Unscented Information Filtering for Distributed Estimation and Multiple Sensor Fusion Deok-Jin Lee∗ Naval Postgraduate School, Monterey, CA, 93943, U.S.A. This paper represents distributed estimation and multiple sensor information fusion using an unscented information filtering algorithm. The proposed information fusion algorithm is developed by embedding the unscented transformation method used in the sigma point filter into the extended information filtering architecture, and its algorithm is further extended for distributed estimation in hierarchical sensor networks. The new information fusion filter achieves not only the accuracy and robustness of the sigma point filter, but also the flexibility of the information filter for multiple sensor estimation in distributed sensing networks. Performance comparison of the proposed sensor fusion filter with the extended information filter is demonstrated through a simple target-tracking simulation study.

I.

Introduction

Multiple sensor data fusion techniques are widely used in various applications such as target tracking, surveillance, robot navigation, and large-scale systems.1 The advantages of multiple sensor fusion algorithms lie on the fact that the multiple sensor fusion offers complementary characteristics among different kinds of sensors that have different coverage capability, and provides higher robustness due to the inherent redundancy. Sensor fusion can be loosely defined as how to best extract useful information from multiple sensor observations. The main components of multiple sensor fusion consist of validation of sensor data, fusion architectures, and estimation techniques.2 The sensor data validation is achieved by comparing observations obtained from sensors with the predicted observations from measurement model, and accepting only those that lie within a predetermined error bound. In sensor fusion for multiple target tracking problems, data association is required in addition to the validation processes to obtain improved estimates, which is well described in literatures3, 4 and will not be the subject in this paper. Instead, sensor fusion architectures and estimation techniques will be discuss mainly. The fusion architectures for the multiple sensor data integration can be loosely categorized into three types;5 centralized, hierarchically distributed, and fully decentralized, as shown in Fig. 1. A centralized architecture processes all the measurement data provided from the sensors by using a single fusion node with direct connections to all sensor devices. In a hierarchically distributed architecture the fusion nodes are arranged in a hierarchy with the lowest level nodes processing sensor data and sending results to higher level nodes. In a fully decentralized architecture there is no central fusion center, but each node can communicate with any other node subject to communication connectivity. Although centralized multisensor systems provide improvement over single sensor systems, the computational load imposed on the central processor is severe and the failure of any one of the sensor nodes results in a whole system failure. In recent years, sophisticated functional requirements and complexity have increased demands on distributed fusion systems. In distributed sensor fusion systems, each sensor extract useful information first by processing raw sensor data prior to communication, which results in less computational load on the central fusion processor and less communication bandwidth. If output data from each senor node is communicated to the higher levels at a rate slower than the sensor observation rates, considerable mitigation in communication can be achieved in distributed fusion architectures.2 Moreover, decentralized fusion architectures including the hierarchically distributed and fully decentralized ones are highly survivable and robust to the sensor or processor failures since each sensor node deals with its own processing and ∗ NRC Research Associate/Adjunct Research Professor, Unmanned Systems Lab., Department of Mechanical and Astronautical Engineering, 699 Dyer Rd. Ha144, Naval Postgraduate School, Monterey, 93943, AIAA Member, e-mail: [email protected]/[email protected]

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Figure 1. Multiple sensor fusion architectures

estimation problems. Detailed descriptions of these architectures and their advantages and limitations are found in the literature.2, 5, 6 For the processing of multiple sensor data the Kalman filter (KF) has been used extensively in both the central and distributed architectures.7 The KF, however, gives rise to a high computational load when all sensor measurements are processed centrally. On the other hand, information filtering, which is essentially a Kalman filter expressed in terms of the inverse of the covariance matrix, has been widely used in multiple sensor fusion and decentralized estimation applications7 due to its advantages over the standard Kalman filter (KF); the structure of the information estimation is computationally simpler than the KF update equations, and it is easily initialized compared to the KF algorithms without knowing a priori information of the state of the systems. For nonlinear estimation problems, the information filter can be extended by applying a linearized estimation algorithm used in the extended Kalman filter (EKF), which is called the extended information filter (EIF).7, 8 Some of the drawbacks, however, inherited from the EKF still affect the EIF in terms of the truncation errors due to the approximation in the first and second order moments. The drawbacks due to the nonlinearities mentioned above can be compensated for by utilizing the sigma point filtering (SPF) algorithms including the unscented Kalman filter9 and the divided difference filter.10 Since the advent of the SPFs, many applications and extensions of the sigma point filtering also have been made in order to enhance the performance of the nonlinear estimation and filtering problems.11–14 For decentralized estimation, the sigma point information filter15 was proposed by utilizing the statistical linear regression Kalman filtering methodology16 for the linearization of system and measurement equations. On the other hand, a number of preprocessing stages are constructed to transform observation data into the liner form before it is fused in the sensor nodal filter, and the linear information filter is applied for the transformed observations.2 This, however, causes additional computational load due to the applied coordinate transformations. In this paper, the unscented information filtering (UIF)17 algorithm derived by embedding a statistical linear error propagation technique, based on the unscented transformation used in the SPF, into the EIF architecture for nonlinear estimation and multiple sensor fusion problems is illustrated first. Then, with an extension to the unscented information filter for the centralized multiple sensor estimation, distributed 2 of 15 American Institute of Aeronautics and Astronautics

