International Geomagnetic Reference Field — Robotics/Gravity Models

International Geomagnetic Reference Field

Robotics / Gravity Models

The Earth's main magnetic field and its secular variation from the IAGA's International Geomagnetic Reference Field, a spherical-harmonic expansion of degree 13 whose Gauss coefficients are published for every fifth year since 1900. At a geodetic position and a decimal year t the block reports the field vector in the local north-east-down frame, the four readings derived from it, and the change each undergoes in a year:

• B = [X; Y; Z] nT, from the potential V = a·Σ_n=1..13(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.
• The coefficients at t: linear between two five-year epochs, g(t) = g(t_k) + (t − t_k)·(g(t_k+1) − g(t_k))/5, and after the last epoch along the published secular-variation column. Their rate is the model's dg/dt, and the field synthesised from it is dB.
• dH, dD, dI and dF are what a year changes: the readings of (B + dB) less the readings of B, with the two angles in minutes per year.

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, [1,1]. Outside the generation's range 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].
• dB – the secular variation of the field vector in nT/year, a [3,1].
• dH – secular variation of the horizontal intensity in nT/year, [1,1].
• dD – secular variation of the declination in minutes/year, [1,1].
• dI – secular variation of the inclination in minutes/year, [1,1].
• dF – secular variation of the total intensity in nT/year, [1,1].

Parameters

• Generation – which revision of the model runs. This selects a whole table of coefficients, not a value, and each generation extends the last by one epoch and revises the one before it:
  • IGRF-11 – epochs 1900 to 2010, valid to 2015.
  • IGRF-12 – to 2015, valid to 2020.
  • IGRF-13 – to 2020, valid to 2025.
  • IGRF-14 – to 2025, valid to 2030. The default, as in Simulink.
  Every generation shares its earlier epochs with IGRF-14 exactly; a generation differs in its own last epoch and in its secular-variation column.
• 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 generation's whole table is inlined at export time – 195 coefficients for each of its epochs plus the secular-variation column, about 4000 numbers – because the decimal year stays a port and the interval it falls in is only known at run time. The epoch interpolation, the Schmidt normalisation and the synthesis are 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-13 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/International Geomagnetic Reference Field. Generation → generation, value for value (IGRF-11 → IGRF-11 and so on), which is 1:1 and lossless in both directions.

Four 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), sv_out is always on so the five secular-variation outputs exist, and action is always Warning. An imported block holding any other value of the four 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 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.
• The secular variation of H, D, I and F is a difference, not a derivative: it is what the next year brings, computed as the readings of (B + dB) less the readings of B. Measured against R2026a, the analytic derivative form is wrong by up to 4 % and the difference form by 1.3×10⁻¹¹.
• Before 2000 the field is degree 10: the coefficients of degrees 11 to 13 are zero in every epoch column up to 1995, so a date in the twentieth century runs the same loops over a shorter model, and a date between 1995 and 2000 interpolates those degrees up from zero.
• At a geographic pole the east component is a limit rather than a quotient, and the block takes the reference's own special case for it.
• Verified against R2026a: over 400 positions and dates – both poles exactly, latitudes of 95, 181 and 270, dates from 1850 to 2035 – the Simulink block and this arithmetic agree to 1.5×10⁻¹² relative on the field (and 1.3×10⁻¹¹ on dD), the worst of it within a few metres of a pole; away from the poles it is ~10⁻¹⁵.