To appear in AIAA J. Guidance, Control, and Dynamics

Multi-vehicle Control in a Strong Flowfield

with Application to Hurricane Sampling

Levi DeVries∗ and Derek A. Paley†

University of Maryland, College Park, Maryland, 20742, U.S.A

A major obstacle to path-planning and formation-control algorithms in

multi-vehicle systems are strong flows in which the flow speed is greater

than the vehicle speed relative to the flow. This challenge is especially per-

tinent in the application of unmanned aircraft used for collecting targeted

observations in a hurricane. The presence of such a flowfield may inhibit

a vehicle from making forward progress relative to a ground-fixed frame,

thus limiting the directions in which it can travel. This paper presents

results for motion coordination in a strong, known flowfield using a self-

propelled particle model of vehicle motion in which each particle moves

at constant speed relative to the flow. A new formulation of the particle

model with respect to a rotating reference frame is provided along with

a set of conditions on the flowfield that ensures trajectory feasibility. Re-

sults from the Lyapunov-based design of decentralized control algorithms

are presented for circular, folium, and spirograph trajectories, which are

selected for their potential use as hurricane sampling trajectories. The the-

oretical results are illustrated using numerical simulations in an idealized

hurricane model.

Nomenclature

N Number of particles in the system

k Particle index k = 1, . . . , N

rk/O Position of particle k with respect to O

∗Graduate Student, Department of Aerospace Engineering, University of Maryland, and AIAA StudentMember.†Assistant Professor, Department of Aerospace Engineering, University of Maryland, and AIAA Associate

Fellow.

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rk Position coordinates of kth particle with respect to inertial frame

rk Position coordinates of kth particle with respect to rotating frameIvk/O Inertial velocity of kth particle with respect to O

rk Velocity coordinates of kth particle with respect to inertial frame

˙rk Velocity coordinates of kth particle with respect to rotating frame

fk Flow velocity coordinates at rk

fk Flow velocity coordinates at rk relative to rotating frame

θk Orientation of kth flow-relative velocity with respect to inertial frame

θk Orientation of kth flow-relative velocity with respect to rotating frame

γk Orientation of kth total velocity with respect to inertial frame

γk Orientation of kth total velocity with respect to rotating frame

α Orientation of rotating reference frame with respect to inertial reference frame

uk Flow-relative steering control of kth particle with respect to inertial frame

uk Flow-relative steering control of kth particle with respect to rotating frame

νk Total steering control of kth particle with respect to inertial frame

νk Total steering control of kth particle with respect to rotating frame

sk Total speed of kth particle with respect to inertial frame

sk Total speed of kth particle with respect to rotating frame

hk Flow-relative speed of kth particle with respect to inertial frame

hk Flow-relative speed of kth particle with respect to rotating frame

ck Center of the kth particle’s trajectory

ρk Position coordinates of kth particle in ck-centered reference frame

φk Polar angle of kth particle’s position in ck-centered reference frame

κk Curvature of kth particle’s trajectory

ω0 Constant curvature of circle of radius |ω0|−1

K Control gain

n Number of lobes in folium figure

a Maximum radius of folium lobe

i Imaginary unit

vmax Maximum wind speed in idealized hurricane model

rmax Radius of maximum wind speed in idealized hurricane model

µ Decay constant of idealized hurricane model

I Inertial reference frame

B Rotating reference frame

Ck Path frame of kth particle with respect to frame IDk Path frame of kth particle with respect to frame BL Desired trajectory

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ej jth unit vector of frame IIωB Angular velocity of frame B with respect to frame IΩ Constant angular rate of rotating frame

d Distance of circle center from origin

d Non-dimensional distance of circle center from origin

I. Introduction

Unmanned vehicles have shown great value in their ability to explore harsh physical

environments that are too dangerous for manned platforms. For example, autonomous un-

derwater vehicles have shed light on some of the deepest trenches of the sea1 and robotic

rovers have explored expanses of the desolate martian landscape. Recently, autonomous ve-

hicles have been proposed as a safe and effective way to penetrate extreme weather systems

such as hurricanes and tornados.2 In 2006, an Aerosonde unmanned aircraft flying at low

altitude successfully penetrated a typhoon eyewall while streaming temperature, pressure,

and windspeed data to a remote command center.3 In 2010, the Global Hawk UAV flying

at high altitude passed over Hurricane Earl collecting temperature, convection, and precipi-

tation measurements.a In addition, smaller unmanned aircraft are being utilized to sample

within pre-tornadic supercell thunderstorms.4

The assimilation of data from previously inaccessible regions of a storm presents the pos-

sibility of more accurate forecasts of both the intensity and trajectory of a weather system.

For hurricanes, existing data-sampling methods include the use of satellite imagery, manned

flights, and more recently, remotely piloted unmanned flights. New data assimilation tech-

niques are in development for remotely piloted Global Hawk flights,5 which loiter above the

winds associated with a hurricane at altitudes of up to 65,000 ft.6 Manned aircraft eject

one-time-use dropsondes to collect data at low altitudes where it is too dangerous to fly.

By deploying hundreds of dropsondes, meteorologists obtain vertical profiles of atmospheric

data in the storm.

Autonomous unmanned aircraft have been proposed as a safe, efficient alternative to

manned flights for low-altitude penetration of hurricanes,3 collecting observations of temper-

ature, humidity, and wind at altitudes lower than manned aircraft can safely fly. Replacing

dropsondes with a smaller number of unmanned, reusable aircraft flying in a moving forma-

tion would provide continuous, dynamic data from within the hurricane eyewall. Unmanned

aJenner, L. and Dunbar, B. “NASA Science Aircraft Monitor Hurricane Earl on Sept. 2,”http://www.nasa.gov/mission pages/hurricanes/missions/grip/main/GRIP status 09 03 10.html, 11 Oct.,2011

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platforms are designed to withstand the severe winds within the storm and may operate

autonomously using automatic control algorithms.

Sampling a circulating flowfield such as a hurricane for forecasting purposes motivates the

desire to regulate azimuthal and radial sampling densities. In the case of hurricane sampling,

manned flights traverse figure-four patterns in order to sample each quadrant of the storm

while minimizing the overall track length.b Emulating this sampling strategy requires the

development of control algorithms that drive sampling vehicles in non-convex formations.

In addition, these trajectories must be feasible for the sampling platform to achieve in the

presence of a strong wind.

In order to design steering algorithms for unmanned aircraft in a hurricane, the authors

study a planar, self-propelled Newtonian particle model in a strong flowfield (i.e., with flow

speed equal to or exceeding the vehicle speed relative to flow). Decentralized Lyapunov-based

control algorithms that stabilize moving formations of multiple vehicles are derived, extend-

ing previous work for moderate7 (flow speeds between 10% and 99% of vehicle speed) and

flow-free models.8,9,10,11 Using the flow-free model, Sepulchre et al.10 provided theoretically

justified algorithms for stabilization of parallel, circular, and circular symmetric formations.

Arranz et al. expanded this result to the family of radially contracting and expanding9 as

well as spatially translating8 circular formations. Moreover, with the use of curvature func-

tions, Paley et al.12 extended the multi-vehicle formation result to convex loops; Zhang and

Leonard11 used orbit functions to stabilize formations on smooth curves.

