A State-Domain Robust Chi-Square Test Method for GNSS/INS Integrated Navigation

Aiming at abrupt faults in GNSS/INS integrated systems in complex environments, classical fault detection algorithms are mostly developed from the measurement domain. A robust chi-square test method based on the state domain is proposed in this paper. The fault detection statistic is built based on the difference between the prior state estimation and the posterior state estimation in Kalman filtering. To improve the calculation stability, singular value decomposition (SVD) is used to factor the covariance matrix of the difference. The relevant formulas of the proposed method were theoretically derived, and the relationship between the proposed method and the existing innovation chi-square test method was revealed. The proposed method was compared with state-of-the-art chi-square test methods and verified by GNSS/INS integrated navigation experiments using simulation data and real data. The experimental results show that the proposed method (a) directly works in the state domain, (b) does not require the known real system state, (c) has computational efficiency and good robustness, and (d) accurately detects abrupt faults.


Introduction
To take full advantage of each individual navigation system and overcome their shortcomings, integrated navigation systems based on Global Navigation Satellite System (GNSS) and Inertial Navigation System (INS) are widely used in positioning and attitude determination applications [1]. Kalman filtering plays an important role in the integration of GNSS and INS data. A basic assumption for applying standard Kalman filtering is that both the dynamic model and the stochastic information provided to the filter are accurate [2]. If this assumption is not valid or there are many outliers, the Kalman filter may result in poor performance such as abrupt faults in the filtering result [3,4]. Therefore, it is necessary to carry out realtime fault detection to ensure the reliability and precision of the integrated navigation system [5,6].
In the field of integrated navigation fault detection, chisquare test methods are the classical methods and are widely used because they can detect the faults caused by outliers in the measurements and inaccurate dynamic models or random information of the Kalman filter. The methods determine the fault detection threshold by the probability of a false alarm (FA) specified in the integrated navigation system, so they do not require any user interaction. The methods can be principally divided into two groups [7][8][9]: (i) State chi-square test methods: detect faults through the state estimate error. A state chi-square test method, which directly detects faults based on the difference between the true state and the posterior state estimate, was proposed in [10]; however, it requires knowledge of the true state and is thus suitable for simulation-based fault detection, not for real-time fault detection. The state chi-square test method, which is based on a state propagator and detects faults based on the difference between the estimate of the state propagator and the posterior state estimate, is affected by the initial system value error, the system noise, and the modelling error as it works, and the state propagator value will deviate increasingly from the true value because there is no measurement update [11]. The double-state chisquare test method uses two state propagators that work alternately and correct each other, which can avoid the problem of no measurement update, but the fault detection rate and calculation efficiency will decrease over time [12][13][14][15][16][17] (ii) Innovation chi-square test methods: detect faults through the innovation (i.e., the measurement prediction error). The innovation chi-square test method, which indirectly detects faults based on the difference between the true measurement and the predictive measurement, can detect abrupt faults in time with a small amount of calculation [18][19][20][21][22].
In [23], the Federal Kalman filter based on the innovation chi-square test method is used in suborbital vehicles, and the results show that the proposed method can quickly correct the errors caused by GPS abrupt faults within 1 s. However, this method provides a direct assessment of the measurement prediction error but only an indirect assessment of the filtering state estimate error. In fact, accurate and consistent measurement predictions do not necessarily result in accurate and consistent state estimates [24] The goal of the paper is to propose a state-domain robust fault detection method addressing the aforementioned limitations of the state-of-the-art tests, i.e., to design a method that (a) directly works in the state domain, (b) does not require the known real system state, (c) has computational efficiency (i.e., detect faults without state propagators) and good robustness, and (d) accurately detects the abrupt fault.
The remainder of this paper is organized as follows. Section 2 lists the formulas of the dynamic and observation equations of the GNSS/INS system. In Section 3, the innovation chi-square test method is reviewed and analysed. The state-domain robust fault detection method is described in Section 4. The experimental description and analysis are given in Section 5. Finally, the paper is concluded in Section 6.

The Dynamic and Observation Equations of the GNSS/INS System
A loosely coupled GNSS/INS system is adopted. The state vectors are composed of the position and velocity error in an earth-centered and earth-fixed frame (ECEF frame, e frame), the attitude error is described between the computer e frame and the platform e ′ frame, and the gyro and accelerometer drift error is in the body frame (b frame), which can be expressed as [25][26][27] The nonlinear differential error model of a low-cost INS is as follows: where ΔR e and ΔV e are the position and velocity error in the e frame, respectively; _ φ e is the attitude error between the computer e frame and the platform e′ frame; I 3×3 is a 3 × 3 unit matrix; C e e ′ is the rotation matrix between the computer e frame and the platform e ′ frame; C e ′ b is the rotation matrix between the body frame and the platform e ′ frame; Ω e ie is the skew symmetric matrix of the earth rotation rate ω e ie ; and ε b and ∇ b are gyro and accelerometer drift errors in the body frame, respectively.
The dynamic model of a loosely coupled GNSS/INS system can be expressed as follows: where the vector x k represents the state of the system, f k is the system function, and w k is the process noise vector, which is commonly assumed to be zero-mean Gaussian white noise with covariance matrix Q k . Generally, the measurement model can be expressed as where the vector z k represents the measurement,R GPS are the GNSS vector outputs, h k is the measurement function, and v k is the measurement noise vector, which is commonly assumed to be zero-mean Gaussian white noise with covariance matrix R k .

