Rigid Body Transformations in 2D

Provides a brief review of 2D rigid body transformations, from the perspective of a programmer who just wants to use them. See (Lynch, Park Modern Robotics Chapter 3) for more details.

A 2D coordinate frame consists of 2 orthogonal unit vectors located somewhere in space

• Whenever I discuss frames in these notes, they are stationary. The body frame of a moving robot is assumed to be instantaneously fixed to the location of the robot at any given time.

A rotation matrix describes the rotation between two coordinate frames. If there is a pure rotation (i.e., no translation) between frames \(\{a\}\) (with coordinate vectors \(\hat{x}_a\) and \(\hat{y}_a\)) and \(\{b\}\) (with coordinate vectors \(\hat{x}_b\) and \(\hat{y}_b\)) the rotation matrix \(R_{ab}\) lets you find the coordinates of a point \(p\) in frame \(a\) given coordinates in frame \(b\) using the equation

\begin{equation} p_a = R_{ab} p_b \end{equation} \begin{equation} R_{ab} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \end{equation}

• Rotations can be composed: \(R_{ac} = R_{ab}R_{bc}\).
• Rotations can always be inverted and this reverses the rotation:

\begin{equation} R_{ba} = R_{ab}^{-1} = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \end{equation}

If the origin of frame \(\{j\}\) is displaced from frame \(\{i\}\) by \(x \hat{x}_i + y \hat{y}_i\) and the angle between the frames is \(\theta\), then to convert a point \(p_j\) in frame \(j\) to frame \(i\) coordinates:

\begin{equation}\label{eq:homtran} \begin{bmatrix}p_i \\ 1\end{bmatrix} = T_{ij}\begin{bmatrix}p_j \\ 1 \end{bmatrix}, \end{equation}

where

\begin{equation} T_{ij} = \begin{bmatrix} \cos\theta & -\sin\theta & x \\ \sin\theta & \cos\theta & y \\ 0 & 0 & 1 \end{bmatrix} \end{equation}

The inverse is also well defined and allows you to reverse the transformation

\begin{equation} T_{ij}^{-1} = T_{ji} = \begin{bmatrix} \cos\theta & \sin\theta & -x\cos\theta -y\sin\theta \\ -\sin\theta & \cos\theta & -y\cos\theta +x\sin\theta \\ 0 & 0 & 1 \end{bmatrix} \end{equation}

In Equation\eqref{eq:homtran} the points \(p_i\) and \(p_j\) are written in homogeneous coordinates, that is a two-dimensional point \((x, y)\) is written as \((x, y, 1)\).

• There is intuition and geometric understanding you can learn to understand the reasoning behind the use of homogeneous coordinates (discussed later)
• You can also, for the purposes of this class, view homogeneous coordinates as a mechanical rule: append a 1 to the point/vector you are transforming, then use only the first two elements of the result

A 2D twist consists of an angular velocity and two translational velocities:

\begin{equation} V = \begin{bmatrix} \dot{\theta} \\ \dot{x} \\ \dot{y} \end{bmatrix} \end{equation}

• A twist represents a rigid body motion.
• For a moving rigid body, each point on that body may have a different velocity.
• For example, on a spinning disc, the points farther from the center of rotation have a higher translational velocity.

The 2D adjoint converts a twist in one frame to a twist in another frame.

• These are just different coordinate representations for what is the same rigid body motion.

The 2D Adjoint is

\begin{equation} \mathrm{Ad}_{ij} = \begin{bmatrix} 1 & 0 & 0 \\ y & \cos\theta & -\sin\theta \\ -x & \sin\theta & \cos\theta \end{bmatrix}. \end{equation}

To convert a twist from one frame to another

\begin{equation} V_i = \mathrm{Ad}_{ij} V_j \end{equation}

• These are some thought exercises designed to help build intuition about frames
• They are designed as a guide for thinking about these concepts
• Use interactively: read a line, then try to visualize. It may help to close your eyes
• This section is experimental: everyone thinks differently and I tried to put in words how I think about these concepts. I'm not sure how helpful it will be to others: let me know!