pic: 2826 Halonomic Drive Prototype

[Jared Russell]

I think you have that backwards - an O pattern on the ground provides more efficient turning. In fact, I don't believe that a square robot with an X on the ground can turn at all.

To turn clockwise with an O pattern on the ground (or an X pattern looking at the top of the robot):
The front left wheel provides equal torque in the FORWARD and RIGHT directions.
The front right wheel provides equal torque in the BACKWARD and RIGHT directions.
The rear left wheel provides equal torque in the FORWARD and LEFT directions.
The rear right wheel provides equal torque in the BACKWARD and LEFT directions.

Hence 100% of the torque (for a square robot) is around the axis of rotation. For an elongated drive base with 45 degree offset rollers, the percentage is less than 100%.

Now consider a clockwise turn for the opposite configuration - X on the ground (looks like an O looking at the top of the robot): None of the four wheels can supply torque around the axis of rotation (only towards or away from it). For a non-square drive base this isn't the case, but the percentage of output torque that aids in rotation is still going to be less than 100%.

* All of this assumes 45 degree offset rollers with negligible friction. If the rollers are sufficiently sticky, you can achieve an increased turning effect with the X on the ground (at the cost of decreased strafing efficiency).