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SGP4/SDP4

This package implements the interface to the SGP4/SDP4 propagator provided by SatelliteToolboxSgp4.jl.

Algorithm

The SGP4/SDP4 implementation was built using [1, 2, 3].

Initialization

We must initialize the SGP4/SDP4 propagator with a two-line element set (TLE) using the following function:

Propagators.init(Val(:SGP4), tle::TLE; kwargs...) -> OrbitPropagatorSgp4

which creates a SGP4/SDP4 propagator structure OrbitPropagatorSgp4 with the tle. The following keyword selects the gravitational constants for the propagation algorithm:

  • sgp4c::Sgp4Constants: SGP4 orbit propagator constants (see Sgp4Constants). (Default = sgp4c_wgs84)

The package [SatelliteToolboxSgp4.jl] contains some pre-build constants for this propagator:

SGP4/SDP4 Propagator ConstantsDescriptionType
sgp4c_wgs84Constants based on WGS-84Float64
sgp4c_wgs84_f32Constants based on WGS-84Float32
sgp4c_wgs72Constants based on WGS-72Float64
sgp4c_wgs72_f32Constants based on WGS-72Float32
Note

The type used in the propagation will be the same as used to define the constants in the structure sgp4c.

Note

The package SatelliteToolboxTle.jl defines the type TLE, which is re-exported here. It contains some useful functionalities, such as TLE fetching from online services. For more information, refer to the package documentation.

julia> tle = tle"""
              AMAZONIA 1
              1 47699U 21015A   23083.68657856 -.00000044  10000-8  43000-4 0  9990
              2 47699  98.4304 162.1097 0001247 136.2017 223.9283 14.40814394108652"""TLE:
                      Name : AMAZONIA 1
          Satellite number : 47699
  International designator : 21015A
        Epoch (Year / Day) : 23 /  83.68657856 (2023-03-24T16:28:40.388)
        Element set number : 999
              Eccentricity :   0.00012470
               Inclination :  98.43040000 deg
                      RAAN : 162.10970000 deg
       Argument of perigee : 136.20170000 deg
              Mean anomaly : 223.92830000 deg
           Mean motion (n) :  14.40814394 revs / day
         Revolution number : 10865
                        B* :      4.3e-05 1 / er
                     ṅ / 2 :     -4.4e-07 rev / day²
                     n̈ / 6 :        1e-09 rev / day³
julia> Propagators.init(Val(:SGP4), tle)OrbitPropagatorSgp4{Float64, Float64}:
   Propagator name : SGP4 Orbit Propagator
  Propagator epoch : 2023-03-24T16:28:40.388
  Last propagation : 2023-03-24T16:28:40.388

We also support initializing an SGP4/SDP4 propagator passing the TLE information in individual terms:

Propagators.init(Val(:SGP4), epoch::Number, n₀::Number, e₀::Number, i₀::Number, Ω₀::Number, ω₀::Number, M₀::Number, bstar::Number; kwargs...) -> OrbitPropagatorSgp4

where:

  • epoch::Number: Epoch of the orbital elements [Julian Day].
  • n₀::Number: SGP type "mean" mean motion at epoch [rad/s].
  • e₀::Number: "Mean" eccentricity at epoch.
  • i₀::Number: "Mean" inclination at epoch [rad].
  • Ω₀::Number: "Mean" longitude of the ascending node at epoch [rad].
  • ω₀::Number: "Mean" argument of perigee at epoch [rad].
  • M₀::Number: "Mean" mean anomaly at epoch [rad].
  • bstar::Number: Drag parameter (B*).

The keywords kwargs... are the same as in the first version of this function.

Fitting TLEs

We can use the function:

Propagators.fit_mean_elements(::Val{:SGP4}, vjd::AbstractVector{Tjd}, vr_teme::AbstractVector{Tv}, vv_teme::AbstractVector{Tv}; kwargs...) -> KeplerianElements{Float64, Float64}, SMatrix{6, 6, Float64}

to fit a Two-Line Element set (TLE) for the SGP4 orbit propagator using the osculating elements represented by a set of position vectors vr_teme [m] and a set of velocity vectors vv_teme [m / s] represented in the True-Equator, Mean-Equinox reference frame (TEME) at instants in the array vjd [Julian Day].

It returns the fitted TLE and the final covariance matrix of the least-square algorithm.

This algorithm was based on [4].

Note

This algorithm version will allocate a new SGP4 propagator with the default constants sgp4c_wgs84. If another set of constants are required, use the function Propagators.fit_mean_elements! instead.

