arXiv:1308.6250v1  [cs.SY]  28 Aug 2013
Circumnavigation of an Unknown Target Using UAVs with
Range and Range Rate Measurements
Yongcan Cao, Jonathan Muse, David Casbeer, and Derek Kingston
Abstract— This paper presents two control algorithms en-
abling a UAV to circumnavigate an unknown target using range
and range rate (i.e., the derivative of range) measurements.
Given a prescribed orbit radius, both control algorithms (i)
tend to drive the UAV toward the tangent of prescribed orbit
when the UAV is outside or on the orbit, and (ii) apply zero
control input if the UAV is inside the desired orbit. The
algorithms differ in that, the first algorithm is smooth and
unsaturated while the second algorithm is non-smooth and
saturated. By analyzing properties associated with the bearing
angle of the UAV relative to the target and through proper
design of Lyapunov functions, it is shown that both algorithms
produce the desired orbit for an arbitrary initial state. Three
examples are provided as a proof of concept.
Index Terms— Circumnavigation, UAV
I. INTRODUCTION
The use of Unmanned Aerial Vehicles (UAVs) in both
civilian and military applications has blossomed due to the
advancements in aerospace technology. Because they can
potentially be built smaller, lighter, and cheaper, UAVs have
promising benefits over traditional manned aircraft, yet a key
technological challenge remains in designing proper control
strategies that will provide a specific degree of autonomy [1].
The promise of autonomy has benefits of reduced costs in
human management and maintaining a robust and stable
performance.
One typical application of UAVs is the surveillance and
reconnaissance mission [2]–[5]. The objective is to gather
and manage information from various sensors. In a broad
sense, a successful surveillance and reconnaissance mission
is able to improve situation awareness in an unknown envi-
ronment through information acquisition. For instance, from
the information perspective, a UAV can be deployed to obtain
valuable information regarding a target if it can orbit around
this target at a desired distance. This type of circular motion
around a target is called circumnavigation [4], [5]. In [4],
the circumnavigation problem using range measurements was
solved under a unified localization-and-control framework,
where two algorithms, namely localization and control, were
proposed to guarantee stability. In particular, the concept of
persistent excitation (p.e.) plays a crucial role in the stability
analysis. When the p.e. condition is satisfied, exponential
convergence of the localization algorithm and the control
algorithm is always guaranteed. In [5], the circumnavigation
problem was studied by assuming the availability of bearing
The authors are with the Control Science Center of Excellence, Air
Force Research Laboratory, Wright-Patterson AFB, OH 45433. Approved
for public release; distribution unlimited, 88ABW-2013-1085.
Corresponding Author: Yongcan Cao (yongcan.cao@gmail.com)
measurements. A similar localization-and-control framework
was used with the aid of the p.e. concept. In addition to
the different measurement types, another major difference
between [4] and [5] is the dynamics used to model the agent.
In particular, the single-integrator kinematics was used in [4]
while the unicycle model was used in [5]. One common
feature between [4] and [5] is that the (accurate) location
of the UAV is needed to circumnavigate the target.
In an ideal situation, various measurements, such as bear-
ing and location, can be used in the controller design for the
circumnavigation mission. However, in some adversarial situ-
ations, such as GPS-denied environments due to jamming [6]
or spoofing [7], measurements become more limited. For
instance, under GPS-denied environments, range measure-
ments are possible [8] while other measurements such as
bearing and location are not possible. It is thus challenging
to design proper controller algorithms with measurement
limitations. This is the main motivation of the paper. Upon
solving the problem successfully, our subsequent objective
is to consider more general navigation and control scenario
under GPS-denied environments.
This paper considers the circumnavigation problem when
(1) the unicycle model is used to model the dynamics of
UAVs and (2) only range and range rate measurements are
available. The consideration of the unicycle model is more
appropriate considering the dynamics of UAVs, at the cost
of complex control algorithm design and stability analysis
due to the nature of nonlinearity and the underactuated
system dynamics. The consideration of range and range
rate measurements are meaningful when the UAVs have
limited information regarding the target and itself due to
adversarial environments, such as GPS-denied environments.
Some related work was reported in [9], [10] under these
two assumptions. In particular, the target following problem,
whose objective is to have a robot orbit around a moving
target with some desired radius, was considered in [9].
