Multi-Sensor Optimal Data Fusion Based on the Adaptive Fading Unscented Kalman Filter

This paper presents a new optimal data fusion methodology based on the adaptive fading unscented Kalman filter for multi-sensor nonlinear stochastic systems. This methodology has a two-level fusion structure: at the bottom level, an adaptive fading unscented Kalman filter based on the Mahalanobis distance is developed and serves as local filters to improve the adaptability and robustness of local state estimations against process-modeling error; at the top level, an unscented transformation-based multi-sensor optimal data fusion for the case of N local filters is established according to the principle of linear minimum variance to calculate globally optimal state estimation by fusion of local estimations. The proposed methodology effectively refrains from the influence of process-modeling error on the fusion solution, leading to improved adaptability and robustness of data fusion for multi-sensor nonlinear stochastic systems. It also achieves globally optimal fusion results based on the principle of linear minimum variance. Simulation and experimental results demonstrate the efficacy of the proposed methodology for INS/GNSS/CNS (inertial navigation system/global navigation satellite system/celestial navigation system) integrated navigation.


Introduction
With the rapid development of electronics technologies, various sensors have been developed and applied to many engineering fields such as integrated navigation, target tracking, signal processing, and networked communications [1][2][3][4]. Since the use of multiple sensors provides more accurate and reliable information than that of a single sensor alone, multi-sensor data fusion has received considerable attention in recent years. The basic idea of multi-sensor data fusion is to combine the data obtained from multiple sensors and further associate them with database information to achieve improved estimation accuracy compared to the use of a single sensor [5,6].
The existing studies on multi-sensor data fusion can be divided into two categories [4,6]. One is the centralized fusion method, which can achieve a globally optimal state estimation by directly combining local measurement models to form an augmented measurement model [7,8]. This method has minimal information loss. However, it causes large computation and communication burdens in the fusion center due to high-dimension computations and large data memories. Further, it has a poor robustness and reliability when one sensor is faulty [1,9]. The other is the decentralized fusion method, which can yield globally optimal or suboptimal state estimations according to certain criteria [9,10]. This method has small computation and communication burdens in the fusion center due to the decentralized structure. Furthermore, the decentralized structure also enables easy fault detection and Han presented an adaptive UKF to inhibit the process-modeling error by updating the process noise covariance via minimization of the difference between computed and actual innovation covariances [29]. However, this method requires the calculation of partial derivatives, causing a relatively large computation burden. Meng et al. reported a covariance matching-based adaptive UKF to make online estimates of and adjust process noise covariance using innovation and residual sequences [30]. However, this covariance matching technique yields a steady-state estimation error, leading to a limited improvement in the filtering accuracy [20]. Compared to the above two adaptive UKFs, the adaptive fading UKF (AFUKF) can effectively inhibit the influence of process-modeling error on the filtering solution under a modest computation load [27,28]. In this method, when process-modeling error is detected, an adaptive fading factor is constructed based on innovation vector and its corresponding statistical information. By using the adaptive fading factor to scale process noise covariance or predicted state covariance, historical information is less weighted while current measurement information is more weighted to tune UKF recursively to alleviate the influence of process-modeling error on the estimation solution [19,28]. Therefore, the introduction of AFUKF into multi-sensor optimal data fusion provides a solution to improve the adaptability and robustness of UKF-MODF.
This paper focuses on improvement of our previous work (UKF-MODF) for multi-sensor nonlinear stochastic systems. It presents a novel multi-sensor optimal data fusion methodology based on AFUKF to address the UKF-MODF problems, leading to improved adaptability and robustness of data fusion for multi-sensor nonlinear stochastic systems. This methodology is in a two-level fusion structure: at the bottom level, an AFUKF based on the Mahalanobis distance is developed and serves as local filters to improve the adaptability and robustness of local state estimations against process-modeling error; at the top level, an UT-based multi-sensor optimal data fusion for the case of N local filters is established according to the principle of linear minimum variance to calculate globally optimal state estimation via fusion of local estimations. The proposed methodology not only improves the robustness of multi-sensor data fusion against process-modeling error, but it also achieves globally optimal fusion results based on the principle of linear minimum variance. Simulations and practical experiments as well as comparison analysis with FKF, UKF-FKF and UKF-MODF have been conducted to comprehensively evaluate the performance of the proposed multi-sensor optimal fusion methodology for INS/GNSS/CNS (inertial navigation system/global navigation satellite system/celestial navigation system) integrated navigation system.