In the moderate flow regime, Frew et al.13 utilized Lyapunov analysis to generate guidance

vector fields that globally stabilize circular loitering patterns. Paley and Peterson7 extended

the stability of parallel and circular formations to known, spatially-varying flowfields as well

as circular symmetric formations in uniform flows. Peterson and Paley14 also extended par-

allel and circular formation control to unknown and time-varying flowfields by implementing

onboard flowfield estimators. In the strong flow regime, Kwok and Hernandez15,16 charac-

terized minimum-time trajectories in a flowfield with magnitude strictly greater than the

vehicle speed, deriving cooperative control algorithms to maximize total vehicle coverage

area rather than coordinating vehicle motion with respect to a predefined formation.

The presence of a strong flow inhibits a vehicle from making forward progress in some

areas of the flowfield, thus limiting the range of directions in which it can travel. This presents

an obstacle to multi-vehicle control because the desired formation may not be feasible due

to spatial or temporal variability in the flow. The results of this paper rely on the use

of Lyapunov-based control design, which utilizes scalar artificial potential functions whose

value is minimized by a desired formation see, e.g. [17, p. 111–133]. State-feedback control

laws that ensure negative semi-definiteness of the potential function’s time derivative are

bWilloughby, H. and Majumdar, S., Personal Communication, July 14, 2011

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derived, invoking the invariance principle to establish the stability properties of the closed-

loop system [17, p. 126–133]. The results of this work have additional applications in path

planning for autonomous underwater gliders navigating strong currents,18 navigation of fixed-

wing aircraft in significant and unknown windfields,19,20 and in the navigation and path

planning of micro air vehicles in wind.21

The contributions of the paper are as follows: (1) an idealized model of the dynamics of

unmanned aircraft in strong winds using a self-propelled particle formulation, described with

respect to inertial and rotating reference frames; (2) kinematic conditions that guarantee a

particle can follow a desired trajectory in a strong, known flowfield; and (3) theoretically

justified multi-vehicle control laws stabilizing multi-vehicle formations including circular,

folium, and spirograph patterns, which are selected for potential use as hurricane sampling

patterns. These contributions extend previous results for convex loops in a moderate flow

to closed curves with non-zero curvature in a spatially varying strong flow.

The paper is organized as follows. Section II presents a dynamic model of planar, self-

propelled particle motion, first with respect to an inertial reference frame and then in a

reference frame rotating at a constant angular rate. Section III provides the kinematic

constraints a strong flowfield must satisfy such that a desired trajectory is feasible. Section IV

contains Lyapunov-based decentralized multi-vehicle control algorithms for vehicles traveling

in a strong flow. Section V assesses the feasibility of sampling trajectories in an idealized

hurricane model and illustrates the main results using simulations of the particle motion

under closed-loop control. Section VI summarizes the paper and ongoing work.

II. Self-Propelled Particle Dynamics in a Strong Flowfield

This section reviews the inertial motion of a self-propelled particle subject to a strong

external flowfield and derives a model of self-propelled particle motion with respect to a

reference frame rotating at a constant angular rate relative to an inertial frame. The rotating-

frame dynamics form the foundation for analysis of prospective hurricane sampling patterns

discussed in Section V.

A. Particle Motion With Respect to an Inertial Reference Frame

This paper extends an existing dynamic model common to many works on collective mo-

tion.7,9,12,22,23 The model consists of a collection of N self-propelled Newtonian particles.

Each particle is subject to a control force normal to its velocity and travels at a constant

speed relative to an external flowfield. It is assumed without loss of generality that the

flow-relative speed is unity. Given a ground-fixed inertial reference frame I with origin O

(identified with the complex plane), the kth particle’s position is represented by the vector

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eiθk

fk

rk

rk

Rerk

Imrk

γk θk

I

(a)

eiθk

fk

rk

rk

Rerk

Imrk

γk θk

I

(b)

Figure 1. Illustration of particle motion in a (a) moderate flowfield and (b) strong flowfield.

rk/O, or in complex coordinates [rk/O]I , rk = xk + iyk ∈ C. Given the unit-speed par-

ticle assumption, the kth particle’s velocity orientation can be represented by θk, a point

on the unit circle. Using complex coordinates, the kth particle’s flow-relative velocity is

cos θk + i sin θk = eiθk .

The state-feedback control uk determines the kth particle’s rate of change of velocity

orientation (steering control). The equations of motion for the kth particle in the absence of

a flowfield arec

rk = eiθk

θk = uk (r,θ) , k = 1, . . . , N.(1)

State-feedback control algorithms utilizing r and θ drive each particle to the desired collective

motion. Control algorithms producing collective motion with respect to particle model (1)

have been extensively studied in recent years.8,9,10,12,22,24,25

In the presence of a flowfield fk = f (rk) ∈ C each particle’s velocity is represented by

the vector sum of its velocity relative to the flow and the flow velocity relative to the inertial

frame I, as illustrated in Figure 1. In this case, the kth particle’s motion is governed by the

equations

rk = eiθk + fk

θk = uk (r,θ) , k = 1, . . . , N.(2)

Here the flow magnitude is allowed to exceed the speed of the particle relative to the flow.

Let γk represent the orientation of the kth particle’s inertial velocity and sk its magnitude.

Previous work on motion coordination in a moderate flow,7 i.e., where |fk| < 1, utilized the

cWe drop the subscript and use bold fonts to represent an N × 1 matrix, e.g., r = [r1 r2 ... rN ]T

and

θ = [θ1 θ2 ... θN ]T

.

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following coordinate transformation:

γk , arg(eiθk + fk) (3)

sk , |eiθk + fk|. (4)

Under this transformation, the equations of motion (2) become

rk = skeiγk

γk = νk,(5)

where νk is the control relative to the fixed inertial frame I. Controls uk and νk are related

byd

uk =νk − 〈f ′k, i〉

1− s−1k 〈eiγk , fk〉

=skνk − sk〈f ′k, i〉sk − 〈eiγk , fk〉

, (6)

where f ′k = ∂fk∂rk

. Figure 1 illustrates the particle model in both the moderate (Fig. 1(a)) and

strong flow (Fig. 1(b)) regimes. Note that a unit circle centered at the tip of fk represents

the possible values of rk = eiθk + fk.

Paley and Peterson7 showed that in a moderate flow the transformation (6) defines a one-

to-one mapping between uk (control of the flow-relative velocity orientation) and νk (control

of the velocity relative to I), since the denominator of (6) is never zero. This result allowed

development of control laws for (2), extending those developed for the flow-free model (1).

However, for flow speeds greater than or equal to the vehicle speed, the transformation (6)

from models (2) to (5) is no longer one-to-one. To see this, consider a particle at position

rk subject to a strong flow as shown in Figure 1(b). A unit circle drawn about the tip

of fk defines the possible orientations of the total velocity rk = eiθk + fk. The unit circle

rk = eiθk + fk in a strong flow is not guaranteed to enclose rk, which implies that the set of

possible inertial velocity orientations is a subset of the circle (called the cone of admissable

directions by Bakolas and Tsiotras19). In a moderate flow as illustrated in Figure 1(a) the

unit circle encloses rk, implying that any given θk corresponds to a single γk. Thus, in any

moderate flow a given θk maps to a single γk, but in a strong flow a given γk maps to one of

two possible θk’s. The speed sk is required to determine θk as shown in Figure 1(b). Lemma

1 identifies two singularities in the transformation (6), the proof of which is provided in the

appendix.

