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Minimum time orbit transfer

Introduction

Minimum time control of the Kepler equation (CNES / TAS / Inria / CNRS collaboration, check [1] and [2]):

\[\begin{cases} t_f \to \min,\\ \ddot{q} = -\mu\frac{q}{|q|^3} + \frac{u}{m}\,,\quad t \in [0,t_f],\\ \dot{m} = -\beta|u|,\quad |u| \leq T_{\mathrm{max}}. \end{cases}\]

Fixed initial and final Keplerian orbits (free final longitude).

using OptimalControl
using NLPModelsIpopt
using OrdinaryDiffEq
using Plots
using MINPACK
using ForwardDiff
using LinearAlgebra

Problem definition

Tmax = 60                                  # Maximum thrust in Newtons
cTmax = 3600^2 / 1e6; T = Tmax * cTmax     # Conversion from Newtons to kg x Mm / h²
mass0 = 1500                               # Initial mass of the spacecraft
β = 1.42e-02                               # Engine specific impulsion
μ = 5165.8620912                           # Earth gravitation constant
P0 = 11.625                                # Initial semilatus rectum
ex0, ey0 = 0.75, 0                         # Initial eccentricity
hx0, hy0 = 6.12e-2, 0                      # Initial ascending node and inclination
L0 = π                                     # Initial longitude
Pf = 42.165                                # Final semilatus rectum
exf, eyf = 0, 0                            # Final eccentricity
hxf, hyf = 0, 0                            # Final ascending node and inclination

asqrt(x; ε=1e-9) = sqrt(sqrt(x^2+ε^2))     # Avoid issues with AD

function F0(x)
    P, ex, ey, hx, hy, L = x
    pdm = asqrt(P / μ)
    cl = cos(L)
    sl = sin(L)
    w = 1 + ex * cl + ey * sl
    F = zeros(eltype(x), 6) # Use eltype to allow overloading for AD
    F[6] = w^2 / (P * pdm)
    return F
end

function F1(x)
    P, ex, ey, hx, hy, L = x
    pdm = asqrt(P / μ)
    cl = cos(L)
    sl = sin(L)
    F = zeros(eltype(x), 6)
    F[2] = pdm *   sl
    F[3] = pdm * (-cl)
    return F
end

function F2(x)
    P, ex, ey, hx, hy, L = x
    pdm = asqrt(P / μ)
    cl = cos(L)
    sl = sin(L)
    w = 1 + ex * cl + ey * sl
    F = zeros(eltype(x), 6)
    F[1] = pdm * 2 * P / w
    F[2] = pdm * (cl + (ex + cl) / w)
    F[3] = pdm * (sl + (ey + sl) / w)
    return F
end

function F3(x)
    P, ex, ey, hx, hy, L = x
    pdm = asqrt(P / μ)
    cl = cos(L)
    sl = sin(L)
    w = 1 + ex * cl + ey * sl
    pdmw = pdm / w
    zz = hx * sl - hy * cl
    uh = (1 + hx^2 + hy^2) / 2
    F = zeros(eltype(x), 6)
    F[2] = pdmw * (-zz * ey)
    F[3] = pdmw *   zz * ex
    F[4] = pdmw *   uh * cl
    F[5] = pdmw *   uh * sl
    F[6] = pdmw *   zz
    return F
end

Direct solve

tf = 15                                      # Estimation of final time
Lf = 3π                                      # Estimation of final longitude
x0 = [P0, ex0, ey0, hx0, hy0, L0]            # Initial state
xf = [Pf, exf, eyf, hxf, hyf, Lf]            # Final state
x(t) = x0 + (xf - x0) * t / tf               # Linear interpolation
u = [0.1, 0.5, 0.]                           # Initial guess for the control
nlp_init = (state=x, control=u, variable=tf) # Initial guess for the NLP

ocp = @def begin
    tf ∈ R, variable
    t ∈ [0, tf], time
    x = (P, ex, ey, hx, hy, L) ∈ R⁶, state
    u ∈ R³, control
    x(0) == x0
    x[1:5](tf) == xf[1:5]
    mass = mass0 - β * T * t
    ẋ(t) == F0(x(t)) + T / mass * (u₁(t) * F1(x(t)) + u₂(t) * F2(x(t)) + u₃(t) * F3(x(t)))
    u₁(t)^2 + u₂(t)^2 + u₃(t)^2 ≤ 1
    tf → min
end
Abstract definition:

