I don't know if there is a straightforward answer to this question. I could not find much with google.

To give context, I am building a self balancing robot. It balances pretty well, but I am struggling to eliminate small oscillations around the balanced point. Looking at plots of PWM voltage and motor speed shows the culprit is that small voltage commands from the controller have no effect when the motors are stationary. The integral term in the PID increases the command until the motor starts, but then it goes too fast causing overshoot of the balanced position. The same process then repeats with a small negative command. I use 12V motors and they do not start below 6V.

Since the robot mainly operates near its balanced point, the controller spends most of its time issuing small fine control commands, which are completely ignored. The plot shows that the integral term causes the controller to ramp to the breakaway torque and then shortly after switch to the negative breakaway torque. The fancy pants PID is acting like it has just 3 voltages; 0, +breakaway and -breakaway !

I believe that the current gear ratio of 75:1 is far too high, and believe that using a lower ratio will reduce the starting voltage. Is that correct ?

Since the current motors seem to have more than enough torque, I am considering switching to a 34:1 gearbox, and I am hoping that might roughly halve the starting voltage, but without knowing the relationship between gear ratio and breakaway torque I am just guessing.

 

... Would it be possible to implement a small amplitude oscillation about the static position?
This technique, known as 'dithering', may have a beneficial effect.
See also:
Dithering

 

The mechanical break-away torque is always present in any system that has to transition from static to dynamic friction states.
Less gearing = less sticktion - give the lower ratio a try.
Dithering might work, but will increase motor coil heating and brush wear substantially.

 

Just a thought…
Can you give the motor a longer pulse or two to get it started and then go to the desired PWM?

 

Maybe your motors are too new and not broken in yet. Let them run for a couple days.
If there is grease in the gearbox, clean it out and use sewing machine oil.

 

The problem most likely isn´t breakaway torque but backlash. It takes some amount of rotation of the input shaft for the gears in the garbox to rearrange the mesh and start rotating the output in the opposite direction.

 

Just to follow up:

Thank you to everybody who offered advice.

I bought a gear motor with a lower gear ratio (34:1). It improved matters considerably. The motor does not start to turn until it receives at least 6V. However since the gear ratio is less, it has produces less torque, and so the start up accelerates the robot less and produces less overshoot of the desired position. I feel I would be better off lowering the gear ratio even further.