Fuzzy strong tracking unscented Kalman filter

The EKF uses first order Taylor series expansion, which can be improved by higher order approximations at the expense of computational burden.

The unscented Kalman filter (UKF) (Wan and van der Merwe 2000; Wan and van der Merwe 2001; Simon 2006) has been developed in the context of state estimation of dynamic systems as a nonlinear distribution (or densities in the continuous case) approximation method.
The UKF is superior to EKF both in theory and in many practical situations (Jwo and Lai 2008). The algorithm performs the prediction of the statistics with a set of carefully chosen sample points, referred to as the sigma points employed to propagate the probability of state distribution, for capturing mean and covariance of the system (Julier et al. 2000; Julier and Uhlmann 2002). The series approximations in the EKF algorithm can lead to poor representations of the nonlinear functions and probability distributions of interest. The UKF can capture the states up to at least second order approximation.

The adaptive Kalman filter (AKF) (Mehra 1972; Mohamed and Schwarz 1999; Ding, et al. 2007; Hide, et al. 2003) has been one of the strategies to prevent divergence problem due to modeling errors. A relatively new adaptive filter proposed by Zhou and Frank (1996) is called the strong tracking Kalman filter (STKF), which has several important merits: (1) strong robustness against model uncertainties; (2) good real-time state tracking capability, no matter whether the system has reached steady state or not; (3) moderate computational load. In the same way, a filter called the strong tracking unscented Kalman filter (STUKF) is developed based on the combination of UKF and STKF, leading to the STUKF.

The conventional approach for determining the softening factors relies on personal experience or computer simulation using a heuristic searching scheme. A new approach called the fuzzy strong tracking unscented Kalman filter (FSTUKF) is suggested. In the algorithm, the fuzzy logic reasoning system (FLAS) (Sasiadek, et al. 2000; Jwo and Lai 2009) based on the Takagi-Sugeno (T-S) model (Takagi and Sugeno 1985) is incorporated into the STUKF. The fuzzy reasoning system is constructed for obtaining suitable softening factors according to the time-varying change in dynamics. By monitoring the innovation information, the FLAS, as the filter’s internal mode, is employed for dynamically adjusting the softening factors based on the fuzzy rules so as to enhance the estimation accuracy and tracking capability.

The proposed FSTUKF scheme is applied to the loosely-coupled GPS/INS navigation processing to improve the navigation estimation accuracy at the high dynamic regions while preserving/without sacrificing the precision at the lower dynamic regions. The performance of the proposed FSTUKF method is compared to those of the EKF, UKF, and STUKF approaches.

The Unscented Kalman Filter

The nonlinear filters deal with the case governed by the nonlinear stochastic difference equations:

where the state vector x_k∈ℜⁿ, process noise vector w_k∈ℜⁿ, measurement vector z_k∈ℜ^m, and measurement noise vector v_k∈ℜ^m. The vectors w_k and v_k are zero mean Gaussian white sequences having zero crosscorrelation with each other:

The discrete-time extended Kalman filter algorithm is summarized:

In the EKF, the state distribution is approximated by a Gaussian random variable (GRV), which is then propagated analytically through the first-order linearization of the nonlinear system. Wan and van der Merwe (2000) pointed out that this will introduce large errors in the true posterior mean and covariance of the transformed GRV and lead to suboptimal performance and sometimes filter divergence. The basic premise behind the UKF is it is easier to approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function. Instead of linearizing using Jacobian matrices as in the EKF and achieving first-order accuracy, the UKF uses a deterministic sampling approach to capture the mean and covariance estimates with a minimal set of sample points.

The UKF was first proposed by Julier, et al. (1995) to address nonlinear state estimation in the context of control theory. The state distribution is also approximated by a GRV, but is represented using a minimal set of sample points. These sample points are carefully chosen so as to completely capture the true mean and covariance of the GRV. When the sample points are propagated through the true nonlinear system, the posterior mean and covariance can be captured accurately to the 2nd order of Taylor series expansion for any nonlinear system.
One of the remarkable merits is that the overall computational complexity of the UKF is the same as that of the EKF (Wan and van der Merwe 2000).

The first step in the UKF is to sample the prior state distribution, i.e., generate the sigma points through the unscented transformation (UT) (Julier, et al. 2000; Julier, 2002; Julier and Uhlmann 2002),which is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation. The basic premise is that to approximate a probability distribution is easier than to approximate an arbitrary nonlinear transformation. The mapping of the UKF versus that of the EKF, through the transformation of nonlinear function f(.) and its Jacobian/Hessian, can be illustrated by Figure 1 (Li et al. 2006). The dot-line ellipse represents the true covariance; the solid-line ellipse represents the  calculated covariance. The UKF approach estimates are expected to be closer to the true values than the EKF approach.

Consider an n dimensional random variable x, having the mean  and covariance P, and suppose that it propagates through an arbitrary nonlinear function f. The unscented transform creates 2n+1 sigma vectors x (a capital letter) and weighted points W, given by

The implementation algorithm of UKF is summarized as follows:

The Strong Tracking Algorithm

Zhou and Frank (1996) proposed the concept of strong tracking Kalman filter. In the algorithm, the time-varying suboptimal scaling factor is incorporated, for on-line tuning the covariance of the predicted state. As a type of AKF, the STF is essentially a nonlinear smoother algorithm that employs suboptimal multiple fading factors (Xia,1994), in which the softening factors are involved.

