From Code to Motion: Simulating and Deploying Robots

📘 What is the Lie Derivative?

The Lie Derivative describes how a scalar function changes along the flow of a vector field. It is fundamental in nonlinear control, especially in feedback linearization and geometric control.

🧠 Intuition

Imagine a scalar function h(x) defined over a space, like temperature over a field. If you walk along a vector field f(x) (like wind direction), the Lie Derivative tells you how fast the temperature is changing in the direction of the wind.

🧮 Mathematical Definition

    Lfh(x) = ∇h(x) · f(x)
  

• h(x): Scalar function (e.g., output of the system)
• f(x): Vector field (e.g., system dynamics)
• ∇h(x): Gradient of h(x)

The Lie derivative is the directional derivative of h along the vector field f.

⚙️ In Control Systems

For a nonlinear system:

    ẋ = f(x) + g(x)·u
    y = h(x)
  

The Lie derivatives tell how the output y depends on the state x and input u.

• Lie Derivative of h along f: Lfh(x)
• Lie Derivative of h along g: Lgh(x) = ∇h(x) · g(x)

🔄 Higher-Order Lie Derivatives

Sometimes one derivative is not enough. For example:

    Lf2h(x) = Lf(Lfh(x)) = ∇(Lfh(x)) · f(x)
  

These are used to determine relative degree and to construct feedback linearizing controllers.

🤖 Application in Robotics

• Used in feedback linearization to design nonlinear controllers
• Helps express system output dynamics in terms of input u
• Used in planning and stability analysis of robotic systems

📌 Summary

• Lie Derivative measures how a scalar function changes along a vector field
• Essential for nonlinear system analysis and control
• Used heavily in feedback linearization, backstepping, and differential geometry-based control