about the torque calculation of a mecanum wheel

[GeeTwo]

The original article begins with the same fallacy, and presents additional fallacies as it goes along.

Originally Posted by zhy557

http://researchspace.csir.co.za/dspa...dle/10204/2771
or
http://researchspace.csir.co.za/dspa...lale2_2008.pdf

In the introduction, Tlale defines "effective" variables in equations 1 and 3, but does not describe the derivation, or sufficiently define the meaning to be entirely certain that he is following the same "lost force" fallacy. He never again uses these definitions, though he does use the word effective several times with similar meaning.

Figure 4 and the discussion between equations 4 and 5 clearly depict the fallacy that the "applied torque" must be decomposed into an effective andi an ineffective part and that the "ineffective" part simply disappears. To what object was this force applied? What object received the reaction (Newton's third law) force? If the answer to these is "nothing", then did that force ever exist? If not, how can it have been part of the "applied torque"? The answer, of course, is that it never did exist and the force countering the "applied torque" is not in the direction perpendicular to the wheel axle.

As shown here and in Ether's paper, the frictional force at the roller/carpet interface is essentially parallel to the roller axis. However, ignoring roller friction, the only three forces acting on the wheel are this contact force, the applied torque of the motor, and the forces necessary to constrain the wheel to stay on its axle. In the steady-state case (that is, all wheel speeds, the motion of the CoM and the rate of rotation about the CoM are all constant), The forces and torques on the wheel relative to its axis must add up to zero (otherwise the wheel speed would change). Presuming essentially frictionless wheel bearings, the torque in the plane of the wheel applied by the floor to the wheel must be equal and opposite to the torque applied by the motor/gearbox to the wheel. Since the force is not applied in a purely rotational direction, it must be greater than rF, and in particular rF/sinα, not rFsinα. As Tlale does not present any quantitative results, there are again no numerical opportunities for this inconsistency to be realized within the scope of Tlales paper.

Section IV (turning) is IMO an even worse job of physics. While there is a clear reference to a "pure turning motion" in the second sentence, only one of the cases presented (4 wheels) is at all likely to be about the center of mass, and therefore "pure" rotation.

FWIW, there are two cases which may produce pure rotation with two driven wheels: When wheels on the opposite corners are spun in opposite directions. Presuming that the other two wheels are at least somewhat free to rotate, the robot will rotate as well. These would be expressed as "+ 0 - 0", "- 0 + 0", "0 - 0 +", and "0 + 0 -" in the tables of either paper.

Finally, do not think that because the force at the floor is greater than rF we are getting something for nothing. Since this force is still a frictional force limited to be μW_i, a small value for sinα reduces the amount of torque which may be applied at the motor/gearbox without losing traction.