diving to 4 directions

[Cory]

Originally Posted by Lil' Lavery

There are several ways to achieve "omni-directional" motion (motion on both the x and y axis, as well as the ability to rotate). The most common methods are holonomic (and its subset mecanum) and swerve (crab). There of course are other methods (such as "ball drives" or hovercraft), but aren't typically very effective on the FIRST scale.
Holonomic Drives, sometimes referred to as killough drives or kiwi drives, are a vector-based drive system. They typically consist of 3 or 4 (typically) "omni-wheels" placed radially around the robot (although alternate configurations are possible with as few as 2 powered wheels). More specific math can be found in other posts, and in a link I will provide later, but essentially each wheel contributes a composite victor and the robot moves in the resultant vector (and is also effected by any torque created on the robot). The most common supplier of omni-wheels in FIRST is http://andymark.biz/
A subset of the holonomic that has become very popular in FIRST over the past few years (particularly in 2007 with the release of the AndyMark mecanum wheel) is Mecanum Drives. They function under the same idea as standard holonomic drives, but rather than have straight rollers and angeled wheels, they have angeled rollers and straight wheels.
Swerve Drives, also known as crab drives, are an omni-directional system in which the wheels themselves actually rotate to achieve motion in different directions. Wheels are typically built into "modules" which are rotated by one or more steering motors. In Co-axial swerves, all drive wheel modules are powered by a single transmission, but in most FIRST swerves each wheel is powered individually.

Since I have taken a considerable chunk of time to (distractedly) write this, I'm sure most of this information has already been covered. But you can get a more in-depth description of each, and the math involved by searching chief delphi. Consider searching for "omni-drives", "holonomic", "mecanum", "kiwi", or "swerve". Also, read this paper by Ian Mackenzie of 1114 dealing with the kinematics (math of motion) of omni-directional systems.

Sean lists pretty much everything you need to know. I have one minor nitpick to make though.

Traditionally, a true omnidirectional drivetrain allows translation in the x and y axes as well as rotation about it's center point, at the same time. A swerve drive will not allow you to do this. I don't believe a mechanum drive will either, but to be honest I haven't paid an ounce of attention to them beyond that cool Airtrax video.