Library

abstract type AstroCoordTransformation <: AstrodynamicsTransformation

An abstract type representing a Transformation Between Astrodynamics Coordinates with no Time Regularization

abstract type AstrodynamicsTransformation <: Transformation

An abstract type representing a Transformation Between Astrodynamics Coordinates

Cartesian{T} <: AstroCoord

Cartesian Orbital Elements. 6D parameterziation of the orbit. x - X-position y - Y-position z - Z-position ẋ - X-velocity ẏ - Y-velocity ż - Z-velocity

Constructors Cartesian(x, y, z, ẋ, ẏ, ż) Cartesian(X::AbstractArray) Cartesian(X::AstroCoord, μ::Number)

abstract type AstroCoord{N, T} <: StaticMatrix{N, 1, T}

An abstract type representing a N-Dimensional Coordinate Set

Cylindrical{T} <: AstroCoord

Cylindrical Orbital Elements. 6D parameterziation of the orbit ρ - in-plane radius θ - in-plane angle z - out-of-plane distance ρdot - instantaneous rate of change of in-plsne radius θdot - instantaneous rate of change of θ ż - instantaneous rate of change of z

Constructors Cylindrical(r, θ, z, ṙ, θdot, ż) Cylindrical(X::AbstractArray) Cylindrical(X::AstroCoord, μ::Number)

Delaunay{T} <: AstroCoord

Delaunay Orbital Elements. 6D parameterziation of the orbit.

Fields

L - Canonical Keplerian Energy G - Canonical Total Angular Momentum H - Canonical Normal Angular Momentum (Relative to Equator) M - Mean Anomaly ω - Argument of Periapsis Ω - Right Ascension of the Ascending Node

Additional Properties

E - Eccentric Anomaly (computed from mean anomaly and eccentricity extracted from L and G) f - True Anomaly (computed from mean anomaly and eccentricity extracted from L and G)

Constructors

• Delaunay(L, G, H, M, ω, Ω)
• Delaunay(X::AbstractVector{<:Number})
• Delaunay(X::AstroCoord, μ::Number)

EDromo{T} <: AstroCoord

EDromo state vector. ζ₁ - In-plane element 1 ζ₂ - In-plane element 2 ζ₃ - Corresponds to orbital energy ζ₄ - Quaternion element 1 ζ₅ - Quaternion element 2 ζ₆ - Quaternion element 3 ζ₇ - Quaternion element 4 ζ₈ - Time element

Constructors EDromo(ζ₁, ζ₂, ζ₃, ζ₄, ζ₅, ζ₆, ζ₇, ζ₈) EDromo(X::AbstractArray)

GEqOE{T} <: AstroCoord

Generalized Equinoctial Orbital Elements (GEqOE). 6D parametrization of the orbit that generalizes the equinoctial elements when perturbing forces are derived from a disturbing potential.

Reference: Baù, G., Hernando-Ayuso, J., & Bombardelli, C. (2021). "A generalization of the equinoctial orbital elements." Celestial Mechanics and Dynamical Astronomy, 133(9), 1-32.

Fields

• ν::T - Generalized mean motion: ν = (1/μ)(-2E)^(3/2)
• p₁::T - Generalized eccentricity component: p₁ = g sin(Ψ)
• p₂::T - Generalized eccentricity component: p₂ = g cos(Ψ)
• L::T - Generalized mean longitude
• q₁::T - Inclination component: q₁ = tan(i/2) sin(Ω)
• q₂::T - Inclination component: q₂ = tan(i/2) cos(Ω)

Where:

• E is the total energy (including disturbing potential U)
• g is the generalized Laplace vector magnitude
• Ψ is the generalized longitude of pericenter
• i is the orbital inclination
• Ω is the longitude of the ascending node

Constructors

• GEqOE(ν, p₁, p₂, L, q₁, q₂)
• GEqOE(X::AbstractArray)
• GEqOE(X::Cartesian, μ::Number, config::RegularizedCoordinateConfig)

Computed Properties

• g: Generalized eccentricity magnitude √(p₁² + p₂²)
• Ψ: Generalized longitude of pericenter atan(p₁, p₂)
• L₀(geq, t): Generalized mean longitude at epoch L - νt

Notes

• The perturbing potential W is passed via RegularizedCoordinateConfig. For Keplerian orbits use W = 0; for perturbed orbits precompute W = V_total - V_keplerian externally.
• Non-singular for circular and equatorial orbits
• Singular for retrograde equatorial orbits (i = π) and rectilinear motion
• Defined for negative total energy (E < 0)

The IdentityTransformation is a singleton Transformation that returns the input unchanged, similar to identity.

J2EqOE{T} <: AstroCoord

Modified Equinoctial Orbital Elements. 6D parameterziation of the orbit. n - mean motion h - eccetricity projection onto longitude of perigee k - eccetricity projection onto ⟂ longitude of perigee p - projection of half inclination onto RAAN q - projection of half inclination onto ⟂ RAAN L - true longitude

Constructors J2EqOE(n, h, k, p, q, L) J2EqOE(X::AbstractArray) J2EqOE(X::AstroCoord, μ::Number)

Keplerian{T} <: AstroCoord

Keplerian Orbital Elements. 6D parameterziation of the orbit.

Fields

a - semi-major axis e - eccentricity i - inclination Ω - Right Ascension of Ascending Node ω - Argument of Perigee f - True Anomaly

Exotic Properties

M - Mean Anomaly (computed from true anomaly and eccentricity) E - Eccentric Anomaly (computed from true anomaly and eccentricity)

Constructors

• Keplerian(a, e, i, Ω, ω, f)
• Keplerian(X::AbstractArray)
• Keplerian(X::AstroCoord, μ::Number)

KustaanheimoStiefel{T} <: AstroCoord

Kustaanheimo-Stiefel (KS) state vector. The KS transformation regularizes the two-body problem by transforming the 3D Cartesian coordinates into a 4D space, which eliminates the singularity at r=0.

