Rotation matrix

The columns describe a frame#

One interpretation places the three unit axes of a body frame into the columns of a matrix, expressed in a reference frame. Multiplying this matrix by a vector expressed in body coordinates expresses that vector in the reference coordinates.

Modern Robotics derives the constraints: each column has unit length, distinct columns are perpendicular, and the determinant is positive one. Together these define the group called SO(3).

Composition and inversion#

Multiplying two rotation matrices composes their rotations. The inverse is the transpose, which makes reversing a frame relationship straightforward. The order of composition matters because three-dimensional rotations generally do not commute.

A homogeneous transformation adds translation to this orientation representation, as the transformation-matrix lesson explains. Rotation alone cannot describe where a robot's hand is located.

Nine values represent three freedoms#

Although the matrix stores nine numbers, its constraints leave three degrees of freedom. MIT's discussion of orientation representations explains why matrices avoid the coordinate singularities of a minimal angle representation. An arbitrary nine-number array is not a valid rotation.

Quaternions and Euler angles encode the same orientation with different storage and mathematical properties. Choose the representation and frame convention explicitly when passing orientation data between systems.