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MEMS/GPS integrated navigation system has been widely used for land-vehicle navigation. This system exhibits large errors because of its nonlinear model and uncertain noise statistic characteristics. Based on the principles of the adaptive Kalman filtering (AKF) and unscented Kalman filtering (AUKF) algorithms, an adaptive unscented Kalman filtering (AUKF) algorithm is proposed. By using noise statistic estimator, the uncertain noise characteristics could be online estimated to adaptively compensate the time-varying noise characteristics. Employing the adaptive filtering principle into UKF, the nonlinearity of system can be restrained. Simulations are conducted for MEMS/GPS integrated navigation system. The results show that the performance of estimation is improved by the AUKF approach compared with both conventional AKF and UKF.

Microelectromechanical systems (MEMS) and Global Positioning System (GPS) integrated navigation system have the advantages of small size, light weight, and low cost, but, because of its low accuracy, it can only be applied in low accuracy navigation fields such as unmanned aircrafts and land-vehicles [

The classical Kalman filter (KF) provides a recursive solution for estimation of linear dynamic systems. The optimality of the KF algorithm is mainly dependent on a priori statistic of the process and measurement noise and the linear system model. However, if the priori information is insufficient or biased, the precision of the estimated states will be degraded, even leading to divergences [

By utilizing the innovation and residual information, the AKF could adapt the filter stochastic properties online to accommodate itself to changes in vehicle dynamics. Thus, this technique could reduce the reliance of filter on the prior statistical information and obtain the noise statistic parameters of the dynamic system. The essence of AKF is to adapt the filter weights, so as to restrain model errors and improve the accuracy of filters. It is showed that applying AKF to the INS/GPS integrated navigation system could obtain better estimated performance than by using conventional KF, especially less than 20% root mean square error in attitude estimation [

UKF, which is another extension of Kalman filter, could give reliable estimates even if the nonlinearities of system are quiet severe [

As a combination of AKF and UKF, the adaptive UKF has been developed and applied to nonlinear joint estimation of both time-varying states and parameters [

In the integrated navigation field, almost all of the systems are nonlinear. The general nonlinear discrete system model is given as

The initial state

Assuming that

According to system model of (

Choose

Then, computing the predicted state

Parameter

Then, propagating the sigma points

Computing the predicted measurement vector

Aiming at the uncertainty of process and measurement noise statistic properties, the measurement information are used to real time estimate and update the means and covariances of noises. Assume that

The noise parameters

Because

The original problem has changed to calculate the conditional probability distributions

According to the Gaussian distributions of

Moreover, assuming that the measurements

Substituting (

Logarithm on both sides of (

By the logarithmic nature,

Then, the noise statistic estimator can be derived, which is defined by

In (

From the consideration for the nonlinear purposes, the noise statistic estimator derived above should be modified. In the linear applications, the term of

Similarly,

Submitting (

The unbiased properties of the noise estimates for UKF are proved in the appendix.

Based on the UKF and its noise statistic estimator, the prediction and update steps of AUKF algorithm are as follows.

Then, according to (

With updated process noise parameters, compute the predicted state

According to (

With real-time measurement noise parameters, compute the predicted measurement

Estimate the cross-covariance

Then compute the filter gain

Because of the highly nonlinear characteristic of MEMS/GPS, the conventional AKF based on small angle approximations is limited. Meanwhile, due to the time-varying noise stochastic properties for land-vehicle, the standard UKF in Section

In the simulation, the parameters of sensor errors are shown in Table

Parameters of sensor errors.

Sensor | Characteristic | Value |
---|---|---|

MEMS Gyro | Drift | 5°/h |

MEMS Gyro | Measuring white noise | 0.5°/h |

MEMS accelerometer | Bias | 2 mg |

MEMS accelerometer | Measuring white noise | 50 |

GPS (position) | Measuring white noise | 6 m |

GPS (velocity) | Measuring white noise | 0.01 m/s |

Figure

The track of land vehicle.

The architecture of MEMS/GPS integrated navigation with AUKF is shown in Figure

The architecture of MEMS/GPS integrated system with AUKF.

As shown in Figures

Comparison of position errors.

Comparison of velocity errors.

Comparison of attitude errors.

Figure

The simulation results indicate that if the AUKF scheme is considered the filtering solution, there are only small variations impacting on the performance of MEMS/GPS integrated navigation system, and this system has an excellent robustness. However, long processing time would cause slight divergence of the attitude errors, sequentially the velocity and position errors.

This study has developed an AUKF approach to improve the navigation performance of MEMS/GPS integrated system for land-vehicle applications. By treating this problem within conventional UKF framework, the noise estimator is adopted and could effectively estimate the process and measurement noise characteristics online. The results indicate that the proposed AUKF algorithm could efficiently improve the navigation performance of land-vehicle integrated navigation system. By comparing with AKF and UKF methods, the AUKF solution has a more stable and superior performance.

The process noise and measurement noise statistic properties are computed by estimator in (

According to (

Hence the mean of the process noise can be expressed as

From (

Substituting (

When the posteriori mean and covariance are known, the output residual vector of UKF is zero-mean Gaussian white noise and we see

From (

Thus, the estimate of the process noise noted by (

Similarly, the estimate of the measurement noise is

The unbiased properties for the estimates of noises are proved.

The authors declare that they have no conflict of interests regarding the publication of this paper.

Funding for this work was provided by the National Nature Science Foundation of China under Grant nos. 61374007 and 61104036. The authors would like to thank all the editors and anonymous reviewers for improving this paper.