PYTHON · NUMPY

The plant that would not stand still

An arm's joint tone is built so that tone alone balances gravity at its working posture. You put the arm exactly there, at rest, and command a PD policy to hold that same posture. What does the arm do over the next forty seconds?

Measure what posture authority costs a policy on a gravity-loaded arm, and certify the plant that measures it, by requiring the body to hold a posture it is already in before trusting any steady-state number it produces.

01Challenge

One two-link arm held up by joint tone, one PD policy riding on it, one commanded offset. Find the policy stiffness that closes half the offset. Run the same measurement at four control rates and decide which of the answers, if any, is about the arm.

Predict firstAn arm's joint tone is built so that tone alone balances gravity at its working posture. You put the arm exactly there, at rest, and command a PD policy to hold that same posture. What does the arm do over the next forty seconds?

02Model

The first three modules made one case: a body's substrate carries competence that never appears in the observations. This lesson pays the other half of the ledger, because the substrate is not free. Hold your arm out and gravity is handled for you by tone you never command. Now try to place your hand somewhere your posture disagrees with, precisely, and feel the tone push back. That push is a price, it is paid in stiffness, and it is measurable.

The measurement is simple to state. The arm's tone is a spring toward a rest posture, arranged so the tone alone balances gravity at the working posture. A PD policy rides on top. Command the shoulder some offset away and the arm stops short, because the tone pulls one way and the policy the other. Sweep the policy's gain until it closes half the offset. That gain, divided by the tone's stiffness, is what half a posture costs.

There is a second lesson folded into the first, and it is the more valuable one. When the Institute first ran this sweep it published an answer that was wrong: two to five times the tone's stiffness, and worse for fine corrections than coarse ones. A tidy, memorable, counter-intuitive result. It was an artifact, it was live for a day, and it was retracted. The defect was not in the analysis. It was in the plant, and every gate the lab had pointed at the analysis.

So this lesson runs the certification first and the measurement second, which is the order the lab now uses. Before asking a body a question about equilibrium, ask whether it has one.

03Guided practice

Run the cell top to bottom. Read the certification table before you look at the sweep: two of those four control rates disqualify every steady-state number measured at them, and the sweep below shows you what those numbers look like anyway.

04Feedback

PASS WHEN You certified a plant, then measured it, and saw the same question return 1.35x and 88.58x depending only on whether the certification passed.

If your run fails, check:

The certification is one line of physics: the tone is built to balance gravity at this posture, so the commanded torque is exactly zero and the arm cannot move. At 20 ms it moves anyway, to 0.231 rad, and stays there carrying 21.6 rad/s. An equilibrium with a little error would sit still. This is an oscillation.
At a certified tick the answer is 1.34x, 1.35x, 1.36x for offsets of 0.30, 0.20 and 0.10 rad. Flat. The price of half a posture is a property of the posture, not of how large a correction you want.
At the uncertified tick the same sweep returns 2.11x, 61.16x, 88.58x: an enormous dependence on offset size, which is exactly the shape of a fixed oscillation amplitude divided by a shrinking offset. A constant over D looks like a law about D.
The cause is the zero-order hold on the damping term, not the spring. The elbow's effective inertia gives the damping a 3.4 ms time constant, held across a 20 ms tick, so the torque is applied long past the motion it was computed for. The criterion is b*T/m < 2; at 20 ms this plant runs 5.86.
Note which check is the strong one. Settle time barely helps: at the bad tick, waiting ten times longer moves the answer only from 82.97x to 88.58x, a wobble you could talk yourself out of. The hold-still test moves it from 0.000 to 0.231 rad. Prefer the test that fails loudly.

05Retrieve & space

The arm is commanded to hold the posture it is already standing in, and the net force on it is verified at 4e-16. Why does that make its motion a decisive test rather than a small numerical complaint?
A residual is identical at 200, 400, 800 and 1600 ticks. Name a state of the system, other than equilibrium, that produces exactly that, and name the quantity that separates the two.
At the uncertified tick, half authority costs 2.11x at an offset of 0.30 rad and 88.58x at 0.10 rad. Explain how a constant, offset-independent defect produces that pattern.

06Mastery & project

Given a simulated body and a steady-state claim measured on it, state the certification you would demand before accepting the claim, the quantity that certification must inspect beyond position, and the parameter you would refine to distinguish a plant defect from a discretisation artifact.

Take a simulator you use. Find its equilibrium, put the system there at rest, command it to stay, and run it for as long as your longest experiment. Report the drift and the velocity. If it moves, work out which of its terms is being held across a tick longer than its own time constant, and report the ratio.

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PYTHON · NUMPY

The plant that would not stand still

The arm, its tone, and its policy are given. The commanded torque is HELD across each control tick, the way a real servo loop holds it, and the control rate is the only thing that changes.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you