A Robust Cubature Kalman Filter with Abnormal Observations Identification Using the Mahalanobis Distance Criterion for Vehicular INS/GNSS Integration

INS/GNSS (inertial navigation system/global navigation satellite system) integration is a promising solution of vehicle navigation for intelligent transportation systems. However, the observation of GNSS inevitably involves uncertainty due to the vulnerability to signal blockage in many urban/suburban areas, leading to the degraded navigation performance for INS/GNSS integration. This paper develops a novel robust CKF with scaling factor by combining the emerging cubature Kalman filter (CKF) with the concept of Mahalanobis distance criterion to address the above problem involved in nonlinear INS/GNSS integration. It establishes a theory of abnormal observations identification using the Mahalanobis distance criterion. Subsequently, a robust factor (scaling factor), which is calculated via the Mahalanobis distance criterion, is introduced into the standard CKF to inflate the observation noise covariance, resulting in a decreased filtering gain in the presence of abnormal observations. The proposed robust CKF can effectively resist the influence of abnormal observations on navigation solution and thus improves the robustness of CKF for vehicular INS/GNSS integration. Simulation and experimental results have demonstrated the effectiveness of the proposed robust CKF for vehicular navigation with INS/GNSS integration.


Introduction
The Internet of Things (IoT) is a burgeoning concept of connected objects operating together to exchange information with each other [1]. As a prototypical IoT application, intelligent transportation has attracted great research interest during the past several decades [2,3]. Currently, global navigation satellite system (GNSS)-based navigation technology plays an important role in intelligent transportation systems to provide location information for vehicles. GNSS can output a seamless positioning solution for vehicles in an open sky environment with good satellite visibility [4,5]. However, it is difficult for GNSS-alone positioning to satisfy the stringent requirements of vehicle positioning due to the vulnerability to signal blockage in many urban/suburban areas [5][6][7]. Moreover, due to their inherent low power, GNSS signals are susceptible to interference, leading to the problems of deliberate spoofing and jamming [6][7][8].
To address the limitations of GNSS, a commonly used strategy is to augment GNSS with a complementary navigation system which is insusceptible to external signal interruptions or interferences [9][10][11]. For this purpose, the inertial navigation system (INS), a self-contained system However, the forgetting factors used in this filter have to be determined empirically for the case of time-variant noises.
Mahalanobis distance is a criterion to identify outliers for multivariate data in statistics [35,37]. Chang established a robust KF by combining the Mahalanobis distance criterion with the linear KF to improve the filtering robustness [35]. However, as stated previously, due to the theoretical limitation, this method can only be used for a linear system, unsuitable for a nonlinear system such as INS/GNSS integration. To the best of our knowledge, there has been very limited research on combining the Mahalanobis distance criterion with the nonlinear CKF for nonlinear INS/GNSS integration. This paper presents a novel robust CKF with scaling factor to improve the robustness of nonlinear vehicular INS/GNSS integration. The proposed method establishes novel Mahalanobis distance criterion theories to improve the robustness of the standard CKF. It employs the concept of the Mahalanobis distance criterion to identify abnormal observations involved in nonlinear INS/GNSS integration. Based on this, a robust factor (scaling factor) is developed via the Mahalanobis distance criterion and is introduced into the standard CKF to inflate the observation noise covariance, thus decreasing the filtering gain to resist the influence of abnormal observations on dynamic state estimate. The proposed robust CKF can accommodate the influence of abnormal observations on navigation solution, resulting in the improved robustness for vehicular INS/GNSS integration. Simulations and practical experiments have been conducted to comprehensively evaluate the performance of the proposed robust CKF for INS/GNSS integrated vehicle navigation.

System Model of Vehicular INS/GNSS Integration
Vehicular INS/GNSS integration allows us to completely exploit the individual advantages of INS and GNSS by using high-precision GNSS position and velocity to correct the INS drift errors, leading to a feasible solution to enhance the accuracy of vehicle navigation.

