World Magnetic Model — Robotics/Gravity Models

World Magnetic Model

Robotics / Gravity Models

The Earth's main magnetic field from the joint NOAA/BGS World Magnetic Model, a spherical-harmonic expansion of degree 12 whose Gauss coefficients g_n^m, h_n^m move linearly with time from the model's epoch. At a geodetic position and a decimal year t the block reports the field vector in the local north-east-down frame and the four readings derived from it:

• B = [X; Y; Z] nT, from the potential V = a·Σ_n=1..12(a/r)ⁿ⁺¹Σ_m=0..n (g_n^mcos mλ + h_n^msin mλ)·P_n^m(cosθ), evaluated in geocentric spherical coordinates and rotated into the geodetic frame.
• H = √(X² + Y²), F = √(H² + Z²), D = atan2(Y, X) and I = atan2(Z, H), the last two in degrees.

Ports

• h – height above the WGS84 ellipsoid in metres, [1,1].
• lat – geodetic latitude in degrees, north positive, [1,1]. A latitude past a pole is folded back (±180 − φ) and its longitude turned half a revolution, as the reference does.
• lon – geodetic longitude in degrees, east positive, [1,1]; wrapped to [−180, 180].
• dyear – the time as a decimal year (2025.5 is halfway through 2025), [1,1]. Outside the model's five-year window the coefficients are extrapolated rather than clamped.
• B – the field vector [X; Y; Z] in nT, a [3,1]: north, east and down.
• H – horizontal intensity in nT, [1,1].
• D – declination in degrees, east positive, [1,1].
• I – inclination in degrees, down positive, [1,1].
• F – total intensity in nT, [1,1].

Parameters

• Model – which release's coefficients run. This selects a whole coefficient set, not a value, and each is valid for the five years from its epoch:
  • WMM2000 (2000-2005), WMM2005 (2005-2010), WMM2010 (2010-2015) – the superseded releases, kept so an old recording can be reproduced.
  • WMM2015 V1 (2015-2020) – deprecated by NOAA: it degrades in the Arctic. WMM2015 V2 (2015-2020) is its 2019 correction and the one to use.
  • WMM2020 (2020-2025) and WMM2025 (2025-2030) – the current pair; WMM2025 is the default, as in Simulink.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text.

The chosen model's coefficients are inlined at export time as two tables of 169 numbers (the field and its yearly rate), together with the Schmidt factors and the recursion constants of the Legendre functions; the decimal year stays a port, so one exported core covers every date. The synthesis itself is emitted once, as loops, so all ten targets run the same statements in the same order.

The three hardware targets are simulation-only: sines, arctangents, square roots and a degree-12 recursion do not belong in a Q16.16 datapath, so the arithmetic runs in real and only the ports are fixed point. Q16.16 also bounds every port to about ±32767, which for this block means a height below about 32.7 km and a field below 32767 nT – the Earth's field exceeds that over most of the globe, so a hardware core is honest only where the field is weak (the South Atlantic, where it falls to ~22000 nT). PLC Structured Text has neither floor nor atan2 and rebuilds both from TRUNC and ATAN.

Simulink bridge

Import and export, mapped to Aerospace Blockset's aerolibgravity2/World Magnetic Model. Model → model, value for value (WMM2025 (2025-2030) → WMM2025 (2025-2030) and so on for all seven), which is 1:1 and lossless in both directions.

Seven Simulink parameters are always implied and carry no configuration here: units is always Metric (MKS) (English units – feet and nanogauss – do not cross), time_in is always on so the decimal year arrives on the fourth port rather than as a month/day/year dialog (no configuration here can add or remove a port), action is always Warning, and h_out, dec_out, inc_out and ti_out are always on so all five outputs exist. An imported block holding any other value of the seven is reported rather than silently accepted. action only decides whether Simulink reports a position or date outside the model's range; the values are the same either way, and this block computes them without a report. The Custom model, which reads a NOAA .COF file at model load, does not cross: an exported core carries its coefficients as numbers and has no file to read. The Simulink block defines no SampleTime (measured), so the rate stays on the ICore side.

Notes

• Algebraic and stateless: the output depends only on this sample's inputs.
• Not linear, so the block carries no state space and model reduction correctly reports it as unmergeable.
• At a geographic pole the east component is a limit rather than a quotient (sinθ is zero there), and the block takes the reference's own special case for it. Declination is still defined, but it changes by a whole revolution as the pole is crossed.
• Verified against R2026a: over 400 positions and dates – both poles exactly, latitudes of 95, 181 and 270, a longitude of −370, heights from −1 km to 850 km, dates inside and outside the five-year window – the Simulink block and this arithmetic agree to 1.3×10⁻¹⁵ relative.
• The model is a coefficient set, not an approximation order: WMM2020 and WMM2025 differ by about 100 nT at the same place and date, which is the field's own drift, not an error in either.