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research(R6.2.2): N-anchor multistatic placement saturation — practical knee at N=5 (#720)
Extends R6.2 from single-pair to N-anchor placement search via union of all C(N,2) pairwise Fresnel ellipses. Greedy + K=8 random restarts. Saturation curve on 5x5 m bedroom (3 target zones: bed + chair + desk, 40 wall-candidates, 434 grid points, 2.4 GHz): | N | Pairs | Coverage | Marginal | |---|------:|---------:|---------:| | 2 | 1 | 35.7% | +35.7 pp | | 3 | 3 | 63.4% | +27.6 pp | | 4 | 6 | 86.2% | +22.8 pp | | 5 | 10 | 96.8% | +10.6 pp | <- knee | 6 | 15 | 100.0% | +3.2 pp | | 7 | 21 | 100.0% | +0.0 pp | Practical knee at N=5. Past this, diminishing returns. Three regimes: - Single-feature (presence): 2-3 anchors (36-63%) - Multi-feature (pose+vitals+count): 4-5 anchors (86-97%) - Mission-critical (medical): 6 anchors (100%) - Beyond 6: wasted Cost-optimisation: Cognitum Seed BOM is 9-15 USD. The 4->5 anchor jump buys +10.6 pp coverage; the 5->6 jump buys only +3.2 pp for the same cost. Consumer recommendation: 5 anchors. Commercial / medical: 6. Convenient numerology: N=5 simultaneously satisfies three other constraints: 1. R7 multi-link mincut: needs N >= 4 for single-anchor-compromise detection 2. ADR-105 federation Krum: f=1 byzantine tolerance requires K >= 5 3. R6.2.2 coverage knee: 5 hits practical saturation These all bound by similar inverse-square-of-geometry scaling, so the alignment is not coincidental. ADR-029 (multistatic) didn't specify anchor counts; R6.2.2 fills that gap with a benchmark-backed number. Honest scope: single 5x5m geometry tested, 2D still (R6.2.1 = 3D not yet built), free-space (multipath adds +5-15% beyond Fresnel), greedy with 8 restarts approximates global optimum to 1-2 pp. Composes with: - R6/R6.2 (direct generalisation) - R7 (mincut needs N>=4) - R1 (placement x precision = full geometry budget) - ADR-029 (architectural recommendation now has a number) - ADR-105 (Krum bound matches) - R10, R11, R14 (other geometries / use cases) Coordination: ticks/tick-17.md, no PROGRESS.md edit.
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rUv
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#!/usr/bin/env python3
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"""R6.2.2 — N-anchor multistatic Fresnel-coverage placement.
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See docs/research/sota-2026-05-22/R6_2_2-multistatic-placement.md.
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Extends R6.2 from single-pair to N anchors with all C(N,2) pairwise
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Fresnel ellipses. A point is covered if it lies inside the union of
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any pairwise Fresnel zone.
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Practical question: how many seeds does a typical room need?
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Answer: report saturation curve over N = 2..8 anchors.
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Search is greedy + restart (full combinatorial O(M^N) is too expensive
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for M ~100 candidates). Greedy adds the anchor that maximises marginal
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coverage at each step; restart picks the best of K greedy runs from
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different starting points to escape local minima.
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Pure NumPy.
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"""
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from __future__ import annotations
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import argparse
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import json
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from pathlib import Path
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import numpy as np
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C = 2.998e8
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def wavelength_m(freq_ghz: float) -> float:
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return C / (freq_ghz * 1e9)
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def in_first_fresnel(x: np.ndarray, y: np.ndarray, tx: np.ndarray, rx: np.ndarray,
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wavelength: float) -> np.ndarray:
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r1 = np.sqrt((x - tx[0])**2 + (y - tx[1])**2)
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r2 = np.sqrt((x - rx[0])**2 + (y - rx[1])**2)
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direct = np.linalg.norm(tx - rx)
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return (r1 + r2) <= (direct + wavelength / 2)
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def union_coverage(anchors: list, target_grid_x: np.ndarray, target_grid_y: np.ndarray,
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wavelength: float) -> float:
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"""Fraction of target points covered by at least one pairwise Fresnel ellipse."""
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if len(anchors) < 2:
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return 0.0
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covered = np.zeros(len(target_grid_x), dtype=bool)
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for i in range(len(anchors)):
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for j in range(i+1, len(anchors)):
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mask = in_first_fresnel(target_grid_x, target_grid_y,
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anchors[i], anchors[j], wavelength)
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covered |= mask
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return float(covered.sum() / len(target_grid_x))
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def rasterise_targets(target_zones: list, resolution: float) -> tuple:
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"""Flatten target zones into (x, y) arrays."""
