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 深空探测学报  2020, Vol. 7 Issue (1): 102-108  DOI: 10.15982/j.issn.2095-7777.2020.20171117002 0

### 引用本文

XU Hao, PEI Fujun, JIANG Ning. An Attitude Estimation Method Based on LIE Group Representation for Deep Space Probe[J]. Journal of Deep Space Exploration, 2020, 7(1): 102-108. DOI: 10.15982/j.issn.2095-7777.2020.20171117002.

### 文章历史

1. 北京工业大学 信息学部，北京 100124;
2. 计算智能与智能系统北京市重点实验室，北京 100124

An Attitude Estimation Method Based on LIE Group Representation for Deep Space Probe
XU Hao1,2, PEI Fujun1,2, JIANG Ning1,2
1. Faculty of Information Technology，Beijing University of Technology, Beijing 100124，China;
2. Beijing Key Laboratory of Computational Intelligence and Intelligent System，Beijing 100124，China
Abstract: A novel algorithm for attitude estimation of deep spacecraft based on star sensor is proposed. The Lie group is used to describe the attitude, which avoids the non-uniqueness and complex calculation of the quaternion conversion into the attitude matrix. A description of the motion model based on star sensor and the kinematics model of space rigid body in Lie group is given. A method based on Lie group is proposed to determine the dynamic attitude of deep spacecraft. The linearization model solves the errors generated by the traditional nonlinear model in the filtering process, and eliminates the steps of converting the quaternion into the attitude matrix and reduces the computational complexity. In the simulation, the proposed method is compared with the traditional quaternion attitude determination algorithm. The results prove that the algorithm has better stability and accuracy.
Key words: deep space probe    star sensor    attitude estimation    LIE group filter

Hight lights:

1. ●　The innovative introduction of Lie group structure describes the rigid body attitude, which overcomes the problems of singularity and non-uniqueness in traditional methods.
2. ●　Using the measurement information of the star sensor and the dynamic equations of the spacecraft, the system equations described by Lie Group are constructed to estimate the position of the spacecraft in space.
3. ●　Based on the mapping relationship between Lie group and Lie algebra, the steps of Lie group filtering are deduced and the spacecraft attitude matrix is directly estimated.
4. ●　Through the simulation and analysis of traditional quaternion filtering algorithm and Lie group filtering algorithm, the feasibility and superiority of the attitude estimation filtering algorithm based on Lie group description is proved.

1 星敏感器 1.1 星敏感器基本工作原理

 图 1 星敏感器的基本工作原理 Fig. 1 The working principle of star sensor
1.2 星敏感器观测模型

 ${{l}} = {[{p_x}, {p_y}, f]^{\rm{T}}}$ (1)

 ${{{l}}_i}(k) = {{X}}(k){{{r}}_i} + {{{v}}_i}$ (2)

 $\begin{array}{l} E\{ {{{v}}_i}\} = 0\\ E\{ {{{v}}_i}{{{v}}_i}^{\rm{T}}\} = {\sigma _i}^2{{{I}}_3} \end{array}$ (3)

 图 2 星敏感器测量原理 Fig. 2 The measurement principle of star sensor
2 基于李群的航天器姿态模型

 ${{SO}}(n) = \{ {{R}}|{{R}} \in {{{R}} ^{n \times n}}，{{R}}{{{R}}^{\rm{T}}} = {{I}}，\det ({{R}}) = 1\}$ (4)

 $\dot {{R}} = {{RO}}$ (5)

 ${{Q}}({{k}}){\rm{ = }}\left[ {{{k}} \times } \right] = \left[ {\begin{split} 0& \quad{ - {k_3}}&{{k_2}}\\ {{k_3}}&\quad\;\; 0&{ - {k_1}}\\ { - {k_2}}&\;\;\quad{{k_1}}&0\;\; \end{split}} \right]$ (6)

 ${{\omega }} = {[{\omega _x}{\text{，}}{\omega _y}\text{，}{\omega _z}]^{\rm{T}}}$ (7)

 $\dot {{R}} = {{R}}[{{\omega}} ] \times$ (8)