Problem Formulation
Fault detection is a vital and safety-related task for integrated navigation systems. In this section, the Kalman filter for GNSS/INS integrated navigation is formulated and the innovation chi-square test method, which is a popular and widely discussed test used for several decades, is introduced and analysed.

Kalman Filter for GNSS/INS Integrated Navigation.
The methods purposely designed for abrupt fault detection in the integrated navigation system are based on the Gaussian assumption and the statistical hypothesis testing. Therefore, 2 Journal of Sensors let a state variable x with a known prior Gaussian probability density function (PDF) be considered, where the notation N fx ;x ′ , P xx ′ g stands for the normal distribution with the meanx ′ = E½x and the covariance matrix P xx ′ = cov ½x; meanwhile, the time update and measurement update of the Kalman filter for GNSS/INS integrated navigation can be expressed as follows: wherex k ′ is the prior state estimate, P xx,k ′ is the covariance matrix ofx k ′, K k is the gain matrix,x k is the posterior state estimate, P xx,k is the covariance matrix ofx k ,ẑ k ′ is the predictive measurement, P zz,k ′ is the covariance matrix ofẑ k ′ , and 3.2. The Innovation Chi-Square Test Method. In principle, the detection of the abrupt faults can be formulated using statistical hypothesis testing. The null hypothesis H 0 : assuming no fault, i.e., assumingx k and P xx,k , is accurate enough, which is tested against the alternative hypothesis H 1 : assuming there is a fault.
The innovation chi-square test method that detects abrupt faults by predictive measurementẑ k ′ and measurement z k is described as follows: Step 1. Define a required (or allowed) probability of false alert P FA .
Step 2. Compute statistic If the null hypothesis H 0 is valid, then the PDF pðZ k Þ is (approximately) a chi-squared distribution with n z degrees of freedom (DOF), where e k = z k −ẑ k ′ and n z is the dimension of the measurement domain.
Step 3. Compute the corresponding P FA quantile where FðZ k Þ is the cumulative distribution function with respect to pðZ k Þ and the operator inf stands for the infimum. The quantile q Z FA,k is further denoted as the fault detection threshold.
Step 4. Compare the statistic Z k (14) with the threshold q Z FA,k (15). If Z k ≤ q Z FA,k , then it is considered to be no fault. Otherwise, it would be considered to have a fault.
However, this method provides a direct assessment of the measurement prediction error e k but just an indirect assessment of the filtering state estimate error. In fact, accurate and consistent measurement predictions do not necessarily result in accurate and consistent state estimates. The measurement domain test assumes the Gaussian of the measurement prediction PDF and therefore cannot consider the error in computation of the "crosscovariance" matrix P xz,k ′ [24].

The State-Domain Robust Chi-Square Test Method
In this section, the state-domain robust chi-square test method is proposed.

Derivation.
The proposed test is based on a statistical analysis of the difference between prior and posterior state estimates defined as Its statistical properties are summarized in the following theorem.

Theorem 1.
Under valid hypothesis H 0 , the difference d k is a random variable witĥ where n x is the dimension of the state domain.

Journal of Sensors
Proof. The proof of (17): both the prior and posterior mean estimates are unbiased, i.e.,x k ′ =x k = E½x k . The proof of (18): the difference d k (16) can be written as wherex k ′ is the prior state estimate error andx k is the posterior state estimate error.
where the (cross-)second-order moment of the posterior and prior state estimate error is defined as M x ′ x,k = E½x k ′x k T , and Then, by substituting (21)      Step 1. Define a required (or allowed) probability of false alert P FA .
Step 2. Compute statistic If the null hypothesis H 0 is valid, then the PDF pðC k Þ is (approximately) a chi-squared distribution with n x DOF. To improve the calculation stability, singular value decomposition (SVD) is used to factor P dd,k .
Step 3. Compute the corresponding P FA quantile where FðC k Þ is the cumulative distribution function with respect to pðC k Þ.
Step 4. Compare the statistic C k (22) with the threshold q C FA,k (23). If C k ≤ q C FA,k , then it is considered to be no fault. Otherwise, it would be considered to have a fault.

Relationship between the Proposed Method and the
Innovation Chi-Square Test Method. The innovation chisquare test method can be seen as a special case of the proposed method under the consideration of certain simplifications.
With respect to (6) and (8), the difference d k and P dd,k can be written as Assuming n x = n z , K k is invertible, and faults caused by the outliers in the measurements and inaccurate dynamic models or random information of the Kalman filter, the statistic C k (22) can be further treated as In this case, the proposed method is equivalent to the innovation chi-square test method.

