A Fast Covariance Union Algorithm for Inconsistent Sensor Data Fusion

We consider a challenging scenario in this research, where the sensors may receive spurious sensor data, potentially causing inconsistent state estimates. Covariance union (CU) is a fault-tolerant algorithm that can deal with inconsistent state estimation fusion. However, existing CU algorithms suffer from high computational costs due to optimizing nonlinear cost functions when generating fusion weights. To overcome this deficiency, an efficient CU algorithm named fast covariance union (FCU) is developed. We have proved that the fusion weight of FCU can be optimally generated by a closed-form algorithm without optimizing any nonlinear cost function, leading to better fusion efficiency. In addition, the FCU algorithm ensures the fused estimate be consistent as long as one of the estimates is consistent. Finally, the Monte Carlo simulation results show that the FCU algorithm has higher computational efficiency than the existing CU algorithms and handles the spurious sensor data fusion effectively.


I. INTRODUCTION
Multisensor fusion is an effective way to improve the reliability, robustness, and accuracy of the estimate system by combining information from different individual sensors. Multisensor systems are widely used in many areas, such as robotics [1], [2], and sensor networks [3], [4]. Generally, depending on whether the raw data are processed, the architecture of existing multisensor systems can be classified into two groups: centralized and distributed. In the former, raw data are directly sent to a central node and then are fused for state estimates, leading to optimal global estimates. In the latter, each sensor only handles local raw data and communicates with neighbors or a fusion center. Compared with the centralized fusion architecture, the distributed fusion architecture has less computation and communication costs, and it is robust to a single point failure and enjoys good scalability.
Research on distributed data fusion has increasingly gained attention in recent years. It can be classified into two main types [5]: 1) fusion for deciding on a hypothesis, such as detecting the presence of targets [6], [7] or classifying a signal [8]; 2) fusion for target tracking [9]- [12].
The associate editor coordinating the review of this manuscript and approving it for publication was Dusmanta Kumar Mohanta .
In the distributed target tracking fusion problem, consistency is one of the most important criteria for evaluating the reliability of an estimator [13], [14]. However, getting a consistent estimate in a distributed manner is challenging, owing to unknown inter-estimate correlations and inconsistent estimates. Covariance intersection (CI) [15] is a well-known consistent fusion approach, which can obtain consistent estimates even facing an unknown degree of inter-estimate correlation. It has been widely used in many fields [12], [16]- [18]. Practically, sensors produce spurious data because the sensor measurements are corrupted by unexpected uncertainties, such as permanent failures, spike faults, slowly developing failures, and sensor glitches. In these cases, the estimates provided by the sensor may be inconsistent due to unmodeled faults [10], [19]. However, fusing inconsistent estimates with the consistent estimates in the CI algorithm may leads to severely inaccurate results [11].
Compared with CI, covariance union (CU) is more faulttolerant. The idea of the CU algorithm was first proposed by Uhlmann [11] in 2003. Although the spurious estimates are unidentified, the consistency can be guaranteed by extending the fused covariance to cover all local means and covariances. The specific applications of the CU algorithm were proposed in [20] and [21]. A generalized CU algorithm was proposed in [22], which is an amalgamation of standard mixture VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ reduction and CU. The generalized CU gives a tighter covariance matrix than that in CU. Bochardt and Uhlmann [23] proposed a general CU algorithm by applying a covariance addition to the CU algorithm. The solution of this general CU is equivalent to the minimum enclosing ellipsoid. However, these CU algorithms suffer from high computational costs due to optimizing nonlinear cost functions when generating fusion weights [10].
In this paper, a fast covariance union (FCU) algorithm is proposed to solve the multisensor fusion problem with spurious sensor data. The main innovations of this work are outlined as follows: 1) A computationally efficient CU algorithm called FCU is proposed. 2) We have proved that the fusion weight of FCU can be optimally generated by a closed-form algorithm without optimizing any nonlinear cost function.
3) The FCU algorithm ensures the fused estimate be consistent as long as one of the estimates is consistent.