Dynamic Model
Vehicular INS/GNSS integration uses the standard inertial navigation equations as the dynamic model. Define the inertial frame as i, the body frame as b, and the Earth frame as e. The E-N-U (East-North-Up) geography frame (g) is selected as the navigation frame (n). Then, the standard inertial navigation equations using quaternion parameterization are given by [5,18]: . v n = C(q n b )f b − (2ω n ie + ω n en ) × v n + g n (2) .
In (1), q n b is the attitude quaternion from the b-frame to n-frame for the vehicle, ⊗ denotes the quaternion multiplication, and ω b nb is the body angular rate with respect to the n-frame and is described as: where ω b ib is the body angular rate measured by gyroscopes in the b-frame; ω n ie is the earth rotation rate in the i-frame; ω n en is the angular rate of the n-frame with respect to the e-frame; and C(q n b ) denotes the attitude matrix corresponding to the attitude quaternion q n b . In (2) and (3), v n = [v n E , v n N , v n U ] T represents the vehicular velocity relative to the Earth; g n is the gravity vector; f b denotes the specific force measured by accelerometers in the b-frame, p n = [λ, L, h] T is the vehicle position in longitude, latitude and altitude; and M is represented as: where R M and R N are the radii of curvature in meridian and prime vertical, respectively. The measurement model of the gyro is defined as: where ε b is the gyro bias; and η gv and η gu obey the zero-mean Gaussian white noise processes with spectral densities.
Similarly, the measurement model of the accelerometer is described as: where ∇ b is the accelerometer bias; and η av and η au are the zero-mean Gaussian white noise processes with spectral densities. It should be noted that the attitude quaternion in (1) must obey a normalization constraint, which may be violated due to the inherent averaging operation of CKF. This problem can be addressed by utilizing three-dimensional Generalized Rodrigues Parameters (GRPs) to represent the quaternion error vector and update it via quaternion multiplication. Based on this, the normalization constraint can be maintained [5,38].
Represent the attitude error quaternion as δq = [δq 0 , δu T ] T , where δu is the vector part of the quaternion δq. The GRP corresponding to the error quaternion δq is defined as: where s is a constant within [0,1], and l is a scale factor [5].
Denote the system state x(t) as: The dynamic model for vehicular INS/GNSS integration can be described as: .
where f (·) is the nonlinear function describing the dynamic model in continuous form, and w(t) is the process noise vector. By discretizing (12) via the improved Euler method [39], the discrete-time dynamic model of vehicular INS/GNSS integration is obtained by: where f (·) is a nonlinear function describing the dynamic model in discrete form; and w(k) is the discrete-time process noise vector.

Observation Model
The observation model for vehicular INS/GNSS integration is described as [8]: where H k = 0 6×3 I 6×6 0 6×6 , v k is the observation noise vector, and the observation vector z k is selected as: is the velocity vector output by the GNSS receiver and (λ G , L G , h G ) is the position vector given by the GNSS receiver.

Robust Cubature Kalman Filter Based on the Mahalanobis Distance Criterion
In this section, a new approach of abnormal observation identification is established using the concept of the Mahalanobis distance criterion. Based on this, a novel robust CKF with scaling factor is further developed to resist the influence of abnormal observations on system state estimation.

