PYTHON · NUMPY

Impedance control and contact: staying inside the disk

Discover the safety condition for touching an unknown world — and find that it is the Smith chart's unit disk.

01Challenge

Fly a tool into a digitally rendered wall and measure the exact stiffness at which the wall starts handing energy back.

Predict firstYou render a virtual wall in software and make it stiffer and stiffer. What happens?

02Model

A robot touching a stiff surface chatters. A haptic wall buzzes. Both are the standing wave that burned Smith's line: energy reflecting at a mismatched boundary faster than it can be absorbed. Colgate and Hogan proved the condition in 1988 — a robot is stable against every passive environment if and only if its own interaction port is passive, Re Z(jω) ≥ 0 at all frequencies. Network theory says that is exactly when the scattering operator S = (Z−1)(Z+1)⁻¹ stays inside the unit disk. That is the Smith chart's Möbius map, character for character. Robots do not fail contact by matching imperfectly. They fail by leaving the disk.

03Guided practice

1. Where the energy comes from. A digital controller samples, then HOLDS its force until the next sample. The hold delays the spring by about half a period, and a delayed spring is, to first order, a negative damper of −K·T/2. 2. The measurement. Fly a tool in at a known speed and see whether it comes back faster. 3. The boundary. It flips exactly at b + B = K·T/2. Sample rate literally buys you damping — halve the period and you can render twice the stiffness.

04Feedback

PASS WHEN PASS when your measured passivity boundary matches b + B = K·T/2.

If your run fails, check:

A cell errored — run it again.
The measured boundary should track 2(b+B)/T across sample rates.

05Retrieve & space

What is the passivity condition for safe contact? (Re Z(jω) ≥ 0 at every frequency — equivalently |Γ| ≤ 1).
Why does a zero-order hold destabilise a stiff virtual wall? (it delays the spring force by ~T/2, which acts as a negative damper of −KT/2).
How do you render a stiffer wall safely? (sample faster, or add real damping — b + B must exceed KT/2).

06Mastery & project

You can state the passivity condition, explain why digital control threatens it, and predict the stiffness limit from the sample rate.

The habit: before trusting a controller near contact, ask whether its port is passive.

GO FURTHER · THE LIVE PLAYGROUND

The passivity disk — live ↗ — Watch the impedance locus leave the unit disk and the wall start chattering at the same instant.

← Learned matching: when AI designs the interface Morphological computation: the body as computer →

PYTHON · NUMPY

Impedance control and contact: staying inside the disk

Measure, from scratch, the stiffness at which a digital wall starts creating energy.

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