Kinematics

Kinematics: Teaching Your Robot Where to Go

Imagine you want to grab a coffee cup on your desk. Your brain instantly calculates how to move your shoulder, elbow, and wrist to reach exactly the right spot. That's essentially what kinematics does for robots – it's the math that turns "go there" into precise motor commands.

The Challenge: From Motors to 3D Space

Your robot has actuators (motors) that can move to specific angles when you send them commands. But here's the tricky part: you don't think in joint angles. You think in 3D space – "move to that position over there."

So how do we bridge this gap? How do we translate between "move your joints to 45°, 30°, 60°" and "pick up that object at coordinates (10, 5, 8)"?

That's where kinematics comes in – it's the mathematical translator between joint space and real-world positions.

Forward Kinematics: From Joint Angles to Position

Forward kinematics asks: "If I set my joints to these specific angles, where will my robot's hand end up?"

Think of it like following a recipe. You know all the ingredients (joint angles), and you want to see what the final dish (end effector position) looks like.

$$\mathbf{x} = \mathbf{f}(\mathbf{q}) = \begin{bmatrix} x \\ y \\ z \\ \phi \\ \theta \\ \psi \end{bmatrix} = \begin{bmatrix} f_1(q_1, q_2, \ldots, q_n) \\ f_2(q_1, q_2, \ldots, q_n) \\ f_3(q_1, q_2, \ldots, q_n) \\ f_4(q_1, q_2, \ldots, q_n) \\ f_5(q_1, q_2, \ldots, q_n) \\ f_6(q_1, q_2, \ldots, q_n) \end{bmatrix}$$

Where:

• q = joint angles (what you control)

• x = end effector pose (where your robot hand ends up)

• x, y, z = position in 3D space

• φ, θ, ψ = orientation (how the hand is rotated)

For simple robots: If you have just one joint (like a single rotating arm), this is just trigonometry – the kind you learned in high school.

For complex robots: Multiple joints make this much trickier. That's where the Denavit-Hartenberg convention comes in – a systematic way to handle all those interconnected rotations and translations. It's like having a standardized coordinate system for each joint.

Inverse Kinematics: The Real Challenge

Inverse kinematics asks the harder question: "I want my robot's hand at this specific position – what joint angles do I need?"

This is like reverse-engineering a recipe. You know what the final dish should look like, but you need to figure out the exact ingredients and proportions.

$$\mathbf{q} = \mathbf{f}^{-1}(\mathbf{x})$$

Why is this so much harder?

Multiple Solutions: Just like you can reach the same coffee cup by bending your elbow up or down, robots often have multiple ways to reach the same point. Which one should it choose?

Unreachable Positions: Try touching the back of your head with your elbow – some positions are just physically impossible. Robots have similar limitations.

Complex Math: Unlike forward kinematics, there's often no simple formula. You're solving systems of nonlinear equations that can be quite nasty.

The Solution: Treat it as an optimization problem. Instead of trying to solve it exactly, we iteratively adjust joint angles until we get "close enough" to the target position. It's like playing a game of "hot and cold" until you find the treasure.

Making the Robot Move: Putting It All Together

Now that you understand the theory, here's how you actually make your robot move to a specific position:

1. Know Where You Are Read the current joint angles from your motors. This tells you the robot's starting pose.

2. Calculate Where to Go Use inverse kinematics to figure out what joint angles will get you to your target position. This often involves iterative solving – making small adjustments until you converge on the right answer.

3. Send the Commands Tell each motor to move to its calculated angle. The robot smoothly transitions from its current pose to the desired one.

Pro tip: You rarely jump directly to the target. Instead, you plan a smooth path with intermediate waypoints, solving inverse kinematics for each step along the way.

Simulation: Practice Before You Break Things

Before commanding a real robot to move, smart developers test everything in simulation first. Think of it as a physics-accurate video game where you can:

• Test your kinematics without risk of damaging expensive hardware

• Iterate rapidly on motion planning algorithms

• Replicate real-world physics while running much faster than real-time

• Debug weird edge cases that might crash a real robot

Simulation is especially crucial for inverse kinematics, where a small math error could send your robot arm flying in the wrong direction. Better to discover that bug in the virtual world first!

Why This Matters for You

Whether you're building a simple pick-and-place robot or training the next generation of intelligent manipulators, kinematics is your foundation. Every time you want your robot to "go there" or "do that," you're using these concepts.

Master kinematics, and you'll have robots that move with precision and purpose. Skip it, and you'll have expensive paperweights that wave around randomly.

Ready to see this in action? Let's start with some simple forward kinematics examples...