Physics Simulation Principles

Introduction to Physics Simulation in Robotics

Physics simulation in robotics involves creating virtual environments that accurately model the physical laws governing real-world interactions. This enables robotic systems to be tested, trained, and validated in a safe, controlled, and cost-effective manner before deployment in the real world.

Core Physics Principles

1. Gravity Modeling

Gravity is the fundamental force that affects all physical interactions in terrestrial robotics. In simulation:

• Standard Gravity: Typically set to 9.81 m/s² (Earth's gravity)
• Direction: Usually in the negative Z direction in most simulation environments
• Effects: Influences robot balance, locomotion, and object interactions
• Variations: Can be modified to simulate different planetary environments

2. Collision Detection

Collision detection is critical for realistic physical interactions:

Broad Phase Collision Detection

• Uses spatial partitioning to quickly eliminate non-colliding pairs
• Common techniques: bounding volume hierarchies (BVH), spatial grids
• Optimizes performance by reducing the number of detailed checks

Narrow Phase Collision Detection

• Performs precise collision detection between potentially colliding objects
• Uses algorithms like GJK (Gilbert-Johnson-Keerthi) or Minkowski Portal Refinement
• Calculates contact points, normals, and penetration depths

3. Contact Models

Contact models determine how objects interact when they collide:

Penalty-Based Methods

• Treat contacts as spring-damper systems
• Objects slightly penetrate before experiencing reaction forces
• Computationally efficient but may introduce numerical artifacts

Constraint-Based Methods

• Formulate contacts as mathematical constraints
• Prevent any penetration between objects
• More accurate but computationally intensive

Common Contact Parameters

• Friction: Static and dynamic friction coefficients
• Restitution: Bounciness of collisions (0 = no bounce, 1 = perfectly elastic)
• Stiffness: How rigid the contact response is
• Damping: Energy dissipation during contact

Simulation Algorithms

1. Forward Dynamics

Forward dynamics calculates the motion of a system given applied forces:

F = ma (Newton's second law)
τ = Iα (Rotational equivalent)

Where:

• F = applied force
• m = mass
• a = acceleration
• τ = torque
• I = moment of inertia
• α = angular acceleration

2. Integration Methods

Various numerical integration methods solve the equations of motion:

Euler Integration

• Simplest method: x_new = x_old + v * dt
• Fast but can be unstable for stiff systems
• Energy tends to increase over time

Runge-Kutta Methods (RK4)

• More accurate but computationally expensive
• Better energy conservation properties
• Commonly used in high-fidelity simulations

Semi-Implicit Euler

• Compromise between stability and computational cost
• Better stability than explicit Euler
• Common in real-time robotics simulation

Real-Time vs. Non-Real-Time Simulation

Real-Time Simulation

• Simulation time matches real-world time (1:1 ratio)
• Essential for hardware-in-the-loop testing
• Constrained by computational resources
• May sacrifice accuracy for performance

Non-Real-Time Simulation

• Simulation can run faster or slower than real-time
• Allows for detailed analysis and debugging
• Enables accelerated training of AI systems
• Can achieve higher fidelity physics

Accuracy vs. Performance Trade-offs

Model Simplification

• Visual vs. Collision Models: Use detailed meshes for visualization, simplified shapes for collision detection
• Reduced DOF: Simplify complex systems to essential degrees of freedom
• Proxy Objects: Replace complex geometries with simpler approximations

Adaptive Time Stepping

• Use smaller time steps during complex interactions
• Increase time steps during stable periods
• Balance accuracy and performance dynamically

Simulation Fidelity Considerations

1. The Reality Gap

The difference between simulation and reality that must be addressed:

• Visual Fidelity: How closely the visual representation matches reality
• Physical Fidelity: How accurately physics are modeled
• Sensor Fidelity: How well simulated sensors match real sensors
• Domain Randomization: Introducing variations to improve transfer learning

2. System Identification

• Calibrating simulation parameters to match real-world behavior
• Tuning mass, inertia, friction, and other physical properties
• Validating simulation outputs against real-world data

Application to Humanoid Robotics

Physics simulation is particularly important for humanoid robotics because:

Balance and Locomotion

• Humanoid robots must maintain balance in dynamic environments
• Walking, running, and other locomotion patterns require precise physics modeling
• Center of mass calculations are critical for stable movement

Contact-Rich Tasks

• Manipulation tasks involve complex contact interactions
• Grasping requires accurate friction and contact modeling
• Multi-contact scenarios (e.g., walking on uneven terrain) are common

Safety Considerations

• Testing dangerous behaviors in simulation first
• Validating control algorithms before real-world deployment
• Protecting expensive hardware from potential damage

Simulation Software Architecture

Physics Engines

• ODE (Open Dynamics Engine): Traditional choice for robotics simulation
• Bullet Physics: Popular for both gaming and robotics applications
• DART (Dynamic Animation and Robotics Toolkit): Advanced multi-body dynamics
• MuJoCo: High-fidelity physics simulation (commercial)

Sensor Simulation

• Camera Simulation: Modeling pinhole camera models, distortion, noise
• LiDAR Simulation: Ray-casting algorithms for distance measurement
• IMU Simulation: Modeling accelerometer and gyroscope measurements with noise
• Force/Torque Sensors: Simulating contact forces and moments

The next section will explore the comparison between Gazebo and Unity as simulation platforms for robotics applications.