Being functional and task-driven is something that we assume a robot is... But as we all know, technology can fail, miserably, rendering it helpless. However, the robots that we consider in this category are purposefully helpless:
MANIC/LOVE, a puppet boy in a somewhat uncomfortable position
Norman White's "Helpless Robot"
PARO the helpless seal
Tweenbots that need some help in crossing a park or public square
hitchBOT the hitchhiking robot
Rendering a robot helpless can also learn us about our own emotions towards robots:
upside down Furbies describes such an experiment
Helplessness is instantiated in various other robotic forms, such as in
Leonard van Munster's Dancing White Man, and
even homeless robots.