marine locomotion: a tethered uav-buoy system with surge
TRANSCRIPT
Marine Locomotion: A Tethered UAV-Buoy System with Surge Velocity Control
Ahmad Kourania, Naseem Daherb,∗
aVision and Robotics Lab, Department of Mechanical Engineering, American University of Beirut, Riad El-Solh, Beirut, 1107 2020, LebanonbVision and Robotics Lab, Department of Electrical and Computer Engineering, American University of Beirut, Riad El-Solh, Beirut, 1107 2020, Lebanon
Abstract
Unmanned aerial vehicles (UAVs) are reaching offshore. In this work, we formulate the novel problem of a marine locomotivequadrotor UAV, which manipulates the surge velocity of a floating buoy by means of a cable. The proposed robotic system canhave a variety of novel applications for UAVs where their high speed and maneuverability, as well as their ease of deployment andwide field of vision, give them a superior advantage. In addition, the major limitation of limited flight time of quadrotor UAVs istypically addressed through an umbilical power cable, which naturally integrates with the proposed system. A detailed high-fidelitydynamic model is presented for the buoy, UAV, and water environment. In addition, a stable control system design is proposedto manipulate the surge velocity of the buoy within certain constraints that keep the buoy in contact with the water surface. Polarcoordinates are used in the controller design process since they outperform traditional Cartesian-based velocity controllers whenit comes to ensuring correlated effects on the tracking performance, where each control channel independently affects one controlparameter. The system model and controller design are validated in numerical simulation under different wave scenarios.
Keywords: Aerial Systems: Applications, Marine Robotics, Motion Control, Locomotive UAV, Floating Buoy Manipulation.
1. Introduction
Aerial drones are finding their way into different sectorsof the industry, including construction [1, 2], agriculture [3],package delivery [4], inspection and maintenance [5], to namea few, in which drones not only independently fly in the air, butalso physically interact with the environment. In terms of ac-tivities, unmanned aerial vehicles (UAVs) can move slung pay-loads in solo [6] or cooperatively [7] for transportation tasks,they can interact and collaborate with unmanned ground vehi-cles (UGVs), and they can be equipped with robot manipulatorsto achieve different geometric configurations [8] or to coopera-tively manipulate other objects [1, 9, 10].
A common medium for UAVs to interact with their envi-ronment is through a tether [11, 12], as it can have a varietyof interesting applications including the transmission of power,forces, and data. Tethered UAVs were studied for stability andcontrol while maintaining positive cable tension in [13]. In fact,the tether can enhance the stability and performance of an aerialvehicle [14, 15], and can enable estimating the UAV’s relativepose to the tether anchor using on-board inertial sensors only[11], which can be upgraded with additional sensors, such asforce sensors and encoders, for better performance [16]. A ma-jor contribution to this field was presented in [17], which provedthat tethered aerial robotic systems are deferentially flat withrespect to two outputs: the link’s elevation angle and either thevehicle’s attitude or the longitudinal link force.
∗Corresponding author.Email addresses: [email protected] (Ahmad Kourani),
[email protected] (Naseem Daher)
Aerial drones are well-suited for applications that meet the4D criteria: dull, dirty, distant, and dangerous [18]. As such,the offshore oil industry and offshore wind-farms are excellentcandidates for their adoption, given the potential of drones tobecome the go-to technology in assets inspection and infras-tructure maintenance [18]. For example, traditional offshoresolutions such as inspecting offshore wind-farms entail movinga vessel, which is expensive and requires a human crew, un-like the deployment of drones that can significantly save costand time [5]. UAVs can also perform sensing jobs, place sen-sors, and perform on-sight repairs and maintenance [5]. Fur-thermore, offshore applications of UAVs make it more likelyfor aviation authorities to permit their utilization, since they aredeployed faraway from human populations [19].
Due to their limited power capacity and flight time, the in-teraction of UAVs with the marine environment is still in itsearly stages. Current uses are mainly limited to informationgathering such as transmitting visual feedback to human op-erators, targeting the locations of floating objects for their re-trieval [20], and generating and transmitting full area maps andpath-planning for other agents to perform rescue missions [21].Physical interaction is limited to low-power applications suchas landing assistance on a rocking ship [22], power-feeding theUAV through a cable [23, 24], and sensing jobs [25].
