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3D Trajectory Optimization in Rician Fading for UAV-Enabled Data Harvesting

Changsheng You, Rui Zhang

arXiv:1901.04106v5cs.IT

TL;DR

The paper addresses design issues from angle-dependent Rician fading and the lack of a closed-form effective fading parameter. It approximates that parameter and uses block coordinate descent to design 3D UAV trajectories, showing benefits from altitude variation and joint angle-aware design.

  • Problem

    Angle-dependent Rician fading introduces design issues, including the lack of a closed-form expression for the effective fading parameter.

  • Method

    The effective fading parameter is approximated with a logistic model, then an efficient block coordinate descent algorithm optimizes the trajectory.

  • Results

    The proposed 3D trajectory can enhance data collection through altitude variation for a single SN and joint angle-aware horizontal and vertical design for multiple SNs.

  • Takeaways & Limitations

    Altitude variation is especially beneficial with high maximum vertical speed and stringent outage probability requirements, while angle-aware 3D design improves multi-SN collection.

Abstract

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In this paper, we consider a UAV-enabled WSN where a flying UAV is employed to collect data from multiple sensor nodes (SNs). Our objective is to maximize the minimum average data collection rate from all SNs subject to a prescribed reliability constraint for each SN by jointly optimizing the UAV communication scheduling and three-dimensional (3D) trajectory. Different from the existing works that assume the simplified line-of-sight (LoS) UAV-ground channels, we consider the more practically accurate angle-dependent Rician fading channels between the UAV and SNs with the Rician factors determined by the corresponding UAV-SN elevation angles. However, the formulated optimization problem is intractable due to the lack of a closed-form expression for a key parameter termed effective fading power that characterizes the achievable rate given the reliability requirement in terms of outage probability. To tackle this difficulty, we first approximate the parameter by a logistic ('S' shape) function with respect to the 3D UAV trajectory by using the data regression method. Then the original problem is reformulated to an approximate form, which, however, is still challenging to solve due to its non-convexity. As such, we further propose an efficient algorithm to derive its suboptimal solution by using the block coordinate descent technique, which iteratively optimizes the communication scheduling, the UAV's horizontal trajectory, and its vertical trajectory. The latter two subproblems are shown to be non-convex, while locally optimal solutions are obtained for them by using the successive convex approximation technique. Last, extensive numerical results are provided to evaluate the performance of the proposed algorithm and draw new insights on the 3D UAV trajectory under the Rician fading as compared to conventional LoS channel models.

I. INTRODUCTION

The paper develops 3D UAV trajectory design for data collection in WSNs using angle-dependent Rician fading rather than simplified LoS channels. It jointly optimizes communication scheduling and UAV movement to maximize minimum collection rates under outage constraints.

  • System motivation: UAV-enabled WSNs use a mobile data collector to harvest sensing data from distributed sensor nodes, which can remain silent for energy saving.SNs transmit when awakened and scheduled by the UAV.
  • 3D trajectory design: Most prior UAV trajectory designs assume fixed altitude and optimize only a 2D trajectory, whereas this paper exploits vertical movement through a 3D framework.The proposed design targets improved rate performance in UAV-enabled WSNs.
  • Channel modeling: Simplified LoS channel models are convenient for optimization but may be inaccurate in urban and suburban areas because they neglect shadowing and small-scale fading.Obstacles can block propagation, creating LoS or NLoS links with location-dependent characteristics.
  • Channel modeling: Angle-dependent Rician fading captures deterministic LoS and random multipath components, with the Rician factor varying with elevation angle and environment.Higher elevation angles tend to increase the Rician factor because they are associated with less reflection, scattering, and obstruction.
  • Optimization objective: The paper maximizes the minimum average data collection rate across SNs while guaranteeing that each SN meets a prescribed outage-probability constraint.Rician fading requires outage-aware adaptive-rate transmission and joint optimization of horizontal and vertical trajectories.
  • Solution approach: The effective fading power lacks a closed-form expression, so the paper approximates it with a logistic function and solves the resulting non-convex problem using block coordinate descent and successive convex approximation.The algorithm iteratively handles scheduling, horizontal trajectory, and vertical trajectory optimization.
  • Numerical insights: For one SN, altitude variation can enhance collection rate, especially with high maximum vertical speed and stringent outage requirements.For multiple SNs, angle-aware joint horizontal and vertical design significantly enhances performance over the conventional simplified-LoS trajectory.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models a UAV collecting data from multiple ground sensor nodes over a discretized 3D trajectory. It jointly designs communication scheduling and UAV movement to maximize the minimum average collection rate while maintaining reliable reception.

