The Physics of Simulation: Gazebo

The Digital Twin Concept

Before building a $50,000 humanoid robot, you need to test it millions of times in software. Enter Gazebo—an open-source robot simulator that models the laws of physics with stunning accuracy.

Gazebo simulates:

Gravity (9.81 m/s² pulling objects down)
Inertia (massive objects resist acceleration)
Friction (surfaces resist sliding motion)
Collisions (objects can't occupy the same space)
Joint Dynamics (motors apply torque, joints have limits)

Why Simulation Matters

Boston Dynamics trained Atlas to do backflips by running 100 million simulated jumps before attempting one in reality. Simulation is where robots learn to walk, run, and manipulate—without breaking hardware.

The Physics Loop: How Simulation Works

Every simulation runs a physics loop at 1000 Hz (1ms per iteration). Here's what happens in each step:

flowchart TD
    A[1. Read Sensor Data] --> B[2. Apply Control Forces]
    B --> C[3. Compute Rigid Body Dynamics]
    C --> D[4. Detect Collisions]
    D --> E[5. Resolve Contact Forces]
    E --> F[6. Update State: Position, Velocity, Acceleration]
    F --> G[7. Render Frame: Visual + Sensor Data]
    G --> H{Time < Max?}
    H -->|Yes| A
    H -->|No| I[End Simulation]
    
    style A fill:#00FFD4,stroke:#00F0FF,stroke-width:2px,color:#000
    style C fill:#FF006B,stroke:#FF0080,stroke-width:2px,color:#fff
    style D fill:#FF006B,stroke:#FF0080,stroke-width:2px,color:#fff
    style F fill:#8B5CF6,stroke:#A78BFA,stroke-width:2px,color:#fff
    style G fill:#00FFD4,stroke:#00F0FF,stroke-width:2px,color:#000

Step-by-Step Breakdown:

Read Sensor Data (cyan): Get camera images, LIDAR distances, IMU readings
Apply Control Forces (blue): Your controller commands motors to apply torque
Compute Rigid Body Dynamics (magenta): F = ma, calculate accelerations
Detect Collisions (magenta): Check if any objects overlap
Resolve Contact Forces (purple): Apply reaction forces (e.g., ground pushes up on foot)
Update State (purple): New positions, velocities, accelerations
Render Frame (cyan): Draw 3D scene and publish sensor data on ROS 2 topics

This loop runs 1000 times per second, creating the illusion of continuous motion.

Physics Fundamentals

1. Gravity

Gravity pulls all objects toward the ground at 9.81 m/s² (Earth standard).

<!-- In Gazebo world file -->
<physics type="ode">
  <gravity>0 0 -9.81</gravity>  <!-- X Y Z components -->
  <max_step_size>0.001</max_step_size>  <!-- 1ms time step -->
  <real_time_factor>1</real_time_factor>  <!-- 1x real-time speed -->
</physics>

What this means:

An object dropped from 1 meter hits the ground in 0.45 seconds
Terminal velocity depends on mass and air resistance (not modeled by default)
You can change gravity to simulate Mars (3.71 m/s²) or the Moon (1.62 m/s²)

2. Inertia

Inertia is an object's resistance to changes in motion. Heavy objects are harder to accelerate.

<!-- In URDF link definition -->
<inertial>
  <mass value="3.0"/>  <!-- 3 kg thigh -->
  <inertia ixx="0.04" ixy="0.0" ixz="0.0"
           iyy="0.04" iyz="0.0"
           izz="0.001"/>  <!-- kg·m² -->
</inertial>

Inertia Tensor Explained:

ixx, iyy, izz: Moment of inertia around X, Y, Z axes
Higher values = harder to rotate around that axis
For a cylinder (like a thigh): Higher inertia along length axis

Critical for Stability

Incorrect inertia values cause simulation instability. A robot might:

Spin uncontrollably (inertia too low)
Move sluggishly (inertia too high)
Fall through the floor (conflicting physics constraints)

Always use realistic values from CAD software or physics calculators.

3. Friction

Friction resists sliding motion between surfaces.

<!-- In SDF/Gazebo collision definition -->
<surface>
  <friction>
    <ode>
      <mu>1.0</mu>   <!-- Coefficient of friction (0 = ice, 1 = rubber) -->
      <mu2>1.0</mu2> <!-- Secondary direction friction -->
    </ode>
  </friction>
  <contact>
    <ode>
      <kp>1000000.0</kp>  <!-- Contact stiffness (N/m) -->
      <kd>1.0</kd>        <!-- Contact damping (N·s/m) -->
    </ode>
  </contact>
</surface>

Friction Coefficients:

μ = 0.0: Frictionless (ice skating)
μ = 0.5: Low friction (wet floor)
μ = 1.0: High friction (rubber on concrete)
μ = 1.5: Very high friction (climbing shoe on rock)

Why it matters: A humanoid's foot needs high friction (μ ≈ 1.0) to push off the ground without slipping.

