[DonRotolo]

For an 'order of magnitude' calculation (usually written on a napkin), just take the width of the wheel, divide it in half. That's your 'torque arm'.

Then take the force needed to slide the wheel (say, in Pounds) over the surface.

Multiply the force by the torque arm, that's a close estimate. So, if you need 24 pounds of force to slide the wheel sideways, and it is a 1" wheel, you get a torque of about 1 Ft-lb.

You can measure the sliding force, which depends on the coefficient of friction with the surface and considering the weight bearing on the wheel, with a pull-type scale.

This is purely an estimate, since it does not take into account several factors, including:
1. The friction in the shaft & bearing assembly
2. The force resisting the wheel from turning varies by the distance from the axis, we are assuming a point-load at the furthest edge only.
3. we are considering the static case only (the robot is not moving). With movement the forces are reduced, so we can consider only the worst case.
4. We assume the axis of the wheel turn is the centerline of the wheel. The force is significantly reduced if the axis is offset - look at a lawn tractor front wheel to see the idea.

No, it's not a scientific or engineering calculation, but it'll get you close. You probably would need about 2 or 3 FtLb for a 1" wheel, double that for a 2" wheel, 160 lb robot with wedgetop on carpet. But in the engineering world, this kind of guesstimate is usually sufficient.

You CAN also build a model and measure the torque required! That the "empirical" method.

I'd like to see someone answer the original question though with a more rigorous treatment.