The proposed sliding mode controller can guarantee stability
when the robot is out of some non-empty domain. In other
words, no global stability is guaranteed. In [10], the problem
of detecting a target was considered where the objective
is to drive the robot close to a stationary target. Since no
special motion of the robot is needed upon approaching
the target, the detecting problem considered in [10] is less
challenging than the circumnavigation problem. The novel
contributions of the paper include a complete global stability
analysis for circumnavigation of an unknown target using
only range and range rate measurements. Related work
includes algorithms for reaching targets with using only
range and range rate measurements (but no circumnaviga-
tion), or circumnavigation using bearing measurements, or
circumnavigation without a guarantee of global stability.
To our best knowledge, achieving circumnavigation globally
using only range and range rate measurements has not been
demonstrated previously.
The paper is organized as follows. In Section II, the
problem studied in this paper is introduced. A control al-
gorithm is then proposed in Section III. Sections IV and V
analyze, respectively, the stable circular motion using the
proposed control algorithm and the conditions under which
the stable circular motion can be achieved. In Section VI,
another control algorithm is proposed and analyzed. Both
the algorithm presented in Section III and the algorithm
presented in VI have unique features. In Section VII, three
simulation examples are provided as a demonstration of the
effectiveness of the proposed control algorithms. Section VIII
is a brief summarization of the paper with a short discussion
of future research directions.
II. PROBLEM STATEMENT
In this paper, the dynamics of the UAV are modeled as
˙x = V cos(ψ),
˙y = V sin(ψ),
t ≥0
(1)
˙ψ = ω,
where [x, y]T is the location of the UAV, V is the velocity
of the UAV, and ψ is the heading of the UAV. Let there
be a static unknown target T . The objective is to design ω
such that the UAV can circumnavigate the unknown target
at some desired distance rd using range and range rate
measurements. Two assumptions are made regarding the
UAV and its measurement capabilities:
1. The velocity V is constant; and
2. Both range and range rate are measurable.
In [5], the circumnavigation problem was solved by as-
suming the availability of the bearing angle and the location
of the UAV. By estimating the position of the target, the
control algorithm was built based on the estimated position
of the target. In Section III, a novel control algorithm
is proposed based on range and range rate measurements
without estimating the position of the target.
III. ALGORITHM
Before describing the proposed control algorithm, let’s
first take a look at a typical scenario in Fig. 1 where the UAV
is outside the black solid circle with a radius rd centering at
the target T . The proposed control algorithm is a feedback
control law by comparing the difference between the desired
change rate of r2(t) and the actual change rate of r2(t). The
desired change rate of r2(t) is the change rate of r2(t) when
the UAV moves towards the tangent point on the black solid
circle. By computation, the desired change rate of r2(t) is
given by 2r(t)V cos(π−sin−1( rd
r(t))). The actual change rate
of r2(t) is the change rate of r2(t) when the UAV moves
along its current heading. Note that the actual change rate
ra
rd
T
r(t)
θb
V
Fig. 1. An illustration of variables and notations used in the paper. The blue
triangle denotes the UAV. T denotes the target. The blue arrow denotes the
heading of the UAV. r(t) denotes the range between the UAV and the target.
V denotes the (constant) velocity of the UAV. θb denotes the bearing angle.
rd and ra denotes, respectively, the desired radius and the actual radius.
of r2(t) is given by 2r(t) ˙r(t). Therefore, when the UAV
is outside the black solid circle, ω = k[2r(t)V cos(π −
sin−1( rd
r(t))) −2r(t) ˙r(t)], where k is a positive constant.
When the UAV is inside the black solid circle, no control
action is applied to the UAV. The main purpose is to drive
the UAV outside the black solid circle. As a summarization,
the proposed control algorithm for ω is given by
 k[2r(t)V cos(π −sin−1( rd
r(t))) −2r(t) ˙r(t)],
r(t) ≥rd,
0,
otherwise,
(2)
where k is a positive constant.1 Based on the control algo-
rithm (2), when r(t) < rd, the UAV will keep its current
course until the condition r(t) ≥rd is triggered. When
r(t) < rd, the condition r(t) ≥rd is always triggered after
a finite period of time since the UAV is moving along a line
with a nonzero velocity. As a consequence, the key is to
study the case when r(t) ≥rd, which is the focus of the
stability analysis in Section V.