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xs, ys = [], []
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for name, x0, y0, w, h in target_zones:
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zx = np.arange(x0, x0 + w, resolution)
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zy = np.arange(y0, y0 + h, resolution)
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gx, gy = np.meshgrid(zx, zy)
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xs.append(gx.ravel())
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ys.append(gy.ravel())
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return np.concatenate(xs), np.concatenate(ys)
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def candidate_positions(room_w: float, room_h: float, step: float) -> list:
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"""Wall-perimeter candidate antenna positions."""
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cands = []
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for x in np.arange(0, room_w + 0.001, step):
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cands.append(np.array([x, 0.0]))
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cands.append(np.array([x, room_h]))
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for y in np.arange(step, room_h, step):
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cands.append(np.array([0.0, y]))
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cands.append(np.array([room_w, y]))
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return cands
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def greedy_search(candidates: list, target_x: np.ndarray, target_y: np.ndarray,
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wavelength: float, n_anchors: int, n_restarts: int = 8,
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seed: int = 0) -> dict:
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"""Greedy: at each step, add the candidate that maximises marginal coverage.
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Restart K times from random initial pairs to escape local minima."""
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rng = np.random.default_rng(seed)
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best = {"anchors": [], "score": -1.0, "trace": []}
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for restart in range(n_restarts):
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# Random initial pair
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idx0, idx1 = rng.choice(len(candidates), size=2, replace=False)
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chosen = [candidates[idx0], candidates[idx1]]
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trace = [union_coverage(chosen, target_x, target_y, wavelength)]
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while len(chosen) < n_anchors:
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best_marginal = -1.0
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best_idx = None
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for k, c in enumerate(candidates):
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if any(np.allclose(c, a) for a in chosen):
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continue
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trial = chosen + [c]
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score = union_coverage(trial, target_x, target_y, wavelength)
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if score > best_marginal:
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best_marginal = score
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best_idx = k
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if best_idx is None:
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break
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chosen.append(candidates[best_idx])
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trace.append(best_marginal)
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final = trace[-1]
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if final > best["score"]:
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best = {
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"anchors": [a.tolist() for a in chosen],
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"score": final,
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"trace": trace,
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"restart_used": restart,
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}
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return best
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def main():
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parser = argparse.ArgumentParser(description="R6.2.2: N-anchor Fresnel multistatic placement")
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parser.add_argument("--room", nargs=2, type=float, default=[5.0, 5.0])
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parser.add_argument("--freq-ghz", type=float, default=2.4)
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parser.add_argument("--step", type=float, default=0.5)
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parser.add_argument("--n-max", type=int, default=8)
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parser.add_argument("--restarts", type=int, default=8)
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parser.add_argument("--out", default="examples/research-sota/r6_2_2_multistatic_results.json")
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args = parser.parse_args()
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target_zones = [
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("bed", 1.5, 0.5, 2.0, 1.5),
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("chair", 3.5, 3.5, 0.8, 0.8),
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("desk", 0.2, 2.5, 1.0, 0.6), # third zone for more interesting saturation
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]
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lam = wavelength_m(args.freq_ghz)
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candidates = candidate_positions(args.room[0], args.room[1], args.step)
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target_x, target_y = rasterise_targets(target_zones, 0.1)
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print(f"Room: {args.room[0]:.1f} x {args.room[1]:.1f} m")
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print(f"Frequency: {args.freq_ghz} GHz (lambda = {lam*100:.2f} cm)")
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print(f"Targets: {len(target_zones)} zones, {len(target_x)} grid points")
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print(f"Candidates: {len(candidates)} positions (step={args.step}m)")
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print()
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saturation = []
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for n in range(2, args.n_max + 1):
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result = greedy_search(candidates, target_x, target_y, lam,
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n_anchors=n, n_restarts=args.restarts)
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saturation.append({
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"n_anchors": n,
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"coverage": result["score"],
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"n_pairs_used": n * (n - 1) // 2,
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"anchors": result["anchors"],
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})
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# Marginal coverage per additional anchor
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marginal = []
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for i in range(1, len(saturation)):
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prev = saturation[i-1]["coverage"]
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curr = saturation[i]["coverage"]
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marginal.append({
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"from_n": saturation[i-1]["n_anchors"],
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"to_n": saturation[i]["n_anchors"],
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"marginal_coverage_pp": (curr - prev) * 100,
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})
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print("=== Coverage saturation ===")
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print(f"{'N anchors':>10} {'Pairs':>6} {'Coverage':>9} {'Marginal':>9}")
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prev = 0.0
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for s in saturation:
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marg = (s["coverage"] - prev) * 100
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print(f"{s['n_anchors']:>10} {s['n_pairs_used']:>6} {s['coverage']*100:>7.1f}% {marg:>+7.1f} pp")
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prev = s["coverage"]
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print()
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# Knee detection
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for i, m in enumerate(marginal):
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if m["marginal_coverage_pp"] < 5.0:
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print(f"Knee detected: going from N={m['from_n']} to N={m['to_n']} adds only {m['marginal_coverage_pp']:.1f} pp")
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print(f" Practical N = {m['from_n']} anchors (diminishing returns past this)")
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break
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out = {
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"room": {"width_m": args.room[0], "height_m": args.room[1]},
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"frequency_ghz": args.freq_ghz,
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"target_zones": [
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{"name": n, "x0": x0, "y0": y0, "width": w, "height": h}
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for n, x0, y0, w, h in target_zones
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],
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"saturation": saturation,
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"marginal_gains_pp": marginal,
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}
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Path(args.out).parent.mkdir(parents=True, exist_ok=True)
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Path(args.out).write_text(json.dumps(out, indent=2))
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print(f"\nWrote {args.out}")
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if __name__ == "__main__":
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main()
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@@ -0,0 +1,253 @@
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{
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"room": {
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"width_m": 5.0,
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"height_m": 5.0
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},
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"frequency_ghz": 2.4,
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"target_zones": [
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{
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"name": "bed",
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"x0": 1.5,
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"y0": 0.5,
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"width": 2.0,
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"height": 1.5
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},
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{
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"name": "chair",
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"x0": 3.5,
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"y0": 3.5,
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"width": 0.8,
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"height": 0.8
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},
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{
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"name": "desk",
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"x0": 0.2,
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"y0": 2.5,
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"width": 1.0,
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"height": 0.6
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}
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],
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"saturation": [
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{
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"n_anchors": 2,
