UAV-Enabled Wireless Power Transfer: Trajectory
Design and Energy Region Characterization
Jie Xu1 , Yong Zeng2, and Rui Zhang2
of Information Engineering, Guangdong University of Technology
2 Department of Electrical and Computer Engineering, National University of Singapore
E-mail: jiexu@gdut.edu.cn, {elezeng, elezhang}@nus.edu.sg

arXiv:1706.07010v1 [] 21 Jun 2017

1 School

Abstract—This paper studies a new unmanned aerial vehicle
(UAV)-enabled wireless power transfer (WPT) system, where a
UAV-mounted energy transmitter (ET) broadcasts wireless energy
to charge distributed energy receivers (ERs) on the ground. In
particular, we consider a basic two-user scenario, and investigate
how the UAV can optimally exploit its mobility to maximize the
amount of energy transferred to the two ERs during a given
charging period. We characterize the achievable energy region
of the two ERs, by optimizing the UAV’s trajectory subject to
a maximum speed constraint. We show that when the distance
between the two ERs is smaller than a certain threshold, the
boundary of the energy region is achieved when the UAV hovers
above a fixed location between them for all time; while when their
distance is larger than the threshold, to achieve the boundary of
the energy region, the UAV in general needs to hover and fly
between two different locations above the line connecting them.
Numerical results show that the optimized UAV trajectory can
significantly improve the WPT efficiency and fairness of the two
ERs, especially when the UAV’s maximum speed is large and/or
the charging duration is long.
Index Terms—Wireless power transfer, unmanned aerial vehicle (UAV), energy region, trajectory design.

I. I NTRODUCTION
Radio frequency (RF) transmission enabled wireless power
transfer (WPT) has been regarded as a promising technique to
provide perpetual and cost-effective energy supplies to lowpower wireless networks (see, e.g., [1], [2] and the references
therein). In conventional WPT systems, dedicated energy
transmitters (ETs) are usually deployed at fixed locations to
charge distributed energy receivers (ERs) such as low-power
sensors and Internet-of-things (IoT) devices, etc.
However, due to the severe propagation loss of RF signals
over distance, the performance of practical WPT systems is
constrained by the low end-to-end WPT efficiency and the
short power coverage range. As a result, in order to provide
ubiquitous wireless energy supply to massive low-power ERs,
such fixed ETs need to be deployed in an ultra-dense manner.
This, however, would tremendously increase the cost, and
hinder the large-scale implementation of WPT systems. In the
literature, different approaches have been proposed aiming to
resolve this issue by enhancing the WPT efficiency at the link
level, including e.g. multi-antenna beamforming [3]–[5] and
waveform optimization [6]. Different from the prior studies,
in this paper we tackle this problem more cost-effectively from
a new design perspective at the system level, and propose
a radically novel architecture for WPT systems by utilizing

UAV-ET

Energy flow

ER

Fig. 1.

Illustration of the UAV-enabled WPT system.

unmanned aerial vehicles (UAVs) as mobile ETs.
UAV has recently drawn significant interests in many applications, such as weather monitoring, forest fire detection,
communication relaying, etc. Particularly, by mounting communication transceivers to low-altitude UAVs, they can be
used as aerial mobile base stations or relays to help enhance
the performance of terrestrial wireless communication systems
(see, e.g., [7] and the references therein). By optimally designing the UAV’s trajectory and employing distance-based
user scheduling, the communication link distance between the
UAV and each of its served ground users can be effectively
shortened, thus significantly improving the system throughput
[8].
Motivated by UAV-assisted wireless communications, in this
paper we study a new UAV-enabled WPT system as illustrated
in Fig. 1. Specifically, with the proposed architecture, a group
of UAVs are deployed as ETs that fly above a large area
to cooperatively charge distributed ERs on the ground. By
exploiting the fully controllable mobility introduced by UAVs
via proper trajectory design, the new system is expected to
significantly improve the WPT efficiency while reducing the
number of required ETs as compared to the conventional
WPT system with ETs deployed at fixed locations on the
ground. Notice that there have been some prior works (e.g.,
[9], [10]) that proposed to use moving ground vehicles as
mobile charging stations to wirelessly charge sensor nodes.
Different from ground vehicles that can only move along a set
of fixed trajectories in a two-dimensional (2D) area, UAVs can
be more flexibly deployed and moved in the three-dimensional
(3D) free space. Furthermore, compared to terrestrial wireless
channels that typically suffer from various impairments such
as shadowing and fading in addition to path loss, UAVs usually

