Homogeneous Transformation: Rotation and Translation – Fusion of Engineering, Control, Coding, Machine Learning, and Science

In our previous post, we have explained rotation matrices. In this post, we explain how to perform rotation and translation as a single matrix operation.

A YouTube video accompanying this post is given below.

Consider the figure below.

The coordinate system $B$ is translated from the coordinate system $A$ , and after that it has been rotated for the angle $\theta$ . The location of the coordinate system $B$ with respect to the coordinate system $A$ is represented by the vector $\leftindex^{A}{\mathbf{d}}$ . The notation $\leftindex^{A}{\mathbf{d}}$ means that the vector $\mathbf{d}$ is represented in the coordinate system $A$ . That is, its components (projections) are represented in the coordinate system $A$ . Similarly, the notation $\leftindex^{B}{\mathbf{p}}$ means that the vector $\mathbf{p}$ is represented in the coordinate system $A$ . Let the coordinates of the vector $\mathbf{p}$ expressed in the coordinate system $B$ be given as follows:

(1) $\begin{align*}\leftindex^{B}{\mathbf{p}} =\begin{bmatrix} p_{x_{B}} \\ p_{y_{B}} \end{bmatrix}\end{align*}$

where $p_{x_{B}}$ and $p_{y_{B}}$ are the coordinates of the vector $\mathbf{p}$ expressed in the coordinate system $B$ .

Problem 1: Given the coordinates of the vector $\leftindex^{B}{\mathbf{p}}$ , translation vector $d$ , and the angle of rotation $\theta$ , find the coordinates of the vector $\leftindex^{A}{\mathbf{p}}$ .

Solution:

(2) $\begin{align*} \leftindex^{A}{\mathbf{p}} = \leftindex^{A}{\mathbf{d}} + \leftindex^{A}_{B}{R} \cdot \leftindex^{B}{\mathbf{p}} \end{align*}$

where $\leftindex^{A}_{B}{R}$ is the rotation matrix that transforms vectors from $B$ to $A$ coordinate systems.

That is

(3) $\begin{align*}\begin{bmatrix} p_{x_{A}} \\ p_{y_{A}} \end{bmatrix} = \begin{bmatrix} d_{x_{A}} \\ d_{y_{A}} \end{bmatrix} + \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix} \begin{bmatrix} p_{x_{B}} \\ p_{y_{B}} \end{bmatrix} \end{align*}$

If you do not remember how the rotation matrix $\leftindex^{A}_{B}{R}$

(4) $\begin{align*} \leftindex^{A}_{B}{R} = \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix}\end{align*}$

is constructed, see our previous post.

By multiplying vectors and matrices, and by adding the results, from (3), we have

(5) $\begin{align*} p_{x_{A}} = d_{x_{A}} + cos(\theta) p_{x_{B}} - sin(\theta) p_{y_{B}} \\ p_{y_{A}} = d_{y_{A}} + sin(\theta) p_{x_{B}} + cos (\theta) p_{y_{B}} \end{align*}$

The tranformation (3), can be written as a single vector matrix multiplications. Namely, we can formally write

(6) $\begin{align*}\begin{bmatrix} \leftindex^{A}{\mathbf{p}} \\ 1 \end{bmatrix} = \begin{bmatrix} \leftindex^{A}_{B}{R} & \leftindex^{A}{\mathbf{d}} \\ \mathbf{0}_{2\times 1} & 1 \end{bmatrix} \begin{bmatrix} \leftindex^{B}{\mathbf{p}} \\ 1 \end{bmatrix} \end{align*}$

where $\mathbf{0}_{2\times 1}$ is 2 times 1 matrix of zeros. By expanding the last equation, we obtain

(7) $\begin{align*}\begin{bmatrix} p_{x_{A}} \\ p_{y_{A}} \\ 1 \end{bmatrix} = \begin{bmatrix} cos(\theta) & -sin(\theta) & d_{x_{A}} \\ sin(\theta) & cos(\theta) & d_{y_{A}} \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} p_{x_{B}} \\ p_{y_{B}} \\ 1 \end{bmatrix} \end{align*}$

The matrix

(8) $\begin{align*} \begin{bmatrix} cos(\theta) & -sin(\theta) & d_{x_{A}} \\ sin(\theta) & cos(\theta) & d_{y_{A}} \\ 0 & 0 & 1 \end{bmatrix} \end{align*}$

is called a homogeneous transform.

In the next post, we we will see how to use this transform to solve the forward kinematics problem of robotic manipulators.

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