北京航空航天大学学报 ›› 2017, Vol. 43 ›› Issue (12): 2539-2546.doi: 10.13700/j.bh.1001-5965.2016.0882

• 论文 • 上一篇    下一篇

基于三元角的坐标旋转变换方法

苗继松, 邵琼玲, 任元   

  1. 装备学院, 北京 101416
  • 收稿日期:2016-11-17 修回日期:2017-02-15 出版日期:2017-12-20 发布日期:2017-04-07
  • 通讯作者: 任元 E-mail:renyuan_823@aliyun.com
  • 作者简介:苗继松,男,硕士研究生。主要研究方向:捷联惯性导航与快速发射技术;任元,男,博士,副教授,硕士生导师。主要研究方向:先进惯性测量与导航控制技术。
  • 基金资助:
    国家“863”计划(2015AA7026083)

Coordinate rotation transformation method based on ternary angle

MIAO Jisong, SHAO Qiongling, REN Yuan   

  1. Equipment Academy, Beijing 101416, China
  • Received:2016-11-17 Revised:2017-02-15 Online:2017-12-20 Published:2017-04-07
  • Supported by:
    National High-tech Research and Development Program of China (2015 AA7026083)

摘要: 坐标旋转变换常用方法有四元数和欧拉角。欧拉角需3次转位,3个参数,有12种转位次序;四元数需一次转位,4个参数。欧拉角因转位次序固定,极易出现万向节锁现象。四元数虽可避免万象节锁现象,但比欧拉角多一个维度,在数据存储上要多33%的数据量,且易因浮点数舍入误差累积而导致不合法现象。为避免上述方法的缺陷,提出一种新的坐标旋转变换方式,引入偏矢轴和偏矢角等全新概念,并严格推导了基于三元角的坐标旋转变换矩阵。在描述上,该方法仅需2次转位,比欧拉角转位次数少,且避免了万象节锁现象;比四元数参数少,且更形象直观,易理解,在对复合运动的描述上更为方便。所提方法对惯性导航、旋转调制等相关领域中姿态变换的设计与分析提供了更加方便的数学手段。

关键词: 四元数, 欧拉角, 三元角, 坐标变换, 偏矢轴, 偏矢角

Abstract: Quaternion and Euler angle are used to describe coordinate transformation. Euler angle is characterized by three-time rotation and three parameters, and there are 12 kinds of rotation order. The characteristics of the quaternion are described by one rotation and four parameters. Using Euler angle is easy to cause gimbal lock phenomenon. Although it can avoid gimbal lock phenomenon, quaternion is more than Euler angles with one dimension and 33% amount of data. It may be illegal due to the accumulation of rounding error of floating point. To avoid the defects of the above methods, a new coordinate transformation method was proposed and two new concepts of deflection-vector axis and deflection-vector angle were introduced. The coordinate rotation transformation matrix based on the ternary angle was strictly deduced. Compared with the Euler rotation transformation, this method needs less rotation and avoids gimbal lock phenomenon; compared with the quaternion, it needs less parameters and is easy to understand. This method is more convenient for the description of the compound rotation. The proposed method provides more convenient mathematical means for the design and analysis of attitude transformation in related fields, such as inertial navigation and rotation modulation.

Key words: quaternion, Euler angle, ternary angle, coordinate transformation, deflection-vector axis, deflection-vector angle

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