Article Figures & Data
Figures
- FIGURE 1
In the proposed framework, we generate orbit data for an ELFO and LLO to test different orbit parameterization methods and obtain a parameterized predicted orbit; we then evaluate the performance of the methods with regard to three criteria.
- FIGURE 2
3-1-3 Euler angle transformation from an inertial (red) to rotational (blue) frame of reference
We employ a Euler angle frame rotation to enable ephemeris representation in both inertial and rotational Moon frames of reference (figure adapted from Folta et al. (2022)).
- FIGURE 3
(Left) x-position of a satellite in an ELFO near perilune approximated with a standard polynomial model obtained using three different sampling strategies and (right) the absolute truncation error resulting from parameterization
- FIGURE 4
Overview of the scaling and bit quantization process
The left figure shows the original curve, and the right figure shows the scaled curve. The curve is first scaled with the magnitude of the curve and then quantified using Nb + 1 bits (1 is for the sign). The maximum quantization error is
, which should be smaller than the LSB. Therefore, the required number of bits is calculated as Equation (18). - FIGURE 5
Optimization pipeline for ephemeris parameters
After propagating the orbit of choice in MATLAB, we set up the surrogate model and perform optimization to find all necessary parameters, storing the parameterization metrics. These metrics include starting time, parameterization interval, and message length (in bits and number of parameters).
- FIGURE 6
(Left) LLO and (right) ELFO as viewed from the MI frame
- FIGURE 7
Feasible SISE-compliant solutions found in orbit with respect to the parameterization interval (agnostic to the choice of surrogate model)
- FIGURE 8
Maximum parameterization interval at different true anomaly initializations for (left) polynomial and (right) Chebyshev surrogate models in an ELFO
- FIGURE 9
Maximum parameterization interval at different true anomaly initializations for (left) polynomial and (right) Chebyshev surrogate models in an LLO
- FIGURE 10
Surrogate model message length for (left) ELFO and (right) LLO
The Chebyshev basis surrogate model consistently outputs a smaller message bit length than the polynomial basis surrogate model. We also observe increasing message length with increasing basis order and fit interval, although the latter does not always occur (note the ELFO orbit for higher n).
Tables
- TABLE 1
Ephemeris Parameterization Methods for Terrestrial Satellite Navigation
ICD: interface control document, LEO: low Earth orbit, OE: orbital elements
Constellation Source Orbit Fitting Parameter Parameterization Method # of Params Fit Interval GPS (Default) ICD MEO Classical OE linear drift + sinusoidal 15 2 h Galileo (Default) ICD MEO Classical OE linear drift + sinusoidal 15 3 h BeiDou (Default) ICD MEO Classical OE linear drift + sinusoidal 18 - GEO IGSO L5 SBAS Reid et al. (2013) GEO Classical OE linear drift + sinusoidal (argument of latitude only) 9 10 min BeiDou (Modified) Xiaogang et al. (2017) GEO Non-singular OE linear drift + sinusoidal (longitude & orbital radius) 13 2 h GLONASS ICD MEO PZ-90 Cartesian numerical propagation 9 30 min L1 SBAS Reid (2017) GEO ECEF Cartesian second-order polynomial extrapolation 9 6 min GNSS (alt.) Dobbin and Axelrad (2023) LEO GEO ECEF Cartesian B-spline >10 <40 min Parameters Scaling Factor LSB Precision Unit αx, αy, αz 21 10−8 km/sk 21 10−7 rad 2−5 10−15 rad/s Contribution Model Parameters Point mass Sun, Earth, Jupiter High-order gravity Moon: 50 x 50 Solar radiation pressure Mu: 50 kg, Au: 1m2, CR: 1.5 - TABLE 4
LLO and ELFO Initial Conditions (in a J2000 Oriented Moon-Centered Frame) and Orbital Period
RAAN: right ascension of the ascending node
Orbit Semimajor Axis Ecc. Inclination RAAN Arg. of Perilune Mean Anomaly Orbital Period LLO 1850 km 0.05 99° 82.5° 294° 0° 1.98 h ELFO 6540 km 0.6 56.2° 180° 90° 0° 13.18 h - TABLE 5
SISE Performance of Different Surrogate Models for an ELFO for Select Parameterization Intervals
Fit Interval Highest 3σ SISEpos [m] Highest 3σ @ 10 s SISEvel [mm/s] Polynomial Chebyshev Polynomial Chebyshev 1 h 0.65 0.66 1.08 1.08 2 h 1.14 1.14 1.02 1.02 4 h 1.41 1.41 1.00 1.00 8 h 3.67 3.67 0.65 0.65 SISE Requirement <13.34 m (3σ) <1.2 mm/s (3σ @ 10 s) - TABLE 6
SISE Performance of Different Surrogate Models for an LLO for Select Parameterization Intervals
Fit Interval Highest 3σ SISEpos [m] Highest 3σ @ 10 s SISEvel [mm/s] Polynomial Chebyshev Polynomial Chebyshev 5 min 0.13 0.13 0.93 0.93 15 min 0.16 0.16 1.10 1.10 30 min 0.12 0.12 1.08 1.08 SISE Requirement <13.34 m (3σ) <1.2 mm/s (3σ @ 10 s) - TABLE 7
Orbital Coverage of Polynomial and Chebyshev Surrogate Models at Different Basis Orders for LLO and ELFO
Basis order (n) Polynomial Model Chebyshev Model 6 8 10 12 14 6 8 10 12 14 ELFO 27% 43% 60% 77% 100% 27% 43% 60% 77% 100% LLO 67% 100% 100% 100% 100% 67% 100% 100% 100% 100% - TABLE 8
Receiver-Side Algorithm for Obtaining the Navigation Satellite Position and Velocity in the Lunar
Fixed Frame from the Coordinate-Based Parameter Set at Time t
Operation Description Normalize time (t) from reference epoch time (toe) and known validity interval (Δt) Obtain satellite position in MI frame Obtain satellite velocity in MI frame Calculate Euler angles at t for frame transformation ψ rotation matrix θ rotation matrix ϕ rotation matrix Rotation matrix from MI frame to ME frame Satellite position in ME frame Satellite velocity in ME frame
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