Adaptive Fading UKF Based Multi-Sensor Optimal Data Fusion
The framework of the proposed AFUKF-based multi-sensor optimal data fusion methodology (AFUKF-MODF) is shown in Figure 1. It has a two-level fusion structure: at the bottom level, an AFUKF based on the Mahalanobis distance is developed and serves as local filters to improve the robustness of local state estimations against process-modeling error; at the top level, according to the principle of linear minimum variance, the UT-based multi-sensor optimal data fusion for the case of N local filters is established to calculate the globally optimal state estimation by fusion of local estimations. The framework of the proposed AFUKF-based multi-sensor optimal data fusion methodology (AFUKF-MODF) is shown in Figure 1. It has a two-level fusion structure: at the bottom level, an AFUKF based on the Mahalanobis distance is developed and serves as local filters to improve the robustness of local state estimations against process-modeling error; at the top level, according to the principle of linear minimum variance, the UT-based multi-sensor optimal data fusion for the case of N local filters is established to calculate the globally optimal state estimation by fusion of local estimations.  Figure 1. The framework of the proposed AFUKF-based multi-sensor optimal data fusion.

Local State Estimation
Consider the ith (i = 1, 2, · · · , N) local filter with the following dynamic model where x(k) ∈ R n is the system state vector; f (·) is the nonlinear state function; w(k) is the process noise which is commonly assumed as a zero-mean Gaussian white noise with covariance Q ≥ 0; z i (k) ∈ R m i is the measurement of the ith local filter; h i (·) is the nonlinear measurement function of the ith local filter; and v i (k) is the measurement noise of the ith local filter and is commonly assumed as a zero-mean Gaussian white noise with covariance R i > 0.

Classical UKF
In order to describe the Mahalanobis distance-based AFUKF clearly, let us briefly review the classical UKF at first. For the nonlinear system model given by (1), the procedure of the classical UKF can be summarized as: Step 1: Initialization Step 2: Time Update Assume that the state estimatex i (k − 1) and the corresponding error covariance matrix P i (k − 1) are given, the sigma points are selected as where a is a tuning parameter to determine the spread of the sigma points aroundx i (k − 1) and it is commonly set to a small positive value; and nP i (k − 1) j denotes the jth column of the square root of the matrix nP i (k − 1). Calculate the predicted state mean and covariance as where .
Step 3: Sigma Point Update Select a set of new sigma points with the meanx i (k/k − 1) and the covariance P i (k/k − 1) Step 4: Measurement Update The weighted mean and covariance of the predicted measurement are computed by Subsequently, the state estimate and associated error covariance matrix are updated as Step 5: Get back to Step 2 for the next sample until all samples are processed.

Mahalanobis Distance Based Adaptive Fading UKF
The Mahalanobis distance of innovation vector, which is based on hypothesis testing theory, is commonly used as a measure to identify system modeling error for Gaussian systems [27,31]. This paper adopts the Mahalanobis distance into AFUKF and further develop a Mahalanobis distance-based AFUKF to improve the adaptability and robustness of the classical UKF against process-modeling error for multi-sensor nonlinear stochastic systems.
Define the innovation vector of the ith local filter as For the nonlinear Gaussian system given by (1), z i (k) should obey the zero-mean Gaussian distribution with the covariance Thus, the square of the Mahalanobis Distance of the innovation vector should obey the chi-square distribution with m i degrees of freedom [27,31], i.e., where m i is the dimension of the measurement in the ith local filter. According to the hypothesis testing theory, for a given significance level α, we have where P(·) represents the probability of a random event.
If (18) holds, this means the system works under the optimal condition, i.e., the nonlinear multi-sensor system described by (1) does not have process-modeling error. However, if (18) does not hold, it can be concluded with high probability that there exists process-modeling error in the multi-sensor system (1). In this case, AFUKF incorporates a time-varying adaptive fading factor into the predicted state covariance matrix to refrain from the influence of prior knowledge of the current state estimate. Thus, the predicted state covariance matrix in AFUKF is modified as Introduce the adaptive fading factor λ i k to the predicted state covariance matrix such that (18) holds. Thus, we have the following equation where P * i (ẑ i (k/k − 1)) is the measurement prediction covariance matrix calculated by using It can be seen from (20) that the determination of the adaptive fading factor λ i k is a problem of solving the nonlinear equation. Since the derivative of g i (λ i k ) with respect to λ i k is difficult to calculate, the traditional Newton's method cannot be used to solve the nonlinear Equation (20). In this paper, a chord secant method [32,33] is adopted to iteratively determine the adaptive fading factor λ i k by solving the nonlinear Equation (20). Thus, we have where t represents the iteration time.
Substituting (20) into (21) yields The above iterative process to determine the adaptive fading factor λ i k is initialized as λ i k (1) = 1 and λ i k (0) = 0. It will be terminated when the criterion M i k [λ i k (t)] ≤ χ 2 m i ,α is satisfied. The calculation process of the Mahalanobis distance-based AFUKF can be summarized as: Step 1: Initialization Initiate the Local filters by presetting thex i (0) and P i (0) as (2).
Step 2: Local Filtering • Perform the classical UKF procedures (11)- (14) to compute the local state estimations. Else, • Determine the adaptive fading factor λ i k by iteratively computing (22) Then, replace the predicted state covariance matrix P i (k/k − 1) of the classical UKF with the modified type P * i (k/k − 1) as described by (19).
(iii) Repeat Steps (i) and (ii) for the next sample.
Step 3: Complete the local state estimations until all samples are processed.
Each local filter is performed in a parallel manner to generate the local state estimationx i (k) and the corresponding error covariance P i (k) (i = 1, 2, · · · , N) by using the Mahalanobis distance-based AFUKF. After this, these local state estimations will be further fused by the UT-based data fusion method, which is to be described in the next section, to obtain the globally optimal state estimation.

