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Energy-Efficient UAV Communication with Trajectory Optimization

Yong Zeng, Rui Zhang

arXiv:1608.01828v1cs.IT

TL;DR

The paper studies energy-efficient UAV communication with a ground terminal by jointly considering communication throughput and propulsion energy through trajectory optimization. It derives an energy-consumption model, analyzes unconstrained and constrained trajectories, and reports higher energy efficiency for its proposed designs than benchmark schemes.

  • Problem

    The paper studies energy-efficient point-to-point communication between a UAV and ground terminal while accounting for the UAV’s propulsion energy consumption.

  • Method

    The paper derives a theoretical UAV propulsion-energy model, defines communication energy efficiency, and optimizes circular and generally constrained trajectories.

  • Results

    Both rate-maximization and energy-minimization designs have vanishing energy efficiency under unconstrained trajectory optimization, while the proposed designs achieve significantly higher energy efficiency than benchmark schemes.

  • Takeaways & Limitations

    Trajectory designs that explicitly account for propulsion energy are needed for energy-efficient UAV communication.

Abstract

from arXiv · show

Wireless communication with unmanned aerial vehicles (UAVs) is a promising technology for future communication systems. In this paper, we study energy-efficient UAV communication with a ground terminal via optimizing the UAV's trajectory, a new design paradigm that jointly considers both the communication throughput and the UAV's energy consumption. To this end, we first derive a theoretical model on the propulsion energy consumption of fixed-wing UAVs as a function of the UAV's flying speed, direction and acceleration, based on which the energy efficiency of UAV communication is defined. Then, for the case of unconstrained trajectory optimization, we show that both the rate-maximization and energy-minimization designs lead to vanishing energy efficiency and thus are energy-inefficient in general. Next, we introduce a practical circular UAV trajectory, under which the UAV's flight radius and speed are optimized to maximize the energy efficiency for communication. Furthermore, an efficient design is proposed for maximizing the UAV's energy efficiency with general constraints on its trajectory, including its initial/final locations and velocities, as well as maximum speed and acceleration. Numerical results show that the proposed designs achieve significantly higher energy efficiency for UAV communication as compared with other benchmark schemes.

I. INTRODUCTION

The paper formulates energy-efficient UAV communication as trajectory optimization that jointly balances throughput and propulsion energy. It develops fixed-wing energy modeling and trajectory designs, showing that unconstrained rate-maximization and energy-minimization can be energy-inefficient while proposed designs improve energy efficiency.

  • Motivation: UAV communication is constrained by finite onboard energy, making information bits per unit energy a central design objective.UAV size and weight limit available energy, while propulsion consumption can greatly exceed communication power.
  • Problem formulation: The proposed framework maximizes energy efficiency in bits/Joule by jointly optimizing communication throughput and UAV propulsion energy through trajectory design.The setting is a point-to-point link between a UAV and a ground terminal over a finite time horizon.
  • Energy model: The paper derives a fixed-wing propulsion-energy model depending on flying velocity, direction, and acceleration, then defines UAV communication energy efficiency from it.Existing heuristic models are described as primarily accounting for speed, whereas this model also relates energy consumption to velocity and acceleration.
  • Unconstrained trajectories: Both unconstrained rate-maximization and energy-minimization designs lead to vanishing energy efficiency and are therefore energy-inefficient in general.The result motivates balancing throughput against propulsion energy rather than optimizing either objective alone.
  • Circular trajectory: A circular UAV trajectory centered at the ground terminal jointly optimizes flight radius and speed to maximize communication energy efficiency.The design addresses the fixed-wing constraint that strictly zero-speed hovering can be inefficient or impossible.
  • General constraints: For general trajectory constraints, an efficient algorithm approximately optimizes energy efficiency subject to endpoint locations and velocities, maximum speed, and maximum acceleration.The algorithm uses linear state-space approximation and sequential convex optimization techniques.

II. SYSTEM MODEL AND UAV ENERGY EFFICIENCY

The paper models point-to-point UAV communication with a ground terminal and optimizes the UAV trajectory to maximize transmitted information while accounting for energy consumption. The channel uses a free-space path-loss model under fixed-altitude horizontal flight assumptions.

