From Code to Motion: Simulating and Deploying Robots

🧭 What is Affine Control Form?

A nonlinear system is in Affine Control Form if it can be written as:

      𝑥̇ = f(x) + g(x)·u
    

Where:

• x ∈ ℝⁿ is the state vector
• u ∈ ℝᵐ is the control input
• f(x) is the drift (uncontrolled) dynamics
• g(x) is the input matrix

This form is required by many nonlinear control methods like backstepping, CLF design, and feedback linearization.

✅ Why Use Affine Form?

• It separates dynamics that you can and cannot control
• Allows explicit design of u using Lyapunov or other control laws
• Essential for backstepping, sliding mode, adaptive, and nonlinear feedback control

📘 Example 1: Already in Affine Form

      𝑥̇₁ = x₂
      𝑥̇₂ = -x₁ + sin(x₂) + u
    

This is affine-in-control:

      f(x) = [x₂; -x₁ + sin(x₂)]
      g(x) = [0; 1]
    

So the system becomes:

      𝑥̇ = f(x) + g(x)·u
    

📘 Example 2: Not in Affine Form

      𝑥̇₁ = x₂
      𝑥̇₂ = u²
    

Here, u appears nonlinearly (u²), so this is not affine. You cannot extract u linearly.

Control methods requiring affine form won't work here unless the system is transformed.

⚠️ Key Conditions

• Control input must appear linearly: u, not u², sin(u), exp(u)
• Write equations so that: 𝑥̇ = f(x) + g(x)·u
• All uncontrolled terms go into f(x); all control-dependent terms into g(x)

📚 Summary Table