I understand that there is thinking to grab an asteroid to be able to use it for study and resource. When/where is the best place to do this? From its own orbit? From an Earth/Moon close encounter? (Context: I thought of this question as I read of a close encounter upcoming this weekend).
One option might be to rendezvous with a temporarily captured orbiter (TCO) which makes a close encounter by passing by the Sun-Earth Lagrange point 1 or 2 (the Moon is not involved!) It is about as easy to reach in terms of delta-v as the easiest to reach known Near Earth Asteroid in stable orbit. But the Lagrange point has launch window everyday, not just once per asteroid orbit, and I think also some benefits for maneuvering because of the gravitational instability of the Lagrange point. And it is only 1% of an AU away, NEA's have much longer perigeo maximum distance from Earth. Since (small) asteroids are believed to pass by there regularly, maybe one mission could "grab" several of them to get a more representative sample, with huge benefits to science about them?
I've recently asked related questions about this here: https://astronomy.stackexchange.com/questions/6184/can-the-gaia-telescope-detect-small-temporarily-captured-asteroids-near-its-lagr
A challenge for this is that one probably won't be able to observe the target asteroid from Earth before the mission launches. They are too small, only a 5 meter TCO has ever been discovered, the rest are obviously smaller. But one could turn that challenge into a reason to develop the ability to react flexibly to previously unknown information when on site in space. Especially the military should be interested.
I've played with this. It seems to me there are two major delta Vs:
1) Delta V to nudge an asteroid's heliocentric orbit so the rock passes through earth's sphere of influence.
2) Once in earth's sphere of influence, delta V to shed enough velocity so that the hyperbolic orbit with regard to earth becomes an elliptical capture orbit.
As a matter of course we already have many rocks that pass through our sphere of influence. And there are far more that pass near the earth's sphere influence and need only a small nudge to alter its pass so it passes through.
Keeping a rock within our sphere of influence is tougher. It helps if the rock has an earth like orbit: low inclination, semi major axis close to 1 A.U. and a low eccentricty (nearly circular). However a rock with an earth like orbit would have a large synodic period with regard to earth. It would only rarely come near earth's neighborhood. But there since there's a lot of rocks with earth like orbits, earth approaches from these more accessible rocks aren't exceedingly rare.
If the rock flies by the moon's path while passing through our neighborhood, the moon can change the rock's velocity with regard to earth. Sometimes the moon can change a hyperbolic path to a capture orbit. It's thought this occurs naturally. If humans were to give a rock a strategic nudge here and there, it could occur much more often.
The Keck Report outlines how a robotic craft with robust solar electric propulsion might retrieve an asteroid like 2008 HU4 and park it in lunar orbit. It is interesting the many of the Keck Report authors are prominent asteroid guys like J. S. Lewis (Author of Mining the Sky) or Chris Lewicki (of Planetary Resources).
I also took a crack at speculating how 2008 HU4 might be captured when it nears the earth in 2015. Of course we don't have time to build a retrieval vehicle in time for 2008 HU4's 2015 fly by, but more chances will comes along.
Grabbing it is easy - approach in a gently intersecting orbit and attach yourself with any convenient method. When and where is a question of convenience. We have done this many times with satellites, the moon, Mars, and Venus.
Moving it somewhere else is a different (but also simple) question. Attach a rocket of suitable size, point, and fire.