possess better channels to ground ERs due to the higher chance
of having line-of-sight (LOS) links with them.
A fundamental question to be addressed in UAV-enabled
WPT systems is as follows: how to jointly design the optimal
trajectories of multiple UAVs to maximize the energy transferred to all ERs in a fair manner? This question, however, has
not yet been studied in the literature to our best knowledge,
and it is also non-trivial even for the simplest case with one
UAV and two ERs. Notice that at each time in this case, the
transferred powers from the UAV to the two ERs critically
depend on the UAV’s location, and when the UAV moves from
one ER to the other, the received power will decrease/increase
at the two ERs, respectively, thus resulting in a power trade-off
between them.
To gain the most essential insights on the optimal UAV
trajectory design, in this paper we focus on the basic UAVenabled WPT system with two users and characterize their
achievable energy region over a given charging period, which
constitutes all the achievable energy pairs of the two ERs
over all possible UAV trajectories subject to maximum speed
constraints. In order to reveal the optimal performance tradeoff between the two ERs, we aim to characterize the Pareto
boundary of the energy region and their corresponding optimal
UAV trajectory design. To this end, we adopt an energy-profile
technique as in [11], and maximize the total energy transferred to the two ERs, subject to different fairness constraints
between them.
First, we consider the ideal case by ignoring the UAV’s
maximum speed constraint, under which we find the optimal
UAV trajectory to solve the energy-profile constrained total
energy maximization problem. Based on this solution, we
then propose an efficient solution for this problem in the
general case with the UAV’s speed limit considered. It is
shown that when the distance between the two ERs is smaller
than a certain threshold, the boundary of the energy region is
achieved when the UAV hovers above a fixed location between
them during the whole charging period; while when their
distance is larger than the threshold, to achieve the boundary
of the energy region, the UAV in general needs to hover and
fly between two different locations above the line connecting
the two ERs. Numerical results show that the optimized UAV
trajectory can significantly improve the WPT efficiency as well
as the performance fairness of the two ERs, especially with
large values of the UAV’s maximum speed and/or the charging
duration.
II. S YSTEM M ODEL
We consider a two-user UAV-enabled WPT system, where
a UAV broadcasts wireless energy to charge two separated
ERs on the ground. We consider a given charging period with
duration T , denoted by T , [0, T ]. Each ER k ∈ {1, 2} has a
fixed location on the ground, denoted by (xk , 0) in 2D. Let D
denote the distance between the two ERs, then without loss
of generality we assume x1 = −D/2 and x2 = D/2. The
UAV is assumed to fly at a fixed altitude H > 0, whose timevarying location is denoted as (x(t), H), t ∈ T . Denote by

V in meter/second (m/s) the maximum possible speed of the
UAV. We then have the constraint |ẋ(t)| ≤ V, ∀t ∈ T , with
ẋ(t) denoting the time-derivative of x(t).
As the wireless channel between the UAV and each ER is
normally LOS-dominated, we adopt the free-space path loss
model similarly as in [7], [8]. At time t ∈ T , the channel
power gain from the UAV to ER k p
∈ {1, 2} is modeled as
(x(t) − xk )2 + H 2 is
hk (t) = β0 d−2
(t),
where
d
(t)
=
k
k
their distance and β0 denotes the channel power gain at a
reference distance of d0 = 1 m. Assuming that the UAV has
a constant transmit power P , the harvested power by ER k at
time t is thus given by
ηβ0 P
Qk (x(t)) = ηhk (t)P =
,
(1)
(x(t) − xk )2 + H 2
where 0 < η < 1 denotes the energy conversion efficiency of
the rectifier at each ER. Thus, the total energy harvested by
each ER k ∈ {1, 2} over the total duration is a function of the
UAV trajectory {x(t)}, which can be written as
Z T
Ek ({x(t)}) =
Qk (x(t))dt.
(2)
0

Next, we define the achievable energy region for the twouser UAV-enabled WPT system, which constitutes all the
achievable energy pairs of the two ERs under all feasible UAV
trajectories {x(t)} satisfying the maximum speed constraint,
i.e.,
E=

[

|ẋ(t)|≤V,∀t∈T

{(e1 , e2 ) : 0 ≤ ek ≤ Ek ({x(t)}), ∀k ∈ {1, 2}}.

(3)
Note that this region is generally not a convex set, due to
the non-concavity of Ek ({x(t)}) with respect to {x(t)}. Our
objective is thus to characterize the Pareto boundary of the
energy region, which consists of all energy pairs at each
of which it is impossible to improve one ER’s harvested
energy without simultaneously decreasing the other’s. Towards
this end, we maximize the total energy transferred to both
ERs, subject to different energy fairness constraints based
on the technique of energy-profile [11].1 Mathematically, we
formulate the following optimization problem with a particular
energy-profile vector α = [α1 , α2 ]:
(P1) : max

{x(t)},E

s.t.

E
Z

0

T

Qk (x(t))dt ≥ αk E, ∀k ∈ {1, 2}

|ẋ(t)| ≤ V, ∀t ∈ T ,

(4)
(5)

where the variable E denotes the total energy harvested by
the two ERs, and the constraints in (4) specify the energy
fairness between the two ERs. Here, αk ≥ 0, k ∈ {1, 2},
1 Another commonly adopted approach to characterize the Pareto boundary
of the energy region is via maximizing the weighted sum of the harvested
energy of the ERs with different user weights. However, as the energy region
E is generally a non-convex set, this approach may fail to characterize all the
Pareto boundary points (see, e.g., [12] and [13, Chapter 4.7.3]).

denotes the target ratio of ER k’s harvested energy over E
with α1 + α2 = 1.
Note that problem (P1) is difficult to solve optimally, since
it involves an infinite number of variables {x(t)} and the
constraints in (4) are non-convex due to the non-concavity
of Qk (x(t)). To tackle this difficulty, in Section III we first
consider the ideal case without the speed constraints in (5) and
solve this problem optimally. Note that the speed constraints
in (5) can be approximately ignored in practice if the product
of the charging duration T and the maximum UAV speed V
is much larger than the distance D between the two ERs. For
ease of presentation, we rewrite problem (P1) without (5) in
the following problem denoted by (P2).

