The exponential coordinates parametrize a rotation matrix in terms of a rotation axis (represented by a unit vector  ) and an angle of rotation θ about that axis; the vector then serves as the three-parameter exponential coordinate representation of the rotation.

（1）如果定系s绕  轴旋转θ,新的位置相对于定系将由旋转矩阵R表达

（2）the angular velocity  expressed in {s} such that, if a frame initially coincident with {s} followed  for one unit of time (i.e.,  is integrated over this time interval), its final orientation would be expressed by R.

(3)the angular velocity  expressed in {s} such that, if a frame initially coincident with {s} followed  for θ units of time (i.e., θ is integrated over this time interval) its final orientation would be expressed by R.

（1）绕某轴    ）旋转

（2）将（1）中两者相乘形成的三维向量

时，引入矩阵的迹：

时，  ,

（1）如果  ,意味着  且旋转轴  没有被定义

（2）如果  ，意味着  ,旋转轴  的表示为下式三者任意一个：

（3）如果  ,那么：