Keyboard shortcuts

Press ← or → to navigate between chapters

Press S or / to search in the book

Press ? to show this help

Press Esc to hide this help

Reusable Booster Autonomous Landing

This live example starts a reusable orbital booster on a moving drone ship. Press Space to ignite the center engine. The stage performs a six-second ascent burn, coasts through apogee, captures its descending state, and plans a return to the deck. A differential-flatness planner generates the landing reference, an SE_2(3) GPS/IMU error-state Kalman filter estimates the vehicle state, and a geometric controller drives an illustrative hybrid allocation between the center-engine gimbal and RCS.

// rumoca-live-scenario: ../repo-examples/interactive/reusable_booster/rumoca-scenario.toml

The standard live widget opens the Modelica source and TOML scenario in its editor. Open Tune environment and controller over the scene to change wind, gust intensity, deck heave/roll/pitch, or the four controller-gain multipliers while the simulation is running. Those sliders write scenario locals directly to Modelica inputs, so they do not recompile or reset the run. This makes it possible to search for the landing-failure boundary. The editor’s Settings panel remains available for mission constants stored in TOML [parameters].

Press R at any time, including after the run reports Finished, to restore the vehicle and restart at t = 0. The upper-right badge shows mission phase, wind and gust speed, and deck roll/pitch. T switches between realtime and fast pacing, F toggles fullscreen, and Q stops the run.

For a native run:

cargo xtask repo modelica-deps ensure
cargo run -p rumoca --release -- \
  sim -c examples/interactive/reusable_booster/rumoca-scenario.toml

Reference Planning

Position is the translational flat output. For each axis, the Modelica planner constructs a quintic from the position, velocity, and acceleration captured when the coasting stage reaches 8 m/s downward speed. Its terminal state is the predicted deck CG target along the tilted deck normal, including the target point’s translational and rotational velocity and acceleration:

p(t) = c0 + c1 t + c2 t^2 + c3 t^3 + c4 t^4 + c5 t^5
(p, v, a)(0) = (p0, v0, a0),  (p, v, a)(T) = (pf, vf, af)

Analytic derivatives provide reference velocity and acceleration. The vector a_ref + g e3 determines the flatness-derived reference thrust direction and attitude. The amber Three.js line is the complete polynomial reference; its marker advances along the path while the cyan line records the simulated trajectory. The green line is the navigation estimate and the magenta marker is the latest GPS fix consumed by the filter. Green and magenta wireframes show their respective axis-aligned 3-sigma marginal-variance envelopes at 4x visual scale so the sub-metre posterior remains legible beside a 40.9 m stage. They show per-axis marginal variance, not covariance orientation.

The nominal landing-plan duration is 18 seconds. Guidance continues to hold the terminal reference until a safe physical deck contact is detected; reaching the polynomial end does not shut down the engine above the ship.

A 21-point sampled preflight check covers throttle, tilt, positive vertical thrust, deck clearance, and approximate propellant budget. The scene explicitly reports REFERENCE INFEASIBLE when the unconstrained quintic exceeds those modeled limits; this warning is not a constrained trajectory optimizer.

GPS/IMU SE_2(3) Kalman Filter

The navigation state uses the same group representation as the controller:

Xhat = (phat, vhat, Rhat) in SE_2(3)

At 10 Hz, piecewise-constant gyroscope and accelerometer measurements drive LieGroups.SE23.Quat.exp_mixed. The function implements the closed-form mixed invariant propagation described in Purdue’s On Closed-Form Preintegration for a Class of Mixed-Invariant Systems in SE_n(3):

Xhat(k+1) = exp(M dt) Xhat(k) exp(N dt)

The call includes gravity as a world-frame right increment and the nilpotent position-velocity coupling matrix. This is the library implementation from the pinned CogniPilot modelica_models checkout; the example does not duplicate the exponential.

A 5 Hz synthetic differential-GPS position fix performs the Kalman correction. The filter propagates a coupled 9 x 9 position, velocity, and attitude-error covariance. Its inertial transition includes position-velocity coupling and the sensitivity of specific-force integration to attitude error, allowing a position fix to correct velocity and attitude through cross-covariance. The GPS update uses Joseph form for covariance robustness and a multiplicative attitude correction. The model intentionally omits sensor biases, time delay, outlier rejection, Earth rotation, and geodetic effects.

Geometric Tracking

The controller represents estimated pose and velocity together on SE_2(3):

Xhat = (phat, vhat, Rhat)
Xref = (pref, vref, Rref)
xi   = Log(Xhat^-1 Xref)
delta = J_left(xi) K xi

inverse, product, log_map, and left_jacobian come directly from the pinned CogniPilot modelica_models LieGroups.SE23.Quat package. Translation feedback corrects the flatness acceleration, and an SO(3) inner loop aligns the booster axis with the resulting force vector. The inner loop tracks the geodesically rate-limited attitude command with per-axis gain vectors and includes rigid-body gyroscopic and rotating-reference feedforward terms.