    tf ∈ R, variable
    t ∈ [0, tf], time
    x = ((P, ex, ey, hx, hy, L) ∈ R⁶, state)
    u ∈ R³, control
    x(0) == x0
    (x[1:5])(tf) == xf[1:5]
    mass = mass0 - β * T * t
    ẋ(t) == F0(x(t)) + (T / mass) * (u₁(t) * F1(x(t)) + u₂(t) * F2(x(t)) + u₃(t) * F3(x(t)))
    u₁(t) ^ 2 + u₂(t) ^ 2 + u₃(t) ^ 2 ≤ 1
    tf → min

The (non autonomous) optimal control problem is of the form:

    minimize  J(x, u, tf) = g(x(0), x(tf), tf)

    subject to

        ẋ(t) = f(t, x(t), u(t), tf), t in [0, tf] a.e.,

        ψ₋ ≤ ψ(t, x(t), u(t), tf) ≤ ψ₊, 
        ϕ₋ ≤ ϕ(x(0), x(tf), tf) ≤ ϕ₊, 

    where x(t) = (P(t), ex(t), ey(t), hx(t), hy(t), L(t)) ∈ R⁶, u(t) ∈ R³ and tf ∈ R.

nlp_sol = OptimalControl.solve(ocp; init=nlp_init, grid_size=100)
▫ This is OptimalControl version v1.1.3 running with: direct, adnlp, ipopt.

▫ The optimal control problem is solved with CTDirect version v0.17.3.

   ┌─ The NLP is modelled with ADNLPModels and solved with NLPModelsIpopt.
   │
   ├─ Number of time steps⋅: 100
   └─ Discretisation scheme: midpoint

▫ This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.1.

Number of nonzeros in equality constraint Jacobian...:     9211
Number of nonzeros in inequality constraint Jacobian.:      303
Number of nonzeros in Lagrangian Hessian.............:     9027

Total number of variables............................:      907
                     variables with only lower bounds:        0
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        0
Total number of equality constraints.................:      611
Total number of inequality constraints...............:      101
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:      101

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.5000000e+01 2.14e-01 2.75e-01   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  1.5630325e+01 1.74e-01 6.57e-01  -6.1 7.40e+00    -  5.82e-01 1.87e-01h  3
   2  1.6098934e+01 1.52e-01 1.17e+00  -6.4 3.75e+00   0.0 4.78e-01 1.25e-01h  4
   3  1.9047770e+01 6.23e-02 2.76e+00  -1.1 4.01e+00  -0.5 7.61e-01 5.98e-01H  1
   4  2.0500845e+01 2.93e-02 2.04e+00  -1.7 1.63e+00  -0.1 9.99e-01 8.93e-01h  1
   5  2.0655039e+01 1.41e-04 3.32e-01  -3.4 1.54e-01   0.4 9.98e-01 1.00e+00h  1
   6  2.0463684e+01 3.13e-04 1.53e-01  -3.6 1.91e-01  -0.1 1.00e+00 1.00e+00h  1
   7  1.9885423e+01 2.40e-03 1.55e-01  -4.5 5.79e-01  -0.6 1.00e+00 9.99e-01f  1
   8  1.9329009e+01 5.94e-03 1.23e-01  -2.7 1.05e+00  -1.1 1.00e+00 5.28e-01f  1
   9  1.8485264e+01 5.41e-02 7.87e-02  -2.4 1.43e+00  -1.5 1.00e+00 5.92e-01f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  1.7319517e+01 2.39e+00 2.16e-01  -2.2 3.16e+00  -2.0 1.00e+00 6.78e-01f  1
  11  1.5566289e+01 8.34e+00 3.67e-01  -2.3 2.05e+01  -2.5 7.86e-02 1.71e-01h  1
  12  1.3881805e+01 9.47e+01 5.74e-01  -2.3 1.16e+01  -2.1 3.94e-01 7.14e-01h  1
  13  1.4987784e+01 2.26e+01 3.72e-01  -1.9 3.88e+00  -0.7 7.95e-01 1.00e+00h  1
  14  1.5395019e+01 4.56e+00 4.52e-01  -1.7 1.80e+00  -0.3 1.00e+00 1.00e+00h  1
  15  1.5327104e+01 3.85e+00 2.90e-01  -1.9 4.30e+00    -  1.00e+00 3.84e-01h  1
  16  1.5671211e+01 6.37e-01 3.15e-01  -1.7 3.49e+00    -  1.00e+00 9.99e-01h  1
  17  1.5808227e+01 3.19e-03 5.77e-02  -1.7 7.93e-01    -  1.00e+00 1.00e+00h  1
  18  1.5206980e+01 1.62e-02 4.61e-02  -7.7 6.01e-01    -  1.00e+00 1.00e+00h  1
  19  1.4783741e+01 3.15e-01 9.44e-03  -3.1 5.59e-01    -  1.00e+00 9.87e-01h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  20  1.4796879e+01 2.26e-02 3.61e-04  -4.7 2.00e-01    -  1.00e+00 1.00e+00h  1
  21  1.4796618e+01 6.50e-05 8.32e-06  -5.7 1.94e-02    -  1.00e+00 1.00e+00h  1
  22  1.4796431e+01 2.54e-08 3.90e-09 -11.0 2.54e-04    -  1.00e+00 1.00e+00h  1
  23  1.4796431e+01 1.42e-14 6.38e-15 -11.0 1.29e-07    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 23

