# Quadrotor Accelerometer and Gyroscope Sensor Fault Diagnosis Using Nonlinear Adaptive Estimation Methods

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3 tween the inertial velocity and body velocity is given by: v E = R EB v B. As in (Freddi et al., 29; Ireland & Anderson, 212; Lantos & Marton, 211), it is assumed that Euler angles measurements are available. For instance, these measurements can be generated by a camera-based motion capture system, a technology commonly employed for in-door UAV flight (Leishman et al., 214; Ireland & Anderson, 212; Guenard, Hamel, & Mahony, 28; Dydek, Annaswamy, & Lavretsky, 213). MEMS sensors, such as accelerometers and gyroscopes, measure forces and moments acting in the body frame. By considering IMU measurements susceptibility to bias faults, the accelerometer and gyroscope sensor measurements are given by: y a = a + β a (t t a )b a + n a (6) y ω = ω + β ω (t t ω )b ω + n ω (7) where y a R 3 and y ω R 3 are the accelerometer and gyroscope measurements, respectively, b a R 3 and b ω R 3 represent sensor bias faults in accelerometer and gyroscope measurements, respectively, the terms n a and n ω represent the measurement noise, and: a = 1 m U f D. (8) represents the nominal acceleration measurement without bias and noise. Additionally, β a ( ) and β ω ( ) are fault time profile functions with unknown fault occurrence times t a and t ω, respectively. In this paper, they are modeled as step functions given by: {, when t < ta β a (t t a ) = 1, when t t a {, when t < tω β ω (t t ω ) = 1, when t t ω It is assumed that the position measurements in the Earth frame are available. Hence, the system model is augmented by the following output equation: y p = p E + d p (9) where d p represents zero mean position measurement noise. Assumption 1. The sensor biases b a and b ω in Eq. (6) and Eq. (7) are assumed to be constant and bounded. Assumption 2. The sensor measurement noises, denoted by n a, n ω and d p in Eq. (6), Eq. (7), and Eq. (9), respectively, are assumed to be bounded zero mean signals. That is: E(n a ) =, E(n ω ) =, E(d p ) =, where E represents the expectation operator. Remark 1. It is worth noting that, in practical applications, after the occurrence of an IMU sensor bias, its magnitude may be time-varying and grow slowly over time. However, the change in the bias is often small over a short time duration. Therefore, the bias may be assumed to be constant on the short time duration under consideration. The objective of this research focuses on the design, analysis, and experimental demonstration of a robust fault detection, isolation, and estimation scheme for sensor bias faults in accelerometer and gyroscope measurements described by Eq. (6) and Eq. (7). 3. FAULT DETECTION AND ISOLATION METHOD This section presents the proposed method for detecting and isolating sensor faults in accelerometer and gyroscope measurements. Substituting the sensor model given by Eq. (6) - Eq. (7) into the systems dynamics Eq. (1) - Eq. (4), we obtain: ṗ E = v E (1) v E = R EB (η)y a + R EB (η)β a b a R EB n a (11) g η = R η (φ, θ)y ω R η (φ, θ)β ω b ω R η (φ, θ)n ω (12) y z 1 x (y q β ω b q d q )(y r β ω b r d r ) ω = z x y (y r β ω b r d r )(y p β ω b p d p ) x τ φ + 1 y τ θ x y 1 z (y p β ω b p d p )(y q β ω b q d q ) z τ ψ (13) where R η (φ, θ) is the rotation matrix relating angular rates to Euler angle rates and is given by: 1 sin φ tan θ cos φ tan θ R η (φ, θ) = cos φ sin φ. sin φ sec θ cos φ sec θ As can be seen from Eq. (1) - Eq. (13), a bias in accelerometer measurements affects only the position and velocity states. On the other hand, gyroscope measurements affect only Euler angles and angular rate states. Based on this observation, the effect of a bias in either sensor measurements can be treated as a virtual actuator fault. In addition, due to the decoupling of the two sensor faults in the quadrotor state equations, it follows naturally to also divide the fault diagnosis tasks of these two types of sensor faults. The proposed fault diagnosis architecture is shown in Figure 1. As can be seen, under normal operating conditions, two FDI estimators monitor the system for detecting and isolating fault occurrences in accelerometer and gyroscope measurements. Once a fault is detected and isolated, the corresponding nonlinear adaptive estimator 3