The geometric controller runs at 20 Hz with a zero-order hold, while the rigid-body, engine, drag, mass, and contact dynamics remain continuous. This matches a sampled flight-control implementation and avoids reevaluating the Lie-group maps at every adaptive integrator stage.

The control allocator prioritizes center-engine thrust vectoring for pitch/yaw moment. RCS receives only the saturated residual and the requested axial moment. RCS availability fades in between 80 and 120 m engine-plane altitude, its aggregate force/moment has an 80 ms first-order response, and its representative cold-gas mass flow contributes to vehicle mass depletion. Engine and RCS plume geometry, light, and opacity follow actual actuator state. Landing-foot tangential damping is regularized by a Coulomb cone, so friction is capped by normal load and vanishes for an unloaded foot.

Embeddable eFMI Control Law

The continuous vehicle and the CMM-based Lie-group preprocessing remain in the simulation model. The force/moment feedback law behind them is a separate deep module with four vector inputs: translational terms, rotational terms, gain scales, and principal inertia. The 20 Hz simulation controller and the export model call the same functions, so the generated code does not maintain a second copy of the control equations.

The export model is fixed-sample and discrete, which makes it admissible for Rumoca’s galec-production target. Select Generate .alg below to inspect the eFMI Algorithm Code, then Generate C/H to inspect the corresponding C99 Production Code.

// rumoca-live-scenario: ../repo-examples/interactive/reusable_booster/rumoca-scenario.controller-galec-production.toml

The native packaging command emits both a directory-form eFMU and a matching .efmu archive with Algorithm Code and Production Code representations:

cargo xtask repo modelica-deps ensure
cargo run -p rumoca -- \
  compile examples/interactive/reusable_booster/ReusableBoosterLanding.mo \
  --model ReusableBoosterEmbeddedControlLaw \
  --source-root target/cmm/CMM-a642c381 \
  --target galec-production \
  --output examples/interactive/reusable_booster/gen/control_law_efmu

This exported seam starts after the geometric adapter has computed the world-frame translational correction, attitude error, and reference body rate. Exporting the full estimator and Lie-group adapter would additionally require GALEC projection support for the indexed array outputs and matrix operations used by those library functions.

Dryden Wind and Moving Deck

The default mean wind is 5 m/s. Its direction swings by 15 degrees on a 10-second period, while longitudinal, lateral, and vertical deterministic inputs pass through standard Dryden-shaped filters:

Hu(s)   = sigma sqrt(2 Lu / (pi V)) / (1 + Lu s / V)
Hv,w(s) = sigma sqrt(L / (pi V)) (1 + sqrt(3) L s / V) / (1 + L s / V)^2

Band-limited multi-sine inputs replace ideal white noise so a browser run is deterministic and repeatable. The resulting three-axis gust is rotated with the mean-wind direction, added to the mean wind, and subtracted from inertial velocity before the body-axis drag calculation. The numeric indicator, windsock, and deck arrow use this same changing total wind vector.

The drone ship has analytic heave, roll, and pitch kinematics. The planner predicts the terminal CG position along the tilted deck normal and matches that point’s velocity and acceleration, including rotational motion. For each landing foot, the contact model transforms its position into the deck frame and computes

penetration = max(0, -z_foot_deck)
vrel = vfoot - (vdeck + omega_deck x (pfoot - pdeck))

Normal spring/damping acts along the tilted deck normal and tangential damping uses the relative deck-point velocity. Contact is limited to the physical deck bounds. The same Modelica deck position and quaternion drive the Three.js ship, so visible wave motion and collision geometry remain synchronized. The telemetry badge reports deck roll and pitch beside the changing wind and gust values.

Touchdown and Failure State

At landing-phase contact, at least three feet must carry meaningful normal load and the CG projection must remain at least 0.25 m inside their support polygon. The maximum foot speed normal to the deck must be at most 2.0 m/s, tangential speed at most 1.5 m/s, and booster tilt at most 10 deg from the local deck normal. Safe first contact cuts the engine and starts a bounded suspension- settling interval. Stable support during that interval latches LANDED; unsafe kinematics or failure to establish support latches a crash. The scene reports foot count, support margin, and separate normal/tangential speeds.

Vehicle and Contact Model

The visual and physical model use a representative 40.9 m stage and 3.66 m diameter. The plant includes an illustrative landing mass and fixed landing inertia/CG, a generic center-engine propulsion model, propellant depletion, thrust response, anisotropic aerodynamic drag, quaternion 6-DOF dynamics, and four spring-damper landing contacts. The scene includes an interstage, nine-engine mount, deployed grid fins, articulated landing legs, and a scale-matched autonomous drone ship.

After pressing Capture, mouse movement orbits and tilts the scene, the middle or right mouse button pans, and the wheel zooms. Press Escape to release capture.

The geometry, landing mass, inertia, RCS force, throttle floor, gains, and contact coefficients are mutually consistent educational values. They are not intended to reproduce any particular operational vehicle.