Formelsammlung


I. Coordinate Transformation, DCM

: Vector in Euklidian space
: Coordinates of a vector measured in the frame
: Coordinates of a vector measured in the frame


Coordinates of basis vector measured in Frame :

   

       

: -th coordinate of vector measured in K

: -th coordinate of vector measured in K‘

Transformation of coordinates of a vector :

    


II. Rigid Body

: Orientation of body-fixed coordinate frame as measured in the earth coordinate frame
: Coordinates of a vector measured in the earth frame
: Coordinates of the same vector measured in the body frame
: Rotation rate vector of body measured in the body frame
: Unit quaternion associated to the rotation matrix


Definition Pitch-Roll:

Definition Yaw-Pitch-Roll:

Definition II Pitch-Roll:

Definition II Yaw-Pitch-Roll:


III. Rotations

Rotation Matrices

     




Generators

     


Definition: 

,    ,   


  


Rotations and Generators


Rotation Derivatives

Definition: 


IV. Quaternions

: Quaternion
: Real part of quaternion
: Imaginary part of quaternion



Quaternion Multiplications


Unit Quaternions



Unit Quaternions and Rotations

: Unit quaternion associated to the rotation matrix
: Rotation matrix associated to the unit quaternion


,     where

,        where    (for rigid body kinetics it’s )


Note: For numerical implementation see [1].


Spherical Linear Interpolation, Slerp

,     where


V. General





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