Lemma 1. For |fk| ≥ 1, the transformation (6) is singular when θk = arg(fk)±cos−1(|fk|−1).

Lemma 1 shows that strong flows introduce singularities in the coordinate transform (6),

d〈x, y〉 , Re (x∗y), where x∗ is the complex conjugate of x, denotes the inner product of complex numbersx and y.

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resulting in an unbounded turn rate. However, the design of the vehicle and the medium

in which it travels dictate the maximum controlled rate of turn a priori. The turn-rate

constraint can be modeled as saturation of uk, in which case (2) becomes

rk = eiθk + fk

θk = sat (uk (r,θ) ;umax) ,(7)

where

sat (u;umax) =

umax, u > umax

u, −umax ≤ u ≤ umax

−umax, u < −umax.(8)

The saturation model (7) allows use of model (5) to design νk and (6) to map νk to

uk. Unless restricted by the vehicle dynamics, umax must be chosen large enough so that

νmax = νmax(umax) > |κmaxsmax|, where κmax is the maximum desired curvature of the

particle trajectory and smax is the maximum particle speed along the trajectory. (The relation

between umax and νmax is discussed by Peterson and Paley.14) The upper bound on νk is

represented using γk = sat (ν (r, θ) ; νmax). As θk approaches either singularity provided in

Lemma 1, θk remains bounded, passing through the singular point at a constant rate ±umax.

B. Particle Motion with Respect to a Rotating Reference Frame

To achieve adequate radial and azimuthal sampling within a circulating flowfield, a proposed

sampling formation is that of a spirograph. When viewed with respect to a frame rotating

at constant angular rate, a spirograph becomes a circular trajectory. Control algorithms

producing circular trajectories are discussed in Section IV. Here, a dynamic model of self-

propelled particle motion in a rotating reference frame is derived.

Let I = (O, e1, e2, e3) represent an inertial reference frame with origin O and B =

(O,b1,b2,b3), where b3 = e3, represent a rotating reference frame with angular velocityIωB = Ωe3. (Complex notation is used to represent Cartesian coordinates in I and B.) The

orientation α = Ωt+ α(0) of frame B with respect to I satisfies e1 · b1 = cosα, as shown in

Figure 2(a). The inertial kinematics of the kth particle with respect to O are described by

rk , [rk/O]I and Ivk/O ,Iddt

(rk/O). Let the path frame Ck = (k, c1, c2, c3) be a frame with

origin at the kth particle’s position, where c3 = e3, c1 = Ivk/O/hk, and hk , ‖Ivk/O‖ is the

speed of the particle (equal to one when no flow is present). Let θk be the orientation of the

(flow-relative) velocity so that e1 · c1 = cos θk.

Since frame B shares its origin with frame I, the kinematics of the kth particle in frame

B are given by rk , [rk/O]B and Bvk/O ,Bddt

(rk/O). As in frame I, this gives rise to

the path frame Dk = (k,d1,d2,d3), with origin at rk, where d3 = e3, d1 = Bvk/O/hk,

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I

rk

α

B

b1

b2

e2

e1

θk

θk

O

Ω

(a)

Imrk

rk

θk

hkeiθk + fk = skeiγk

γk

B

hkeiθk

fk

RerkO

(b)

Figure 2. The orientation of the kth particle’s velocity (a) in no flow and (b) in flow f .

and hk , ‖Bvk/O‖ represents the speed of the kth particle relative to frame B. In B-

frame coordinates, (xk, yk)B, the position and velocity are given by rk/O = xkb1 + ykb2 andBvk/O = xkb1 + ykb2, respectively, which implies hk =

√x2k + y2

k. Let θk be the orientation

of Bvk/O relative to b1, so that b1 · d1 = cos θk.

The transport equation [26, p. 433–435], Ivk/O = Bvk/O+IωB×rk/O, gives the kinematic

relationship

hkc1 = hkd1 + Ωb3 × (xkb1 + ykb2) = hkd1 + Ω(xkb2 − ykb1). (9)

Noting that c1 = cos θke1 + sin θke2, (9) can be written

hk cos (θk − α) b1 + hk sin (θk − α) b2 =(hk cos θk − Ωyk

)b1 +

(hk sin θk + Ωxk

)b2. (10)

The inertial derivative of (9), assuming hk is constant, gives

hkθkc2 = ˙hkd1 + hk

(˙θk + Ω

)d2 −

(Ωyk + Ω2xk

)b1 +

(Ωxk − Ω2yk

)b2. (11)

By utilizing c2 = − sin (θk − α) b1 + cos (θk − α) b2, (11) results in the following scalar

equations of motion relative to reference frame B:

−hkuk sin(θ − α) = ˙hk cos θk − hk( ˙θk + Ω) sin θk − xkΩ2 − ykΩhkuk cos(θk − α) = ˙hk sin θk + hk(

˙θk + Ω) cos θk − ykΩ2 + xkΩ.(12)

Note that substituting xk = hk cos θk and yk = hk sin θk into (12) and using (10) to

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eliminate θk − α and hk gives

−ukhk sin θk − ukΩxk = ˙hk cos θk − hk( ˙θk + Ω) sin θk − xkΩ2 − hk sin θkΩ

ukhk cos θk − ukΩyk = ˙hk sin θk + hk(˙θk + Ω) cos θk − ykΩ2 + hk cos θkΩ.

(13)

Solving (13) for ˙hk and ˙θk, respectively, with complex notation rk = xk + iyk, results in the

equations of motion of the kth particle relative to frame B:

˙rk = hkeiθk

˙hk = (Ω2 − ukΩ)〈rk, eiθk〉˙θk = uk − 2Ω + h−1

k (Ω2 − uΩ)〈rk, ieiθk〉 , uk.

(14)

Let uk be defined as the flow-relative control with respect to rotating frame B. The

mapping from uk to uk is

uk =ukhk + 2Ωhk − Ω2〈rk, ieiθk〉

hk − Ω〈rk, ieiθk〉. (15)

Equations (14) and (15) are the equations of motion of a self-propelled particle represented

in coordinates relative to a rotating frame with constant angular rate Ω. Note that the

mapping from uk to uk is singular when hk = Ω〈rk, ieiθk〉. Lemma 2 establishes the singular

conditions with respect to the inertial speed hk. (The proof is provided in the appendix.)

Lemma 2. The control transform (15) is singular when hk = Ω〈rk, eiθk〉.

The singularity identified in Lemma 2 occurs when the speed of the vehicle is zero with

respect to frame B. For example, when eiθk is normal to rk and Ω > 0, the singular condition

simplifies to Ω−1 = |rk|.Next, augment the dynamics presented in (14) to include flow terms represented by fk.

Note fk = f(rk) ∈ C, is assumed to be time invariant with respect to frame B, which implies

that the corresponding flowfield with respect to frame I is azimuthally symmetric about O.

