Neural Network Aided Adaptive Extended Kalman Filtering Approach for DGPS Positioning

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1 THE JOURNAL OF NAVIGATION (4), 7, f The Royal Institute of Navigation DOI: 1.117/ Printed in the United Kingdom Neural Network Aided Adaptive Extended Kalman Filtering Approach for DGP Positioning Dah-Jing Jwo and Hung-Chih Huang (National Taiwan Ocean University) ( djjwo@mail.ntou.edu.tw) The extended Kalman filter, when employed in the GP receiver as the navigation state estimator, provides optimal solutions if the noise statistics for the measurement and system are completely known. In practice, the noise varies with time, which results in performance degradation. The covariance matching method is a conventional adaptive approach for estimation of noise covariance matrices. The technique attempts to make the actual filter residuals consistent with their theoretical covariance. However, this innovation-based adaptive estimation shows very noisy results if the window size is small. To resolve the problem, a multilayered neural network is trained to identify the measurement noise covariance matrix, in which the back-propagation algorithm is employed to iteratively adjust the link weights using the steepest descent technique. Numerical simulations show that based on the proposed approach the adaptation performance is substantially enhanced and the positioning accuracy is substantially improved. KEY WORD 1. GP.. Extended Kalman filter. 3. Adaptive. 4. Neural network. 1. INTRODUCTION. The well-known Kalman filter (Gelb, 1974; Brown and Huang, 1997), which provides optimal (from the viewpoint of minimum mean square error) estimate of the system state vector, has been widely applied to the fields of navigation such as GP receiver position/velocity determination, and the integrated navigation system design. As for the GP navigation schemes, the least squares and Kalman filtering approaches have been commonly used to estimate the user position as well as the velocity. In general, results based on the Kalman filter, due to its characteristics that attempt to mitigate high frequency noise, shows better performance than those based on the least squares approach. In practice, the Kalman filter will provide the optimal result if the complete a priori knowledge of process noise covariance matrix and measurement noise covariance matrix are available. Therefore, a lot of effort has been made to improve the estimation of the covariance matrices. Mehra (197) classified the adaptive approaches into four categories: Bayesian, maximum likelihood, correlation and covariance matching. These methods can be applied to the Kalman filtering algorithm to realize adaptive Kalman filtering (Mehra, 197, 1971, 197; Mohamed and chwarz, 1999;

2 4 DAH-JING JWO AND HUNG-C HIH HUANG VOL. 7 Hide, Moore and mith, 3). However, the first two of the above-mentioned methods are computationally demanding so that their practical applications are limited. As for the correlation methods, a set of equations is derived to relate the functions to the unknown parameter. The popular innovation-based adaptive estimation method, called the covariance matching technique, attempts to make the filter residuals consistent with their theoretical covariances. Results from such innovationbased adaptive estimation are very noisy if the window size is small. On the other hand, the transient time needed to reach the converged value will increase if the window size is increased. Artificial neural networks (Rosenblatt, 196; Widrow and Lehr, 199; Chester, 1993; Haykin, 1999), or simply neural networks (NNs) are trainable, dynamic systems that can estimate input-output functions. The NN is motivated by their ability to approximate an unknown nonlinear input-output mapping through supervised training. They have been applied to a wide variety of problems since they are a modelfree estimator, i.e., without a mathematical model. The back-propagation (BP) neural network has been the most popular learning algorithm throughout all neural applications. BPNN is a neural system with a back-propagation algorithm that can learn input-output functions from a series of samples. It is a gradient-based algorithm, in the sense that the weight update is performed along the direction of the gradient of an appropriate error function. The BPNN is simple and requires a minimal amount of storage. The neural network approach will be employed to aid the Kalman filter for estimating the time varying variances. The noise covariance is a complicated mapping with the innovation. The innovation produced by the Kalman filter is used as inputs of the neural network, and the desired outputs are the corresponding noise spectral strength. The neural network is then trained off-line using the steepest descent (D) technique to minimize the differences between the outputs of NN and the desired outputs. Consequently, the estimation accuracy of the noise parameters can be substantially improved when the NN is utilized to correctly estimate the noise covariance matrices in the adaptive Kalman filter mechanism. This paper is organized as follows. In ection, the preliminary background on GP navigation using extended Kalman filter (EKF) is briefly reviewed. The conventional adaptive extended Kalman filter (AEKF) approaches are introduced in ection 3. In ection 4, the proposed neural network aided AEKF algorithm is presented and in ection, the performance by applying NN aided AEKF to DGP positioning solution is presented. The conclusion is given in ection 6.. GP NAVIGATION UING EXTENDED KALMAN FILTER. The most commonly used approaches for the GP navigation solutions are the least squares and the extended Kalman filtering approaches (Axelrad, 1996). The Kalman filtering is recognised as one of the most powerful state estimation techniques. The purpose of the Kalman filter is to provide the estimation with minimum error variance. It has been successfully applied to the integrated GP/IN navigation system design, stand-alone GP receiver position/velocity determination, and the radar target tracking. GP navigation algorithms using extended Kalman filter (EKF) is briefly reviewed for convenience.

