Lesson 02 · 15 minutes · Preparation

A position needs
a point of view.

Your next win: explain why “move to x = 0.4” is incomplete, and why a visible object can still be unreachable.

A coordinate frame is a measuring convention

A frame specifies an origin and axis directions. The same cup can have one position relative to the table and another relative to an arm's base. Before a controller acts on a position, it needs the frame and units. A camera pixel is not automatically a position in metres.

We will eventually use two arm-base frames, camera frames, and a world frame. For now, explore a two-link drawing in one plane. This is an educational sketch, not the SO-101 geometry, an IK solver, or a physics engine.

Turn the joints; watch the tip

xy base

Link lengths: 0.3 m and 0.2 m. The dashed circle marks the maximum geometric reach in this simplified sketch.

Forward and inverse questions

Forward kinematics asks where the tip is when the joint angles are known. For our two-link sketch, adding the two link vectors gives the tip position:

x = 0.3 cos(shoulder) + 0.2 cos(shoulder + elbow)
y = 0.3 sin(shoulder) + 0.2 sin(shoulder + elbow)

Inverse kinematics asks which joint angles could place the tip at a desired target. There may be several answers, or none. Actual robot joint limits, orientation requirements and obstacles remove additional possibilities. Seeing a target is not evidence that an arm can grasp it.

What is missing from “move to x = 0.4, y = 0.1”?

Try three configurations

  1. Set both angles to 0°. Read the tip position. It should be 0.5 m along the x axis.
  2. Leave the shoulder at 0° and fold the elbow to 180°. The tip is now 0.1 m from the base.
  3. Predict whether this drawing can reach x = 0.6 m, y = 0.0 m. Explain why without moving a slider.
For the future two-arm scene, “reachable” must be checked for both arms and their shared space. A language model proposing a target cannot waive the robot's geometry or collision constraints.

Read a little deeper

Read the coordinate-frame and kinematics material in MIT's Robotic Manipulation notes. Connect the diagrams to the sketch before worrying about the course's software stack.