 ${{\dot \omega }} = {{{J}}^{ - 1}}\{ {{N}} - {{\omega }} \times {{J\omega }}\}$ (9)

3 姿态估计李群滤波算法

3.1 空间飞行器姿态动力学系统模型

 ${{{\dot X}}_t} = {{{X}}_t}[{{{\omega }}_t} + {{{w}}_t}] \times$ (10)
 ${{{l}}_t} = {{{X}}_t}{{{r}}_t} + {{{v}}_t}$ (11)

 ${{{X}}_{k + 1}} = {{{W}}_k}{{{X}}_k}{{{A}}_k}$ (12)
 ${{{l}}_{k + 1}} = {{{X}}_{k + 1}}{{{r}}_{k + 1}} + {{{v}}_{k + 1}}$ (13)

3.2 离散时间的李群滤波

 ${{{L}}_{k + 1}} = {{{L}}_k} + {{{W}}_k}$ (14)
 ${{{Y}}_k} = {{{H}}_k}{{{L}}_k} + {{{V}}_k}$ (15)

 ${\hat {{L}}_{k + 1\left| k \right.}} = {\hat {{L}}_k}$ (16)
 ${\hat {{L}}_{k + 1}} = {\hat {{L}}_{k + 1\left| k \right.}} + {{{K}}_{k + 1}}({{{Y}}_{k + 1}} - {{{H}}_{k + 1}}{\hat {{L}}_{k + 1\left| k \right.}})$ (17)

 ${{{e}}_{k + 1\left| k \right.}} = {{{e}}_{k + 1}} + {{{W}}_n}$ (18)
 ${{{e}}_{k + 1}} = {{{e}}_{k + 1\left| k \right.}} - {{{K}}_{k + 1}}({{{H}}_{k + 1}}{{{e}}_{k + 1\left| k \right.}} + {{{V}}_{k + 1}})$ (19)

 ${\hat {{X}}_{k + 1\left| k \right.}} = {\hat {{X}}_k}{{{A}}_k}$ (20)
 ${\hat {{X}}_{k + 1}} = {\hat {{X}}_{k + 1\left| k \right.}}{J_{k + 1}}({{{K}}_{k + 1}}({{{l}}_{k + 1}} - {{{X}}_{k + 1\left| k \right.}}{{{r}}_{k + 1}}))$ (21)

 ${{{\eta}} _{k + 1}} = {{{X}}_{k + 1}}{\hat {{X}}_{k + 1}^{\rm{T}}}$ (22)
 ${{{\eta}} _{k + 1\left| k \right.}} = {{{X}}_{k + 1}}{\hat {{X}}_{k + 1\left| k \right.}^{\rm{T}}}$ (23)

 ${\hat {{X}}_{k + 1\left| k \right.}} = {\hat {{X}}_k}{{{A}}_k}$ (24)

 ${{{P}} _{k|k - 1}} = {{{A}}_k}{{{P}} _{k - 1}}{{{A}}^{\rm{T}}}_k + {{{Q}} _w}$ (25)

 ${{{\varepsilon}} _{k + 1}} = \left\{ {\begin{split} {{{{l}}_{k + 1}} - {{{X}}_{k + 1\left| k \right.}}{{{r}}_{k + 1}}} \\ {{{{l}}^{'}}_{k + 1} - {{{X}}_{k + 1\left| k \right.}}{{{r}}^{'}}_{k + 1}} \end{split}} \right\}$ (26)

 ${{{S}}_k} = {{{H}}_k}{{{P}}_{k|k - 1}}{{H}}_k^{\rm T} + {{{Q}}_v}$ (27)

 ${{{K}}_k} = {{{P}} _{k|k - 1}}{{H}}_k^{\rm{T}}{{{S}} _k}_{}^{ - 1}$ (28)

 ${{{P}} _k} = ({{I}} - {{{K}}_k}{{{H}}_k}){{{P}} _{k|k - 1}}{({{I}} - {{{K}}_k}{{{H}}_k})^{\rm{T}}} + {{{Q}} _v}$ (29)