3 NO. 3 KALMAN FILTERING APPROACH FOR DGP P OITIONING 41 The process model and measurement model are represented as Process model: _x=fx+gu (1a) Measurement model: z=hx+v (1b) where the vectors u(t) and v(t) are both white noise sequences with zero means and mutually independent: E[u(t)u T (t)]=qd(txt) E[v(t)v T (t)]=rd(txt) E[u(t)v T (t)]= (a) (b) (c) where d(t) is the Dirac delta function, E[. ] represents expectation, and superscript T denotes matrix transpose. Expressing Equations (1a) and (1b) in discrete-time equivalent form leads to x k+1 =W k x k +G k w k (3a) z k =H k x k +v k (3b) where the state vector x k s< n, process noise vector w k s< n, measurement vector z k s< m, and measurement noise vector v k s< m. In Equation (3), both the vectors w k and v k are zero mean Gaussian white sequences having zero cross correlation with each other: E[w k w T i ]= Q k, i=k (4a), ilk E[v k v T i ]= R k, i=k (4b), ilk E[w k v T i ]= for all i and k (4c) where Q k is the process noise covariance matrix, R k is the measurement noise covariance matrix, and W k =e FDt is the state transition matrix. The Kalman filter algorithm is summarized as follow: Prediction steps/time update equations: ^x x =W k+1 k^x k () P x k+1 =W kp k W T k +G kq k G T k (6) Correction steps/measurement update equations: K k =P x k HT k [H kp x k HT k +R k] x1 (7) ^x k =^x x k +K k[z k xh k^x x k ] (8) P k =[IxK k H k ]P x k (9) Equations ( 6) are the time update equations of the algorithm from k to step k+1, and Equations (7 9) are the measurement update equations. These equations

4 4 DAH-JING JWO AND HUNG-C HIH HUANG VOL. 7 incorporate a measurement value into a priori estimation to obtain an improved a posteriori estimation. In the above equations, P k is the error covariance matrix defined by E[(x k xx^k) (x k xx^k) T ], in which x^k is an estimation of the system state vector x k, and the weighting matrix K k is generally referred to as the Kalman gain matrix. The Kalman filter algorithm starts with an initial condition value, x^x and P x. When new measurement z k becomes available with the progression of time, the estimation of states and the corresponding error covariance would follow recursively ad infinitum. The extended Kalman filtering is a nonlinear version of Kalman filtering, which deals with the case described by the nonlinear stochastic differential equations: _x=f(x, t)+u(t) (1a) z=h(x, t)+v(t) (1b) The algorithm for the extended Kalman filtering is essentially similar to that of Kalman filtering, except that some modifications are made. Firstly, the state update equation becomes ^x k =^x x k +K k[z k x^z x k ] (11) where and ^x x k =f(^x x k, k) (1) ^z x k =h(^x x k, k) (13) econdly, the linear approximation equations for process and measurement are obtained through the relations x=^x x k x=^x x k More detailed discussion can be referred to Gelb (1974) and Brown and Huang (1997). The flow chart for the GP navigation solutions using extended Kalman filter approach is shown inside the right-hand-side block of Figure. (1) 3. CONVENTIONAL ADAPTIVE EXTENDED KALMAN FILTER (AEKF). The implementation of Kalman filter requires the a priori statistical knowledge of the process noise and measurement noise. Poor knowledge of the noise statistics may seriously degrade the Kalman filter performance and even provoke the filter divergence. To fulfil the requirement, an adaptive Kalman filter can be utilized as the noise-adaptive filter to estimate the noise covariance matrices. Mehra (197) classified the adaptive approaches into four categories: Bayesian, maximum likelihood, correlation and covariance matching. The innovation sequences have been utilized by the correlation and covariance-matching techniques to

