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Robust Variable-Horizon MPC for Landing a Multirotor UAV on a Moving Platform
Sander Doodeman, Niels Berkers, Paula Chanfreut, Elena Torta, Duarte Antunes
TL;DR
Robust variable-horizon landing control under bounded disturbances is computationally complex, while disturbances can undermine recursive feasibility. The method searches near the previously selected horizon and uses a fixed RPI tube for horizon-independent constraint tightening in a differentially flat, decoupled UAV model. Physical experiments demonstrate the feasibility of the complete onboard architecture and robust control behavior.
Problem
Robust variable-horizon landing control under bounded disturbances is computationally complex, while disturbances can undermine recursive feasibility.
Source: §I · p. 2Method
The method searches near the previously selected horizon and uses a fixed RPI tube for horizon-independent constraint tightening in a differentially flat, decoupled UAV model.
Source: §I · p. 2Results
Physical experiments demonstrate the feasibility of the complete onboard architecture and robust control behavior.
Source: E. Discussion · p. 7Takeaways & Limitations
Experiments show robust VH-MPC can run on a lightweight Raspberry Pi onboard computer, controlling body rates and thrust reliably at 20 Hz.
Source: §VI · p. 8Physical experiments used only harmonic pitch motion, not simultaneous 6-DOF vessel motion.
Source: E. Discussion · p. 8
Abstract
from arXiv · showhide
Landing a multirotor Unmanned Aerial Vehicle (UAV) on a moving platform is challenging because a UAV is underactuated and the desired landing state is generally a non-equilibrium state. Shrinking- and variable-horizon approaches are promising for reaching such non-equilibrium targets, but often lack robustness to disturbances. Robust Variable-Horizon Model Predictive Control (VH-MPC) addresses this limitation but is computationally complex for a high-dimensional system such as a multirotor UAV. This paper presents a computationally efficient robust Variable-Horizon MPC method for reaching non-equilibrium targets under bounded disturbances. The online horizon search is restricted to a neighborhood of the previously selected horizon, while maintaining recursive feasibility under bounded disturbances. A fixed robust positively invariant tube provides horizon-independent constraint tightening. By exploiting the differential flatness property of a UAV, this approach is applied to a decoupled linearized system, enabling real-time implementation on an onboard Raspberry Pi 5 at 20 Hz. Simulations show that restricting the horizon search substantially reduces computation with limited impact on the objective, while real-time Gazebo simulations demonstrate the robustness of the method. Physical experiments demonstrate successful landing of a multirotor UAV on a moving Stewart platform.
I. INTRODUCTION
Landing a multirotor on a moving vessel is difficult because the target is non-equilibrium and disturbances can undermine feasibility. The paper proposes robust, computationally efficient variable-horizon control and an onboard UAV architecture, then validates it experimentally.
I. INTRODUCTION
- Landing on nearby vessels can support UAV recharging during offshore inspections, where limited flight time and hazardous manual inspection make fast, reliable landings valuable.The introduction also notes that UAV-based inspection can reduce inspection time and worker exposure to hazards.Source: §I · p. 1
- Underactuation makes the vessel’s moving 6-DOF landing state a non-equilibrium target that trajectory tracking cannot feasibly reach, while wind and ground effect add disturbances.Vessel roll and pitch further complicate landing, especially in harsh sea conditions.Source: §I · p. 1
- Existing MPC and shrinking-horizon landing methods do not account for disturbances, leaving recursive feasibility unassured.Position-tracking approaches also incur error at a non-equilibrium target; deep reinforcement learning lacks robustness guarantees.Source: §I · pp. 1–2
- The paper restricts online horizon search to a neighborhood of the previous choice and uses a fixed RPI tube for horizon-independent constraint tightening while retaining recursive feasibility.This addresses the computational burden of evaluating multiple candidate horizons online.Source: §I · p. 2
- Differential flatness decouples the linearized UAV model, enabling fixed-tube calculation and high-rate robust VH-MPC inputs in a practical landing architecture.The experiments validate real-time computation and robust reachability of a non-equilibrium landing state.Source: §I · p. 2
II. PRELIMINARIES
The preliminaries define the set operations and tube-MPC model used later. They state bounded-disturbance and controllability assumptions, then describe the nominal trajectory and error tube underlying robust constraints.
II. PRELIMINARIES
- Minkowski sum combines elements from two sets, while the Pontryagin set difference retains points whose translation by the second set remains within the first.The operations are denoted ⊕ and ⊖, respectively.Source: §II · p. 2
- The model assumes bounded disturbances in a compact convex set containing zero, with state and input constraints represented by convex sets.The system is discrete linear time-invariant, with state, input, and disturbance vectors and dynamics defined by A and B.Source: §II · p. 2
- Tube MPC optimizes a nominal trajectory and derives the applied control from its first nominal input, with feedback based on the real-to-nominal state error.The formulation assumes controllability and a feedback gain K for which A+BK is Schur.Source: §II · p. 2
- An RPI error set contains the next error under feedback and bounded disturbances; the tightened state and input constraint sets are assumed nonempty.The stated condition is (A+BK)E ⊕ W ⊆ E.Source: §II · pp. 2–3
III. ROBUST VARIABLE-HORIZON MPC
Robust VH-MPC combines variable-horizon planning with tube MPC to reach non-equilibrium targets under bounded disturbances while preserving recursive feasibility. Its horizon search is restricted around the previously selected horizon, and the cost balances state, input, and horizon reduction.
