6.3. Geometric Transforms

We now discuss an application of matrix multiplication related to robotics, machine design, image processing, animation, and any other field of study concerned with the geometry of items that move in space. The discussion here is directed toward machines, especially robots, but the concepts can be directly applied to other domains. Geometric transformations determine the coordinate location and orientation of items in space. At the heart of geometric transformations is matrix multiplication.

A linear transformation can be regarded as a function. However, a matrix usually changes a vector or matrix when the two are multiplied. We might say that we apply a transform to a vector, which means that matrix to vector multiplication occurs. A linear transformation \(\bf{T}\) must satisfy the two linear criteria.

(6.7)\[\begin{split}\begin{aligned} \mathbf{T}(\bm{u} + \bm{v}) &= \mathbf{T}(\bm{u}) + \mathbf{T}(\bm{v}) \\ \mathbf{T}(c\,\bm{u}) &= c\,\mathbf{T}(\bm{u}) \end{aligned}\end{split}\]

The requirements of equation (6.7) are clearly satisfied if \(\bf{T}\) is a matrix and \(\bf{T}\) is applied to a vector or matrix with matrix multiplication, \(\mathbf{T}(u) = \mathbf{T}\,\bm{u}\).

A detailed discussion of geometric transformations is found in chapters 2 and 7 of Peter Corke’s text on robotics control [CORKE17].

A coordinate frame in three-dimensional space consists of three perpendicular (orthogonal) axes. Points have a \((x, y, z)\) position relative to the origin of the coordinate frame. We often use multiple coordinate frames to simplify the expression of point locations. For example, we usually have a world coordinate frame. Stationary robots use the robot’s base as the origin of its world coordinate frame. There might also be coordinate frames for the workpiece the robot is working on, the robot’s end effector, or a camera if we use a vision system. The location and orientation of each coordinate frame is called the pose of the coordinate frame, which is expressed by a matrix. We use matrix multiplication to express the pose of the coordinate frame and associated points in other coordinate frames.

We will discuss rotation and translation of points and coordinate frames in two dimensions. The concepts described in two dimensions can then be extended to three dimensions.