Again, adapting the kinematic model of the effect of the flow that is used to derive (2), the

flowfield is incorporated into (14) so that

˙rk = hkeiθk + fk

˙hk = (Ω2 − ukΩ)〈rk, eiθk〉˙θk = uk.

(16)

Note uk is still calculated from uk using (15).

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Path-planning applications desire control of the total velocity of the particle rather than

the flow-relative velocity. For this reason let

sk , |hkeiθk + fk| (17)

γk , arg(hkeiθk + fk), (18)

such that ˙rk = skeiγk . Figure 2(b) illustrates the following relations

sk cos γk = hk cos θk + 〈fk, 1〉sk sin γk = hk sin θk + 〈fk, i〉,

(19)

which after division yield

tan γk =hk sin θk + 〈fk, i〉hk cos θk + 〈fk, 1〉

. (20)

Taking the time derivative of (20) and using (19) gives

˙γk =〈 ˙fk, ie

iγk〉 − (Ω2 − ukΩ)[〈rk, skeiγk〉 − 〈rk, fk〉

]〈fk, ieiγk〉h−2

k +(sk − 〈fk, eiγk〉

)uk

sk, νk,

(21)

where νk represents control of the total velocity orientation with respect to the rotating

frame and ˙fk = ∂fk∂rk

˙rk. Using (21) with (15) yields the following mapping between uk and νk

necessary for implementing a control designed in the rotating frame for use in the inertial

frame:

uk =skνk−〈 ˙

fk,ieiγk 〉+2Ω(sk−〈fk,eiγk 〉)+Ω2〈rk,skeiγk−fk〉〈fk,ieiγk 〉h−2

k −Ω2(sk−〈fk,eiγk 〉)〈rk,iskeiγk−ifk〉h−2k

sk−〈fk,eiγk 〉−Ω|fk|2h−2k 〈rk,ie

iγk 〉+Ωskh−2k (〈rk,ifk〉+〈rk,eiγk 〉〈fk,ieiγk 〉+〈fk,eiγk 〉〈rk,ieiγk 〉)−Ωskh

−2k 〈rk,ie

iγk 〉.

(22)

The equations of motion of the kth particle with respect to a rotating frame subject to a

flowfield are:˙rk = ske

iγk

˙hk = (Ω2 − ukΩ) 〈rk, skeiγk − fk〉h−1k

˙γk = νk,

(23)

where uk is given by (22). Similar to the inertial-frame dynamics, saturation of uk is used

to avoid singularities in the transformations (15) and (22).

For a known, spatially varying flowfield, sk can be calculated as follows.7 Let fk = ηk+iβk

be the flowfield at rk, where ηk = 〈fk, 1〉 and βk = 〈fk, i〉 represent the real and imaginary

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fk

Leiγk

ieiγk

rk

Imrk

Rerk

fk

eiγk

I ieiγk

(a)

L

rk

Imrk

Rerk

eiθk

B

|Ω|−1

rkeiθk

α∗

(b)

Lrk

Imrk

Rerk

eiγk

B

fk

ieiγk

eiγk

ieiγk

fk

(c)

Figure 3. Feasibility constraint for (a) inertial frame, (b) rotating flow-free frame, and (c)rotating with-flow reference frame.

parts of the flow respectively. By definition (17) gives

sk =√h2k − η2

k − β2k + 2sk (ηk cos γk + βk sin γk). (24)

Squaring this result and utilizing the quadratic formula to solve for sk gives

sk = 〈eiγk , fk〉+√h2k − 〈ieiγk , fk〉2, (25)

where the positive root is taken since sk > 0.

III. Trajectory Feasibility in a Strong Flow

Strong flows present the possibility that a desired trajectory or formation is not feasible.

This section derives the kinematic conditions a flowfield must satisfy to ensure trajectory

feasibility. Conditions are derived for trajectories relative to both inertial and rotating

reference frames. These results have similarities to that of Bakolas and Tsiotras in assessing

the reachability of two points within a flowfield for a kinematic model of an aircraft,19 but this

paper considers flowfields that can vary continuously through space rather than flowfields

that are assumed to be regionally uniform.

A. Feasibility With Respect to an Inertial Reference Frame

A strong flowfield presents a challenge to coordinated motion if an individual vehicle is

unable to reach the desired trajectory. Moreover, the desired trajectory itself may not

be achievable if even a portion of the trajectory opposes the flow. The following analysis

describes a set of constraints the flow must satisfy such that a given desired trajectory is

feasible. (Equivalently, one can view the constraints as limiting the possible trajectories in

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(a) (b)

Figure 4. Feasible regions for (a) counterclockwise circular trajectories with radius |ω0|−1 = 20and (b) quadrifolium trajectories with maximum lobe length a = 20.

a given flowfield.)

For a vehicle to travel along curve L, the flow at every point on the path must be such

that the vehicle can maintain velocity tangent to L. That is, for every point on the desired

trajectory, the component of the flow normal to L must be less than the vehicle speed relative

to the flow. For a unit speed particle and tangent vector eiγk , this implies that the absolute

value of the inner product between the normal vector ieiγk and the flow fk must satisfy

|〈ieiγk , fk〉| ≤ 1. If 〈eiγk , fk〉 ≤ 0, then the flow opposes (or is normal to) the direction of the

trajectory and must therefore have magnitude less than one. If 〈eiγk , fk〉 > 0, then the flow

must only satisfy the normal constraint. This result is summarized as follows:

Theorem 1. Trajectory L is feasible in flowfield f if, for all rk ∈ L, fk = f (rk) satisfies

〈eiγk , fk〉 ≤ 0 and |fk| < 1, or

〈eiγk , fk〉 > 0 and |〈ieiγk , fk〉| ≤ 1,(26)

where eiγk is tangent to L at rk.

Theorem 1 implies that the flow vector at a given point on the trajectory must lie within a

U-shaped envelope oriented along the trajectory tangent, shown by the green-shaded region

in Figure 3(a). Trajectories that do not satisfy (26) are not feasible. Theorem 1 allows

one to quantify regions of a known, time-invariant flowfield in which parametric families

of feasible trajectories are found. Analysis over the entire space of candidate trajectory

centers produces a map of regions in which feasible trajectories can be achieved. Figure 4(a)

illustrates the feasible regions for fixed-size circular trajectories of radius |ω0|−1 = 20 in a

randomly generated strong flowfield. Figure 4(b) illustrates quadrifolium trajectories with

maximum lobe length a = 20 (folium trajectories are discussed in Section IV.C). A trajectory

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whose center lies in the shaded region is not feasible; unshaded regions are feasible. Red

arrows along the sampling trajectories indicate where the flow fails to satisfy the conditions

in Theorem 1.

B. Feasibility with Respect to a Rotating Reference Frame

Section II derives the dynamics of a self-propelled particle with respect to a rotating reference

frame. The equations of motion in (14) reveal that the speed of the particle in the rotating

frame is a dynamic variable whose time derivative depends on the position and velocity of the

particle. This subsection defines kinematic constraints on motion with respect to a rotating

reference frame.

In the absence of an external flow, the kinematic terms arising from the rotating reference

frame B affect the speed hk of the vehicle with respect to B. For a constant-speed particle

traveling in the direction of rotation, the speed of the particle with respect to B is a decreasing

function of the rotation rate and the distance to the center of rotation. For a given trajectory

to be feasible with respect to the rotating frame, a particle must maintain forward progress

over the entire trajectory, which implies the following result. It is assumed that a fixed point

in the rotating frame is not a feasible trajectory.