 ${\hat {{X}}_{k + 1}} = {\hat {{X}}_{k + 1\left| k \right.}}\exp ({{{K}}_{k + 1}}{{{\varepsilon}} _{k + 1}})$ (30)

 $\exp (\theta ) = {{I}} + \frac{{\sin (\left\| \theta \right\|)}}{{\left\| \theta \right\|}}\theta \times + \frac{{1 - \cos (\left\| \theta \right\|)}}{{{{\left\| \theta \right\|}^2}}}{(\theta \times )^2}$ (31)

 $\left\{ \begin{split} & {{\hat {{X}}}_{k + 1\left| k \right.}} = {{\hat {{X}}}_k}{{{A}}_k} \\ & {{{P}} _{k|k - 1}} = {{{A}}_k}{{{P}} _{k - 1}}{{{A}}_k^{\rm{T}}} + {{{Q}} _w} \\ & {{{S}} _k} = {{{H}}_k}{{{P}} _{k|k - 1}}{{H}}_k^{\rm{T}} + {{{Q}} _v} \\ & {{{K}}_k} = {{{P}} _{k|k - 1}}{{H}}_k^{\rm{T}}{{{S}} _k}^{ - 1} \\ & {{{P}} _k} = ({{I}}- {{{K}}_k}{{{H}}_k}){{{P}} _{k|k - 1}} \\ & {{\hat {{X}}}_{k + 1}} = {{\hat {{X}}}_{k + 1\left| k \right.}}\exp ({{{K}}_{k + 1}}{{{\varepsilon}} _{k + 1}}) \end{split} \right.$
4 仿真结果与分析

 $\begin{split} {{{v}}_1} = {[1,0,0]^{\rm T}}\\ {{{v}}_2} = {[0{\rm{,1,0}}]^{\rm T}} \end{split}$

X-Y-Z三轴姿态初值分别为0，–90°，90。

 ${{X}} = \left[ {\begin{split} \; 0 \quad\quad & 1 \quad &0 \;\\ \; 0 \quad\quad & 0 \quad &1 \;\\ \; 1 \quad\quad & 0 \quad &0 \; \end{split}} \right]$

 $\begin{split} {{{w}}_1} = {[0,0,1]^{\rm T}}\\ {{{w}}_2} = {[1{\rm{,0,0}}]^{\rm T}} \end{split}$

 ${{\omega }} = {10^{ - 4}} \times {[\cos (0.1t),\cos (0.08t),\cos (0.06t)]^{\rm{T}}}{\rm rad}/{\rm s}$

 ${{J}} = \left[ {\begin{split} \;\; {200} \!\!\!\!\! \quad\quad & \quad\quad 0& \quad\quad 0\;\;\;\;\;\;\;\; \\ 0 \quad\quad & \quad\quad\!\!\!\!\! {300}& \quad\quad 0\;\;\;\;\;\;\;\; \\ 0 \quad\quad & \quad\quad 0&{100}\;\;\;\;\;\; \end{split}} \right]$

 $t = 1{\rm s}$

 {{{X}}_0} = \left[ {\begin{align} \; 1 &\quad \quad 0 \quad &0 \;\\ \; 0 &\quad \quad 1 \quad &0 \;\\ \; 0 &\quad \quad 0\quad &1 \; \end{align}} \right]
 ${{{P}} _0} = \left[ {\begin{split} \; 5 \quad \quad & 0 \quad &0 \;\\ \; 0 \quad \quad & 5 \quad &0 \;\\ \; 0 \quad \quad & 0 \quad &5 \; \end{split}} \right]$
 ${{{Q}} _w} = 5 \times {10^{ - 5}}\left[ {\begin{split} \; 1\quad \quad &0 \quad &0\; \\ \; 0\quad \quad &1 \quad &0 \;\\ \; 0\quad \quad &0 \quad &1 \; \end{split}} \right]$
 ${{{Q}} _v} = {0.125^2}{I _6}$

 图 3 李群滤波三轴误差角 Fig. 3 The angle errors in three axis using Lie group filter
 图 4 四元数滤波三轴误差角 Fig. 4 The angle errors in three axis using quaternion filter
5 结　论

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