Fused MEO and LEO positioning, navigation and timing

Three scenario kinds provide positioning, navigation and timing (PNT) with satellites in low Earth orbit (LEO) alone or fused with the medium-Earth-orbit (MEO) global navigation satellite systems (GNSS): leo-pvt (position, velocity and time), leo-ppp (precise point positioning) and ntn-positioning (a 5G non-terrestrial network). Source: src/leo_fusion/.

The engine is system-agnostic. Every constellation is Walker shells, explicit orbital elements or a GNSS preset of the constellation-design kind; every signal is a carrier, a chip rate and a carrier-to-noise density (C/N0) envelope; every error budget is an explicit one-sigma. Named systems are optional presets, one file each, and every capability, test and bundled scenario runs without any particular one of them.

Systems and presets#

A scenario lists its systems as [[system]] tables. This complete scenario joins the GPS baseline constellation with a generic LEO layer and runs as written (kshana <file>.toml; engine 0.29.0 gives a median position dilution of precision of 2.43 for GPS alone and 1.17 fused, and a root-mean-square 3D error of 2.06 m and 0.47 m):

kind = "leo-pvt"
mode = "joint"
seed = 3
duration_s = 600.0
step_s = 60.0
[user]
lat_deg = 40.42
lon_deg = -3.70
height_m = 650.0
[[system]]
name = "GPS"
preset = "gps-baseline"
cn0_dbhz = [38.0, 48.0]
sisre_m = 0.6

[[system]]
name = "Any LEO layer"
carrier_hz = 1.5e9          # any band
chip_rate_hz = 10.23e6      # omit with doppler_only = true for a signal of opportunity
cn0_dbhz = [45.0, 55.0]     # at the elevation mask and at the zenith
sisre_m = 0.3               # signal-in-space range error of orbit and clock, one sigma
clock = "estimated"         # its own receiver clock (inter-system bias solved) or "known"
isb_ns = 40.0               # true inter-system bias used to simulate its pseudoranges
[[system.shell]]
total = 240
planes = 12
phasing = 1
altitude_km = 1000.0
inclination_deg = 60.0

preset = "gps-baseline" (the Global Positioning System, GPS; or galileo, beidou-meo, glonass) takes a published GNSS constellation; leo_preset = "..." fills whatever the table leaves out from a LEO preset. The C/N0 runs linearly in the sine of the elevation from the mask to the zenith (a modelling choice: a real pass envelope depends on the transmit antenna pattern). The pseudorange one-sigma is the engine's delay-lock-loop (DLL) thermal noise (navsignal::dll_code_jitter_chips) at that C/N0 combined with the signal-in-space range error (SISRE), unless sigma_pr_m fixes it; the range-rate one-sigma is the wavelength times sigma_doppler_hz. Because the sigma is an input, a separate signal model can feed it without the positioning code knowing where it came from.

Every workshop-derived number is in the fusion module of src/celeste_iod.rs, used by scenarios/celeste-iod-fused-pvt.toml (tests/workshop_preset_isolation.rs enforces where they may appear). How the preset is withheld: every workshop-derived number of the engine lives in one file, src/celeste_iod.rs, which build.rs compiles in only when it exists, and in the repository-only scenarios scenarios/*celeste-iod*.toml. Deleting that file and those scenarios withholds the preset with no source edit; every other test, oracle and scenario runs without it, and the README's scenario-file count still counts the withheld files.

leo-pvt#

One kind, four modes (mode = "doppler" | "joint" | "polar" | "timing").

Doppler. A receiver measuring Doppler f_D at wavelength λ observes the range rate ρ̇ = −λ f_D = u·(v_s − v_u) + ḋ, with u the line of sight and ḋ the receiver clock drift. A batch Gauss-Newton fit solves the position, the drift and optionally a constant velocity, with analytic partials ∂ρ̇/∂r = −(I − u uᵀ)(v_s − v_u)/ρ: only the satellite velocity across the line of sight carries position information, so the geometry is how the lines of sight turn during the window. The report gives the error against window length, the Doppler, Doppler-rate and jerk envelope of each system, and a single-pass table. One pass locates the user along the track (the time of zero Doppler) and across it (the steepness of the Doppler curve); the cross-track error grows without bound as the pass goes overhead, and the mirror image across the ground track fits almost as well (only the Earth's rotation separates them). A Doppler-only system (doppler_only = true, the starlink-sop preset) runs in this mode.

