And you thought you had cool omni-wheels

[Max Lobovsky]

I've been having a hard time proving this, but I believe this is the way both holonomic systems work:

efficiency as compared with wheels facing forward

100% speed
100% power output
1/sqrt(2) max force (obviously, at stall)

It is pretty clear that force is 1/sqrt(2) because when the robot is pushing against something fixed, the force vector that the motor is creating is 45deg to the direction force is being applied.

The other two claims aren't as simple. The only way I can attempt to prove them is through conservation of energy stuff, when I try to look at force vectors, they don't make sense.

Anyway, assuming lossless omniwheels (no friction in lateral movement), 100% of power output goes in the forward direction. Even though there should be a force vector perpendicular to direction of movement, it does not use any energy because it the lateral movement is lossless. If power output is constant, speed must be also constant because max speed is is just the point where f_friction(v) = f_motor(v). (I think equivalent holonomic and straight wheels systems would require different gearing, though, because even if wheels are same sized, a revolution of a straight wheel covers further linear distance than what at 45deg)