Rotation vector (rotation matrix) and Euler angle

Rotating vector （ Rotation matrix ） And the Euler angle , The method of mutual transformation here is ：

Rotating vector --> Rotation matrix --> Euler Angle

Rotation matrix --> Euler Angle

One 、 Rotating vector --> Rotation matrix --> Euler Angle

SO(3) The rotation matrix of has 9 Quantity , But only 3 A degree of freedom , Empathy SE(3) Yes 16 Quantity , But only 6 A degree of freedom . In actual rotation , Any rotation can be represented by a rotation axis and a rotation angle , We use a vector , The direction is consistent with the rotation axis , The length is equal to the rotation angle , In this way, only a three-dimensional vector is needed to describe the rotation . about SE(3), It can be expressed by a rotation vector and a translation vector , It happens that the degree of freedom is 6. If you use a rotation vector to describe R： The axis of rotation is a vector of unit length n, The angle is 𝜃θ, that 𝜃nθn It can represent this rotation . Rotation matrix R Rotation vector 𝜃nθn The transformation process of is Rodriguez transformation ：

At the end of here n∧n∧  As shown above , Represents a vector represented by a matrix . Then, in turn, the rotation angle is obtained through the rotation matrix  𝜃θ; 

tr(R) For matrix R The trace of . For the shaft n,Rn=n; It means that the rotation of the rotating shaft around itself does not change , Mathematically speaking n It's a matrix R The characteristic value is 1 The eigenvector corresponding to . From now on, the rotation represented by the rotation axis and rotation angle is compact , No redundancy , But Euler angle RPY In the space of , When a rotation reaches +⎯⎯⎯90∘+_90∘ Yes, there is singularity . It is equivalent to the longitude and latitude of the earth when the latitude is +⎯⎯⎯90∘+_90∘ when , Longitude meaningless .

Two 、 Rotation matrix --> Euler Angle

These are some problems encountered at work , Simply record  , More abundant 3D conversion can understand .

Euler Angle 、 Four yuan number 、 Rotation matrix 、 Conversion between shaft angles

  Three dimensional rotation ： Euler Angle 、 Four yuan number 、 Rotation matrix 、 Conversion between shaft angles - You know (zhihu.com)