Rigid-Body Dynamics

The study of how forces and torques cause acceleration and rotation in one or more connected rigid bodies.

Rigid-body dynamics studies how one or more rigid bodies, connected by joints, move under applied forces and torques. A single rigid body obeys the Newton-Euler equations: F=ma governs translation (F is net force, m is mass, a is the center of mass's acceleration), and τ=Iα+ω×(Iω) governs rotation (τ is net torque, I is the inertia tensor, α is angular acceleration, ω is angular velocity). A multi-body system like a robot arm is usually written M(q)q̈+c(q,q̇)+g(q)=τ: q is the joint angle vector, M is the mass matrix, c is the Coriolis and centrifugal term, g is the gravity term, and τ is joint torque. Given torques, solving for motion is forward dynamics, computed at every step by a physics simulator; given a desired motion, solving for the required torques is inverse dynamics, used for torque control and gravity compensation. Roy Featherstone's book Rigid Body Dynamics Algorithms is the standard reference, and MuJoCo's relevant algorithms are based on it.

ExampleWhen an arm holds a cup of water perfectly still, q̇ and q̈ are both zero, so the equation reduces to g(q)=τ — the result is exactly the torque each joint needs to output to counteract gravity, which is gravity compensation.

Related

Newton-Euler Equations · Euler-Lagrange Equations · Mass Matrix · Forward Dynamics · Inverse Dynamics · Physics Engine

Sources

Wikipedia: Rigid body dynamics
Modern Robotics (Lynch & Park), Ch.8 Dynamics of Open Chains
MuJoCo Documentation: Computation