## Note: Minimum Motor Torque for Turning

I wanted to do a quick calculation for a student Vex robot that has 5 in wheels and is having one rail stalling during a turn.    The stalling could have been predicted with some basic torque calculations.

Using static moment equations you can easily derive the minimum torque to just turn  for a symmetric robot with four direct drive  motors on wheels with radius r .  Legacy three-wire 6.5 inlb motors are assumed.

Define the half spacing between axles as “l” and the half width between wheels as “w”   , the robot weight as W_lbs and the coefficient of friction for a tire side force as u.  Also assume that the weight is evenly distributed on the four wheels and the center of turn is at the cg.  Also w > l so turning can occur.

Then

Input turning moment per wheel = w* torque/r and this must be greater than the moment resisting the turn caused by the side friction forces on the wheel.

Resistance moment = l*W_lb*u/4  .    The factor of 4 converts the W_lbs to normal force on the wheel.

So torque > r*(l/w)*W_lb*u/4

Lets plug in some typical numbers.  r = 2.5 in, l=3.5, w = 7

This gives a required motor torque of  2.5*3.5/7/4*W_lbs*u

or torque >( 2.5/8) * W_lbs* u

or W_lbs <  torque*8/2.5/u

Without omni wheels, u is typically  around 1 or higher and if the robot W_lb is 12 lbs  then the minimum torque

torque_min > 3.75 in lbs

This represents about 60% of the old three wire motors 6.5 in lb  spec torque .       Clearly there is not much margin for any torque losses in the drive train or extra side friction on the wheels or extra weight from a heaver robot or from game objects.  Also any contact forces resisting the turn will easily stall the turn.

Using a 4 in omni wheel in place of the 5 in traction wheels increases the input torque by  5/4 and reduces the u by about 1/3   so we get a combined improvement of  15/4 or 3.75 factor on the required torque.

min_torque_ 4 in omni =  3.75 / 3.75 = 1 in lb or about 15% of the available torque of an old motor.   This is where you want to be in your design.