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Homogeneous Transformation

A homogeneous transformation matrix combines a rotation and a translation in one operation, making it easy to convert coordinates from one frame to another. When you chain several frames, link by link along a robot arm, you multiply the transformation matrices in the right order.

T=(Rp⃗01)T=\begin{pmatrix}R&\vec p\\0&1\end{pmatrix}homogeneous transform: rotation R and translation \vec p
x⃗G=Rx⃗L+p⃗\vec x_G=R\vec x_L+\vec pglobal coordinates from local ones

Symbols

RRrotation matrix
p⃗\vec porigin of the new frame, in the old one
x⃗L, x⃗G\vec x_L,\ \vec x_Glocal and global coordinates

Example

Frame B rotated 90°90° and shifted (5,0)(5,0); the point (1,0)(1,0) locally:

x⃗G=(cos⁡90∘⋅1,sin⁡90∘⋅1)+(5,0)=(5,1)\vec x_G=(\cos90^\circ\cdot1,\sin90^\circ\cdot1)+(5,0)=(5,1).

Rotation and translation in one matrix – remember to rotate first, then add p⃗\vec p.
Practise rotations and transformations for free →

← Rotation Matrix · Euler Angles and Quaternions →

Part of Robotics: Rotations and transformations.