A. Obtaining f (λ1 , λ2 ) by Solving Problem (7)
For any given (λ1 , λ2 ) ∈ X , problem (7) can be decomposed into different subproblems as follows.
2
X
λk Qk (x(t)), ∀t ∈ T
(9)
max ψλ1 ,λ2 (x(t)) ,
x(t)

k=1



2
X
αk λk E
max 1 −
E

(10)

k=1

Here, (9) consists of an infinite number of subproblems, each
corresponding to a time instant t. Let the optimal solutions to
(9) and (10) be denoted by x∗λ1 ,λ2 (t)’s, ∀t ∈ T , and Eλ∗1 ,λ2 ,
respectively. Note that each subproblem in (9) is irrespective
of the time index t; thus in this subsection we can denote the
(P2) : max E
{x(t)},E
function ψλ1 ,λ2 (x(t)) and the solution x∗λ1 ,λ2 (t) as ψλ1 ,λ2 (x)
s.t. (4)
and x∗λ1 ,λ2 , respectively, by dropping the index t.
P
As for problem (10), since 1 − 2k=1 αk λk = 0 holds for
In Section IV, we proceed to consider the general case of (P1)
with the UAV speed constraints in (5), and propose an efficient any given (λ1 , λ2 ) ∈ X , the objective value is always zero.
solution to (P1) based on the optimal solution obtained for In this case, we can choose any arbitrary real number as the
optimal solution Eλ∗1 ,λ2 for the purpose of obtaining the dual
(P2).
function f (λ1 , λ2 ).
Next, we only need to consider problem (9) under
III. O PTIMAL S OLUTION TO P ROBLEM (P2)
(λ1 , λ2 ) ∈ X . However, the objective function ψλ1 ,λ2 (x) in
Though problem (P2) is still non-convex, it can be easily (9) is non-convex, thus making this problem difficult to solve.
shown that it satisfies the so-called time-sharing condition Fortunately, there is only one single variable in (9). Therefore,
[14]. Therefore, the strong duality holds between (P2) and it can be solved by first deriving the first-order derivative
its dual problem. As a result, we can solve (P2) by using the of ψ
′
λ1 ,λ2 (x), denoted by ψλ1 ,λ2 (x) given in (11) at the top
Lagrange dual method [13].
of next page, and then comparing all the solutions to the
Let λk ≥ 0, k ∈ {1, 2}, denote the dual variable associated equation ψ ′
(x) = 0. As the numerator in the right-handwith the kth constraint in (4). The Lagrangian of (P2) is thus side (RHS)λ1of,λ2(11) is a polynomial function with the highest
!
Z T
2
X
power being five, it is evident that there are at most five real
L({x(t)}, E, λ1 , λ2 ) = E +
λk
Qk (x(t))dt − αk E . solutions to ψ ′
λ1 ,λ2 (x) = 0. By comparing all these solutions,
0
k=1
the optimal solution x∗λ1 ,λ2 to problem (9) can be obtained
(6)
as the one with the largest objective value. If there are more
Accordingly, the dual function of (P2) is
than one solution achieving the same largest objective value,
f (λ1 , λ2 ) = max L({x(t)}, E, λ1 , λ2 ),
(7) then we can arbitrarily choose any one of them as x∗λ1 ,λ2 .2
{x(t)},E
By substituting x∗λ1 ,λ2 to problem (9) and Eλ∗1 ,λ2 to problem
for which the following lemma holds.
(10), the function f (λ1 , λ2 ) is finally obtained.
bounded from
Lemma 3.1: In order for f (λ1 , λ2 ) to be P
In the following, we provide insights on the optimal solution
2
above (i.e., f (λ1 , λ2 ) < ∞), it must hold that k=1 αk λk = x∗
λ1 ,λ2 to problem (9). When λ1 = λ2 , we have the closed1.
form solution to (9) as follows.
P
P
Proof: Suppose that 2k=1 αk λk > 1 (or 2k=1 λk αk <
Lemma 3.2: In the case with λ1 = λ2 = λ, the optimal
1). Then by setting E → −∞ (or E → ∞), we have solution to problem (9) is given by
√

f (λ1 , λ2 ) → ∞. Therefore, this lemma is proved.
±ξ, if D > 2H/√3,
∗
x
,
(12)
λ,λ
Based on Lemma 3.1, the dual problem of (P2) is given by
0,
if D ≤ 2H/ 3,
where s
(D2) : min f (λ1 , λ2 )
p
λ1 ≥0,λ2 ≥0
D4 /4 + H 2 D2
2
X
ξ , −(D2 /4 + H 2 ) +
< D/2. (13)
2
s.t.
αk λk = 1.
(8)
√
In other words, if D > 2H/ 3, then√problem (9) has two
k=1
Then, we can solve problem (P2) by equivalently solving optimal solutions; while if D ≤ 2H/ 3, then problem (9)
has only one unique optimal solution, which is zero.
its dual problem (D2). Let the feasible
P2 set of λ1 and λ2
characterized by λ1 ≥ 0, λ2 ≥ 0, and k=1 αk λk = 1 as X .
∗
2 Note that the optimal solution E ∗
λ1 ,λ2 and xλ1 ,λ2 are not unique in
In the following, we first solve problem (7) to obtain f (λ1 , λ2 )
general, and thus they may not be feasible for the primal problem (P2). As a
under any given (λ1 , λ2 ) ∈ X , and then find the optimal λ1 result, an additional step is required to obtain the primal feasible and optimal
solution of E and x(t)’s to (P2), as will be shown in Section III-C later.
and λ2 to minimize f (λ1 , λ2 ).