  • The system comprises a UAV communicating with a ground terminal over a point-to-point wireless link.
  • The design objective is to optimize the UAV trajectory for maximum information transmission to the ground terminal.
  • Energy efficiency is defined as aggregated transmitted information bits normalized by total UAV energy consumption over duration T.
  • The UAV flies horizontally at constant altitude H, with horizontal trajectory q(t)=[x(t), y(t)]^T over 0≤t≤T.
  • The UAV-to-terminal channel is assumed LoS-dominated with perfectly compensated Doppler and free-space path loss h(t)=β0d^-2(t).
  • With constant UAV transmit power P, the instantaneous capacity depends on bandwidth B, receiver noise power σ2, and reference SNR γ0=β0P/σ2.

B. UAV Energy Consumption Model and Energy Efficiency

The energy model separates communication-related and propulsion energy, then characterizes fixed-wing propulsion consumption through speed, acceleration, and flight conditions. For level flight, energy depends on motion rather than the UAV’s location, while zero speed is infeasible for remaining airborne.

  • The UAV’s total energy consumption includes communication-related energy and propulsion energy.
  • The paper ignores communication-related energy because it is usually much smaller than propulsion energy in practice.
  • For fixed-wing UAVs under normal operation, propulsion energy is modeled as a function of trajectory-derived motion variables.
  • For level flight at fixed altitude, energy consumption depends on velocity v(t) and acceleration a(t), not actual location q(t).
  • The propulsion-energy expression combines drag work, which depends on speed and heading changes, with kinetic-energy change determined by initial and final speeds.
  • If the UAV speed reaches zero, required energy tends to infinity because a fixed-wing aircraft must maintain forward motion to remain aloft.
  • In steady straight-and-level flight, power varies with speed through cubic parasitic-power and inverse-speed induced-power terms.
  • Energy efficiency is obtained by combining the transmitted-information model with the UAV propulsion-energy model.

III. ENERGY EFFICIENCY WITH UNCONSTRAINED TRAJECTORY

The paper analyzes unconstrained trajectory optimization as benchmarks and finds that optimizing throughput alone or propulsion energy alone yields generally vanishing energy efficiency. The energy-minimization solution is steady straight-and-level flight at a specific minimum-power speed.

  • Unconstrained rate maximization and energy minimization are studied to provide insight and benchmark designs.
  • A. Rate-Maximization Trajectory: Rate maximization keeps the UAV stationary above the ground terminal to maintain the best communication channel.
  • A. Rate-Maximization Trajectory: Because stationary fixed-wing flight has zero speed, its energy consumption diverges and the resulting energy efficiency is zero.
  • B. Energy-Minimization Trajectory: The unconstrained energy-minimization solution is steady straight-and-level flight at the power-minimum speed Vem.
  • B. Energy-Minimization Trajectory: Energy-minimization trajectories are non-unique because the initial location and constant flying direction can be arbitrary.
  • B. Energy-Minimization Trajectory: Among these trajectories, flights symmetric around the ground terminal provide the highest information throughput.
  • B. Energy-Minimization Trajectory: For sufficiently long operation, throughput remains finite while minimum energy grows linearly with T, causing energy efficiency to vanish.

IV. ENERGY-EFFICIENCY MAXIMIZATION WITH CIRCULAR TRAJECTORY

The paper introduces a practical circular trajectory centered at the ground terminal and jointly optimizes its radius and speed to balance throughput against propulsion cost. This design achieves strictly positive energy efficiency for any positive duration, unlike the unconstrained benchmarks.

  • The circular-trajectory design addresses the tradeoff between communication throughput and UAV propulsion energy.
  • The UAV follows a constant-speed circular path centered at the ground terminal, parameterized by radius r and speed V.
  • Smaller radius improves information throughput but requires more power for sharper heading changes.
  • Rate-maximization circular flight reduces to stationary hovering at r=0, while energy-minimization circular flight approaches straight flight as r→∞ and V=Vem.
  • For circular flight, acceleration is perpendicular to velocity and has magnitude V^2/r, linking propulsion cost to speed and radius.
  • The circular energy-efficiency objective is independent of T and reduces to a one-dimensional numerical optimization after fixing the optimal speed for each radius.
  • Because efficiency is zero at radius zero and approaches zero as radius grows without bound, a finite optimal radius exists.
  • The optimized circular trajectory achieves strictly positive energy efficiency for every T>0, outperforming the rate-maximization and energy-minimization designs.

