Workspace and Reachability
The workspace of an arm is every point its tool can reach. Joint limits, link lengths, and the robot's own base all shape it. Inside it sits a smaller region, the dexterous workspace, where the tool can arrive from every direction.
Two workspaces
The reachable workspace is every point the tool tip can touch with at least one orientation. The dexterous workspace is every point it can touch with every orientation. For a planar arm with links \(L_1, L_2\) and tool \(L_3\) and no joint limits, both are rings:
What to notice
Tighten joint 2's range and the reachable area loses its inner part, since the elbow can no longer fold back. Tighten joint 1 and the workspace becomes a wedge. A longer tool grows the reachable area but shrinks the dexterous one, because the wrist needs room to swing the tool all the way around. Tasks that need the tool to approach from many angles, like welding around a pipe, must sit in the dexterous region.