5 NO. 3 KALMAN FILTERING APPROACH FOR DGP P OITIONING 43 estimate the noise covariances. The basic idea behind the covariance-matching approach is to make the actual value of the covariance of the residual consistent with its theoretical value. From the incoming measurement z k and the optimal prediction x^kx obtained in the previous step, the innovations sequence is defined as n k =z k xh k^x x k (16) The innovation represents the additional information available to the filter as a consequence of the new observation z k. The weighted innovation, K k (z k xh k x^kx ), acts as a correction to the predicted estimate x^kx to form the estimation x^k. ubstituting the measurement model, Equation (3b), into Equation (16) yields n k =H k (x k x^x x k )+v k (17) which is a zero-mean Gaussian white noise sequence. By taking variances on both sides of Equation (17), we have the theoretical covariance C k =H k P x k HT k +R k (18) This leads to an estimate of R k : ^R k =^C k xh k P x k HT k (19) where ^C k is the statistical sample variance estimate of C k. Matrix ^C k can be computed through averaging inside a moving estimation window of size N (Mohamed and chwarz, 1999) X k ^C k = 1 n k n T k () N j=j where j =kxn+1 is the first sample inside the estimation window. If the window size is too small, the estimation of measurement covariance will be very noisy. On the other hand, if a large window size is utilized, the estimation of measurement covariance will be smoother, however, at the expense of long transient time. Usually, the window size N is chosen empirically to give some statistical smoothing. In some practical applications, there are instances in which the noise spectral amplitudes rapidly change; in those cases the conventional approach will not suffice the adaptation requirement. 4. THE PROPOED NEURAL NETWORK AIDED AEKF CHEME. Neural networks have been applied to a wide variety of problems. They have been studied for more than three decades since Rosenblatt first applied single-layer perceptrons to pattern classification learning in the late 19s. NN is a network structure consisting of a number of nodes connected through directional links. Each node represents a process unit, and the links between nodes specify the casual relationship between the connected nodes. The learning rule specifies how these parameters should be updated to minimize a prescribed error measure, which is a mathematical expression that measures the discrepancy between the network s actual output and a desired output. The multi-layered neural network is a well-known neural model. The D technique is employed to adjust the link weights so that the differences between the NN outputs

6 44 DAH-JING JWO AND HUNG-C HIH HUANG VOL. 7 and the desired outputs are minimized. In the forward pass, the link weights are fixed and the response of the NN is computed by subjecting the network to a prescribed set of input patterns. In the backward pass, the adjustments to the link weights are computed for the purpose of minimizing a cost function defined as the sum of squared errors Back-propagation algorithm. The error signal at the output of neuron j at iteration n (i.e., presentation of the nth training example) is defined by e j =d j (n)xy j (n) where neuron j is an output node. The instantaneous value of the error energy for neuron j is defined as 1 e j (n). The instantaneous value of total error energy value, E(n), is obtained by summing 1 e j (n) over all neurons in the output layer: E(n)= 1 X e j (n) (1) where the set L includes all the neurons in the output layer of the network. Let P denote the total number of patterns contained in the training set, the average percentage error is defined as Ean= 1 X P d j xy j () P d j j=1 The induced local field n j (n) with neuron j is n j (n)= Xm i= jl w ji (n)y i (n) (3) where m is the total number of inputs (including the bias) applied to neuron j. Hence the function signal y j (n) appearing at the output of neuron j at iteration n is y j (n)= j (n j (n)) (4) The correction Dw ji (n) applied to w ji (n) is defined by the delta rule: Dw ji (n)=gd j (n)y i (n)+adw ji (nx1) () where g is the learning rate; a is a positive number called the momentum coefficient; y i (n) is the output of the ( jx1)th layer and the local gradient d j (n) is defined as d j (n)=e j (n) _ j (n j (n)) (6) When neuron j is a hidden node, we may redefine the local gradient d j (n) for hidden j as d j (n)= _ j (n j (n)) X k d k (n)w kj (n) (7) As for the activation function, the sigmoidal (or logistic) function is selected. This form of sigmoidal nonlinearity in its general form is defined as 1 j (n j (n))= (8) 1+ exp [xan j (n)]

