Locomotion & Balance

Why this matters: Walking is not falling down. It's controlled falling, caught just in time. This chapter covers the art and science of making robots walk.

Introduction: The Art of Not Falling

Humans learn to walk in about a year. Engineers have spent 50+ years trying to replicate it in robots. Walking is hard because it requires:

• Balance: Keeping the center of mass supported
• Dynamics: Managing momentum and impact
• Adaptability: Handling uneven terrain, stairs, and surprises

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Balance Fundamentals

Center of Mass (CoM)

The point where all mass can be considered concentrated:

r_CoM = Σ(m_i × r_i) / Σ(m_i)

Zero Moment Point (ZMP)

The point on the ground where reaction forces create zero moment:

graph TD
    A[Center of Mass] --> B[Gravity Force]
    B --> C[Ground Reaction]
    C --> D[ZMP]
    D --> E{Inside Support<br/>Polygon?}
    E -->|Yes| F[Stable]
    E -->|No| G[Falling]

The ZMP Rule

If ZMP is inside the support polygon (foot outline), the robot won't tip over. If it's outside, gravity wins.

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Walking Gaits

Phases of Walking

Phase	Description	Duration
Double Support	Both feet on ground	10-15%
Single Support	One foot on ground	35-40%
Swing	Foot in air	35-40%

Gait Generation Methods

1. ZMP-Based Planning

Classic approach (Honda ASIMO, early humanoids):

def plan_zmp_trajectory(footsteps, duration):
    zmp_traj = []
    for step in footsteps:
        # ZMP moves from one foot to the other
        zmp_traj.extend(interpolate(
            current_foot.center,
            next_foot.center,
            duration
        ))
    return zmp_traj

2. Capture Point / DCM

More dynamic approach, used in modern robots:

ξ = x + ẋ / ω_0

Where ω_0 = sqrt(g / z_CoM) is the natural frequency.

The capture point is where the robot would stop if it put its foot there immediately.

3. Reinforcement Learning

Train end-to-end policies in simulation:

reward = (
    forward_velocity_reward
    - energy_penalty
    - falling_penalty
    + alive_bonus
)

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Dynamic Maneuvers

Running

Running differs from walking in that both feet leave the ground:

Walking	Running
Always one foot down	Flight phase
CoM nearly constant height	CoM bounces
Lower impact forces	2-3x body weight impact

Jumping

Requires explosive actuation and precise timing:

sequenceDiagram
    participant C as Controller
    participant A as Actuators
    participant B as Body

    C->>A: Crouch (load springs)
    A->>B: Store elastic energy
    C->>A: Launch command
    A->>B: Full extension
    B->>B: Flight phase
    B->>A: Impact detection
    C->>A: Absorb landing

Push Recovery

When pushed, the robot must react:

1. Ankle strategy: Small pushes, stiff ankles
2. Hip strategy: Medium pushes, bend at hip
3. Step strategy: Large pushes, take a step

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Terrain Adaptation

Uneven Ground

Robots must handle:

• Slopes
• Steps and stairs
• Loose surfaces (gravel, grass)
• Unexpected obstacles

State Machine Approach

class TerrainStateMachine:
    states = {
        "FLAT_GROUND": FlatGroundController(),
        "STAIRS_UP": StairsUpController(),
        "STAIRS_DOWN": StairsDownController(),
        "SLOPE": SlopeController(),
        "ROUGH": RoughTerrainController()
    }

    def update(self, perception):
        detected_terrain = self.classify_terrain(perception)
        self.transition_to(detected_terrain)
        return self.current_state.compute_action()

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Case Studies

Boston Dynamics Atlas

The most dynamic humanoid in the world:

• Hydraulic actuation
• ZMP + MPC control
• Can run, jump, do backflips
• 80kg, 1.5m tall

Tesla Optimus

Electric actuation, designed for manufacturing:

• 28 actuators
• Target: Safe human-speed walking
• Focus on manipulation over acrobatics

Agility Robotics Digit

Purpose-built for logistics:

• Electric series-elastic actuators
• 4 DoF per leg
• Can carry boxes, climb stairs

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Key Takeaways

Summary

1. Balance is about keeping ZMP in the support polygon
2. Walking is controlled falling
3. Running adds a flight phase
4. Capture point methods enable dynamic walking
5. Terrain adaptation requires perception + control

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Further Reading

• Chapter 2.1: Kinematics & Dynamics
• Chapter 2.2: Actuation & Control
• Chapter 3.1: ROS 2 Concepts