This package is part of the SatelliteToolbox.jl ecosystem.

This package contains the implementation of the SGP4/SDP4 orbit propagator for the Julia language.

julia> using Pkg
julia> Pkg.add("SatelliteToolboxSgp4")

First, we need to initialize the structure that contains the information to propagate the orbit using the function sgp4_init. Usually, we pass a TLE to initialize the SGP4 algorithm:

julia> using SatelliteToolboxTle

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
       """

julia> sgp4d = sgp4_init(tle)

sgp4_init supports the keyword sgp4c to select the constants used to propagate the orbit. It must be an object of type Sgp4Constants. The following constants are already defined in this package:

• sgp4c_wgs84: (DEFAULT) Constants based on WGS84 using Float64.
• sgp4c_wgs84_f32: Constants based on WGS84 using Float32.
• sgp4c_wgs72: Constants based on WGS72 using Float64.
• sgp4c_wgs72_f32: Constants based on WGS72 using Float32.

Note The propagator will use the same type of object sgp4c to propagate the orbit. Hence, if one selects sgp4c_wgs84_f32, the SGP4 will compute everything considering Float32 numbers.

The SGP4 can also be initialized by passing the mean elements directly. For more information, see the documentation of the function sgp4_init.

Afterward, we can propagate the orbit using the function sgp4!(sgp4d, t) that propagates the mean elements defined in sgp4d by t minutes. This function returns the position [km] and velocity [km/s] vectors represented in the True Equator, Mean Equinox (TEME) reference frame.

# Propagate the orbit for 10 minutes.
julia> r_teme, v_teme = sgp4!(sgp4d, 10)
([-5300.1473032595195, 2356.780136349037, 4149.0611906521035], [4.464838382952148, -0.5106103512199875, 5.9760603775620815])

Warning We do not use SI units here to keep consistency with the original SGP4/SDP4 algorithms.

The function sgp4(t, args...) creates the propagator and propagates the orbit defined in args... by t minutes. It returns the same information as the function sgp4! and the initialized propagator structure. args... must be the same arguments supported by sgp4!.

julia> r_teme, v_teme, sgp4d = sgp4(10, tle)

julia> r_teme
3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
 -5300.1473032595195
  2356.780136349037
  4149.0611906521035

julia> v_teme
3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
  4.464838382952148
 -0.5106103512199875
  5.9760603775620815

sgp4 also supports the same keywords arguments as sgp4!.

We also have the function sgp4_init! that initializes a SGP4 propagator structure in-place, avoiding unnecessary allocations in some cases. For more information, see the function documentation.

This package also provides a way to fit a TLE for the SGP4 algorithm given a set of osculating state vectors through the following function:

fit_sgp4_tle(vjd::AbstractVector{Tepoch}, vr_teme::AbstractVector{Tv}, vv_teme::AbstractVector{Tv}; kwargs...) where {Tepoch<:Number, Tjd<:Number, Tv<:AbstractVector} -> TLE

where the osculating elements are given by a set of position vectors vr_teme [km] and a set of velocity vectors vv_teme [km / s] represented in the True-Equator, Mean-Equinox reference frame (TEME) at instants in the array vjd [Julian Day].

The algorithm performs a least-square fitting to minimize the residue between the osculating elements provided by the SGP4 propagator and the input data. It was based on [4].

This function returns the fitted TLE and the last covariance matrix obtained from the least-square algorithm.

Note This algorithm version will allocate a new SGP4 propagator with the default constants sgp4c_wgs84. If another set of constants are required or if the user wants to reduce the allocations, use the function fit_sgp4_tle! instead.

The following keywords are avaible:

• 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)

This algorithm uses a least-square algorithm to fit a TLE based on a set of osculating state vectors. Since the system is chaotic, a good initial guess is paramount for algorithm convergence. We can provide an initial guess using the keyword initial_guess.

If initial_guess is a TLE, we update the TLE epoch using the function update_sgp4_tle_epoch! to the desired one in mean_elements_epoch. Afterward, we use this new TLE as the initial guess.

