In one line
A homogeneous transformation matrix is a 4×4 matrix that rotates and translates a point (or frame) in one multiplication, so successive motions can be chained by multiplying matrices.
Explained simply
Telling a friend how to find your seat in a cinema: 'turn right, then walk 5 steps'. Each instruction is a turn or a move. A homogeneous matrix packs one turn and one move together, and you can multiply several of them to get the combined instruction.
Technically, \( T = \begin{bmatrix} R & p \\ 0\;0\;0 & 1 \end{bmatrix} \), where R is the 3×3 rotation matrix and p the 3×1 position vector of the new frame's origin in the old frame. Points are written as \( [x\; y\; z\; 1]^T \).
How it works
- Rotation about z by \( \theta \): \( R_z = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \).
- Pure translation by (a, b, c): R = identity, p = (a, b, c).
- Transform a point: \( P_0 = T\,P_1 \), where \( P_1 \) is the point in the moving frame, \( P_0 \) in the base frame.
- Chain frames: \( {}^0T_2 = {}^0T_1\,{}^1T_2 \). Order matters: matrix multiplication is not commutative.
- Inverse: \( T^{-1} = \begin{bmatrix} R^T & -R^Tp \\ 0 & 1 \end{bmatrix} \).
Key formula
\( T = \begin{bmatrix} R_{3\times3} & p_{3\times1} \\ 0_{1\times3} & 1 \end{bmatrix} \); R is orthonormal (\( R^{-1} = R^T \), det R = 1); p in m.
Quick example
A frame is rotated 90° about z and its origin placed at (2, 0, 0). A point at (1, 0, 0) in the moving frame: rotation gives (0, 1, 0), then translation gives (2, 1, 0) in the base frame. In matrix form, T has first row [0, -1, 0, 2], second row [1, 0, 0, 0], third row [0, 0, 1, 0].
Common mistakes
- Using a 3×3 matrix and forgetting the translation column, or forgetting the 1 at the end of the point vector.
- Multiplying in the wrong order when chaining frames.
- Using the transpose of R (inverse rotation) by mistake.
How MU asks it
Theory 3 to 4 marks: 'What is a homogeneous transformation matrix? Write its general form and explain each part'. Numerical 5 to 6 marks: find the new coordinates of a point after given rotations and translations, or build T for a 2-DOF arm. Write each matrix, multiply step by step, and state the final coordinates.