Maximum correntropy-based pseudolinear Kalman filter for passive bearings-only target tracking

Abstract

This paper proposes a new approach for solving the bearings-only target tracking (BoT) problem by introducing a maximum correntropy criterion to the pseudolinear Kalman filter (PLKF). PLKF has been a popular choice for solving BoT problems owing to the reduced computational complexity. However, the coupling between the measurement vector and pseudolinear noise causes bias in PLKF. To address this issue, a bias-compensated PLKF (BC-PLKF) under the assumption of Gaussian noise was formulated. However, this assumption may not be valid in most practical cases. Therefore, a bias-compensated PLKF with maximum correntropy criterion is introduced, resulting in two new filters: maximum correntropy pseudolinear Kalman filter (MC-PLKF) and maximum correntropy bias-compensated pseudolinear Kalman filter (MC-BC-PLKF). To demonstrate the performance of the proposed estimators, a comparative analysis assuming large outliers in the process and measurement model of 2D BoT is conducted. These large outliers are modeled as non-Gaussian noises with diverse noise distributions that combine Gaussian and Laplacian noises. The simulation results are validated using root mean square error (RMSE), average RMSE (ARMSE), percentage of track loss and bias norm. Compared to PLKF and BC-PLKF, all the proposed maximum correntropy-based filters (MC-PLKF and MC-BC-PLKF) performed with superior estimation accuracy.

Appendix A. Derivation of pseudolinear measurement equation

The bearing measurement model given in Eq. (3) is

$$\begin{aligned} \tilde{z}_k= \beta _k+v_k, ~~ \text{ where } ~~ \beta _k=\text{ tan}^{-1}({\Delta x_k,}{\Delta {y_k}}). \end{aligned}$$

(A1)

The term $\beta _k$ is the true measurement and it is also represented as $z_k$. Hence, the measured bearing is given as

$$\begin{aligned} \tilde{z}_k=z_k+v_k, ~~ \text{ where } ~~ z_k=\beta _k=\text{ tan}^{-1}({\Delta x_k,}{\Delta {y_k}}). \end{aligned}$$

(A2)

To obtain the linearized model of the measurement equation,

$$\begin{aligned} \sin {v_k}\!=\!\sin {(\tilde{z}_k-z_k)}\!=\!\sin {\tilde{z}_k}\cos {z_k}\!-\!\cos {\tilde{z}_k}\sin {z_k}. \end{aligned}$$

Then, using the trigonometric relations from Eq. (A2) as $\cos {z_k}=\Delta y/\Vert d_k\Vert$, $\sin {z_k}=\Delta x/\Vert d_k \Vert$ and ${\Delta x_k,}= x^t_k-x^o_k$, ${\Delta {y_k}}= y^t_k-y^o_k$

$$\begin{aligned}&\sin {v_k}=\sin {\tilde{z}_k}(\Delta y/\Vert d_k \Vert )-\cos {\tilde{z}_k}(\Delta x/\Vert d_k \Vert ), \\&\Vert d_k \Vert \sin {v_k}=\sin {\tilde{z}_k}(y^t_k-y^o_k)-\cos {\tilde{z}_k}(x^t_k-x^o_k), \\&-x^t_k\cos {\tilde{z}_k} +y^t_k\sin {\tilde{z}_k} \\&\quad =-x^o_k\cos {\tilde{z}_k}+y^o_k\sin {\tilde{z}_k}+\Vert d_k \Vert \sin {v_k}. \end{aligned}$$

With further simplifications, the pseudolinear measurement equation can be written as

$$\begin{aligned} \begin{aligned}&u_k' X^o_k=u_k' M X^t_k+\eta _k, \\&Z_k=H_k X^t_k+\eta _k, \end{aligned} \end{aligned}$$

(A3)

where

$$\begin{aligned} u_k'=[-\cos (\tilde{z}_k)~~\sin (\tilde{z}_k)],~~ M=\begin{bmatrix} 1 &{}0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 \end{bmatrix} \end{aligned}$$

such that $H_k=u_k'M= [-\cos (\tilde{z}_k) \ \sin (\tilde{z}_k) \ 0 \ 0],$ $\eta _k=$ $\Vert d_k\Vert \sin (v_k).$