V. ENERGY EFFICIENCY MAXIMIZATION WITH GENERALLY CONSTRAINED TRAJECTORY

The paper develops an efficient energy-efficiency maximization method for UAV trajectories subject to practical state, speed, and acceleration constraints. Discretization, convex lower bounds, and sequential optimization yield a tractable algorithm with local optimality guarantees.

  • Proposed method: The discretized formulation accommodates initial and final locations and velocities, together with maximum speed and acceleration constraints.These constraints correspond to the discrete equivalents of the practical trajectory requirements C1–C6.
  • Motivation: General trajectory optimization is difficult because it involves infinitely many position, velocity, and acceleration variables and a ratio of integrals without closed forms.The energy-efficiency objective is therefore not directly tractable in its continuous formulation.
  • Proposed method: The proposed solution combines discrete linear state-space approximation with sequential convex optimization.The trajectory is represented using discrete location, velocity, and acceleration states, while convex approximations handle the objective and constraints.
  • Optimization procedure: The original problem is neither convex nor quasi-convex because its objective has a non-concave numerator over a non-convex denominator.Sequential convex optimization replaces this formulation with fractional problems having concave numerators, convex denominators, and convex constraints.
  • Convergence: Algorithm 1 monotonically improves the energy-efficiency lower bound and converges to a point satisfying the original problem’s KKT conditions.The resulting point is guaranteed to be at least a local optimal solution.
  • Extensions: Algorithm 1 can also optimize unconstrained energy efficiency and serve rate-maximization or energy-minimization special cases.For unconstrained energy-efficiency maximization, discarding C1–C6 provides an alternative design without restricting the trajectory to a circle.

VI. NUMERICAL RESULTS

The numerical evaluation fixes representative communication and flight parameters to assess the proposed UAV trajectory designs. Under these settings, the UAV’s maximum SNR is 30dB and its transmission power is much smaller than minimum propulsion power.

  • Simulation setup: The simulations use fixed altitude H = 100m, communication bandwidth B = 1MHz, and receiver noise power spectral density N0 = −170dBm/Hz.The corresponding receiver noise power is σ2 = −110dBm.
  • Simulation setup: The UAV transmission power is P = 10dBm (0.01W), while the reference channel power is β0 = −50dB.These parameters determine the communication link conditions used in the numerical results.
  • Simulation setup: 30dB is the maximum SNR achieved when the UAV is directly above the ground terminal.This is the peak SNR under the stated channel and power assumptions.
  • Flight-energy setup: The assumed propulsion parameters give minimum-energy speed Vem = 30m/s and minimum propulsion power Pem = 100W.The UAV transmission power satisfies P ≪ Pem.

A. Unconstrained Trajectory Optimization

For unconstrained trajectories, rate maximization and energy minimization produce stationary hovering or minimum-power straight flight, whereas energy-efficiency designs achieve substantially better efficiency. The optimized circular trajectory closely matches sequential convex optimization.

  • Algorithm behavior: Algorithm 1 converges monotonically, and its adopted lower bounds become especially tight as the algorithm converges.The convergence behavior is evaluated using three energy-efficiency curves, including the accurate model and two lower-bound formulations.
  • Trajectory designs: With T = 60s, rate maximization yields stationary hovering with V = 0, while energy minimization yields straight flight at Vem = 30m/s.These designs optimize only communication rate or propulsion energy rather than their ratio.
  • Trajectory designs: The energy-efficiency-maximizing circular trajectory uses optimal radius r⋆ = 158m and speed V⋆ = 25.67m/s.The converged sequential-convex trajectory is reported as similar to this circular trajectory.
  • Performance comparison: Table I compares the four designs using average speed, acceleration, communication rate, power consumption, and overall energy efficiency.The comparison covers both communication and flight-energy metrics.
  • Performance comparison: The circular trajectory with optimized radius and speed performs comparably to sequential convex optimization in the unconstrained setup.Both energy-efficiency designs notably outperform rate-maximization and energy-minimization designs on energy efficiency.

B. Constrained Trajectory Optimization

The constrained designs compare rate, propulsion power, and energy efficiency under endpoint, velocity, speed, acceleration, and time constraints. The proposed EE-maximization trajectory balances communication quality and power consumption, achieving higher energy efficiency than the benchmark designs.