This state vector contains 10 elements. u₁ - First component of the KS position vector u₂ - Second component of the KS position vector u₃ - Third component of the KS position vector u₄ - Fourth component of the KS position vector u₁prime - First component of the KS velocity vector u₂prime - Second component of the KS velocity vector u₃prime - Third component of the KS velocity vector u₄prime - Fourth component of the KS velocity vector h - The negative of the total orbital energy (-E) τ - The time element, which can be either physical time t or a linear time element.

Constructors KustaanheimoStiefel(u₁, u₂, u₃, u₄, u₁prime, u₂prime, u₃prime, u₄prime, h, τ) KustaanheimoStiefel(X::AbstractArray)

Milankovich{T} <: AstroCoord

Milankovich Orbital Elements. 7D parameterziation of the orbit. hx - X-component of Angular Momentum Vector hy - Y-component of Angular Momentum Vector hz - Z-component of Angular Momentum Vector ex - X-component of Eccentricity Vector ey - Y-component of Eccentricity Vector ez - Z-component of Eccentricity Vector L - True Longitude

Constructors Milankovich(hx, hy, hz, ex, ey, ez, L) Milankovich(X::AbstractArray) Milankovich(X::AstroCoord, μ::Number)

ModEq{T} <: AstroCoord

Modified Equinoctial Orbital Elements. 6D parameterziation of the orbit.

Fields

p - semi-parameter f - eccentricity projection onto longitude of perigee g - eccentricity projection onto ⟂ longitude of perigee h - projection of half inclination onto RAAN k - projection of half inclination onto ⟂ RAAN L - true longitude

Constructors

• ModEq(p, f, g, h, k, L)
• ModEq(X::AbstractArray)
• ModEq(X::AstroCoord, μ::Number)

See Also

• ModEqN: Modified Equinoctial coordinates with mean motion instead of semi-parameter

ModEqN{T} <: AstroCoord

Modified Equinoctial Orbital Elements with Mean Motion. 6D parameterziation of the orbit.

Fields

η - mean motion (rad/s) f - eccentricity projection onto longitude of perigee g - eccentricity projection onto ⟂ longitude of perigee h - projection of half inclination onto RAAN k - projection of half inclination onto ⟂ RAAN L - true longitude

Constructors

• ModEqN(η, f, g, h, k, L)
• ModEqN(X::AbstractArray)
• ModEqN(X::AstroCoord, μ::Number)

See Also

• ModEq: Modified Equinoctial coordinates with semi-parameter instead of mean motion

Poincare{T} <: AstroCoord

Poincaré Canonical Orbital Elements. 6D parametrization of the orbit using Cartesian-style canonical coordinate-momentum pairs derived from Delaunay variables.

These elements are non-singular for circular orbits (e → 0) and equatorial orbits (i → 0), making them particularly useful in perturbation theory and Hamiltonian celestial mechanics.

Supports both elliptic (a > 0) and hyperbolic (a < 0) orbits. For hyperbolic orbits, the eccentricity action is defined as P = Λ + G (instead of Λ - G for elliptic) to keep all quantities real-valued. The orbit type is recovered unambiguously during inversion: P ≤ Λ → elliptic, P > Λ → hyperbolic.

Fields

Λ - Canonical action for semi-major axis: Λ = √(μ|a|) λ - Mean longitude: λ = M + ω + Ω ξ - Eccentricity cosine component: ξ = √(2P) cos(ω̃) η - Eccentricity sine component: η = -√(2P) sin(ω̃), where ω̃ = ω + Ω p - Inclination cosine component: p = √(2Q) cos(Ω), where Q = G(1 - cos(i)) q - Inclination sine component: q = -√(2Q) sin(Ω)

where P = Λ - G (elliptic) or P = Λ + G (hyperbolic), and G is the Delaunay angular momentum.

Constructors

• Poincare(Λ, λ, ξ, η, p, q)
• Poincare(X::AbstractVector{<:Number})
• Poincare(X::AstroCoord, μ::Number)

References

• Murray, C.D. and Dermott, S.F. "Solar System Dynamics." Cambridge University Press (1999).
• Laskar, J. and Robutel, P. "Stability of the Planetary Three-Body Problem." Celestial Mechanics and Dynamical Astronomy 62 (1995): 193-217.

RegularizedCoordinateConfig{DT,TT,WT,TT2,FT}

Configuration struct for regularized coordinate transformations (EDromo, GEqOE, Kustaanheimo-Stiefel, Stiefel-Scheifele).

This struct encapsulates the parameters required for regularized coordinate transformations:

• DU: Reference distance unit for non-dimensionalization
• TU: Reference time unit for non-dimensionalization
• W: Perturbing potential energy (unified for all coordinate systems)
• t₀: Initial physical time
• flag_time: Time element formulation (PhysicalTime, ConstantTime, or LinearTime)

Spherical{T} <: AstroCoord

Spherical Orbital Elements. 6D parameterziation of the orbit r - radius θ - in-plane angle ϕ - out-of-plane angle ṙ - instantaneous rate of change of radius θdot - instantaneous rate of change of θ ϕdot - instantaneous rate of change of ϕ

Constructors Spherical(r, θ, ϕ, ṙ, ω, Ω) Spherical(X::AbstractArray) Spherical(X::AstroCoord, μ::Number)

StiefelScheifele{T} <: AstroCoord

Stiefel-Scheifele orbital elements. This is a 10-dimensional parameterization of a regularized orbit.

