· Hankyu Kim · Robotics  · 2 min read

Transformation Matrix

The core idea of transformation matrices is that a point represented in one coordinate frame (B) can be expressed in another frame (A) by combining rotation and translation into a single operator.

The core idea of transformation matrices is that a point represented in one coordinate frame (B) can be expressed in another frame (A) by combining rotation and translation into a single operator.

Recall) Rotation Matrix

A rotation matrix is a 3x3 matrix that tells you how one coordinate system is rotated relative to another. It’s made of cosines of angles between the axes of the two coordinate systems and has the neat property that its inverse is just its transpose, making calculations easier. In robotics, rotation matrices help represent and compute how robot parts or tools are oriented in space, which is essential for tasks like moving a robot arm to pick up objects correctly.

By combining rotation matrices with position vectors into a transformation matrix, robots can precisely control both where and how they move in three-dimensional space. This concept forms a mathematical foundation for robot kinematics and manipulation.

Rotation, Translation, and Transformation Matrices

A 3×3 rotation matrix RR describes the orientation change between frames, and a 3×1 translation vector tt represents the shift of origins. These are combined into a 4×4 homogeneous transformation matrix TT:

T=[Rt01]T = \begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix}

This allows simultaneous handling of orientation and position changes. Points are expressed in homogeneous coordinates by appending a 1:

ph=[xyz1]p_h = \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}

The transformed point in another frame is:

pA=TAB pBp_A = T_{A}^{B} \, p_B

Dual Interpretation of Transformation Matrices

  1. Coordinate Mapping: Expresses the same physical point in a different frame.
  2. Operator: Physically moves points from one frame to another.

The mathematics is identical; only the interpretation differs.

Rotations About Axes

Rotations about x, y, z axes can be represented individually:

Rx(θ)=[1000cos⁡θ−sin⁡θ0sin⁡θcos⁡θ],Ry(θ)=[cos⁡θ0sin⁡θ010−sin⁡θ0cos⁡θ],Rz(θ)=[cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001]R_x(\theta) = \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos\theta & -\sin\theta\\ 0 & \sin\theta & \cos\theta \end{bmatrix}, \quad R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\\ -\sin\theta & 0 & \cos\theta \end{bmatrix}, \quad R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1 \end{bmatrix}

Composite Transformations

Multiple frames can be linked by multiplying transformation matrices:

TAC=TAB TBCT_{A}^{C} = T_{A}^{B} \, T_{B}^{C}

This is crucial in robotics, e.g., computing the end-effector position relative to the base through a kinematic chain.

Practical Applications

  • Determining a robot end-effector’s position and orientation relative to objects
  • Moving points through multiple coordinate frames
  • Planning and simulating robot trajectories with chains of transformations
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