Landing a 6-DOF rocket with robust & nonlinear MPC

Year: 2025
Context: EPFL · Model Predictive Control course
Team: Team of 3
Links: Code (opens in a new tab)

TL;DR

Problem: Land a simulated 6-DOF rocket (12 states) precisely within actuator and tilt limits, despite model mismatch, unknown mass and disturbances.
What we built: A dual-layer flight control system (PID and MPC) for highly accurate speed and trajectory tracking, an offset-free observer, a robust tube MPC for a safety-first landing, and a custom nonlinear MPC in CasADi.
Result: The nonlinear MPC recovers from a 30° roll and lands on target in under 4 s; the tube MPC keeps the rocket above ground under worst-case disturbance.
Stack: Python, CasADi + IPOPT, CVXPY + PIQP, mpt4py, python-control

How do you land precisely when your model is wrong and the world pushes back?

30° ROLL → LANDED < 4 S
MPC
Nonlinear MPC
Tube MPC
CasADi

Simulated rocket at the end of its curved flight path above a green ground grid, in the course's 3D visualizer

The approach

Decouple. Linearize around hover and split the rocket into x, y, z and roll subsystems. Each gets an MPC with an LQR terminal cost and a maximal invariant terminal set, which keeps the controller feasible over time.
Dual layer. An outer PID position loop sends velocity setpoints to the MPCs. Soft constraints keep the problem feasible on the full nonlinear simulator.
Offset-free tracking. A disturbance-augmented model and a Luenberger observer remove the steady-state error when the rocket is 1.5× heavier than modeled, and while it burns fuel.
Safety-first landing. A robust tube MPC on altitude, built from a minimal robust invariant set and tightened constraints, guarantees the rocket never goes below the ground for any disturbance within its bounds.
Custom nonlinear MPC. One controller for the full nonlinear dynamics in CasADi: RK4 discretization, 3.5 s horizon, LQR terminal cost and warm-started IPOPT solves.

What it achieved

30° roll → landed in under 4 s. The custom nonlinear MPC corrects complex aerodynamic drag and prevents orientation errors during landing, starting 10 m up with a 30° roll.
Guaranteed stability under unpredictable wind disturbances. Even with a constant worst-case disturbance, the tube MPC keeps the rocket above ground and the throttle within limits.
Fast linear MPCs. The per-axis controllers stabilize from 5 m/s in under 7 s and track a 35° roll setpoint in about 3 s.
Offset-free at 1.5× mass. The disturbance observer cancels the steady-state velocity error of a rocket 50% heavier than its model.

Figures

Fig. 01Nonlinear MPC, closed loop. From 10 m up with a 30° roll, the rocket reaches its landing target point in under 4 s, with every input inside its limits.Team figure · course reportFull size image: NMPC closed loop: x, y, z converge to the landing target and the roll angle goes from 30 degrees to zero in under 4 s, with all inputs inside their limits (opens in a new tab)

Fig. 02Tube MPC on altitude: the minimal robust invariant set (the tube the true state stays in around the plan) and the terminal set.Team figure · course reportFull size image: Minimal robust invariant set and terminal set used by the tube MPC on altitude (opens in a new tab)

Fig. 03Worst-case disturbance on every step: altitude never drops below the ground and throttle stays within its limits.Team figure · course reportFull size image: Tube MPC under worst-case disturbance: vertical speed, altitude staying above ground, and throttle within limits (opens in a new tab)

Fig. 04Offset-free MPC: the observer's disturbance estimate settles, which removes the tracking offset caused by the heavier-than-modeled rocket.Team figure · course reportFull size image: The offset-free observer's disturbance estimate converging to its steady-state value (opens in a new tab)

Notes

The rocket model, simulator and 3D visualizer were provided by the course. We closed the project with a comparison of nonlinear MPC and robust linear MPC on safety guarantees, computational cost and axis coupling.