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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。
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.
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Key words:
- quaternion /
- Euler angle /
- ternary angle /
- coordinate transformation /
- deflection-vector axis /
- deflection-vector angle
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图 1 三元角坐标变换方式示意图
Figure 1. Schematic diagram of ternary angle coordinate transformation
图 2 末系到初系先偏转过程简图
Figure 2. Process diagram of deflection first from last coordinate system to initial coordinate system
图 3 末系到初系先旋后偏过程图
Figure 3. Process diagram of rotation first deflection second from last coordinate system to initial coordinate system
图 4 末系到初系先旋转过程图
Figure 4. Process diagram of rotation first from last coordinate system to initial coordinate system
图 5 末系到初系后偏转过程图
Figure 5. Process diagram of post deflection from last coordinate system to initial coordinate system
图 6 坐标系的三维变换效果图
Figure 6. Three-dimensional effect graphs of coordinate transformation
图 7 常值漂移单轴旋转调制效果分析
Figure 7. Analysis on effect of single-axis rotation modulation with constant drift
图 8 常值漂移复合旋转调制效果分析
Figure 8. Analysis on effect of compound rotation modulation with constant drift
表 1 航向角真值
Table 1. Truth value of course angle
T22 T21 ψ →0 + 90° →0 - -90° + + ψt + - ψt - + ψt+180° - - ψt-180° 
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