Lemma 3. Trajectory L is a feasible solution to (14) if |rk| < |Ω|−1, or if |rk| ≥ |Ω|−1 and

Ω〈rk, ieiθk〉 > 0 and |Ω〈rk, eiθk〉| ≤ 1, (27)

where eiθk is tangent to L at rk ∈ C.

Proof. Summing the squared components of (10) and using hk = 1 yields

1 = h2k + Ω2

(x2 + y2

)+ 2Ωhk(x sin θk − y cos θk).

Completing the square to solve for hk and simplifying with rk = xk + iyk gives

hk = Ω〈rk, ieiθk〉+√

1− Ω2〈rk, eiθk〉2, (28)

where the positive root is taken since hk > 0 is required to maintain forward progress. Note

that the inner products 〈rk, ieiθk〉 and 〈rk, eiθk〉 differ in phase by π/2 which implies that

for |rk| < |Ω|−1, (28) is real and positive. A trajectory is infeasible if (28) is negative or

complex, implying that for |rk| ≥ |Ω|−1 a feasible trajectory must satisfy (27).

Lemma 3 is illustrated for Ω > 0 in Figure 3(b). Feasible trajectories with |rk| ≥ |Ω|−1

must travel in a direction opposing the rate of rotation in order to maintain forward progress

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with respect to the rotating frame. Where the trajectory becomes infeasible is shown shown

as a dashed line. The range of feasible directions of travel is determined by the position

of the vehicle and rate of rotation of the rotating frame. Notice that in complex notation,

(10) can be written ei(θk−α) = hkeiθk + iΩrk. Therefore, one can represent the velocity with

respect to the rotating frame as hkeiθk = ei(θk−α)− iΩrk. The range of feasible velocities with

respect to the rotating frame is thus defined by a unit circle drawn about the tip of −iΩrk.In a similar manner to a vehicle’s range of travel in a strong flowfield,15 it can be shown that

for |rk| ≥ |Ω|−1, the angular range α∗ of feasible directions of travel about arg(−iΩrk) is

given by α∗ = 2 sin−1 (|Ωrk|−1). α∗ can be calculated by replacing the vector fk with −iΩrkin Figure 1(b) (also see Figure 10 in Appendix) and noting that α∗ is the angle formed

between the two lines tangent to the unit circle. Figure 10 denotes this angle as α∗ = 2ξ,

where ξ = sin−1(|Ωrk|−1).

When an external flowfield is present as in (16), the feasibility constraint on the flow-

field in a rotating frame is similar to Theorem 1. The transport equation [26, p. 433-435]

represented in complex coordinates with respect to frame B gives

ei(θk−α) + fke−iα = ske

iγk + iΩrk. (29)

An external flowfield is represented with respect to the rotating frame via the transport

equation such that in complex coordinates

fk = fkeiα + iΩrke

iα, (30)

where fk represents the flowfield relative to the rotating frame. Rearranging (30) reveals

fke−iα = fk + iΩrk, (31)

which when substituted into (29) gives

skeiγk = ei(θk−α) + fk. (32)

Equation (32) shows that the total velocity of the kth particle with respect to frame Bcan be represented as a unit circle drawn about the tip of fk. Similar to Theorem 1, for a

particle to travel along curve L in rotating frame B, the component of the flow normal to

the trajectory must be less than the vehicle speed with respect to the flow in the inertial

frame. For an inertially unit-speed particle and tangent angle eiγk in the rotating frame, this

implies that the absolute value of the inner product between the normal vector ieiγk and the

flow must satisfy |〈ieiγk , fk〉| ≤ 1. If 〈eiγk , fk〉 ≤ 0 then the flow is normal to or opposes the

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direction of travel and must therefore have magnitude less than one. If 〈eiγk , fk〉 > 0 only

the normal constraint must be satisfied. This implies the following result.

Theorem 2. Trajectory L in rotating reference frame B is a feasible solution to (23) with

flowfield f if for all rk ∈ L, fk = f (rk) is such that

〈eiγk , fk〉 ≤ 0 and |fk| < 1 or

〈eiγk , fk〉 > 0 and |〈ieiγk , fk〉| ≤ 1,(33)

where eiγk is tangent to C at rk.

Note that Theorem 2 has no apparent dependence on rk and Ω as in Lemma 3. The

dependence is implicit in fk, which is a function of both the external flow fk and the kinematic

term iΩrk as shown in (31).

IV. Multi-vehicle Control in a Strong Flowfield

This section reviews the design of multi-vehicle controls for a moderate flowfield and

derives circular and folium control laws for the particle model (7) in a strong flowfield. The

results of this section are also applicable to the model (16) for a rotating reference frame as

discussed in Section V.

A. Review of Curvature Control in a Moderate Flowfield

This section describes individual particle curvature control in a moderate flow, following

the work of Paley et al.,12 who produced decentralized algorithms to stabilize the family of

convex loops called superellipses in a flow-free environment and Techy et al.,23 where the

convex-loop results were extended to the vehicle model with (uniform) flow (5).

Our goal is to drive the kth particle about a smooth, closed curve L with definite, bounded

curvature, κk. To do so, L is parameterized by its center ck. In a ck-centered coordinate

system the particle position ρk is parameterized by the polar angle φk. For a closed curve

L, completion of one rotation about L sweeps through 2π radians. Thus, ρk (φ) ∈ R where

φk ∈ [0, 2π). The orientation of the particle’s inertial velocity relative to the ck centered

coordinate system is γk and it is assumed that a smooth mapping γk 7→ φ (γk) exists. The

derivative dρkdφk

is tangent to L, implying the constraint eiγk =∣∣∣ dρkdφk

∣∣∣−1dρkdφk

.

The local curvature is12

κk (φk) = ±dγkdσ

, (34)

where

σ (φk) =

∫ φk

0

∣∣∣∣dρdφ∣∣∣∣ dφ (35)

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is the arc length along the curve. Note that the sign of κk determines the direction of

rotation about the curve, which is either clockwise or counterclockwise. Equations (34) and

(35) imply12

κ−1k = ±dσk

dφk

dφkdγk

= ±∣∣∣∣dρkdφk

∣∣∣∣ dφkdγk. (36)

Under the tangent constraint, (36) can be written12

eiγkκ−1k = ±dρk

dφk

dφkdγk

=dρkdγk

.