Although unmanned surface vehicles (USVs) are naturally-suited robots in marine environments, UAVs can outperformthem in certain aspects that make it more practical to adoptUAV-based marine solutions and applications. First, UAVsare advantageous over USVs in terms of their field of vision(bird’s-eye view), ease of deployment, and maneuverability, allof which give UAVs the advantage while performing tasks in
Preprint submitted to Robotics and Autonomous Systems August 2, 2021
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unstructured and hard-to-reach areas, and tasks that require pre-cision and quick deployment. In addition, UAVs are especiallyadvantageous in rivers since they can follow shorter paths aboveland and avoid in-water obstacles and waterfalls. Furthermore,it is more challenging to deploy and retrieve USVs since theyrequire direct access to the water surface, whereas UAV’s ben-efit from their vertical takeoff and landing (VTOL) capabilitiesto be independently deployed from anywhere. This fact high-lights the advantage that UAVs have in addressing the issue ofthe limited and expensive free-space on offshore structures, ve-hicles, and coastal strips that stand to benefit from deployingaerial robotic system solutions.
From the above literature survey and discussion, it is evi-dent that having an integrated system that incorporates an um-bilical power cable can open the door in front of a whole newlevel of UAV marine applications. An analogous system wasinvestigated in [26], in which an unmanned ground vehicle car-ries the power source while following a tethered UAV. Further-more, an optimal length and tension design of a cable that linksa UAV and USV was provided in [23]. The optimization prob-lem minimizes both fouling (cable entanglement or jamming)and excessive downforce on the UAV during dynamic heaves,which boosts the power capacity of the UAV and simultane-ously optimizes the dynamic performance of the coupled sys-tem. In addition, the employment of USVs as a landing plat-form has been studied in the literature; for instance, a coupledUAV−USV system was proposed in [27], where the USV isequipped with an expendable landing deck for additional safety,and the system serves as a foundation for further collaborativetasks.
Leveraging the technological advances in UAVs technol-ogy in terms of robustness, accuracy, operational cost, andlately, power efficiency, and motivated by applications requir-ing fast action with minimal water surface disruption [28], weare proposing the employment of a quadrotor UAV to manipu-late a passive floating object via a cable, whereby the quadrotorperforms the function of a locomotive. The umbilical powerline solution naturally integrates into this system, where the ca-ble can be used for both force and power transmission. Thehereby proposed problem generalizes the fixed-point tether de-scribed in [12] to a moving-frame tether, namely planar motionin the horizontal and vertical directions, and is subject to ad-ditional constraints such as maintaining contact with the watersurface. The formulated problem and proposed solution pavethe way in front of numerous UAV−USV interaction applica-tions, some of which are described next.
The proposed marine locomotive UAV system can be usedin coordination with nearby ships and marine structures to in-crease their maneuverability and decrease their response time,as well as nearshore and other water surfaces such as rivers andacross waterfalls. The proposed system can help in performinga variety of tasks including rescue operations, floating objectsrecovery, building and inspecting marine structures, water sam-ples collection, delicately placing and relocating marine sensorsand buoys with minimal water surface disruption, fishing activ-ities, and water surface clean-up efforts, to name a few. In thiscontext, we are motivated by the marine application in [28],
which proposed a sensor for measuring oil slick thickness dur-ing marine oil spill events. The proposed sensor is fixed to afloating buoy that is pulled by another vessel to skim the watersurface. One main challenge in the proposed solution lies in thevessel’s motion ahead of the sensor, which tends to disturb theoil layer and thus reduces the measurement’s accuracy.
This paper offers several technical contributions. First, thenovel problem of the marine locomotive UAV is formulated,which paves the way for further research into the interactionbetween UAVs and the marine environment. Second, the sys-tem is defined in a sea/ocean environment that accounts forthe presence of gravity waves and surface current, which nat-urally extends to wave-free water mediums. Third, the buoyand quadrotor UAV coupled dynamics are modelled with highfidelity using the Lagrangian formulation with appropriate con-straints for the tethered UAV−buoy system. Forth, the attain-able equilibrium states are derived with a proper definition ofthe system’s operational limits and constraints in terms of ca-ble tension, water surface contact, and buoy velocity. Fifth,we design and validate a buoy surge velocity control system,supervised by a state machine that switches between opera-tional modes, which results in accurate tracking performanceeven in the presence of disturbing waves, water currents, andfeedback noise, while reducing the system’s energy consump-tion by maintaining a constant UAV altitude. The controller re-lies on polar coordinates with respect to the buoy’s referenceframe to realize correlated tracking, which outperforms tra-ditional Cartesian-based and unsupervised UAV-only velocitycontrollers that do not lend themselves well to this application.Lastly, we make available a physics engine that can be used forsimulating tethered UAV−buoy locomotives via a custom-builtsimulator1.