unscented information filters are developed by embedding the unscented information filter into the hierarchical estimation architecture in distributed sensing networks. The motivation behind this paper comes from the fact that the unscented filtering provides a more accurate estimate than the EIF for nonlinear estimation problems, and the simplicity of the information filtering structure makes it suitable for multiple sensor estimation applications in distributed sensing nodes. The remainder of this paper is organized as follows. Section II presents an overview of the decentralized extended information filter. In section III, the unscented information filtering algorithm for nonlinear estimation is discussed, and then distributed unscented information filtering algorithms based on the unscented information filter are developed for multiple sensor estimation. Finally, simulation results are presented in Section V.

II.

Distributed Extended Information Filtering

In this section, multiple sensor fusion algorithms for both centralized and distributed estimation within the extended information filter framework1 is discussed. A.

Multiple Sensor Estimation

Consider discrete-time nonlinear dynamic and measurement equations xk+1 = f (xk , wk , k) ˜ zk = h(xk , k) + vk

(1)

where xk ∈
Yk|k = P−1 k|k = J k ∆

y ˆk|k = P−1 ˆk|k = Yk|k x ˆk|k k|k x

(2)

The update equations for the information matrix and the information state vector are obtained by T −1 Yk|k = P−1 k|k−1 + Hk Rk Hk

= Yk|k−1 + Ik y ˆk|k = y ˆk|k−1 + HTk R−1 ˆk|k−1 ] k [υ k + Hk x =y ˆk|k−1 + ik

(3)

(4)

where ik is the information state contribution and Ik is its associated information matrix defined by ∆

ik = HTk R−1 ˆk|k−1 ] k [υ k + Hk x ∆

Ik = HTk R−1 k Hk

(5) (6)

and Hk is the partial of the nonlinear measurement equation, and υ k is the innovation vector, υ k = ˜ zk − h(ˆ xk|k−1 , k). The predicted information state vector and covariance are obtained from y ˆk|k−1 = (Pk|k−1 )−1 x ˆk|k−1

(7)

Yk|k−1 = (Pk|k−1 )−1 = [Fk Pk−1|k−1 FTk + Qk ]−1

(8)

where Fk is a partial of the nonlinear dynamic equation. The extended information filter is further extended to multiple sensor estimation to increase the accuracy and reliability of the estimation. Suppose an observation vector ˜ zi,k is available from N different sensor sites and each sensor observes a common state according to the local observation model expressed by ˜ zi,k = hi,k (xk , k) + vi,k , i = 1, . . . , N 3 of 15 American Institute of Aeronautics and Astronautics

(9)

where the noise vector vi,k ∼ N (0, Ri,k ) is assumed to be white Gaussian and uncorrelated between sensors. The variance of a composite observation noise vector vk comprising a stacked vector of observations is T expressed in terms of the block diagonal matrix, Rk = diag ([R1,k , . . . , RN,k ]) . In the multiple sensor estimation, sensor failure detection and isolation capability to cut any anomalous sensor data resulting from sensor failure or from corruption of the sensor signals is important for reliable sensor fusion systems. The sensor data validation is achieved by comparing observations obtained from sensors with the predicted observations from measurement model, and accepting only those that lie within a predetermined error bound. A normalized innovation is defined as −1 d2k ≡ υ Tk (Pυυ υk (10) k ) υυ T where υ k is the innovation vector and Pυυ k is the innovation covariance matrix given by Pk = Hk Pk|k−1 Hk + Rk . This normalized innovation describes a quadratic ellipsoidal volume centered on the observation prediction, and if the innovations are zero mean and white then the normalized innovation is a χ2 random variable with nυ degrees of freedom. If an observation falls within this volume, then it is considered valid. For this test the equivalent statistical χ2 test is expressed by18