Joint. Each epoch, weighted least squares over the pseudoranges of the GNSS systems alone, the LEO systems alone and all of them, one receiver clock per system (the inter-system bias, ISB, is estimated) or a known broadcast offset on the reference time scale. It reports the geometric, position, horizontal, vertical and time dilution of precision (GDOP, PDOP, HDOP, VDOP, TDOP), the errors, the estimated biases, and the DOP against the number of LEO satellites added to the GNSS geometry.

Polar. Satellites in view, median DOP and availability against latitude for MEO GNSS, LEO and both.

Timing. At a known position every LEO pseudorange is a clock measurement. The receiver runs a two-state (phase, frequency) Kalman filter whose process noise is the oscillator class's white and random-walk frequency noise (the slot-timing and clock models), holds time between passes, and converts system time to Coordinated Universal Time (UTC) with the IS-GPS-200 expression Δt_UTC = Δt_LS + A0 + A1 (t_E − t_ot + 604800 (WN − WN_t)); the uncertainty of the broadcast offset is a per-run bias.

leo-ppp#

A float precise point positioning (PPP) extended Kalman filter (EKF) on ionosphere-free code and phase: static coordinates, a white receiver clock, one inter-system bias per extra system, a random-walk zenith wet delay, and a float ambiguity per satellite arc. The precise orbit-and-clock error is common to a satellite's code and phase, so each satellite is one correlated two-row update, in a Joseph-stabilised form that keeps the covariance positive definite over thousands of millimetre-level phase updates. Measurements are simulated from the same model, so the normalised estimation error squared (NEES) of the position must follow a chi-square law with three degrees of freedom; the test checks it over Monte Carlo seeds. Convergence is the first epoch after which the horizontal and vertical errors stay below the thresholds (10 cm) until the end of the run.

A LEO satellite crosses the sky in about ten minutes, a MEO satellite moves a few degrees, so LEO arcs separate the ambiguities from the position far sooner.

ntn-positioning#

A 5G non-terrestrial network (NTN) downlink in the mobile-satellite service (MSS) S band, 3rd Generation Partnership Project (3GPP) band n256 (2170 to 2200 MHz). The Cramér-Rao bound (CRB) on time of arrival is c / (2π β sqrt(2 (C/N0) T)), with β the root-mean-square (Gabor) bandwidth (B/√12 for a flat OFDM spectrum), and on the frequency of a tone sqrt(3 / (2π² (C/N0) T³)). The kind compares channels (by default a 5 MHz New Radio channel and a 200 kHz narrowband channel) through downlink time-of-arrival fixes with an unknown receiver clock and a single-satellite Doppler fix over a pass.

Bundled scenarios#

The kinds also run inside other scenarios: leo-pvt is a member of the leo-focus-fused-pnt-sisre sweep and of the leo-vertical-autonomous-vehicle, leo-vertical-polar-arctic, leo-vertical-critical-infrastructure-timing and leo-vertical-5g-network-timing campaigns; leo-ppp of the leo-focus-ppp-altitude sweep; ntn-positioning of the leo-focus-ntn-bandwidth sweep; and leo-pvt and leo-ppp are the fused-fix and PPP stages of the leo-pnt-chain kind. Their results are on LEO-PNT.md.

Published figures and labels#

Everything else is MODELLED with its rationale in the verification matrix (docs/VERIFICATION-MATRIX.md): the DOP is checked against a hand-derived four-satellite case, the fixes against their covariance, the Doppler model against the range-acceleration identity, the PPP filter against the chi-square NEES band, and the bounds against quadrature.

Limits, stated plainly#

• Two-body orbits with the secular drift from J2, the Earth's second zonal harmonic; no broadcast-ephemeris fit error beyond the SISRE one-sigma, no satellite antenna pattern.
• No ionospheric or tropospheric delay in leo-pvt (the PPP filter carries a wet delay state; its ionosphere-free combination removes the first-order ionosphere), no scintillation, multipath, cycle slips or integer ambiguity resolution.
• The NTN accuracy is a bound, not an achieved result.
• Preset values marked representative are illustrative and describe no operator's design.