Kinematics - Rigid Body Transformation

Core Math in Robotics Kinematics following course MEAM 620 at Penn

January 29, 2017 - 3 minute read -
robotics notes

Rigid Body Transformations


  • 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 (right-hand rule) by angle


Composite Transformations

  • Post-multiply successive transformations about intermediate frames
  • Pre-multiply 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

Rodrigues’ Formula

Rodrigues’ formula gives us an decomposition of the rotation matrix into axis and angle.


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 angle-axis

    • Not unique

    • Commutativity

    • Conjugate

    • Inverse

    • Norm

    • Multiplication

    • Operation on a vector ( is representing rotation matrix )