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 Roll-Pitch:
Definition II Yaw-Roll-Pitch:
III. Rotations
Rotation Matrices
Generators
Definition:
,
,
Rotations and Generators
Rotation Derivatives
Definition:
,
,
Skew Symmetric Matrices
,
Rotations and Skew Symmetric Matrices
,
,
Exponential Map
with
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