The elements are: α1 - The first component of the Stiefel-Scheifele position vector α2 - The second component of the Stiefel-Scheifele position vector α3 - The third component of the Stiefel-Scheifele position vector α4 - The fourth component of the Stiefel-Scheifele position vector β1 - The first component of the Stiefel-Scheifele velocity vector β2 - The second component of the Stiefel-Scheifele velocity vector β3 - The third component of the Stiefel-Scheifele velocity vector β4 - The fourth component of the Stiefel-Scheifele velocity vector ω - Related to the total energy of the orbit t - Time or a time element, depending on flag_time

Constructors: StiefelScheifele(α1, α2, α3, α4, β1, β2, β3, β4, ω, t) StiefelScheifele(v::AbstractVector)

The Transformation supertype defines a simple interface for performing transformations. Subtypes should be able to apply a coordinate system transformation on the correct data types by overloading the call method, and usually would have the corresponding inverse transformation defined by Base.inv(). Efficient compositions can optionally be defined by compose() (equivalently ∘).

USM6{T} <: AstroCoord

Unified State Model Orbital Elements. 6D parameterziation of the orbit using Velocity Hodograph and MRP's C - Velocity Hodograph Component Normal to the Radial Vector Laying in the Orbital Plane Rf1 - Velocity Hodograph Component 90 degrees ahead of the Eccentricity Vector - Along the Intermediate Rotating Frame X-Axis Rf2 - Velocity Hodograph Component 90 degrees ahead of the Eccentricity Vector - Along the Intermediate Rotating Frame Y-Axis σ1 - First Modified Rodriguez Parameter σ2 - Second Modified Rodriguez Parameter σ3 - Third Modified Rodriguez Parameter

Constructors USM6(C, Rf1, Rf2, σ1, σ2, σ3) USM6(X::AbstractArray) USM6(X::AstroCoord, μ::Number)

USM7{T} <: AstroCoord

Unified State Model Orbital Elements. 7D parameterziation of the orbit using Velocity Hodograph and Quaternions C - Velocity Hodograph Component Normal to the Radial Vector Laying in the Orbital Plane Rf1 - Velocity Hodograph Component 90 degrees ahead of the Eccentricity Vector - Along the Intermediate Rotating Frame X-Axis Rf2 - Velocity Hodograph Component 90 degrees ahead of the Eccentricity Vector - Along the Intermediate Rotating Frame Y-Axis ϵO1 - First Imaginary Quaternion Component ϵO2 - Second Imaginary Quaternion Component ϵO3 - Third Imaginary Quaternion Component η0 - Real Quaternion Component

Constructors USM7(C, Rf1, Rf2, ϵO1, ϵO2, ϵO3, η0) USM7(X::AbstractArray) USM7(X::AstroCoord, μ::Number)

USMEM{T} <: AstroCoord

Unified State Model Orbital Elements. 6D parameterziation of the orbit using Velocity Hodograph and Exponential Mapping C - Velocity Hodograph Component Normal to the Radial Vector Laying in the Orbital Plane Rf1 - Velocity Hodograph Component 90 degrees ahead of the Eccentricity Vector - Along the Intermediate Rotating Frame X-Axis Rf2 - Velocity Hodograph Component 90 degrees ahead of the Eccentricity Vector - Along the Intermediate Rotating Frame Y-Axis a1 - First Exponential Mapping Component a2 - Second Exponential Mapping Component a3 - Third Exponential Mapping Component

Constructors USMEM(C, Rf1, Rf2, a1, a2, a3) USMEM(X::AbstractArray) USMEM(X::AstroCoord, μ::Number)

EDromo2cart(u, μ, ϕ::Number, config::RegularizedCoordinateConfig)

Converts an EDromo state vector to a Cartesian state vector.

This is the backend implementation for the transformation from EDromo to Cartesian. It requires the fictitious time ϕ and a RegularizedCoordinateConfig with the same parameters used in the forward transformation.

Arguments

• u::AbstractVector: EDromo state vector [ζ₁, ζ₂, ζ₃, ζ₄, ζ₅, ζ₆, ζ₇, ζ₈].
• μ::Number: Gravitational parameter of the central body.
• ϕ::Number: Fictitious time parameter.
• config::RegularizedCoordinateConfig: Configuration parameters for the transformation.

Returns

• SVector{6, RT}: The 6-element Cartesian state vector [x, y, z, ẋ, ẏ, ż].

EP2MRP(β::AbstractVector{<:Number})

Converts Euler Parameter rotation description into Modified Rodriguez Parameters.

Arguments

• β::AbstractVector{<:Number}: The Euler Parameter description of a rotation.

Returns

• σ::AbstractVector{<:Number}: The Modified Rodriguez Parameter description of a rotation.

function IOE2J2IOE(u::AbstractVector{T}, μ::V) where {T<:Number, V<:Number}

Computes the J2 Perturbed Intermediate Orbit Elements from a Intermediate Orbit Element set.

Arguments

-u::AbstractVector{<:Number}: The Intermediate Orbit Element vector [I1; I2; I3; I4; I5; I6]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_J2IOR::SVector{6, <:Number}: The J2 Perturbed Intermediate Orbit Element vector [a; e; i; Ω(RAAN); ω(AOP); f(True Anomaly)].

function IOE2koeM(u::AbstractVector{T}, μ::V) where {T<:Number, V<:Number}

Computes the Keplerian orbital elements from a Intermediate Orbit Element set.

