Understanding Robot Kinematics

In this section, we’ll build intuition for kinematics. Which is the math that links joint angles to end‑effector positions i.e. where is the hand given the joints? We will do this by walking through a concrete example.

Let’s explore how traditional robotics approaches robot control through a concrete example.

From Complex to Simple: The SO-100 Example

We start with a familiar robot and simplify it so we can reason step by step without losing the key ideas.

The SO-100 arm simplified to a 2D planar manipulator by constraining some joints.

The SO-100 is a 6-degree-of-freedom (6-DOF) robot arm. To understand the principles, let’s simplify it to a 2-DOF planar manipulator by constraining some joints.

The Simplified Robot

To keep the math clear, our simplified robot has:

• Two joints with angles θ₁ and θ₂
• Two links of equal length l
• Configuration q = [θ₁, θ₂] ∈ [-π, +π]²

Forward Kinematics: From Joints to Position

Question: Given joint angles θ₁ and θ₂, where is the end-effector?

Answer: We can calculate the end-effector position mathematically: $$p(q) = \begin{pmatrix} l \cos(\theta_1) + l \cos(\theta_1 + \theta_2) \\ l \sin(\theta_1) + l \sin(\theta_1 + \theta_2) \end{pmatrix}$$

This is called Forward Kinematics (FK) - mapping from joint space to task space.

Inverse Kinematics: From Position to Joints

Now let’s try to invert the mapping: given a desired hand position, what joints achieve it?

Question: Given a desired end-effector position p*, what joint angles should we use?

Answer: This is Inverse Kinematics (IK) - much harder to solve!

We need to solve: $p(q) = p^*$

In general, this becomes an optimization problem:$$\min_{q \in \mathcal{Q}} \|p(q) - p^*\|_2^2$$

The Challenge of Constraints

Free to move

Constrained by floor

Multiple obstacles

Real robots face constraints:

• Physical limits - Can’t move through the floor
• Obstacles - Must avoid collisions
• Joint limits - Finite range of motion

These constraints make the feasible configuration space $\mathcal{Q}$ much more complex!

Key Insight

Even for this simple 2-DOF robot, solving IK with constraints is non-trivial. Real robots have:

• 6+ degrees of freedom
• Complex geometries
• Dynamic environments
• Uncertain models

Traditional approaches require extensive mathematical modeling and expert knowledge for each specific case.