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The landing procedure of Curiosity on Mars has been described as a real challenge in part because of the powered descent stage and sky crane. Indeed the separation of Curiosity with the backshell + parachute occurred at an altitude of 1.8km at a velocity of 100m/s. The powered descent stage has then brought the sky crane with Curiosity to a stop at an altitude of a couple of meters and then hovered for about 15 seconds.

Has there been any tests on Earth of the powered descent stage's drop and hovering before sending it to Mars ? and powered descent stage + sky crane tests ? Are there any videos of these tests?

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    $\begingroup$ It never hovered, in the sense of trying to get to zero velocity relative to the surface. It targeted about a 0.75 m/s constant speed descent. Hovering would just be a waste of fuel. $\endgroup$ – Mark Adler Feb 13 '14 at 1:30
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No, there were no tests of a descent stage with rockets firing on (or slightly above) Earth. That may have been one part of the reason that there was so much cheering, jumping, hugging, and crying when the thing actually worked.

The system as a whole was validated and verified through many subsystem tests and analyses, tied together in a detailed computer simulation of the event. @Maxpm's answer has links for two of those. Others such tests include radar tests on aircraft in the desert, heatshield coupon testing in arcjets, various separation tests, etc.

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If the sky crane was ever tested on Earth, there's no public footage of it.

However, there are test videos for other parts of the descent phase:

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