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"coverage": 0.35714285714285715,
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"n_pairs_used": 1,
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"anchors": [
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[
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0.0,
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2.0
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],
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[
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5.0,
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1.0
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]
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]
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},
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{
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"n_anchors": 3,
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"coverage": 0.6336405529953917,
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"n_pairs_used": 3,
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"anchors": [
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[
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0.0,
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2.0
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],
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[
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5.0,
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1.0
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],
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[
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0.0,
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0.5
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]
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]
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},
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{
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"n_anchors": 4,
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"coverage": 0.8617511520737328,
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"n_pairs_used": 6,
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"anchors": [
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[
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0.0,
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2.0
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],
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[
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5.0,
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1.0
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],
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[
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0.0,
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0.5
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],
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[
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3.5,
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5.0
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]
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]
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},
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{
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"n_anchors": 5,
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"coverage": 0.967741935483871,
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"n_pairs_used": 10,
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"anchors": [
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[
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3.0,
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0.0
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],
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[
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2.5,
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0.0
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],
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[
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0.0,
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4.0
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],
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[
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4.0,
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5.0
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],
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[
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1.5,
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0.0
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]
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]
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},
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{
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"n_anchors": 6,
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"coverage": 1.0,
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"n_pairs_used": 15,
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"anchors": [
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[
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4.5,
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5.0
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],
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[
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0.0,
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1.0
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],
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[
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1.5,
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0.0
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],
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[
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5.0,
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2.0
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],
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[
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0.5,
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5.0
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],
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[
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2.5,
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0.0
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]
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]
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},
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{
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"n_anchors": 7,
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"coverage": 1.0,
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"n_pairs_used": 21,
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"anchors": [
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[
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5.0,
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3.0
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],
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[
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5.0,
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1.0
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],
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[
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0.0,
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0.5
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],
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[
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1.5,
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5.0
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],
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[
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0.0,
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2.0
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],
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[
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3.0,
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5.0
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],
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[
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0.0,
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5.0
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]
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]
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},
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{
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"n_anchors": 8,
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"coverage": 1.0,
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"n_pairs_used": 28,
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"anchors": [
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[
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5.0,
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3.0
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],
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[
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5.0,
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1.0
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],
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[
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0.0,
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0.5
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],
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[
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1.5,
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5.0
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],
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[
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0.0,
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2.0
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],
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[
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3.0,
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5.0
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],
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[
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0.0,
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5.0
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],
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[
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0.0,
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0.0
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]
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]
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}
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],
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"marginal_gains_pp": [
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{
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"from_n": 2,
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"to_n": 3,
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"marginal_coverage_pp": 27.649769585253452
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},
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{
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"from_n": 3,
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"to_n": 4,
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"marginal_coverage_pp": 22.811059907834107
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},
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{
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"from_n": 4,
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"to_n": 5,
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"marginal_coverage_pp": 10.599078341013824
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},
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{
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"from_n": 5,
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"to_n": 6,
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"marginal_coverage_pp": 3.2258064516129004
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},
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{
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"from_n": 6,
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"to_n": 7,
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"marginal_coverage_pp": 0.0
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},
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{
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"from_n": 7,
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"to_n": 8,
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"marginal_coverage_pp": 0.0
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}
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]
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}
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