Modeling the static relationship of commands to torques.

Static and dynamic friction make the actual torque the motor applies different from the command. For very small commands, static friction is strong enough to prevent the motor from moving, so there is no output torque from the motor. This causes a dead zone where small commands have no effect. How can we measure how much friction there is? A fixed command results in a steady state velocity where the back EMF balances the motor torque. Measuring this steady state velocity approximates measuring the torque coming out of the motor.

I wrote a program, steps_sweep8, the steadily increases the command from its minimum, or steadily decreases the command from its maximum. An upward sweep looks like this:

Note that this involved about 1 million samples, and since we sampled at 1kHz, that took about 1000 seconds or about 17 minutes (2 seconds per command magnitude from -255 to 255). The raw velocity plot from the left wheel looked like this

After filtering the velocity plot looks like this

Let's now plot the filtered velocity versus the command rather than samples. I use the Matlab program find_cmd_and_vel to extract the asymptotic velocity after the command is changed. The data is saved in the file zzz. Here is what the commands to make the next plot look like:

load f0.dat
load f1.dat
find_cmd_and_vel(f0)
in shell: mv zzz f0u.dat
find_cmd_and_vel(f1)
in shell: mv zzz f1d.dat
load f0u.dat
load f1d.dat
hold off
plot(f0u(:,2),f0u(:,3))
hold on
plot(f1d(:,2),f1d(:,3))
plot(f0u(:,2),f0u(:,4))
plot(f1d(:,2),f1d(:,4))
title('left and right: command to velocity')
xlabel('command')
ylabel('radians/second after about 2 seconds')
legend('left increasing at 8.28V', ...
        'left decreasing at 8.22V', ...
	'right increasing at 8.28V', ...
        'right decreasing at 8.22V', ...	
	'Location','NorthWest')

I see 3-5 areas where the velocity does not change as the command changes. The big one at the center is the dead zone for small commands. The left half of the blue line is the wheel slowing down, and it stops moving (is captured by static friction) at a command of -10. The right half of the red line is the wheel slowing down (it is sweeping downwards from right to left) and it stops moving at a command of 10. So static friction can stop the wheel moving around a command between +/-10. For the upward sweep (the blue line) the wheel started moving again at a command of around 15. For the downward sweep (the red line) the wheel started moving again (breaks static friction) at around -15. These numbers vary with different motors, and when moving in different directions, so you may have to adjust them for your motor.

There are several other flat spots, which I can't explain. They may be due to how the electronics of the motor driver works. Since I can't explain them, I plan to ignore them.

We can fit these lines with equations. I will just fit the left wheel. The left wheel has the same slopes, but slightly different intercepts. The left and right halves of each line are slightly offset, so we will give them different equations:

if command > 12 when starting or if command > 9 when stopping:
  velocity = 0.0159*command - 0.0784;
else if command < -10 when stopping or if command < -15 when starting:
  velocity = 0.0157*command + 0.0850;
else
  velocity = 0;					    

Presumably this relationship depends linearly on the battery voltage, because all the motor driver does is apply the battery directly to the motor for a variable amount of time (PWM). To test this, I depleted the battery by doing about 20 of these sweeps, and got this set of plots [This data was taken from a different robot with a 13 count encoder]:

Each pair of lines were done at a different voltage (around 6.5V vs. 8.1V) and have different slopes. We are getting a different steady state velocity and thus a different torque at the same command.

We can also notice the the motor stops reliably at a command of +/-11, but the start command varies.

The flat spots of constant non-zero velocity versus changing command are weird. They happen at a particular velocity level (or amount of back EMF), not at a particular command. Even more mysterious.

The modified model to take this battery voltage (V) effect into account is: [Back to the equations I got from the 11 encoder count robot]

if command > 15 when starting or if command > 10 when stopping:
  velocity = 0.0159*command*V/8.25 - 0.0784;
else if command < -10 when stopping or if command < -15 when starting:
  velocity = 0.0157*command*V/8.25 + 0.0850;
else
  velocity = 0;					    

For quasistatic control (slow movements), we have a desired torque we want to apply, and we need to estimate the command that will achieve it. If the robot is moving and the command we get from solving the equations above for command is more than +/-10, this is easy. If the robot is moving and the command we get is smaller in magnitude than 10, we have three cases: If we want to keep moving in the same direction, we should increase the magnitude of the command to more than 10. If we want the wheel to stop, we should use the smaller command. If we want the wheel to reverse direction, we should reverse the sign of the command and apply a magnitude of 15 or so. If the robot is not moving and we want to start it moving, we should increase the magnitude of the command to bigger than 15, with the sign given by the direction we want to start moving in. This strategy leads to a discontinuous command when the robot starts moving (hit the motor with a command of +/-15 to get it moving), and when the robot reverses velocity. Starting with a positive command, the command decreases until it is 10, and then it jumps to -15 to get it moving in the reverse direction.

Faster movements are more complicated, as we shall see later.