ΑETHERION — Flight Dynamics Library

AETHERION is a high-precision, research-grade C++ library for rocket and aerospace flight dynamics, built on:

• Featherstone 6-D Spatial Vector Algebra (SVA)
• ECI / WGS84 frame transformations
• Algorithmic Differentiation (AD-friendly)
• Clean ODE formulations for high-fidelity simulation
• Lie-group state estimation (invariant EKF on SE₂(3))

The name ΑΙΘΕΡΙΟΝ (Aetherion) is inspired by the Greek word Αιθήρ, referring to the upper atmosphere — the pristine realm of celestial motion.

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Two-Stage Rocket to Orbit — NASA Validation

The flagship simulation is a full 6-DOF two-stage rocket gravity-turn ascent from the equator, validated against NASA TM-2015-218675 Atmospheric Scenario 17. The trajectory below was produced entirely by Aetherion using Featherstone SVA dynamics, the US1976 atmosphere, WGS84 gravity, and an implicit RADAU-IIA integrator.

Through the first-stage burn and the coast, altitude and speed stay within 0.2 % of the NASA reference runs (Δt = 1 ms). After the second-stage burn the two NASA reference implementations themselves differ by 17 km in altitude and 10° in pitch; Aetherion ends between them: 248 km against 251 km (Sim 04) and 234 km (Sim 06), 8 363 m/s against 8 336 and 8 381 m/s.

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Features

Dynamics Core

• Full 6-DOF rigid-body dynamics using Featherstone Spatial Vector Algebra
• ECI equations of motion — no pseudo-forces, no approximations
• Air-relative velocity for aerodynamics; rotating atmosphere handled correctly
• Variable-mass bodies — propellant burn, staging events, CG travel

Numerical Methods

• Lie-structure-preserving Runge-Kutta-Munthe-Kaas (RKMK) integrators on SE(3) and product SE(3) × ℝⁿ manifolds
• Explicit RK4 and fully implicit RADAU-IIA schemes (Newton-solved stages) for stiff ascent phases
• Two-stage Newton-Raphson trim solver for straight-and-level fixed-wing flight, with aerodynamic moment transfer from the reference centre to the CG

State Estimation

• Reference invariant EKF (InvariantEKF<S, C>) on SE₂(3) × ℝ⁷: attitude, velocity and position as one SE₂(3) element, with gyro, accelerometer and barometer biases kept Euclidean
• Left- or right-invariant error convention chosen by template parameter, so both filters run on the same models
• Launch-centred inertial (LCI) navigation frame with WGS-84 J2 gravity; strapdown process model with a trapezoidal transition matrix
• GNSS position, GNSS velocity and barometer measurement models; Joseph-form updates; NEES/NIS consistency metrics
• Composable state blocks and a hashed vehicle configuration (kNavBaroGnss16), plus a log contract for replaying co-simulation logs
• (Planned) replay harness, tuning on the F-16 check cases, float32 rerun, and C code generation of f, h and their Jacobians

See doc/sensor_fusion.rst and TODO-sensor-fusion.md for the design and roadmap.

Algorithmic Differentiation

• AD-friendly formulations throughout — compatible with CppAD, dual numbers, and similar libraries
• CppAD-taped Jacobians in the trim solver and the estimator: the process and measurement models tape through Eigen and are checked against finite differences
• (Planned) CppADCodeGen export of the estimator Jacobians as plain C

Environment Models

• US1976 Standard Atmosphere (temperature, pressure, density, speed of sound), with ISA + dT and sea-level pressure offsets re-integrated hydrostatically
• Steady wind and Dryden turbulence (MIL-F-8785C) on the F-16 and rocket plants, exposed as FMU ports
• WGS84 gravity model with ECI/ECEF/geodetic coordinate conversions
• Branch-free formulations designed for AD and Kalman filtering

