Robotics 1 - Rigid-Body Transformations

Rigid-body transformations describe the same point from different coordinate frames. The calculation always reduces to rotation, translation, and the order in which they are composed.

2D Transformations

Translation and Frames

A frame has an origin and orthonormal axes. If a point is written in frame, denote its coordinates by.

The notes write translation in its dimension-independent 3D form:

In 2D, simply remove thecoordinate. Translation is affine because it moves the origin.

Rotation Matrix

A counterclockwise rotation byis

Its columns are the rotated coordinate axes. They are orthonormal, so

Planar rotations commute:

Nested Frames

Let framehave positionand orientationrelative to. Lethave positionand orientationrelative to.

Rotateinto framebefore adding it to.

Then

The key rule is that vectors must be expressed in the same frame before addition.

Homogeneous Transform

Appending aturns the affine transformation into one matrix multiplication:

Thus: rotate first, then translate.

Composition Order

Forand,

Reversing the order changes the translation term, so.

Rotate first, then translate.

Translate first, then rotate.

With column vectors, the rightmost matrix acts first.

3D Transformations

Principal-Axis Rotations

Using the right-hand convention,

As in 2D,and.

Right-Hand Rule and Order

Point the right thumb along the positive axis; the curled fingers give the positive rotation direction.

Unlike 2D rotations, rotations about different 3D axes do not commute:

Therefore a rotation sequence must specify both the order and whether the axes are fixed or moving.

Homogeneous Transform

A 3D rigid-body transform is

Applied to a point,

Again, rotation happens first and translation second.

Frame-Label Composition

Letmap coordinates from frameinto frame:

Valid chains cancel their adjacent frame labels:

• A new motion expressed in the fixed/world frame is pre-multiplied.
• A new motion expressed in the current/body frame is post-multiplied.

Camera–Robot–World Example

Letbe the camera,the robot body, andthe world.

The matching inner frame labels determine the multiplication order.

The lecture gives

The adjacentlabels match, so

This is the practical workflow for every frame problem: label each transform, cancel adjacent frames, and then multiply in that order.

Cover photo: Ajay Suresh / Wikimedia Commons , licensed under CC BY 2.0 . The cover has been resized for layout.