Module 08: Locomotion

Introduction

Bipedal locomotion is perhaps the most challenging aspect of humanoid robotics. This module covers the dynamics, stability analysis, and control strategies for walking robots.

Section 1: Stability Criteria

1.1 Zero Moment Point

Zero Moment Point (ZMP): The point on the ground where the sum of moments due to gravity and inertial forces equals zero. When ZMP is within the support polygon, the robot is dynamically stable.

$\mathbf{p}_{ZMP} = \frac{\sum_i m_i(\ddot{z}_i + g)p_i - \sum_i m_i\ddot{p}_i z_i}{\sum_i m_i(\ddot{z}_i + g)}$

1.2 Support Polygon

The convex hull of contact points defines stable ZMP locations.

Section 2: Gait Generation

2.1 Central Pattern Generators

Neural-inspired oscillators for rhythmic movement:

class CPGOscillator:
    def __init__(self, frequency, amplitude):
        self.freq = frequency
        self.amp = amplitude
        self.phase = 0

    def step(self, dt):
        self.phase += 2 * np.pi * self.freq * dt
        return self.amp * np.sin(self.phase)

2.2 Preview Control

Using ZMP trajectory preview for stable walking.

ZMP-based control assumes flat ground. On uneven terrain, additional measures (foot placement adaptation, compliant control) are needed.

Section 3: Whole-Body Control

3.1 Task-Space Inverse Dynamics

Hierarchical control framework:

$\boldsymbol{\tau} = \mathbf{M}\ddot{\mathbf{q}}_d + \mathbf{h} - \mathbf{J}_c^T\mathbf{f}_c$

3.2 Operational Space Control

Decoupling tasks in Cartesian space.

Section 4: Capture Point Dynamics

4.1 Linear Inverted Pendulum Model

Simplified dynamics for walking:

Capture Point: The location where the robot must step to come to rest without falling. Enables push recovery and dynamic balance.

4.2 Step Planning

Real-time foot placement based on capture point.

Summary

Key takeaways:

1. ZMP within support polygon ensures stability
2. Gait generation can use CPGs or optimization
3. Whole-body control coordinates all DOFs
4. Capture point enables reactive balance control

Key Concepts

• ZMP: Zero Moment Point stability criterion
• Support Polygon: Convex hull of contact points
• Whole-Body Control: Coordinating all robot DOFs
• Capture Point: Reactive balance metric

Further Reading

1. Vukobratovic, M. & Borovac, B. (2004). "Zero-Moment Point—Thirty Five Years of Its Life"
2. Kajita, S. et al. (2014). "Introduction to Humanoid Robotics"