II. SYSTEM MODEL AND PROBLEM STATEMENT
Consider a sensor network composed of M nodes, where each node can communicate with neighboring nodes. At time k +1, node i takes measurement of the target which the sensor network interests. A state-space model associated with the target and the measurement is given as follows: whereX k+1 is the target state to be estimated, Z i k+1 the measurement of the target taken by node i, f and g are nonlinear functions, w k and v i k+1 are zero-mean Gaussian noise with covariance Q k and R i k , respectively. We denote the estimate of target obtained by node i based on local measurement Z i k+1 as X i k+1 , P i k+1 , where X i k+1 denotes the estimated state vector and P i k+1 denotes the estimated error covariance matrix.
T , the X i k+1 , P i k+1 is a consistent estimate [24].
In many practical situations, sensor noise may be corrupted by unexpected uncertainties. In these cases, the sensor noise is biased [10], [26]. Since the unexpected uncertainties are not attributable to the inherent noise, they are difficult to model. Subsequently, the target state estimated by node i may be inconsistent.
Given two mean and covariance estimates (X a , P a ) and (X b , P b ), with X a = [a 1 , . . . , a n ] T , where n is the dimension of X.
When (X a , P a ) and (X b , P b ) are all consistent estimates, Julier and Uhlmann [15] introduced CI algorithm, which can yield a consistent estimate even fusing estimates are with unknown correlations. However, when either (X a , P a ) or (X b , P b ) is an inconsistent estimate of the target, the fusion of CI cannot guarantee consistency [25]. To resolve this problem, Uhlmann [11] proposed a CU algorithm. No matter which estimate is spurious, the CU combines these estimates to form a mean and covariance estimate (u, U), which is guaranteed to be consistent.

III. FAST COVARIANCE UNION ALGORITHM
Matzka and Altendorfer [21] proposed a CU algorithm, referred to as MCU in this paper. The algorithm is given as follows: where max(U a , U b ) is the element-wise maximum value of the matrices U a and U b .
The second derivative of ω is ,ω ρ(n) . Furthermore, the positive second derivative indicates that the extreme is a local minimum value.
ω =ω ρ(1) 11. end if 12. Calculate (u, U) using ω and (3)-(6) Based on this algorithm, we can get an exact value of the nonlinear-convex optimization problem. Moreover, since the nonlinear-convex optimization problem has been transformed into a linear function, the computation time has been greatly reduced.

IV. SIMULATIONS
The performance of the FCU algorithm was evaluated by numerical simulation under two conditions: 1) normal sensor data, and 2) abnormal sensor data. We compared the performance of FCU with the following four algorithms: 1) MCU [21]; 2) fast fault-tolerant convex combination (FFCC) algorithm, which is an extension of the CI algorithm and can handle inconsistent sensor data fusion [26]; 3) the CU algorithm proposed by Julier (JCU) [20], and 4) CI [15].
In this simulation, two sensors S 1 and S 2 were used to track a moving target with constant-velocity. We refer to the moving target as O, the state of the target at time k is denoted T denote target's position and velocity, respectively. The state transition model of the moving target is given by (17).
where ζ k is a zero-mean white Gaussian process noise with covariance ε k . The measurements of the two sensors at time k + 1 are modeled as: with where γ i k+1 is a zero-mean white Gaussian process noise with covariance σ i k+1 . At time k, the state and covariance of the target estimated by S i are denoted as . When sensor S i obtains its measurement Z S i k+1 at time k + 1, we first use the state transition model which is given in (17) to predict the state of target as follows: The covariance term of the target estimated by S i is propagated as follows: Using the measurement Z S i k+1 , the state and covariance of the target are updated by the Kalman filter as follows: (25) with Finally, S i 's local estimate as follows: where j ∈ {1, 2} \i, max(T i , T j ) is the element-wise maximum value of the matrices T i and T j . The fused results X O k+1 ,P O k+1 are the final state and covariance of the target estimated by the two sensors at time k + 1. Then the fused results stored in S 1 and S 2 , respectively, i.e., In this simulation, we use a computer with i7 CPU 3.6GHz, RAM 8GB, and Matlab 2008a. TABLE 1 lists the simulation parameters. We chose root mean square error (RMSE), normalized estimation error squared (NEES) [27] and time to measure the accuracy, consistency and computational efficiency of the five algorithms, respectively. VOLUME 9, 2021

A. SIMULATION WITH NORMAL SENSOR DATA
In this simulation, the measurements of the two sensors are affected by normal sensor noise, which is given in (20). Figs. 1 and 2 show the RMSE and NEES over time of the position estimation of the target, respectively. TABLE 2 shows the average results over the whole simulation time. It is noteworthy that the accuracy and consistency of the MCU and FCU algorithms are very close. This implies no significant difference between the results obtained by the minimization trace optimization and minimization determinant optimization in weight ω calculation. From Fig. 2, we can see that the NEES values are around the ideal value of 2 in the CI algorithm. For MCU, JCU, FCU, and FFCC algorithms, the NEES values are much smaller than 2. This is because that using these fusion algorithms, the covariance is enlarged to cover all local means and covariances.
The average computational time of the five fusion algorithms in each Monte Carlo run is given in TABLE 2. The weights ω in MCU, CI, and JCU algorithms are computed by the ''fminbnd'' optimization function in the Matlab. From TABLE 2, it can be seen that the FCU algorithm has the shortest calculation time, while the JCU algorithm has the largest calculation time among the five algorithms.

