| This paper studies motion planning from one zero-velocity state to another for a three-joint robot in a horizontal plane with a passive revolute third joint. Such a robot is small-time locally controllable on an open subset of its zero-velocity section, allowing it to follow any path in this subset arbitrarily closely. However, some paths are "preferred" by the dynamics of the manipulator in that they can be followed at higher speeds. In this paper, the authors describe a computationally efficient trajectory planner that finds fast, collision-free trajectories among obstacles. They are able to decouple the problem of planning feasible trajectories in the robot's six-dimensional state space into the computationally simpler problems of planning paths in the three-dimensional configuration space and time scaling the paths according to the manipulator dynamics. This decoupling is made possible by the existence of velocity directions, fixed in the passive link frame, which can be executed at arbitrary speeds. The authors have demonstrated the motion planner on an experimental underactuated manipulator. To the authors' knowledge, it is the first implementation of a collision-free motion-planning algorithm for a manipulator subject to a second-order nonholonomic constraint.