Arguments

-u::AbstractVector{<:Number}: The Intermediate Orbit Element vector [I1; I2; I3; I4; I5; I6]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_koeM::SVector{6, <:Number}: The Keplerian state vector [a; e; i; Ω(RAAN); ω(AOP); M(Mean Anomaly)].

function IOE2modEq(u::AbstractVector{T}, μ::V) where {T<:Number, V<:Number}

Computes the Modified Equinoctial orbital elements from a Intermediate Orbit Element set.

Arguments

-u::AbstractVector{<:Number}: The Intermediate Orbit Element vector [I1; I2; I3; I4; I5; I6]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_modEq::SVector{6, <:Number}: The Modified Equinoctial state vector [n; f; g; h; k; L].

function J2EqOE2cart(u::AbstractVector{<:Number}, μ::Number)

Computes the Cartesian state vector from a J2 Perturbed Equinoctial Orbit Elements.

Arguments

-u::AbstractVector{<:Number}: The J2 Perturbed Equinoctial Orbit Element vector [n; h; k; p; q; L]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_cart::SVector{6, <:Number}: The Cartesian state vector [x; y; z; ẋ; ẏ; ż].

function J2IOE2IOE(u::AbstractVector{T}, μ::V) where {T<:Number, V<:Number}

Computes the Intermediate Orbit Elements from a J2 Perturbed Intermediate Orbit Element set.

Arguments

-u::AbstractVector{<:Number}: The J2 Perturbed Intermediate Orbit Element vector [a; e; i; Ω(RAAN); ω(AOP); f(True Anomaly)]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_IOE::SVector{6, <:Number}: The Intermediate Orbit Element vector [I1; I2; I3; I4; I5; I6].

KS2cart(u_ks, μ, config::RegularizedCoordinateConfig)

Converts a Kustaanheimo-Stiefel (KS) state vector to a Cartesian state vector.

Arguments

• u_ks::AbstractVector: 10-element KS state vector.
• μ::Number: Gravitational parameter (unused in this direction, kept for API consistency).
• config::RegularizedCoordinateConfig: Configuration parameters.

Returns

• SVector{6, RT}: The 6-element Cartesian state [x, y, z, ẋ, ẏ, ż].

KeplerSolver(M::T, e::Number; tol::Float64=10 * eps(T), max_iters::Int=100) where {T<:Number}

Solves for true anomaly given the mean anomaly and eccentricity of an orbit.

Arguments

-M::Number: Mean Anomaly of the orbit [radians]. -e::Number: Eccentricity of the orbit.

Keyword Arguments

-tol::Float64: Convergence tolerance of Kepler solver. [Default=10*eps(T)] -max_iters::Int: Maximum number of iterations. [Default=100]

Returns

-f::Number`: True Anomaly of the orbit [radians]

L₀(geq::GEqOE, t::Number; t₀::Number=0.0)

Generalized mean longitude at epoch: L₀ = L - νt

MRP2EP(σ::AbstractVector{<:Number})

Converts Modified Rodriguez Parameters rotation description into Euler Parameter.

Arguments

• σ::AbstractVector{<:Number}: The Modified Rodriguez Parameter description of a rotation.

Returns

• β::AbstractVector{<:Number}: The Euler Parameter description of a rotation.

Mil2cart(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Milankovich state vector into the Cartesian state vector.

Arguments

-u::AbstractVector{<:Number}: The Milankovich state vector [H; e; L]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_cart::SVector{6, <:Number}: The Cartesian state vector [x; y; z; ẋ; ẏ; ż].

ModEq2ModEqN(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Modified Equinoctial elements to Modified Equinoctial with mean motion.

Arguments

-u:AbstractVector{<:Number}: The Modified Equinoctial state vector [p; f; g; h; k; L]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_ModEqN::SVector{6, <:Number}: The Modified Equinoctial (mean motion) state vector [η; f; g; h; k; L].

ModEq2koe(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Modified Equinoctial elements into the Keplerian elements.

Arguments

-u:AbstractVector{<:Number}: The Modified Equinoctial state vector [p; f; g; h; k; l]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_koe::SVector{6, <:Number}: The Keplerian state vector [a; e; i; Ω(RAAN); ω(AOP); ν(True Anomaly)].

ModEqN2ModEq(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Modified Equinoctial with mean motion to Modified Equinoctial elements.

Arguments

-u:AbstractVector{<:Number}: The Modified Equinoctial (mean motion) state vector [η; f; g; h; k; L]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_ModEq::SVector{6, <:Number}: The Modified Equinoctial state vector [p; f; g; h; k; L].

StiefelScheifele2cart(ss_vec, μ, ϕ::Number, config::RegularizedCoordinateConfig)

Converts a Stiefel-Scheifele state vector to a Cartesian state vector.

Arguments

• ss_vec::AbstractVector: The Stiefel-Scheifele state vector.
• μ::Number: The gravitational parameter of the central body.
• ϕ::Number: Fictitious time parameter.
• config::RegularizedCoordinateConfig: Configuration parameters.

Returns

• Cartesian: The Cartesian state vector.

USM62USM7(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts USM with Modified Rodrigue Parameters to USM with quaternions.

Arguments

-u::AbstractVector{<:Number}: The USM6 vector [C; Rf1; Rf2; σ1; σ2; σ3]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_USM::SVector{6, <:Number}: The Unified State Model vector [C; Rf1; Rf2; ϵO1; ϵO2; ϵO3; η0].

USM72USM6(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts USM with quaternions to USM with Modified Rodrigue Parameters.

Arguments

-u::AbstractVector{<:Number}: The Unified State Model vector [C; Rf1; Rf2; ϵO1; ϵO2; ϵO3; η0]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_USM6::SVector{6, <:Number}: The USM6 State vector [C; Rf1; Rf2; σ1; σ2; σ3].