  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The UAV collects data from N ground SNs during a fixed duration T, with each SN represented by a 3D Cartesian location.
  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The UAV starts and ends at predetermined 3D locations, whose horizontal coordinates and altitudes are specified.
  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The optimization jointly designs communication scheduling and the UAV’s 3D trajectory to maximize the minimum average data collection rate.
  • A. UAV Trajectory Model: The time horizon is divided into M equal slots, and the UAV position is approximated as a sequence of horizontal and vertical coordinates.
  • A. UAV Trajectory Model: Independent horizontal and vertical speed limits impose per-slot travel bounds Sxy = Vxyδ and Sz = Vzδ.
  • A. UAV Trajectory Model: The trajectory must satisfy horizontal and vertical movement constraints between consecutive slots.
  • A. UAV Trajectory Model: The UAV must follow its predetermined endpoints and maintain altitude z[m] ≥ H to avoid obstacles and preserve LoS paths.

B. UAV-Ground Channel Model

The channel model combines distance-dependent attenuation with angle-dependent Rician small-scale fading. Adaptive transmission rates, scheduling, and outage constraints connect channel reliability to the UAV’s 3D position.

  • Each UAV-SN link is modeled as a block-fading channel with large-scale gain βn[m] and small-scale fading coefficient gn[m, ℓ].
  • The channel distributions across time slots depend on the 3D UAV trajectory, making the resulting optimization challenging.
  • The large-scale channel gain accounts for pathloss and shadowing, with pathloss governed by exponent α.
  • Rician fading includes a deterministic LoS component and a random zero-mean unit-variance CSCG scattered component.
  • Rician factors are assumed identical across fading blocks within a slot but may differ between time slots.
  • The Rician factor increases with elevation angle because higher angles yield less scattering and a larger LoS component.
  • Each scheduled SN transmits at a fixed slot-specific rate, while unscheduled SNs remain silent; at most one SN is scheduled per slot.
  • Rates are selected to satisfy a maximum outage probability ε, but the effective fading power is an implicit trajectory-dependent function without an explicit form.

D. Problem Formulation

The paper formulates a max–min reliable data-collection problem with binary scheduling and 3D trajectory constraints. Because the effective-fading-power function lacks a closed form, it is approximated before solving the resulting problem algorithmically.

  • D. Problem Formulation: The objective is to maximize the minimum average data collection rate while satisfying scheduling, trajectory, and reliability constraints.
  • D. Problem Formulation: The original problem is intractable because the effective-fading-power function has no closed-form expression and involves the inverse Marcum-Q function.
  • D. Problem Formulation: The formulation contains binary scheduling variables, creating integer constraints in addition to the trajectory optimization.
  • D. Problem Formulation: The paper derives a suboptimal solution by reformulating the problem with a tractable approximation of the effective-fading-power function.
  • III. APPROXIMATION FOR EFFECTIVE FADING POWER AND PROBLEM REFORMULATION: The effective-fading-power function is approximated using logistic regression, and the optimization problem is reformulated accordingly.

A. Approximation for Effective Fading Power

The paper approximates the effective-fading-power function, which lacks an explicit form and depends intricately on the 3D UAV trajectory, using a logistic model fitted to numerical data. The resulting model matches the numerical data in most cases and exposes how outage tolerance, Rician factors, and elevation angle affect fading power.

