The matching principle.
A wave meets a boundary, and whatever does not match comes back. That single condition is the quantitative form of this lab's oldest question, why systems fail at their seams. Where a robot touches an unknown world it is a theorem rather than a figure of speech: contact is stable against every passive environment exactly when the interaction port's impedance stays inside the unit disk, and that disk is the one Phillip Smith drew for antennas in 1939. The same geometry prices what a channel can carry, caps how well any network can be matched across a band, and sets what erasing a bit costs in heat. This track is the research into how far that condition carries, which of its links are identities and which are only useful analogies, and what it buys a machine that has to touch things.
Take the course · PAI-101 →The Smith Chart Navigator ↗Raise the stiffness and the impedance locus leaves the disk; the wall starts handing energy back at the same instant. A digital controller holds each force for one sample period, which delays the spring by half of it, and a delayed spring is a negative damper. Sample rate buys stiffness.
In the field · the condition is already what safe contact is built on. Collaborative arms such as KUKA's LBR iiwa and Franka sense torque at every joint so the port they present to a person can be commanded rather than merely reacted to, and haptic hardware has long been specified by its Z-width, the span of impedance it can render while staying passive. One layer down, the transistors doing the sensing are etched behind live RF matching networks that retune as the plasma's impedance moves mid-pulse. The research here is what changes when the thing commanding that impedance is learned: a policy that emits stiffness is choosing a point on this disk, and a contact task is a trajectory across it.
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