IV. STABLE MOTION
Since the focus is on the circular motion of the UAV
around some unknown target, it is assumed that such a stable
circular motion exists. Here the stable circular motion is
defined as follows.
Definition 4.1: A stable circular motion refers to the be-
havior that the UAV, with dynamics (1), moves around a
target with a constant speed and a constant radius.
In order to characterize a circular motion, three elements,
namely, the center, the direction of rotation, and the radius
are needed. The location of the target is the center of the
orbit. In the following of the section, the focus is on deriving
the direction of rotation and the radius.
Let’s first derive the radius. Note that the radius cannot
be smaller than rd due to the control algorithm for the case
r(t) < rd. Let the radius of the stable circular motion be
given by ra(≥rd). Then the magnitude of the nominal
1In practical implementation, ˙r(t) can be estimated effectively using a
stable linear filter.
angular velocity is given by
ω⋆= V
ra
.
(3)
In other words, the direction of rotation is either clockwise
or counterclockwise. Since ra ≥rd, it follows from (2) that
ω = 2kraV cos(π −sin−1(rd
ra
)),
(4)
where ˙rd = 0 since the radius is equal to ra, which is a
positive constant. Under the assumption that a stable circular
motion exists, it follows that the magnitude of ω should
match ω⋆. Equivalently, it can be obtained that
ω⋆= |ω| = 2kraV
cos(π −sin−1(rd
ra
))
 .
(5)
By definition,
cos(π −sin−1(rd
ra
))

= cos(sin−1(rd
ra
))
=
s
1 −
r2
d
(ra)2 .
(6)
By substituting (6) into (5), one can obtain
V
ra
= 2kraV
s
1 −
r2
d
(ra)2 ,
(7)
which implies that
1
4k2 = (ra)4

1 −
r2
d
(ra)2

.
(8)
By computation, ra is given by
ra =
s
r2
d +
p
r4
d + 1/k2
2
.
(9)
Therefore, if a stable circular motion exists, the radius is
given by ra defined in (9).
Next, the direction of rotation is derived. Note that the
magnitude of ω, i.e., V
ra , denotes the nominal angular velocity
and the sign of ω when r(t) ≡ra denotes the direction
of rotation. More specifically, if the sign of ω is positive,
the heading angle increases, indicating that the UAV rotates
counter clockwise. Analogously, if the sign of ω is negative,
the UAV rotates clockwise. From (4), the sign of ω is
determined by the sign of cos(π−sin−1( rd
ra )). Since ra > rd
and sin−1( rd
ra ) ∈(0, π
2 ), π −sin−1( rd
ra ) ∈( π
2 , π). It follows
that cos(π −sin−1( rd
ra )) < 0. Combining with the previous
analysis implies that the UAV rotates clockwise when k > 0.
As a summarization, the following theorem illustrates the
properties of the stable circular motion.
Theorem 4.2: Consider system dynamics (1) subject to
control input (2). If a stable circular motion exists, the radius
is given by ra in (9). In addition, the UAV rotates clockwise
when k > 0.
From the previous analysis, it can be observed that the
actual radius ra is greater than the desired radius rd. In order
to guarantee that ra matches the desired distance, one can
choose rd as
˜rd
△= rd
s
1 −
1
4k2r4
d
.
(10)
By using ˜rd to replace rd in (2), it can be computed from (9)
that ra = rd, which indeed meets our objective as mentioned
in Section II. To guarantee the existence of a positive ˜rd, it
is required that k >
1
2r2
d .
Up to now, the property of the stable circular motion
using (2) was analyzed under the assumption that the stable
circular motion does exist. However, it remains unclear if
the assumption stands. In Section V, it is shown that this
assumption always holds for any nonzero k satisfying k >
1
2r2
d .
V. MAIN RESULT
This section proves the main result of the paper given in
the following theorem:
Theorem 5.1: Consider the UAV dynamics in (1) subject
to the control policy in (2). If k >
1
2r2
d , then r(t) →ra as
t →∞, where ra is defined in (9).
Before proving Theorem 5.1, a few definitions and lemmas
are necessary.
