Fixed Timestep

The primary advantage of a fixed timestep is deterministic execution. By advancing the simulation state by a constant Δt (e.g., 1ms or 0.001 seconds) for every update cycle, numerical integrators for physics (like Symplectic Euler or Runge-Kutta 4) produce identical results given identical initial conditions. This is critical for:

• Reproducible reinforcement learning experiments.
• Debugging control policies and physical behaviors.
• Ensuring that a policy trained in simulation behaves identically when replayed, eliminating variability from variable frame rates.

Fixed timesteps provide guaranteed stability for simulating stiff systems, which are common in robotics. Stiff systems have components that react at vastly different time scales (e.g., a fast-responding motor controller and a slow-moving heavy arm). A variable timestep can become unstable when it automatically grows too large for the fast dynamics. A fixed timestep, chosen to be smaller than the fastest dynamics' stability threshold, ensures:

• Stable simulation of contact forces and joint constraints.
• Accurate integration of PID controller feedback loops.
• Prevention of energy explosion in spring-damper systems and actuator models.

Real-world robot control systems operate on fixed-frequency control loops (e.g., 1kHz for high-performance arms). Fixed timestep simulation directly mirrors this hardware reality, allowing a one-to-one mapping between simulation cycles and real control cycles. This enables:

• Direct porting of low-level controllers (torque control, impedance control) from sim to real.
• Accurate modeling of sensor update rates and actuator command latency.
• Hardware-in-the-Loop (HIL) testing where the simulator provides sensor data at the exact rate the real controller expects.

In a fixed timestep architecture, the physics update runs at a constant rate independent of the graphical rendering frame rate. This is often managed via a game loop pattern that accumulates time and runs multiple physics steps per rendered frame if needed. This ensures:

• Physics accuracy is not tied to visual performance or GPU load.
• The simulation can run faster than real-time for rapid parallelized training on headless servers.
• Visuals can be rendered at a variable rate for a smooth user interface without affecting the underlying rigid body dynamics or contact resolution.

The choice of timestep size (Δt) is a critical engineering trade-off. A smaller timestep increases accuracy and stability but requires more computational cycles per second of simulated time.

• Typical Values: 1ms (0.001s) for high-fidelity robotic manipulation, 2-5ms for legged locomotion, 10-20ms for coarse vehicle dynamics.
• Error Accumulation: Truncation error from the numerical integrator accumulates over time. For long simulations, higher-order integrators (RK4) are used with a fixed timestep to minimize this drift.
• The timestep must be small enough to accurately sample the highest-frequency dynamics being modeled, such as fast contact events or high-bandwidth motor dynamics.

Fixed timestep is contrasted with variable timestep (or adaptive timestep) simulation, where Δt changes based on system stiffness or error estimates. Key differences:

• Variable Timestep: Can be more computationally efficient for simple systems by taking larger steps when possible, but introduces non-determinism and can become unstable for robotic systems with frequent contacts.
• Fixed Timestep: Guarantees consistency and is the standard for industrial-grade physics engines (e.g., NVIDIA PhysX, Bullet, MuJoCo) used in Sim-to-Real Transfer Learning. It provides the predictable, stable environment required for training robust reinforcement learning policies.

Game engines are the most ubiquitous implementers of fixed timestep for real-time interactive physics. They separate the variable frame rate of rendering from the constant physics update rate.

• Unity: Uses FixedUpdate() for physics calculations, running at a default rate of 50 Hz (0.02-second intervals), independent of Update().
• Unreal Engine: Employs a tick group system where physics simulations are assigned to a fixed-interval tick, ensuring stable collision detection and rigid body dynamics.
• This separation prevents simulation jitter and ensures that in-game physics (e.g., object collisions, character movement) behave identically across different hardware performance levels.

High-fidelity robotics simulators rely on fixed timestep for accurate, reproducible training of control policies and sensor models.

• NVIDIA Isaac Sim: Operates on a fixed timestep (e.g., 1ms) to provide deterministic physics and sensor data generation, which is essential for training robust reinforcement learning policies via Isaac Gym.
• Gazebo/Ignition: Uses a fixed simulation step for its ODE, Bullet, or Simbody physics engine backends. This guarantees that robot dynamics, contact forces, and forward/inverse dynamics calculations are numerically stable and repeatable across training runs, a prerequisite for sim-to-real transfer.

The underlying physics engines that power simulators are designed around fixed-timestep numerical integration of equations of motion.

• Bullet Physics: Designed for fixed-step simulation, using discrete-time solvers for rigid body dynamics and contact constraints.
• NVIDIA PhysX: Provides a fixed substep model for stability, critical for complex collisions and joint limits.
• MuJoCo: Employs a fixed, user-defined timestep for its implicit integration solver, which is particularly stable for systems with contacts and actuators. Its deterministic output is a key feature for reinforcement learning research.