4 3.2. Accelerometer Fault Diagnostic Estimator The translational dynamics of the quadrotor described by Eq. (1) and Eq. (11) can be rewritten as follows: ẋ = Ax + f(η, y a ) + G a (η)β a b a + D a (η, t) y = Cx + d p (18) Figure 1. Sensor fault diagnosis architecture. is activated for sensor bias estimation Gyroscope Fault Diagnostic Estimator Based on Eq. (12), the fault diagnostic estimator for the gyroscope bias can be designed as follows: ˆη = Λ 1 (ˆη η) + R η (φ, θ)y ω, (14) where ˆη R 3 are the Euler angle estimates, and Λ 1 R 3 3 is a positive-definite diagonal design matrix. Let the Euler angle estimation error be defined as: η η ˆη. (15) Based on Eq. (12), Eq. (14) and Eq. (15), the dynamics of the attitude angle estimation error are given by: η = η ˆη = Λ 1 η R η (φ, θ)β ω b ω R η (φ, θ)n ω. (16) In the absence of a gyroscope fault (i.e. for t < t ω ), the attitude angle estimation error is given by: η(t) = e Λ1(t tω) η() t e Λ1(t τ) R η (φ, θ)n ω dτ = r ω (t) + e ω (t) (17) where r ω (t) e Λ1(t tω) η() converges exponentially to zero, and e ω (t) t e Λ1(t τ) R η (φ, θ)n ω dτ represents an additive zero mean noise term generated by filtering the measurement noise n ω through the following stable linear filter: ė ω = Λ 1 e ω R η (φ, θ)n ω. Therefore, E( η) converges exponentially to zero in the absence of faults. In addition, based on Eq. (16), it can be seen that the residual η is only sensitive to gyroscope sensor bias b ω. Thus, if E( η) is significantly different from zero, it can be concluded that a fault in the gyroscope measurements has occurred. where x = [p T E, vt E ]T, y = p E, [ ] 3 3 I A = 3, G 3 3 a (η) = f(η, y a ) = R EB y a +, D a(η, t) = g [ 3 3 R EB ], [ ] 3 1, R EB n a and C = [I 3, 3 3 ], where I 3 is a 3 3 identity matrix, 3 3 is a 3 3 matrix with all entries zeros, and 3 1 is a 3 1 zero vector. Based on Eq. (18), the following fault diagnostic observer is chosen: ˆx = Aˆx + f(η, y a ) + L 1 (y ŷ) ŷ = C ˆx (19) where ˆx R 6 represents the inertial position and velocity estimation, ŷ R 3 are the estimated position outputs, L 1 is a design matrix chosen such that the matrix Ā1 (A L 1 C) is asymptotically stable. Note that (A, C) is an observable pair. Let us define the position estimation error as: and the state estimation error as: ỹ y ŷ, (2) x x ˆx. (21) By using equations Eq. (18) - Eq. (19), the estimation error dynamics are given by: x = Ā1 x + G a (η)β a b a + D a (η, t) L 1 d p ỹ = C x + d p. (22) In the absence of accelerometer bias (i.e. for t < t a ), the position estimation error is given by: ỹ =CeĀ1(t ta) x()+c t eā1(t τ) (D a (η, t) L 1 d p )dτ +d p = r a (t) + e a (t) + d p, (23) where r a (t) CeĀ1(t ta) x() converges exponentially to zero, and e a (t) C t eā1(t τ) (D a (η, t) L 1 d p )dτ 4

5 represents an additive zero mean noise term generated by filtering n a and d p through the following stable linear filter: ) ė a = Ā1e a + (D a (η, t) L 1 d p. Clearly, E(ỹ) reaches zero exponentially in the absence of the accelerometer bias b a. Furthermore, it can be seen from Eq. (22), the residual ỹ is only sensitive to the bias b a. Therefore, if any component of E(ỹ) deviates significantly from zero, it can be concluded that a fault in the accelerometer sensor measurement has occurred Fault Detection and Isolation Decision Scheme As described in Sections 3.1 and 3.2, the two fault diagnostic estimators are designed such that each of them is only sensitive to one type of sensor faults. Based on this observation, the residuals η and ỹ generated by Eq. (17) and Eq. (23) can also be used as structured residuals for fault isolation. More specifically, the following fault detection and isolation decision scheme is formulated: In the absence of any faults, all components of E( η) and E(ỹ) should be close to zero. If all components of E( η) remain around zero, and at least one component of E(ỹ) is significantly different from zero, then it can be concluded that an accelerometer fault has occurred. If all components of E(ỹ) remain around zero, and at least one component of E( η) is significantly different from zero, then it can be concluded that a gyroscope fault