This paper considers a shortcutting heuristic to smooth jerky trajectories for many-DOF robot manipulators subject to collision constraints, velocity bounds, and acceleration bounds. The heuristic repeatedly picks two points on the trajectory and attempts to replace the intermediate trajectory with a shorter, collision-free segment. Here, we construct segments that interpolate between endpoints with specified velocity in a time-optimal fashion, while respecting velocity and acceleration bounds. These trajectory segments consist of parabolic and straight-line curves, and can be computed in closed form. Experiments on reaching tasks in cluttered human environments demonstrate that the technique can generate smooth, collision-free, and natural-looking motion in seconds for a PUMA manipulator and the Honda ASIMO robot. ©2010 IEEE.