B. SIMULATION WITH ABNORMAL SENSOR DATA
In TABLE 1, it can be seen that S 1 and S 2 are heterogeneous sensors, and the sensor noise of S 1 is smaller than S 2 . To verify   the influence of inconsistent estimates on the fusion results, three conditions are considered: 1) the sensor data of S 1 is abnormal; 2) the sensor data of S 2 is abnormal, and 3) S 1 and S 2 have the same processing sensor noise, and the sensor data of S 2 is abnormal.

1) THE SENSOR DATA OF S 1 IS ABNORMAL
We assume that the measurements of S 1 are corrupted by a biased Gaussian noise η k+1 , i.e., where η k+1 = N Rv * (2, 2) , (2m) 2 , (2m) 2 , Rv is the coefficient. while σ 1 k+1 is still used to compute V S 1 k+1 in (27). Then, the model mismatch of the Kalman filter in S 1 leads to an inconsistent state estimation of the target.
When Rv = 1, the simulation results are shown in Figs. 3, 4, and TABLE 3. We can see that the CI algorithm gets disappointing results in both RMSE and NEES  compared with the MCU, JCU, FCU, and FFCC algorithms. The NEES values for the CI algorithm are around 5, which indicates that the state estimation of the target is inconsistent. In MCU, JCU, and FCU algorithms, consistent state estimates are obtained. This is because in the CU algorithm, even if S 1 's data is abnormal, the consistent estimate of the target can be achieved with the help of S 2 . Besides, the MCU, JCU, and FCU algorithms have similar performance regarding RMSE and NEES. Additionally, the average computational time of the five fusion algorithms is similar to the results in TABLE 2. The FCU algorithm has the highest computational efficiency.  In the FFCC algorithm, as Rv increases, the average NEES value gradually exceeds the ideal value of 2. This is due to the fact that when Rv is much larger, the belief δ will not decrease proportionally [26].

2) THE SENSOR DATA OF S 2 IS ABNORMAL
We assume that the measurements of S 2 are corrupted by a biased Gaussian noise τ k+1 , i.e., where τ k+1 = N Rv * (2, 2) , (3m) 2 , (3m) 2 . The S 1 is affected by the normal sensor noise which is given in (20). Fig. 6 displays the influence of different Rv values on different algorithms when S 2 's data is abnormal. It is noteworthy that the average RMSE and NEES values in the CI algorithm do not change with an increase in Rv. This is because when the determinant or trace of the covariance matrix is smaller, the weight ω in the CI algorithm is larger [28]. In this simulation, no matter what Rv is, the weight ω in the CI algorithm is always ω = 0.9999, which means that S 1 will play a dominating role in the fusion.

3) S 1 AND S 2 HAVE THE SAME PROCESSING SENSOR NOISE, AND THE SENSOR DATA OF S 2 IS ABNORMAL
In this simulation, we study the two sensors have the same processing noise, i.e., σ 1 k+1 = σ 2 k+1 . The sensor noise of S 1 is given in TABLE 1 and S 2 is affected by an abnormal sensor data as follows: The results are shown in Fig. 7 and are similar to those in Fig. 5. The CI algorithm gets disappointing results in RMSE and NEES compared with the MCU, JCU, FCU, and FFCC algorithms. Besides, the MCU and FCU get the same estimated results. This is because when σ 1 k+1 = σ 2 k+1 , we can get , then whether using the minimization trace optimization or minimization determinant optimization in weight ω calculation, the weight ω is always 0.5.

V. CONCLUSION
This paper has considered the estimation fusion problem under spurious sensor data. We have presented an FCU algorithm to solve this problem. The FCU algorithm ensures the fused estimate be consistent as long as one of the estimates is consistent. In addition, we have proved that the fusion weight of FCU can be optimally generated by a closed-form algorithm without optimizing any nonlinear cost function. Thus, the computational complexity of the FCU algorithm is reduced significantly compared with the CU algorithm under the determinant minimization criterion. The performance of the proposed FCU algorithm has been verified by comparing it with state-of-the-art algorithms in simulation. The results show that the FCU algorithm can efficiently handle the spurious sensor data fusion and has higher computational efficiency than the existing CU algorithms.