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.

Before you can ask that question you have to know the body can hold still. A simulated body integrated at too coarse a control rate can sustain an oscillation forever, and every steady-state number you read off it is then the amplitude of that oscillation rather than an equilibrium. It is not a subtle failure once you look for it, and the test costs one line: put the body at a posture where the forces already balance, command it to stay, and require that it does. A body at rest has no velocity. A cycling one does, and that is the quantity that separates them.

So the cell certifies the plant first and measures second, which is the right order for any question about equilibrium. You will see the same sweep return a clean answer at one control rate and a meaningless one at another.

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 a stable 1.35x at one control rate and no well-defined answer at all at another.

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, and it is the same to three digits however long you run. 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 numbers tens of times larger, and they are not a measurement of anything. The search assumes the residual falls as gain rises; on an oscillating plant it does not, so the bisection lands wherever the cycle's phase happens to put it. Change the run length or the number of search steps and the answer moves. That is what a quantity with no fixed point underneath it looks like.
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.
Step 3 is the tell you can use anywhere: run the same question twice at different settle times. The certified tick gives 1.36x and 1.36x. The uncertified one gives 5.15x and 81.57x. A real steady state does not care how long you waited for it.

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 a certified tick, half authority costs 1.34x, 1.35x and 1.36x for offsets of 0.30, 0.20 and 0.10 rad. At an uncertified one the same search returns numbers tens of times larger that move when you change the run length or the number of search steps. Say why the second set is not a measurement, and what property of the residual the search was assuming.

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.

GO FURTHER · THE LIVE PLAYGROUND

Read TR-2026-43 ↗ · The full argument, and the posture price measured here in its place among the other three.
Run the live instrument ↗ · The companion bench for module 3: what an observer's reconstruction costs a held body.

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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