Bias-Compensated Pseudo-measurement Tracking Filter Design in Line-of-Sight Coordinates

Abstract

Target tracking using pseudo-measurement is widely used because of its simple implementation and computational advantages despite biased behavior. The biasedness of the pseudo-measurement tracking filter originates from a measurement-dependent Kalman gain and a state-dependent non-Gaussian biased noise. This paper proposes a bias-compensated pseudo-measurement filter considering a range rate measurement in the line-of-sight Cartesian coordinate system. Firstly, a pseudo-measurement model incorporating range rate measurements in line-of-sight coordinates is derived. The noise and measurement covariance of a proposed de-biased pseudo-measurement model are shown to be statistically consistent for highly noisy measurements. Secondly, a bias-compensated pseudo-measurement filter that adopts modified gain to suppress the estimation bias is formulated. The asymptotic stability of a proposed filter is further discussed. Lastly, simulation results show that a proposed de-biased pseudo-measurement filter is very effective in considering a range rate measurement.

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Appendix: Noise Covariance Matrix of Pseudo-masurement Model

Appendix: Noise Covariance Matrix of Pseudo-masurement Model

This appendix provides a detailed expression of the measurement noise covariance matrix given in Eq. (12). The measurement noise covariance matrix is obtained from the pseudo-measurement model derived in (4).

$$\begin{aligned} R_{k} & = E\left[ {\left( {{\mathbf{V}}({\mathbf{x}}_{k} ,{\mathbf{v}}_{k} ) - E[{\mathbf{V}}({\mathbf{x}}_{k} ,{\mathbf{v}}_{k} )]} \right)^{2} } \right] \\ \,\,\,\,\, & = E[{\mathbf{V}}({\mathbf{x}}_{k} ,{\mathbf{v}}_{k} )^{2} ] - E[{\mathbf{V}}({\mathbf{x}}_{k} ,{\mathbf{v}}_{k} )]^{2} \\ \,\,\,\,\, & \cong E[{\mathbf{V}}({\hat{\mathbf{x}}}_{k|k - 1} ,{\mathbf{v}}_{k} )^{2} ] - {\varvec{\upmu}}_{p}^{2} . \\ \end{aligned}$$
(49)

The expectation of squared noise is given in (50). Note that the square of bias in (9) is subtracted from the expectation of squared noise. The definition of range \(r\), range rate \(\dot{r}\), azimuth \(\phi\) and elevation \(\theta\) can be found in Eq. (5).