λ1 (2x(t) + D)(x2 + D2 /4 + H 2 − Dx)2 + λ2 (2x − D)(x2 + D2 /4 + H 2 + Dx)2
.
(x2 + D2 /4 + H 2 − Dx)2 (x2 + D2 /4 + H 2 + Dx)2

Proof: See Appendix A.
When λ1 6= λ2 , we have the following remark.
Remark 3.1: In the case with λ1 6= λ2 , the optimal solution
x∗λ1 ,λ2 to problem (9) is unique, which satisfies
√

[−D/2, −ξ] , if D > 2H/√3 and λ1 > λ2 ,



[−D/2, 0] ,
if D ≤ 2H/√3 and λ1 > λ2 ,
x∗λ1 ,λ2 ∈

[0,
D/2]
,
if D ≤ 2H/√3 and λ1 < λ2 ,


[ξ, D/2] ,
if D > 2H/ 3 and λ1 < λ2 .
(14)
Note that Remark 3.1 is validated via extensive simulations
with different values of D and H, although it is difficult to
be rigorously proved due to the complication of the first-order
derivative ψλ′ 1 ,λ2 (x) in (11). Lemma 3.2 and Remark 3.1 are
essential to draw insights on the optimal solution to the primal
problem (P2) later.

(11)

0.2
Average harvested power at ER 2 (mW)

ψλ′ 1 ,λ2 (x) = −ηβ0 P

0.18

(Q (D/2),Q (D/2))
1

2

(Q1(0),Q2(0))

0.16
0.14
0.12
0.1
0.08

(Q (−D/2),Q (−D/2))
1

2

0.06
0.04
0.04

0.06

0.08
0.1
0.12
0.14
0.16
0.18
Average harvested power at ER 1 (mW)

0.2

Fig. 2. Energy region of the two-user UAV-enabled WPT system, where the
distance between the two ERs is D = 5√m.

In the case with D ≤ 2H/ 3, we have x⋆ (t) = 0, ∀t ∈
B. Finding Optimal Dual Solution to (D2)
T . Accordingly, the energies transferred to the two ERs
are E1⋆ = E2⋆ = T Q1 (0) = T Q2 (0), and we have E ⋆ =
With f (λ1 , λ2 ) obtained, we then solve the dual problem
2T Q1(0). This case only occurs
when α1 = α2 = 1/2.
(D2) to find the optimal λ1 and λ2 to minimize f (λ1 , λ2 ).
√
• In the case with D > 2H/ 3, the charging period T is
Note that the dual function f (λ1 , λ2 ) is always convex but
divided into two phases T1 = [0, τ ] and T2 = (τ, T ], and
generally non-differentiable [13]. As a result, problem (D2)
we have x⋆ (t) = −ξ, ∀t ∈ T1 , and x⋆ (t) = ξ, ∀t ∈ T2 ,
can be solved by subgradient based methods such as the ellipwith ξ given in (13). Accordingly, the energy transferred
soid method [15]. Note that the subgradient of the objective
to each ER k ∈ {1, 2} is Ek⋆ = τ Qk (−ξ)+(T −τ )Qk (ξ).
function f (λ1 , λ2 ) is
E⋆
1
s0 (λ1 , λ2 )
Here, τ is a constant that is chosen such that E1⋆ = α
α2 .
2
Z T
‡
⋆
⋆
⋆
Z T
In this case, we have E = E1 + E2 .
=
Q1 (x∗λ1 ,λ2 )dt − α1 Eλ∗1 ,λ2 ,
Q2 (x∗λ1 ,λ2 )dt − α2 Eλ∗1 ,λ2
Proof: See Appendix B.
0
0
‡