USM72USMEM(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts USM with quaternions to USM with exponential mapping.

Arguments

-u::AbstractVector{<:Number}: The Unified State Model vector [C; Rf1; Rf2; ϵO1; ϵO2; ϵO3; η0]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_USMEM::SVector{6, <:Number}: The USMEM State Vector [C; Rf1; Rf2; a1; a2; a3].

USM72koe(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Unified State Model elements into the Keplerian orbital element set. Van den Broeck, Michael. "An Approach to Generalizing Taylor Series Integration for Low-Thrust Trajectories." (2017). https://repository.tudelft.nl/islandora/object/uuid%3A2567c152-ab56-4323-bcfa-b076343664f9

Arguments

-u::AbstractVector{<:Number}: The Unified State Model vector [C; Rf1; Rf2; ϵO1; ϵO2; ϵO3; η0]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_koe:SVector{6, <:Number}: The Keplerian State vector [a; e; i; Ω(RAAN); ω(AOP); f(True Anomaly)].

USMEM2USM7(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts USM with exponential mapping to USM with quaternions.

Arguments

-u::AbstractVector{<:Number}: The USMEM vector [C; Rf1; Rf2; a1; a2; a3]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_USM::SVector{7, <:Number}: The Unified State Model vector [C; Rf1; Rf2; ϵO1; ϵO2; ϵO3; η0].

_solve_kepler_generalized(L, p₁, p₂; tol=1e-14, maxiter=200)

Solve the generalized Kepler equation (Eq. 25): L = K + p₁ cos(K) - p₂ sin(K)

Uses Newton-Raphson iteration.

angle_between_vectors(
    v1::AbstractVector{T1}, v2::AbstractVector{T2}
) where {T1<:Number,T2<:Number}

Computes the angle between two vectors in a more numerically stable way than dot product.

Arguments

-v1::AbstractVector{<:Number}: The first vector of the computation -v2::AbstractVector{<:Number}: The second vector of the computation

Returns

-angle::Number: The angle between the two vectors

angularMomentumQuantity(u::AbstractVector{<:Number})

Computes the instantaneous angular momentum.

Arguments

-u::AbstractVector{<:Number}: The Cartesian state vector [x; y; z; ẋ; ẏ; ż].

Returns

-angular_momentum::Number: Angular momentum of the body.

angularMomentumQuantity(X::AstroCoord, μ::Number)

Computes the instantaneous angular momentum.

Arguments

-X::AstroCoord: An coordinate set describing the orbit. -μ::Number: Standard graviational parameter of central body.

Returns

-angular_momentum::Number: Angular momentum of the body.

angularMomentumVector(u::AbstractVector{<:Number})

Computes the instantaneous angular momentum vector from a Cartesian state vector.

Arguments

-u::AbstractVector{<:Number}: The Cartesian state vector [x; y; z; ẋ; ẏ; ż].

Returns

-'angular_momentum::Vector{<:Number}': 3-Dimensional angular momentum vector.

angularMomentumVector(X::AstroCoord, μ::Number)

Computes the instantaneous angular momentum vector from a Cartesian state vector.

Arguments

-X::AstroCoord: An coordinate set describing the orbit. -μ::Number: Standard graviational parameter of central body.

Returns

-angular_momentum::Vector{<:Number}: 3-Dimensional angular momentum vector.

cart2EDromo(u, μ, ϕ::Number, config::RegularizedCoordinateConfig)

Converts a Cartesian state vector to an EDromo state vector.

This is the backend implementation for the transformation from Cartesian to EDromo. It requires the fictitious time ϕ and a RegularizedCoordinateConfig with necessary parameters.

Arguments

• u::AbstractVector: Cartesian state vector [x, y, z, ẋ, ẏ, ż] in [L] and [L/T].
• μ::Number: Gravitational parameter in [L³/T²].
• ϕ::Number: Fictitious time parameter.
• config::RegularizedCoordinateConfig: Configuration parameters for the transformation.

Returns

• SVector{8, RT}: The 8-element EDromo state vector.

function cart2J2EqOE(u::AbstractVector{<:Number}, μ::Number)

Computes the J2 Perturbed Equinoctial Orbit Elements from a Cartesian state vector.

Arguments

-u::AbstractVector{<:Number}: The Cartesian state vector [x; y; z; ẋ; ẏ; ż]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_J2EqOE::SVector{6, <:Number}: The J2 Perturbed Equinoctial Orbit Element vector [n; h; k; p; q; L].

cart2KS(u, μ, config::RegularizedCoordinateConfig)

Converts a Cartesian state vector to a Kustaanheimo-Stiefel (KS) state vector.

Arguments

• u::AbstractVector: Cartesian state [x, y, z, ẋ, ẏ, ż].
• μ::Number: Gravitational parameter.
• config::RegularizedCoordinateConfig: Configuration parameters (uses config.W for perturbing potential).

Returns

• SVector{10, RT}: The 10-element KS state vector.

cart2Mil(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Cartesian state vector into the Milankovich state vector.

Arguments

-u::AbstractVector{<:Number}: The Cartesian state vector [x; y; z; ẋ; ẏ; ż]. -μ::Number: Standard graviational parameter of central body.

Keyword Arguments

-equatorial_tol::Float64: The tolerance on what is considered an equatorial orbit (no inclination). [Default=1e-15] -circular_tol::Float64: The tolerance on what is considered a circular orbit (no eccentricity). [Default=1e-15]

Returns

-u_Mil::SVector{7, <:Number}: The Milankovich state vector [H; e; L].

cart2StiefelScheifele(cart_vec, μ, ϕ::Number, config::RegularizedCoordinateConfig)

Converts Cartesian coordinates to Stiefel-Scheifele coordinates.