Experiment Description
In this section, the proposed method, called the statedomain robust chi-square test method (SRCTM), is compared with the innovation chi-square test method (ICTM) and the state chi-square test method (SCTM) and verified  Journal of Sensors by GNSS/INS integrated navigation experiments using simulation data and real data. To objectively compare the efficiency of the different methods, the probability of false alerts P FA is set as 10 -6 and n x = n z = 6. Meanwhile, the same hardware and software are adopted. The hardware is a PC with a Core i5-10500 CPU and 16 GB RAM. The R2019b version of MATLAB software running on a Windows 10 system is used to design the different algorithms.

GNSS/INS Integrated Navigation Experiment
Using the Simulation Data. The GNSS/INS integrated navigation experiment using simulation data utilized a simulation framework for low-cost integrated navigation systems called NaveGo, which simulates measurements from inertial sensors and a GNSS receiver. It is found that the absolute differences between real and simulated systems are under 23    Journal of Sensors centimeters for 2D position and under 10 centimeters for vertical position [28].
To verify the effectiveness of the SRCTM and compare it with the ICTM and the RCTM, two abrupt faults based on the standard deviation (STD) as 9STDs were added in the dynamic model (3) artificially at 10 s and 100 s, and two abrupt faults based on the STD as 5STDs were added in the posterior state estimate (6) artificially at 300 s and 400 s. Figure 1 shows the true trajectory and the estimated trajectory, which contains the abrupt faults. Figure 2 contains the position error of the GNSS/INS and confidence region based on the STD as ±3STDs, and the error exceeds 3STDs at the time when the abrupt fault is added. Figure 3 shows the statistics of the different methods, and both the ICTM and the SRCTM are very sensitive to abrupt faults in the dynamic model. Only the SRCTM can detect faults in the posterior state estimate (because the SRCTM is based on the difference between the prior and posterior state estimates). In addition, the SRTM can hardly detect faults, and the statistics change only in the initial period, because the state propagator value will deviate increasingly from the true value without measurement update. Note that the SRTM is clearly more time-consuming than the ICTM and the SRCTM because of the extra state propagator. In Figure 4, the position error using the simulation data and the confidence region of the state propagator defined by 3STDs are provided.

GNSS/INS Integrated Navigation Experiment
Using the Real Data. The GNSS/INS integrated navigation experiment using real data was carried out at the University of Nottingham, UK, in November 2013. A GNSS antenna, a GNSS receiver, and a SPAN-LCI IMU were mounted in a van, and data were logged from the receiver's serial ports to a laptop for storage and processing. The vector between the IMU center and GNSS antenna was accurately surveyed using a total station and was considered accurate within 1 cm. A base station was set up on the roof of the Nottingham Geospatial Building to provide DGPS and RTK corrections. The update rate of INS is 200 Hz, and that of the GNSS is 1 Hz. The average baseline length was less than 3 km for the test. When driving on the roads of the University of Nottingham, the test vehicle performed accelerations and braking at an average speed of at least 20 km/h. Figure 5 is a photograph of the van, and Figure 6 is the test trajectory. The high accuracy real-time output results of the SPAN system are used as the reference value, and the double-difference code GNSS position and velocity results are used as the input measurements.
To verify the effectiveness of the SRCTM and compare it with the SCTM and the ICTM, an abrupt fault based on the STD as 5STDs was artificially added in the measurement model (4) at 540 s, and an abrupt fault based on the STD as 5STDs was added in the posterior state estimate (6) artificially at 570 s. In addition, the results contain some outliers due to the vehicle passing over a speed bump or the data was lost. Figure 7 contains the position error of the GNSS/INS using real data and the confidence region given by ±3STDs, and the error exceeds 3STDs at the time when the abrupt fault is added or there are many outliers. Figure 8 shows the statistics of the different methods using the real data, and it can be seen that both the ICTM and the SRCTM are very sensitive to abrupt faults in the measurement model. Only the SRCTM can detect the fault in the posterior state estimate (because the SRCTM is based on the difference between the prior and posterior state estimates). As in Section 5.1, the SRTM can hardly detect the faults, and the statistics change only in the initial period because the state propagator value will deviate increasingly from the true value without measurement update. In Figure 9, the position error using the real data and the confidence region of the state propagator defined by 3STDs are provided.

Conclusion
This paper focuses on detecting abrupt faults in integrated navigation systems. A state-domain robust chi-square test method was proposed. Compared to the state-of-the-art method, the proposed method (a) directly works in the state domain, (b) does not require the known real system state, (c) has computational efficiency and good robustness, and (d) can accurately detect abrupt faults. The performance and properties of the proposed method are illustrated in the theoretical derivation and experimental description.

Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
The authors declare no conflict of interest.