Global State Estimation
Suppose that the local state estimation achieved by the ith (i = 1, 2, · · · , N) local filter isx i (k). According to the linear weighting fusion criterion, the globally optimal state estimation can be described as the following formx where α i (i = 1, 2, · · · , N) is the weighting matrix to be determined; and I n represents an n-dimensional identity matrix. Based on the principle of linear minimum variance, the globally optimal state estimationx * (k) should satisfy the following conditions [14,24]: Condition 2.x * (k) makes tr{P * (k)} minimum, where P * (k) is the error covariance matrix ofx * (k), and tr{·} is the trace operator. Theorem 1. Denote the local state estimation achieved by the ith (i = 1, 2, · · · , N) local filter byx i (k), and its corresponding error covariance matrix by P ii (k) (i.e., P i (k), i = 1, 2, . . . , N in the local filters). The cross-covariance matrix between the state estimation errors ofx i (k) andx j (k) is P ij (k) (i, j = 1, 2, . . . , N; i = j). Define (Σ(k)) ij = P ij (k) ∈ R n×n (i, j = 1, 2, . . . , N), E = [I n , I n , . . . , I n ] T ∈ R nN×n and α = [α 1 , α 2 , . . . , α N ] T ∈ R nN×n . Based on the principle of linear minimum variance, the globally optimal state estimationx * (k) is obtained asx where α i (i = 1, 2, · · · , N) is given by Proof. As E{x i (k) − x(k)} = 0, taking the expectation of (25) and considering the relation we can obtain It can be seen from (27),x * (k) obtained by (25) is the unbiased estimation of x(k). Now let us consider Condition 2 to derive the globally optimal state estimationx * (k). Define the estimation error of the local state estimationx i (k) (i = 1, 2, . . . , N) as Thus, the estimation error ofx * (k) is described as According to (29), the error covariance matrix ofx * (k) is Applying the Lagrange multiplier to solve the minimum of tr{P * (k)}, the Lagrange function is defined as where Λ ∈ R n×n is the Lagrange multiplier.
Taking the partial derivative of (31) with respect to α and letting it to be zero, we have By solving (33), it is verified that The proof of Theorem 1 is completed.
It can be seen from (25) and (26), in order to calculate the global optimal state estimationx * (k), it is necessary to predetermine the matrix Σ(k). In the matrix Σ(k), P ii (k)(i = 1, 2, . . . , N) can be directly determined by the error covariance matrix of the state estimation in the ith local filter. However, it is difficult to calculate the cross-covariance matrix P ij (k) between the state estimation errors ofx i (k) and x j (k). This paper adopts the UT concept to approximate P ij (k), and the derivation process is shown in Theorem 2.

