06 / 06 · Riemannian geometry · closed form, no solver
APF pushes the drone away from obstacles. WARP curves the space it lives in instead, so that every obstacle surface is infinitely far away. Hitting one would take an infinitely long path.
What the controller computes
01 · Curve the space
Obstacles enter a conformal metric g = Φ(h)² I, where h is the distance to the nearest inflated surface. Within 1.2 m of a surface Φ = 1 + β(h_s − h)³ / (h_s² h); beyond it Φ = 1 and space is flat.
Near a surface Φ grows like 1/h, so every surface sits at infinite geodesic distance, like the edge of a Poincaré disk. Safety becomes topology, not force balance.
02 · Distance in the curved space
The guidance needs the geodesic distance to the goal, which solves |∇d| = Φ. A fast-sweeping eikonal solver in plain numpy does it on a grid, about 0.2 s for 50k cells, and always terminates. No CasADi, no IPOPT, no sampling.
Watch the fronts slow down where Φ is large and flow around the pillars.
03 · Fly downhill
a = −2(v·∇u)v + |v|²∇u − (k·d_g/Φ)·ĝ − c·v, with u = ln Φ. The first two brake and bend near surfaces, the third pulls along the geodesic, and damping is the only energy sink. A fourth, magnetic term v × B handles moving obstacles and can never do work.
The orange arrow is the command. It shrinks near pillars: the opposite of APF.
04 · Nothing to get stuck in
Obstacles contribute no equilibria at all, so there is no balance point to stall in. Started inside the trap pocket, the closed loop still reaches the goal: offline it took 22 s, where APF stays stuck forever.
The price: it is expensive, conservative, and converges softly at the end.
metric Φ · commandgeodesic distanceflown
Try it
Click or tap to move the goal; the eikonal field is re-solved on the spot. Raise β to warp harder and give the pillars a wider berth. A 2D slice of warp_core.py at the 2 m cruise height, same defaults, ideal closed loop.
Closed form, evaluated every tick. Numbers are the defaults in warp_core.py; h is the distance to the nearest surface inflated by 0.5 m, dg the geodesic distance to the goal, ĝ its gradient direction.
command
a = −2(v · ∇u) v + |v|2 ∇u − (k dg / Φ) ĝ − c v + v × B, |a| ≤ 2.5
with
what it guarantees
The repo validates a theory document, core.md. Offline, on the exact closed loop with RK4 integration:
| theorem | prediction | observed |
|---|---|---|
| T1 forward invariance | h(t) ≥ h_min > 0, no exceptions | min h 1.04 m (bound 2.2e-3 m), no collision in any scenario |
| Cor. 1.1 geometric speed limit | Φ(h)·|v| ≤ √(2E₀) | 1.74 against 5.78, never approached |
| (E) energy monotone | dE/dt = −c|v|²_g ≤ 0 | max dE per step −5e-5, strictly decreasing |
| Thm 4 bounded control | |a| ≤ ā, |a| → 0 at surfaces | max 2.24 < 2.5; mean |a| near walls smaller than in free space |
| Prop 2 no deadlock | no spurious equilibria | trap pocket start: reached in 22 s (APF: stuck forever) |
| Thm 5 dynamic safety | no finite-time collision | head-on pedestrian: 1.38 m clearance; crossing: 2.70 m |
| flight | result | clearance | theorem checks |
|---|---|---|---|
| fixed obstacles | reached, 112 s | 0.59 m, no margin penetration | all pass |
| 2D lidar pillars | reached, 124 s | 0.82 m | pass except Cor. 1.1 +6% (field-discovery bookkeeping) |
| moving pedestrian, 1.2 m/s head-on | reached, 50 s | 0.38 m to the mover | magnetic steering active in the encounter |
It took five fixed-mode flights to get there; each exposed a mechanism the ideal loop cannot show (grid escape, an unanchored z-channel, anchor positive feedback, a boundary slingshot through tracking lag, a mirage sink onto the low cylinder). All fixes live at the velocity-command interface; the control law never changed. The T1 margin is exponentially conservative: non-zero, so collision is topologically ruled out, but real clearances are about three orders of magnitude larger.
Python · numpy · ROS 2 Humble · PX4 v1.14 · Gazebo Garden · Ambient Riemannian sources ↗