III. ROBUST VARIABLE-HORIZON MPC
- Robust VH-MPC combines VH-MPC with tube MPC so the controller can reach a non-equilibrium target under bounded disturbances while maintaining recursive feasibility.Nominal MPC cannot attain the target under the stated full-horizon objective, while VH-MPC can become infeasible under disturbances.Source: §III · p. 3
- The nominal cost penalizes state and input sequences, with γ ≥ 0 weighting horizon reduction.Q and R are positive definite state and input cost matrices; the selected nominal control is the first input in the optimal sequence.Source: §III · p. 3
- The online search considers horizons near the previous optimum, reducing the candidate set while retaining recursive feasibility.The search range is defined around the previous optimal horizon, and the paper proves recursive feasibility for the modified formulation.Source: §III · p. 3
- A feasible solution can be shifted to satisfy the next-step initial, state, input, and terminal constraints, supporting feasibility at k + 1.The shifted horizon N*_k − 1 remains in the feasible horizon set for any ΔN ≥ 0.Source: §III · p. 3
- Higher γ enforces faster arrival, while lower γ can produce smoother trajectories or input profiles through Q or R.With ΔN = 0, the approach is identical to a standard shrinking-horizon approach.Source: §III · p. 3
IV. ROBUST VH-MPC-BASED UAV CONTROL ARCHITECTURE
The UAV architecture applies robust VH-MPC to a decoupled, linearized multirotor model and runs its control loop at 20 Hz. The supplied figures illustrate constrained trajectories and horizon-cost selection.
IV. ROBUST VH-MPC-BASED UAV CONTROL ARCHITECTURE
- The robust VH-MPC loop calculates thrust and body-rate inputs using a decoupled, linearized multirotor model and runs at 20 Hz.A higher-rate inner loop maintains thrust and body rates and generates direct PWM motor signals.Source: §IV · pp. 3–4
- Fig. 2 compares robust VH-MPC, VH-MPC, and nominal MPC trajectories under bounded disturbances, marking constraints, the shrunken set, tube, and an infeasible state.The robust method is presented as suitable for bounded-disturbance landing, while nominal MPC cannot attain the target and disturbed VH-MPC becomes infeasible.Source: Figure 2 · p. 4
- Fig. 3 plots costs for evaluated horizons at each timestep, marking the chosen horizon in red and infeasible horizons with magenta crosses.Higher γ enforces faster arrival; lower γ can yield smoother trajectories or input profiles.Source: Figure 3 · p. 4
A. Modeling and OCP for the multirotor UAV
The UAV model is decoupled into small linearized subsystems, making robust VH-MPC practical at high control rates while horizon selection still considers the complete UAV objective.
A. Modeling and OCP for the multirotor UAV
- Decoupling reduces RPI-set calculations and MPC subproblems to systems of at most three states while retaining horizon selection over the complete UAV objective.The decomposition uses differential flatness; yaw is controlled separately with a PD controller.Source: A. Modeling and OCP for the multirotor… · pp. 4–5
- The controller solves each subsystem’s MPC problem for the selected horizon range, then derives the control inputs from the resulting optimal horizon.The UAV state includes position, velocity, roll, and pitch; yaw is omitted because translational dynamics are independent of it.Source: A. Modeling and OCP for the multirotor… · pp. 4–5
- The linearized model is valid around hover, while discrepancies at higher touchdown angles are treated as disturbances.The approximation is less accurate for touchdowns with higher angles.Source: A. Modeling and OCP for the multirotor… · p. 5
B. Tube calculation
A fixed robust positively invariant tube makes constraint tightening independent of the candidate horizon, trading potential conservatism for lower computational cost.
B. Tube calculation
- A single fixed RPI set serves as the tube cross-section across candidate horizons and can be calculated offline.Decoupling limits the RPI calculation to systems of at most three dimensions.Source: B. Tube calculation · p. 5
- The tube cross-section is represented as a polytope, and optimizing its offsets for prescribed facet normals yields an RPI set through an efficient linear program.The ellipsoid-based approximation supplies facet normals; optimizing the offsets guarantees the resulting set is RPI.Source: B. Tube calculation · p. 5
- The fixed tube trades potential reductions in conservatism for lower computational complexity, which matters for high-rate implementation.Dynamic tube shapes can be less conservative and improve control performance but carry high computational cost.Source: B. Tube calculation · p. 5
V. EXPERIMENTAL VALIDATION
The validation evaluates the controller’s computational feasibility, robustness, and landing performance, and analyzes the range of feasible initial conditions.