Definition 5.2: The circle centered at the unknown target
with a radius rd is defined as Cd. The UAV is inside
(respectively, outside) Cd if r(t) < rd (respectively, r(t) ≥
rd).
Definition 5.3: Denote the reference vector as the vector
from the current location of the UAV to the target. The
bearing angle θb(t) ∈[0, 2π) at time t is defined as the
angle from the reference vector to the current heading of the
UAV measured counterclockwise.
As an example, in Fig. 1, the black solid circle denotes
Cd as defined in Definition 5.2. An illustration of the bearing
angle θb is also shown in this figure.
Lemma 5.1: Consider the UAV dynamics in (1) subject
to the control policy in (2). Let there be t0 ≥0 such that
r(t0) ≥rd and θb(t0) ∈(sin−1(
rd
r(t0)), 2π −sin−1(
rd
r(t0))),
then r(t) ≥rd, ∀t ≥t0.
Proof: The proof of the lemma can be divided into the
following two steps:
Step 1: θb(t) ∈[sin−1( rd
r(t)), 2π −sin−1( rd
r(t))) holds
for all t ≥t0. Based on the proposed algorithm (2),
ω < 0 at t = t0 since 2r(t) ˙r(t) > 2r(t)V cos(π −
sin−1( rd
r(t))) when t = t0. As a consequence, the UAV
will rotate clockwise initially. Note that both 2r(t) ˙r(t) and
2r(t)V cos(π −sin−1( rd
r(t))) are continuous with respect
to t. Therefore, 2r(t) ˙r(t) > 2r(t)V cos(π −sin−1( rd
r(t)))
always holds before 2r(t) ˙r(t) = 2r(t)V cos(π−sin−1( rd
r(t)))
happens. Consequently, the UAV will stop rotating clockwise
once 2r(t) ˙r(t)
= 2r(t)V cos(π −sin−1( rd
r(t))). Indeed,
2r(t) ˙r(t) = 2r(t)V cos(π −sin−1( rd
r(t))) if and only if
θb(t) = sin−1( rd
r(t)) or θb(t) = 2π −sin−1( rd
r(t)). Thus,
θb(t) ∈[sin−1( rd
r(t)), 2π −sin−1( rd
r(t))]. To prove Step 1,
it suffices to show that θb(t) = 2π −sin−1( rd
r(t)) cannot
hold. When θb(t0) ∈(sin−1(
rd
r(t0)), 2π −sin−1(
rd
r(t0))) and
ω = 0, θb(t) = 2π −sin−1( rd
r(t)) never holds for all t ≥t0
because the UAV moves along a straight line and the straight
line is always outside the circle Cd. As a consequence,
θb(t) = 2π −sin−1( rd
r(t)) never holds for all t ≥t0 when
ω ≤0 and θb(t0) ∈[sin−1(
rd
r(t0)), 2π −sin−1(
rd
r(t0))).
Combining the previous arguments completes the proof of
Step 1.
Step 2: r(t) ≥rd for all t ≥t0. From Step 1, it is known
that θb(t) ∈[sin−1( rd
r(t)), 2π−sin−1( rd
r(t))). When r(t) = rd,
it follows that sin−1( rd
r(t)) = π
2 . Combining with the previous
two sentences indicates that θb(t) ∈( π
2 , 3π
2 ] at the time when
r(t) = rd. Noticing that
˙r = −V cos(θb(t)),
(11)
it then follows that ˙r ≥0 at the time when r(t) = rd. This
implies that the UAV cannot get any closer to the target if
r(t) = rd. At the time t⋆when r(t⋆) becomes larger than rd,
the bearing angle has to be in the set ( π
2 , 3π
2 ), which satisfies
the condition that θb(t⋆) ∈[sin−1(
rd
r(t⋆)), 2π−sin−1(
rd
r(t⋆))).
By repeating the analysis in Steps 1 and 2, it is clear that
the UAV will never move inside Cd.
Lemma 5.2: Consider the UAV dynamics in (1) subject to
the control policy in (2). The UAV can only move inside Cd
at most once.
