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 武汉大学学报·信息科学版  2017, Vol. 42 Issue (8): 1035-1039

#### 文章信息

WEI Erhu, WANG Lingxuan, ZHANG Shuai, LIU Jingnan

Simulation Analysis of Mars Probe Autonomous Positioning and Timing Based on X-Ray Pulsars

Geomatics and Information Science of Wuhan University, 2017, 42(8): 1035-1039
http://dx.doi.org/10.13203/j.whugis20150437

### 文章历史

1. 武汉大学测绘学院, 湖北 武汉, 430079;
2. 武汉大学卫星导航定位中心, 湖北 武汉, 430079

Simulation Analysis of Mars Probe Autonomous Positioning and Timing Based on X-Ray Pulsars
WEI Erhu1, WANG Lingxuan1, ZHANG Shuai1, LIU Jingnan2
1. School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China;
2. GNSS Research Center, Wuhan University, Wuhan 430079, China
First author: WEI Erhu, PhD, professor, specializes in space geodesy and geodynamics. E-mail: ehwei@sgg.whu.edu.cn
Foundation support: The National Natural Science Foundation of China, No. 41374012
Abstract: Measurement based on X-Ray Pulsars is not only able to provide stable and reliable position, velocity and attitude information for deep space probe, but also can correct the long term bias of the atomic clock in the probe. Especially for the Mars exploring missions, the flight time of Mars probe is fairly long so that the clock bias is accumulating continuously, which will greatly damage the accuracy of the Mars probe autonomous positioning and timing system. In this paper, a novel synchronous Positioning/Timing method for Mars probe is proposed based on the principle of synchronous positioning and timing by X-Ray pulsars. And the clock correction is modeled accordingly. Pulsar optimization factor is used to conduct the optimal selection of pulsars. Then the Positioning/Timing model is presented and the clock corrections based on the new clock model are estimated together with the position and velocity of the Mars probe by using the adaptive extended Kalman filter. So that the position, velocity and clock model parameters of Mars probe could be calculated synchronously. Simulation experiments are carried out based on the proposed method. Results show that the clock bias can maintain in 300ns, while the position and velocity accuracy of Mars probe is better than 200m and 0.03m/s after corrected, respectively. Conclusions could be drawn that the proposed method of Mars Probe Autonomous Positioning and Timing Based on X-Ray Pulsars could overcome the initial state error effectively and obtain more accurate filter value of position, velocity and clock parameters for Mars probe.
Key words: X-ray pulsars     Mars probe     adaptive extended Kalman filter     positioning     timing

X射线脉冲星自主导航技术具有可靠性高、自主性强、精确性良好和使用范围广等优点，在火星探测器自主导航方面极具应用潜力。国内外一些学者在利用X射线脉冲星观测量为执行火星探测任务的飞行器提供自主导航服务方面开展了相关研究，包括相关时空基准的定义和转换、适用于火星探测器的脉冲星测量模型的建立、火星探测器轨道动力学研究和导航滤波算法等内容[2, 3]。本文在此基础上，针对初始状态误差较大会导致普通的滤波算法收敛较慢甚至发散，以及火星探测任务周期长，探测器上搭载的原子钟的钟差随时间累积，严重损害了自主定位系统的精度等问题，提出了利用X射线脉冲星观测量同步滤波估计探测器的位置、速度和钟差模型参数的同步定位和授时方法，并利用设计的自适应扩展卡尔曼滤波器进行了仿真实验。

1 脉冲星同步定位和授时模型

 $\tau = \frac{1}{c}\mathit{\boldsymbol{n}} \cdot \mathit{\boldsymbol{r + }}\Delta {t_S} + v$ (1)

1.1 轨道动力学模型

 $\mathit{\boldsymbol{\ddot r = }} - \frac{{{\mu _M}\mathit{\boldsymbol{r}}}}{{{r^3}}} + {\mathit{\boldsymbol{a}}_{{\mathit{\boldsymbol{J}}_2}}} + \mathit{\boldsymbol{\omega }}$ (2)

