Math & Kinematics

Degrees of Freedom (DOF)

Degrees of Freedom (DOF) are the number of independent parameters required to fully specify the configuration of a mechanical system. A rigid body in space has six degrees of freedom, three translational and three rotational, so a manipulator needs at least six joints to reach arbitrary end-effector poses; a seventh adds kinematic redundancy for obstacle and joint-limit avoidance. Grübler's formula relates the DOF of a linkage to its joints and links.

Why it matters for physical AI

DOF count determines what poses a robot can reach, how it handles singularities, and the dimensionality of its action space. Learned policies must scale with DOF, which is why humanoids and dexterous hands are markedly harder than 6-DOF arms.