Φferromotion · textbook · chapter 16 rust → wasm · on-device
Modern robots increasingly see the world as a cloud of little 3D Gaussians — the native output of splat reconstruction. FOCI checks collision directly on that map: the overlap between two Gaussians has a closed form, and because the robot's own Gaussians rotate with it, an elongated body can turn to slip through a gap it would smash into head-on. This page runs that overlap-integral collision on your device.
A 3D Gaussian-splat reconstruction represents a scene as thousands of small anisotropic blobs. Rather than meshing that into boxes and spheres — throwing away the very shape information the splats encode — FOCI keeps the Gaussians and asks a cleaner question: how much do two Gaussian density fields overlap? That overlap integral has an exact closed form, a single Gaussian in the separation of their means under their summed covariance. It is smooth, cheap, and — the key property — it knows about shape and orientation, not just distance between centers.
Below, two obstacle Gaussians wall off a corridor, leaving a narrow slot in the middle. The robot is a single Gaussian, deliberately long and thin. Drag it toward the slot and rotate it with the slider. Its colour is its collision cost — green is clear, red is jammed. Pushed in broadside, the long axis spans the walls and the cost flares red; rotate it to line up with the corridor and it threads through, green.
a conservative box or sphere around this robot could never fit — only its true orientation-aware shape does
The usual shortcut is to wrap the robot in a bounding box or a sphere and keep that clear of obstacles. But a bounding box of a long thin robot is nearly as wide as it is long — it can never fit through a slot narrower than the robot's length, no matter how you turn it. The robot physically fits; the conservative model says it doesn't. FOCI avoids that by scoring the actual Gaussian overlap, so the planner is free to use orientation as a way through — exactly what lets a legged robot slip sideways between two rocks.
On load, this page measured the collision cost of entering the slot broadside (yaw 0°) versus turned to align with it (yaw 90°), at the slot's center:
| collision cost head-on (yaw 0°) | … |
| collision cost turned to align (yaw 90°) | … |
| turning cuts the cost by | … |
| verdict | … |
Same position, same robot, same obstacles — only the orientation changed, and the collision cost collapsed. A model that scored collision from position alone would report the same value for both and never find the way through.
Score collision as the overlap of the actual Gaussians — the map's and the robot's — and orientation stays in the cost where a planner can exploit it. A tight slot is not a wall; it is an invitation to rotate.
FOCI closes the geometry side of the book the way the planning chapters closed the optimization side: keep the real structure of the problem instead of a conservative surrogate, and the solutions hiding in that structure become reachable. On a real Gaussian-splat map of a room, the same overlap integral scores a whole robot against hundreds of thousands of splats — and the robot turns to fit.
What you just drove: the foci overlap-integral collision from ferromotion-core, compiled to WebAssembly — the same code the native tools link against. Every frame sums the closed-form overlap of the robot's Gaussian against each obstacle Gaussian at the current pose; nothing precomputed.
Verified in the library: the closed-form overlap integral matches Monte-Carlo integration of the product of the two Gaussian densities (rel err <1%); the kernel is 1 at coincident means and decays monotonically to ~0; its gradient matches finite differences; and turning an elongated robot to align with a slot cuts its collision cost to under a quarter of the head-on value. Each is a test in cargo test, not a claim in prose. See also ch.15 — landing a rocket · the full textbook.
Institute for Physical AI · the Rust library · crates.io