Grübler–Kutzbach mobility criterion

The Grübler–Kutzbach criterion is a simple counting formula that tells you how many independent ways a machine made of rigid bars and joints can move, before you build it or even pick up a wrench. Imagine a folding chair, a robot arm, or a car suspension as a set of stiff parts (called links) pinned together at connections (called joints). Each loose part, floating freely, could move in several ways; each joint then ties parts together and takes some of those freedoms away. The criterion adds up the freedoms and subtracts the restrictions to give one number, called the mobility, which is how many separate motions you must control to fully command the mechanism.

The recipe is the same every time. You start by giving every moving link its full quota of freedoms (three in a flat, drawing-board world; six in real 3-D space), but you do not count the fixed base, since it never moves. Then, for every joint, you subtract the motions it forbids: a hinge, for example, allows only turning and blocks everything else. Add up what is left and you have the mobility. If the answer is one, the machine has a single degree of freedom and one motor or handle controls all of it, like a pair of scissors. If it is two or three, you need that many independent drives. If it comes out as zero or less, the parts are locked into a rigid frame that cannot move at all.

Engineers reach for this criterion at the very first sketch, because it catches expensive mistakes early. It instantly reveals whether a proposed linkage will actually move, whether it has just enough motors, or whether it is over-constrained and will jam or bind. It works for everything from a humble door hinge to a six-legged Stewart platform. One caveat: it counts only the general case and can be fooled by special geometries (perfectly parallel bars, shared pivot lines) that grant a sneaky extra motion the formula misses, so a clear-eyed designer always sanity-checks the number against the actual shape of the machine.