∗
∗
Proposition
3.2: If λ⋆1 6= λ⋆2 , the optimal solution to (P2)
= T Q1 (xλ1 ,λ2 ), T Q2 (xλ1 ,λ2 ) ,
⋆
∗
the energy
where we choose Eλ∗1 ,λ2 = 0 for simplicity, though we can is given as x (t) = xλ⋆1 ,λ⋆2 , ∀t ∈ T . Accordingly,
⋆
∗
transferred
to
each
ER
k
∈
{1,
2}
is
E
=
T
Q
(x
⋆ ,λ⋆ ), and
∗
k
k
λ
also choose any other real numbers for Eλ1 ,λ2 . Here, the
1
2
⋆
⋆
⋆
we
have
E
=
E
+
E
.
1
2
superscript ‡ denotes the transpose. Furthermore, the equality
Proof: This proposition follows based on the fact that
constraint
two inequality constraints
P2 in (8) can be viewed as P
2
when
λ⋆1 6= λ⋆2 , the optimal solution x∗λ⋆ ,λ⋆ to problem (9) is
1 − k=1 αk λk ≤ 0 and −1 + k=1 αk λk ≤ 0, whose
1
2
subgradients are given by s1 (λ1 , λ2 ) = [−α1 , −α2 ]‡ and unique as shown in Remark 3.1.
Propositions 3.1 and 3.2 provide interesting structures for
s2 (λ1 , λ2 ) = [α1 , α2 ]‡ , respectively. We denote the obtained
the
optimal solution to problem (P2).√If the distance between
⋆
⋆
dual solution to (D2) as λ1 and λ2 .
the two ERs is small (i.e., D ≤ 2H/ 3), then the UAV only
C. Constructing Optimal Solution to (P2)
needs to hover above a fixed ground location to achieve the
Based on the dual optimal solution λ⋆1 and λ⋆2 to (D2), we corresponding Pareto-optimal point, regardless of the values
need to obtain the primal optimal solution to (P2), denoted of λ⋆1 and λ⋆2 . The ground location becomes the middle point
by {x⋆ (t)} and E ⋆ . It is worth noting that when using the between the two ERs when λ⋆1 = λ⋆2√. If the distance between
Lagrange dual method to solve problem (P2) via the dual the two ERs is large (i.e., D > 2H/ 3), then in the case with
problem (D2), the optimal solution to problem (7) under λ⋆1 , λ⋆2 λ⋆1 = λ⋆2 , the UAV needs to hover at two symmetric locations
(i.e., x∗λ⋆1 ,λ⋆2 (t) and Eλ∗⋆1 ,λ⋆2 ) is the primal optimal solution to over the middle point of the two ERs successively, with one
(P2), if and only if such a solution is unique and primal at (−ξ, H) closer to ER 1 and the other at (ξ, H) closer to
feasible [13]. On the other hand, when x∗λ⋆ ,λ⋆ (t) and Eλ∗⋆ ,λ⋆ ER 2, each for a certain portion of the total time; while in the
1
2
1
2
for problem (7) are non-unique and infeasible to (P2), they case with λ⋆1 6= λ⋆2 , the UAV only needs to hover at one fixed
are not the optimal solution to (P2) in general. In this case, location to achieve the corresponding Pareto-optimal point.
Example 3.1: For better illustration, Figs. 2 and 3 show
we need some additional steps to construct the primal optimal
⋆
⋆
the
Pareto boundary of the energy region with H = 5 m,
solution {x (t)} and E to (P2).
P = 40 dBm, and η = 50%.
We consider two examples
We have the following two propositions.
√
√
Proposition 3.1: If λ⋆1 = λ⋆2 = λ⋆ , the optimal solution of when D = 5 m < 2H/ 3 and D = 8 m > 2H/ 3,
{x⋆ (t)} and E ⋆ to (P2) is given as follows:
respectively. Here, the amount harvested energy at the two
•

Static
Mobile, T = 1s, V = 10m/s
Mobile, T → ∞

0.2
Average harvested power at ER 2 (mW)

(Q1(ξ),Q2(ξ))
0.18
(Q (D/2),Q (D/2))
0.16

1

2

0.14
0.12
0.1

(Q1(0),Q2(0))

(Q1(−ξ),Q2(−ξ))

0.08
0.06

(Q (−D/2),Q (−D/2))
1

0.04
0.04

0.06

2

0.08
0.1
0.12
0.14
0.16
0.18
Average harvested power at ER 1 (mW)

0.2

Fig. 3. Energy region of the two-user UAV-enabled WPT system, where the
distance between the two ERs is D = 8 m.

ERs is normalized by the charging duration T , and thus the
average harvested power αk E/T is considered for each ER
k ∈ {1, 2}. In both figures, the solid red curve corresponds
to the Pareto boundary with the maximum speed constraints
ignored (i.e., V → ∞), which is obtained by optimally solving
(P2) under different energy-profile values of α. We have the
following observations. First, it is observed that for both cases
with D = 5 m and D = 8 m, the energy regions of the two
ERs are convex sets. This is due to the fact that problem (P2)
satisfies the time-sharing condition and thus the strong duality
holds between (P2) and its dual problem (D2). Next, it is
also observed that when the UAV hovers at the location above
ER 1 (with x(t) = −D/2, ∀t ∈ T ) or ER 2 (with x(t) =
D/2, ∀t ∈ T ), the boundary point (Q1 (−D/2), Q2 (−D/2))
or (Q1 (D/2), Q2 (D/2)) is achieved, in which the energy
transferred to ER 1 or ER 2 is maximized, respectively.
Furthermore, when D = 5 m, it is observed from Fig. 2
that when the UAV hovers above the middle location between
the two ERs (with x(t) = 0, ∀t ∈ T ), (Q1 (0), Q2 (0))
corresponds to one boundary point of the energy region.
In this case, the boundary points between (Q1 (0), Q2 (0))
and (Q1 (−D/2), Q2 (−D/2)) (or (Q1 (D/2), Q2 (D/2))) are
obtained via the UAV hovering at one fixed location between (0, H) and (−D/2, H) (or (D/2, H)). By contrast,
when D = 8 m, it is observed from Fig. 3 that when the
UAV hovers at the middle location between the two ERs,
(Q1 (0), Q2 (0)) is not on the Pareto boundary of the energy
region; while the boundary points between (Q1 (−ξ), Q2 (−ξ))
and (Q1 (ξ), Q2 (ξ)) are obtained via the UAV time-sharing
between the two different hovering locations (−ξ, H) and
(ξ, H). These observations are consistent with Propositions
3.1 and 3.2.
IV. P ROPOSED S OLUTION

