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3D Euler-Angle Orientation Control for Two-Ray Fading Mitigation in Maritime Air-to-Sea Communications
Mohammed Bajja, Abdoul Karim A. H. Saliah, Hajar El Hammouti, Daniel Bonilla Licea, Giuseppe Silano
TL;DR
Maritime Air-to-Sea links suffer deep two-ray fades, while existing mitigation largely leaves UAV attitude unused. The paper controls full 3D attitude through antenna phase-center displacement, using closed-form orientation candidates, a smooth reference, and constrained NMPC. The method increases cumulative throughput by 11.4% over pitch-only control and 22.2% over zero orientation while preserving trajectory tracking and actuator limits.
Problem
Maritime Air-to-Sea links experience deep fades from destructive interference, while existing mitigation primarily optimizes UAV position or trajectory rather than attitude.
Method
The paper models attitude-dependent antenna phase-center displacement, derives closed-form minimum-norm Euler-angle candidates under small-angle and far-field assumptions, and tracks a soft-minimum reference with constrained NMPC.
Results
11.4% more cumulative throughput than pitch-only control and 22.2% more than zero orientation are achieved while preserving trajectory tracking and respecting actuator limits.
Takeaways & Limitations
Full 3D attitude can serve as a physical-layer control variable for proactively steering maritime two-ray interference toward constructive conditions.
Abstract
from arXiv · showhide
Maritime Air-to-Sea links are dominated by a line-of-sight ray and a sea-surface reflected ray whose destructive combination produces deep fades. Existing mitigation strategies optimize Unmanned Aerial Vehicle position or trajectory but leave attitude unexploited. This paper treats the full three dimensional attitude as a physical-layer control variable that shapes the two-ray interference through antenna phase-center displacement. Under small-angle and far-field assumptions, the constructive-interference condition reduces to an affine constraint in the Euler angles and admits a closed-form family of minimum-norm attitude candidates. A differentiable soft-minimum rule yields a smooth reference tracked by a constrained Nonlinear Model Predictive Control controller on a fully-actuated tilting multirotor. The proposed scheme increases cumulative throughput by 11.4% over a pitch-only benchmark and 22.2% over a zero-orientation baseline, while preserving trajectory tracking and respecting actuator limits.
I. INTRODUCTION
Maritime two-ray fading is highly sensitive to geometry, yet prior mitigation primarily optimizes UAV position and trajectory rather than attitude. This paper makes full 3D attitude a communications control variable and combines closed-form orientation references with constrained tracking.
- I. INTRODUCTION: Maritime links combine a strong line-of-sight ray with a sea-surface reflection, producing deep fades and large instantaneous SNR fluctuations.Small changes in platform position or orientation can substantially alter the relative phase of the two rays.
- I. INTRODUCTION: 3D attitude is treated as a controllable variable that displaces the antenna phase center and shapes the instantaneous two-ray interference condition.The approach targets deterministic phase control rather than modeling attitude variation only as random jitter.
- I. INTRODUCTION: Under small-angle and far-field assumptions, constructive interference becomes an affine Euler-angle constraint with closed-form minimum-norm attitude candidates.The candidates are governed by a geometry-dependent phase-sensitivity vector.
- I. INTRODUCTION: A differentiable soft-minimum rule selects a smooth feasible attitude reference instead of discontinuously switching between candidate solutions.This formulation is designed for real-time control compatibility.
- I. INTRODUCTION: A constrained NMPC controller tracks the communications-aware reference on a fully actuated tilting multirotor without modifying the position reference.The controller jointly addresses attitude tracking, trajectory tracking, and actuator constraints.
II. SYSTEM MODEL
The system model represents the maritime link as a direct ray plus a specular sea-surface reflection. Its analytical assumptions isolate geometric two-ray interference while neglecting diffuse and hardware-related propagation effects.
- II. SYSTEM MODEL: The received signal is modeled as the superposition of a direct LoS ray and one sea-surface specular reflection.The reflection has magnitude factor γ ∈(0, 1) and an additional π phase shift.
- II. SYSTEM MODEL: Sea-state diffuse scattering, wave and superstructure shadowing, polarization mismatch, and Doppler are neglected, while both endpoints use isotropic antennas.These simplifications isolate the geometric mechanism governing deep fades and enable closed-form orientation laws.
- II. SYSTEM MODEL: The geometry is defined by UAV antenna altitude H, ship antenna altitude h, and horizontal separation L, with LoS and reflected path lengths determined from these quantities.The UAV frame places the sea surface at z = −20 m in the stated scenario.
- II. SYSTEM MODEL: Deep fades occur when the two contributions combine destructively, causing near-cancellation despite both rays being individually strong.The reflection phase shift is central to this cancellation condition.