Arguments

• cart_vec::AbstractVector: The Cartesian state vector.
• μ::Number: The gravitational parameter of the central body.
• ϕ::Number: Fictitious time parameter.
• config::RegularizedCoordinateConfig: Configuration parameters.

Returns

• StiefelScheifele: The Stiefel-Scheifele state vector.

cart2cylind(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Computes the cylindrical orbital elements from a Cartesian set.

Arguments

-u::AbstractVector{<:Number}: Cartesian state vector [x; y; z; ẋ; ẏ; ż]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_cylind::SVector{6, <:Number}`: The cylindrical orbital element vector [r; θ; z; ṙ; θdot; ż].

cart2delaunay(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Computes the Delaunay orbital elements from a Cartesian set. Laskar, Jacques. "Andoyer construction for Hill and Delaunay variables." Celestial Mechanics and Dynamical Astronomy 128.4 (2017): 475-482.

Arguments

-u::AbstractVector{<:Number}: The Cartesian orbital element vector [x; y; z; ẋ; ẏ; ż]. -μ::Number: Standard graviational parameter of central body.

Returns

-'u_cart::SVector{6, <:Number}': Delaunay Orbital Element Vector [L; G; H; M; ω; Ω]

cart2geqoe(u, μ, config::RegularizedCoordinateConfig)

Transform Cartesian coordinates to Generalized Equinoctial Orbital Elements (GEqOE).

The perturbing potential W is taken from the config struct. For a pure Keplerian orbit, set W = 0. For perturbed orbits, precompute the disturbing potential (Vtotal - Vkeplerian) externally and pass it via the config.

Reference: Baù, G., Hernando-Ayuso, J., & Bombardelli, C. (2021). "A generalization of the equinoctial orbital elements." Celestial Mechanics and Dynamical Astronomy, 133(9), 1-32.

Arguments

• u::AbstractVector{<:Number}: Cartesian state vector [x; y; z; ẋ; ẏ; ż]
• μ::Number: Gravitational parameter
• config::RegularizedCoordinateConfig: Configuration containing the perturbing potential W

Returns

• SVector{6}: GEqOE state vector [ν; p₁; p₂; L; q₁; q₂]

function cart2koe(
    u::AbstractVector{T}, μ::V; equatorial_tol::Float64=1E-15, circular_tol::Float64=1E-15
) where {T<:Number,V<:Number}

Computes the Keplerian orbital elements from a cartesian set.

Arguments

-u::AbstractVector{<:Number}: The Cartesian state vector [x; y; z; ẋ; ẏ; ż]. -μ::Number: Standard graviational parameter of central body.

Keyword Arguments

-equatorial_tol::Float64: The tolerance on what is considered an equatorial orbit (no inclination). [Default=1e-15] -circular_tol::Float64: The tolerance on what is considered a circular orbit (no eccentricity). [Default=1e-15]

Returns

-u_koe::SVector{6, <:Number}`: Keplerian orbital element vector [a; e; i; Ω(RAAN); ω(AOP); f(True Anomaly)].

cart2sphere(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Computes the spherical orbital elements from a spherical set.

Arguments

-u::AbstractVector{<:Number}: The Cartesian state vector [x; y; z; ẋ; ẏ; ż]. -μ::Number: Standard graviational parameter of central body.

Returns

-'u_sphere::SVector{6, <:Number}': Spherical Orbital Element Vector [r; θ; ϕ; ṙ; θdot; ϕdot]

compute_characteristic_scales(state, μ)

Computes characteristic distance and time scales for regularized coordinate transformations.

The scales are computed as:

• DU: Distance unit based on the position magnitude
• TU: Time unit derived from DU and μ using Kepler's third law: TU = sqrt(DU³/μ)

Arguments

• state::AbstractVector: Cartesian state vector [x, y, z, ẋ, ẏ, ż]
• μ::Number: Gravitational parameter

Returns

• (DU, TU): Tuple of characteristic distance and time scales

compute_initial_phi(state, μ, config)

Computes the initial value of the fictitious time ϕ for regularized coordinates (EDromo, Stiefel-Scheifele).

This calculation is based on Baù, G., Bombardelli, C., Peláez, J., and Lorenzini, E., "Nonsingular orbital elements for special perturbations in the two-body problem". MNRAS 454(3), pp. 2890-2908. 2015.

Arguments

• state::AbstractVector: Cartesian state vector [x, y, z, ẋ, ẏ, ż]
• μ::Number: Gravitational parameter
• config::RegularizedCoordinateConfig: Configuration struct containing DU, TU, and W

Returns

• Number: The computed value of ϕ₀

computed_a(geq::GEqOE, μ::Number)

Generalized semi-major axis: a = (μ/ν²)^(1/3)

computed_ρ(geq::GEqOE, μ::Number)

Generalized semi-latus rectum: ρ = a(1 - g²)

cylind2cart(u::AbstractVector{T}, μ::Number) where {T<:Number}

Computes the Cartesian orbital elements from a cylindrical set.

Arguments

-u::AbstractVector{<:Number}: The cylindrical state vector [r; θ; z; ṙ; θdot; ż]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_cart::SVector{6, <:Number}: The Cartesian orbital element vector [x; y; z; ẋ; ẏ; ż].

delaunay2cart(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Computes the Cartesian orbital elements from a Delaunay set. Laskar, Jacques. "Andoyer construction for Hill and Delaunay variables." Celestial Mechanics and Dynamical Astronomy 128.4 (2017): 475-482.