Creating Your First Gazebo World

A world file (.world or .sdf) defines the simulation environment. Here's a minimal world:

Complete World File: `empty_world.world`

<?xml version="1.0"?>
<sdf version="1.6">
  <world name="empty_world">
    
    <!-- ========================================
         PHYSICS ENGINE CONFIGURATION
         ======================================== -->
    <physics type="ode">
      <max_step_size>0.001</max_step_size>  <!-- 1ms (1000 Hz) -->
      <real_time_factor>1.0</real_time_factor>
      <real_time_update_rate>1000.0</real_time_update_rate>
      
      <gravity>0 0 -9.81</gravity>  <!-- Earth gravity -->
      
      <ode>
        <solver>
          <type>quick</type>  <!-- Quick solver (fast, less accurate) -->
          <iters>50</iters>   <!-- Constraint solver iterations -->
          <sor>1.3</sor>      <!-- Successive over-relaxation -->
        </solver>
        <constraints>
          <cfm>0.0</cfm>      <!-- Constraint force mixing -->
          <erp>0.2</erp>      <!-- Error reduction parameter -->
          <contact_max_correcting_vel>100.0</contact_max_correcting_vel>
          <contact_surface_layer>0.001</contact_surface_layer>
        </constraints>
      </ode>
    </physics>

    <!-- ========================================
         LIGHTING (SUN)
         ======================================== -->
    <light name="sun" type="directional">
      <cast_shadows>true</cast_shadows>
      <pose>0 0 10 0 0 0</pose>  <!-- 10m above origin -->
      <diffuse>0.8 0.8 0.8 1</diffuse>  <!-- Bright white light -->
      <specular>0.2 0.2 0.2 1</specular>
      <direction>-0.5 0.1 -0.9</direction>  <!-- Angled downward -->
    </light>

    <!-- ========================================
         GROUND PLANE
         ======================================== -->
    <model name="ground_plane">
      <static>true</static>  <!-- Doesn't move under physics -->
      <link name="link">
        <collision name="collision">
          <geometry>
            <plane>
              <normal>0 0 1</normal>  <!-- Surface normal (points up) -->
              <size>100 100</size>    <!-- 100m x 100m plane -->
            </plane>
          </geometry>
          <surface>
            <friction>
              <ode>
                <mu>1.0</mu>   <!-- High friction (rubber) -->
                <mu2>1.0</mu2>
              </ode>
            </friction>
          </surface>
        </collision>
        
        <visual name="visual">
          <cast_shadows>false</cast_shadows>
          <geometry>
            <plane>
              <normal>0 0 1</normal>
              <size>100 100</size>
            </plane>
          </geometry>
          <material>
            <ambient>0.3 0.3 0.3 1</ambient>  <!-- Dark gray -->
            <diffuse>0.7 0.7 0.7 1</diffuse>
            <specular>0.01 0.01 0.01 1</specular>
          </material>
        </visual>
      </link>
    </model>

    <!-- ========================================
         SCENE PROPERTIES
         ======================================== -->
    <scene>
      <ambient>0.4 0.4 0.4 1</ambient>  <!-- Ambient light (soft global lighting) -->
      <background>0.25 0.25 0.25 1</background>  <!-- Gray sky -->
      <shadows>true</shadows>
      <grid>true</grid>  <!-- Show grid lines -->
    </scene>

    <!-- ========================================
         GAZEBO GUI SETTINGS
         ======================================== -->
    <gui>
      <camera name="user_camera">
        <pose>5 -5 2 0 0.275 2.356</pose>  <!-- Camera position and orientation -->
        <view_controller>orbit</view_controller>
      </camera>
    </gui>

  </world>
</sdf>

Breaking Down the World File

Physics Configuration (Lines 8-26)

<physics type="ode">
  <max_step_size>0.001</max_step_size>
  <real_time_factor>1.0</real_time_factor>

Key Parameters:

max_step_size: Time step size (1ms = 1000 Hz simulation rate)
real_time_factor: 1.0 = real-time, 0.5 = half-speed (more accurate), 2.0 = 2x speed
type="ode": Open Dynamics Engine (Gazebo's default physics solver)

Performance Tip

For humanoid robots, use 0.001s (1ms) time steps. Smaller = more accurate but slower. Larger = faster but unstable for complex robots.

Sun Light (Lines 31-37)

<light name="sun" type="directional">
  <direction>-0.5 0.1 -0.9</direction>

Why directional light?

Simulates sunlight (rays are parallel)
Creates realistic shadows
Direction vector points from sky toward ground

Alternative: type="point" creates a light bulb (rays radiate from a point).