χ2 =

υ Tk (Pυυ k ) nυ

−1

υk

≥0

(11)

where nυ is the dimension of the innovation vector. This has non-negative value with a minimum value of zero, thus an upper limit threshold value on χ2 can be used to detect anomalous sensor information. A threshold value is chosen empirically such that the sensor data is rejected when χ2 > χ2max . With the validated sensor observations, the information contribution terms can be expressed by a linear combination of each local information state contribution ii and its associated information matrix Ii at the i sensor site N X ik = HTi,k R−1 ˆk|k−1 ] (12) i,k [υ i,k + Hi,k x i=1

Ik =

N X

HTi,k R−1 i,k Hi,k

(13)

i=1

The update equations for the multiple sensor estimation and data fusion are expressed by y ˆk|k = y ˆk|k−1 +

N X

ˆk|k−1 ] HTi,k R−1 i,k [υ i,k + Hi,k x

(14)

i=1

Yk|k = Yk|k−1 +

N X

HTi,k R−1 i,k Hi,k

(15)

i=1

Note that the multiple estimation derived based on the extended information filter has a centralized sensor fusion architecture where a master fusion filter produces a global estimate by using all the sensor information provided from each local filter. In the next subsection, a distributed extended information filtering algorithm is introduced. B.

Distributed Extended Information Filtering

A basic principle of the distributed approach lies on the fact that each of the local filters operates autonomously and independently with its own measurements. It is assumed that there is no sharing of measurement information among the local filters and a master fusion filter does not have direct access to the raw measurement feeding the local filters.19 The decentralized architecture shown in Fig. 1 has inherently a cascaded mode of operation, because the outputs of one or more of the local filters are acting as inputs to the master filter fusion filter. The outputs of the local filters are treated as measurements that are fed into the master fusion filter, and designing the integration filter using the estimation theory, i.e., a Kalman filter, plays an important role of the decentralized multiple sensor estimation to produce global estimates. If the local filters do not have access to each other’s measurements given in Eq.(9) and can do their own local predictions using Eqs. (7) and (8), each local filter is expressed by its respective error covariance and estimate in terms of information contributions as −1 −1 T P−1 i,k|k = Pi,k|k−1 + Hi,k Ri,k Hi,k

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(16)

Figure 2. A hierarchically distributed sensor fusion architecture without feedback from fusion filter

³ ´ −1 T x ˆi,k|k = Pi,k|k P−1 x ˆ + H R [υ + H x ˆ ] i,k i,k i,k|k−1 i,k|k−1 i,k i,k i,k|k−1

(17)

Note that the local estimates will be suboptimal, conditioned on their respective measurements, but not with respective to the combined measurements, and it is assumed that the local estimates are conditionally independent given their measurements. The global estimate and associated error covariance for the master fusion filter will be rewritten in terms of the computed estimates and covariances from the local filters by using the relations19 −1 −1 HTi,k R−1 (18) i,k Hi,k = Pi,k|k − Pi,k|k−1 ˆi,k|k−1 ] = P−1 HTi,k R−1 ˆi,k|k − P−1 ˆi,k|k−1 i,k [υ i,k + Hi,k x i,k|k x i,k|k−1 x

(19)

If the distributed filtering architecture is considered for a general N number of local filters and sensor measurements, the global estimates are computed by19 −1 P−1 k|k = Pk|k−1 +

N h i X −1 P−1 i,k|k − Pi,k|k−1

(20)

i=1

" x ˆk|k = Pk|k

P−1 ˆk|k−1 k|k−1 x

+

N ³ X

P−1 ˆi,k|k i,k|k x

−

P−1 ˆi,k|k−1 i,k|k−1 x

´

# (21)

i=1

Note that the global state update equation in the above decentralized filter can be written in terms of the information state vector and information covariance matrix as y ˆk|k = y ˆk|k−1 +

N X ¡

y ˆi,k|k − y ˆi,k|k−1

¢

(22)

i=1

Yk|k

N X £ ¤ = Yk|k−1 + Yi,k|k − Yi,k|k−1

(23)

i=1

Note that as the local filters can do their own local predictions and then repeat the cycle at step k + 1, the master filter can predict its global estimate and get new state x ˆk+1|k+1 and covariance Pk+1|k+1 for the next step.