 ${\mathit{\boldsymbol{a}}_{{\mathit{\boldsymbol{J}}_2}}} = \left[ {\begin{array}{*{20}{c}} { - \frac{{{\mu _M}x}}{{{\mathit{\boldsymbol{r}}^3}}}{\mathit{\boldsymbol{J}}_2}{{\left( {\frac{{{R_M}}}{\mathit{\boldsymbol{r}}}} \right)}^2}\frac{3}{2}\left( {1 - 5\frac{{{z^2}}}{{{\mathit{\boldsymbol{r}}^2}}}} \right)}\\ { - \frac{{{\mu _M}y}}{{{\mathit{\boldsymbol{r}}^3}}}{\mathit{\boldsymbol{J}}_2}{{\left( {\frac{{{R_M}}}{\mathit{\boldsymbol{r}}}} \right)}^2}\frac{3}{2}\left( {1 - 5\frac{{{z^2}}}{{{\mathit{\boldsymbol{r}}^2}}}} \right)}\\ { - \frac{{{\mu _M}z}}{{{\mathit{\boldsymbol{r}}^3}}}{\mathit{\boldsymbol{J}}_2}{{\left( {\frac{{{R_M}}}{\mathit{\boldsymbol{r}}}} \right)}^2}\frac{3}{2}\left( {1 - 5\frac{{{z^2}}}{{{\mathit{\boldsymbol{r}}^2}}}} \right)} \end{array}} \right]$ (3)

1.2 钟差模型

 $\begin{array}{l} {x_1}\left( {{t_{k + 1}}} \right) = {x_1}\left( {{t_k}} \right) + \tau \cdot {x_2}\left( {{t_k}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{\tau ^2}}}{2}{x_3}\left( {{t_k}} \right) + \omega \left( k \right) \end{array}$ (4)

 ${\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right]_{k + 1}} = {\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_{k + 1,k}}{\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right]_k} = {\mathit{\boldsymbol{\omega }}_k}$ (5)

 ${\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_{k + 1,k}} = \left[ {\begin{array}{*{20}{c}} 1&\varepsilon &{{\varepsilon ^2}/2}\\ 0&1&\varepsilon \\ 0&0&1 \end{array}} \right]$ (6)

 $\begin{array}{*{20}{c}} {{Q_k}\left( \varepsilon \right) = E\left[ {{\mathit{\boldsymbol{\omega }}_k}\mathit{\boldsymbol{\omega }}_k^{\rm{T}}} \right] = }\\ {\left[ {\begin{array}{*{20}{c}} {{q_1}\varepsilon + \frac{1}{3}{q_2}{\varepsilon ^3} + \frac{1}{{20}}{q_3}{\varepsilon ^5}}&{\frac{1}{2}{q_2}{\varepsilon ^2} + \frac{1}{8}{q_3}{\varepsilon ^4}}&{\frac{1}{6}{q_3}{\varepsilon ^3}}\\ {\frac{1}{2}{q_2}{\varepsilon ^2} + \frac{1}{8}{q_3}{\varepsilon ^4}}&{{q_2}\varepsilon + \frac{1}{3}{q_3}{\varepsilon ^3}}&{\frac{1}{2}{q_3}{\varepsilon ^2}}\\ {\frac{1}{6}{q_3}{\varepsilon ^3}}&{\frac{1}{2}{q_3}{\varepsilon ^2}}&{{q_3}\varepsilon } \end{array}} \right]} \end{array}$ (7)

1.3 状态方程

 $\mathit{\boldsymbol{x}}\left( t \right) = {\left[ {x\;y\;z\;{v_x}\;{v_y}\;{v_z}\;{x_1}\;{x_2}\;{x_3}} \right]^{\rm{T}}}$ (8)