has occurred. If at least one component of E( η) and at least one component of E(ỹ) are both significantly different from zero, then it can be concluded that both a gyroscope and an accelerometer sensor measurement fault has occurred. The above FDI decision scheme is summarized in Table 1, where represents residuals with zero mean, and 1 represents significantly non-zero residuals. Table 1. Fault Isolation Decision Truth Table. No Fault Gyro Bias Accel Bias Accel & Gyro Bias E( η) 1 1 E(ỹ) FAULT ESTIMATION As shown in Figure 1, once a sensor fault is detected and isolated, the corresponding nonlinear adaptive estimator is activated for estimating the unknown fault magnitude in the accelerometer and/or gyroscope measurements. In this section, the design of nonlinear adaptive estimators for sensor bias estimation is described Accelerometer Fault Estimation Based on Eq. (18), the adaptive observer for estimating the accelerometer bias magnitude is chosen as: ˆx = Aˆx + f(η, y a ) + L 2 (y ŷ) + G a (η)ˆb a + Ω ˆba (24) Ω = (A L 2 C)Ω + G a (η) (25) ŷ = C ˆx, (26) where ˆx is the estimated position and velocity vector, ŷ is the estimated position output, ˆb a is the estimated sensor bias, and L 2 is the observer gain matrix. The adaptation in the above adaptive estimator arises due to the unknown bias b a. The adaptive law for updating ˆb a is derived using Lyapunov synthesis approach (Bastin & Gevers, 1988; X. Zhang, 211) and is given by: ˆb a = ΓΩ T C T ỹ, (27) where Γ > is a symmetric and positive-definite learning rate matrix, and ỹ a y a ŷ a is the output estimation error. Let us also define the state estimation error as x x ˆx, and the parameter estimation error as b a ˆb a b a. The stability and performance properties of the above adaptive scheme are described below. Theorem 1. Suppose that an accelerometer sensor bias fault occurs at t a and is detected at some time T a > t a. Then, if there exists constants α 1 α > and T >, such that t > T a α 1 I 1 t+t Ω T C T CΩdτ α I, (28) T t the adaptive fault estimation scheme described by Eq. (24) - Eq. (27) guarantees that: 1. all signals in the adaptive estimator remain bounded; 2. E( x) and E( b a ) converge exponentially to zero. Proof. Based on Eq. (18) and Eq. (24), the dynamics governing the state estimation error dynamics are given by x = Ā2 x G a (η) b a Ω ˆba + D a (η, t) L 2 d p, (29) where Ā2 A L 2 C and b a ˆb a b a is the parameter estimation error. By substituting G a (η) = Ω (A L 2 C)Ω (see Eq. (25)) into Eq. (29), we have x = Ā2 x ( Ω Ā2Ω) b a Ω ˆba + D a (η, t) L 2 d p = Ā2( x + Ω b a ) Ω b a Ω ba + D a (η, t) L 2 d p. (3) Note that, based on Asumption 1, we have ba = ˆba. By defin- 5

6 ing x x + Ω b a, the above equation can be rewritten as: x = Ā2 x + D(η, t) L 2 d p, (31) where Ā2 is asymptotically stable by design (note that the pair (A, C) is observable by design). By noting that the terms D(η, t) and L 2 d p in Eq. (31) are bounded (see Assumption 2), it follows that x is also bounded. By using the adaptive parameter estimation algorithm Eq. (27) and ỹ = C( x Ω b) + d p, we obtain b a = ΓΩ T C T ỹ = ΓΩ T C T CΩ b a + ΓΩ T C T C x + ΓΩ T C T d p. (32) Note the condition given by Eq. (28) provides the required persistent excitation condition for parameter convergence (Ioannou & Sun, 1996). Thus, using this property in conjunction with Theorem 2.2 from (Anderson et al., 1986), it follows that the homogeneous part of Eq. (32) is exponentially stable. In addition, as can be seen from Eq. (25), Ω is also bounded, which along with Assumption 2 implies that all signals in Eq. (32) are bounded. Because x, Ω and b a are bounded, it then follows that x is also bounded. This concludes the proof of the first part of the theorem. By taking the expectation of Eq. (31) and by using E(n a ) = and E(d p ) =, we have: d ( ) ) E( x) = dt Ā2E( x) + E (D a (η, t) L 2 d p = Ā2E( x). (33) Clearly, E( x) converges to zero exponentially because of the stability properties of Ā2. By taking the expectation of Eq. (32), we have d ( ) E( b a ) = ΓΩ T C T CΩE( b a ) + ΓΩ T C T CE( x) dt + ΓΩ T C T E(d p ) (34) = ΓΩ T C T CΩE( b a ). Thus, based on the persistent of excitation condition Eq. (28), it follows from Eq. (34) that E( b a ) converges to