spaxel/docs/research/05-mesh-topology.md

Multi-Node Mesh Topology

Link Count

A network of N nodes creates N×(N−1) directed links (each ordered TX→RX pair). Every bidirectional link is two directed links.

Nodes (N)	Directed links	Unique bidirectional links
3	6	3
4	12	6
5	20	10
6	30	15
8	56	28

Each link is a bistatic radar with the two nodes as TX and RX. A person's position creates an ellipsoidal constraint for each link. More independent links → tighter constraint intersection → better localization.

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Why Link Diversity Matters

Each link's first Fresnel zone ellipsoid has a different orientation and coverage area. A person at position P may be:

• Inside the FFZ of links whose paths cross through P
• Outside the FFZ of tangential links

Using links with many different orientations means some will always be sensitive to any given person location. This reduces blind spots — positions where no single link would show perturbation.

Localization accuracy improvement scales roughly as √N_links for independent measurements (analogous to averaging independent noise sources).

Coverage per Link

One link's effective sensing volume is approximately its first Fresnel zone: for a 5 m link at 2.4 GHz, this is an ellipsoid with semi-major axis 2.5 m and semi-minor axis ~0.4 m at the midpoint.

Rule of thumb: one node per 50–70 m² for room-scale presence detection. For sub-metre localization: 4–8 nodes per sensing region.

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Optimal Node Placement

Principles

• Non-collinear: All nodes on one line → degenerate geometry; poor localization of targets along the line
• Distributed perimeter: Nodes at room corners/walls create links crossing the interior from many angles
• Angular diversity: Prefer a uniform distribution of link directions (0°, 45°, 90°, 135°…)
• Avoid clustering: Two nodes close together provide nearly identical angular information

Height Variation

Nodes at different heights are essential for meaningful Z-axis detection:

• Most indoor links are horizontal (nodes mounted at similar heights) → poor elevation angle diversity
• Deploying some nodes high (2.0–2.5 m) and some low (0.3–1.0 m) creates vertical Fresnel zone components
• A person standing vs. crouching will show differently across high/low node pairs
• Z-accuracy with mixed heights: ~0.5–1.5 m (significantly worse than XY at ~0.5 m)

Minimum Counts by Task

Task	Minimum nodes	Recommended
Presence detection (binary)	1 link (2 nodes)	2–3 links
Coarse 2D localization	3 nodes	4–6 nodes
Sub-metre 2D localization	4–6 nodes	6–8 nodes
3D localization	6+ nodes	8+ nodes
Multiple people tracking	4 per person	6–12 nodes per person

Geometric Dilution of Precision (GDOP)

Borrowed from GPS — quantifies how geometry affects localization accuracy for a given target location:

• GDOP < 2: good geometry, accurate localization
• GDOP > 5: poor geometry, avoid this target location if possible
• GDOP varies across the room; worst near corners (far from all nodes) and along node-to-node lines

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TX/RX Role Assignment

Multistatic Mesh Modes

Mode	Description
Pure TX	Node transmits probe packets; does not capture CSI
Pure RX	Node captures CSI from TX nodes; does not transmit probes
Both	Alternates TX and RX; used when node count is low

The mothership assigns roles dynamically via MQTT config push:

{
  "role": "rx",
  "listen_macs": ["aa:bb:cc:dd:ee:ff", "11:22:33:44:55:66"]
}

Probe Packet Strategy

TX nodes send probe requests or null data packets on a schedule. Rate: 20–50 Hz is sufficient for motion detection; higher rates improve DFS resolution but increase channel contention.

Coordination: the mothership can stagger TX schedules across nodes to prevent simultaneous transmissions (which would corrupt CSI at the RX nodes).

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Node Self-Positioning via MDS

If node positions are not known a priori, Multidimensional Scaling (MDS) can recover relative coordinates from pairwise distance estimates.

MDS-MAP Algorithm

1. Measure pairwise ToF (or RSSI-based ranging) between all node pairs → distance matrix D_ij
2. Construct double-centred Gram matrix: B = −½ · H · D² · H, where H = I − (1/N)·11^T
3. Eigendecompose B; take top 2 (or 3) eigenvectors × √eigenvalues as relative coordinates
4. Anchor to ≥3 reference points with known positions to get absolute coordinates

Ranging Accuracy on ESP32

Method	Range resolution
RSSI path loss model	~1–3 m (very noisy)
ToF from CIR peak (20 MHz BW)	c/(2B) = 7.5 m — too coarse
ToF from CIR peak (40 MHz BW)	c/(2B) = 3.75 m — still coarse
ToF + DFS refinement (Widar2.0)	~0.05 m — requires post-processing

Practical recommendation: Measure node positions manually with a tape measure for initial deployment. MDS self-positioning is a future enhancement once the system is validated. Use the mothership floor plan editor to pin nodes to known coordinates.

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Apartment Perimeter Deployment

For a typical apartment deployment with nodes at perimeter walls and mixed heights:

Top-down view (8 nodes):         Side view (one wall):

  N1 ──── N2 ──── N3              2.0m  N1
  │                │
  │    interior    │              1.0m
  │                │
  N8              N4              0.3m        N3
  │                │
  N7 ──── N6 ──── N5

• 4 corner nodes at 2.0 m height → good horizontal coverage of interior
• 4 mid-wall nodes at 0.3–1.0 m → elevation diversity, improves Z detection
• 56 directed links from 8 nodes → excellent interior coverage from all angles
• Blind spots: corners very close to nodes (over-perturbation, hard to localise precisely)
• Coverage: reliable detection throughout the interior; sub-metre 2D localisation feasible