ΦInstitute for Physical AI· the Charlot Lab

The Ferromotion Textbook

Sixteen ideas that make machines move — and each one runs, in full, on your device as you read.

This is not a book about robotics that shows you finished results. Every chapter drives the same Rust library the Institute's tools are built on, compiled to WebAssembly, and computes its central claim live in front of you — no recordings, no precomputed plots. Drag the pieces, push the limits, and watch the numbers hold. Each chapter also verifies its own headline on load, against an analytic answer or a physical invariant, so you can trust that the thing you are playing with is the real thing.

runs on your device self-verifying the real Rust library no install
Chapter 1Actuation
The body is the controller
A muscle rejects a disturbance before a nerve could carry the news. When the body is fast enough, it is the controller — and a loop cannot compete.
governed by the delay margin
Chapter 2Coordination
Agreement, and the one number behind it
Agents that see only their neighbours still come to agree — and how fast is set entirely by a single eigenvalue of the network.
governed by the Fiedler value λ₂
Chapter 3Safety
The command it will not obey
A filter that minimally corrects any command so a robot physically cannot be driven into a hazard — whatever it is told.
governed by the barrier h(x)
Chapter 4Manipulation
When touching becomes holding
A grip holds against a push from any direction only when its friction cones surround the origin. Drag the fingers and read the grasp by eye.
governed by the Ferrari–Canny Q1
Chapter 5Learning
Show it once
Teach a motion a single time; replay its shape to any new goal, always arriving. The learning carries the style; the arrival is guaranteed.
governed by the structural attractor
Chapter 6Perception
The estimator that stays honest
Measure error on the group instead of by subtraction, and the model of how it grows becomes exact — for any error, however large.
governed by the invariant error-model
Chapter 7Planning
As fast as the motors allow
The fastest way to follow a path is bang-bang: crawl the corners, floor the straights, always pinned to a limit. Race it against one safe speed.
governed by the velocity ceiling
Chapter 8Locomotion
Where to put your foot
Walking is a fall, caught over and over. There is exactly one point on the ground where a step brings the body to rest — steer that, and the body follows.
governed by the capture point ξ
Chapter 9Modeling
Make it linear
A nonlinear system moves in a straight line after all — in the right coordinates. Add one observable and a linear model predicts it exactly, learned from data.
governed by the lifted operator
Chapter 10Composition
Do everything at once
Reach the goal, avoid the obstacle, stay smooth — fused into one reactive motion by a metric that weights each behavior by how much it matters right now.
governed by the Riemannian metric
Chapter 11Compliance
The robot that bends
A soft arm has no joints — it curves everywhere. Describe it by its strain along the body, and its bending under load matches beam theory exactly.
governed by the strain field
Chapter 12Mapping
Revising the past
A drifting loop never closes — until one late "I've been here before" re-optimizes the whole trajectory and snaps every past pose into place.
governed by the pose graph
Chapter 13Cables
Held by cables
A cable only pulls, never pushes. Holding a platform is finding positive, in-range tensions that balance the load — and the workspace ends where a cable goes slack.
governed by the tension distribution
Chapter 14Contact
Planning through contact
Contact puts a kink in the dynamics, and gradients die at a kink. Smooth the force, then step only as far as the smoothing stays honest — a trust region shaped by the contact.
governed by the contact trust region
Chapter 15Guidance
Landing a rocket
A non-convex landing is too hard to solve directly — so solve a sequence of convex pictures of it. Watch a straight-line guess bend into a real descent as the defect collapses superlinearly.
governed by successive convexification
Chapter 16Geometry
Turning to fit
Check collision on a map of Gaussians by their overlap, and orientation stays in the cost — so an elongated robot can turn to slip through a slot a bounding box could never clear.
governed by the Gaussian overlap integral

Every chapter runs ferromotion — a pure-Rust library for the kinematics, dynamics, and control of physical AI — compiled to WebAssembly and executed in your browser. The interactive claims are computed on demand and each chapter re-derives its headline result on load; the same properties are checked in the library's cargo test suite. Nothing here is a screenshot of an answer. Institute for Physical AI · the Charlot Lab · the Rust library · crates.io