Theorem 2.
For the multi-sensor system given by (1), the cross-covariance matrix P ij (k) between the estimation errors of local state estimationsx i (k) andx j (k) (i, j = 1, 2, . . . , N; i = j) is computed as where χ (i) s,k/k−1 denotes the sigma point transformed by the nonlinear function f (·) in (4) for the ith local filter; γ (i) s,k/k−1 denotes the sigma point transformed by the nonlinear function h i (·) in (8) for the ith local filter; and s = 0, 1, · · · , 2n represents the order of the transformed sigma points.
Proof. Define the estimation error of the ith local filter as Then, it is verified from (37) Based on the tranformated sigma points χ (i) s,k/k−1 and γ (i) s,k/k−1 (i = 1, 2, · · · , N), P ij (k) can be approximated by using UT The proof of Theorem 2 is completed.

Performance Evaluation and Discussion
A prototype system of INS/GNSS/CNS integration was implemented using the proposed AFUKF-MODF. Simulations and experiments as well as comparison analysis with FKF, UKF-FKF [13], and UKF-MODF [14] were conducted to comprehensively evaluate the performance of the proposed AFUKF-MODF. The navigation frame (n-frame) is selected as the E-N-U (East-North-Up) geography frame (g-frame). Denote the inertial frame by i, the earth frame e, the body frame b and the INS simulated actual platform frame n . The system state vector is defined as

System Model of INS/GNSS/CNS Integration
z the accelerometer zero-bias. The nonlinear attitude and velocity error equations are given as [34,35] T is the velocity of the vehicle in n-frame; C n n , C n b and C n b are the rotation matrices; δg n is the gravity error;f b is the measured specific force in the b-frame, and it is composed of accelerometer zero-bias ∇ b and its white noise ω b a ; δf b is the corresponding error; δω b ib is the measurement error of the gyro, which is composed of gyro constant drift ε b and its white noise ω b g ; ω n ie is the rotational angular velocity of the earth; ω n en is the angular velocity of the vehicle relative to the earth; ω n in = ω n ie + ω n en is the relative rotational angular velocity between the n-frame and i-frame;ω n ie ,ω n en andω n in are the actual values of ω n ie , ω n en and ω n in in the n -frame; and δω n ie , δω n en and δω n in represent the corresponding errors. The above parameters can be calculated as where L and h represent the latitude and altitude of the vehicle; and R M and R N are the median radius and normal radius. C −1 ω is computed as The position error equation of INS is described by [36]  The gyro constant drift ε b and accelerometer zero-bias ∇ b are commonly described as random constants [14,30], i.e., .
According to the selected system state vector, the process model of the INS/GNSS/CNS integration can be established by combining (41)-(46) .
where f (·) is a nonlinear function describing the system state equation in continuous form; and T is the process noise vector.
By discretizing (47) with the improved Euler formulation [37], the discrete-time process model of the INS/GNSS/CNS integration is obtained as where f (·) is a nonlinear function describing the system state equation in discrete form; and w(k) is the discrete-time process noise vector.

Measurement Model of INS/GNSS Subsystem
Take the difference between INS and GNSS in terms of velocity and position as the measurement of the INS/GNSS subsystem, i.e., Then, the measurement model of the INS/GNSS subsystem can be described as [11,24] where and v v (k) and v P (k) are the measurement noises, which correspond to the velocity and position errors of GNSS, respectively.

Measurement Model of INS/CNS Subsystem
CNS provides high-precise attitude information for a vehicle through the star sensor. Take the difference between INS and CNS in terms of attitude as the measurement of the INS/CNS subsystem, i.e., where (φ EI , φ N I , φ U I ) T and (φ EC , φ NC , φ UC ) T are the attitude information obtained by INS and CNS, respectively. Accordingly, the measurement model of the INS/CNS subsystem is constructed as [11,24] where H 2 (k) = [I 3×3 , 0 3×12 ], and v 2 (k) is the measurement noise corresponding to the measurement error of the star sensor. The INS/GNSS/CNS integrated navigation system is a multi-sensor nonlinear system with 2 local filters (N = 2). The nonlinear system model, which is generally described by (1)

