I have some
recent experience in the world of killough drives. Here's some of what I learned:
Less wheels are better than more
Each wheel needs to be individually powered and controlled. You can't couple any of the wheels together and still retain holonomic motion. If you're going to spend resources (time, money, anguish) on an 8-motor omnidirectional drivetrain, why not make a fully independent swerve instead?
For number of wheels N > 3, you must design your chassis to be flexible enough to load the wheels approximately evenly. For N = 3, the chassis can be as rigid as you wish.
(N = 2 is a bicycle, N = 1 is a unicycle)
Odd values of N are better than even values of N
What's interesting about omni and mecanum wheels is that the rollers resist turning. This small but significant amount of friction seems to be worsened when there is an axial thrust on the roller.
When you arrange 4 omni wheels in a rectangle, you get a drivetrain that prefers to travel in the cardinal directions and dislikes traveling in arbitrary, off-cardinal directions. This is because when the robot travels in the cardinal directions, the rollers on the pair of parallel wheels that are pointed in the direction of travel are not spinning. Turning to an arbitrary, off-cardinal direction means overcoming the nonnegligible friction on these rollers. If the robot is accelerating along a cardinal direction, then the addition of axial thrust on the rollers will effectively lock the robot's orientation until the acceleration stops! I have video of this phenomenon that I can share.
This issue is minimized when N = [3, 5, 7, ...] because when the robot is moving in an arbitrary direction, there can never be more than one wheel with motionless rollers.