Tuning a PID

A PID controller drives a joint to a target by feeding back its error. The proportional term pushes in proportion to how far off it is. The derivative term resists how fast it is moving. The integral term adds up the error over time, so a steady push from a load, like gravity on a held joint, is cancelled instead of leaving a gap.

The plant here is a rotor with inertia and friction, sampled every half a millisecond, with derivative taken on the measurement so a step in the target does not kick the output. Without the integral term, load or limits it is almost exactly a second-order system, and the simulation is held to its closed-form response. Its inertia, friction, load and the size of the step are made up, so the response is a shape, not a measurement of any real joint.

Turn on the torque limit and give it some integral gain. The output saturates while the error is still large, the integral keeps adding up anyway, and when the joint arrives the stored integral drives it past the target. That is windup; the plot shows the integral’s share of the torque. Sensor noise shows the other trap: the derivative term amplifies it.

It matters beyond the textbook because almost every learned policy still outputs targets for a PD loop. Asimov’s walking policy emits 23 joint-position offsets, scaled by 0.25 around a default pose, and per-joint PD gains track them: stiffness 150 on the hips and knees, damping capped at 5, under torque limits.

The “Asimov knee” preset loads Kp 150 and Kd 5 with a 45 N·m torque limit, the simulation values from the policy’s contract. The inertia, friction and load stay the page’s illustrative ones, so it shows what those gains and that limit do to a joint like this, not how Asimov’s knee moves.