In a reference frame not centered at ck, L has center ck = rk ∓ ρk. The time derivative

along solutions of (5) gives the velocity of the center7,12

ck = skeiγk ∓ dρk

dγk

dγkdt

= eiγk[sk ∓ κ−1

k νk]. (37)

From (37) note that the curvature control algorithm7

νk = ±κksk (38)

forces ck = 0, implying that the kth particle drives about a stationary C.This work uses Lyapunov analysis to design decentralized controls to collectively steer

the particles such that they achieve coincident centers, i.e., c = [c1 c2 ... cN ]T = c01, where

c0 ∈ C and 1 ∈ RN . Consider the potential function23

S (r,γ) =1

2〈c,Pc〉 (39)

where P is an N × N projection matrix, P = IN×N − 1N

11T . P defines the communica-

tion topology of the system and in this paper corresponds to all-to-all communication.27

(Extensions of the flow-free model to limited communication topologies is possible,25,28 but

outside the scope of this paper.) Equation (39) is positive semi-definite with equality to zero

occurring when c = c01, c0 ∈ C. Using (37), the time derivative of (39) is7,12,23

S (r,γ) =N∑k=1

〈ck, Pkc〉 =N∑k=1

〈eiγk ,Pkc〉(sk ∓ κ−1

k νk), (40)

where Pk is the kth row of P. The control12,23

νk = ±κk(sk +K〈eiγk ,Pkc〉

)(41)

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substituted into (40) gives12,23

S (r,γ) = −KN∑k=1

〈eiγk ,Pkc〉2 ≤ 0. (42)

The invariance principle stipulates that solutions of (5) with control (41) converge to the

largest invariant set Λ where12

〈eiγk ,Pkc〉 ≡ 0. (43)

Since eiγk 6= 0, then Pkc = 0 in Λ, which is satisfied when the centers are coincident.

Moreover, when (43) is satisfied the control in (41) simplifies to (38) which implies c = 0.

(Note that this framework does not incorporate collision avoidance between particles.)

A simple example of this control strategy is that of a circular formation. A circle has

constant curvature, κk = ω0 for all k, which defines a radius of |ω0|−1. A particle traversing

a circular trajectory has total velocity tangent to the radius of the circle. Thus, the center

of the kth particle’s trajectory with control νk = ω0sk is7

ck = rk + ω−10

rk|rk|

= rk + ω−10 eiγk . (44)

The vector c is then used to calculate the control of kth particle according to (41) with

κk = ω0.

B. Decentralized Multi-vehicle Control in a Strong Flowfield

This section describes the design of a multi-vehicle control law that steers individual particles

to a desired feasible formation in a strong flowfield. Lyapunov analysis is used to establish

the stability of multi-vehicle formations in a strong flow when the initial formation centers lie

within a feasible ball about a feasible steady-state reference center c0. The notation B (x, a)

is used to represent a ball of radius a centered about x ∈ C.

Theorem 3. Consider a strong flow fk. Let F be a feasible region and c0 ∈ C. Let

νmax > maxk,t|ω0sk(t)|

and B (c0, |ck (0)− c0|) ⊂ F . The model (7) with the mapping (6) and

νk = sat(ω0

(sk +K〈eiγk , ck − c0〉

); νmax

)(45)

forces convergence of the kth particle to a circular formation of radius |ω0|−1 centered at c0.

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Proof. Consider the candidate potential function

V =1

2|ck − c0|2, (46)

where ck is given by (44). The time derivative of (46) is

V = 〈ck − c0, ck〉 = 〈ck − c0, eiγk〉

(sk − ω−1

0 νk). (47)

Note from (45) that for |νk| < νmax

〈ck − c0, eiγk〉 =

νk − ω0skKω0

. (48)

Substituting this result into (47) gives

V =

(νk − ω0skKω0

)(sk − ω−1

0 νk)

= −(νk − ω0sk)2

Kω20

≤ 0. (49)

When |νk| = νmax, V < 0 since νmax > maxk,t |ω0sk(t)|. Consequently, ck is contained by

B (c0, |ck (0)− c0|) ⊂ F . Moreover, solutions converge to the largest invariant set in which

V = 0. From (45) and (47) this set contains solutions for which νk = ω0sk, which implies

ck = c0.

Theorem 3 shows that an individual particle will converge to a reference center provided

that the reference center and a ball containing the initial particle centers lie within a feasible

region. We call this reference center a prescribed center and any particle with information of

the prescribed center an informed particle. The following result shows that a multi-vehicle

system having initial centers in a feasible region and at least one informed particle will

converge to a prescribed center in the same feasible region.

Theorem 4. Consider a strong flowfield fk. Let F be a feasible region and c0 ∈ C. Also, let

νmax > maxk,t|ω0sk(t)| ,

and

B(c0,maxk|ck (0)− c0|) ⊂ F . (50)

The model (7) with the mapping (6) and

νk = sat(ω0

[sk +K

(〈eiγk ,Pkc〉+ ak0〈eiγk , ck − c0〉

)]; νmax

), K > 0, (51)

where ak0 = 1 for at least one particle and zero otherwise, forces convergence to the circular

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formation of radius |ω0|−1 centered at c0.

Proof. Consider the composite potential function10

S = 12〈c,Pc〉+ 1

2

∑Nk=1 ak0 |ck − c0|2 . (52)

The time derivative of (52) along solutions of (7) is

S =N∑k=1

(〈eiγk , Pkc〉+ ak0〈eiγk , ck − c0〉

) (sk − ω−1

0 νk).

For νk < νmax notice from (51)

νk − ω0skKω0

= 〈eiγk , Pkc〉+ ak0〈eiγk , ck − c0〉

giving

S = −∑Nk=1

(νk−ω0sk)2

Kω20≤ 0. (53)

When νk = νmax, S < 0 since νmax > maxk,t |ω0sk(t)|. Therefore the collection of particle

centers is bounded within the ball (50). Solutions converge to the largest invariant set for

which S = 0, which occurs when νk = ω0sk for all k, implying from (51)

〈eiγk ,Pkc〉+ ak0〈eiγk , ck − c0〉 = 0 (54)

for all k.

If ak0 = 0 for at least one and at most N − 1 particles, then (54) is satisfied only if

Pkc = 0, which occurs when ck = cj for all k and j. For k with ak0 = 1, (54) is satisfied

only if ck = c0. Therefore, c must satisfy c = c01.7

Note that the control is calculated based on the inertial variable γk, which corresponds to

one of two values of θk. This implies that the control does not differentiate between aligning

of opposing the flowfield. In circulating flowfields, simulation results show that particles with

random initial conditions converge to a circular formation in which particles aligned with

the flow travelled faster than those anti-aligned. Simulations also showed that increasing the

gain K in (51) tends to align particles with the flow.

Figure 5(a) shows simulation results in the randomly generated strong flowfield previ-

ously shown in Figure 4. Particle trajectories are shown in light grey and the final particle

formation is shown in dark grey. The circular control parameters were |ω0|−1 = 20, K = 0.01,

and c0 = −40 + 30i. Note c0 was chosen to be consistent with the feasibility analysis shown

in Figure 4(a). The value of νmax was calculated from umax = π/2 rad/s. Note from Figures

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60 40 20 0 20 40 60

60

40

20

0

20

40

60

Re(r)

Im(r)

(a)

60 40 20 0 20 40 60

60

40

20

0

20

40

60

Re(r)

Im(r)

(b)

Figure 5. Simulation of N = 10 particles in an arbitrary strong flow (a) with circular controlalgorithm (51) and (b) with curvature control (59) to a quadrifolium formation.

4(a) and 5(a) that B(maxk |ck(0) − c0|) does not lie entirely in a feasible region, yet the

particle centers still converge to c0. Simulation results also indicate that a circular formation

converges even when a formation center is not prescribed.