Based on the idea as in the AKF, the synthesis of UKF and strong tracking filter leads to the strong tracking unscented Kalman filter (STUKF). As in the STKF, suboptimal fading factors are introduced into the nonlinear smoother algorithm:

The fuzzy Logic Adaptive System (FLAS)

A FLAS mechanism can be incorporated for determining the softening factor, leading to the fuzzy strong tracking unscented Kalman filter (FSTUKF). The FSTUKF is composed of the FLAS, the strong tracking mechanism, and the UKF. The characteristics of the fuzzy adaptive system depend on the fuzzy rules and the effectiveness of the rules directly influences its performance. The defuzzification is used to provide the deterministic values of a membership function for the output. Using fuzzy logic to infer the consequent of a set of fuzzy production rules invariably leads to fuzzy output subsets. Fuzzy modeling is the method of describing the characteristics of a system using fuzzy inference rules.

The parameters for checking the degree of divergence (DOD) to identify the degree of change in vehicle dynamics need to be defined. The innovation information at the present epoch is employed for timely reflecting the change in vehicle dynamics.
The averaged magnitude of innovation at the present epoch (i.e., the window size is one) can be used as the first DOD parameter:

Furthermore, the other DOD parameter ξ can be defined as the trace of innovation covariance matrix at present epoch divided by the number of measurements employed for navigation processing:

An Experiment on GPS/INS Navigation State Estimation

The differential equations describing the two-dimensional inertial navigation state, where two accelerometers and one gyroscope are involved, are (Farrell and Barth 1999):

where[ a_u , a_v] are the measured accelerations in the body frame, ω_r is the measured yaw rate in the body frame. The error model for INS is constructed by the navigation states augmented by the accelerometer biases and gyroscope drift:

which were utilized in the integration Kalman filter as the inertial error model. In (28), δn and δe represent the east, and north position errors; δv_n and δv_e represent the east, and north velocity errors; δψ represents yaw angle; δa_u, δa_v , and δω_r represent the accelerometer biases and gyroscope drift, respectively. The measurement model is written as

Assume that the differential GPS mode is used, and only the multipath and receiver thermal noise are included. Figure 2 shows the architecture of the loosely-coupled GPS/INS integrated navigation based on the FLAS-coupled STUKF mechanism. The dynamic characteristics of the vehicle can be approximately divided into two categories: (a) constant-velocity straightline motion during four time intervals, 1~180, 361~450, 496~630 and 901~1080s, all at a speed of m/s; (b) circular motion during 181~360, 451~495 and 631~900s. The trajectory for the simulated vehicle is shown as in Figure 3. The parameters utilized in the STUKF are as follows: α=1e-4, β=2, k=0, η=0.25, ρ=0.1. The numerical efficient and stable method such as the Cholesky factorization has been used in obtaining the sigma points.

In the FSTUKF, the softening factor ε is determined by the FLAS. The philosophy for defining the rules is straightforward:
(a) for the case that the DOD parameter is small, our objective is to obtain results with better estimation accuracy, and a larger softening factor ε should be applied; (b) for the case that the DOD parameter is increased, our objective is to increase the tracking capability, and a
smaller softening factor should be applied. The membership functions (MFs) of input fuzzy variable DOD parameters as shown in Figure 4 are triangle MFs:

Performance comparison between UKF and EKF is shown in Figure 5;Navigation accuracy comparison for FSTUKF, STUKF and UKF is illustrated in Figure 6. A single fixed value of ε, whether larger or smaller, is not easy to fit the accuracy requirement in all types of environments and is therefore unrealistically in practical applications. Figure 7 shows the fading factors resulting from STUKF and FSTUKF, respectively. For the three regions of high dynamic environments, the fading factors using FSTUKF have been amplified at a higher rate as compared to the STUKF.

Two remarks are given concerning the experiment.

(1) In the four time intervals, 1~180, 361~450, 496~630 and 901~1080s, the vehicle is not maneuvering and is conducting constant-velocity straightline motion. For this case, all the filters provide equivalently good results. The navigation accuracies based on the four approaches have relatively small difference. By use of the T-S fuzzy logic, the FLAS senses smaller values of DOD parameters, and gives a larger softening factor. With large softening factors, the UKF, STUKF, and FSTUKF deteriorate to the EKF, and the navigation accuracies based on all the filters are equivalent.

(2) In the three time intervals, 181~360, 451~495 and 631~900s, the vehicle is maneuvering. The mismatch of the model leads the STUKF to larger navigation error while the FLAS timely detects the change of DOD parameter, and then reduces softening factor so as to maintain good tracking capability.

Conclusions

An alternative state estimation technique called the fuzzy strong tracking unscented Kalman filter has good potential as the GPS/INS navigation state estimation technique. The approach is designed so as to improve the navigation accuracy at the high dynamic regions while preserving/without sacrificing the precision at the lower dynamic regions.

Traditional strong tracking Kalman filter for determining the softening factors heavily relies on personal experience or computer simulation using a heuristic searching scheme. In this article, the fuzzy system has been employed to improve the STUKF performance. Through the use of fuzzy logic, the FLAS has been incorporated into the STUKF as a mechanism for timely detecting the dynamical changes and implementing the on-line tuning of the softening factors by monitoring the innovation information so as to maintain good estimation accuracy and tracking capability. Performance comparisons on EKF, UKF, STUKF and STUKF have been conducted. As the improved version of STUKF, the FSTUKF algorithm leads to very promising results in both navigational accuracy and tracking capability and has very god potential as an alternative navigation state estimator.

Acknowledgements

This work has been supported in part by the National Science Council of the Republic of China under grant NSC 96-2221-E-019-007 and NSC 97-2221-E-019-012.

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