reachable workspace
The reachable workspace of a robot arm is the full region of space its hand (the end-effector — the gripper or tool at the tip) can touch, in at least one way, by some combination of its joint movements. It is the answer to a very physical question: stand the robot down and let it flail through every pose it can adopt — what is the complete set of points the very tip of it can reach?
Crucially, reachable means it can get the tip there somehow, even if only at an awkward angle. It does not promise the arm can approach that point from any direction it likes — only that the point is touchable at all. So the reachable workspace tends to be a generous, blobby volume, often shaped like a thick shell or a partial sphere around the arm's base, bounded on the outside by how far the arm stretches and on the inside by parts it simply cannot fold back to (and by its own body, the table, or obstacles in the way).
Knowing this volume is the first thing you check when placing a robot in a factory or a kitchen: is the object I want it to grab actually inside the region it can reach? Anything outside is, quite literally, out of reach, no matter how cleverly you command the joints.
Sit at a table and keep your shoulder fixed. The whole curved sweep of spots your fingertip can touch — near, far, high, low, even if you have to twist your wrist oddly to reach the edges — is your arm's reachable workspace.
Every point your fingertip can touch with a fixed shoulder, however awkwardly, is in the reachable workspace.
The dexterous workspace — where the hand can also point in any orientation — is always a subset, often a much smaller core, of the reachable workspace.