7 NO. 3 KALMAN FILTERING APPROACH FOR DGP P OITIONING 4 function signals error signals bias Cv(t) w 11 1 w 1N 1 w 1(N+1) 1 ˆR 11 Cv(t-1) ˆR w j1 w jn w j(n+1) Cv(t-p) j j w j(p+)1 w jjn w jj(n+1) j Rˆ jj Input layer 1st hidden layer Nth hidden layer Output layer Figure 1. The topology of a multilayered neural network. where a> and x <n j (n)<. Differentiating Equation (8) with respect to n j (n) gives _ j (n j (n))=y j (n)[1xy j (n)] (9) igmoid hidden and output units usually use a bias or threshold term in computing the net input to the unit. A bias term can be treated as a connection weight from a special unit with a constant activation value. The topology of a multi-layered neural network is shown in Figure Input-output functional mapping. Off-line training of neural network is conducted using the D technique to minimize the differences between the outputs of NN and the desired outputs. During the training phase, the innovation C k produced by the Kalman filter is employed as the input to the NN. Referring to Figure 1, the inputs of neural network are the innovations from the present instant to time (txp). The neural network output vectors ideally describe the actual measurement noise strength. The NN employed in the present work is made of five layers: one input layer, three hidden layers, and one output layer. Due to the complexity of the present problem, five layers are required to accomplish the mapping. The topology has 1 neurons in the input layer. Three hidden layers of sigmoid transfer function are composed of 3 neurons in the first hidden layer, 18 neurons in the second layer, and 9 neurons in the third hidden layer. The bias (or threshold), as one of the inputs, is added into both the hidden neurons and output neurons. More inputs for the NN may be used at the expense of large time consumption for convergence. In each layer of the NN, the actual outputs are calculated using the sigmoidal nonlinear function and used as inputs to the next layer. At the time of recall, when the AEKF receives the measurement z k, it provides the estimations of the state vector and the z k xz^k to calculate the innovation. When the input nodes receive the innovation, the

8 46 DAH-JING JWO AND HUNG-C HIH HUANG VOL. 7 Figure. Flowchart for the neural network aided adaptive extended Kalman filter. appropriate covariance R^ k is obtained. Thus, the extended Kalman filter is provided with the adaptive capability for estimation by combining the filter and the NN. A flow chart of the NN aided AEKF is presented as in Figure, in which there are two main blocks. The right-hand side block represents the covariance identification loop using NN, while the left-hand side block is the standard navigation loop using EKF A simple example for algorithm validation. For simplicity, yet without loss of generality, a simple double-integrator model is employed to test the adaptive capability. This simple example can be applied to the state estimation of one-dimensional trajectory with a constant-velocity model such as in the radar target tracking. Consider the continuous-time double integrator model governed by Equations (1a) and (1b), it is seen that F= 1 ; G= ; H=[1 ] 1

