Principle:Google deepmind Mujoco Physics Stepping

Knowledge Sources	MuJoCo Computation MuJoCo: A physics engine for model-based control MuJoCo
Domains	Physics_Simulation, Numerical_Integration, Robotics
Last Updated	2026-02-15 06:00 GMT

Overview

Mechanism that advances the physics simulation by one timestep by computing forward dynamics and integrating the equations of motion.

Description

Physics Stepping is the core operation of any MuJoCo simulation loop. It combines forward dynamics computation (collision detection, constraint solving, force evaluation) with numerical time integration to advance the simulation state from time t to time t + dt. The timestep dt is specified in the model options.

MuJoCo supports multiple integration schemes:

• Euler: Semi-implicit Euler (fast, first-order)
• RK4: Fourth-order Runge-Kutta (accurate, four evaluations per step)
• Implicit/ImplicitFast: Implicit integration for improved stability with stiff systems

Usage

Use this principle in the main simulation loop to advance time. Call mj_step repeatedly to simulate physics. Set control signals (d->ctrl) before each call to apply actuator commands.

Theoretical Basis

The stepping process consists of two phases:

Phase 1 — Forward Dynamics: $\ddot{q}=M(q{)}^{-1}(\tau +J^{T}f-c(q,\dot{q}))$

Phase 2 — Time Integration:

For Euler integration: $v_{t+1}=v_t+\ddot{q}\cdot dt$ $q_{t+1}=q_t+v_{t+1}\cdot dt$

The semi-implicit scheme uses updated velocities for position integration, improving energy conservation.

Related Pages

Implemented By

• Implementation:Google_deepmind_Mujoco_mj_step

Uses Heuristic

• Heuristic:Google_deepmind_Mujoco_Solver_Selection_Guide