Periodic Orbit Computation

GeneralPeriodicOrbit

A general representation of a periodic orbit.

Fields

• x0::SVector{6, Float64}: The initial conditions for the periodic orbit.
• P::Float64: The period of the periodic orbit.
• S::Float64: The arc length along the periodic orbit.
• mu::Float64: The gravitational parameter of the CRTBP.

TypeAPeriodicOrbit(
    rx, rz, vy, mu[, TU, DU];
    N_cross = 1,
    constraint = (:x_start_coordinate, 0.8),
    ode_solver = Vern9(),
    ode_reltol = 1e-12,
    ode_abstol = 1e-12,
    nl_solver  = SimpleTrustRegion(autodiff = nothing, nlsolve_update_rule = Val(true)),
)

TypeAPeriodicOrbit

A representation of a CRTBP periodic orbit of Type A.

Fields

• u0::SVector{3, Float64}: The initial conditions for the periodic orbit.
• P::Float64: The period of the periodic orbit.
• S::Float64: The arc length along the periodic orbit.
• N_cross::Int: The number of crossings through x-z plane in half period.
• mu::Float64: The gravitational parameter of the CRTBP.

TypeAPeriodicOrbit(
    orbit::TypeAPeriodicOrbit;
    constraint = (:x_start_coordinate, 0.8),
    ode_solver = Vern9(),
    ode_reltol = 1e-12,
    ode_abstol = 1e-12,
    nl_solver  = SimpleTrustRegion(autodiff = nothing, nlsolve_update_rule = Val(true)),
)

generate_periodic_orbit_cheb_approximation(
    orbit::AbstractPeriodicOrbit, τ1_npoints::Int, τ1_order::Int, PT::ParameterizationType;
    solver = Vern9(),
    reltol = 1e-14,
    abstol = 1e-14,
) -> FastChebInterpolation{FastChebInterp.ChebPoly{1, SVector{6, Float64}, Float64}}

Generate a Chebyshev approximation of the periodic orbit in terms of PT employing τ1_npoints fit to a Chebyshev polynomial of order τ1_order.

Arguments

• orbit::AbstractPeriodicOrbit: The periodic orbit.
• τ1_npoints::Int: The number of points to fit the Chebyshev polynomial.
• τ1_order::Int: The order of the Chebyshev polynomial.
• PT::ParameterizationType: The type of parameterization to use (Time or ArcLength).

Keyword Arguments

• solver::OrdinaryDiffEq.AbstractODESolver: The ODE solver to use for the integration.
• reltol::Real: The relative tolerance for the ODE solver.
• abstol::Real: The absolute tolerance for the ODE solver.

Returns

• FastChebInterpolation{FastChebInterp.ChebPoly{1, SVector{6, Float64}, Float64}}: The Chebyshev approximation.

generate_periodic_orbit_cheb_interpolation(
    orbit::AbstractPeriodicOrbit, τ1_order::Int, PT::ParameterizationType;
    solver = Vern9(),
    reltol = 1e-14,
    abstol = 1e-14,
) -> FastChebInterpolation{FastChebInterp.ChebPoly{1, SVector{6, Float64}, Float64}}

Generate a Chebyshev interpolation of the periodic orbit in terms of PT employing a Chebyshev polynomial of order τ1_order.

Arguments

• orbit::AbstractPeriodicOrbit: The periodic orbit.
• τ1_order::Int: The order of the Chebyshev polynomial.
• PT::ParameterizationType: The type of parameterization to use (Time or ArcLength).

Keyword Arguments

• solver::OrdinaryDiffEq.AbstractODESolver: The ODE solver to use for the integration.
• reltol::Real: The relative tolerance for the ODE solver.
• abstol::Real: The absolute tolerance for the ODE solver.

Returns

• FastChebInterpolation{FastChebInterp.ChebPoly{1, SVector{6, Float64}, Float64}}: The Chebyshev interpolation.

get_full_orbit(orbit::AbstractPeriodicOrbit)

Get the full orbit of the periodic orbit.

Arguments

• orbit::AbstractPeriodicOrbit: The periodic orbit.

correct_typeA_initial_conditions(
    rx_guess, rz_guess, vy_guess, mu;
    N_cross     = 1,
    constraint  = (:x_start_coordinate, 0.0),
    ode_solver  = Vern9(),
    ode_reltol  = 1e-14,
    ode_abstol  = 1e-14,
    nl_solver   = SimpleTrustRegion(autodiff = nothing, nlsolve_update_rule = Val(true)),
)

Correct the initial conditions of a type-A periodic orbit using a forward shooting method.

Arguments

• rx_guess::Real: Initial guess for the x-coordinate of the initial state.
• rz_guess::Real: Initial guess for the z-coordinate of the initial state.
• vy_guess::Real: Initial guess for the y-velocity of the initial state.
• mu::Real: Gravitational parameter of the CRTBP.

Keyword Arguments

• N_cross::Int = 1: Number of crossings of the xz-plane to consider in the half period.
• constraint::Tuple{Symbol,Real}: Tuple of the constraint type and value corresponding to the final constraint considered to provide a fully defined system of nonlinear eqautions. The constraint type can be either :x_start_coordinate or :jacobi_integral. The constraint value is the value that the constraint should be equal to for the desired periodic orbit.
• ode_solver::OrdinaryDiffEq.AbstractODESolver: The ODE solver to use for the integration.
• ode_reltol::Real: The relative tolerance for the ODE solver.
• ode_abstol::Real: The absolute tolerance for the ODE solver.
• nl_solver::NonlinearSolve.AbstractNLsolveSolver: The nonlinear solver to use for the root-finding problem. Note, there seems to be a strange bug with the NonlinearSolve.jl TrustRegion and NewtonRaphson solvers when using out-of-place Jacobian defined to return an SMatrix. Therefore, it is recommened to use either the default SimpleTrustRegion solver or the SimpleNewtonRaphson solver (which are specifically tailored towards StaticArrays.jl arrays).