open vs closed kinematic chain

A kinematic chain is just a string of rigid parts (links) hooked together by joints that let them move. The big question is how those parts are wired up: does the chain have loose ends, or does it close back on itself into a loop? An open chain is like your own arm. Starting at your shoulder, the parts run one after another out to your fingertips, and the far end is free to wave around in the air. Nothing connects the hand back to the shoulder, so each joint can be moved on its own without fighting the others.

A closed chain has at least one loop, where the parts connect back around to form a ring. Think of your two legs while you stand with both feet planted on the floor: hip to thigh to shin to foot on the left, the same on the right, and the floor ties the two feet together, closing the loop through the ground. Now the joints are no longer free agents. Because the loop has to stay closed, bending one knee forces the other joints to adjust to compensate; their motions are tied together by constraints, like people holding hands in a circle who can't all step in different directions.

This single difference shapes a robot's whole personality. Open chains are easy to command and can reach far and into awkward spots, but they tend to be floppier and weaker because each joint alone bears the load. Closed chains are stiffer, stronger, and more precise because several parts share the work, but their tangled constraints make them trickier to control and they usually can't reach as far. Most robot arms are open chains; many fast, powerful, or high-precision machines are built as closed ones.