algorithm is investigated for typical fault scenarios based on parameters of one of AR-CAA UASs described in Ref. [18]. The nonlinear UAS model is built by using the AeroSim simulation blockset [21] which provides a complete set of tools for high-fidelity development of 6-DOF aircraft dy-namic models for flight control and data analysis.
A simplified block diagram of simulation struc-ture for fault detection is shown in Fig. 2. The nonlinear UAS dynamics are constructed by fol-lowing the model establishment process described in Section II. In simulations, the UAS is con-trolled to achieve steady-state flight conditions in consideration of actuator deflection limits (δa ∈ [−15o, 15o], δ
e ∈ [−25o, 25o]). Lateral dynamics are stabilized through designing a proportional-integral controller from sideslip angle to the aileron. Control gains are chosen to be kpa = 0.2279 and kia = 0.0027 after a few trials. Meanwhile, longitudinal dynamics are controlled through ap-plying a proportional-integral-derivative controller from airspeed to the elevator which aims to stabi-lize airspeed and pitch to desired values. Control gains for the elevator are chosen to be kpe = −0.055, kie = −0.01 and kde = −0.1 and airspeed is stabilized to 27.5m/s. To make the investigation more reliable, white noise with standard deviation of 0.3m, 0.3m and 1.2m is added to longitudinal, lateral and vertical velocities to test the performance of the UKF. Also, attitudes and angular rates are contaminated by white noise with standard devia-tions of 0.1o.
Simulations are conducted with the purpose of testing scenarios where actuator faults happen indi-vidually and simultaneously. Sampling time is set to be 0.01s and all simulations are conducted for 100s. Performance of the UKFs when applied to conduct fault detection is shown in Fig. 3 when floating faults occur. Here, floating faults refer to any fault fluctuating with time. It is assumed that floating actuator faults happen during the flight: Aileron faults take place during time intervals [20s, 30s] and [40s, 50s], elevator faults happen between [40s, 50s] and [70s, 80s]. It is seen that transient response occurs during an initial period of time, and it takes
Fig. 2. Simulation structure for fault detection
0 10 20 30 40 50 60 70 80 90 100 −20 −10 0 10 20 30 Ail. (deg) 0 10 20 30 40 50 60 70 80 90 100 −10 0 10 20 Ele. (deg) Time (s) True Estimated Floating Floating Floating Floating
Fig. 3. Actuator fault detection for floating faults
around 5s to capture system dynamics. It is observed that aileron and elevator faults are detected during [20s, 30s] and [70s, 80s] with the maximum estima-tion delay of 0.6s, which is considered acceptable for the controller design to handle these faults. In the worse case when both faults occur at the same time, which is more likely to cause a fatal crash in a short time, the UKF still estimates both faults with acceptable time delay, thus providing timely status information for fault-tolerant control. For this scenario, estimated system states (velocities, angular rates and attitudes) are shown in Fig 4. It is seen that the UKF can consistently estimate local velocities and angular rates with a high accuracy. Roll and pitch are also estimated accurately when floating
ac-tuator faults occur. It is seen that when faults occur, system states deviate significantly from steady-state values. Thus, faults are considered to occur when estimated system states or actuators exceed steady-state values for a period of time. A fault isolation method will be developed which justifies occurrence of a fault when occurrence probability exceeds a certain value for a sufficient period of time.
The detection ability of the UKF is also evaluated for lock-in-place faults as shown in Fig. 5, it is noticed that lock-in-place faults are estimated during time intervals [15s, 25s], [50s, 60s] for aileron and [22s, 29s] and [58s, 63s] for elevator. The maximum detection delay is around 0.7s. System states are also estimated with sufficient accuracy, as shown in Fig. 6. The standard deviations of estimated states for the two scenarios are shown in Table I. It is seen that the UKF can smooth out the noisy mea-surements and estimate system states effectively.
V. CONCLUSION ANDFUTURE WORK This paper presents a UKF-based recursive al-gorithm for aileron and elevator fault detection for a small UAS. A proper augmented system model is set up to capture characteristics of system dy-namics when accidental faults occur. Simulation re-sults demonstrated that the proposed algorithm can monitor the actuator faults with adequate efficiency. Future work includes the design of a fault isolation procedure to justify occurrence of actuator faults and flight testing of our algorithms. Gust effects will also be taken into account to evaluate performance of the proposed detection method.
VI. ACKNOWLEDGMENT
The research presented in this paper forms part of Project ResQu led by the Australian Research Centre for Aerospace Automation (ARCAA). The authors gratefully acknowledge the support of AR-CAA and the project partners, Queensland Univer-sity of Technology (QUT); Commonwealth Scien-tific and Industrial Research Organisation (CSIRO); Queensland State Government Department of Sci-ence, Information Technology, Innovation and the Arts; Boeing and Insitu Pacific.
0 10 20 30 40 50 60 70 80 90 100 22 24 26 28 30 32 u (m/s) measured true UKF 0 10 20 30 40 50 60 70 80 90 100 −2 0 2 v (m/s) 0 10 20 30 40 50 60 70 80 90 100 −5 0 5 w (m/s) 0 10 20 30 40 50 60 70 80 90 100 −100 10 20 p (deg/s) 0 10 20 30 40 50 60 70 80 90 100 −40 −200 20 40 q (deg/s) 0 10 20 30 40 50 60 70 80 90 100 −100 10 20 r (deg/s) 0 10 20 30 40 50 60 70 80 90 100 −20 0 20 (deg) 0 10 20 30 40 50 60 70 80 90 100 −20 0 20 (deg) Time (s)
Fig. 4. Estimate of states of floating faults
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0 10 20 30 40 50 60 70 80 90 100 −20 −10 0 10 20 30 Ail. (deg) 0 10 20 30 40 50 60 70 80 90 100 −30 −20 −10 0 10 20 Ele. (deg) Time (s) True Estimated Lock−in−place Lock−in−place Lock−in−place Lock−in−place
Fig. 5. Actuator fault detection for lock-in-place faults
0 10 20 30 40 50 60 70 80 90 100 20 30 40 u (m/s) measured true UKF 0 10 20 30 40 50 60 70 80 90 100 −2 0 2 v (m/s) 0 10 20 30 40 50 60 70 80 90 100 −5 0 5 w (m/s) 0 10 20 30 40 50 60 70 80 90 100 −10 0 10 20 p (deg/s) 0 10 20 30 40 50 60 70 80 90 100 −40 −200 20 40 q (deg/s) 0 10 20 30 40 50 60 70 80 90 100 −20 0 20 Time (s) r (deg/s) 0 10 20 30 40 50 60 70 80 90 100 −100 −50 0 (deg) 0 10 20 30 40 50 60 70 80 90 100 −200 20 40 (deg)
Fig. 6. Estimate of states of lock-in-place faults
TABLE I
STANDARD DEVIATIONS OF THE ESTIMATED STATES
States Unit Std. Dev. (FL) Std. Dev. (LIP)
u m/s 0.0646 0.1203 v m/s 0.0412 0.0674 w m/s 0.3634 0.5068 p deg/s 0.6892 1.1382 q deg/s 5.0234 2.8900 r deg/s 0.7236 1.1959 φ deg 0.9348 1.0170 θ deg 0.9610 1.5909
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