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III.

Distributed Unscented Information Filtering

In this section, the unscented information filtering (UIF)17 algorithm is first derived by embedding a statistical linear error propagation approach based on the unscented transformation into the extended information filtering structure, and then it is further extended to distributed unscented information filtering algorithms which are designed by combing the unscented information filter with hierarchically distributed fusion architectures. A.

Unscented Information Filtering

An augmented state vector xak−1|k−1 ∈
³q ´ (na + λ) Pak−1|k−1 ³q ´i =x ˆak−1|k−1 − (na + λ) Pak−1|k−1

a Xi,k−1 =x ˆak−1|k−1 + a Xi,k−1

(25)

i

2

where λ = α (na + κ) − na is a scaling parameter with the constant parameters 0 ≤ α ≤ 1 and κ.9 The corresponding weights for the mean and covariance are defined by (m)

W0

= λ/(na + λ)

(m) Wi (c) W0 (c) Wi

= 1/ {2(na + λ)} ,

i = 1, . . . , 2na

= λ/(na + λ) + (1 − α2 + β) = 1/ {2(na + λ)} ,

(26)

i = 1, . . . , 2na

where β is a third parameter for incorporating extra higher order effects.20 Now, the information prediction equations are derived by implementing the unscented transformation9 into (7) and (8) as y ˆk|k−1 = Yk|k−1

2na X

(m)

Wi

x Xi,k

(27)

i=0

¡ ¢−1 Yk|k−1 = Pk|k−1

(28)

x x x w where Xi,k is the predicted sigma point vector, obtained from Xi,k =f (Xi,k−1 , Xi,k−1 , k − 1), and the predicted state covariance matrix is computed by

Pk|k−1 =

2na X

(c)

x x Wi [Xi,k −x ˆk|k−1 ][Xi,k −x ˆk|k−1 ]T

(29)

i=0

Recall that the derivation of the update equations in the EIF is based on the inverse of the covariance matrix and the linearized measurement equation. The UKF update equation, however, is not an explicit function of the linearized measurement matrix Hk , thus the UKF can not be directly embedded into the extended information update equations. Instead, based on the assumption that the nonlinear measurement equation in (1) can be mapped into a function of its statistical estimates such as mean and variances, the information update equations of the EIF are reformulated by utilizing the statistical linear error propagation methodology.14 First, using the error propagation the observation covariance and its cross-correlation covariance are approximated by PYY zk|k−1 )(zk − ˆ zk|k−1 )T ] k|k−1 = E[(zk − ˆ ' Hk Pk|k−1 HTk 6 of 15 American Institute of Aeronautics and Astronautics

(30)

Y PX ˆk|k−1 )(zk − ˆ zk|k−1 )T ] k|k−1 = E[(xk − x

(31)

' Pk HTk

where zk =h(xk ), and Hk is the linearized measurement matrix. Now, multiplying the predicted covariance and its inverse term on the right side of the information matrix equation in (6) and replacing Pk HTk with Y 17 PX k|k−1 leads to the following Ik = HTk R−1 k Hk T −T = (Pk|k−1 )−1 Pk|k−1 HTk R−1 k Hk|k−1 (Pk|k−1 ) (Pk|k−1 ) −1

= (Pk|k−1 )

(32)

Y −1 XY T −1 PX k Rk (Pk ) (Pk|k−1 )

Y where Pk|k−1 is obtained from (29), and the cross-correlation matrix PX k|k−1 is obtained by 2na X (c) x Y −x ˆk|k−1 ][Yi,k − ˆ zk|k−1 ]T PX = Wi [Xi,k k|k−1

(33)

i=1 x where Yi,k =h(Xi,k ), and the predicted measurement vector ˆ zk|k−1 is obtained by ˆ zk|k−1 = Similarly, the information state contribution ik can be rewritten by17

P2na i=0

(m)

Wi

Yi,k .

ik = HTk R−1 ˆk|k−1 ] k [υ k + Hk x T −T = (Pk|k−1 )−1 Pk|k−1 HTk R−1 x ˆk|k−1 ] k [υ k + Hk (Pk|k−1 ) (Pk|k−1 ) −1

= (Pk|k−1 )

−1 Y PX k|k−1 Rk [υ k

+

(34)

Y T −T (PX x ˆk|k−1 ] k|k−1 ) (Pk|k−1 )

