GNSS and NMEA

Use a 3.3 V USB–serial adapter (CP2102, CH340, FT232) or the module’s own USB port. Module TX to adapter RX, module RX to adapter TX (only needed to send commands), ground to ground, and power as the breakout board says — most have a regulator and take 3.3–5 V. Put the antenna where it can see the sky; the first fix from cold takes about 30 s outdoors.

A sentence is $ + a five-letter address + comma-separated fields + + a checksum + CR LF. The first two letters are the talker: GP GPS, GL GLONASS, GA Galileo, GB (or BD) BeiDou, GQ QZSS, GI NavIC, and GN for a solution that combines several. Proprietary sentences start with P and a maker code (PUBX, PMTK, PAIR, PCAS). The checksum is the exclusive-OR of every character between $ and , as two hex digits. Positions are ddmm.mmmm — degrees and decimal minutes, not decimal degrees, a common source of 30 km mistakes.

The parsers cover GGA, RMC, GSA, GSV, VTG, GLL, ZDA, GNS, GST and TXT, including the NMEA 4.10 additions: the system ID at the end of GSA (so a GNGSA per constellation can be told apart), the signal ID at the end of GSV (L1 C/A, L5, E1, B1I…), and the navigational status in RMC and GNS. Without a system ID, satellites are assigned from their numbers (GLONASS 65–96, SBAS 33–64 and 120–158, and the 3xx/4xx u-blox extended numbering for Galileo and BeiDou). The receiver model groups sentences into epochs by their time field, collects a GSV cycle before replacing a constellation’s satellites, and takes the satellites used in the fix from the GSA sentences of the same epoch.

The tests check each parser against the example sentences printed in the common references — Dale DePriest’s NMEA data pages, the NovAtel OEM7 and Trimble NMEA references, the u-blox receiver description and the MediaTek PMTK manual — and every command builder against published strings such as $PMTK220,100*2F (10 Hz), $PMTK251,115200*1F and u-blox’s $PUBX,41,1,0007,0003,19200,0*25.

u-blox’s binary protocol frames a message as B5 62, a class, an ID, a little-endian 16-bit length, the payload, and two checksum bytes: for each byte from the class to the end of the payload, CK_A += byte and CK_B += CK_A (an 8-bit Fletcher checksum). CFG-RATE for 10 Hz is B5 62 06 08 06 00 64 00 01 00 01 00 7A 12. NAV-PVT (class 01, ID 07, 92 bytes) carries the time, fix type and flags (including the RTK carrier solution), latitude and longitude in 10⁻⁷ degrees, heights in millimetres, accuracy estimates, velocity and PDOP. NAV-SAT lists each satellite’s constellation, number, C/N₀, elevation, azimuth and whether it is used. The M9 and M10 replaced most CFG messages with key/value pairs: CFG-VALSET writes 32-bit keys whose size is encoded in bits 28–30 of the key (0x30210001 CFG-RATE-MEAS is two bytes, 0x40520001 CFG-UART1-BAUDRATE four) to RAM, battery-backed RAM and/or flash. Key IDs and message layouts are from the u-blox M8, M9 and M10 interface descriptions.

Each satellite in use gives a row of the geometry matrix G: the unit vector from the satellite towards you in east/north/up, −(cos el sin az, cos el cos az, sin el), and a 1 for the receiver clock. Q = (G^TG)⁻¹, and the DOPs are square roots of sums of its diagonal: HDOP = √(q_EE + q_NN), VDOP = √q_UU, PDOP = √(q_EE + q_NN + q_UU), TDOP = √q_tt, GDOP all four. The tests check it against the case that can be done by hand: one satellite overhead and three on the horizon 120° apart give HDOP = VDOP = √(4/3) ≈ 1.155, PDOP = √(8/3) ≈ 1.633 and GDOP = √3. The east/north block of Q is drawn as the DOP ellipse. Receivers usually weight satellites by elevation or C/N₀, so their figures differ a little from the unweighted values here.

A pseudorange is ρ_i = |s_i − r| + c·Δ_t_: the true range plus the receiver clock error expressed as a distance, the same for every satellite. With four or more satellites the solver linearises about a guess — row i of G is the unit vector from satellite to guess plus a 1 — solves G^TG Δ = G^T(ρ − ρ̂), steps, and repeats (Gauss–Newton). From the Earth’s centre it converges in four to six steps, and the tests show it recovering a known position to under a millimetre and a 1.2 ms clock error exactly. The demonstration satellites are circular orbits at the real constellations' radii and inclinations (GPS 26 560 km, 55°; GLONASS 25 508 km, 64.8°; Galileo 29 600 km, 56°; BeiDou MEO 27 906 km, 55°) rotated into the Earth-fixed frame — realistic geometry, not real ephemerides. WGS-84 conversions use Bowring’s iteration.

The scatter is taken about the samples' mean in local metres. CEP50 and CEP95 are the median and 95th percentile of the radial errors (with linear interpolation between ranks); DRMS = √(σ_E² + σ_N²) and 2DRMS twice that (95–98 % of fixes, depending on the ellipse’s shape); 0.59(σ_E + σ_N) is the textbook approximation to CEP50. For a circular normal distribution CEP50 = 1.1774 σ and CEP95 = 2.4477 σ per axis, which the tests check on 20 000 simulated fixes.