9 NO. 3 KALMAN FILTERING APPROACH FOR DGP P OITIONING Error average epochs x 1 4 Figure 3. Convergence history of error average for the double integrator model. and these signals satisfy the following: E[u(t)u T (t)]=qd(txt); E[v(t)v T (t)]=rd(txt); E[u(t)v T (t)]=. Expressing the models in discrete-time equivalent form, the corresponding W k, Q k and H are found to be W k = 1 Dt Dt 3 ; Q 1 k = Dt Dt 3 7 q; H=[ 1 ] (3) Dt In this example, q value is assumed known and the measurement noise variance r is to be identified. The r values selected for testing adaptation capability covers four different values, which are, 8, 6, and 1 m, respectively. The NN has a structure (where the numbers represent the numbers of input layer neuron, three hidden layers, and output layer, respectively) and the following parameter values are used: bias=x1; learning rate g=. 3; momentum coefficient a=. 3. The convergence history of error average for the double integrator example is shown in Figure 3. Comparison of the adaptation results between conventional and proposed approaches for the double integrator example is provided in Figure 4. It is seen that substantial improvement on noise identification capability has been achieved by using the proposed approach as compared to the conventional covariance-matching method.. APPLYING NN AIDED AEKF TO DGP POITIONING OLUTION. When selecting the EKF as the navigation state estimator in the

10 48 DAH-JING JWO AND HUNG-C HIH HUANG VOL. 7 1 Covariance matching method Noise variance (m ) 1 Exact value The proposed approach Time (s) Figure 4. Comparison of measurement noise (r) identification results for the double integrator example. GP receiver, and using b and d to represent the GP receiver clock bias and drift, the differential equation for the clock error is written as _b=d+u b _d=u d (31) where u b yn(, f ) and u d yn(, g ) are independent Gaussianly distributed white sequences. The dynamic process of the GP receiver in medium dynamic environment can be represented by the PV model (Brown and Huang, 1997; Axelrad, 1996): _x 1 1 _x _x 3 1 _x 4 _x = 1 6 _x _x _x 8 x 1 x x 3 x 4 x x 6 x 7 x 8 u u u u 7 u 8 where x 1, x 3, x represent the east, north, and vertical position; x, x 4, x 6 represent the east, north, and vertical velocity; and x 7 and x 8 represent the receiver clock offset (3)

11 NO. 3 KALMAN FILTERING APPROACH FOR DGP P OITIONING 49 and drift errors, respectively. The state transition matrix for the model can be found to be 3 1 Dt 1 1 Dt 1 W k = 1 Dt (33) Dt The process noise covariance matrix is as follows: Dt 3 p Dt p 3 Dt p p Dt Dt 3 p Dt p 3 Dt p p Dt Q k = Dt 3 p Dt p 3 Dt p p Dt f Dt+ Dt 3 6 g 4 3 Dt g 3 Dt g 7 g Dt (34) Let each of the white-noise spectral amplitudes that drive the random walk position states be p =1. (m/sec )/rad/sec. Furthermore, let the clock model spectral amplitudes be f =. 4(1 x18 )sec and g =1. 8(1 x18 )sec x1. These spectral amplitudes are used to evaluate the Q k parameters in Equation (34). In the case that only the pseudorange observables are available, the measurement equation based on n observables leads to 6 4 r 1 r. r n ^r 1 h x (1) h y (1) h z (1) 1 7 x ^r h x () h y () h z () = ^r n h (n) x h (n) y h (n) z 1 6 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4 H k fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} Z k x 1 x x 3 x 4 x x 6 x 7 x Assuming measurement errors among satellites are uncorrelated, we have 3 r r1 r r R k = r rn n r1 n r.. n rn 3 7 (3) (36)

12 46 DAH-JING JWO AND HUNG-C HIH HUANG VOL. 7 Reference measurement prediction zˆ k = h(ˆ x k ) xˆ GP pseudorange z k + - AEKF Rˆ k z k zˆ k NN-based noise Covariance estimate C υk Innovation average computation Figure. GP navigation solutions using neural network aided adaptive extended Kalman filter Error average epochs x 1 4 Figure 6. Convergence history of error average for the DGP positioning example. Based on the discussion provided above, the neural network aided adaptive extended Kalman filter as a navigation state estimator can now be established. Figure illustrates the system architecture for performing GP navigation solutions using NN aided adaptive EKF.