Now, to make the information contribution equations compatible to those of the EIF, a pseudo measurement matrix Hk is defined as ¢−1 X Y ∆ ¡ Pk|k−1 (35) HTk = Pk|k−1 Then, in term of the pseudo measurement matrix Hk the information contribution equations are expressed by ik = HTk R−1 ˆk|k−1 ] k|k [υ k + Hk x

(36)

Ik = HTk R−1 k|k Hk

(37)

Based on the above results, it is seen that there exists a mapping which can approximate the nonlinear measurement equation in (1) in terms of the statistical error variances and its mean by zk = h(xk , k) ' Hk xk + u ˜k

(38)

where u ˜ k = h(ˆ xk|k−1 ) − Hk x ˆk|k−1 is a measurement residual term. The mapping in (38) is verified by showing that the information contribution terms in (36) and (37) are obtained directly by implementing the transformed function of (38) into (8) and (6). Finally, the update equations for the unscented information filtering is obtained by applying (3) and (4) in terms of the information terms, and the updated state estimate x ˆk|k is computed by x ˆk|k = Pk|k y ˆk|k using (2). B.

Multiple Sensor Estimation

The unscented information filtering can be further extended to multiple sensor estimation in a centralized fusion architecture to increase the reliability of the estimation. Suppose a observation vector ˜ zi,k+1 is available from N different sensors, and each sensor observes a common state according to the local observation model expressed by ˜ zi,k = Hi,k xk + u ˜ i,k + vi,k (39) where the noise vector vi,k is assumed to be white Gaussian and uncorrelated between sensors. The variance of a composite observation noise vector vk comprising a stacked vector of observations is expressed in terms 7 of 15 American Institute of Aeronautics and Astronautics

of the block diagonal matrix, Rk = diag([RT1,k , . . . , RTN,k ])T . Then, each local information state contribution ii,k and its associated information matrix Ii,k at the i sensor site are computed by ˆk|k−1 ] ii,k = HTi,k R−1 i,k [υ i,k + Hi,k x

(40)

Ii,k = HTi,k R−1 i,k Hi,k

(41)

Since the information contribution terms have group-diagonal structure in terms of the innovation and measurement matrix, the update equations for the multiple sensor estimation and data fusion are expressed by a linear combination of the local information contribution terms as N X

y ˆk|k = y ˆk|k−1 +

ii,k

(42)

i=1

Yk|k = Yk|k−1 +

N X

Ii,k

(43)

i=1

The prediction equations are calculated by using (27) and (28) for the multiple senor estimation problem. C.

Distributed Unscented Information Filtering

In this subsection, a distributed unscented information filtering (DUIF) for a hierarchical sensor fusion with no feedback from the master fusion filter is developed by embedding a statistical linear error propagation approach based on the unscented transformation into the decentralized extended information filtering architecture. In addition, the DUIF is further extended to a closed-loop distributed sensor fusion algorithm where the information from the global sensor fusion filter is shared with each local filter. Suppose that the local filters do not have access to each other’s measurements given in Eq. (39) and can do their own local predictions computed from Eqs. (27) and (29), and a master fusion filter does not have direct access to the raw measurement feeding the local filters. Each local filter is expressed by its respective error covariance and estimate in terms of each local information state contribution ii and its associated information matrix Ii at the i sensor site as T −1 −1 P−1 i,k|k = Pi,k|k−1 + Hi,k Ri,k Hi,k

(44)

³ ´ T −1 x ˆi,k|k = Pi,k|k P−1 x ˆ + H R [υ + H x ˆ ] i,k i,k i,k|k−1 i,k i,k i,k|k−1 i,k|k−1

(45)

From the above relation, each local information state contribution ii and its associated information matrix Ii at the i sensor site are rewritten in terms of the computed estimates and covariances from the local filters −1 −1 HTi,k R−1 i,k Hi,k = Pi,k|k − Pi,k|k−1

(46)

HTi,k R−1 ˆi,k|k−1 ] = P−1 ˆi,k|k − P−1 ˆi,k|k−1 i,k [υ i,k + Hi,k x i,k|k x i,k|k−1 x

(47)

Finally, the global estimates of the decentralized unscented filtering architecture for a general number of local filters and sensor measurements are computed by −1 P−1 k|k = Pk|k−1 +