The rest of this paper is structured as follows. A detaileddescription of the tethered UAV−buoy system model is pre-sented in Section 2. The designed control system is detailed inSection 3. Section 4 presents numerical simulation results thatdemonstrate the validity of the derived system model and the ef-fectiveness of the designed controller. Section 5 discusses somepractical considerations for the implementation of the proposedsystem, and finally Section 6 concludes the paper and providesan outlook into future work.
2. Tethered UAV-Buoy System Model
The dynamic model of the tethered UAV−buoy system re-quires the integration of multiple domains including the fluidmedium; the dynamics of the floating buoy, the UAV, and thecable; and the combined system of rigid bodies. In this section,we introduce the required subsystems to formulate the problemon hand.
2.1. PreliminariesThis section introduces some of the critical notations that
are used throughout the paper. We let the set of positive-real
1github.com/KouraniMEKA/Marine-Locomotive-UAV
2
wave profile
g
θ
u1
f1
f2u2
mT
TΔhru
r
u
zx
wIo
lα
brmb
FB
ob
u
'wg
ou
Figure 1: Planar model of a quadrotor UAV pulling a floating buoy through atether.
numbers x ∈ R |x > 0 be denoted as R>0, and the set ofnon-negative real numbers x ∈ R |x ≥ 0 be denoted as R≥0.Also, let s•, c•, and t• respectively be the sine, cosine, andtangent functions for some angle (•). In addition, let ‖·‖ denotethe L2 norm.
2.2. Problem DefinitionConsider the two-dimensional (2D) space in the water ver-
tical plane where the problem is set up as shown in Fig. 1, andletW = x, y represent the inertial frame of reference whoseorigin, OI, is at the local mean sea level horizontal line. Con-sidering the tethered UAV−buoy system depicted in Fig. 1, thebuoy is physically connected to the UAV by means of a ca-ble of length l ∈ R>0, forming an angle α ∈ (0, π) with thepositive x-axis, which is defined as the elevation angle. Letrb = xb, zb ∈ R2 and ru = xu, zu ∈ R2 respectively bethe coordinates of the buoy’s center of mass, (Ob), and that ofthe UAV, (Ou), inW; for ease of use, we set V := xb to depictthe buoy’s horizontal velocity. Let Bb and Bu be the body-fixedreference frames of the buoy at Ob, and of the quadrotor atOu, respectively. The floating buoy has a volume gb ∈ R>0, abounded massmb ∈ (0, ρwgb) moment of inertia Jb ∈ R>0 inBb; the quadrotor UAV has a mass mu and a moment of inertiaJu ∈ R>0 in Bu. Also let the orientation, measured clockwise,of Bb and Bu with respect toW be described by the angles θb
and θu ∈ (−π, π], respectively. Let Vb = ub, wb ∈ R2 andΩb ∈ R be the linear and angular velocities of the buoy in Bb,respectively. Furthermore, let the translational rotation matrixfrom any body frame toW be described as:
R• =
[c• −s•s• c•
]. (1)
Both the buoy and the UAV are subject to gravitational ac-celeration, g, and cable tension, T,∈ R≥0. Moreover, the buoyis subjected to hydrostatic and hydrodynamic forces that are de-scribed later, and the UAV propulsion can be simplified to onlyinclude the total thrust u1 ∈ R≥0, and a single torque that in-duces a pitch motion u2 ∈ R since the motion of the systemis restricted under the scope of this work to the vertical plane.Considering the relatively faster response of the UAV actuatorsas compared to the UAV itself, their dynamics are neglected inmodeling.
Remark 1. The scope of this work covers the manipulation ofa floating buoy, thus the buoy’s average density should not ex-ceed the density of water, which is achieved with the constraintmb ∈ (0, ρwgb).
Assumption 1. The cable is inextensible; it is attached to thebuoy’s center of mass at one end and to the UAV’s center ofmass at the other via revolute joints to prevent moment trans-mission; and for relatively small systems considered in thiswork, it can be of negligible mass. Considerations for heavycables (slung payload) are provided in Section 5.