Standard Cubature Kalman Filter
To show the derivation of the proposed robust CKF more clearly, the concept of the standard CKF is briefly reviewed at first. Consider the nonlinear discrete-time system consisting of (13) and (14) x where x k ∈ R n and z k ∈ R m denote the state and measurement at time k of the dynamic system; w k and v k are the uncorrelated zero-mean Gaussian white noises with covariances E w k w T k = Q and E v k v T k = R; and f (·) is the nonlinear function describing the dynamic model and H k is the observation matrix.
CKF is a deterministic sampling filter based on the third-degree spherical-radical cubature rule for nonlinear systems. The computational process of the standard CKF for the nonlinear system described by (16) can be summarized as: Step 1: Initialization.
Initialize the state estimatex 0 and its error covariance P 0 with: Step 2: Time update.
Assume state estimatex k−1 and its error covariance matrix P k−1 are given. The cubature points can be calculated by: where S k−1 is computed by the Cholesky decomposition of P k−1 ; and ξ i is selected as: where e i denotes the ith column vector of the n × n identity matrix. Each of the cubature points is transformed through the dynamic model to yield a new set of points Subsequently, the predicted state mean and covariance are calculated as: Step 3: Observation update.
Since the observation model is a linear model, the observation update process can be conducted in a similar way as KF.ẑ Step 4: Repeat Steps 2 and 3 for the process of all the samples.
It is obvious that if the system model (16) involves abnormal observations, it will directly decrease the estimation accuracy of the system state from (28), deteriorating the navigation performance for vehicular INS/GNSS integration. Therefore, it is necessary to establish a way to resist the influence of abnormal observations on the navigation solution.

Mahalanobis Distance Criterion for Abnormal Observation Identification
The Mahalanobis distance is a criterion to detect outliers for multivariate data in statistics. For a multi-dimensional vector x = (x 1 , x 2 , · · · , x p ) T with the mean µ = (µ 1 , µ 2 , · · · , µ p ) T and the covariance Σ, its Mahalanobis distance is defined as [16,39] To identify abnormal observations involved in vehicular INS/GNSS integration using the Mahalanobis distance criterion, we first denote the innovation vector as: For a Gaussian system without abnormal observations, z k should obey the zero-mean Gaussian distribution with the covariance: Following the Mahalanobis distance's definition given in (30), we have: It can be obtained from statistical analysis that the square of the Mahalanobis distance of the innovation vector should obey the chi-square distribution with m degrees of freedom [35,39], i.e., According to the hypothesis testing theory, for a given significance level α(0 < α < 1), there has a threshold χ 2 m,α to make the following hold: where P(·) represents the probability of a random event.
Thus, we can establish the following criterion to identify abnormal observation: where the threshold χ 2 m,α can be achieved from the χ 2 distribution table. Upon the identification of abnormal observations, we further develop a novel robust CKF with scaling factor based on the Mahalanobis distance criterion to improve the performance of vehicular INS/GNSS integration by resisting the influence of abnormal observations on system state estimate.

Remark 1.
In (35), if α is chosen as a very small value, the judging index γ k will be smaller than χ 2 m,α . This means the system works under the normal condition, i.e., the nonlinear system described by (16) does not involve any abnormal observation. However, if z k and Pẑ k/k−1 do not meet the condition given by (35), it can be concluded with high probability (1 − α) that there are violations to the "a priori" assumption, e.g., there may be an unknown abnormal observation involved in the observation model.

Determination of the Robust Factor Based on the Mahalanobis Distance Criterion
In this paper, a novel robust CKF with scaling factor is developed to address the influence of abnormal observations on system state estimate. If the judging index γ k is not greater than χ 2 m,α , the standard CKF will be carried out, otherwise a scaling factor κ k , which is also called the robust factor, will be introduced to inflate the observation noise covariance R. Accordingly, the predicted innovation covariance for the proposed robust CKF should be changed to: Substituting (37) into (33) and making (35) hold, the following equation can be obtained: It can be seen from (38) that the determination of the robust factor κ k is a problem of solving the nonlinear equation, and it can be solved iteratively by using Newton's method [16,39]: where i represents the i-th iteration. Substituting (38) into (39), we have: According to the following derivative formula for the inverse matrix [39]: it can be easily achieved that: where A in (41) is an invertible matrix as a function of t.
The above iterative process to determine the robust factor κ k is initialized as 1, e.g., κ k (0) = 1, and it will be terminated when the judging index satisfies γ * k (i) ≤ χ 2 m,α , or the iteration index is greater than the predetermined threshold.
For the proposed robust CKF, when γ * k > χ 2 m,α , because of κ k , R will be "enlarged". Accordingly, the actual observation will be less weighted, while the predicted state will be more weighted in the observation update procedure of CKF.