Aerodynamics & Propulsion

• DAVE-ML (DML) aerodynamic model loader — table-driven aero/inertia/propulsion data
• Multi-axis force and moment models; CG-offset spatial inertia
• Stage-aware propellant tracking with automatic separation logic

Interoperability

• FMI/FMU export via fmu4cpp — drop Aetherion plants into any FMI-compliant simulator. Shipped FMUs: F16Plant, F16Autopilot, TwoStageRocket (steady wind, Dryden turbulence and atmosphere offsets exposed as ports)
• Simulink models built from the FMUs by matlab/build_f16_simulink.m (run in CI) and matlab/build_twostagerocket_simulink.m
• JSON configuration files for initial conditions and simulation parameters
• CSV output compatible with NASA reference data formats

Code Quality

• Post-modern C++23 design throughout
• Enforces a practical subset of the JSF AV C++ coding standard
• CI gates: Clang-Format · CMake-Format · CMake-Lint · IWYU · Clang-Tidy · Metrix++ cyclomatic complexity · AddressSanitizer / UBSan

The two CMake gates share their configuration in .cmake-format and cover every CMakeLists.txt and *.cmake outside the pruned build/, out/, vendor/ and .venv/ trees. Reproduce them locally with:

pip install cmakelang

bash .run-cmake-format   # rewrites listfiles in place; CI fails if this leaves a diff

mapfile -t listfiles < <(
  find . \( -path ./build -o -path ./out -o -path ./vendor -o -path ./.venv \) -prune \
    -o -type f \( -name CMakeLists.txt -o -name "*.cmake" \) -print
)
cmake-lint "${listfiles[@]}" --config-files .cmake-format

--config-files takes a variable number of arguments, so the listfiles have to come before it: piping them in as xargs cmake-lint --config-files .cmake-format makes them look like extra config files and the run checks nothing at all.

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SVA → AD → EKF Pipeline

The library is architected so that smooth, scalar-templated models feed auto-Jacobians into the filter linearisation, producing a single consistent GNC pipeline from plant model to state estimator. The plant supplies the truth; the estimator's process and measurement models are taped with CppAD, and the reference EKF is the specification that generated flight code will be checked against.

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Mathematical Foundations

ΑETHERION implements the full suite of Featherstone spatial constructs:

Symbol	Meaning
v	Spatial velocity (twist)
a	Spatial acceleration
I	Spatial inertia
v×	Motion cross-product operator
v×*	Force cross-product operator (adjoint)

Equation of motion:

Control surfaces and payload stages can be modelled as rigidly attached bodies within the same spatial algebra.

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Frames & Conventions

Label	Frame
W	Inertial (ECI)
E	Earth-fixed (ECEF)
B	Rocket/aircraft body

All transforms follow:

notation for spatial transforms, keeping the frame algebra explicit and AD-friendly.

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Validated Examples

Each example below runs an atmospheric check case from NASA TM-2015-218675 (NESC 6-DOF flight simulation check cases). How closely it reproduces the reference runs -- per scenario, per quantity, and against the spread among the NASA implementations themselves, which for some scenarios exceeds Aetherion's difference from any one of them -- is tabulated in the validation chapter; scripts/checkpoint_tables.py regenerates those tables from a run. (EkfReplay, a sensor-fusion tool, is not a check case.)

Example	Vehicle	Scenario
DraglessSphere	Dragless sphere, gravity only	1
TumblingBrickNoDamping	Free tumbling brick	2
TumblingBrickWithDamping	Tumbling brick with aerodynamic damping	3
SphereWithAtmosphericDrag	Sphere with atmospheric drag	6
DroppedSphereSteadyWind	Sphere in steady crosswind	7
DroppedSphere2DWindShear	Sphere in 2-D wind shear	8
EastwardCannonball	Eastward ballistic cannonball	9
NorthwardCannonball	Northward ballistic cannonball	10
F16SteadyFlight	F-16 subsonic trimmed flight	11
F16SupersonicTrim	F-16 supersonic trimmed flight (Mach ≈ 2)	12
F16AltitudeChange	F-16 altitude change	13.1
F16AirspeedChange	F-16 airspeed change	13.2
F16HeadingChange	F-16 heading change	13.3
F16LateralSideStep	F-16 lateral side-step	13.4
CircleNorthPole	F-16 North Pole circumnavigation	15
CircleEquatorDateLine	F-16 circle around equator / date line	16
TwoStageRocket	Two-stage rocket to orbit	17

