Omni-directional motion, resolving three or more vectors? In robotics, there exists drive-trains that can move omni-directionally (that is in any direction). These come in many shapes and sizes, but most come in a three wheel or four wheel configuration, to keep things simple lets assume three wheels. 

These drives are made possible by what is know as an omni-wheel, that is a wheel with casters along it which allows power to be applied in the direction the wheel spins, but not in the direction perpendicular to the wheel. 

You can think of the "power" direction as having a lot of friction, and of course the part that rolls with the wheel, while the "slip" of the casters provides a very low friction, non-powered direction.
I come from a programming background, not physics but I want to better understand the physics at work here. So let's look at a simplified picture:

I know that friction and the center of mass are important factors. That's about it.
So my question is, given a direction (green arrow), can a set of forces (red arrows) be calculated such that they equal the overall resulting force. IE, given blue and green can you calculate the various red forces?
 A: This shows your system and the forces acting:

I've chosen my axes to make $F_A$ act horizontally; in general your object won't be aligned like this, but you just have to rotate your axes before doing the calculation. The requirements are that the forces $F_A$, $F_B$, and $F_C$ add up to the required force $F$, and that the object doesn't rotate as it moves i.e. that the net torque is zero. This gives us three equations in the three unknowns:
$$ \begin{align}
F_x &= F_A - F_B cos(60) - F_C sin(30) \\
F_y &= -F_B sin(60) + F_C cos(30) \\
0 &= F_A + F_B + F_C
\end{align} $$
or more simply:
$$ \begin{align}
F_x &= F_A - \frac{1}{2} F_B - \frac{1}{2} F_C \\
F_y &= -\frac{\sqrt{3}}{2} F_B + \frac{\sqrt{3}}{2} F_C \\
0 &= F_A + F_B + F_C
\end{align} $$
So for any given force $F$ you just have to solve those three simultaneous equations to get the forces on the drive wheels. After some quick scribbling I get (I don't guarantee there are no errors in this!):
$$ \begin{align}
F_A &= \frac{2}{3} F_x \\
F_B &= -\frac{1}{3} F_x - \frac{1}{\sqrt{3}} F_y \\
F_C &= -\frac{1}{3} F_x + \frac{1}{\sqrt{3}} F_y
\end{align} $$