                                   (scaled)                 (unscaled)
Objective...............:   1.4796431325861663e+01    1.4796431325861663e+01
Dual infeasibility......:   6.3837823915946501e-15    6.3837823915946501e-15
Constraint violation....:   1.4210854715202004e-14    1.4210854715202004e-14
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   1.0000007287831476e-11    1.0000007287831476e-11
Overall NLP error.......:   1.0000007287831476e-11    1.0000007287831476e-11


Number of objective function evaluations             = 35
Number of objective gradient evaluations             = 24
Number of equality constraint evaluations            = 35
Number of inequality constraint evaluations          = 35
Number of equality constraint Jacobian evaluations   = 24
Number of inequality constraint Jacobian evaluations = 24
Number of Lagrangian Hessian evaluations             = 23
Total seconds in IPOPT                               = 5.182

EXIT: Optimal Solution Found.
tf = variable(nlp_sol)
p  = costate(nlp_sol)
p0 = p(0)
plot(nlp_sol)
Example block output

Shooting (1/2), Tmax = 60 Newtons

function ur(t, x, p, tf) # Regular maximising control
    H1 = p' * F1(x)
    H2 = p' * F2(x)
    H3 = p' * F3(x)
    u = [H1, H2, H3]
    u = u / sqrt(u[1]^2 + u[2]^2 + u[3]^2)
    return u
end

fr = Flow(ocp, ur) # Regular flow (first version)

function shoot(ξ::Vector{T}) where T
    tf, p0 = ξ[1], ξ[2:end]
    xf_, pf = fr(0, x0, p0, tf)
    s = zeros(T, 7)
    s[1:5] = xf_[1:5] - xf[1:5]
    s[6] = pf[6]
    s[7] = p0[1]^2 + p0[2]^2 + p0[3]^2 + p0[4]^2 + p0[5]^2 + p0[6]^2 - 1
    return s
end

