The big idea: push harder the further you are

Imagine a rover that must drive 5 m and stop exactly on a mark. A position sensor tells it where it is. How hard should the motors push?

The simplest sensible answer: push in proportion to how far away you are. Far from the mark, push hard. Nearly there, push gently. On the mark, stop pushing.

The distance still to go is called the error, written \(e\):

The setpoint is where you want to be (5 m). The measurement is where the sensor says you are.

One number, one rule

Say the rover is at 3 m, so \(e = 5 – 3 = 2\ \text{m}\). With a gain of \(K_p = 60\ \text{N/m}\) the motor force is

At 4.5 m the error is 0.5 m and the force drops to 30 N. That is proportional control, the P in PID. It already works surprisingly well, but it has two problems you will see in the simulation:

• The rover arrives with speed, overshoots the mark and swings back and forth.
• On a ramp, gravity pulls it back. Near the mark the error is small, so the push is small, and the rover stops short: 17 cm short in our example.

Three questions, three terms

A PID controller fixes both problems by asking three questions every time step:

1. P: how far off am I now? Push in proportion.
2. I: how long have I been off? Add up the error over time and push harder if it will not go away. This removes the 17 cm.
3. D: how fast am I closing in? If the error is shrinking quickly, ease off early. This stops the overshoot.

The loop

Every controller in this course runs the same feedback loop, many times a second:

1. Measure where you are.
2. Compare with the setpoint to get the error.
3. Compute the command from P, I and D.
4. Act: send it to the motor, then repeat.

Keep this picture in mind. Every equation in Lesson 4 is one line of step 3.