TO

P ROBLEM (P1)

In this section, we consider problem (P1) in the general case
when the UAV’s maximum speed constraints in (5) are taken
into account. However, due to such constraints, it is difficult
to use the Lagrange dual method to find the optimal solution
to problem (P1). Thus, we solve (P1) based on the solution to
(P2) obtained in the previous section. Let the obtained solution
to (P1) be denoted by {x⋆⋆ (t)} and E ⋆⋆ . We first present the

following proposition.
√
Proposition 4.1: When D ≤ 2H/ 3 or λ⋆1 6= λ⋆2 for (D2),
the optimal solution to (P1) is identical to that to (P2), i.e.,
x⋆⋆ (t) = x⋆ (t), ∀t ∈ T , and E ⋆⋆ = E ⋆ . √
Proof: In the case with D ≤ 2H/ 3 or λ⋆1 6= λ⋆2 , it
is evident that the optimal solution to problem (P2) is also
feasible to (P1), as the UAV only needs to hover at one single
location without violating the speed constraints. As the optimal
value achieved by (P2) serves as an upper bound on that by
(P1), the optimal solution to (P2) is also that to (P1).
Next,
√ it only remains to consider the case when D >
2H/ 3 and λ⋆1 = λ⋆2 for (D2), in which case the UAV needs to
time-share between two hovering locations at the optimality of
problem (P2) that ignores the UAV speed constraints, and thus
the optimal solution of {x⋆ (t)} to (P2) is generally not feasible
to (P1) with the UAV speed constraints considered. As a result,
optimally solving (P1) becomes quite difficult in this case. In
the following, we first solve (P1) when α1 = α2 = 1/2 to
obtain the boundary point when the two ERs need to harvest
the same amount of energy,3 and then propose an efficient
method to obtain a suboptimal solution to (P1) when α1 6= α2 .
√
A. Optimal Solution to (P1) when D > 2H/ 3 and α1 =
α2 = 1/2
With α1 = α2 = 1/2, problem (P1) can be re-expressed as
(P3) : max E
{x(t)},E
T

s.t.

Z

0

Qk (x(t))dt ≥ E/2, ∀k ∈ {1, 2}

|ẋ(t)| ≤ V, ∀t ∈ T .
(15)
Proposition 4.2: The optimal solution to (P3) has the following hover-fly-hover structure based on two symmetric locations (−x̂, H) and (x̂, H) (with 0 < x̂ ≤ ξ): First, the UAV
hovers at the location (−x̂, H) for time t ∈ [0, T /2 − x̂/V ];
next, the UAV flies from (−x̂, H) to (x̂, H) with the maximum
speed V during the time interval t ∈ (T /2− x̂/V, T /2+ x̂/V );
finally, the UAV hovers at the location (x̂, H) for remaining
time t ∈ [T /2 + x̂/V, T ]. In other words, the optimal UAV
trajectory is
t ∈ [0, T /2 − x̂/V ]
 −x̂,
x⋆⋆ (t) =
V t − V T /2, t ∈ (T /2 − x̂/V, T /2 + x̂/V )

x̂,
t ∈ [T /2 + x̂/V, T ].
(16)
Here, we have x̂ = ξ if T ≥ 2ξ/V ; Rwhile x̂ = V T /2
T
if T < 2ξ/V . Accordingly, E ⋆⋆ = 0 Q1 (x⋆⋆ (t))dt +
RT
⋆⋆
0 Q2 (x (t))dt.
Proof: See Appendix C.
Proposition 4.2 provides key insights on how to maximize
the equal energy transferred to the two ERs
√ when the two ERs
are distributed far apart with D > 2H/ 3. If the charging
duration is long (i.e., T > 2ξ/V ), then the UAV should
hover at two symmetric locations (−ξ, H) and (ξ, H) with
equal time and travel from one location to the other with
3 Note that if α = α = 1/2, then we have λ⋆ = λ⋆ for (D2). But the
1
2
1
2
reverse is not true in general.