B. Orientation-dependent channel
The orientation-dependent channel incorporates the UAV antenna phase-center offset into the world-frame geometry, making path lengths and phase depend on roll, pitch, and yaw. The resulting received-power optimization is non-convex and is approximated later under stated operating assumptions.
- B. Orientation-dependent channel: The antenna phase center is obtained by rotating its fixed body-frame offset with the UAV attitude η = [ϕ, ϑ, ψ]⊤.The rotation matrix R(η) maps the body-frame offset into the world frame.
- B. Orientation-dependent channel: Using the sea-surface image source, the orientation-dependent LoS and reflected path lengths determine the attitude-dependent phase difference and channel coefficient.The virtual receiver is reflected across the horizontal sea surface.
- B. Orientation-dependent channel: In the considered far-field maritime regime, attitude enters through both path length and phase, with the phase term dominating.At zero attitude, the formulation recovers the static channel model.
- B. Orientation-dependent channel: Maximizing received power over admissible Euler angles is non-convex because rotation and path lengths depend trigonometrically on attitude.The admissible set reflects mechanical limits, flight safety, and the small-angle operating regime.
III. CLOSED-FORM 3D ORIENTATION OPTIMIZATION
Under small-angle and far-field assumptions, the two-ray phase becomes affine in the Euler angles, enabling closed-form minimum-norm attitudes that enforce constructive interference. The formulation also identifies when attitude cannot first-order control the phase.
- A. Far-field small-angle regime: Small-angle and far-field assumptions reduce antenna-displacement effects to an affine two-ray phase model.The path-length variations are obtained by projecting first-order antenna displacement onto locally parallel ray directions.
- B. Closed-form minimum-norm candidates: The orientation dependence is dominated by the phase term, so leading-order SNR maximization selects constructive-interference phases despite the reflection phase shift.Constructive interference occurs when ∆Φ(η) belongs to {(2m + 1)π : m ∈ Z}.
- B. Closed-form minimum-norm candidates: Theorem 1 represents constructive-interference attitudes as hyperplanes k^Tη = C_m, indexed by the fringe integer m.The phase-sensitivity vector k captures first-order sensitivity to roll, pitch, and yaw.
- B. Closed-form minimum-norm candidates: Each hyperplane has a closed-form minimum-norm candidate aligned with k, and a feasible candidate is selected when it lies within the admissible attitude set.The selected candidate enforces constructive interference at leading order while minimizing Euclidean norm.
- B. Closed-form minimum-norm candidates: When k = 0, no infinitesimal rotation displaces the antenna along the LoS–reflection distinguishing direction, so attitude is not first-order phase-controllable.The controller defaults to zero attitude correction, leaving translational motion as the route for link improvement.
C. Smooth attitude reference law
The smooth attitude reference law replaces discontinuous minimum-norm candidate selection with a differentiable soft minimum. Its temperature controls the trade-off between selectivity and smoothness while handling feasibility edge cases.
- C. Smooth attitude reference law: The feasible candidate of smallest norm preserves the small-angle regime but can jump as link geometry changes.Discontinuous arg-min switching can create transients for the tracking controller.
- C. Smooth attitude reference law: A differentiable soft-minimum rule produces a smooth attitude reference from the finite feasible candidate set.The rule is defined for a finite nonempty feasible set with a unique minimum-norm element.
- C. Smooth attitude reference law: The temperature parameter trades selectivity against smoothness: large values approach hard minimum selection, while small values average neighboring candidates and may miss the optimal fringe.The reference remains aligned with the phase-sensitivity vector and varies smoothly with link geometry.
- C. Smooth attitude reference law: The method tunes temperature to geometric variation timescales and sizes the candidate and admissible sets so feasible candidates remain available throughout the mission.If the feasible set is empty or has non-unique minima, continuous fallback references are used.
IV. OPTIMAL CONTROL PROBLEM
A fully actuated tilting multirotor tracks both mission state and the communication-optimal attitude reference through constrained NMPC. Full actuation supports attitude regulation without sacrificing translational authority.
- IV. OPTIMAL CONTROL PROBLEM: The fully actuated GTMR model includes position, Euler angles, translational velocity, angular velocity, and squared rotor-speed inputs.Its geometry decouples attitude regulation from translational authority.
- IV. OPTIMAL CONTROL PROBLEM: NMPC embeds the communication-optimal attitude reference into the tracking objective at each control update.The controller simultaneously accounts for mission-state tracking and communication-driven orientation.