2.3. Water Medium ModelThe water medium under consideration here is the
sea/ocean, where the main aspects of interest are gravity wavesand water surface current.
2.3.1. Gravity Wave ModelAssumption 2. In the considered problem environment, thewater depth is assumed to be much larger than the wavelengthof gravity waves, which are assumed to be of moderate height.This permits adopting linear wave theory in this work [29].In addition, the wave direction is limited to be in the vertical(x− z) plane.
Based on Assumption 2, the water elevation variation, ζ, attime t and horizontal position x due to gravity waves is statisti-cally described as:
ζ(x, t) =
N∑n
An sin(dnωnt− knx+ σn), (2)
where An, ωn, kn ∈ R≥0, dn ∈ −1, 1, and σn ∈ (−π, π] arerespectively the wave amplitude, circular frequency, wave num-ber, wave direction coefficient, and random phase angle of wavecomponent number n ∈ Sn with Sn = 1 ≤ n ≤ N |N ∈ N.Furthermore, based on Assumption 2, the wave number in deepwater is given by the dispersion relation as kn = ω2
n/g. Thehorizontal and vertical fluid particles’ wave-induced velocitiescan be prescribed as [29]:
vwx (x, z, t) =
N∑n
dnωnAneknz sin(dnωnt− knx+ σn),
vwz (x, z, t) =
N∑n
dnωnAneknz cos(dnωnt− knx+ σn).
(3)
where ωn relates to the wave period, Tn, via ωn = 2π/Tn.
2.3.2. Water CurrentFor brevity purposes, a simple yet comprehensive model of
the water current is adopted. The water surface current, actingin the horizontal x-direction, is given as:
Uc = Ul + Us, (4)
where Ul ∈ R is the lumped sum of different water currentcomponents, and Us ∈ R is the component generated from
3
Stokes drift [30]. The Stokes drift velocity is one componentthat emerges from nonlinear wave theory, and is defined as theaverage transport velocity of a wave over one period:
Us(z) =
N∑n
dnA2nωnkne
2knz. (5)
2.4. Buoy’s Dynamic ModelA floating buoy is subjected to different types of forces,
with the main ones being radiation, damping, and restorationforces. The radiation forces consist of the added mass andadded damping. In this section, a detailed description of thebuoy model is presented after introducing the following as-sumptions.
Assumption 3. The axes of the buoy’s body frame coincidewith its principle axes of inertia, which is a common practice tosimplify the modeling of marine vehicles [30].
Assumption 4. The water-buoy friction dominates the energydissipation in the system, and the system is assumed to operateunder moderate weather conditions, thus energy losses due toair drag are neglected.
Considering the buoy dynamics in Bb with the state vector,νb = [ub, wb,Ωb]ᵀ, and applying Newton’s second law of mo-tion yields:
M′bνb + C′bνb + D′bνb + G′b = τ ′b, (6)
where M′b, C′b, and D′b ∈ R3×3 are respectively the buoy’s
inertia, Coriolis, and damping matrices expressed in Bb; therelative velocity vector is defined as νb = νb − [Uc + vwx cθb −vwz sθb , v
wz cθb +vwx sθb , 0]ᵀ; G′b ∈ R3 is the gravitational forces
and moments vector; and τ ′b ∈ R3 includes external forces andmoments.
The inertia matrix is M′b = diag(mb +a11,mb +a33, Jb +
a55), where a11, a33, and a55 ∈ R≥0 are the surge, heave, andpitch rate components of the generalized added mass matrix.The added mass can be described as the amount of fluid that isaccelerated with the body, and can be written as a function ofthe buoy’s mass and moment of inertia. Furthermore, for lowfrequency motion, a11 ≈ 0.05mb, and a33 ≈ mb [30]. GivenM′
b, the Coriolis matrix is calculated as follows:
C′b =
0 0 (mb + a33)wb
0 −(mb + a11)ub 0−a33wb a11ub 0
. (7)
The total damping term of the buoy in Bb is expressed as:
D′b = DP + DS + DW, (8)
where DP = diag(b11, b33, b55) ∈ R3×3 is the radiation in-duced potential damping matrix with surge, heave, and pitchcomponents, and DS = diag(DS,1, DS,2, DS,3) ∈ R3×3 is theskin friction matrix, calculated as:
DS,i = CS,iAwt1
2ρw|ν|, i = 1, 2, (9)
where CS,i ∈ R>0 is a constant, Awt ∈ R≥0 is the buoy’swetted area, and DS,3 ∈ R≥0. DW ∈ R3×3 is the wave driftdamping matrix, which will be dropped from (8) since its ef-fect is already included in the Stokes drift velocity in (5). As-suming that the buoy, with a mean immersed height him, ver-tically oscillates in the x − z plane at ωo,3 ∈ R≥0, such thatωo,3 < 0.2
√g/him, which is practical for the problem on hand,
we have b33 ≈ 2mbωo,3. Moreover, the potential damping co-efficients in the horizontal plane vanish at both limits 0 and∞of the oscillation frequency, thus the potential damping in thex-direction is b11 ≈ 0 [30].
