I am assuming that the desired application would be to control a quadrotor drone or similar platform.
The scalar representations of distance shown in the paper above can be represented in 2 ways (that I see). You could extend from the 2d plane to the 3d plane using the same vector norm function as before. This would mean that the mathematical operations would not change, just the low level data structure would be a 3 element vector instead of a 2 element vector.
You could also simply do all of those equations separately on the X, Y, and Z components of the pose vector at the same time. This could be implemented with 3 different threads running the same mathematical process. This is probably safer since you can then prioritize each axis exclusive of the others in case you have to navigate areas where one axis of motion is not going to require large position deltas.
Does that help?