Remark 3.
In practical applications, a simplified way to determine the robust factor can also be considered to improve the computational efficiency. An inflated observation noise covariance results in an inflated innovation vector covariance, thus the robust factor can be directly used to adjust the innovation vector covariance for the filtering robustness, i.e.,: In this case, the robust factor κ k can be solved analytically rather than iteratively as in (42). It is directly calculated as: Remark 4. The judging index γ * k and the robust factor κ k in the proposed robust CKF are computed using the information at the current epoch only. For many other methods, the robust factors are determined via both current and historical information. Thus, the proposed method has a smaller computational load and is more sensitive to the abnormal observation at the present epoch, leading to the effectiveness in responding to dynamically changing abnormal observations.

Proposed Robust Cubature Kalman Filter
For the vehicular INS/GNSS integrated system, if the judging index γ k is not greater than χ 2 m,α the standard CKF will be carried out, otherwise the robust factor κ k will be introduced to inflate the observation noise covariance R. The procedure of the proposed robust CKF is shown in Figure 1, which can be summarized as: Sensors 2019, 19, x 9 of 20 • Change the innovation covariance as (37).

•
Execute the standard CKF procedure (26)-(29) to update the system state estimate.
Step 4: Repeat Steps 2 and 3 for the process of all the samples.

Performance Evaluation and Analysis
Simulations and practical experiments were conducted to comprehensively evaluate the proposed Mahalanobis Distance Criterion-based robust CKF (MDC-RCKF) for vehicular INS/GNSS integration. Comparison analysis of the proposed MDC-RUKF with the standard CKF, and H-infinity strategy based robust CKF (HI-RCKF) in [29] was also conducted to demonstrate the performance improvement.

Simulations and Analysis
Monte Carlo simulations were carried out for the performance evaluation of the proposed MDC-RCKF in vehicular INS/GNSS integration with abnormal observations. The two typical cases of abnormal observations, i.e., outliers in observation and contaminated Gaussian noise distribution, are considered in this section. Step 1: Initialization.
(i) Compute the innovation vector z k and its covariance Pẑ k/k−1 ; (ii) Set κ k (0) = 1 and obtain the judging index γ * k ; • Perform the standard CKF procedure (26)-(29) to obtain the system state estimate.Else, • Determine the robust factor κ k by iteratively solving (42) until the judging index satisfies Change the innovation covariance as (37).

•
Execute the standard CKF procedure (26)-(29) to update the system state estimate.
Step 4: Repeat Steps 2 and 3 for the process of all the samples.

Performance Evaluation and Analysis
Simulations and practical experiments were conducted to comprehensively evaluate the proposed Mahalanobis Distance Criterion-based robust CKF (MDC-RCKF) for vehicular INS/GNSS integration.
Comparison analysis of the proposed MDC-RUKF with the standard CKF, and H-infinity strategy based robust CKF (HI-RCKF) in [29] was also conducted to demonstrate the performance improvement.

