If we think of gravity as being the consequence of the curvature of 'space-time' then the idea of 'anti-gravity' makes no sense. It is only if the gravitational field is the result of some kind of force that we can make sense out of such ideas as 'attraction' and 'repulsion' when thinking about gravity.
In order to explain how an object can be 'repulsed' by a gravitational field it is required that we introduce the idea of a gravitational 'charge'. The gravitational force would parallel the behavior of the electric force with the difference being that gravitation would be 'anti-symmetrical', in that in a gravitational field like charges would attract while unlike charges would repel. Now it is said that gravitation parallels the electromagnetic field phenomenon in many ways, with some noticeable differences, among them being that there is no 'negative charge' and that gravitation is always attractive and never repulsive for that reason. However saying this is much like saying 'there is no anti-matter in the universe' or 'there are no positrons'. We don't see anti-matter in the universe and we don't see many positrons, which is fortunate given that positrons and electrons, which we do see, annihilate each other and release high energy gamma rays in the process. It has been said that the 'Law of the Conservation of Charge' is invariant, but it is obvious that this is not true under a certain specific set of circumstances, for while it is true that electrons can only be created when at the same time a corresponding positron is created (thus preserving the zero charge of the energetic photon) it is also true that electrons exist and positrons do not exist, at least not for very long in this universe, and the fact that the universe exists at all is evidence that the Law of Conservation of Charge is not invariant. We do not see 'anti-gravity' or a 'negative gravitational charge' in the universe for if we did we would not see much of a universe, and the sublimation of this negative charge is required if this particular universe is to exist at all, just as positrons or anti-matter must not exist if the universe as we know it is to exist.
I say that the 'negative gravitational charge' is 'sublimated' because I believe that we do see it but we do not recognize that we are seeing it, because it is weak. Weak as this charge might be it is still required if we are to see any universe at all. If a mass is to have a 'positive gravitational charge' this would suggest that the 'charge' is a characteristic of the mass. As we know the gravitational force increases with the size of the mass and so therefore it would seem that the same force that holds a small together is the same force that holds a large mass together, and so therefore the forces that operate on the subatomic level and are holding together sub-atomic particles with a small mass, such as protons in a nucleus, are cumulative and so as mass increases so do the cumulative forces. At the same time we can see evidence for an 'anti-gravitational force' in that there is also a weaker force at work which prevents collapse under the positive gravitational force, and we could say that it is this stronger gravitational force which is responsible for the fact that matter exists while it is this weaker negative force which is responsible for the fact that space exists. As mass increases it must be true that the stronger attractive force and the weaker repulsive force increase in such a fashion as to be nonlinear for at some point gravitational collapse does occur.
Here I am assuming that 'gravitational charge' is an inherent property of 'mass', and I am aware that physicists consider the 'strong force' and the 'weak' force to be forces separate from the force of gravity which is a very weak force in quantum physics. The concept of 'negative mass' seems meaningless and nonsensical and this then leads me to consider the following hypothetical scenario.
If the gravitational charge is an inherent property of a mass then we can imagine the creation of a type of 'anti-mass'. This would be a 'negative mass' only in the sense that the 'charge' of the mass would be 'negative'. So we would imagine that we would have a nucleus composed of 'anti-protons' with a net negative charge bound together with the ubiquitous neutrons surrounded by a cloud of positrons and this would be an 'anti-atom'. My assumption here is that we only see mass in this universe and we do not see 'anti-mass' just as we see electrons and we do not see positrons (even though a positron is not an imaginary particle). If 'charge' is a property of mass then we would expect that an 'anti-mass' must also be possible and that the given that 'opposite charges' should be repulsive as far as it concerns gravity we might for the first thing see something which 'falls up and away' rather than what we always see in this universe which is something which falls down.
My observations of floating craft maneuvering within a gravitational field suggest that three vectors are required to explain the range of possible movement, and here what comes to mind is the picture given to explain angular momentum. The index finger is pointed straight ahead, the middle finger is then moved to become perpendicular to the palm (and at a right angle to the index finger) and then the thumb is pointed upwards in the 'thumbs up' position. The thumb corresponds to movement in the up down direction, the index finger forward and back and the middle finger left to right. Movement up and down within a gravitational field makes sense to me, and is easy to picture. Now as for movement left or right or forward or back or combinations of pairs giving a full range of motion, the only possible explanation I can come up with that makes any sense to me is the notion of 'differentials'. So if it was true that our craft was 'neutral' with respect to gravitational charge, then perhaps we could imagine ourselves hovering (or we could be 'negative' with equality and our velocity would therefore be zero). If we wanted to move to the left then perhaps we could be 'neutral' on the right but positive on the left and thus 'fall to the left'.