B. Hover-Fly-Hover
Trajectory Design for (P1) when D >
√
2H/ 3, λ⋆1 = λ⋆2 , and α1 6= α2
√
Last, we consider the case when D > 2H/ 3, λ⋆1 = λ⋆2 ,
and α1 6= α2 . In this case, we propose an efficient hoverfly-hover design based on the optimal solution to (P3) in
Proposition 4.2. In particular, denote the two hovering locations as x̂1 and x̂2 , where we have −ξ ≤ x̂1 ≤ ξ and
x̂1 ≤ x̂2 ≤ ξ. Then, the proposed hover-fly-hover design is
described as follows: First, the UAV hovers at the location
(x̂1 , H) for the time interval t ∈ T̂1 , [0, t̂]; then, the UAV
flies from (x̂1 , H) to (x̂2 , H) with the maximum speed V for
t ∈ T̂2 , (t̂, t̂ + (x̂2 − x̂1 )/V ); finally, the UAV hovers at the
location (x̂2 , H) for t ∈ T̂3 , [t̂ + (x̂2 − x̂1 )/V, T ]. In other
words, we have

t ∈ T̂1
 x̂1 ,
(17)
x̂(t) =
x̂1 + V (t − t̂), t ∈ T̂2

x̂2 ,
t ∈ T̂3 .
Here, the two hovering locations x̂1 and x̂2 as well as the
hovering duration t̂ are design variables. Accordingly, the
transferred energy to each ER k ∈ {1, 2} is given by
Z T
Êk (x̂1 , x̂2 , t̂) =
Qk (x̂(t))dt
0

=t̂Qk (x̂1 ) + (T − τ̂ − (x̂2 − x̂1 )/V )Qk (x̂2 )





x̂2 − xk
x̂1 − xk
ηβ0 P
arctan
− arctan
,
+
VH
H
H
and the total energy harvested by the two ERs is
Ê(x̂1 , x̂2 , t̂) = Ê1 (x̂1 , x̂2 , t̂) + Ê2 (x̂1 , x̂2 , t̂).
To find the optimal x̂1 , x̂2 , and t̂, we employ a twodimensional search over x̂1 ∈ [−ξ, ξ] and x̂2 ∈ [x̂1 , ξ].
Under each given pair of x̂1 and x̂2 , t̂ should be chosen such
that Ê1 (x̂1 , x̂2 , t̂)/Ê2 (x̂1 , x̂2 , t̂) = α1 /α2 . By comparing the
values of Ê(x̂1 , x̂2 , t̂), we can find the variables x̂1 , x̂2 , and
t̂ achieving the maximum total transferred energy, which are
⋆⋆
⋆⋆
denoted as x̂⋆⋆
1 , x̂2 , and t̂ , respectively. By substituting them
in (17), an efficient solution {x⋆⋆ (t)} and the corresponding
E ⋆⋆ to (P1) are finally obtained.
Remark 4.1: Although it is difficult to rigorously prove
its optimality, we expect that such a hover-fly-hover design
is quite efficient for problem (P1). For example, when the
charging duration T becomes sufficiently large, the objective
value of (P1) obtained by the hover-fly-hover design can be
shown to converge to the optimal value of (P2) asymptotically
as the effect of the finite traveling time between the two ERs
diminishes as T → ∞.
V. N UMERICAL R ESULTS
In this section, we present numerical results to evaluate the
performance of UAV-enabled WPT in a two-user setup. The

0.22
Average harvested power at each ER (mW)

the maximum speed. Otherwise, if the charging duration is
short (i.e., T ≤ 2ξ/V ), then the UAV should keep flying at
its maximum speed from one ER to the other by following a
symmetric trajectory around their middle point (0, H), without
hovering over any of them due to the insufficient charging
time.

Static
Mobile,T = 10s,V = 5m/s
Mobile,T = 10s,V = 10m/s
Mobile,T → ∞

0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

0

5

10
15
20
Distance between two ERs, D (m)

25

30

Fig. 4. Average harvested power at each ER versus the distance D between
the two ERs.

parameters are set to be the same as for Example 3.1. First,
Figs. 2 and 3√show the (normalized) energy
√ region when D =
5 m < 2H/ 3 and D = 8 m > 2H/ 3, respectively. It is
observed from Fig. 2 that when D = 5 m, the Pareto boundary
of the energy region does not depend on the charging duration
T and the UAV’s speed constraint V . This is because in this
case, the optimal solution to problem (P1) is the same as that
to (P2), in which the UAV hovers at a fixed location during
the whole charging period (see Proposition 4.1). It is observed
from Fig. 3 that when D = 8 m, the achievable energy region
under finite T and V is a non-convex set in general, and it is
smaller than that under infinite T and/or V (i.e., with the UAV
speed constraints in (5) ignored). Furthermore, the case with
T = 1 s and V = 10 m/s achieves a larger energy region than
that of the benchmark case with V = 0, which is obtained by
varying the UAV location above the line between the two ERs.
This shows that the UAV mobility is beneficial for enlarging
the energy region for ERs.
Fig. 4 shows the common average harvested power at
each ER in the case with α1 = α2 = 1/2 versus the
distance D
√ between the two ERs. It is observed that when
D ≤ 2H/ 3 = 5.77m, the static UAV design (or equivalently
with V = 0) achieves the same performance as the mobile
UAV design, regardless of the speed constraint V of the UAV
and the charging duration T . By contrast, when D > 5.77m,
the proposed mobile UAV design achieves larger average
harvested energy than the static UAV design (with the UAV
fixed at (0, H)), and the performance gain becomes more
pronounced as D increases. Furthermore, it is observed that as
the UAV’s speed limit V becomes large, the average harvested
power by the ERs is closer to the upper bound with T → ∞ or
V → ∞, as the traveling time between the two ERs becomes
insignificant.
VI. C ONCLUSION
This paper studied a two-user UAV-enabled WPT system.
We characterized the Pareto boundary of the achievable energy
region of the two ERs by optimizing the UAV’s trajectory
under the maximum speed constraints. It was shown that when
√
the distance D between the two ERs is no larger than 2/
√ 3
times of the UAV’s flying altitude H (i.e., D ≤ 2H/ 3),
each Pareto-boundary energy pair is achieved when the UAV