Simulations and Analysis
Monte Carlo simulations were carried out for the performance evaluation of the proposed MDC-RCKF in vehicular INS/GNSS integration with abnormal observations. The two typical cases of abnormal observations, i.e., outliers in observation and contaminated Gaussian noise distribution, are considered in this section.
The simulated movement trajectory for the vehicle is shown in Figure 2, which includes various maneuvers such as uniform linear motion, accelerated linear motion, turning and so on. The simulation parameters are listed in Table 1. The simulation time lasts 1000 s and the filtering period is 1 s. To identify abnormal observations, χ 2 m,α for the proposed MDC-RCKF was chosen as 12.592, which was derived from the χ 2 distribution with the confidence level of 95% (α = 0.05) and 6 degrees of freedom (m = 6). Monte Carlo simulations were conducted at the same conditions to compare the performance of the proposed MDC-RCKF with the standard CKF and HI-RCKF. The root mean squared errors (RMSE) is adopted to evaluate the navigation performance of the proposed MDC-RCKF for INS/GNSS integration. It is defined as: where M is the Monte Carlo runs, which is set as 100 for this simulation. The simulated movement trajectory for the vehicle is shown in Figure 2, which includes various maneuvers such as uniform linear motion, accelerated linear motion, turning and so on. The simulation parameters are listed in Table 1

1( )
where M is the Monte Carlo runs, which is set as 100 for this simulation.   Figure 2. The movement trajectory of the vehicle.

Navigation Accuracy Evaluation
In order to evaluate the performance of the proposed MDC-RCKF in terms of abnormal observations, the following two typical cases are considered in the simulation analysis: where ε is the ratio of the perturbing distribution, which is selected as 0 < ε ≤ 0.5. The ρ perturbing obeys the Gaussian distribution with a larger standard deviation. In this section, the standard deviation of the perturbing distribution is assumed to be 5 times larger than that of the nominal distribution, and ε is set to 0.2.  However, the navigation accuracy achieved by HI-RCKF is slightly lower that of the standard CKF and proposed MDC-RCKF. This is because HI-RCKF has no identification process for abnormal observations, i.e., when the observation is accurate, HI-RCKF can only obtain the suboptimal results for the navigation solution.       Further, for the time points of 200 s, 400 s, 600 s, 800 s and 1000 s with outliers in observation, the navigation accuracy of the standard CKF severely declines due to the influence of abnormal observations. HI-RCKF can weaken the outliers in observation to some extent by using the H-infinity strategy. The achieved mean RMSEs in both horizontal velocity and position are 23.4% and 20% smaller than those of the standard CKF, respectively. However, this method still has pronounced errors in the navigation solution. In contrast, the proposed MDC-RCKF achieves the highest navigation accuracy for these time points by adjusting the robust factor calculated from Mahalanobis distance criterion. Its mean RMSEs in both horizontal velocity and position are at least 13% and 23.7% smaller than those of HI-RCKF, respectively.
The intuitive comparison of the standard CKF, HI-RCKF and proposed MDC-RCKF for the mean RMSEs of both horizontal velocity and position are described in Figures 5 and 6. The relevant results also illustrate that the proposed MDC-RCKF has better robustness to inhibit the outliers in observation comparing to the other filters. Thus, the proposed method can obtain a superior performance for vehicular INS/GNSS integration. However, the navigation accuracy achieved by HI-RCKF is slightly lower that of the standard CKF and proposed MDC-RCKF. This is because HI-RCKF has no identification process for abnormal observations, i.e., when the observation is accurate, HI-RCKF can only obtain the suboptimal results for the navigation solution.
Further, for the time points of 200 s, 400 s, 600 s, 800 s and 1000 s with outliers in observation, the navigation accuracy of the standard CKF severely declines due to the influence of abnormal observations. HI-RCKF can weaken the outliers in observation to some extent by using the H-infinity strategy. The achieved mean RMSEs in both horizontal velocity and position are 23.4% and 20% smaller than those of the standard CKF, respectively. However, this method still has pronounced errors in the navigation solution. In contrast, the proposed MDC-RCKF achieves the highest navigation accuracy for these time points by adjusting the robust factor calculated from Mahalanobis distance criterion. Its mean RMSEs in both horizontal velocity and position are at least 13% and 23.7% smaller than those of HI-RCKF, respectively.
The intuitive comparison of the standard CKF, HI-RCKF and proposed MDC-RCKF for the mean RMSEs of both horizontal velocity and position are described in Figures 5 and 6. The relevant results also illustrate that the proposed MDC-RCKF has better robustness to inhibit the outliers in observation comparing to the other filters. Thus, the proposed method can obtain a superior performance for vehicular INS/GNSS integration.