V. EXPERIMENTAL VALIDATION
- The validation assesses computational feasibility, robustness, and the landing performance of a real multirotor UAV on a moving platform.The section uses the parameters in Table I and chooses K as the LQR feedback gain.Source: §V · p. 5
- The experiments analyze the range of initial conditions for which the landing problem is feasible.The passage does not report the resulting range.Source: §V · p. 5
- The validation uses the parameters in Table I and an LQR feedback gain K.Source: §V · p. 5
A. Computational performance and horizon selection
Restricting the online horizon search reduces computation with limited increase in cumulative objective cost; at ΔN = 4, the controller remains within the 50 ms period on every iteration.
A. Computational performance and horizon selection
- The study runs 1,000 simulations per horizon set on a Raspberry Pi 5, with bounded noise and specified initial offsets and platform angles.Initial vertical and horizontal deviations are each 1.0 ± 0.5 m; the target platform roll is 15 degrees and pitch is 30 degrees.Source: A. Computational performance and… · p. 6
- At ΔN = 4, nine candidate horizons keep every iteration below 50 ms, with total cost almost equal to considering all horizons.The passage identifies this as computationally suitable for 20 Hz real-time operation.Source: A. Computational performance and… · p. 6
- Restricting the horizon search substantially reduces computation time with only a limited increase in cumulative objective cost.Source: A. Computational performance and… · pp. 5–6
B. Initial feasibility
The section characterizes where landing can start, evaluates robust VH-MPC under real-time Gazebo delays, and reports physical landing behavior with onboard control. The experiments also show adaptive horizon selection and measured-state containment within the projected tube.
B. Initial feasibility
- For a hovering UAV and flat platform, landing can start when p0 lies within px, py ∈ [−2.93, 2.93] and pz ∈ [−0.10, 3.47].For the stated moving-platform and moving-UAV configuration, the feasible ranges are px ∈ [−3.74, 1.31], py ∈ [−2.93, 1.92], and pz ∈ [−0.10, 3.35].Source: B. Initial feasibility · p. 6
C. Robustness
- Real-time Gazebo simulations use periodic platform motion, computation delays, and EKF-based platform-motion prediction.Ground-truth UAV and platform states are available to the robust VH-MPC implementation in the simulations.Source: C. Robustness · p. 6
- With real-time computation delays, ΔN = 2 provides the best combined cost-function and success-rate performance, while ΔN = 0 gives the highest mission success rate.Around 25 simulations were performed for each ΔN.Source: C. Robustness · p. 6, Table IV · p. 6
D. UAV landing experiment
- During a typical landing, the measured state remains within the RPI tube around a nominal trajectory that reaches the moving platform’s terminal target state.Position deviations permitted by the tube are relatively high, while velocity and tilt deviations are smaller.Source: D. UAV landing experiment · p. 7, Figure 6 · p. 7, Figure 7 · p. 7
- The selected horizon adapts while searching only a small neighborhood around the previously selected horizon.The cost-to-go is reduced at every time step, while horizon length can change to achieve a lower cost-to-go.Source: D. UAV landing experiment · p. 7
E. Discussion
The discussion identifies limits on robustness and landing accuracy from the fixed RPI tube and bounded-disturbance assumption. Physical experiments demonstrate the onboard architecture, but use simpler platform motion than a real vessel.
E. Discussion
- Larger disturbance bounds enlarge the fixed RPI tube and make tightened constraints more conservative, which can limit touchdown accuracy.This is the stated trade-off between robustness and landing accuracy.Source: E. Discussion · p. 7
- Experimental disturbances can exceed the assumed set W and produce infeasible candidate horizons; extending the search is a practical fallback outside the recursive-feasibility guarantee.The experiments report disturbances are never fully bounded within W.Source: E. Discussion · p. 7
- Physical experiments demonstrate the feasibility of the complete onboard architecture and robust control behavior.The Stewart platform reproduces representative periodic motion but is limited to harmonic pitch, so simultaneous 6-DOF vessel motion has not been experimentally evaluated.Source: E. Discussion · pp. 7–8
VI. CONCLUSION
The conclusion presents robust VH-MPC as a method for reaching non-equilibrium targets and landing a multirotor UAV on a moving platform in real time. It reports onboard control at 20 Hz and identifies vision-based platform measurements and richer error modeling as future work.
VI. CONCLUSION
- Robust VH-MPC reaches non-equilibrium targets, and its multirotor extension enables real-time landing on a 6-DOF moving platform.Experiments ran body-rate and thrust control reliably at 20 Hz on an onboard Raspberry Pi.Source: §VI · p. 8
- Future work includes vision-based platform detection to run the algorithm fully onboard without external measurements.The conclusion also identifies measurement and vessel-motion modeling errors for potential incorporation into the framework.Source: §VI · p. 8
- The proposed method can be tested in a real vessel scenario with wind disturbances once the future onboard-measurement work is implemented successfully.Source: §VI · p. 8