Argument

-u::AbstractVector{<:Number}: The Delaunay orbital element vector [L; G; H; M; ω; Ω]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_cart::SVector{6, <:Number}`: The cartesian orbital element vector [x; y; z; ẋ; ẏ; ż].

eccentricAnomaly2MeanAnomaly(E::Number, e::Number)

Converts the true anomaly into the mean anomaly.

Arguments

-E::Number: Eccentric anomaly of the orbit [radians]. -e::Number: Eccentricity of the orbit.

Returns

-'M::Number': Mean anomaly of the orbit [radians].

eccentricAnomaly2TrueAnomaly(E::Number, e::Number)

Converts the eccentric anomaly into the true anomaly.

Arguments

-E::Number: Eccentric anomaly of the orbit [radians]. -e::Number: Eccentricity of the orbit.

Returns

-f::Number: True anomaly of the orbit [radians].

geqoe2cart(u, μ, config::RegularizedCoordinateConfig)

Transform GEqOE to Cartesian coordinates.

The perturbing potential W is taken from the config struct. It must be the same value used in the forward transformation.

Arguments

• u::AbstractVector{<:Number}: GEqOE state vector [ν; p₁; p₂; L; q₁; q₂]
• μ::Number: Gravitational parameter
• config::RegularizedCoordinateConfig: Configuration containing the perturbing potential W

Returns

• SVector{6}: Cartesian state vector [x; y; z; ẋ; ẏ; ż]

get_EDromo_time(u, ϕ::Number, config::RegularizedCoordinateConfig)

Computes the physical time from the EDromo state vector.

Arguments

• u::AbstractVector: EDromo state vector [ζ₁, ζ₂, ζ₃, ζ₄, ζ₅, ζ₆, ζ₇, ζ₈].
• ϕ::Number: Fictitious time parameter.
• config::RegularizedCoordinateConfig: Configuration parameters for the transformation.

Returns

• Number: The computed physical time.

get_KS_time(u, config::RegularizedCoordinateConfig)

Computes the physical time from the KS state vector.

Arguments

• u::AbstractVector: KS state vector [u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, h, τ].
• config::RegularizedCoordinateConfig: Configuration parameters.

Returns

• Number: The computed physical time.

get_stiefelscheifele_time(u::AbstractVector{T}, ϕ::Number, config::RegularizedCoordinateConfig) where {T<:Number}

Computes the time element for Stiefel-Scheifele transformations.

Arguments

• u::AbstractVector{T}: The Stiefel-Scheifele state vector.
• ϕ::Number: Fictitious time parameter.
• config::RegularizedCoordinateConfig: Configuration parameters.

Returns

• The time element.

function koeM2IOE(u::AbstractVector{T}, μ::V) where {T<:Number, V<:Number}

Computes the Intermediate Orbit Elements from a Keplerian set.

Arguments

-u::AbstractVector{<:Number}: The Keplerian state vector [a; e; i; Ω(RAAN); ω(AOP); M(Mean Anomaly)]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_IOE::SVector{6, <:Number}: The Intermediate Orbit Element vector [I1; I2; I3; I4; I5; I6].

koe2ModEq(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Keplerian elements into the Modified Equinoctial elements.

Arguments

-u:AbstractVector{<:Number}: The Keplerian State vector [a; e; i; Ω(RAAN); ω(AOP); ν(True Anomaly)]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_ModEq::SVector{6, <:Number}: The Modified Equinoctial state vector [p; f; g; h; k; l].

koe2USM7(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Keplerian orbital elements into the Unified State Model set. Van den Broeck, Michael. "An Approach to Generalizing Taylor Series Integration for Low-Thrust Trajectories." (2017). https://repository.tudelft.nl/islandora/object/uuid%3A2567c152-ab56-4323-bcfa-b076343664f9

Arguments

-u:AbstractVector{<:Number}: The Keplerian state vector [a; e; i; Ω(RAAN); ω(AOP); f(True Anomaly)]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_USM::SVector{7, <:Number}: The Unified State Model vector [C; Rf1; Rf2; ϵO1; ϵO2; ϵO3; η0].

koe2cart(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Computes the Cartesian orbital elements from a Keplerian set.

Arguments

-u::AbstractVector{<:Number}: The Keplerian state vector [a; e; i; Ω(RAAN); ω(AOP); f(True Anomaly)]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_cart::Vector{6, <:Number}: The Keplerian orbital element vector [x; y; z; ẋ; ẏ; ż].

koe2poincare(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Keplerian orbital elements into Poincaré canonical variables. Poincaré elements use Cartesian-style canonical coordinate-momentum pairs derived from Delaunay variables, and are non-singular for circular (e → 0) and equatorial (i → 0) orbits.

Supports both elliptic (a > 0) and hyperbolic (a < 0) orbits, following the same convention as Delaunay: Λ = √(μ|a|). For hyperbolic orbits, the eccentricity action is defined as P = Λ + G (instead of Λ - G) to keep √(2P) real.

Arguments

-u::AbstractVector{<:Number}: The Keplerian state vector [a; e; i; Ω(RAAN); ω(AOP); ν(True Anomaly)]. -μ::Number: Standard gravitational parameter of central body.

Returns

-u_poincare::SVector{6, <:Number}: The Poincaré state vector [Λ; λ; ξ; η; p; q].