$$\begin{aligned} E[V(&{\mathbf{x}},{\mathbf{v}})^{2} ] = \left[ {\begin{array}{*{20}c} {\gamma_{11} } & 0 & {\gamma_{13} } & {\gamma_{14} } \\ 0 & {\gamma_{22} } & 0 & {\gamma_{24} } \\ {\gamma_{13} } & 0 & {\gamma_{33} } & {\gamma_{34} } \\ {\gamma_{14} } & {\gamma_{24} } & {\gamma_{34} } & {\gamma_{44} } \\ \end{array} } \right], \\ & c_{1\theta } = E\left[ {\cos v_{\theta } } \right] = e^{{ - \frac{{\sigma_{\theta }^{2} }}{2}}} ,\\ & c_{2\theta } = E\left[ {\cos v_{\theta }^{2} } \right] = (1 + e^{{ - 2\sigma_{\theta }^{2} }} )/2,\\ &c_{3\theta } = E\left[ {\sin v_{\theta }^{2} } \right] = (1 - e^{{ - 2\sigma_{\theta }^{2} }} )/2, \\ & c_{1\phi } = E\left[ {\cos v_{\phi } } \right] = e^{{ - \frac{{\sigma_{\phi }^{2} }}{2}}}, \\ & c_{2\phi } \, = E\left[ {\cos v_{\phi }^{2} } \right]\, = (1 + e^{{ - 2\sigma_{\phi }^{2} }} )/2,\\ & c_{3\phi } = E\left[ {\sin v_{\phi }^{2} } \right] = (1 - e^{{ - 2\sigma_{\phi }^{2} }} )/2, \\ \gamma_{11} & = \sigma_{r}^{2} + r^{2} (1 - 2c_{1\theta } c_{1\phi } + c_{2\theta } c_{2\phi } ) \\ \,\,\,\,\,\, & \quad + z^{2} (\sin^{2} \theta c_{2\theta } + \cos^{2} \theta c_{3\theta } )(c_{2\phi } - 2c_{1\phi } + 1)\, \\ \,\,\,\, & \quad + 2rz\sin \theta (c_{1\theta } c_{1\phi } - c_{2\theta } c_{2\phi } - c_{1\theta } + c_{2\theta } c_{1\phi } ), \\ \gamma_{22} & = (x^{2} + y^{2} )c_{3\phi }, \\ \gamma_{33} & = r^{2} c_{3\theta } c_{2\phi } + 2rz\sin \theta c_{3\theta } (c_{1\phi } - c_{2\phi } ) \\ & \quad + z^{2} (\sin^{2} \theta c_{2\theta } + \cos^{2} \theta c_{3\theta } )(c_{2\phi } - 2c_{1\phi } + 1), \\ \gamma_{13} & = rz\cos \theta \{ (c_{3\theta } - c_{2\theta } )(c_{2\phi } - c_{1\phi } ) + c_{1\theta } (c_{1\phi } - 1)\} \\ \,\,\,\,\,\, & \quad \, + z^{2} \cos \theta \sin \theta (c_{2\theta } - c_{3\theta } )(c_{2\phi } - 2c_{1\phi } + 1), \\ \gamma_{34} & = r\dot{x}\sin \theta \cos \phi c_{3\theta } c_{2\phi } + r\dot{y}\sin \theta \sin \phi c_{3\theta } c_{2\phi } \\ &\quad - r\dot{z}\cos \theta c_{3\theta } c_{1\phi } \\ \,\,\,\,\,\,\, & \quad + z\dot{r}\cos \theta \{ c_{1\theta } (c_{1\phi } - 1) + c_{2\theta } (c_{1\phi } - c_{2\phi } )\} \\ \,\,\,\,\,\,\, & \quad + z\dot{x}\sin^{2} \theta \cos \phi c_{3\theta } (c_{1\phi } - c_{2\phi } ) \\ & \quad + z\dot{y}\sin^{2} \theta \sin \phi c_{3\theta } (c_{1\phi } - c_{2\phi } ) \\ \,\,\,\,\,\,\, & \quad + z\dot{z}\cos \theta \sin \theta c_{2\theta } (c_{2\phi } - 2c_{1\phi } + 1) \\ & \quad + z\dot{z}\cos \theta \sin \theta c_{3\theta } (c_{1\phi } - 1), \\ \gamma_{44} & = \sigma_{{\dot{R}}}^{2} + \dot{r}^{2} (1 - 2c_{1\theta } c_{1\phi } + c_{2\theta } c_{2\phi } ) \\ \,\,\,\,\,\,\, & \quad + \dot{x}^{2} (\cos^{2} \theta \sin^{2} \phi c_{2\theta } c_{3\phi } + \sin^{2} \theta \cos^{2} \phi c_{3\theta } c_{2\phi } \\ & \quad + \sin^{2} \theta \sin^{2} \phi c_{3\theta } c_{3\phi } ) \\ \,\,\,\,\,\,\, & \quad + \dot{y}^{2} (\cos^{2} \theta \cos^{2} \phi c_{2\theta } c_{3\phi } + \sin^{2} \theta \sin^{2} \phi c_{3\theta } c_{2\phi } \\ & \quad + \sin^{2} \theta \cos^{2} \phi c_{3\theta } c_{3\phi } ) \\ \,\,\,\,\,\,\, & \quad + \dot{z}^{2} \{ \sin^{2} \theta c_{2\theta } (c_{2\phi } - 2c_{1\phi } + 1) + \cos^{2} \theta c_{3\theta } \} \\ \,\,\,\,\,\, & \quad + 2\dot{r}\dot{z}\sin \theta \{ c_{1\theta } (c_{1\phi } - 1) + c_{2\theta } (c_{1\phi } - c_{2\phi } )\} \\ \,\,\,\,\,\, & \quad + 2\dot{x}\dot{y}\{ \sin^{2} \theta \cos \phi \sin \phi c_{3\theta } (c_{2\phi } - c_{3\phi } ) \\ & \quad - \cos^{2} \theta \cos \phi \sin \phi c_{2\theta } c_{3\phi } \} \\ \,\,\,\,\, & \quad - 2\dot{x}\dot{z}\cos \theta \sin \theta \cos \phi c_{3\theta } c_{1\phi } \\ & \quad - 2\dot{y}\dot{z}\cos \theta \sin \theta \sin \phi c_{3\theta } c_{1\phi } . \\ \end{aligned}$$
(50)

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Cho, M., Tahk, M. & Park, J. Bias-Compensated Pseudo-measurement Tracking Filter Design in Line-of-Sight Coordinates. Int. J. Aeronaut. Space Sci. (2020). https://doi.org/10.1007/s42405-020-00280-9

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Keywords

  • Pseudo-measurement filter
  • Bias compensation
  • Modified gain
  • Line-of-sight Cartesian coordinate system
  • Range rate measurement
  • Radar target tracking