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A spherical simplex-radial cubature quadrature Kalman filter (SSRCQKF) is proposed in order to further improve the nonlinear filtering accuracy. The Gaussian probability weighted integral of the nonlinear function is decomposed into spherical integral and radial integral, which are approximated by spherical simplex cubature rule and arbitrary order Gauss-Laguerre quadrature rule, respectively, and the novel spherical simplex-radial cubature quadrature rule is obtained. Combined with the Bayesian filtering framework, the general form and the specific form of SSRCQKF are put forward, and the numerical simulation results indicate that the proposed algorithm can achieve a higher filtering accuracy than CKF and SSRCKF.

The nonlinear state estimation problem widely exists in signal processing, target tracking, intelligent sensing, and other fields, which is a subject undergoing intense study [

The most widely used nonlinear Kalman filtering algorithms are extended Kalman filter (EKF) [

In order to further improve the nonlinear Kalman filtering accuracy, this paper proposes a novel spherical simplex-radial cubature quadrature Kalman filter (SSRCQKF). The structure of this paper is as follows. The spherical simplex-radial cubature quadrature rule is proposed in Section

Consider the integral

In general, the analytical solutions of above integrals are difficult to obtain, so the numerical approximate method is considered. It is pointed out in [

For the radial integral

And

It can be seen that the approximation accuracy of the above rule depends on the number of quadrature points.

Equation (

The spherical simplex rule (

Due to

Equation (

Consider the following discrete nonlinear system with additive noise:

The matrix

Calculate the following points:

Calculate the nonlinear propagation of the points:

Calculate the prior state estimation and prior error covariance matrix:

Calculate the following points:

Calculate the nonlinear propagation of the points:

Calculate the predicted measurement value:

Calculate the predicted measurement covariance matrix:

Calculate the cross covariance matrix:

Calculate the Kalman filtering gain:

Calculate the a posteriori state estimation:

Calculate the a posteriori error covariance matrix:

It can be seen from the algorithm process that the filtering accuracy depends on the order of the Gaussian-Laguerre quadrature rule; the higher the order is, the higher that estimation accuracy is achieved, but the more the points and weights are required. However, for the identified

The general form of SSRCQKF is presented in Section

The values of

The item

Let

Then,

Furthermore, the weights

The spherical simplex-radial cubature quadrature rule with

Based on (

One has

Cycle

One has

One has

One has

The effectiveness of the proposed SSRCQKF algorithm is verified by the following three-dimensional strong nonlinear system, which includes trigonometric function operation, power operation, and exponential operation.

RMSE of state 1.

RMSE of state 2.

RMSE of state 3.

In order to compare the four filters quantitatively, the average value and variance of RMSE of the four filters in 100 steps are counted and listed in Tables

Average RMSE of state.

Filters | State 1 | State 2 | State 3 |
---|---|---|---|

CKF | 0.8743 | 0.6120 | 0.4928 |

SSRCKF | 0.9540 | 0.7152 | 0.6429 |

SSRCQKF-2 | 0.8621 | 0.5737 | 0.4427 |

SSRCQKF-3 | 0.8570 | 0.5705 | 0.4305 |

Variance RMSE of state 1.

Filters | State 1 | State 2 | State 3 |
---|---|---|---|

CKF | 0.0108 | 0.0108 | 0.0097 |

SSRCKF | 0.0120 | 0.0115 | 0.0217 |

SSRCQKF-2 | 0.0102 | 0.0089 | 0.0060 |

SSRCQKF-3 | 0.0099 | 0.0087 | 0.0056 |

The proposed SSRCQKF is applied in target tracking in this section. The target is assumed to be in constant velocity (CV) motion; the state equation of CV model in two-dimensional case is described as follows:

In target tracking system, the bearings-only measurement equation is written as follows:

In the simulation, the radar location is set to be

The simulation results, including the position RMSE and velocity RMSE of various filters, are shown in Figures

Average RMSE of various filters.

Filters | Position RMSE/m | Velocity RMSE/(m/s) |
---|---|---|

CKF | 3.0372 | 0.1874 |

SSRCKF | 2.4039 | 0.1655 |

SSRCQKF-2 | 2.3591 | 0.1642 |

SSRCQKF-3 | 2.3238 | 0.1630 |

The position RMSE of various filters.

The velocity RMSE of various filters.

In order to further improve the filtering accuracy of the nonlinear system, this paper proposes a SSRCQKF algorithm by combining the spherical simplex-radial cubature rule with arbitrary order Gaussian-Laguerre quadrature rule. The proposed algorithm has a higher filtering accuracy than CKF and SSRCKF. The results of the two numerical simulations, including the three-dimensional strongly nonlinear system and target tracking, verify the validity of the proposed algorithm.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

_{∞}information filter and its extensions