If initial_guess is an AbstractVector, we use this vector as the initial mean state vector for the algorithm. It must contain 7 elements as follows:

┌                                    ┐
│ IDs 1 to 3: Mean position [km]     │
│ IDs 4 to 6: Mean velocity [km / s] │
│ ID  7:      Bstar         [1 / er] │
└                                    ┘

If initial_guess is nothing, the algorithm takes the closest osculating state vector to the mean_elements_epoch and uses it as the initial mean state vector. In this case, the epoch is set to the same epoch of the osculating data in vjd. When the fitted TLE is obtained, the algorithm uses the function update_sgp4_tle_epoch! to change its epoch to mean_elements_epoch.

Note If initial_guess is not nothing, the B* initial estimate is obtained from the TLE or the state vector. Hence, if estimate_bstar is false, it will be kept constant with this initial value.

julia> vjd  = [2.46002818657856e6];

julia> vr_teme = [@SVector [-6792.402703741442, 2192.6458461287293, 0.18851758695295118]];

julia> vv_teme = [@SVector [0.3445760107690598, 1.0395135806993514, 7.393686131436984]];

julia> tle, P = fit_sgp4_tle(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:         47           2.7699e-07           2.5122e-10           2.7699e-07                 -100 %

(TLE: UNDEFINED (Epoch = 2023-03-24T16:28:40.388), [2.5682204663112826 0.5152694462560151 … -1.2385529801114805 0.0; 0.5152694462560317 4.6021551786709445 … -0.1449556257318217 0.0; … ; -1.2385529801114785 -0.14495562573181023 … 0.999604694355324 0.0; 0.0 0.0 … 0.0 0.0])

julia> tle
TLE:
                      Name : UNDEFINED
          Satellite number : 9999
  International designator : 999999
        Epoch (Year / Day) : 23 /  83.68657856 (2023-03-24T16:28:40.388)
        Element set number : 0
              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 : 0
                        B* :            0 1 / er
                     ṅ / 2 :            0 rev / day²
                     n̈ / 6 :            0 rev / day³

We can also update SGP4 TLE epoch using the function:

update_sgp4_tle_epoch(tle::TLE, new_epoch::Union{Number, DateTime}; kwargs...) -> TLE

which returns a new TLE obtained by updating the epoch of tle to new_epoch.

Note This algorithm version will allocate a new SGP4 propagator with the default constants sgp4c_wgs84. If another set of constants are required or if the user wants to reduce the allocations, use the function update_sgp4_tle_epoch! instead.

The following keywords are avaible:

• atol::Number: Tolerance for the residue absolute value. If, at any iteration, the residue is lower than atol, the computation loop stops. (Default = 2e-4)
• rtol::Number: Tolerance for the relative difference between the residues. If, at any iteration, the relative difference between the residues in two consecutive iterations is lower than rtol, the computation loop stops. (Default = 2e-4)
• max_iterations::Int: Maximum number of iterations allowed for the least-square fitting. (Default = 50)
• verbose::Bool: If true, the algorithm prints debugging information to stdout. (Default = true)

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> update_sgp4_tle_epoch(tle, DateTime("2023-05-01"))
ACTION:   Fitting the TLE.
           Iteration        Position RMSE        Velocity RMSE           Total RMSE       RMSE Variation
                                     [km]             [km / s]                  [ ]
PROGRESS:          2          6.78579e-06          2.29683e-08          6.78583e-06             -99.9999 %

TLE:
                      Name : AMAZONIA 1
          Satellite number : 47699
  International designator : 21015A
        Epoch (Year / Day) : 23 / 121.00000000 (2023-05-01T00:00:00)
        Element set number : 999
              Eccentricity :   0.00012481
               Inclination :  98.43040000 deg
                      RAAN : 198.88793445 deg
       Argument of perigee :  24.20140448 deg
              Mean anomaly :  86.69370896 deg
           Mean motion (n) :  14.40824649 revs / day
         Revolution number : 10865
                        B* :      4.3e-05 1 / er
                     ṅ / 2 :            0 rev / day²
                     n̈ / 6 :            0 rev / day³

The code in this package was built using the following 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. American Institute of Aeronautics ans Astronautics.