hovers above a fixed location between them throughout the
charging period, and the energy
√ region is a convex set. On the
other hand, when D > 2H/ 3, the Pareto-boundary of the
energy region is generally achieved when the UAV follows
a hover-fly-hover trajectory, where the UAV hovers at two
different locations in a time-sharing manner, and flies from
one location to the other with the maximum speed. It was
shown by simulations that the proposed UAV-enabled WPT
with the controllable mobile ET is a cost-effective solution to
improve the performance of the conventional WPT with fixed
ET. In future work, we will extend our study to the general
scenario with more than two ERs.
A PPENDIX
A. Proof of Lemma 3.2
In the case with λ1 = λ2 = λ, the first-order derivative
ψλ′ 1 ,λ2 (x(t)) in (11) can be simplified as
′
ψλ,λ
(x(t)) = −4ηβ0 P λx(t)×

x4 (t) + 2(D2 /4 + H 2 )x2 (t) − 3/16D4 + H 4 − H 2 D2 /2
.
2 + Dx(t))2
+ D2 /4 + H 2 − Dx(t))2 (x2 (t) + D2 /4 + H√

(x2 (t)

Then√we consider the two cases with D > 2H/ 3 and D ≤
2H/ 3, respectively.
√
In the first case with
 D > 2H/ 3, it follows that
> 0, if x(t) ∈ [−D/2, −ξ)



≤
0, if x(t) ∈ [−ξ, 0]
′
(18)
ψλ,λ
(x(t))
>
0,
if x(t) ∈ (0, ξ)



≤ 0, if x(t) ∈ [ξ, D/2].
Note that we have ξ < D/2 here, provided that H > 0.
We can accordingly conclude that ψλ,λ (x(t)) is monotonically increasing, decreasing, increasing, and decreasing over
x(t) ∈ [−D/2, −ξ), [−ξ, 0], (0, ξ), and [ξ, D/2], respectively.
Note that ψλ,λ (−ξ) = ψλ,λ (ξ). Then both x∗λ,λ (t) = −ξ and
x∗λ,λ (t) = ξ are the optimal solution √
to problem (9).
In the second casewith D ≤ 2H/ 3, we have
> 0, if x(t) ∈ [−D/2, 0)
′
ψλ,λ
(x(t))
(19)
≤ 0, if x(t) ∈ [0, D/2].
∗
In this case, xλ,λ (t) = 0 is the optimal solution to problem
(9).
B. Proof of Proposition 3.1

√
In the case with D ≤ 2H/ 3, this proposition directly
follows from Lemma 3.2, by noting that the optimal solution
x∗λ⋆ ,λ⋆ (t) = 0 to problem (9) is indeed unique, ∀t ∈ T . Substituting this into (2), we have the energy transferred to each ER
k ∈ {1, 2} to be identical as Ek⋆ = Ek ({x∗λ⋆ ,λ⋆ }) = T Qk (0).
Furthermore, note that the constraints in (4) are met with
1
equality at the optimality, and hence we have α
α2 = 1 and
accordingly α1 = α2 in this case.√
In the other case with D > 2H/ 3, problem (9) under λ⋆1 =
λ⋆2 = λ⋆ has two optimal solutions x∗λ⋆ ,λ⋆ = −ξ and x∗λ⋆ ,λ⋆ =
ξ. In this case, the optimal solution to (P2) should time-share
between the two solutions such that we have x⋆ (t) = −ξ and
x⋆ (t) = ξ for durations τ and T −τ , respectively. Furthermore,
note that the constraints in (4) should be met with equality
at the optimality. By combining the above two results, this
proposition follows.

C. Proof of Proposition 4.2
First, it is evident that at the optimal solution to (P3), the
harvested energy at the two ERs must be identical. Thus, it
is optimal for the UAV to adopt a symmetric trajectory, i.e.,
if the UAV stays in a location (−x, H) for a certain amount
of time, then it will stay in the symmetric location (x, H) for
the same duration, since otherwise, one ER will harvest more
energy than the other.
Suppose that the UAV stays at two symmetric locations
(−x, H) and (x, H) with the same duration with x ≥ 0. By
noting that Q1 (x) = Q2 (−x) and Q2 (x) = Q1 (−x), it is
evident that the harvested powers at the two ERs are identical
and each can be expressed as ψλ,λ (x), with ψλ1 ,λ2 (x) shown
in (9). As shown in Appendix A, ψλ,λ (x) is monotonically
increasing and decreasing over x ∈ [0, ξ) and [ξ, D/2],
respectively. Therefore, to maximize the transferred energy
to the two ERs, the UAV should set x as large as possible
provided that x ≤ ξ. In this case, the solution in (16) is optimal
to problem (P3) subject to the UAV’s speed constraints in (5).
Proposition 4.2 thus follows.
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