Kinematic Chain

A mechanism model made of rigid links strung together by joints, split into open chains and closed chains.

A kinematic chain is a set of rigid bodies (links) connected by joints, used to describe how a mechanism can move; Franz Reuleaux systematized the idea in his 1876 book Kinematics of Machinery. When the links connect one after another with no loop, it's called an open chain (or serial chain) — the typical example is an ordinary robot arm, with a motor driving every joint. When the links form one or more closed loops, it's a closed chain — examples include the Stewart platform, Delta parallel robots, and the parallel ankle joints found in some humanoid robots — and typically only some of a closed chain's joints are actively driven. A mechanism's degrees of freedom can be estimated with the Grübler formula: M = d(N−1−j) + Σf_i, where d is 3 in the plane or 6 in space, N is the number of links (including the ground), j is the number of joints, and f_i is the i-th joint's degrees of freedom. Open chains have simple forward kinematics but hard inverse kinematics; closed chains are usually the other way around.

ExampleA tabletop 6-axis arm, from base to gripper, is an open chain; a Delta parallel robot's three supporting arms all meet at the moving platform, forming a closed chain.

Also called

Open Chain, Closed Chain

Related

Serial Mechanism · Parallel Mechanism · Kinematic Tree · Grübler's Formula · Degrees of Freedom (DoF) · Parallel Ankle Mechanism

Sources

Kinematic chain - Wikipedia
Modern Robotics (Lynch & Park) preprint PDF, Ch. 1–2 与 Ch. 7 Closed Chains (Chinese)