Degrees of Freedom

The degrees of freedom of a mechanism is the number of independent inputs you need to fix its shape completely. A single hinge has one. A human arm has seven. A triangle of bars has none at all.

Mechanism Input 1 Input 2 Animate the inputs

Grübler's formula

For a planar mechanism with $N$ bodies (counting the fixed ground) and $J$ joints, where joint $i$ allows $f_i$ degrees of freedom:

$$M = 3\,(N - 1 - J) + \sum_{i=1}^{J} f_i$$

Each free body in a plane has 3 degrees of freedom (move in x, move in y, rotate). Each hinge or slider removes 2 of them and leaves 1. In 3D the 3 becomes 6.

What to notice

The four-bar linkage has four bars and four hinges, but only one degree of freedom, so turning one crank fixes everything. That is why four-bars are everywhere, from car wipers to excavator arms. The five-bar needs two inputs. The triangle has zero: it is a structure, which is why bridges and bike frames are built from triangles.

A robot arm needs at least six degrees of freedom to put its hand at any position with any orientation in 3D. Arms with seven, like many research arms, have one extra that lets them move the elbow without moving the hand.