Computational Performance Evaluation
Simulation trials were also conducted to investigate the computational performance of the proposed MDC-RCKF for the above two typical cases of abnormal observations. The Monte Carlo simulation trials were conducted with Matlab programs on an Intel(R) Core(TM) i5-9400F 2.9 GHz PC with 8 GB memory. The average computational times (i.e., the filtering times for each Monte Carlo run) of the standard CKF, HI-RCKF and proposed MDC-RCKF are shown in Table 2. In order to eliminate the effect resulted from the difference of computers, the relative efficiencies of HI-RCKF and MDC-RCKF in comparison with the standard CKF are also calculated and listed in Table 2. It

Computational Performance Evaluation
Simulation trials were also conducted to investigate the computational performance of the proposed MDC-RCKF for the above two typical cases of abnormal observations. The Monte Carlo simulation trials were conducted with Matlab programs on an Intel(R) Core(TM) i5-9400F 2.9 GHz PC with 8 GB memory. The average computational times (i.e., the filtering times for each Monte Carlo run) of the standard CKF, HI-RCKF and proposed MDC-RCKF are shown in Table 2. In order to eliminate the effect resulted from the difference of computers, the relative efficiencies of HI-RCKF and MDC-RCKF in comparison with the standard CKF are also calculated and listed in Table 2. It can be seen from Table 2 that the standard CKF has the smallest computational time among the

Computational Performance Evaluation
Simulation trials were also conducted to investigate the computational performance of the proposed MDC-RCKF for the above two typical cases of abnormal observations. The Monte Carlo simulation trials were conducted with Matlab programs on an Intel(R) Core(TM) i5-9400F 2.9 GHz PC with 8 GB memory. The average computational times (i.e., the filtering times for each Monte Carlo run) of the standard CKF, HI-RCKF and proposed MDC-RCKF are shown in Table 2. In order to eliminate the effect resulted from the difference of computers, the relative efficiencies of HI-RCKF and MDC-RCKF in comparison with the standard CKF are also calculated and listed in Table 2. It can be seen from Table 2 that the standard CKF has the smallest computational time among the above three filtering methods. The computational time of HI-RCKF is at least 198.28% larger than that of the standard CKF. This is because the use of the H-infinity strategy to resist the influence of abnormal observations involves a large computational load at each iteration during the HI-RCKF filtering procedure. In contrast, the proposed MDC-RCKF shows a superior computational performance than HI-RCKF. For the case with outliers in observation, due to the use of the identification process of abnormal observations, the computational time of MDC-RCKF is 55.83% smaller than that of HI-RCKF. For the case with contaminated Gaussian noise distribution, the computational time of MDC-RCKF is also 22.14% smaller than that of HI-RCKF, even though the determination of the robust factor is involved in each iteration during the MDC-RCKF filtering procedure. Moreover, the MDC-RCKF computational time is only slightly greater than the standard CKF one due to the determination procedure of the robust factor. The above analysis demonstrates that the proposed MDC-RCKF has a faster computation speed than HI-RCKF, which is sufficient to achieve the real-time performance for vehicular INS/GNSS integration. The above simulations and analysis for navigation accuracy and computational performance verify that the proposed MDC-RCKF can effectively identify abnormal observations and further inhibit their influence on the navigation solution, leading to smaller navigation error than the standard CKF and HI-RCKF for vehicular INS/GNSS integration. In addition, the proposed MDC-RCKF also has a better computational performance than HI-RCKF and is sufficient to achieve the real-time performance for vehicular INS/GNSS integration.

Practical Experiment and Analysis
To further evaluate the performance of the proposed MDC-RCKF, an experiment of terrestrial vehicular navigation was carried out and the comparison with standard CKF and HI-RCKF was also conducted.

