Chebychev–Grübler–Kutzbach criterion

Remarks

Idea: each moving link has $6$ degrees of freedom in space ($3$ in the plane), giving $6(N - 1)$ in total. A joint with $f_i$ freedoms imposes $6 - f_i$ scalar equations ($3 - f_i$ in the plane). Subtracting gives $M$, and the implicit function theorem turns the rank condition into the manifold statement.

Example: a planar four-bar linkage has $N = 4$ links and $j = 4$ revolute joints with $f_i = 1$, so $M = 3(4 - 1 - 4) + 4 = 1$: it moves with one degree of freedom. A triangle of three links pinned at three joints has $M = 3(3 - 1 - 3) + 3 = 0$ and is rigid.

The independence hypothesis is needed. In special geometries the constraints become dependent and the true mobility exceeds $M$. Take a parallelogram four-bar linkage and add a third crank, equal and parallel to the two cranks, joining ground and coupler. It has $N = 5$ and $j = 6$, so $M = 0$ by the count, yet it moves. Such mechanisms are called overconstrained.

Going deeper

For a student

Designers use the formula as a first check: $M = 1$ means one motor can drive the mechanism, $M = 0$ means a structure, and $M < 0$ means a statically indeterminate structure. The count is a statement about generic configurations, in the same spirit as counting equations and unknowns. Overconstrained exceptions in space include the Bennett linkage, a four-bar spatial loop with $M = -2$ by the count that still moves. The finer analysis of such cases uses screw theory.