Article Figures & Data
Figures
- FIGURE 1
Variation of Galileo orbit prediction errors in radial (left) and along-track direction (right) using high-fidelity models in the orbit determination and prediction. The graphs show the distribution (median and 5th/95th percentile) of the constellation-wide RMS error at the 𝑛-th day of prediction over 180 solutions for the six-month test period
- FIGURE 2
Numerical integration error of Galileo orbits using the fifth-order Dormand-Prince method at 200 s step size over a 14-day arc without shadow boundary handling during the eclipse period of orbital plane C
- FIGURE 3
Clock prediction of a Galileo satellite with Rb clock (top) and satellites with H-maser (bottom) based on second- and first-order clock polynomials adjusted over two days. Individual satellites are distinguished by different colors
- FIGURE 4
Clock drift variation of Galileo satellites over the six-month test period
- FIGURE 5
Variation of Galileo clock offset prediction errors. The graph shows the distribution (median and 5th/95th percentile) of the constellation-wide RMS error at the 𝑛-th day of prediction over 180 solutions for the six-month test period excluding individual clock offset errors larger than 100 m
- FIGURE 6
Variation of Galileo orbit prediction errors in radial (left) and along-track direction (right) using the tailored models of Tables 1 and 2 in the orbit determination and prediction. The graph shows the distribution (median and 5th/95th percentile) of the constellation-wide RMS error at the 𝑛-th day of prediction over 180 solutions for the six-month test period
- FIGURE 7
Evolution of Galileo global average RMS SISRE using the tailored models of Tables 1 and 2 in the orbit determination and prediction
- FIGURE 8
Evolution of SIS-related single-point positioning errors for different forecast periods. The graph shows the distribution (median and 5th/95th percentile) of the 3D RMS position error at the 𝑛-th day of prediction over 180 solutions for the six-month test period
Tables
Data, models, algorithms Precise Tailored Observations Dual-frequency code and phase observations from 80 globally distributed monitoring stations of the International GNSS Service (IGS; Johnston et al., 2017) Cartesian satellite positions Data arc 3 d 2 d Sampling 5 min 5 min Reference frame transformations IAU2006/2000 precession/nutation (Petit & Luzum, 2010); USNO Δ UT1, estimated LOD and polar motion, linear prediction IAU 1976 precession, IAU 1980 nutation (106 terms) and sidereal time (Seidelmann, 2006); IGS Earth orientation parameters (linear extrapolation) Earth gravity EIGEN-05C (12 × 12; Förste et al., 2008), solid Earth and pole tides (IERS Conventions 2010, Petit & Luzum, 2010), ocean tides (FES2004; Lyard et al., 2006), post-Newtonian relativity approximation (IERS Conventions 2010, Petit & Luzum, 2010) GGM01S model (9 × 9), k2 tides (Rizos & Stolz, 1985), no relativity Third-body perturbations Sun, Moon, all planets, Pluto; DE405 (Standish, 1998) Sun and Moon; analytical series truncated to 5″ and 2″ (Montenbruck & Pfleger, 2000) Radiation pressure 5-param. ECOM-1 model (estimated; Beutler et al., 1994) with a priori box-wing-model (Steigenberger & Montenbruck, 2017); conical Earth and Moon shadow; Earth radiation pressure (Springer, 2009); antenna thrust (Steigenberger et al., 2018) 3-param. ECOM-1 model (estimated; Beutler et al., 1994) with a priori 2-param. cuboid model (Montenbruck et al., 2015); conical Earth and Moon shadow Numerical integration 8th-order Adams-Bashforth-Moulton multi-step predictor-corrector method with 8th-order Runge-Kutta starting step (Springer, 2009); shadow boundary handling 5th-order Dormand-Prince Runge-Kutta method (Dormand & Prince, 1980) with 4th-order interpolant (Hairer et al., 1987); optional shadow boundary handling - TABLE 2
Performance comparison of model options. Peak errors in 14-day prediction arcs following a two-day orbit determination have been evaluated for eight test dates spread over a three-month period. Median and maximum errors over the tests are provided for various forms of simplifications relative to a comprehensive reference along with the approximate reduction in computational load. Models marked by a × symbol are recommended for use in the long-term propagator
Model Recommended Option Peak error median/max [m] Runtime reduction [%] Nutation IAU980 (106 terms) (ref) × 49 terms > 0.0005″ 3.4 / 21 –1 18 terms > 0.005″ 23/85 –2 EOPs Observed (ref) × Extrapolated 34/59 –2 Earth gravity 12 × 12 model (ref) 10 × 10 model 0.1 / 0.1 –4 × 9 ×9 model 0.3 / 0.3 –5 Relativity Schwarzschild correction (ref) × neglected 2.6/7.8 –1 Tides Solid Earth (ref) × k2 tide 0.5 / 2.4 –5 none 34 / 74 –8 Sun position full series (ref) × terms > 5″ 3.7 / 5.0 –4 Keplerian 13/29 –11 Moon position full series (ref) × terms > 2″ 6.1 / 47 –7 terms > 10″ 39 / 260 –8 Recommended (×) 32/62 –25 Parameter Value [nm/s2] Cubic body acceleration aC +14.5 Stretching parameter aS +5.0 Solar panel acceleration asp +87.0 - TABLE 4
Peak orbit propagation errors (3D, in m) over two weeks for different integration methods, step sizes, and shadow boundary handling
Step [s] RK4R DP5 Notes without with without with boundary handling boundary handling 50 0.7 0.6 0.1 (ref) 100 19 19 0.5 0.5 150 140 140 5.8 2.9 200 610 590 9.8 12 All satellites 2.5 2.5 Plane A and B 4.4 2.6 Plane C 9.8 12 E14/E18 250 - - 37 34 300 - - 84 82 - TABLE 5
Peak orbit propagation errors (3D, in m) over two weeks after a two-day orbit determination for different integration methods, step sizes, and shadow boundary handling
Step [s] RK4R DP5 Notes without with without with boundary handling boundary handling 50 2.4 0.3 0.5 (ref) 100 21 4.7 4.0 0.3 150 110 37 15 0.8 200 1300 156 35 2.6 All satellites 0.1 0.1 Plane A and B 35 2.6 Plane C 34 0.7 E14/E18 250 - - 62 8.5 300 - - 120 22 - TABLE 6
Long-term orbit propagator parameters and recommended numerical representation. The number of bits in column 4 refers to a single parameter of the specified type. Depending on the parameter generation process, common epochs may be adopted for EOP and orbit/clock parameters to reduce the total amount of data transmitted to the user
Parameter Unit No. of bits LSB Epoch Weop 13 teop [s] 8 3600 Pole offset xp, yp [″] 21 2–20 ẋp, ẏp [″/d] 15 2–21 UT1R offset ΔUT1R [s] 31 2–24 ΔUṪ1R [s/d] 19 2–25 Total EOPs 143 Epoch Woe = Woc 13 toe = toc [s] 8 3600 Epoch position X, y, z [m] 34 2–8 Epoch velocity ẋ, ẏ, ż [m/s] 34 2–21 Empirical accels. D0, Y0, 
[nm/s2] 13 2–9 Clock ao [s] 27 2–30 a1 [s/s] 25 2–50 a2 [s/s2] 17 2–70 Total per satellite 333 - TABLE 7
Execution time for 14 d orbit prediction of a single GNSS satellite on two different processors for embedded systems
Processor Frequency Time ARM Cortex A9 800 MHz 9 s ARM1176JZF-S 700 MHz 30 s
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