 $\mathit{\boldsymbol{\dot x}}\left( t \right) = \mathit{\boldsymbol{F}}\left( t \right) \cdot \mathit{\boldsymbol{x}}\left( t \right) + \mathit{\boldsymbol{W}}\left( t \right)$ (9)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{F}}\left( t \right) = }\\ {\left[ {\begin{array}{*{20}{c}} 0&0&0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ {\frac{{\partial {a_x}}}{{\partial x}}}&{\frac{{\partial {a_x}}}{{\partial y}}}&{\frac{{\partial {a_x}}}{{\partial z}}}&0&0&0&0&0&0\\ {\frac{{\partial {a_x}}}{{\partial x}}}&{\frac{{\partial {a_y}}}{{\partial y}}}&{\frac{{\partial {a_z}}}{{\partial z}}}&0&0&0&0&0&0\\ {\frac{{\partial {a_z}}}{{\partial x}}}&{\frac{{\partial {a_z}}}{{\partial y}}}&{\frac{{\partial {a_z}}}{{\partial z}}}&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&0&0&0 \end{array}} \right]} \end{array}$ (10)
1.4 观测方程

 $\mathit{\boldsymbol{z = }}{\left[ {\Delta {t_1} \cdots \cdots \Delta {t_p}} \right]^{\rm{T}}} = \mathit{\boldsymbol{H}} \cdot \mathit{\boldsymbol{x + \eta }}$ (11)

 $\mathit{\boldsymbol{H = }}\left[ \begin{array}{l} \begin{array}{*{20}{c}} {\frac{{{\mathit{\boldsymbol{n}}_1}}}{c}}&0&0&0&1&0&0 \end{array}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \\ \begin{array}{*{20}{c}} {\frac{{{\mathit{\boldsymbol{n}}_p}}}{c}}&0&0&0&1&0&0 \end{array} \end{array} \right]$ (12)

2 仿真实验与结果分析 2.1 导航脉冲星优选

 $\begin{array}{*{20}{c}} {{\rm{Cov}}\left( {\mathit{\boldsymbol{\hat x}}} \right) = E\left( {\mathit{\boldsymbol{\hat x}}{{\mathit{\boldsymbol{\hat x}}}^{\rm{T}}}} \right) = }\\ {{{\left( {{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{H}}} \right)}^{ - 1}}{\mathit{\boldsymbol{H}}^{\rm{T}}}E\left( {\Delta \mathit{\boldsymbol{\rho }}\Delta {\mathit{\boldsymbol{\rho }}^{\rm{T}}}} \right)\mathit{\boldsymbol{H}}{{\left( {{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{H}}} \right)}^{ - 1}}}\\ {{{\left( {{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{H}}} \right)}^{ - 1}}{\mathit{\boldsymbol{H}}^{\rm{T}}}{\rm{diag}}\left( {\sigma _1^2,\sigma _2^2, \cdots \sigma _p^2} \right)\mathit{\boldsymbol{H}} \cdot }\\ {{{\left( {{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{H}}} \right)}^{ - 1}} = {D_p}} \end{array}$ (13)

 $\mathit{\boldsymbol{H = }}{\left[ {{\mathit{\boldsymbol{n}}_1}\;{\mathit{\boldsymbol{n}}_2}\; \cdots \;{\mathit{\boldsymbol{n}}_p}} \right]^{\rm{T}}}$ (14)

 脉冲星编号 测距精度σ/m B1937+21 689.5 B1821-24 495.8 B0531+21 88.3 B0540-69 1 493.0 B1957+20 932.0 B0614+091 282.0
2.2 仿真环境设置

 图 1 模拟钟差累积示意图 Figure 1 Schematic of Simulate Clock Correction Accumulation

2.3 实验结果和分析

 图 2 滤波修正后的钟差示意图 Figure 2 Clock Correction After Filtering

 图 3 滤波修正后的位置误差示意图 Figure 3 Position Error After Filtering
 图 4 滤波修正后的速度误差示意图 Figure 4 Velocity Error After Filtering

3 结语