Ground Plane (Lines 43-76)

<model name="ground_plane">
  <static>true</static>

Critical Properties:

static="true": Object doesn't move (infinite mass)
<plane>: Infinite 2D surface (efficient for large floors)
<mu>1.0</mu>: High friction for robot foot contact

Visual vs Collision:

Collision geometry: What physics engine "feels" (simple shapes for speed)
Visual geometry: What camera sees (can be detailed meshes)

Running Your World in Gazebo

Step 1: Save the World File

mkdir -p ~/gazebo_worlds
cd ~/gazebo_worlds
nano empty_world.world
# Paste the XML content above, then Ctrl+X, Y, Enter

Step 2: Launch Gazebo

# Method 1: Directly with Gazebo
gazebo empty_world.world

# Method 2: With ROS 2 (preferred for robot control)
ros2 launch gazebo_ros gazebo.launch.py world:=/home/YOUR_USERNAME/gazebo_worlds/empty_world.world

Expected Result:

Gray ground plane extending to horizon
Grid lines visible
Shadows from directional light
Orbit camera control (drag with mouse)

Adding Objects to the World

Example: Spawn a Box

<!-- Add inside <world> tag -->
<model name="red_box">
  <pose>0 0 0.5 0 0 0</pose>  <!-- Position: 0.5m above ground -->
  <static>false</static>  <!-- Falls under gravity -->
  
  <link name="link">
    <collision name="collision">
      <geometry>
        <box>
          <size>0.5 0.5 0.5</size>  <!-- 50cm cube -->
        </box>
      </geometry>
    </collision>
    
    <visual name="visual">
      <geometry>
        <box>
          <size>0.5 0.5 0.5</size>
        </box>
      </geometry>
      <material>
        <ambient>1 0 0 1</ambient>  <!-- Red color -->
        <diffuse>1 0 0 1</diffuse>
      </material>
    </visual>
    
    <inertial>
      <mass>10.0</mass>  <!-- 10 kg box -->
      <inertia>
        <ixx>0.416</ixx>
        <iyy>0.416</iyy>
        <izz>0.416</izz>
      </inertia>
    </inertial>
  </link>
</model>

What happens:

Box spawns 0.5m above ground
Gravity pulls it down (9.81 m/s²)
Collides with ground plane
Settles at z=0.25m (half its height)

Hands-On Exercise: Build a Ramp

Challenge: Create a world with:

Ground plane (already done)
A ramp (inclined plane at 30°)
A sphere that rolls down the ramp

Starter Code:

<model name="ramp">
  <static>true</static>
  <pose>0 0 0.5 0 0.523599 0</pose>  <!-- 0.523599 rad = 30° -->
  <link name="link">
    <collision name="collision">
      <geometry>
        <box>
          <size>2 1 0.1</size>  <!-- 2m long, 1m wide, 10cm thick -->
        </box>
      </geometry>
      <!-- ADD FRICTION HERE -->
    </collision>
    <!-- ADD VISUAL HERE -->
  </link>
</model>

<model name="rolling_sphere">
  <pose>0 0 2 0 0 0</pose>  <!-- Starts 2m high -->
  <!-- YOUR CODE: Add sphere geometry (radius 0.2m, mass 1kg) -->
</model>

Verification:

Launch Gazebo with your world
Press Play button
Sphere should roll down ramp due to gravity
Adjust ramp angle or friction to change speed

Key Takeaways

✅ Gazebo simulates realistic physics (gravity, inertia, friction, collisions)
✅ Physics loop runs at 1000 Hz (1ms time steps for accuracy)
✅ World files (.sdf/.world) define simulation environment
✅ Ground plane requires static="true" and high friction (μ=1.0)
✅ Inertia values must be realistic to prevent instability
✅ Lighting affects visual realism (directional sun + ambient light)

What's Next?

Gazebo provides physically accurate simulation but lacks visual fidelity. The next chapter introduces Unity—a game engine that renders photorealistic environments using ray tracing and advanced materials. You'll learn how to bridge ROS 2 and Unity for high-fidelity robot visualization.

The Digital Twin Concept​

The Physics Loop: How Simulation Works​

Physics Fundamentals​

1. Gravity​

2. Inertia​

3. Friction​

Creating Your First Gazebo World​

Complete World File: empty_world.world​

Breaking Down the World File​

Physics Configuration (Lines 8-26)​

Sun Light (Lines 31-37)​

Ground Plane (Lines 43-76)​

Running Your World in Gazebo​

Step 1: Save the World File​

Step 2: Launch Gazebo​

Adding Objects to the World​

Example: Spawn a Box​

Hands-On Exercise: Build a Ramp​

Key Takeaways​

What's Next?​

Further Reading​