41-EFT.WP.Comms.Navigation v1.0 | Chapter 5: Geometry & Observability — Deployment and GDOP

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Chapter 5: Geometry & Observability — Deployment and GDOP

I. Objectives & Applicability

• Provide the geometry & observability framework for positioning/navigation systems: geometry matrix, Fisher information, definitions/properties of GDOP/PDOP/HDOP/VDOP, 2D/3D observability and degeneracies, GDOP heatmap generation over a coverage region, and deployment optimization strategies.
• All formulas/symbols/definitions are in English and wrapped in backticks. If observations include arrival-time terms, use the previously selected T_arr convention and explicitly record gamma(ell) and d ell.

II. Geometry & Information Baseline

• S50-1 (Geometry matrix & Jacobian)
  For observation vector y, let H = ∂h/∂x be the Jacobian w.r.t. the state x = [p_WB, v_WB, q_WB, b_t, …]. Extract columns directly tied to geometry (e.g., ∂/∂p_WB, ∂/∂b_t) as the geometry matrix G = [∂h/∂p_WB, ∂h/∂b_t].
• S50-2 (Fisher information & posterior covariance)
  With measurement covariance Σ_y, F = H^T Σ_y^{-1} H. The position–clock block’s posterior approximation is Q = (G^T Σ_y^{-1} G)^{-1}. Use cond(F) to quantify ill-conditioning.
• S50-3 (Normalization & weighting)
  To compare heterogeneous channels, whiten by a reference std σ_0: W = Σ_y^{-1/2}, G_w = W G, Q_w = (G_w^T G_w)^{-1}. The DOP indices below are defined from Q_w.

III. DOP Definitions & Properties

• S50-4 (DOP definitions)
  GDOP = √( Q_w[xx]+Q_w[yy]+Q_w[zz]+Q_w[tt] )
  PDOP = √( Q_w[xx]+Q_w[yy]+Q_w[zz] )
  HDOP = √( Q_w[xx]+Q_w[yy] ), VDOP = √( Q_w[zz] )
  where t denotes the clock/time-bias dimension (if modeled).
• S50-5 (Invariance & monotonicity)
  DOP is invariant to uniform scaling of all observation noises; adding independent anchors or complementary measurements (e.g., FOA/AOA) makes GDOP monotonically non-increasing.
• S50-6 (Units & dimensions)
  DOP is dimensionless. Mapping to position error uses the reference std σ_0: PEB ≈ σ_0 · PDOP.

IV. 2D/3D Observability & Degeneracies

• S50-7 (Rank conditions)
  With known clock in 2D, at least 2 non-colinear anchors (incl. AOA cases). In 2D unknown-clock or 3D known-clock, at least 3; in 3D unknown-clock, at least 4 and non-coplanar, ensuring rank(G) ≥ dim([p_WB, b_t]).
• S50-8 (Vertical degeneracy & mitigation)
  When anchors are nearly coplanar or elevation angles cluster around the horizontal plane, VDOP inflates. Introduce FOA, high/low anchors, or oblique AOA to raise vertical information.
• S50-9 (Reference & sync)
  TDOA weakly depends on a global clock but is more geometry sensitive; for TOA, the clock column contributes significantly to GDOP. Differential sync or common-view reduces Q_w[tt].

V. Coverage GDOP Heatmaps & Optimization

1. S50-10 (Region discretization & occlusion)
   On a grid 𝒢 = {r_j}, evaluate G(r_j). For occluded or NLOS_flag = true sites, drop the affected rows or inflate their noise.
2. S50-11 (GDOP heatmap)
   Compute PDOP/GDOP at each r_j; output heatmaps/contours and summarize max/mean and quantiles (p50/p95) as coverage metrics.
3. S50-12 (Deployment objectives)
   • MinMax: min_S max_{r_j∈𝒢} PDOP(r_j; S)
   • D-optimal: min_S ∑_{r_j} log det( Q_w(r_j; S) )
   • E-optimal: min_S max_{r_j} λ_max( Q_w(r_j; S) )
     where S is the anchor set.
4. M5-1 (GDOP-map workflow)
   Input anchors, y_channels, Σ_y, 𝒢 → compute G, Q_w, DOP, then output heatmaps and stats.
5. M5-2 (Deployment-optimization workflow)
   Use greedy/swap heuristics or MIP to solve the above objectives; support soft constraints (placement zones/height/power/backhaul) and hard constraints (no-go areas/max count).