zero exponentially. Finally, by applying the above results to the definition of x, the second part of the theorem can be concluded Gyroscope Fault Estimation Based on Eq. (12), after the occurrence of a gyroscope bias fault is detected and isolated, the following adaptive estimator is designed in order to estimate the unknown sensor bias: ˆη = Λ 2 (ˆη η) + R η (φ, θ)y ω R η (φ, θ)ˆb ω (35) ˆb ω = ΓR η (φ, θ) T (ˆη η) (36) where ˆη is the Euler angle estimate, ˆb ω represents the estimation of the sensor bias, Λ and Γ are positive definite design matrices. The adaptive law given by Eq. (35) is derived using Lyapunov synthesis approach (Ioannou & Sun, 1996). In addition, in order to ensure parameter convergence, R η (φ, θ) will need to satisfy the persistence of excitation condition (Ioannou & Sun, 1996), that is: α 1 I 1 t+t R η (φ, θ) T R η (φ, θ)dτ α I (37) T t for some constants α 1 α >, T >, and for all t T ω, where T ω is the gyroscope fault detection time. Let us define the attitude angle estimation error as η η ˆη and the bias estimation error as b ω ˆb ω b ω. The stability and learning performance of the adaptive estimation scheme Eq. (35) - Eq. (36) is given below. Theorem 2. Suppose that a gyroscope sensor bias fault occurs at time t ω and is detected at some time T ω > t ω. Then, if the PE condition given by Eq. (37) is satisfied, the adaptive scheme described by Eq. (35) - Eq. (36) guarantees that E( η) and E( b ω ) converge to zero exponentially. Proof. The state and parameter estimation error dynamics are given by η = Λ 2 η + R η (φ, θ) b ω R η (φ, θ)n ω (38) b ω = ΓR η (φ, θ) T η. (39) Because n ω is bounded (see Assumption 2), it can be easily shown that all signals involved in Eq. (35) - Eq. (36) are bounded by considering R η (φ, θ)n ω as a bounded disturbance term. Additionally, by taking the expectation of Eq. (38) and Eq. (39), and by making use of Assumption 2, we obtain de( η) = Λ 2 E( η) + R η (φ, θ)e( b ω ) dt (4) de( b ω ) = ΓR η (φ, θ) T E( η). dt (41) Defining ɛ 1 E( η) and ɛ 2 E( b ω ), we can rewrite equations Eq. (4) - Eq. (41) into the following compact form: ] [ ] [ ] [ ɛ1 Λ = 2 R η (φ, θ) ɛ1 ɛ 2 ΓR η (φ, θ) T (42) O ɛ 2 where O is a 3 3 zero matrix. By following similar reasoning logics as given in the proof of Theorem and Lemma in (Ioannou & Sun, 1996), it can be shown that (ɛ 1, ɛ 2 ) = (, ) is exponentially stable, hence, concluding the proof. 6

7 5. EXPERIMENTAL RESULTS In this section, real-time experimental results using an indoor quadrotor test environment are described to illustrate the effectiveness of the sensor fault diagnosis algorithm Experimental Setup A block diagram of the experimental system setup is shown in Figure 2. During flight tests, quadrotor position and attitude information is obtained from a Vicon motion capture camera system. Position and Euler angle measurements are collected every 1ms and relayed from a Vicon dedicated PC via TCP/IP connection to a ground station computer. As in (Macdonald, Leishman, Beard, & McLain, 214), position measurements are corrupted with normal noise with standard deviation of.25m. Additionally, position measurements are down sampled to 1Hz, in order to further simulate real-world applications. The fault diagnosis method is evaluated in realtime during autonomous flight of a quadrotor built in-house with off-the-shelf components. The quadrotor is equipped with the Qbrain embedded control module from Quanser Inc. The control module consists of a HiQ acquisition card providing real-time IMU measurements, and a Gumstix Duo Vero microcontroller running the real-time control software. An IEEE connection between the ground station PC and the Gumstix allows for fast and reliable wireless data transmission and on-line parameter tuning. Position and attitude information obtained from the Vicon system along with trajectory commands generated by the ground station are sent to the quadrotor in order to achieve real-time autonomous flight. The control software executes on-board at 5Hz, and accelerometer and gyroscope measurement are sampled at 2Hz. During the experimental stage, the quadrotor is commanded to move in a circular trajectory, while maintaining constant orientation and altitude. As previously shown, the