Rigid Body Transformations
Notations
 represents the orientation of frame with respect to frame .
 represents the translation of frame with respect to frame .
 represents the homogeneous transformations of frame with respect to frame .
Where, in 3d, is a 3 3 matrix, is a vector.
 represent rotations about the axis (righthand rule) by angle
Composite Transformations
 Postmultiply successive transformations about intermediate frames
 Premultiply successive transformations about fixed frames
Composition of homogeneous transformations follows the rules of rotation matrices.
Inverse Transformations
 Rotations
 Homogeneous transformations
Skew Symmetric Matrices
is the skew symmetrix matrix.
For any skew symmetrix matrix .
2D Rotation
The rotation in 2d can be viewed as the complex number
Rodriguesâ€™ Formula
Rodriguesâ€™ formula gives us an decomposition of the rotation matrix into axis and angle.
Quaternions
Quatornion is a 4 dimensional representation of the rotation matrix. The basis are . The quatornion is a generalization of complex number.
 Unit Quatornion Properties

Relation to angleaxis

Not unique

Commutativity

Conjugate

Inverse

Norm

Multiplication

Operation on a vector ( is representing rotation matrix )