Proof: When r(te) = rd, θb(te) ∈[0, π
2 ) ∪( 3π
2 , 2π) in order
to guarantee that the UAV enters Cd. Recall that no control
input is imposed on the UAV when it is inside Cd. Then at
the time tx when it exits Cd,
θb(tx) =
 π −θb(te),
θb(te) ∈[0, π
2 ),
3π −θb(te),
θb(te) ∈( 3π
2 , 2π),
because the UAV moves along a straight line due to the
fact ω = 0. As a consequence, θb(tx) ∈( π
2 , 3π
2 ). When
r(tx) = rd, it follows that sin−1(
rd
r(tx)) =
π
2 . Therefore,
θb(tx) ∈(sin−1(
rd
r(tx)), 2π −sin−1(
rd
r(tx))) when r(tx) =
rd. By considering the current time tx be the time t0 in
Lemma 5.1, it follows from Lemma 5.1 that the UAV will
never move inside Cd again. Therefore, the UAV can only
move inside Cd at most once.
Lemma 5.3: Consider the UAV dynamics in (1) subject to
the control policy in (2). For any θb(0), there exists t⋆≥0
such that θb(t) ∈[0, π] for any t ≥t⋆.
Proof: By Lemma 5.2, the UAV can move inside Cd at most
once. When the UAV never moves inside Cd, let t1 = 0.
When the UAV moves inside Cd once, let t1 be the time
when the UAV moves from inside Cd to outside Cd. It is
clear that t1 is finite. The lemma is proved if the following
two statements are valid:
(1) For any θb(0), there exists t⋆≥t1 such that θb(t⋆) ∈
[0, π]; and
(2) Once θb(t⋆) ∈[0, π] for some t⋆≥t1, θb(t) ∈[0, π] for
any t ≥t⋆.
The first statement is proved by considering the following
three cases:
(i) θb(t1) ∈(π, 2π −sin−1(
rd
r(t1))): As defined in Defini-
tion 5.3, it can be obtained that
˙θb(t) = ω −

−V sin(θb(t))
r(t)

,
(12)
where −V sin(θb(t))
r(t)
is the angular velocity of the refer-
ence vector and ω denotes the angular velocity of the
heading. Therefore, ˙θb(t) < 0, i.e., θb(t) will decrease,
whenever θb(t) ∈(π, 2π −sin−1( rd
r(t))) because ω < 0
while −V sin(θb(t))
r(t)
> 0. When ω is (approximately)
zero, −V sin(θb(t))
r(t)
is (approximately)
V rd
r2(t). Because
both ω and −V sin(θb(t))
r(t)
are continuous with respect to
t, ˙θb(t) is always upper bounded by some negative con-
stant. Similarly, when −V sin(θb(t))
r(t)
is (approximately)
zero, ω is upper bounded by some negative constant,
indicating that ˙θb(t) is always upper bounded by some
negative constant as well. Therefore, θb(t) ≤π in finite
time. Noting that θb(t1) ∈(π, 2π −sin−1(
rd
r(t1))) ∈
(sin−1(
rd
r(t1)), 2π −sin−1(
rd
r(t1))), it follows from Step
1 in the proof of Lemma 5.1 that θb(t) cannot get
smaller than sin−1( rd
r(t)), which implies that θb(t) ≥
sin−1( rd
r(t)).
(ii) θb(t1) = 2π−sin−1(
rd
r(t1)): Under this case, the UAV is
heading towards the tangent point such that the bearing
is 2π −sin−1(
rd
r(t1)). By computation, ω = 0, which
implies that the UAV will move along a straight line
towards the tangent point. As a consequence, the UAV
will move towards the tangent point whenever r(t) >
rd. Given a constant nonzero velocity, it takes a finite
period of time before r(t) = rd happens. Notice that
r(t) = rd cannot hold for an arbitrary period of time
because otherwise a contradiction happens by noting
that (i) ω = 0 based on (2) during that period of time,
indicating that the UAV cannot rotate; and (ii) the UAV
has to rotate such that r(t) = rd holds for that period of
time. This implies that r(t) will increase to be greater
than rd as soon as r(t) = rd happens. The bearing angle
θb(tf) must be in the interval ( π
2 , 3π
2 ) at the time when
r(t) increases to be greater than rd. When θb(tf) ∈
(π, 3π
2 ), it follows from Case (i) that θb(t) will be in
the set [0, π] after a finite period of time. When θb(tf) ∈
( π
2 , π], it is already in the set [0, π].