13 NO. 3 KALMAN FILTERING APPROACH FOR DGP P OITIONING East error (m) Time (sec) (a) East error 1 EKF NN aided AEKF 1 North error (m) - Vertical error (m) -1 NN aided AEKF EKF Time (sec) (b) North error EKF NN aided AEKF Time (sec) (c) Vertical error Figure 7. Positioning errors of the EKF and NN aided AEKF.

14 46 DAH-JING JWO AND HUNG-C HIH HUANG VOL. 7 The scenario for simulation is as follow. The kinematics of the user is assumed to move at a constant velocity with mean value 1m/s to East and 1d3m/s to North (which results in a mean speed of m/s), starting from the position of North.1 degrees, East degrees. In the case that differential GP (DGP) mode is used and most of the errors can be corrected, but the multipath and receiver thermal noise cannot be eliminated. The original measurement error standard deviations for all the pseudorange observables are assumed to be 3m. After a while, however, the standard deviations of measurement noises are then raised by ten times of the original ones. It is expected that the proposed NN aided AEKF to be employed for performing the position estimation. Again, the NN employed in the present case has a structure and the parameter values used are same as those in the double integrator model. Based on the parameter values and scenario, the simulation for positioning accuracy is conducted. The convergence history of the error average for the DGP positioning example is given in Figure 6. Comparison of positioning errors using EKF and proposed NN aided AEKF is presented in Figure 7. It is seen that substantial accuracy improvement is achieved by using the proposed adaptive technique. 6. CONCLUION. This paper has presented a neural network aided adaptive extended Kalman filtering approach for DGP positioning. After being trained, the neural network was employed as a noise identification mechanism to implement the on-line identification of measurement noise covariance matrices. Based on the proposed approach, the noise adaptation capability has been tested on a double integrator model and shows significant improvement as compared to the conventional innovation-based algorithms such as the covariance matching method. The DGP positioning solution using the proposed NN aided AEKF has been conducted and the result shows that substantial accuracy improvement has been obtained. ACKNOWLEDGEMENT The support provided by the National cience Council of the Republic of China is gratefully acknowledged. REFERENCE Axelrad, P. and Brown, R. G. (1996). GP navigation algorithms, in Parkinson, B. W., pilker, J. J., Axelrad, P. and Enga, P. (Ed.), Global Positioning ystem: Theory and Applications, Volume I, AIAA, Washington DC, Chap. 9. Brown, R. and Hwang, P. (1997). Introduction to Random ignals and Applied Kalman Filtering, John Wiley & ons, New York. Chester, M. (1993). Neural networks: a tutorial, Prentice-Hall. Gelb, A. (1974). Applied Optimal Estimation, M. I. T. Press, MA. Haykin,. (1999). Neural networks: a comprehensive foundation, Prentice-Hall. Hide, C, Moore, T., and mith, M. (3). Adaptive Kalman filtering for low cost IN/GP, This Journal, 6 (1), pp Mehra, R. K. (197). On the identification of variance and adaptive Kalman filtering, IEEE Trans. Automat. Contr., AC-1, pp

15 NO. 3 KALMAN FILTERING APPROACH FOR DGP P OITIONING 463 Mehra, R. K. (1971). On-line identification of linear dynamic systems with applications to Kalman filtering, IEEE Trans. Automat. Contr., AC-16, pp Mehra, R. K. (197). Approaches to adaptive filtering, IEEE Trans. Automat. Contr., AC-17, pp Mohamed, A. H. and chwarz, K. P. (1999). Adaptive Kalman filtering for IN/GP, Journal of Geodesy, 73 (4), pp Rosenblatt, F. (196). Principles of aerodynamics: perceptrons and the theory of brain mechanisms, partan, New York. Widrow, B. and Lehr, M. A. (199). 3 years of adaptive neural networks: Perceptron, madaline, and backpropagation, Proc. of the IEEE, 78 (9), pp