基于SO（3）的航天器姿态误差描述方法综述

doi:10.12305/j.issn.1001-506X.2026.04.25

摘要/Abstract

摘要：

针对基于流形的航天器姿态建模方法数学表述复杂、物理意义不明确等问题，本文进行了流形上的航天器姿态误差描述方法与误差特性的综合分析。首先，基于流形三维特殊正交群（three-dimensional special orthogonal group，SO（3））建立航天器刚体姿态模型，接着分别从黎曼流形与李群的角度，给出了SO（3）上的3种常见误差描述方法。然后，通过微分几何理论及数值仿真工具，分析了3种误差描述体系之间的不同代数特性。最后，基于几何比例-微分（proportional-derivative，PD）控制器，仿真验证了3种误差描述体系在同一控制律作用下的响应特性，为姿态控制器的设计与工程应用提供了理论支撑。

关键词: 三维特殊正交群, 黎曼流形, 李群, 姿态控制

Abstract:

To address the issues of mathematical complexity and lack of physical clarity in manifold-based spacecraft attitude modeling methods, this paper conducts a comprehensive analysis of spacecraft attitude error description methods and their characteristics on manifolds. Firstly, a rigid-body spacecraft attitude model is established based on the three-dimensional special orthogonal group （SO（3）） on manifolds. Subsequently, three common error description methods on SO（3） are presented from the perspectives of Riemannian manifolds and Lie groups. The different algebraic characteristics among these three error description frameworks are then analyzed using differential geometry theory and numerical simulation tools. Finally, based on a geometric proportional-derivative （PD） controller, the response characteristics of the three error description methods under the same control law are verified through simulations, providing theoretical support for the design and engineering application of attitude controllers.

Key words: three-dimensional special orthogonal group （SO（3））, Riemannian manifold, Lie group, attitude control

中图分类号:

TP 139

郑重重, 郭杰, 万泱泱, 王喻林, 唐胜景. 基于SO（3）的航天器姿态误差描述方法综述[J]. 系统工程与电子技术, 2026, 48(4): 1370-1377.

Zhongzhong ZHENG, Jie GUO, Yangyang WAN, Yulin WANG, Shengjing TANG. Survey of SO（3）-based spacecraft attitude error description methods[J]. Systems Engineering and Electronics, 2026, 48(4): 1370-1377.

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姿态描述方法	优势	不足
SO（3）	无奇异，能全局描述姿态，无超越函数运算，可用于连续旋转	6个冗余，不直观，计算量大，数学意义复杂，传感器无法直接获取
欧拉角	无冗余参数，物理意义明确，传感器获取方便	在$ \pm \left( {1/2} \right){\text{π rad}} $时存在奇异，存在大量超越函数运算，不可连续旋转
罗格里斯参数	无冗余，较少的超越函数运算，可用于连续旋转	在$ \pm {1/2}{\text{π rad}} $时存在奇异，不直观
修正罗格里斯参数	无冗余，较少的超越函数运算，可用于连续旋转	在$ \pm {\text{π rad}} $时存在奇异，不直观
四元数	无奇异，无超越函数运算，可用于连续旋转表示	在$ \pm {2}{\text{π rad}} $时存在1个冗余，存在模糊性及退绕现象，不直观

状态	姿态/rad	角速度/（rad/s）
初始	$ [0\;\;0\;\;0]^\mathrm{T} $	$ [0\;\;0\;\;0]^\mathrm{T} $
期望	$ [-2.48\;\;1.48\;\;-1.12]^\mathrm{T} $	$ [0\;\;0\;\;0]^\mathrm{T} $

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