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Testing

• Catch2 unit tests covering every module
• Numerical validation against the NASA TM-2015-218675 reference runs, scenario by scenario (see the validation chapter for the quantities, tolerances and reference spread)
• Estimator validation: Jacobians against finite differences and CppAD; filter consistency (NEES/NIS) in both error conventions on an analytic manoeuvring truth
• AddressSanitizer + UBSanitizer run on every CI push
• Clang-Tidy static analysis and IWYU include hygiene enforced on every PR

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Dependencies

Library	Purpose	Provided by
Eigen	Linear algebra	vendored (vendor/eigen)
CppAD	Algorithmic differentiation	vcpkg (vcpkg.json)
nlohmann/json	JSON configuration files	vendored (header-only)
fmu4cpp	FMI/FMU export	vendored (vendor/fmu4cpp)
ecos	FMI co-simulation engine (FMU smoke testing)	submodule (vendor/ecos)
Catch2 v3	Unit testing	fetched at configure time

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Documentation

Full API reference and simulation guides are hosted on GitHub Pages:

• Detailed Documentation

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Build & Installation

Clone the repository and initialise submodules:

git clone https://github.com/onurtuncer/Aetherion.git
cd Aetherion
git submodule update --init --recursive

The only submodule, vendor/ecos, is used by the test suite alone. A source archive without submodules (a GitHub release tarball, git archive) builds the library, examples and FMUs with -DAETHERION_BUILD_TESTS=OFF; with tests on, configure stops and says to initialise the submodule.

CppAD is consumed through vcpkg on every platform, so a vcpkg clone is the only external prerequisite:

git clone https://github.com/microsoft/vcpkg.git ~/vcpkg   # or C:\vcpkg on Windows
~/vcpkg/bootstrap-vcpkg.sh                                 # bootstrap-vcpkg.bat on Windows
export VCPKG_ROOT=~/vcpkg                                  # setx VCPKG_ROOT C:\vcpkg

CMake picks the toolchain up from VCPKG_ROOT (or VCPKG_INSTALLATION_ROOT, which GitHub-hosted runners already set) and installs the dependencies pinned in vcpkg.json at configure time — no separate vcpkg install step. Two caveats:

• vcvarsall.bat, and therefore every Visual Studio developer prompt, overwrites VCPKG_ROOT with the port-less vcpkg stub bundled inside Visual Studio. That stub is detected and skipped; if it is the only candidate, pass -DAETHERION_VCPKG_ROOT=<path> or set VCPKG_INSTALLATION_ROOT, which vcvarsall leaves alone.
• To build against a CppAD that is already installed system-wide, configure with -DAETHERION_USE_VCPKG=OFF and make sure the prefix is on CMAKE_PREFIX_PATH.

Windows — Visual Studio

1. Open Visual Studio
2. File → Open Folder… → select the Aetherion directory
3. Let Visual Studio configure CMake automatically
4. Choose a configuration (e.g. x64-Release)
5. Build → Build All

Linux / macOS — CMake

cmake -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build --parallel
ctest --test-dir build

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Citing

If you use Aetherion in your research, please cite it using the metadata in CITATION.cff (GitHub's Cite this repository button generates BibTeX and APA from it).

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Community

• Bug Tracker: GitHub Issues
• Discussions: GitHub Discussions

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Author

Prof. Dr. Onur Tuncer
Aerospace Engineer, Researcher & C++ Systems Developer
Istanbul Technical University · [email protected]