Simulations and Analysis
Monte Carlo simulations were conducted to comprehensively evaluate the performance of the proposed AFUKF-MODF for the dynamic flight of an unmanned aerial vehicle (UAV) using the INS/GNSS/CNS integration for navigation and positioning. The flight trajectory, which involves various maneuvers such as climbing, pitching, rolling and turning, is shown in Figure 2. The simulation parameters are listed in Table 1. Both filtering period and fusion period were 1 s. The Monte Carlo simulations were carried out 100 times.
Monte Carlo simulations were conducted to comprehensively evaluate the performance of the proposed AFUKF-MODF for the dynamic flight of an unmanned aerial vehicle (UAV) using the INS/GNSS/CNS integration for navigation and positioning. The flight trajectory, which involves various maneuvers such as climbing, pitching, rolling and turning, is shown in Figure 2. The simulation parameters are listed in Table 1. Both filtering period and fusion period were 1 s. The Monte Carlo simulations were carried out 100 times.   The initial state error covariances and process noise covariance for the local filters are set as (1 × 10 −3 g) 2 I 3×3 (i = 1, 2) (53) However, P i (0) (i = 1, 2) and Q were enlarged to the twice of their initial values for both FKF and UKF-FKF to eliminate the correlation between the two local state estimations by the use of covariance upper bound. Moreover, the process model of the INS/GNSS/CNS integration should be linearized in local fitering processes for the FKF.
The measurement noise covariances are set as In order to evaluate the performance of the proposed AFUKF-MODF in terms of process-modeling error, the following process-modeling error is introduced to the process model Accordingly, the process model used in the local filters is , other time intervals To identify the above process-modeling error, χ 2 m 1 ,α and χ 2 m 2 ,α used in the proposed AFUKF-MODF for the INS/GNSS subsystem and INS/CNS subsystem were chosen as 12.592 and 7.815, respectively. These values are resulted from the χ 2 distribution when the reliability level is 95% (α = 0.05) and the degrees of freedom are 6 and 3, respectively.
For comparison analysis, simulation trials were conducted at the same conditions by using FKF, UKF-FKF, UKF-MODF and AFUKF-MODF, respectively. Moreover, the overall estimation error is adopted to evaluate the navigation accuracy in the simulation analysis for the four fusion methods. It is defined as the norm of the navigation parameters estimation error [38] where ∆x E , ∆x N and ∆x U are the components of ∆x in East, North and Up, respectively. Accordingly, its navigation accuracy is close to that of UKF-MODF during these time intervals.
bound in the local filters. Thus, the fusion results obtained by both UKF-MODF and AFUKF-MODF are globally optimal, which are much more accurate than those by FKF and UKF-FKF. The estimation RMSEs in attitude, velocity and position are around 0.1612 , 0.1155 m/s and 10.6099 m for UKF-MODF, and 0.1643 , 0.1176 m/s and 10.7212 m for AFUKF-MODF. Moreover, since there is no process-modeling error identified, the proposed AFUKF-MODF does not incorporate the adaptive fading factor into the local estimation processes. Accordingly, its navigation accuracy is close to that of UKF-MODF during these time intervals.    In contrast, the proposed AFUKF-MODF has higher navigation accuracy than FKF, UKF-FKF and UKF-MODF during this time interval. Its estimation RMSEs in attitude, velocity and position are around 0.1702 , 0.1280 m/s and 11.5798 m. This is because the proposed AFUKF-MODF can identify the process-modeling error according to the hypothesis testing theory and effectively refrain from the influence of process-modeling error on the fusion solution by adjusting the time-varying adaptive fading factor incorporated in the predicted state covariance matrix, leading to a strong adaptability and robustness.  Table 2. The results in Table 2 also verify that the proposed AFUKF-MODF has a better adaptability and robustness than the other methods, thus leading to improved navigation accuracy for the INS/GNSS/CNS integration. The above simulations and analysis demonstrate that the proposed AFUKF-MODF can overcome the limitation of using covariance upper bound and provide globally optimal fusion results. It can also enhance the adaptability and robustness against process-modeling error to overcome the limitation of UKF-MODF, thus leading to higher navigation accuracy than FKF, UKF-FKF and UKF-MODF in the presence of process-modeling error.

