Grübler's Formula

Computes how many degrees of freedom a mechanism has from its number of links, joints, and each joint's own freedom.

Grübler's formula, also known as the Chebychev–Grübler–Kutzbach criterion after its three originators, counts the degrees of freedom of a mechanism built from links and joints: dof = m(N − 1 − J) + Σfᵢ. Here m is the degrees of freedom of a single free rigid body (3 for a planar mechanism, 6 for a spatial one), N is the number of links (the fixed frame counts as one), J is the number of joints, and fᵢ is the number of degrees of freedom joint i provides (1 for a revolute or prismatic joint, 3 for a ball joint). The logic is that each movable link starts out with m degrees of freedom, and each joint removes m − fᵢ of them as a constraint. It's a quick way to figure out how many motors a parallel mechanism, closed-chain leg, or finger mechanism needs. Its limitation is that it assumes all the constraints are independent; when the geometry is special — parallel or equal-length members, for instance — the constraints can become redundant, and the true number of degrees of freedom can exceed what the formula predicts. Such mechanisms are called overconstrained, and in that case the formula only gives a lower bound.

ExampleA planar four-bar linkage: m = 3, N = 4 (including the frame), J = 4 revolute joints each contributing 1 degree of freedom, so dof = 3×(4−1−4) + 4 = 1 — rotating one crank fully determines the configuration of the whole mechanism.

Also called

Chebychev–Grübler–Kutzbach Criterion, Kutzbach-Grübler Formula, Mobility Formula

Related

Degrees of Freedom (DoF) · Four-Bar Linkage · Parallel Mechanism · Kinematic Pair · Configuration Space (C-Space) · Underactuation

Sources

Wikipedia: Chebychev–Grübler–Kutzbach criterion
Lynch & Park, Modern Robotics（预印本 PDF，2.2.2 节 Grübler's Formula） (Chinese)