  • The effective-fading-power function is difficult to characterize because it depends on the 3D trajectory through the Rician factor and elevation angle.This difficulty makes the outage-aware rate optimization challenging.
  • The authors generate numerical data and observe that effective fading power is non-decreasing with the angle indicator and has an S-shaped dependence.For large Kmax, its derivative first increases and then reaches a maximum as the angle indicator increases.
  • The proposed logistic model is fitted by minimizing mean square error and matches the numerical data in most cases.The explicit form is more tractable for characterizing the relationship between effective fading power and the 3D trajectory.
  • The logistic effective fading power rises with elevation angle in an S-shaped manner, with gains diminishing beyond a threshold.It is smallest when horizontal distance greatly exceeds altitude and reaches its maximum when the UAV is directly above the SN.
  • A larger tolerable outage probability increases effective fading power and allows a higher transmission rate, while limiting angle-related improvement.The latter effect occurs because the minimum effective fading power at zero angle is already larger under the more relaxed outage requirement.
  • Larger Kmax causes faster growth and earlier saturation of effective fading power, whereas the LoS and Rayleigh extremes show little angle-related improvement.As Kmax approaches infinity, the model approaches LoS behavior; when Kmax equals Kmin and approaches zero, it reduces to Rayleigh fading.

B. Problem Reformulation

The logistic fading-power model enables reformulation of the outage-constrained rate problem, but the resulting problem remains non-convex because of coupled trajectory variables, a non-affine equality, and integer scheduling. The paper therefore replaces difficult subproblems with convex approximations and obtains lower-bounded solutions.

  • The logistic model approximates the outage-aware achievable rate and reformulates the original optimization problem into Problem P2.The reformulated problem remains difficult because of non-convexity.
  • Problem P2 is non-convex because the rate constraint couples fading, horizontal trajectory, and altitude variables, while scheduling remains integer-constrained.It also contains a non-affine equality constraint linking the angle indicator to trajectory variables.
  • Relaxing the equality constraint produces Problem P4, whose scheduling block reduces to a standard linear program for any feasible 3D trajectory.The LP can be efficiently solved with existing solvers.
  • The horizontal-trajectory subproblem uses successive convex approximation to lower-bound the non-concave rate-related terms at a local trajectory.The resulting Problem P8 is convex and can be solved efficiently.
  • Because Problem P8 uses convex lower bounds, its feasible set is a subset of Problem P6 and its objective value provides a lower bound for Problem P6.This lower-bound property supports the iterative optimization procedure.

C. UAV Vertical Trajectory Optimization

The vertical-trajectory block is optimized after fixing scheduling and the horizontal trajectory. Following the horizontal-trajectory procedure, the paper applies successive convex approximation and solves the resulting convex problem efficiently.

  • Given feasible scheduling and a horizontal trajectory, the paper formulates a separate optimization problem for the UAV vertical trajectory.This problem has a form similar to the horizontal-trajectory subproblem.
  • The vertical-trajectory problem is transformed into an approximate problem by applying successive convex approximation to the rate constraint.The approximation follows the procedure used for the corresponding horizontal-trajectory problem.
  • The resulting Problem P10 is convex and can be solved using existing methods such as the interior-point method.Its coefficients are determined by the local vertical trajectory.

D. Overall Algorithm, Complexity, and Convergence

The proposed algorithm alternates among scheduling and horizontal and vertical trajectory optimization using convex subproblem solvers. Its objective is non-decreasing across iterations and converges to a locally optimal solution, while numerical results reveal trajectory and rate differences between Rician and LoS models.

  • Overall algorithm: Algorithm 1 sequentially optimizes communication scheduling, horizontal trajectory, and vertical trajectory within each block coordinate descent iteration.The scheduling problem is solved first, followed by the two trajectory blocks.
  • Complexity: The total computation complexity is O((M + NM)^3.5 log^2(1/ϵ)) for solution accuracy ϵ > 0.This accounts for the convex subproblem costs and the block coordinate descent iterations.
  • Convergence: The objective value of Problem P2 is non-decreasing over iterations, and Algorithm 1 converges to a locally optimal solution.The argument uses exact scheduling optimization, tight local Taylor approximations, and lower-bounded convex trajectory subproblems.
  • Numerical insights: The RFB trajectory can reduce UAV-SN distance and increase elevation angle, jointly lowering pathloss and increasing effective fading power.With a fixed vertical trajectory, flying closer to scheduled SNs can further exploit the angle gain.
  • Numerical insights: Rician achievable rates decrease faster with horizontal distance than LoS rates because reducing distance also changes the angle-dependent fading contribution.The LoS achievable rate is always larger in the reported comparison because it assumes zero outage probability, whereas Rician fading requires rate reduction to satisfy the outage constraint.
  • Numerical insights: Rician achievable rate first increases and then decreases with altitude, unlike the monotonically decreasing LoS rate.Altitude optimization balances the increased effective fading power from elevation angle against increased pathloss.