Experiments and Analysis
Practical experiments were also conducted to evaluate the performance of the proposed AFUKF-MODF by observing the flight of an UAV. As shown in Figure 6, the UAV uses an INS/GNSS/CNS integration system for navigation. This navigation system includes a MTi-100 IMU, a Hemisphere P307 BDS/GNSS receiver and a SODERN SED26 star sensor. The main parameters of the above three devices are listed in Tables 3-5. Moreover, another Hemisphere P370 BDS/GNSS receiver, which was placed at a local reference station (around 1km from the initial UAV position), was used along with the one mounted on the UAV to provide the differential GPS (DGPS) data. The maximal distance between the UAV and local reference station was less than 60 km to achieve the position accuracy of less than 0.1 m from the DGPS via post difference processing. The DGPS data were used as the reference values to evaluate the positioning error of the INS/GNSS/CNS integrated system.       The UAV flight test was conducted at the City of Yanliang in Shaanxi, China. The experimental navigation data were selected from the UAV flight test within a continuous time period of 1000 s, where different maneuvers were involved. Within the selected UAV flight test, the UAV initial position was at East longitude 109.217 • , North latitude 34.647 • and altitude 3525 m. The UAV initial velocity was 180 m/s, 150 m/s and 50 m/s in East, North and Up, respectively. The other initial parameters were set identically to those in the simulation case. The filtering periods of the local filters, i.e., the INS/GNSS and INS/CNS subsystems, were 1 s. The period for fusion of local state estimations was also 1 s.
Due to the complexity of the dynamic flight environment, the process model of the INS/GNSS/CNS integration involves modeling error during the UAV flight process. Figure 7 illustrates the UAV position errors obtained by FKF, UKF-FKF, UKF-MODF and AFUKF-MODF, respectively. As shown in Figure 7, the position error achieved by FKF is relatively large due to the influence of process-modeling error, the use of the covariance upper bound and the linearization of the process model. .2610 m), which are much smaller than those by FKF, UKF-FKF and UKF-MODF. This is because the proposed AFUKF-MODF has the capability to inhibit the disturbances of process-modeling error. Table 6 lists the mean absolute errors (MAEs) and standard deviations (STDs) of the position errors achieved by FKF, UKF-FKF, UKF-MODF and AFUKF-MODF. It can be seen that the MAE and STD of the position errors by the proposed AFUKF-MODF are also much smaller than those of the other three methods. .2610 m), which are much smaller than those by FKF, UKF-FKF and UKF-MODF. This is because the proposed AFUKF-MODF has the capability to inhibit the disturbances of process-modeling error. Table 6 lists the mean absolute errors (MAEs) and standard deviations (STDs) of the position errors achieved by FKF, UKF-FKF, UKF-MODF and AFUKF-MODF. It can be seen that the MAE and STD of the position errors by the proposed AFUKF-MODF are also much smaller than those of the other three methods.
The above experimental results and analysis demonstrate that the proposed AFUKF-MODF enhances the adaptability and robustness of multi-sensor data fusion in presence of process-modeling errors, leading to improved navigation accuracy for INS/GNSS/CNS integration in comparison with FKF, UKF-FKF and UKF-MODF.

Conclusions
This paper presents a novel AFUKF-MODF to improve the fusion adaptability and robustness for multi-sensor nonlinear stochastic systems against process-modeling error. The contributions of this paper are: (i) an AFUKF based on the Mahalanobis distance is developed to improve the adaptability and robustness of local state estimations against process-modeling error; and (ii) an UT-  The above experimental results and analysis demonstrate that the proposed AFUKF-MODF enhances the adaptability and robustness of multi-sensor data fusion in presence of process-modeling errors, leading to improved navigation accuracy for INS/GNSS/CNS integration in comparison with FKF, UKF-FKF and UKF-MODF.

Conclusions
This paper presents a novel AFUKF-MODF to improve the fusion adaptability and robustness for multi-sensor nonlinear stochastic systems against process-modeling error. The contributions of this paper are: (i) an AFUKF based on the Mahalanobis distance is developed to improve the adaptability and robustness of local state estimations against process-modeling error; and (ii) an UT-based multi-sensor optimal data fusion for the case of N local filters is also established to achieve the globally optimal state estimation via fusion of local state estimations. Simulations and practical experiments as well as comparison analysis demonstrate that the proposed AFUKF-MODF not only resists the disturbance of process-modeling error on the fusion solution, leading to improved adaptability and robustness for multi-sensor data fusion, but it also provides globally optimal data fusion results in the sense of linear minimum variance. The achieved fusion accuracy is much higher than that of FKF, UKF-FKF and UKF-MODF for INS/GNSS/CNS integration.
Future research work will focus on two aspects. One is the improvement of the proposed AFUKF-MODF. It is expected to combine the proposed AFUKF-MODF with artificial intelligence technologies such as machine learning, pattern recognition and neural networking to intelligently identify and compensate for system modeling error. The other is on the applications of the proposed AFUKF-MODF to address the multi-sensor data fusion problems in other fields such as target tracking, fault diagnosis and signal processing.