From Code to Motion: Simulating and Deploying Robots

⚙️ Feedback Linearization

Feedback Linearization is a nonlinear control technique that cancels the nonlinearities of a system by applying a specific transformation and control law, effectively converting the system into an equivalent linear system for the purpose of control design.

🔧 Motivation

Many robotic systems (like manipulators, mobile robots, drones) have nonlinear dynamics, making traditional linear controllers (like PID or LQR) insufficient. Feedback linearization allows us to "flatten" these nonlinear dynamics into a simpler linear form.

📘 Basic Idea

Consider a nonlinear system:

    ẋ = f(x) + g(x)u
    y = h(x)
  

The goal is to design a control input u such that the input-output behavior becomes linear. This is done by applying a control law:

    u = α(x) + β(x)v
  

Where v is the new (linear) control input, and α(x) and β(x) are designed to cancel out the system's nonlinearities.

🚀 Application in Robotics

• Control of robotic arms (trajectory tracking)
• Quadrotor and drone flight control
• Mobile robot path-following
• Exoskeleton or rehabilitation robot control

🧮 Example (Simple SISO)

Suppose you have a second-order system:

    ÿ = f(x) + g(x)u
  

Define the control input as:

    u = (1/g(x)) * [-f(x) + v]
  

Now the system becomes:

    ÿ = v
  

Which is a linear double integrator! You can now apply a linear controller like PD:

    v = -k₁e - k₂ė
  

✅ Benefits

• Transforms nonlinear systems into linear ones for easier control
• Enables use of well-understood linear control techniques
• Effective for systems with known and exact models

⚠️ Limitations

• Requires full knowledge of system dynamics
• May not handle uncertainties or disturbances well
• Not suitable for underactuated systems directly

📌 Summary

• Feedback linearization cancels nonlinearities to create a linear system
• Often used in nonlinear robotic systems
• Powerful but sensitive to model inaccuracies