References

• Murray, C.D. and Dermott, S.F. "Solar System Dynamics." Cambridge University Press (1999).
• Laskar, J. and Robutel, P. "Stability of the Planetary Three-Body Problem." Celestial Mechanics and Dynamical Astronomy 62 (1995): 193-217.

meanAnomaly2EccentricAnomaly(M::T, e::Number; tol::Float64=10 * eps(T), max_iters::Int=100) where {T<:Number}

Converts the Mean Anomaly into the Eccentric Anomaly

Arguments

-M::Number: Mean Anomaly of the orbit [radians] -e::Number: Eccentricity of the orbit

Keyword Arguments

-tol::Float64: Convergence tolerance of Kepler solver. [Default=10*eps(T)] -max_iters::Int: Maximum number of iterations. [Default=100]

Returns

-E::Number: Eccentric Anomaly of the orbit [radians]

meanAnomaly2TrueAnomaly(M::T, e::Number; tol::Float64=10 * eps(T), max_iters::Int=100) where {T<:Number}

Converts the mean anomaly into the true anomaly.

Arguments

-M::Number: Mean anomaly of the orbit [radians]. -e::Number: Eccentricity of the orbit.

Keyword Arguments

-tol::Float64: Convergence tolerance of Kepler solver. [Default=10*eps(T)] -max_iters::Int: Maximum number of iterations. [Default=100]

Returns

-f::Number: Mean anomaly of the orbit [radians].

meanMotion(X::AstroCoord, μ::Number)

Computes the Keplerian mean motion about a central body.

Arguments

-X::AstroCoord: An coordinate set describing the orbit. -μ::Number: Standard graviational parameter of central body.

Returns

• n::Number: The orbital mean motion.

meanMotion(a::Number, μ::Number)

Computes the Keplerian mean motion about a central body.

Arguments

-a::Number: The semi-major axis of the orbit. -μ::Number: Standard graviational parameter of central body.

Returns

• n::Number: The orbital mean motion.

function modEq2IOE(u::AbstractVector{T}, μ::V) where {T<:Number, V<:Number}

Computes the Intermediate Orbit Elements from a Modified Equinoctial set.

Arguments

-u::AbstractVector{<:Number}: The Modified Equinoctial state vector [p; f; g; h; k; L]. -μ::Number: Standard graviational parameter of central body.

Returns

-u_IOE::SVector{6, <:Number}: The Intermediate Orbit Element vector [I1; I2; I3; I4; I5; I6].

orbitalNRG(X::AstroCoord, μ::Number)

Computes the keplerian orbital energy.

Arguments

-X::AstroCoord: An coordinate set describing the orbit. -μ::Number: Standard graviational parameter of central body.

Returns

-NRG::Number: The orbital energy.

orbitalNRG(a::Number, μ::Number)

Computes the keplerian orbital energy.

Arguments

-a::Number: The semi-major axis of the orbit. -μ::Number: Standard graviational parameter of central body.

Returns

-NRG::Number: The orbital energy.

orbitalPeriod(X::AstroCoord, μ::Number)

Computes the Keplerian orbital period about a central body.

Arguments

-X::AstroCoord: An coordinate set describing the orbit. -μ::Number: Standard graviational parameter of central body.

Returns

-T::Number: The orbital period.

orbitalPeriod(a::Number, μ::Number)

Computes the Keplerian orbital period about a central body.

Arguments

-a::Number: The semi-major axis of the orbit. -μ::Number: Standard graviational parameter of central body.

Returns

-T::Number: The orbital period.

poincare2koe(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Converts Poincaré canonical variables into Keplerian orbital elements.

Supports both elliptic and hyperbolic orbits. The orbit type is determined by comparing the eccentricity action P = (ξ² + η²)/2 against Λ:

• P ≤ Λ → elliptic (a > 0)
• P > Λ → hyperbolic (a < 0)

Arguments

-u::AbstractVector{<:Number}: The Poincaré state vector [Λ; λ; ξ; η; p; q]. -μ::Number: Standard gravitational parameter of central body.

Returns

-u_koe::SVector{6, <:Number}: The Keplerian state vector [a; e; i; Ω(RAAN); ω(AOP); ν(True Anomaly)].

References

• Murray, C.D. and Dermott, S.F. "Solar System Dynamics." Cambridge University Press (1999).
• Laskar, J. and Robutel, P. "Stability of the Planetary Three-Body Problem." Celestial Mechanics and Dynamical Astronomy 62 (1995): 193-217.

sphere2cart(u::AbstractVector{T}, μ::V) where {T<:Number,V<:Number}

Computes the Cartesian orbital elements from a spherical set.

Arguments

-u::AbstractVector{<:Number}: The spherical orbital element vector [r; θ; ϕ; ṙ; θdot; ϕdot]. -μ::Number: Standard graviational parameter of central body.

Returns

-'u_cart::SVector{6, <:Number}': The Cartesian orbital element vector [x; y; z; ẋ; ẏ; ż].

trueAnomaly2EccentricAnomaly(f::Number, e::Number)

Converts the true anomaly into the mean anomaly.

Arguments

-f::Number: True anomaly of the orbit [radians]. -e::Number: Eccentricity of the orbit.

Returns

-E::Number: Eccentric anomaly of the orbit [radians].

trueAnomaly2MeanAnomaly(f::Number, e::Number)

Converts the true anomaly into the mean anomaly.

Arguments

-f::Number: True anomaly of the orbit [radians]. -e::Number: Eccentricity of the orbit.

Returns

-M::Number: Mean anomaly of the orbit [radians].

Computed properties for GEqOE coordinate set.

Additional Properties

• a: Generalized semi-major axis: a = (μ/ν²)^(1/3)
• g: Generalized eccentricity magnitude: g = √(p₁² + p₂²)
• Ψ: Generalized longitude of pericenter: Ψ = atan(p₁, p₂)

inv(trans::Transformation)

Returns the inverse (or reverse) of the transformation trans