Introduction to Robotics
Explainer
The 4×4 matrix behind every robot arm
Why does a robot need a 4×4 matrix to describe 3D space?
Quick check · tap an answer to continue
Lesson notes
Key points
- A pose has two parts: where the frame is, its position, and which way it faces, its orientation.
- Orientation is a 3×3 rotation matrix R, and position is a 3×1 translation vector d.
- Stacking them with a bottom row of 0 0 0 1 lets one multiplication rotate and translate a point together.
- Guess the payoff: chaining link matrices, T₀₁ × T₁₂ × T₂₃, gives the tool pose, which is forward kinematics.
Definitions
- Pose
- Position plus orientation of a frame relative to another frame.
Takeaway: A 4×4 transform packs rotation and translation into one multiply.
Quick check: What does the top-right 3×1 block of a homogeneous transform hold?
- The rotation angles
- The scaling factors
- The translation vector
The 3×1 column d stores the position of one frame's origin in the other.
PoseR and dHomogeneous formChaining
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