p0 = p0 / norm(p0) # Normalization |p0|=1 for free final time
ξ = [tf; p0]; # Initial guess
jshoot(ξ) = ForwardDiff.jacobian(shoot, ξ)
shoot!(s, ξ) = (s[:] = shoot(ξ); nothing)
jshoot!(js, ξ) = (js[:] = jshoot(ξ); nothing)
bvp_sol = fsolve(shoot!, jshoot!, ξ; show_trace=true); println(bvp_sol)
Iter     f(x) inf-norm    Step 2-norm      Step time
------   --------------   --------------   --------------
     1     1.416162e+00     0.000000e+00         8.403236
     2     7.695113e-02     5.736564e-05        36.756818
     3     1.727077e-02     1.711084e-05         0.019976
     4     3.809695e-03     5.434911e-07         0.008486
     5     9.728789e-05     9.728304e-09         0.013757
     6     5.922661e-05     1.085583e-10         0.011403
     7     5.786130e-06     3.601035e-12         0.011718
     8     3.130270e-08     4.488488e-14         0.011902
     9     2.178950e-10     1.382101e-18         0.011969
Results of Nonlinear Solver Algorithm
 * Algorithm: Modified Powell (User Jac, Expert)
 * Starting Point: [14.796431325861663, -0.016170953751230413, -0.9051271580757697, -0.3306423479361639, -0.09646507427591802, 0.035163913890055465, 0.24620510621144945]
 * Zero: [14.80036435464976, -0.01618971492139077, -0.9084889547970921, -0.3252598589824457, -0.09447217089084237, 0.034322962839314686, 0.24184432891076493]
 * Inf-norm of residuals: 0.000000
 * Convergence: true
 * Message: algorithm estimates that the relative error between x and the solution is at most tol
 * Total time: 45.249358 seconds
 * Function Calls: 9
 * Jacobian Calls (df/dx): 1

Shooting (2/2), Tmax = 0.7 Newtons

hr = (t, x, p) -> begin # Regular maximised Hamiltonian (more efficient)
    H0 = p' * F0(x)
    H1 = p' * F1(x)
    H2 = p' * F2(x)
    H3 = p' * F3(x)
    mass = mass0 - β*T*t
    h = H0 + T / mass * sqrt(H1^2 + H2^2 + H3^2)
    return h
end

hr = Hamiltonian(hr; autonomous=false)
fr = Flow(hr) # Regular flow (again)

Tmax = 0.7                                 # Maximum thrust (Newtons)
cTmax = 3600^2 / 1e6; T = Tmax * cTmax     # Conversion from Newtons to kg x Mm / h²
tf = 1.210e3; p0 =-[-2.215319700438820e+01, -4.347109477345140e+01, 9.613188807286992e-01, 3.181800985503019e+02, -2.307236094862410e+00, -5.797863110671591e-01] # Tmax = 0.7 Newtons
p0 = p0 / norm(p0) # Normalization |p0|=1 for free final time
ξ = [tf; p0]; # Initial guess

bvp_sol = fsolve(shoot!, jshoot!, ξ; show_trace=true); println(bvp_sol)
Iter     f(x) inf-norm    Step 2-norm      Step time
------   --------------   --------------   --------------
     1     1.073121e+00     0.000000e+00         5.280209
     2     1.431634e-01     1.222058e+01        20.439927
     3     1.090164e-01     1.104763e+01         0.490706
     4     1.848524e-02     2.972697e+00         0.496979
     5     1.996556e-02     4.469494e-01         0.559106
     6     8.609313e-03     2.182014e-02         0.877174
     7     2.826553e-03     2.618818e-03         0.580057
     8     1.241996e-03     7.992737e-05         0.560083
     9     1.556115e-03     6.608209e-05         0.470845
    10     7.600936e-05     2.812075e-05         0.484288
    11     4.207318e-05     1.773916e-07         0.464814
    12     7.677904e-06     3.852982e-07         0.462949
    13     5.384794e-07     7.246546e-09         0.460094
    14     5.893237e-08     4.735934e-11         0.915811
    15     1.937032e-09     4.666954e-13         0.549823
Results of Nonlinear Solver Algorithm
 * Algorithm: Modified Powell (User Jac, Expert)
 * Starting Point: [1210.0, 0.06881811251625697, 0.13504139789537645, -0.0029863026489984875, -0.9884150724670144, 0.007167346236588738, 0.001801085395597836]
 * Zero: [1214.5922191689763, 0.06933409024376648, 0.5161924136822915, -0.00023166163856008998, -0.8536504421932478, 0.004363236382468494, -9.328384202033245e-5]
 * Inf-norm of residuals: 0.000000
 * Convergence: true
 * Message: algorithm estimates that the relative error between x and the solution is at most tol
 * Total time: 33.092993 seconds
 * Function Calls: 15
 * Jacobian Calls (df/dx): 1