(iii) θb(t1) = (2π −sin−1(
rd
r(t1)), 2π): If θb(t) = (2π −
sin−1( rd
r(t)), 2π) always holds, it takes a finite period of
time before r(t) ≤rd happens. Since the UAV never
moves inside Cd for t ≥t1, either Case (i) or Case (ii)
will happen after a finite period of time. By following
the analysis in Cases (i) and (ii), θb(t) will be in the set
[0, π] after a finite period of time.
To prove the second statement, it is essential to study ˙θb(t)
when θb(t) = 0 or θb(t) = π. Whenever θb(t) = 0 or
θb(t) = π, the change rate of the reference vector defined in
Definition 5.3 is zero. Then it follows that ˙θb(t) = ω. When
θb(t) = 0, it can be computed that
ω = k[2r(t)V cos(π −sin−1( rd
r(t))) + 2r(t)V ] > 0,
where (11) was used to derive the equality. This indicates that
θb(t) will increase as soon as θb(t) = 0 happens. Similarly,
when θb(t) = π, one can obtain that
ω = k[2r(t)V cos(π −sin−1( rd
r(t))) −2r(t)V ] < 0,
which indicates that θb(t) will decrease as soon as θb(t) = π
happens. Therefore the second statement holds as well.
With the previous lemmas, we next prove Theorem 5.1
restated as:
Theorem 5.1: Consider the UAV dynamics in (1) subject to
the control policy in (2). If k >
1
2r2
d , then r(t) →ra as
t →∞, where ra is defined in (9).
Proof: Based on Lemmas 5.2 and 5.3, there exists a time
instant t⋆such that θb(t) ∈[0, π] and r(t) ≥rd for any
t ≥t⋆. Therefore, θb(t) is continuously differentiable with
respect to t for t ≥t⋆. In addition, r(t⋆) remains bounded
because the velocity of the UAV is bounded. For t ≥t⋆,
consider the following Lyapunov function candidate given
by
V = 1 −sin(θb(t)) + ϕ,
where
ϕ
=
R r
ra( 1
ra
−
1
z
+ 2kz cos sin−1( rd
z ) −
2kra cos sin−1( rd
ra ))dz
≥0. Note that V is continuous
with respect to t and V ≥0. From (3) and (5), one can
obtain that
1
ra
=
2kra cos sin−1( rd
ra ). Then V can be
rewritten as
V = 1 −sin(θb(t)) +
Z r
ra
(−1
z + 2kz cos sin−1(rd
z ))dz.
For t ≥t⋆, the derivative of V along the solution of (1)
using (2) is given by
˙V = −cos(θb(t)) ˙θb(t) + ˙ϕ
= −cos(θb(t))
"
2krV

cos(π −sin−1(rd
r )) + cos(θb(t))

+ V sin(θb(t))
r
#
−

2kr cos sin−1(rd
r ) −1
r

V cos(θb(t))
= V cos(θb(t))

−2kr cos(θb(t)) −sin(θb(t))
r
+ 1
r

,
where (11) and (12) were used to derive the second equality.
When θb(t) ∈[ π
2 , π],
˙V ≤0 because cos(θb(t)) ≤0
and sin(θb(t)) ≤1. When θb(t) ∈[0, π
2 ),
˙V
< 0 if
−2kr cos(θb(t)) −sin(θb(t))
r
+ 1
r < 0. When k >
1
2r2
d , it
follows from the fact r ≥rd that 2kr2 > 1. Therefore,
−2kr cos(θb(t)) −sin(θb(t))
r
+ 1
r
< −cos(θb(t))
r
−sin(θb(t))
r
+ 1
r ≤0.
Therefore, ˙V ≤0 for θb(t) ∈[0, π]. Because ˙V is uniformly
continuous when r ≥rd and θb(t) ∈[0, π], it follows from
Lemma 4.3 in [11] that ˙V →0 as t →∞. When k >
1
2r2
d ,
˙V = 0 implies that θb = π
2 . It then follows from (11) that
when θb(t) = π
2 , r(t) is constant. It then follows from the
analysis in Section IV that r(t) = ra. Therefore, θb(t) →π
2
and r(t) →ra as t →∞.