The following keywords are available to configure the fitting process:

  • atol::Number: Tolerance for the residue absolute value. If the residue is lower than atol at any iteration, the computation loop stops. (Default = 2e-4)
  • rtol::Number: Tolerance for the relative difference between the residues. If the relative difference between the residues in two consecutive iterations is lower than rtol, the computation loop stops. (Default = 2e-4)
  • estimate_bstar::Bool: If true, the algorithm will try to estimate the B* parameter. Otherwise, it will be set to 0 or to the value in initial guess (see section Initial Guess). (Default = true)
  • initial_guess::Union{Nothing, AbstractVector, TLE}: Initial guess for the TLE fitting process. If it is nothing, the algorithm will obtain an initial estimate from the osculating elements in vr_teme and vv_teme. For more information, see the section Initial Guess. (Default = nothing)
  • jacobian_perturbation::Number: Initial state perturbation to compute the finite-difference when calculating the Jacobian matrix. (Default = 1e-3)
  • jacobian_perturbation_tol::Number: Tolerance to accept the perturbation when calculating the Jacobian matrix. If the computed perturbation is lower than jacobian_perturbation_tol, we increase it until it absolute value is higher than jacobian_perturbation_tol. (Default = 1e-7)
  • max_iterations::Int: Maximum number of iterations allowed for the least-square fitting. (Default = 50)
  • mean_elements_epoch::Number: Epoch for the fitted TLE. (Default = vjd[end])
  • verbose::Bool: If true, the algorithm prints debugging information to stdout. (Default = true)
  • weight_vector::AbstractVector: Vector with the measurements weights for the least-square algorithm. We assemble the weight matrix W as a diagonal matrix with the elements in weight_vector at its diagonal. (Default = @SVector(ones(Bool, 6)))
  • classification::Char: Satellite classification character for the output TLE. (Default = 'U')
  • element_set_number::Int: Element set number for the output TLE. (Default = 0)
  • international_designator::String: International designator string for the output TLE. (Default = "999999")
  • name::String: Satellite name for the output TLE. (Default = "UNDEFINED")
  • revolution_number::Int: Revolution number for the output TLE. (Default = 0)
  • satellite_number::Int: Satellite number for the output TLE. (Default = 9999)
julia> vr_teme = [
           [-6792.402703741442, 2192.6458461287293, 0.18851758695295118] .* 1000,
           [-6357.88873265975, 2391.9476768911686, 2181.838771262736] .* 1000
       ];
julia> vv_teme = [
           [0.3445760107690598, 1.0395135806993514, 7.393686131436984] .* 1000,
           [2.5285015912807003, 0.27812476784300005, 7.030323100703928] .* 1000
       ];
julia> vjd = [
           2.46002818657856e6,
           2.460028190050782e6
       ];
julia> tle, P = Propagators.fit_mean_elements(Val(:SGP4), vjd, vr_teme, vv_teme; estimate_bstar = false)ACTION:   Fitting the TLE.
           Iteration        Position RMSE        Velocity RMSE           Total RMSE       RMSE Variation
                                     [km]             [km / s]                  [ ]                     


PROGRESS:          1               13.589           0.00473939               13.589                  ---

PROGRESS:          2          0.000902827          4.59551e-06          0.000902839             -99.9934 %

PROGRESS:          3          5.79219e-09          4.38304e-07          4.38343e-07             -99.9514 %

(TLE: UNDEFINED (Epoch = 2023-03-24T16:33:40.388), [1.0857516670112908 -0.024565134376204067 … -0.07514785921985671 -5698.500770541173; -0.024565134379496076 1.0179328946623214 … 0.022881996752420863 1735.4397854652639; … ; -0.0751478592201972 0.02288199675149808 … 0.06516943389767876 4946.909491810927; -5698.500770568585 1735.4397853961086 … 4946.909491812222 3.7563825900165176e8])
julia> tleTLE:
                      Name : UNDEFINED
          Satellite number : 9999
  International designator : 999999
        Epoch (Year / Day) : 23 /  83.69005079 (2023-03-24T16:33:40.388)
        Element set number : 0
              Eccentricity :   0.00012463
               Inclination :  98.43040000 deg
                      RAAN : 162.11312239 deg
       Argument of perigee : 136.15040637 deg
              Mean anomaly : 241.97934027 deg
           Mean motion (n) :  14.40814157 revs / day
         Revolution number : 0
                        B* :            0 1 / er
                     ṅ / 2 :            0 rev / day²
                     n̈ / 6 :            0 rev / day³

References

  • [1] Hoots, F. R., Roehrich, R. L (1980). Models for Propagation of NORAD Elements Set. Spacetrack Report No. 3.
  • [2] Vallado, D. A., Crawford, P., Hujsak, R., Kelso, T. S (2006). Revisiting Spacetrack Report #3: Rev1. AIAA.
  • [3] SGP4 Source code of STRF, which the C code was converted by Paul. S. Crawford and Andrew R. Brooks.
  • [4] Vallado, D. A., Crawford, P (2008). SGP4 Orbit Determination. AIAA.