Plots

tf = bvp_sol.x[1]
p0 = bvp_sol.x[2:end]
ode_sol = fr((0, tf), x0, p0)
t  = ode_sol.t; N = size(t, 1)
P  = ode_sol[1, :]
ex = ode_sol[2, :]
ey = ode_sol[3, :]
hx = ode_sol[4, :]
hy = ode_sol[5, :]
L  = ode_sol[6, :]
cL = cos.(L)
sL = sin.(L)
w  = @. 1 + ex * cL + ey * sL
Z  = @. hx * sL - hy * cL
C  = @. 1 + hx^2 + hy^2
q1 = @. P *((1 + hx^2 - hy^2) * cL + 2 * hx * hy * sL) / (C * w)
q2 = @. P *((1 - hx^2 + hy^2) * sL + 2 * hx * hy * cL) / (C * w)
q3 = @. 2 * P * Z / (C * w)

plt1 = plot3d(1; xlim = (-60, 60), ylim = (-60, 60), zlim = (-5, 5), title = "Orbit transfer", legend=false)
@gif for i = 1:N
    push!(plt1, q1[i], q2[i], q3[i])
end every N ÷ min(N, 100)
Example block output

Reproducibility

The documentation of this package was built using these direct dependencies,
Status `~/work/Kepler.jl/Kepler.jl/docs/Project.toml`
  [e30172f5] Documenter v1.14.1
  [f6369f11] ForwardDiff v1.2.1
  [459d104a] Kepler v0.2.3 `~/work/Kepler.jl/Kepler.jl`
  [4854310b] MINPACK v1.3.0
⌅ [f4238b75] NLPModelsIpopt v0.10.4
  [5f98b655] OptimalControl v1.1.3
  [1dea7af3] OrdinaryDiffEq v6.102.1
  [91a5bcdd] Plots v1.41.1
  [37e2e46d] LinearAlgebra v1.11.0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`
and using this machine and Julia version.
Julia Version 1.11.7
Commit f2b3dbda30a (2025-09-08 12:10 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
A more complete overview of all dependencies and their versions is also provided.
Status `~/work/Kepler.jl/Kepler.jl/docs/Manifest.toml`
  [54578032] ADNLPModels v0.8.13
  [47edcb42] ADTypes v1.18.0
  [a4c015fc] ANSIColoredPrinters v0.0.1
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.42
  [79e6a3ab] Adapt v4.4.0
  [66dad0bd] AliasTables v1.1.3
  [4fba245c] ArrayInterface v7.20.0
  [4c555306] ArrayLayouts v1.11.2
  [13072b0f] AxisAlgorithms v1.1.0
  [d1d4a3ce] BitFlags v0.1.9
  [62783981] BitTwiddlingConvenienceFunctions v0.1.6
  [70df07ce] BracketingNonlinearSolve v1.4.0
  [2a0fbf3d] CPUSummary v0.2.7
  [54762871] CTBase v0.16.4
  [790bbbee] CTDirect v0.17.3
  [1c39547c] CTFlows v0.8.8
  [34c4fa32] CTModels v0.6.8
  [32681960] CTParser v0.7.0
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  [5ae59095] Colors v0.13.1
  [38540f10] CommonSolve v0.2.4
  [bbf7d656] CommonSubexpressions v0.3.1
  [f70d9fcc] CommonWorldInvalidations v1.0.0
  [34da2185] Compat v4.18.0
  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.3
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  [187b0558] ConstructionBase v1.6.0
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  [8bb1440f] DelimitedFiles v1.9.1
  [2b5f629d] DiffEqBase v6.190.2
  [163ba53b] DiffResults v1.1.0
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  [e30172f5] Documenter v1.14.1
  [4e289a0a] EnumX v1.0.5
  [f151be2c] EnzymeCore v0.8.14
⌃ [1037b233] ExaModels v0.9.1
  [460bff9d] ExceptionUnwrapping v0.1.11
  [d4d017d3] ExponentialUtilities v1.27.0
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  [f6369f11] ForwardDiff v1.2.1
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  [28b8d3ca] GR v0.73.17
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.

References

  • 1Bonnard, B.; Caillau, J.-B.; Trélat, E. Geometric optimal control of elliptic Keplerian orbits. Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 4, 929-956.
  • 2Caillau, J.-B.; Gergaud, J.; Noailles, J. 3D Geosynchronous Transfer of a Satellite: continuation on the Thrust. J. Optim. Theory Appl. 118 (2003), no. 3, 541-565.