Fixed timestep is the standard in computational fluid dynamics (CFD), finite element analysis (FEA), and orbital mechanics.

• Numerical Stability: Solvers like those in ANSYS or OpenFOAM use fixed time increments to satisfy the Courant–Friedrichs–Lewy (CFL) condition, preventing solution divergence.
• Deterministic Results: For engineering validation, simulations must produce bit-identical results across runs. A fixed timestep eliminates variability introduced by adaptive solvers, ensuring that a simulated crash test or fluid flow is perfectly reproducible.
• Co-simulation: When coupling different simulation domains (e.g., mechanics with hydraulics), a globally synchronized fixed timestep is often required for data exchange between subsystems.

Real-time operating systems (RTOS) and programmable logic controllers (PLCs) execute control loops on a strict, fixed-period schedule.

• Deterministic Latency: Industrial robots, flight controllers, and automotive ECUs rely on hard real-time constraints. Control algorithms for PID controllers, inverse kinematics, and state estimation (Kalman Filters) are executed at a fixed frequency (e.g., 1 kHz).
• Hardware-in-the-Loop (HIL): Testing with HIL requires the simulation to provide sensor data to the real controller at a perfectly periodic rate. Any jitter or delay in the simulated timestep can cause the controller to behave erroneously, making fixed timestep non-negotiable.

Fixed timestep is a foundational technique for deterministic lockstep networking models in multiplayer games.

• State Synchronization: Each simulation frame (a fixed timestep) is treated as a "turn." Only user inputs are transmitted over the network. All clients, running the same simulation logic with the same fixed delta time, will produce identical game states from the same sequence of inputs.
• Rollback Networking: Advanced systems like GGPO still use a underlying fixed-timestep simulation. They enable responsiveness by predicting inputs and rolling back & re-simulating fixed steps when predictions are wrong, relying on the determinism of the fixed-step physics to guarantee correct reconciliation.

A simulation update method where the physics and control calculations are advanced by a variable time interval, typically tied directly to the frame rate or real-world clock. This contrasts with a fixed timestep.

• Pros: Simpler to implement; simulation time matches real time.
• Cons: Non-deterministic; physics stability depends on frame rate, leading to different behavior on different hardware.
• Use Case: Often used in non-critical visual simulations or games where exact reproducibility is not required.

The core software library responsible for calculating the motion and interactions of rigid bodies, soft bodies, and fluids within a simulation. It executes the numerical integration (like RK4) at each fixed timestep.

• Key Functions: Collision detection, constraint solving, and force application.
• Examples: NVIDIA PhysX, Bullet Physics, ODE (Open Dynamics Engine), MuJoCo.
• Integration: The physics engine's solver is called repeatedly within the main simulation loop to advance the world state by the fixed delta time.

The family of mathematical algorithms used to approximate the solution to differential equations that govern motion (e.g., Newton's second law, F=ma). A fixed timestep provides the constant Δt (delta time) input for these solvers.

• Euler Method: Simple but can be unstable with large timesteps.
• Runge-Kutta Methods (RK4): More complex and computationally expensive, but provide higher accuracy and stability for a given timestep.
• Verlet Integration: Often used in molecular dynamics and particle systems for its stability and energy conservation properties.

The main program loop that orchestrates all activities within a simulator. A robust loop decouples the fixed timestep physics update from variable-rate rendering and input processing.

• Standard Pattern:
  1. Process Input (variable rate).
  2. Accumulate Time since last physics update.
  3. Fixed Update: While accumulated time >= fixed timestep:
     • Call physics engine advance(fixed_timestep).
     • Update control/agent logic.
     • Subtract fixed_timestep from accumulator.
  4. Interpolate State for smooth rendering.
  5. Render (variable rate).
• This ensures physics is deterministic and stable, independent of graphical frame rate.

The property of a system where, given the same initial conditions and inputs, it will produce the exact same outputs every time it is run. A fixed timestep is a foundational requirement for deterministic simulation.

• Critical For:
  • Reproducible scientific experiments.
  • Debugging complex robotic behaviors.
  • Training reinforcement learning policies, where consistent environment responses are needed for learning.
• Threats to Determinism: Use of variable timesteps, unseeded random number generators, floating-point non-associativity on parallel systems, and external timing dependencies.

A technique where the core physics engine performs multiple internal solver iterations per fixed timestep to improve stability and accuracy for complex interactions like stacking, cloth, or fluids.

• How it Works: The main simulation advances by a fixed timestep (e.g., 2ms), but the physics solver may break this into smaller substeps (e.g., 4 steps of 0.5ms each).
• Benefit: Allows the use of a larger, more performant main timestep for high-level logic while maintaining physical fidelity for challenging scenarios.
• Trade-off: Increases computational cost proportionally to the number of substeps.