N h i X −1 P−1 i,k|k − Pi,k|k−1

(48)

i=1

" x ˆk|k = Pk|k

P−1 ˆk|k−1 k|k−1 x

+

N ³ X

P−1 ˆi,k|k i,k|k x

−

P−1 ˆi,k|k−1 i,k|k−1 x

´

# (49)

i=1

and in terms of the information state vector and information covariance matrix, they are given by y ˆk|k = y ˆk|k−1 +

N X ¡

y ˆi,k|k − y ˆi,k|k−1

¢

i=1

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(50)

Figure 3. A hierarchically distributed sensor fusion architecture using unscented information filtering with feedback from master fusion filter

Yk|k = Yk|k−1 +

N X £ ¤ Yi,k|k − Yi,k|k−1

(51)

i=1

If output data from each node is only periodically communicated to the higher levels and the master fusion filter at a rate slower than the sensor observation rates, considerable mitigation not only in communication bandwidth but also computational workload can be achieved. In this feedforward distributed sensor fusion architecture, however, estimates from local sensor nodes might be correlated due to common prediction errors resulting from a common process noise model.5, 22 Therefore, it is necessary to remove the common information to minimize the correlation effect between the local estimates, but identifying the common information is not as straightforward. The decorrelation might be achieved by applying the covariance interaction method.23 Note that the above feedforward distributed multisensor fusion algorithm that is based on the unscented information filter can be extended further to the federated multiple sensor fusion architecture21 by letting the global information that is communicated back to the local filter be divided into portions and allowing the divided information is shared along the local filters. In this closed-loop distributed filtering architecture, the master fusion filter maintains global estimates at the fusion center and communicates its global estimates back to the local filters, and the local filters have better estimates than they would have without feedback, which is described in Fig. 3. It requires, however, more high communication bandwidth due to feeding back the prior information x ˆk+1|k and Pk+1|k to the local filters. Since there is feedback, the master fusion node has to remove its own information sent previously before integrating the local estimates. Suppose the local prior information in the local filters is equal to the predicted one communicated from the global filter −1 P−1 i,k|k−1 = Pk|k−1 x ˆi,k|k−1 = x ˆk|k−1

(52)

T −1 −1 P−1 i,k|k = Pk|k−1 + Hi,k Ri,k Hi,k

(53)

h i T −1 x ˆ + H R ˜ z x ˆi,k|k = Pi,k|k P−1 i,k k|k−1 i,k i,k k|k−1

(54)

Then, each local filter is expressed by

Then, the global estimates in the fusion filter are computed by substituting into Eqs. (??) and (51) −1 Pk|k =

N X

−1 P−1 i,k|k − (N − 1)Pk|k−1

i=1

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(55)

" x ˆk|k = Pk|k

N X

# P−1 ˆi,k|k − (N − 1)P−1 ˆk|k−1 i,k|k x k|k−1 x

(56)

i=1

and in terms of the information state vector and information matrix, they are expressed by y ˆk|k =

N X

y ˆi,k|k − (N − 1)ˆ yk|k−1

(57)

Yi,k|k − (N − 1)Yk|k−1

(58)

i=1

Yk|k =

N X i=1

Now consider a special case where the local prior information in the local filters is equal to the predicted one communicated from the global filter −1 P−1 i,k|k−1 = βi Pk|k−1

(59)

x ˆi,k|k−1 = x ˆk|k−1

where βi is the information-sharing factor distributed to each local filter. In this paper, for simplicity, βi = 1/N is considered where N is the number of sensor nodes. Then, each local filter is expressed by 1 −1 P + HTi,k R−1 i,k Hi,k N k|k−1

P−1 i,k|k = · x ˆi,k|k = Pi,k|k

1 −1 P x ˆk|k−1 + HTi,k R−1 zi,k i,k ˜ N k|k−1

(60) ¸

In the fusion filter the global estimates are expressed by ·N ¸ P −1 −1 −1 −1 Pk|k = Pk|k−1 + Pi,k|k − Pk|k−1 =

i=1

N P i=1

(61)

(62)

P−1 i,k|k

· ¸ N P −1 −1 x ˆk|k = Pk|k P−1 x ˆ + P x ˆ − P x ˆ k|k−1 k|k−1 i,k|k i,k|k k|k−1 k|k−1 ·N ¸ i=1 P −1 = Pk|k Pi,k|k x ˆi,k|k

(63)

i=1

Finally, these are rewritten in terms of the information state vector and information matrix as y ˆk|k =

N X

y ˆi,k|k

(64)

