Workspace density and inverse kinematics for planar serial revolute manipulators

The topic of reachable workspaces of robotic manipulators has received considerable attention over the past half century. One approach to generating workspaces is by sampling joint angles and evaluating the boundary of the resulting set in the space of rigid-body motions. In the case when the manipulator has discrete actuation, such as stepper motors or pneumatic cylinders, not only the boundary of the workspace, but also the density of reachable poses within the workspace is important. Following previous efforts that focused on characterizing this workspace density, we show that this density is particularly efficient to evaluate in the special case of planar serial arms with revolute joints. We then show how the resulting density can be used in inverse kinematics algorithms that are equally applicable for discrete-state and continuous-motion robot arms.