Motion planning under gravity for underactuated three-link robots

We present a method for planning motions of threelink planar robots with a passive rotational third joint in the presence of gravity. These underactuated mechanisms can be fully linearized and input-output decoupled by means of a nonlinear dynamic state feedback, provided that a physical singularity is avoided. The linearizing output is the position of the center of percussion of the third link. Based on this, one can plan motions joining in finite time any initial and desired final state; in particular, transfers between inverted equilibria and swing-up maneuvers are easily obtained. Simulation results are reported for a 3R robot.