- IV. OPTIMAL CONTROL PROBLEM: The prediction model discretizes the continuous dynamics over a sampling period and horizon using a one-step state-transition map.Desired position and attitude references are specified at each prediction step.
1) Discrete-time formulation and stage cost:
The discrete-time NMPC repeatedly solves a finite-horizon constrained tracking problem and applies only the first control input. Its objective regularizes tracking and maneuver effort while enforcing operational and actuator limits.
- 1) Discrete-time formulation and stage cost:: The objective combines position and attitude tracking with velocity, angular-rate, and input regularization terms.The weights penalize aggressive maneuvers, smooth actuator commands, and improve numerical conditioning.
- 1) Discrete-time formulation and stage cost:: At each control update, NMPC solves a finite-horizon problem using the current state as its initial condition.The controller operates in receding-horizon fashion by re-solving at the next update.
- 2) Optimal control problem:: The optimization enforces discrete dynamics together with admissible position, Euler-angle, velocity, angular-rate, and actuator constraints.The operational region and Euler-angle set encode safety and mechanical admissibility.
- 2) Optimal control problem:: Sequential quadratic programming with real-time iteration provides the stated implementation route for real-time tractability.The first control input is applied before the problem is solved again at the next sampling instant.
V. NUMERICAL RESULTS
The maritime simulation compares 3D orientation-aware, pitch-only, and zero-orientation controllers on a circular UAV trajectory while the ship moves laterally. The 3D controller improves communication performance and remains feasible under tracking and actuator constraints.
- Scenario and comparison: The simulation compares 3D orientation-aware, pitch-only, and zero-orientation controllers using channel gain, SNR-outage fraction, and cumulative throughput.The pitch-only benchmark fixes roll and yaw at zero, while the zero-orientation baseline sets all Euler angles to zero.
- Communication performance: 11.4% and 22.2% more transmitted data are achieved than with the pitch-only and zero-orientation benchmarks, respectively.The 3D controller also sustains higher channel gain and uniformly dominates outage performance across SNR thresholds.
- Scenario and comparison: All three controllers execute the prescribed circular UAV trajectory while the ship moves laterally.The trajectory is designed to vary the relative bearing and sweep the two-ray interference pattern through constructive and destructive fringes.
- Attitude tracking and feasibility: Partial attitude tracking realizes most of the communication gain despite residual errors caused by reference fluctuations, actuator limits, and closed-loop bandwidth.The controller still steers the antenna phase center along the relevant direction, with a delay.
- Attitude tracking and feasibility: Linear velocities, angular rates, and rotor speeds remain within bounds, with smooth actuator profiles and no chattering.These results support feasibility throughout the mission while preserving the primary motion task.
VI. CONCLUSIONS
The paper uses full UAV attitude to shape two-ray interference in maritime air-to-sea links through a closed-form, smoothly tracked orientation reference. Simulations show higher cumulative throughput while preserving trajectory tracking and actuator feasibility.
- VI. CONCLUSIONS: Cumulative throughput increases by 11.4% over pitch-only and 22.2% over zero-orientation benchmarks.The gains are reported for the maritime two-ray simulation scenario.
- VI. CONCLUSIONS: The method treats roll, pitch, and yaw as communication control variables by modeling antenna phase-center displacement.Under small-angle and far-field assumptions, the constructive-interference condition becomes affine in the Euler angles.
- VI. CONCLUSIONS: A soft-minimum rule produces a smooth reference from closed-form minimum-norm attitude candidates for constrained NMPC tracking.The controller runs on a fully actuated tilting multirotor.
- VI. CONCLUSIONS: The proposed scheme preserves trajectory tracking and respects actuator limits.The conclusion reports these properties alongside the communication gains.
APPENDIX
The appendix derives an affine phase model and uses it to characterize constructive-interference attitudes as minimum-norm solutions. A soft-minimum construction then provides a smooth selection among feasible candidates.
- A. Proof of Theorem 1: The antenna phase-center displacement admits a first-order affine expansion in the Euler angles under the small-angle model.Substituting the distance expansions for the relevant points yields the phase relation used in the affine constraint.
- A. Proof of Theorem 1: Constructive interference is enforced by the affine Euler-angle constraint k^Tη = C_m.Maximizing leading-order channel gain is equivalent to selecting phase conditions indexed by m.
- A. Proof of Theorem 1: The minimum-norm attitude candidate is η_m = (C_m/||k||^2)k on the constructive-interference hyperplane.The solution follows from the strictly convex minimum-norm quadratic program and its KKT conditions.
- A. Proof of Theorem 1: The soft-minimum weights converge to the unique minimum-norm candidate as the temperature parameter grows.For positive temperature, the weighted attitude is smooth in the candidate set.