Module 05: Dynamics and Control

Introduction

Control systems bridge the gap between desired robot behavior and physical actuation. This module covers classical and modern control techniques for robotic systems.

Section 1: PID Control

1.1 The PID Controller

PID Controller: A feedback controller that computes control effort based on proportional, integral, and derivative terms of the tracking error.

$u(t) = K_p e(t) + K_i \int_0^t e(\tau)d\tau + K_d \frac{de(t)}{dt}$

1.2 Tuning Methods

class PIDController:
    def __init__(self, kp, ki, kd):
        self.kp, self.ki, self.kd = kp, ki, kd
        self.integral = 0
        self.prev_error = 0

    def compute(self, error, dt):
        self.integral += error * dt
        derivative = (error - self.prev_error) / dt
        self.prev_error = error
        return self.kp * error + self.ki * self.integral + self.kd * derivative

Section 2: Computed Torque Control

2.1 Model-Based Control

Using the dynamics model:

$\boldsymbol{\tau} = \mathbf{M}(\mathbf{q})(\ddot{\mathbf{q}}_d + \mathbf{K}_d\dot{\mathbf{e}} + \mathbf{K}_p\mathbf{e}) + \mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{g}(\mathbf{q})$

2.2 Stability Analysis

Choosing gains to place poles in the left half-plane ensures stability.

Computed torque control requires accurate dynamics models. Model errors can destabilize the system.

Section 3: Impedance Control

3.1 Interaction Control

Impedance Control: Control strategy that regulates the relationship between robot motion and interaction forces, making the robot behave like a mass-spring-damper system.

$\mathbf{M}_d(\ddot{\mathbf{x}} - \ddot{\mathbf{x}}_d) + \mathbf{B}_d(\dot{\mathbf{x}} - \dot{\mathbf{x}}_d) + \mathbf{K}_d(\mathbf{x} - \mathbf{x}_d) = \mathbf{f}_{ext}$

Section 4: Stability Theory

4.1 Lyapunov Analysis

For a system $\dot{\mathbf{x}} = f(\mathbf{x})$ , if there exists a positive definite function $V(\mathbf{x})$ with $\dot{V}(\mathbf{x}) < 0$ , the equilibrium is asymptotically stable.

Summary

Key takeaways:

PID control provides simple but effective joint-level control
Computed torque achieves linearization and decoupling
Impedance control enables safe physical interaction
Lyapunov theory provides stability guarantees

Key Concepts

PID Controller: Proportional-Integral-Derivative feedback
Computed Torque: Model-based cancellation of dynamics
Impedance Control: Regulating force-motion relationship
Lyapunov Stability: Energy-based stability analysis

Introduction​

Section 1: PID Control​

1.1 The PID Controller​

1.2 Tuning Methods​

Section 2: Computed Torque Control​

2.1 Model-Based Control​

2.2 Stability Analysis​

Section 3: Impedance Control​

3.1 Interaction Control​

Section 4: Stability Theory​

4.1 Lyapunov Analysis​

Summary​

Key Concepts​

Further Reading​