When θb(t) ∈[0, 2π) rather than θb(t) ∈[0, π), ˙V is not
necessarily negative semi-definite. In particular, when θb(t)
is just a bit larger than 3π
2 , ˙V becomes positive for some
bounded k and r. Therefore, Lemma 5.3 is essential in the
proof of Theorem 5.1.
Note from Theorem 5.1 that θb(t) →π
2 and r(t) →ra as
t →∞. In order to guarantee that θb(t) →π
2 and r(t) →rd
as t →∞, one can simply replace rd by ˜rd given in (10).
VI. A NEW ALGORITHM AND ITS STABILITY
Although the algorithm (2) can guarantee a stable circular
motion, the radius of the circular motion might not match the
desired value. In fact, the difference between them is deter-
mined by various parameters. As mentioned in Section IV,
one way to guarantee that the desired radius can be reached
is to intentionally change rd, which requires a design of rd
a priori. We next propose a new algorithm, which does not
require the intentional change of rd. That is, rd is exactly
the desired radius.
The proposed control algorithm ω is given as
 ksign(V cos(π −sin−1( rd
r(t))) −˙r(t)),
r(t) ≥rd,
0,
otherwise,
(13)
where sign(·) is the signum function and k is a positive
constant. It can be considered as a non-smooth version of
the algorithm (2). In addition to the benefit that rd is chosen
as the (exact) desired radius, another (physical) benefit is that
the control input is always saturated, meaning that |ω| < k.
From the application’s perspective, this control algorithm
is implementable even if the control algorithm (2) is not
implementable in certain cases. In the following part of
the section, it is shown that this algorithm can solve the
circumnavigation problem as well.
Theorem 6.1: Consider the UAV dynamics in (1) subject
to the control policy in (13). If k > V
rd , then r(t) →rd as
t →∞. In particular, the UAV will rotate clockwise around
the target ultimately.
Proof: By following a similar analysis to Lemmas 5.1, 5.2,
and 5.3, one can obtain that the results in these lemmas are
still valid under the control algorithm (13). By recalling the
statements in Lemma 5.3, for any θb(0), there exists t⋆≥0
such that θb(t) ∈[0, π] and r(t) ≥rd for any t ≥t⋆. It is
shown next that θb(t) ∈[0, π
2 ] in finite time. Similar to the
analysis in the proof of Lemma 5.3, θb(t) ≥0 for all t ≥t⋆
because ˙θb(t) > 0 once θb(t) = 0. Our focus next is to show
that θb(t) ≤π
2 in finite time. For t ≥t⋆, since r(t) ≥rd, it
follows from (12) that
˙θb(t) =ω + V sin(θb(t))
r(t)
=ksign(cos(π −sin−1( rd
r(t))) + cos(θb(t)))
+ V sin(θb(t))
r(t)
≤ksign(cos(θb(t))) + V
rd
by noting that cos(π −sin−1( rd
r(t))) ≤0 when r(t) ≥rd,
sin(θb(t)) ≤1, and r(t) ≥rd. As a consequence, ˙θb(t) <
−k + V
rd whenever θb(t) > π
2 . Because k > V
rd , it follows
that ˙θb(t) < 0 whenever θb(t) >
π
2 . Therefore, it takes a
finite period of time before θb(t) = π
2 happens. In addition,
θb(t) is always not greater than π
2 afterwards.
Because θb(t) ∈[0, π
2 ] in finite time, there must exist a
positive constant t ≥t⋆such that θb(t) ∈[0, π
2 ] for all t ≥t.
For t ≥t, consider the Lyapunov function candidate given
by
V = r(t) −rd.
Because r(t) ≥rd, V ≥0. In addition, the derivative of V
is given by
˙V = ˙r(t) = −V cos(θb(t)) ≤0.
Note that ˙V is uniformly continuous with respect to t. It
then follows from Lemma 4.3 in [11] that ˙V →0 as t →∞.
That is, cos(θb(t)) →0 as t →∞. Equivalently, θb(t) →π
2
as t →∞by noting that θb(t) ∈[0, π
2 ]. When θb(t) =
π
2 , r(t) remains constant. If the constant is not rd, ˙θb(t) =
−k +
V
r(t) < 0, which contradicts the fact that θb(t) = π
2 .
Therefore, θb(t) →π
2 and r(t) →rd as t →∞. This proves
the first statement.