C. Stabilization of Folium Formations

An existing strategy for data collection in a hurricane is to fly a figure-four pattern through

the center of the hurricane.e A continuously differentiable flight path covering similar spatial

densities is that of the n-petalled folium, or polar rose, where n ≥ 3.f Specifically, the

quadrifolium (n = 4) is a pattern similar to a figure-four with continuous curvature. In

polar coordinates, the equation of the n-folium is ρk = a sin (gk (n)φk), where ρk is the

distance of the kth particle from the ck, n is the number of lobesg, φ ∈ [0, 2π] is the central

or polar angle, a is the maximum radius of the lobes, and

g(n) =

n, n odd

n/2, n even.

Section IV.A reviews how the control νk = κksk, where κk = κk (γk) is the curvature,

drives the particle along the desired curve. Previous work required the curvature to be

nonzero and the figure to be convex.11,12,23 Here this convexity assumption is relaxed. The

n-folium is not convex; however, its curvature is strictly positive (or negative). In traversing

one rotation about the figure the particle velocity rotates through 2π (g (n)− 1) radians for

eWilloughby, H. and Majumdar, S., Personal Communication, July 14, 2011fWeisstein, E. C., “Quadrifolium”, http://mathworld.wolfram.com/Quadrifolium.html, 11 Oct., 2011gn must be odd or divisible by four to be considered a folium.

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n odd and 2π (g (n) + 1) radians for n even. Using curvature control νk = κk(γk)sk requires

γk to be defined over 2π (g (n)± 1) radians. To accomplish this, one can augment the state-

space equations with an integer lobe-counter, lk ∈ N, which represents the number of lobes

a particle has traversed. When a particle’s inertial velocity orientation γk reaches 2π rad, lk

increases by one and the orientation angle γk is reset to zero. Utilizing the curvature control

as a function of γk and lk gives

νk = κk (γk (θk) + 2πlk) sk. (55)

Equation (55) drives the kth particle about a non-convex figure with strictly positive (nega-

tive) curvature.

In the case of the folium, the curvature is the following function of the polar angle φk:f

κk (φk) =|ρ2k + 2ρ′2k − ρkρ′′k|(ρ2k + ρ′2k )

3/2=

n2 + n2 sin2 (nφk) + cos2 (nφk)

a[cos2 (nφk) + n2 sin2 (nφk)

]3/2 . (56)

In order to provide state feedback, one must specify φk as a function of the tangent angle γk

such that the curvature control law is valid. This relation for the quadrifolium isf

γk (φk) =1

2π + φk − tan−1 (cotφk − tanφk) + πb2φk/πc. (57)

Note the tangent angle is a function of the polar angle, not vice versa. To numerically

implement (57) requires use of a look-up table. From (56) and (57) one can calculate the

curvature as a function of the inertial orientation, such that the control (55) drives the

particle around the quadrifolium trajectory. The following result extends Theorem 4 to

curves with definite curvature.

Proposition 1. Consider a strong flow fk. Let F be a feasible region and c0 ∈ C. Also let

νmax > maxk,t|κk(γk(t))sk(t)| ,

and

B(c0,maxk|ck (0)− c0|) ⊂ F . (58)

Consider the quadrifolium with curvature κk given by (56) and γk given by (57). For the

model (7), the mapping (6) with

νk = sat(κk[sk +K

(〈eiγk ,Pkc〉+ ak0〈eiγk , ck − c0〉

)]; νmax

), (59)

where ck = ck(γk) is the instantaneous center of the quadrifolium, forces convergence of the

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(a)

80 60 40 20 0 20 40 60 8080

60

40

20

0

20

40

60

80

Re(r)

Im(r)

(b)

Figure 6. (a) Regions of achievable centers for counterclockwise circular trajectories of radius

|ω0|−1 = 30; (b) simulation of the control (51) for N = 10 particles with c0 = 0.

kth particle about a quadrifolium formation centered at c0.

Simulation results for N = 10 particles in a quadrifolium formation centered at c0 =

−50 + 50i are shown in Figure 5(b) where c0 was chosen consistent with the feasibility

analysis of Figure 4(b). Note from Figure 4(b) that a ball containing initial particle centers

defined in (50) does not entirely lie in a feasible region, implying that Proposition 1 is overly

conservative.

V. Design and Stabilization of Hurricane Sampling Patterns

This section considers multi-vehicle sampling patterns in an azimuthally symmetric vor-

tex model of an idealized hurricane. In a Rankine vortex29 the tangential windspeed varies

linearly with radius to its maximum vmax at the radius of maximum wind rmax and exponen-

tially decreases for radii greater than rmax with exponential decay constant µ. The Rankine

vortex model is arg(fk) = arg(irk) and

|f(rk)| =

vmax (|rk|/rmax) , 0 < |rk| < rmax

vmax (|rk|/rmax)−µ , |rk| > rmax,

where radius r = rmax represents the eyewall of an idealized hurricane. The following analysis

uses the Rankine vortex parameters vmax = 1.5, rmax = 30, and µ = 0.6.

Circular formations are an attractive sampling option in a circulating flowfield because

large regions of the flowfield are feasible centers of rotation. Figure 6(a) shows the result of

evaluating the kinematic constraint (26) for a circular trajectory of radius |ω0|−1 = rmax. A

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(a)

60 40 20 0 20 40 6060

40

20

0

20

40

60

Re(r)

Im(r)

(b)

Figure 7. (a) Regions of achievable centers for quadrifolium trajectories of radius a = 20; (b)simulation of control (51) with N = 10 particles.

drawback of this formation is the resulting limitations of the azimuthal and radial sampling

densities. For example, a circular formation centered at the origin of a vortex provides good

azimuthal sampling but poor radial sampling. A formation not centered at the origin of

the vortex provides measurements at multiple radii at the expense of under-sampling some

quadrants of the vortex. Figure 6(b) shows simulation results of the control algorithm in (51)

with c0 = 0 and |ω0|−1 = rmax, which provides sampling of the idealized hurricane eyewall.

Figure 7(a) illustrates analysis of the feasibility of quadrifolium with a = 20 over a space

of centers contained within the Rankine vortex. Figure 7(b) illustrates simulation of the

control (59) with N = 10 particles and reference center c0 = 0. rmax is depicted by a

black circle. Feasibility analysis reveals that feasible folia are centered close to the origin.

Moreover, quadrifolia centered about the origin with a = 20 are in a moderate flow regime

for the majority of the trajectory. This analysis implies that quadrifolium formations may

be a poor candidate sampling trajectory in strong flows due to their infeasibility over large

areas and lack of sampling both inside and outside rmax.