Yi,k|k

(65)

i=1

Yk|k =

N X i=1

Note that the effects of the information sharing due to the prediction of the common process noise can 21 be minimized by controlling the information-sharing factor βi . In the federated PN filter, the information sharing factor is designed by using the axiom of conservation of information, i=1 βi = 1. In the distributed estimation with feedback architecture, it is also available to save required communication bandwidth by reducing the communication between the local sensor nodes and the master fusion filter, and by allowing the fusion nodes to communicate at certain periodic rates. The proposed distributed multiple sensor fusion architectures based on the unscented information filter are fault-tolerant and computationally efficient compared to the centralized filters and provide a simple and natural means of integrating multiple sensor information with enhanced estimation accuracy.

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IV.

Simulation Example

Consider the target tracking of a re-entry vehicle entering into an atmosphere from space. The state vector x ∈ <5×1 consists of the position, velocity, and a parameter related to the aerodynamic force, x = [x y x˙ y˙ γ]T . The equations of motion of the vehicle are expressed by ¾ ½ (r0 − krk) µ v˙ = βγ exp kvk v − 3 r + wa h0 krk (66) γ˙ = w3 p p where krk = x2 + y 2 is the distance from the center of the earth, kvk = x˙ 2 + y˙ 2 is the speed of the vehicle, and βγ = β0 exp {γ}. The parameter values used in this study are β0 = −0.59783, h0 = 13.406km, µ = 3.9860 × 105 km3 /s2 , and r0 = 6378km. The process noise vector is defined w = [waT , w3 ]T with zeromean white Gaussian processes. The motion of the vehicle is measured by radars located at each position, (xm,s , ym,s ), and the observations of each radar consists of a range and bearing angle obtained at 10Hz. The measurement equations of each sensor are given by q 2 2 rm,s = (x − xm,s ) + (y − ym,s ) + v1,s ¶ µ (67) y − ym,s + v2,s θm,s = tan−1 x − xm,s where vk,s =[v1,s v2,s ]T is the measurement noise vector. The initial true state vector for the target trajectory and the initial estimate for the filters are given T T by x0 = [6500.4, 349.14, −1.8093, −6.7967, 0.6932] and x ˆ0 = [6499.94, 349.11, −1.8091, −6.7962, 0.69315] . The a priori state covariance matrix is given by P0 = diag([10−6 10−6 10−6 10−6 1]), and the process noise variance matrix is set to Q(t) = diag([2.4064 × 10−5 2.4064 × 10−5 10−6 ]). Tracking radar sites are located at (xm,1 = 6374 km, ym,1 = 0.0 km) and (xm,2 = 6375 km, ym,2 = 0.0 km) with each measurement noise variance, R1,k = diag([0.032 km2 0.022 deg 2 ]) and R2,k = diag([0.042 km2 0.022 deg 2 ]), respectively. 3

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Time (sec) Figure 4. Absolute magnitude of position errors from multiple sensor estimation

The performance of the proposed UIF is compared with the EIF in terms of the estimation accuracy for multiple sensor estimation. The parameters for the UIF are α = 10−3 and β = 2. Fig. 4 describes the position estimation errors from the EIF and the UIF, where the convergence of the UIF is faster than the EIF and also the estimation accuracy of the UKF is better than that of the EIF. This is because the UIF is based on the unscented transformation which provides more accurate prediction performance than the first-order Taylor series expansion. A similar result on the velocity estimation is shown in Fig 5. The parameter estimate related to the atmospheric drag is shown is Fig. 6, where accurate and fast converged 11 of 15 American Institute of Aeronautics and Astronautics

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parameter estimation is achieved by the UIF. The simulation results indicate that the performance of the UIF is superior to the EIF in terms of not only the estimation accuracy, but also the fast convergence. Figures 7-9 show the performance comparison between the proposed DUIF is compared with the DEIF in terms of the estimation accuracy for distributed estimation with feedback information from the master fusion filter. The same parameters are used for the UIF as in the previous multiple sensor estimation. Fig. 7 is the plot for the position estimation errors from the DEIF and the DUIF, where it is shown that the convergence of the DUIF is faster than the EIF and also the estimation accuracy of the DUKF is better than that of the DEIF. This comes from the fact that the UIF provides more accurate prediction performance than the EIF. A similar result on the velocity estimation is obtained as shown in Fig 8. The parameter estimation result as to the atmospheric drag is shown is Fig. 9, where more accurate and fast converged parameter estimation is achieved by the DUIF as expected. From the simulation results it is deduced that the DUIF provides enhanced performance over the DDEIF in terms of not only the estimation accuracy, but also the fast convergence in the distributed multiple sensor estimation. The advantages of the UIF make it useful to not only multiple sensor estimation but also distributed sensor network applications. 3

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Figure 8. Absolute magnitude of velocity errors from distributed estimation

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Figure 9. Absolute magnitude of ballistic parameter errors from distributed estimation

V.