The second statement follows from the fact that the
bearing angle θb(t) ∈[0, π
2 ] in finite time (see the second
paragraph of the proof).
Remark 6.2: Until now, it is assumed that k is a positive
constant. When k is a negative constant, stable circular
motions can be obtained for the UAV dynamics in (1) subject
to the control algorithms in (2) and (13). In particular, the
radius of the stable circular motion remains ra under the
control algorithm (2) if k < −1
2r2
d . The radius of the stable
circular motion remains rd under the control algorithm (13)
if k < −V
rd . However, the UAV will ultimately rotate counter
clockwise around the target.
VII. SIMULATION
This section presents three simulation examples. The ve-
locity V is chosen to be 1. The location of the unknown
target is [0, −10]T. The desired radius rd is chosen as 10.
The initial state of the UAV is randomly chosen from the
set [0, 10] × [0, 10] × [0, 2π). For the control algorithm (2),
k is chosen to be 0.01. For the control algorithm (13), k is
chosen to be 0.12. It can be verified that the conditions in
Theorems 5.1 and 6.1 are satisfied.
−15
−10
−5
0
5
10
15
−20
−15
−10
−5
0
x
y
Fig. 2.
The trajectory of the UAV under (2) with rd = 10. The red triangle
represents the starting position of the UAV. The dashed line represents the
desired trajectory.
0
20
40
60
80
100
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
time
Radial error
Fig. 3.
The tracking error of the UAV under (2) with rd = 10.
Figs. 2 and 3 show, respectively, the trajectory of the UAV
and the tracking error (i.e., r(t) −rd) under the control
algorithm (2). By computation from (9), ra = 10.9868,
which is consistent with the simulation example. It can be
observed from Fig. 2 that a circular motion is obtained
ultimately. In addition, the UAV rotates clockwise around
the target. Meanwhile, it can be observed that r(t) ≥rd
always holds.
By replacing rd in (2) by ˜rd in (10), the trajectory of the
UAV and the tracking error under the control algorithm (2)
are given in, respectively, Figs. 4 and 5. It can be noticed
that the actual radius of the circular motion is exactly
10. Therefore, when rd is chosen properly, the UAV will
circumnavigate around the target at the desired distance.
Figs. 6 and 7 show, respectively, the trajectory of the
UAV and the tracking error under the control algorithm (13).
Again, a circular motion is obtained ultimately. In addition,
−15
−10
−5
0
5
10
15
−20
−15
−10
−5
0
5
x
y
Fig. 4.
The trajectory of the UAV under (2) with rd being replaced by ˜rd
in (10). The red triangle represents the starting position of the UAV. The
dashed line represents the desired trajectory.
0
20
40
60
80
100
−1
0
1
2
3
4
5
6
7
8
time
Radial error
Fig. 5.
The tracking error of the UAV under (2) with rd being replaced
by ˜rd in (10).
the final tracking error goes to zero. By comparing the
simulation results using (2) and those using (13), it can be
seen that the convergence speed using (2) is faster than that
using (13). This is mainly due to the fact that the control
algorithm (2) is not saturated while the control algorithm (13)
is saturated.
VIII. CONCLUSION AND FUTURE WORK
In this paper, we considered the circumnavigation of UAVs
around some unknown target at a desired distance using
range and range rate measurements. Two control algorithms
were proposed to guarantee the accomplishment of the
circumnavigation mission. The stability analysis was based
on analyzing the properties associated with the bearing angle
and designing proper Lyapunov functions. Considering the
fact that range and range rate measurements are noisy in
−10
−5
0
5
10
15
20
−20
−15
−10
−5
0
5
10
x
y
Fig. 6.
The trajectory of the UAV under (13). The red triangle represents
the starting position of the UAV. The dashed line represents the desired
trajectory.
0
20
40
60
80
100
−2
0
2
4
6
8
10
12
14
16
time
Radial error
Fig. 7.
The tracking error of the UAV under (13).
practical applications, one future research direction is on ana-
lyzing the effect of measurements noises on the performance
of the proposed control algorithms. Other future research
directions include the study of the circumnavigation mission
in the presence of wind, and the study of cooperative cir-
cumnavigation mission when multiple UAVs circumnavigate
around a target in a cooperative fashion.
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