The plan I found to be the most interesting and that actually had some tests done was the "Tractor Pulling" plan. The plan in itself is simple and almost sounds ludicrous but with some more thought given in to it, might just work but at an extremely high cost. This plan could go two ways, one is cheap and possibly beneficial to the planet but more risky and the other is expensive but overall more safe than the previous. The first plan is not really a defense plan but could act as one if the initial test is successful. NASA plans to sends a probe or probes out to an asteroid headed in our general direction and see if they can alter the gravity of the object and ultimately alter the trajectory of the space rock. To do this, the probe will fly alongside the asteroid and slowly remove chunks of the rock in hope of changing the objects gravitational field. The original application of this plan was to be used to bring small asteroids into orbit of the moon so that astronauts could use it as a pit stop on their way to Mars in 2030. The physics for this plan would be tricky but possible and since this plan is already on schedule to be tested it may very well be the best solution we have to date. Things to consider when going with this plan would be the mass of the object, the force of gravity of the asteroid (which we need the mass and the radius of the asteroid), how dense the asteroid is and how fast it is traveling towards Earth. Given the quantities that are needed to be known we can see that changing the asteroid mass would affect the overall gravitational force of the object, possibly changing its course. For example, all of the planets in the solar system revolve around the sun because of gravity and the huge amount of gravitational pull the sun has, since gravity depends on mass these planets must be at an exact and constant mass to maintain their orbit. If Earth suddenly became less massive its orbit would change. The same could be said about an asteroid going towards Earth, since it is also caught in the sun's gravitational field. The proof is in Newton's equation for the gravity of two objects (Fgrav -proportional to- m1*m2/d^2) whereas the mass of the two objects is directly proportional to the final gravity. The only problem I see with this plan is that it could possibly put the Earth in more danger, especially in its testing phase. This is due to the fact that they would be bringing in an asteroid closer to Earth instead of redirecting it away from Earth. The second plan is quite similar but could prove to be safer, if it is even possible. For the second defense technique NASA wishes to send one big probe or several probes to throw the asteroid off course using gravity. This could be a possible solution but only if they could gather enough force to counteract the asteroid's gravity (which could be large depending on the asteroid's mass and radius) and even if this amount of force needed to transmit the required impulse can be harnessed it still must be directed at the rock and forced upon it for quite sometime for the change to be effective enough. In short, NASA would need to deploy this plan as early as possible to ensure success. Basically this plan is to change the objects momentum in general. Quantities needed in this plan would be the mass, acceleration, radius, and overall velocity.
I do not own these images.https://www.youtube.com/watch?v=TRddhIVvdAY