B. Single SN

The proposed RFB scheme improves single-SN rate performance by jointly optimizing angle-dependent Rician fading and the UAV’s 3D trajectory. Its gains depend on time duration, with the largest improvement at moderate duration.

  • The RFB scheme significantly outperforms the LB scheme in achieved max-min rate by using angle-dependent Rician fading and jointly optimizing the 3D trajectory.
  • Rate performance monotonically decreases as the final location’s y coordinate increases because more flight time leaves less time for data collection.
  • The RFB scheme adaptively adjusts altitude to balance distance and elevation-angle effects, achieving better elevation angles especially at T = 26 s.
  • The RFB scheme’s estimated max-min rate is close to its achieved rate, whereas the LB scheme has a considerably larger gap.
  • RFB gains are marginal for short and long durations because limited mobility or non-dominant rate regimes reduce the benefit of vertical optimization.
  • At T = 26 s, the RFB scheme attains two folds of max-min rate compared with the LB scheme.

3) Effects of UAV maximum vertical speed:

Increasing vertical maneuverability expands the UAV’s ability to balance distance and elevation-angle effects, improving RFB performance until the angle gain is fully exploited. Outage and Rician-factor requirements also reshape the preferred altitude and achievable gain.

  • 3) Effects of UAV maximum vertical speed:: Higher maximum vertical speed gives the UAV more freedom to balance distance and elevation angle, producing performance gains that saturate at high speed.
  • 3) Effects of UAV maximum vertical speed:: The RFB scheme has a negligible estimated-versus-achieved max-min-rate gap, while the LB scheme’s gap remains significantly larger.
  • 4) Effects of maximum tolerable outage probability:: With larger tolerable outage probability, the RFB trajectory tends to fly lower because angle gain is limited while lower altitude reduces pathloss.
  • 4) Effects of maximum tolerable outage probability:: At epsilon = 0.01, the RFB scheme achieves about 1.5 times the LB scheme’s max-min rate, but the gain decreases with tolerable outage probability.
  • 5) Effects of maximum Rician factor:: Max-min rate increases with maximum Rician factor and converges to the estimated LoS-channel performance when the factor is sufficiently large.
  • 5) Effects of maximum Rician factor:: The RFB performance gain first increases and then decreases with maximum Rician factor because both small and infinite-factor trajectories reduce to LB behavior.

C. Multiple SNs

With multiple SNs, the RFB scheme uses joint scheduling and angle-aware 3D trajectory design to balance users’ rates. It changes both horizontal routing and altitude to improve the network max-min rate.

  • Under T = 26 s, all schemes can only travel near each SN rather than sequentially visiting and hovering over every SN.
  • Compared with LB, RFB moves closer to SNs 2 and 3 while remaining farther from SNs 1 and 4.
  • LB’s inaccurate LoS-based trajectory limits the network max-min rate because SNs 2 and 3 have much smaller individual average achievable rates.
  • RFB raises altitude to maintain SNs 1 and 4 rates while approaching SNs 2 and 3 horizontally, thereby balancing rates across SNs.
  • RFLA provides only marginal gain over LB because it lacks vertical trajectory optimization, whereas RFFSA is comparable to RFB but requires huge altitude-search complexity.
  • The proposed angle-aware horizontal and vertical joint design effectively enhances network rate performance by balancing achievable rates among SNs.

APPENDIX

The appendix establishes convexity and successive-convex-approximation bounds used to solve the reformulated trajectory-optimization subproblems.

  • The optimal solution to Problem P4 makes constraints (16a) and (18a) active for all SNs.
  • The function psi(x, y) is convex for x > 0 and y > 0, supporting the convex reformulations used in the algorithm.
  • SCA constructs a first-order lower bound for the transformed function around a given point (x0, y0).
  • The function D/sqrt(X + x) is convex, enabling an SCA approximation for the corresponding trajectory constraint.
  • For z[m] >= H > 0, v_n[m] is concave in z[m], making constraint (26a) convex.
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