Conclusion

A new unscented information fusion algorithm is derived by embedding the unscented transformation into the extended information filtering architecture by utilizing the statistical linear error propagation method. The motivation behind this paper comes from the fact that the unscented filtering provides a more accurate estimate than the extended information filter for nonlinear estimation problems, and the structure simplicity of the information filtering makes it suitable for multiple sensor systems. As extension to the unscented information filter used in the centralized multiple sensor estimation, the distributed unscented information filtering algorithms are developed by embedding the unscented information filter into the hierarchical sensor fusion architectures for distributed sensing networks. The proposed centralized and distributed multiple sensor fusion architectures based on the unscented information filter are computationally efficient compared to the centralized filters and provide a simple and natural means of integrating multiple sensor information with enhanced estimation accuracy for nonlinear systems. The advantages of the unscented information fusion filters make them useful to not only multiple sensor estimation in distributed sensor network applications.

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11 D.-J. Lee, and K. T. Alfriend, “Sigma Point Filtering for Sequential Orbit Estimation and Prediction,” Journal of Spacecraft and Rockets, vol. 44, no. 2, pp. 388–398, March-April 2007. 12 Tang, Z. and Ozguner, U., “Motion Planning for Multitarget Surveillance with Mobile Sensor Agents,” IEEE Transactions on Robotics, vol. 21, no. 5, pp. 898–908, Oct. 2005. 13 D.-J. Lee, and K. T. Alfriend, “Adaptive Sigma Point Filtering for State and Parameter Estimation,” AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence, Rhode Island, Aug. 2004, AIAA 2004–5101. 14 G. Sibley, G. S. Sukhatme, and L. Matthies, “The Iterated Sigma Point Filter with Applications to Long Range Stereo,” Online Proceedings of the 2nd Robotics: Science and Systems Conference, Philadelphia, Pennsylvania, Aug. 16-19, 2006. 15 T. Vercauteren, and X. Wang, “Decentralized Sigma-Point Information Filters for Target Tracking in Collaborative Sensor Networks,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2997–3009, Aug. 2005. 16 T. Lefebvre, H. Bruyninckx, and J. De Schutter, “Comment on ”a new method for the nonlinear transformation of means and covariances in filters and estimators,” IEEE Transactions on Automatic Control, vol. 47, no. 8, Aug. 2002, pp. 1406–1408. 17 D.-J. Lee, “Nonlinear Estimation and Multiple Sensor Fusion Using Unscented Information Filtering,” IEEE Sigal Processing Letters, Dec, 2008 (to be published). 18 Y. Bar-Shalom, X.-R. Li, and T. Kirubarajan, Estimation with Applications to Tracking and Navigation, 1st ed. John Wiley & Sons, Inc., New York, 2001. 19 H. R. Hashemipour, S. Roy, and A. J. Laub, “Decentralized Structures for Parallel Kalman Filtering,” IEEE Transactions on Automatic Control, vol. 33, no. 1, pp. 88-93, 1988. 20 R. van der Merwe, E. A. Wan, and S. J. Julier, “Sigma-Point Kalman Filters for Nonlinear Estimation and Sensor Fusion: Applications to Integrated Navigation,” AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Rhode Island, Aug. 2004, AIAA 2004-5120. 21 N. A., Carlson, “Federated Square Root Filter for Decentralized Parallel Processes,” IEEE Transactions on Aerospace and Electronic Systems, vol. 26, no. 3, pp. 517-525, 1990. 22 Y. Bar-Shalom, L. Campo, “The Effect of the Common Process Noise on the Two-Sensor Fused-Track Covariance,” IEEE Transactions on Aerospace and Electronic Systems, vol. 22, no. 6, pp. 803-805, Nov. 1986. 23 J. K. Uhlmann, “Covariance Consistency Methods for Fault-Tolerant Distributed Data Fusion,” Information Fusion, vol. 4, pp. 201-215, 2003.

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