An alternative to circular and folium trajectories that achieves good sampling density

and is feasible in a strong Rankine vortex is a spirograph trajectory. A spirograph is achieved

by traversing a circle with respect to a rotating frame. The spirograph family of formations

combines the advantages of circular and folia formations in that they generally align with

the vortex while criss-crossing the radius of maximum wind. For a given rate of rotation Ω,

circle radius |ω0|−1, and radius of circle center d, the feasibility of the resulting formation can

be analyzed using Theorem 2. Figure 8 shows the feasibility of various spirograph parameter

sets in the presence of a Rankine vortex. In each figure, the results are generalized by using

non-dimensional parameter sets (Ω, ω−10 , ρ) in terms of the Rankine vortex parameters rmax

and vmax, where Ω , (2πΩrmax)/vmax (the period of revolution of a passive particle at rmax),

d , d/rmax, and |ω0|−1 , |ω0|−1/rmax. In Figure 8(a), the feasibility of rate of rotation Ω

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(a) (b)

Figure 8. (a) Feasibility of spirograph formations for varying non-dimensional Ω and d values

with fixed radius ω−10 = 2/3, (b) spirograph feasibility for varying Ω and ω−10 with d = 1.

is plotted versus the distance of circle center from the origin for a counterclockwise circle of

constant radius, |ω0|−1 = 2/3. In 8(b), the feasibility of Ω is plotted versus the radius of the

circle for a fixed distance, d = 1. Dark areas indicate where the parameter set (Ω, ω−10 , d) is

infeasible.

Figure 9 shows simulations of the steering control (51) used in the equations of motion

(23) for N = 10 particles. The formation parameters are Ω = 0.0125 rad/s, ω−10 = 40, and

ρ = 30 (Ω = 1.57, |ω0|−1 = 4/3 and d = 1). Note from Figure 8(b) that this parameter

set corresponds to a feasible formation as required. A circular formation with respect to a

rotating reference frame produces a spirograph formation in an inertial reference frame. By

manipulating Ω, ω−10 , and d, spirograph trajectories can be made to concentrate radial and

azimuthal sampling densities to areas of interest within the vortex. This technique potentially

provides superior sampling coverage as compared to the circular and quadrifolium formations

of Figures 6 and 7.

VI. Conclusion

A strong flow presents challenges to multi-vehicle autonomous formation control. The

presence of strong flows limit the authority of the vehicle to completely dictate its inertial

direction of travel, making path planning and collective formation control algorithms more

difficult to design. However, decentralized multi-vehicle control algorithms driving particles

to desired formations are feasible in circulating flowfields. This paper utilizes an idealized

vehicle model consisting of identical unit-speed Newtonian particles with gyroscopic steering

control in a flowfield whose magnitude can exceed one. The authors address the properties

a flowfield must satisfy such that given trajectory is feasible and show that circular for-

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80 60 40 20 0 20 40 60 8080

60

40

20

0

20

40

60

80

Re(r)

Im(r)

(a)

80 60 40 20 0 20 40 60 8080

60

40

20

0

20

40

60

80

Re(r)

Im(r)

(b)

Figure 9. Spirograph formation produced by utilizing circular control with respect to rotatingframe with flow; in (a) an inertial reference frame and (b) a rotating reference frame.

mations will converge to a reference center as long as the reference center is feasible. By

using a lobe-counting variable, this work numerically derives and illustrates a control law

driving a collection of particles to a family of non-convex curves called folia, specifically the

quadrifolium. This curve closely matches that of existing hurricane sampling trajectories

of manned, and remotely piloted vehicles. Finally, the dynamics of an inertially unit speed

particle with respect to a rotating frame are derived to produce spirograph sampling forma-

tions. The authors derive the kinematic feasibility constraints of motion with respect to the

rotating frame and simulate vehicle motion in an idealized hurricane model.

This paper not only expands the applicability of multi-vehicle control to strong flow-

fields but also derives control algorithms to steer vehicles to non-convex patterns, namely

the quadrifolium and the spirograph. These formations may produce better data in environ-

mental sampling missions than those currently in use. Coupled with kinematic constraints

analyzing the feasibility of families of sampling trajectories, the control algorithms derived

in this paper can be used to provide vital information for multi-vehicle sampling missions al-

lowing researchers to choose from various families of sampling trajectories based on both the

goals of the sampling mission and the feasibility of the various sampling trajectories. This

concept was applied to an idealized hurricane model which revealed that spirograph trajec-

tories may be superior to both circular and quadrifolium trajectories in terms of feasibility

and radial and azimuthal sampling density.

Ongoing work seeks to derive control laws converging to a wider set of strong flows and

continues investigation of general flowfield properties that dictate the convergence of multi-

vehicle formations. In addition, the authors seek to optimize over the feasible parameter

space of spirograph sampling formations using the anticipated reduction in forecast error as

a performance metric.

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eiθk

fk

Rerk

Imrk

rk

eiθk + fk Rerk

Imrk

φk

φk

θk

λ

ξ

Figure 10. Orientation relations corresponding to singularities of (6).

Appendix

Proof of Lemma 1: Substituting eiγk = |eiθk +fk|−1(eiθk +fk) and (4) into the denominator

of (6) implies that uk is singular when

sk = 〈eiγk , fk〉 = |eiθk + fk|−1〈eiθk + fk, fk〉 = sk−1(〈eiθk , fk〉+ |fk|2

). (60)

If one considers the flow fk in its polar form fk = |fk|eiψk , where ψk = arg (fk), substitution

into (60) and solving for 〈eiθk , eiψk〉 gives

〈eiθk , eiψk〉 = cos(ψk − θk) = |fk|−1(s2k − |fk|2

)= |fk|−1

(|eiθk + fk|2 − |fk|2

). (61)

Figure 10 shows that when eiθk is drawn such that∣∣eiθk + fk

∣∣ is tangent to a unit circle

drawn about fk, a right triangle is formed with hypotenuse |fk| and sides∣∣eiθk + fk

∣∣ and∣∣eiθk∣∣. Since the vector triad forms a right triangle15 let ξ , sin−1(|fk|−1), and since the sum

of the interior angles of a triangle are supplementary,

ξ + λ+π

2= π. (62)

By projecting fk, we observe

θk − ψk + λ = π. (63)

Equating (63) and (62) and taking the cosine of the result implies

cos (θk − ψk) = −|fk|−1, (64)

which when substituted into (61) gives the Pythagorean Theorem, |eiθk + fk|2 + 1 = |fk|2.

Thus, when sk = 〈eiγk , fk〉, θk is such that eiθk , fk, and eiθk +fk form a right triangle. Solving

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(64) for θk while noting cos (θk − ψk) = cos (ψk − θk) completes the proof.

Proof of Lemma 2: The absolute value of (9) squared gives

h2k = h2

k + Ω2〈r, r〉 − 2Ω〈r, ieiθk〉. (65)

Substituting the singular condition hk = Ω〈rk, ieiθk〉 into (65) yields

h2k = Ω2〈rk, ieiθk 〉2 + Ω2〈rk, rk〉 − 2Ω2〈rk, ieiθk〉2

= Ω2〈rk, eiθk〉2,

revealing a relation between the speed of a particle and its position. The mapping can

become singular when

hk = Ω〈rk, eiθk〉. (66)

Specifically, if the vehicle is assumed to travel at unit speed in the inertial frame the mapping

is singular if Ω−1 = 〈rk, eiθk〉.

Acknowledgments

The authors would like to acknowledge discussions with Angela Maki, Doug Koch, and

Sharan Majumdar related to this work. In addition, acknowledgement is owed to Hugh

Willoughby and Sharan Majumdar for personal communications regarding current hurricane

sampling methods of manned aircraft flights. This material is based upon research supported

by the National Science Foundation under Grant No. CMMI0928416 and the Office of Naval

Research under Grant No. N00014-09-1-1058.

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