Experiment Setup
The experimental system of terrestrial vehicular navigation is made up of an ampere-voltage meter, a battery, a controller and a self-made INS/GNSS integration navigation system. This navigation system consists of a NV-IMU300 inertial measurement unit, and a JAVAD Lexon-GGD112T GPS receiver.

Experimental Results and Analysis
Practical experimental data was processed under the same conditions by the standard CKF, HI-RCKF and proposed MDC-RCKF, respectively. 2 , m α χ in the proposed MDC-RCKF was also chosen as 12.592 for identification of abnormal observations. The other initial parameters were set to the same values as those in the simulation analysis. Figure 12; Figure 13 give the position errors in longitude and latitude of the vehicle by the standard CKF, HI-RCKF and proposed MDC-RCKF. During the test process, the GNSS signals were shielded by the tall buildings or tree canopies in the environment, leading to the outliers in observation. As shown in Figure 12; Figure 13

Experimental Results and Analysis
Practical experimental data was processed under the same conditions by the standard CKF, HI-RCKF and proposed MDC-RCKF, respectively. χ 2 m,α in the proposed MDC-RCKF was also chosen as 12.592 for identification of abnormal observations. The other initial parameters were set to the same values as those in the simulation analysis. Figures 12 and 13 give the position errors in longitude and latitude of the vehicle by the standard CKF, HI-RCKF and proposed MDC-RCKF. During the test process, the GNSS signals were shielded by the tall buildings or tree canopies in the environment, leading to the outliers in observation. As shown in Figure 12; Figure    The mean absolute errors (MAE) and the standard deviations (STD) of the position errors in longitude and latitude for the comparison of the standard CKF, HI-RCKF and proposed MDC-RCKF are intuitively described in Figure 14; Figure 15. These results also verify that the proposed MDC-RCKF is capable of enhancing the filtering robustness and outperforms the standard CKF and HI-RCKF in terms of positioning accuracy in the presence of abnormal observations for the vehicular INS/GNSS integration.  The mean absolute errors (MAE) and the standard deviations (STD) of the position errors in longitude and latitude for the comparison of the standard CKF, HI-RCKF and proposed MDC-RCKF are intuitively described in Figure 14; Figure 15. These results also verify that the proposed MDC-RCKF is capable of enhancing the filtering robustness and outperforms the standard CKF and HI-RCKF in terms of positioning accuracy in the presence of abnormal observations for the vehicular INS/GNSS integration. The mean absolute errors (MAE) and the standard deviations (STD) of the position errors in longitude and latitude for the comparison of the standard CKF, HI-RCKF and proposed MDC-RCKF are intuitively described in Figure 14; Figure 15. These results also verify that the proposed MDC-RCKF is capable of enhancing the filtering robustness and outperforms the standard CKF and HI-RCKF in terms of positioning accuracy in the presence of abnormal observations for the vehicular INS/GNSS integration.

Conclusions
This paper presents a novel robust CKF with scaling factor for nonlinear INS/GNSS integration. The contribution of this paper is that novel theories of the Mahalanobis distance are established for identification of abnormal observations and further for improvement of the robustness of the standard CKF. The abnormal observations involved in nonlinear INS/GNSS integration are identified using the Mahalanobis distance criterion. Subsequently, a robust factor (scaling factor) is introduced into the standard CKF to inflate the observation noise covariance and further decreases the filtering gain to resist the influence of abnormal observations on navigation solution. The simulation and practical experimental results as well as comparison analysis demonstrate that the proposed robust CKF has a strong robustness against abnormal observations, resulting in enhanced performance for INS/GNSS integrated vehicle navigation.
Future research work will focus on the improvement of the proposed robust CKF. The proposed robust CKF will be combined with advanced artificial intelligence technologies such as machine learning, deep learning and neural network to automatically identify abnormal observations and resist their influence on navigation solution.