fault diagnosis technique employed in this approach is independent of the structure of the controller. Therefore, for brevity of presentation, the discussion on controller design is purposely omitted in this paper. In order to evaluate the proposed diagnosis method sensor measurements are artificially corrupted by injecting a constant bias into the accelerometer and gyroscope measurements, respectively, while the quadrotor is airborne. Figure 3 shows the fault time profile of the two types of sensor faults under consideration. As can be seen, an accelerometer fault is introduced at time t = 3s until t = 5s. Between time t = 6s and time t = 8s, a gyroscope bias is injected into the sensor measurements. Additionally, in order to evaluate the performance of the FDIE algorithm in the presence of multiple faults, accelerometer and gyroscope faults are simultaneously injected at time t = 1s. Flight data is processed on-line, and real-time sensor fault diagnostic decision is generated by the diagnostic algorithm. In the following sections, the fault Figure 2. Experimental system architecture setup diagnosis results are detailed. Figure 3. Sensor fault time profile 5.2. Case of Accelerometer Bias The case of an accelerometer measurement bias fault is illustrated in this section. At time t = 3s, a constant bias b a = [.15,.2,.75] T m/s 2, is injected into the accelerometer measurements. Figure 4 shows the residuals generated by the two diagnostic estimators described by Eq. (14) and Eq. (19), respectively. In order to enhance the diagnostic decision based on the FDI logic given by Table 1, the two-sided cumulative sum (CUSUM) test is applied to process the diagnostic residuals (Gustafsson, 2). Figure 5 shows the statistic property generated by the CUSUM test. A fixed threshold is chosen for the detection and isolation of sensor faults. As can be seen, shortly after the occurrence of the fault, at least one component of the test statistic corresponding to the residuals generated by the accelerometer diagnostic estimator exceeds the detection threshold, indicating the occurence of a fault in the accelerometr sensor. On the other hand, all components of the test statistic corresponding to the gyroscope bias remain well below the detection threshold. Based on the detection and isolation logic given in Table 1, it can be concluded that a fault has occurred in the accelerometer measurement. In addition, Figure 6 shows the estimation of the bias in the accelerometer for each axis, respectively. As can be seen, 7

8 the estimate of accelerometer converges closely to the actual value. threshold. Thus, it can be concluded that a fault has occurred in the gyroscope measurement. In addition, Figure 9 shows the estimation of the bias in the gyroscope for each axis, respectively. As can be seen, after a short time, the estimate of gyroscope bias is reasonably close to its actual value. Figure 4. Raw diagnostic residuals: accelerometer bias fault Figure 7. Raw diagnostic residuals: gyroscope bias fault Figure 5. Diagnostic residual generated by CUSUM: accelerometer bias fault. Figure 8. Diagnostic residual generated by CUSUM: gyroscope bias fault. Figure 6. Accelerometer bias estimation Case of Gyroscope Bias A gyroscope bias with b ω = [5, 7, 1] T /s is injected into the sensor measurements at time t = 6s. Figure 7 shows the raw diagnostic residuals generated by Eq. (14) and Eq. (19), respectively. The results of the CUSUM test are shown in Figure 8. As can be seen, at least one component of the test statistic corresponding to the gyroscope sensor fault exceeds the detection threshold shortly after fault occurrence. On the other hand, all components of the test statistic corresponding to the accelerometer fault remain well below the detection Figure 9. Gyroscope bias estimation Case of Simultaneous Faults The case of simultaneous accelerometer and gyroscope faults is also considered. Specifically, at time t = 1s, biases b a = [.15,.2,.75] T m/s 2 and b ω = [5, 7, 1] T /s are injected into accelerometer and gyroscope measurements, respectively. Figure 1 shows the statistic property generated by the CUSUM test. As can be seen, shortly after the occur- 8

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