  • Compared designs: The comparison evaluates rate-maximization, energy-minimization, and EE-maximization trajectories using similar sequential convex optimization techniques.The initial points for Algorithm 1 are set to the direct path from q0 to qF.
  • Trajectory behavior: All three trajectories satisfy the endpoint constraints and are tangent to the prescribed initial and final velocity direction.The tangency follows from the imposed initial and final velocity constraints.
  • Trajectory behavior: The rate-maximization design remains near the ground terminal to preserve the best communication channel for as long as possible.By contrast, the energy-minimization trajectory mostly uses a large turning radius, reducing power consumption but increasing distance from the ground terminal.
  • Trajectory behavior: The EE-maximization design follows an approximately 8-shaped path near the ground terminal, maintaining a sufficiently good channel without excessive power consumption.This trajectory balances the competing communication-rate and propulsion-power objectives.
  • Results: The proposed designs achieve significantly higher energy efficiency than heuristic rate-maximization and energy-minimization designs for UAV communications.The constrained-trajectory comparison attributes this improvement to balancing rate maximization and power minimization.

APPENDIX A

Appendix A derives a fixed-wing UAV propulsion-energy model from aerodynamic forces and flight dynamics. It extends straight-and-level flight power analysis to banked turns and integrates instantaneous power over a trajectory.

  • Force model: The propulsion model accounts for weight, drag, lift, and thrust in fixed-wing aircraft flight.The derivation begins from the four forces acting on an aircraft aloft.
  • Force model: For subsonic flight, drag combines parasitic and lift-induced components, with parasitic drag increasing quadratically with speed.The lift-induced component results from redirecting air to generate lift.
  • Straight-and-level flight: Level flight requires lift to balance weight, so the load factor satisfies κ ≥ 1.The minimum drag occurs at κ = 1 and the drag-minimum speed Vdm = (c2/c1)1/4.
  • Straight-and-level flight: Straight-and-level power is derived as a function of flying speed and tangential acceleration by using power equal to force times speed.The formulation distinguishes forward thrust from reverse thrust during sufficiently abrupt deceleration.
  • Banked level turns: For banked level turns, the aircraft rolls so lift supplies the lateral component needed for centrifugal acceleration.The acceleration is decomposed into components parallel and perpendicular to velocity, and the load factor is related to the perpendicular component.
  • Trajectory energy: The total propulsion energy for trajectory q(t) is obtained by integrating instantaneous power determined by the velocity and acceleration vectors.The derivation concludes with the UAV energy-consumption model used in the paper.

APPENDIX B

Appendix B characterizes the relaxed propulsion-energy optimization and establishes its optimal constant-speed, zero-acceleration solution. Convexity and Jensen’s inequality provide the key proof steps.

  • Optimal solution: The resulting feasible solution also attains the optimal value of the original problem.The proof establishes that the relaxed optimum satisfies the original equality constraints.
  • Relaxed problem: The relaxed problem decomposes velocity into a nonnegative speed and a unit direction, reducing the objective to a speed-dependent function.The reduced expression contains c1V^3(t) + c2/V(t).
  • Proof strategy: The speed objective f(V) = c1V^3 + c2/V is convex for V ≥ 0.This convexity enables the Jensen-inequality argument.
  • Proof strategy: Jensen’s inequality shows that average constant speed achieves a lower objective value than any time-varying speed over the same duration.Therefore, constant speed is optimal for the relaxed problem.
  • Optimal solution: The relaxed optimum has zero acceleration and an optimal speed proportional to (c2/3c1)1/4, with arbitrary flying direction.Imposing a time-invariant direction makes the velocity-acceleration equality constraint automatically satisfied.

APPENDIX C

Appendix C proves the lower-bound construction used in the trajectory optimization. It relies on a first-order Taylor under-estimator and verifies matching gradients at the current trajectory point.

  • Lower-bound construction: The proof uses convexity of h(z) and the global under-estimator property of its first-order Taylor expansion.For any z0, h(z) is bounded below by h(z0) + h′(z0)(z − z0).
  • Lower-bound construction: Substituting the current trajectory quantities into the Taylor inequality yields the lower bound in (53).The substitutions use γ = γ0, A = H^2 + ∥qj[n]∥^2, and z = ∥q[n]∥^2 − ∥qj[n]∥^2.
  • Gradient matching: The gradients of the lower-bound expression and the original average-rate expression are identical when evaluated at q[n] = qj[n].This establishes first-order agreement at the current trajectory iterate.
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