VI. Online Reconfiguration & Adaptive Planning

• S50-13 (Runtime reweighting & switching)
  Adjust observation weights or switch channels using live Σ_y and NLOS_flag; trigger reconfiguration by a cond(F) threshold.
• S50-14 (Fleet/roadside cooperation)
  Sharing temporary anchors (V2V/UAV-UAV) reduces PDOP in hotspots; dataset cards must record temporary-anchor lifetime and sync accuracy.
• M5-3 (Online planning)
  In a receding horizon, maximize tr(F) or minimize GDOP via actions/scheduling (moving anchors/beams/altitude/power).

VII. Channel Contributions to Information (Link to Chapter 4)

• S50-15 (TOA/TDOA)
  ∂/∂p_WB = (1/c_ref) n̂^T or (1/c_ref)(n̂_i^T − n̂_0^T), governing radial geometry; reference selection affects column correlation in G.
• S50-16 (AOA)
  Supplies tangential constraints, effectively shrinking HDOP; array geometry (ULA/URA) and calibration errors enter Σ_y.
• S50-17 (FOA)
  Constrains velocity along n̂, boosting vertical observability; fused with TOA, often reduces GDOP markedly.
• S50-18 (RSS/CP)
  RSS is coarse but useful for aiding; CarrierPhase strongly constrains PDOP after integer fixing, while floating ambiguities require uncertainty inflation.

VIII. Data Contract (Required/Recommended Fields for This Chapter)

unit_system: "SI"

geometry:

anchors: [{id: "...", p_S: [x,y,z], cov: "<3x3>"}]

forbidden_zones: ["<polygon|volume id>"]

height_constraints: {min: "<m>", max: "<m>"}

channels:

y_channels: ["TOA","TDOA","AOA","FOA","RSS","CP"]

Σ_y: "<block-diagonal or sparse>"

coverage:

grid: {xmin:..., xmax:..., ymin:..., ymax:..., z: "<optional>", step: "<m>"}

metrics: ["PDOP","GDOP","HDOP","VDOP"]

optimization:

objective: "MinMax|D-opt|E-opt"

budget: {max_anchors: k, wiring: "<m>", power: "<W>"}

runtime:

reconf_trigger: {condF_max: "<threshold>", NLOS_rate: "<threshold>"}

references:

- "EFT.WP.Comms.Navigation v1.0:Ch.4 S40-*"

- "EFT.WP.Comms.Navigation v1.0:Ch.10"

- "EFT.WP.Core.Metrology v1.0:Ch.1–3,5"

IX. Implementation Bindings (Interface Prototypes)

• I5-1 compute_dop_maps(anchors, y_channels, Σ_y, grid) -> {maps, stats}
• I5-2 select_anchors(anchors, grid, objective, budget) -> {selected, score}
• I5-3 plan_anchor_layout(area, constraints) -> {S_positions, GDOP_map}
• I5-4 reconfigure_runtime(state, flags, thresholds) -> {actions, expected_DOP}

X. Quality Gates (This Chapter)

• Q1 Rank & conditioning: ensure rank(G) = dim([p_WB, b_t]). If cond(F) > κ_th, flag geometric ill-conditioning and return deployment advice.
• Q2 Units & dimensions: DOP is dimensionless; all I/O must pass check_dim; angles in rad, distances in m.
• Q3 Coverage statistics: heatmaps must report max/mean/p50/p95; identify worst points and mitigation actions.
• Q4 Convention consistency: if observations use T_arr, the dataset must include convention/delta_form/gamma(ell)/d_ell.
• Q5 Covariance consistency: occlusion/NLOS reweighting must update the corresponding blocks in Σ_y.

XI. Cross-Volume References & Anchors (This Chapter)

1. Cross-volume (fixed style): this volume Ch. 2 (terminology/metrology baseline), Ch. 3 (path & frequency), Ch. 4 (observation models); EFT.WP.Core.Metrology v1.0 Ch.1–3,5 (units/uncertainty).
2. Anchors:
   • Minimal statements: S50-1—S50-18
   • Workflows: M5-1—M5-3
   • Interfaces: I5-1—I5-4

XII. Summary
This chapter quantifies geometry’s decisive role using the G/F/DOP framework, provides coverage assessment and deployment-optimization methods, and outlines runtime adaptive reconfiguration. Its outputs directly drive Chapter 10’s experiment design & measurement-matrix optimization and Chapter 13’s multi-scenario case studies.