Bézier Collocation for Cislunar Trajectory Optimization

Course project for AAE 568 — Applied Optimal Control and Estimation, Purdue University, Spring 2026. Authors: Mustakeen Bari, Zhuorui Li, Advait Jawaji.

We implement and compare two numerical methods for solving the minimum-energy optimal-control problem on three progressively more challenging dynamical regimes:

1. Phase 0 — Earth-to-Mars two-body (planar Keplerian, electric-propulsion analog)
2. Phase 1 — Planar Earth-Moon CR3BP L1↔L2 (low-thrust cislunar)
3. Phase 2 — Artemis II Post-TLI / Full-mission ephemeris (chemical, real flight data)

The two methods compared at each phase are:

• Indirect single-shooting of the Pontryagin two-point BVP (scipy.integrate + scipy.optimize.fsolve).
• Direct Bézier-curve collocation transcribed into an NLP and solved with CasADi/IPOPT (interior-point) or SLSQP (segmented).

A 3D LEO-to-NRHO framework (ThreeD/) is in place but not exercised end-to-end.

The companion final report (PDF, ≤12 pp) lives outside this repository.

Repository layout

common/               schema, IO, timing utilities used by all phases
Earth-Mars/           Phase 0 solvers, runs, figures
Planer/               Phase 1 solvers, runs, figures
Artemis2/             Phase 2 solvers, runs, figures (Ephem_Full, Post-TLI)
ThreeD/               3D framework + LEO-to-NRHO problem setup
results_summary.json  consolidated solver records (138 entries)
results_table.md/.tex tables generated from the JSON
make_results_table.py regenerates the tables
run_constraint_remediation_cases.py
                      generates bounded-control variants for P0/P1
BOUNDED_RESULTS.md    constrained-formulation findings (umax bounds, saturated-PMP law)

Install

git clone https://github.com/z-r-li/collocation.git
cd collocation
python -m venv .venv && source .venv/bin/activate
pip install -r requirements.txt

CasADi pulls in IPOPT and the MUMPS sparse solver via a wheel; no separate system install is needed on macOS or Linux.

Reproducing the headline results

Each phase has a run_phaseN.py that runs both methods, appends records to results_summary.json, and writes figures alongside the script. The runs are pure instrumentation over the underlying solvers — no retuning, no random seeds beyond what the scripts pass explicitly.

# Phase 0: Earth-Mars two-body
python Earth-Mars/run_phase0.py

# Phase 1: planar CR3BP L1↔L2 (Lyapunov-to-Lyapunov)
python Planer/run_phase1.py

# Phase 2: Artemis II
python Artemis2/run_phase2.py

# Bounded-control (umax) variants for P0 and P1
python run_constraint_remediation_cases.py

# Refresh the consolidated tables
python make_results_table.py

Auxiliary sweep / mesh-refinement scripts (run_phaseN_nsweep*.py, run_phaseN_psweep.py) regenerate the figures in Earth-Mars/phase0_nsweep_figures/ and Planer/.

Headline findings

• On the two-body Phase 0 problem, both methods converge to the same cost (J ≈ 0.0149) to machine tolerance. Shooting is faster (sub-second) but collocation matches its accuracy with substantially less initial-guess sensitivity.
• On the planar CR3BP Phase 1 problem, indirect shooting requires near-exact initial costates and frequently diverges in the strongly hyperbolic cislunar regime; direct Bézier collocation reliably converges from a straight-line state-space initial guess.
• The constrained variants (P0 with u_max = 1e-5 km/s², P1 with u_max = 8e-7 km/s²) demonstrate the saturated-PMP law u* = sat(-½ R⁻¹ Bᵀλ). P0's bound is non-binding (cost matches unconstrained); P1's bound binds and produces visible saturation arcs. Detail in BOUNDED_RESULTS.md.

External data

• JPL 9:2 L₂ southern NRHO: initial conditions taken from the JPL Periodic Orbits Database; hard-coded in ThreeD/leo_to_nrho_cr3bp.py.
• Artemis II flight ephemeris (Post-ICPS-Sep through EI): Orion Mission Manager OEM file, cited and consumed by Artemis2/Ephem_Full/artemis2_ephemeris.py. The .asc file is not redistributed in this repo; obtain it from NASA's public Artemis II reference trajectory release.

Contributing / contact

This is a one-shot academic submission, not an actively maintained library. Issues and PRs are welcome but will be triaged best-effort.

License

MIT.