Forward Dynamics

Computing a robot's current acceleration, and hence its next motion, from its current state and the torques applied to it.

Dynamics studies the relationship between force and motion, and splits into forward and inverse directions. Forward dynamics starts from known joint positions q, velocities q̇, and applied torques τ, and solves for acceleration q̈: from M(q)q̈ + c(q,q̇) = τ, q̈ = M⁻¹(τ − c), where M is the mass matrix and c lumps together the Coriolis, centrifugal, and gravity terms. Integrating q̈ forward over time steps produces the robot's future trajectory — exactly what a physics simulator does at every step, with engines like MuJoCo and Isaac Sim also solving for contact forces and friction on top of this. Inverse dynamics runs the opposite direction — given a desired motion, find the torques needed to produce it — and is mostly used in control. The classic efficient algorithm for forward dynamics is Roy Featherstone's articulated-body algorithm. Note that this differs from a “forward dynamics model” in machine learning, which refers to a neural network trained to predict the next state.

ExampleIn MuJoCo, if you apply zero torque to every joint of an arm and repeatedly call mj_step, the simulator computes the acceleration due to gravity via forward dynamics, and the arm sags and swings naturally under its own weight.

Related

Inverse Dynamics · Euler-Lagrange Equations · Mass Matrix · Articulated Body Algorithm · Physics Engine · Forward Dynamics Model

Sources

MuJoCo Documentation - Computation
Inverse dynamics - Wikipedia
Featherstone's algorithm - Wikipedia