Skip to content

Rigid Body Transform Chain

Run the Rigid Body Transform Chain Fullscreen

Edit the Rigid Body Transform Chain with the p5.js editor

About This MicroSim

This MicroSim demonstrates how rigid body transforms compose in a kinematic chain, like a robot arm. See how each joint's rotation and link's translation combine to position the end effector.

How to Use

  1. Adjust Joint Angles: Use sliders to rotate each joint
  2. Show Coordinate Frames: Toggle to see local coordinate systems
  3. Show Transforms: View the transform composition
  4. Drag to Rotate: Change the 3D view angle
  5. Track End Effector: Watch the computed position update

Key Concepts

Each transform in SE(3) combines rotation R and translation t:

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

Transform composition: T_total = T₁ · T₂ · T₃

Joint Rotation Axis Link Length
1 (Base) Y (yaw) 120
2 (Shoulder) X (pitch) 100
3 (Elbow) X (pitch) 80

Learning Objectives

Students will be able to: - Understand SE(3) rigid body transforms - Compose multiple transforms via matrix multiplication - Compute forward kinematics for serial chains - Visualize how local transforms affect global position

Lesson Plan

Introduction (5 minutes)

Robot arms use chains of transforms where each joint adds rotation and each link adds translation. The composition order matters!

Exploration (10 minutes)

  1. Start with all joints at zero
  2. Rotate joint 1 - notice the entire arm rotates
  3. Rotate joint 2 - only links 2 and 3 move
  4. Observe how the end effector position updates

Forward Kinematics

The end effector position is computed by multiplying all transform matrices from base to tip. This is called forward kinematics.

References