Referring to Assumption 1, the buoy dynamics in (6) can beexpressed inW with the state vector, ηb = [xb, zb, θb]ᵀ, as:
Mbηb + Cbηb + Db˜ηb + Gb = τb, (10)
where Db, Cb, and Db ∈ R3×3 are respectively the buoy’sinertia, Coriolis, and damping matrices expressed in W; therelative velocity vector is defined as ˜ηb = [˜ηb,1, ˜ηb,2, ˜ηb,3]ᵀ =ηb−[Uc+vwx , v
wz , 0]ᵀ; Gb and τb are respectively the vectors of
the buoy’s gravitational and other external forces and momentsinW expressed as:
Gb = [0,mbg, 0]ᵀ, τb = [Tcα, FB +Tsα, Frs]ᵀ, (11)
where FB = ρwggim is the buoyancy force, gim ∈ [0,gb] isthe immersed volume of the buoy, and Frs = fpsθu is the pitchrestoring moment with fp ∈ R being the buoy’s pitch restoringmoment coefficient. We also define:
Mb = (R−1θb
)ᵀM′bR−1
θb,
Db = (R−1θb
)ᵀD′bR−1θb,
Cbηb :=1
2Mbηb,
(12)
where Mb = ηᵀb(∂Mb/∂ηb) [30]. An explicit description of
Db and Db is given by:
Mb =
M ′b,11c2θb
+M ′b,22s2θb
s2θb(M ′b,22 −M ′b,11)/2 0s2θb(M ′b,22 −M ′b,11)/2 M ′b,11s
2θb
+M ′b,22c2θb
00 0 M ′b,33
,(13)
where M ′b,ii, i = 1, 2, 3 are elements of the buoy inertia matrixin Bb, M′
b. The buoy’s damping matrix in the inertial frameWis defined as:
Db =
D′b,11c2θb
+D′b,22s2θb
s2θb(D′b,22 −D′b,11)/2 0s2θb(D′b,22 −D′b,11)/2 D′b,11s
2θb
+D′b,22c2θb
00 0 D′b,33
,(14)
where D′b,ii, i = 1, 2, 3 are elements of the buoy damping ma-trix inBb, D′b. We also letMb,ij ,Db,ij , andCb,ij , i, j = 1, 2, 3be elements of Mb, Db, and Cb, respectively.
4
r < l
r = l
T > m g / sb α
Tr
>
(b)(a)
r = lT
(c)
VΔh
Figure 2: Depiction of the UAV−buoy system in violation of three constraints:(a) slack cable, (b) hanging buoy, and (c) ‘fly-over’ phenomenon.
2.5. UAV’s Dynamic Model
Referring to Assumption 1, and applying Newton’s secondlaw of motion on the UAV quadrotor system inW with the statevector, ηu = [xu, zu, θu]ᵀ, yields:
Muηu + Du˜ηu + Gu = τu, (15)
where Mu = diag(mu,mu, Ju) ∈ R3×3>0 and Du =
diag(Du,1, Du,2, Du,3) ∈ R3×3≥0 are the UAV’s inertia and
damping friction matrices, respectively; the UAV’s relative ve-locity vector is ˜ηu = [˜ηu,1, ˜ηu,2, ˜ηu,3]ᵀ = ηu − [uwd, 0, 0]ᵀ,with uwd being the horizontal wind velocity; and Gu andτu ∈ R3 are vectors of the UAV’s gravitational and other exter-nal forces and moments inW , respectively, expressed as:
Gu = [0,mug, 0]ᵀ, τu = [u1sθu−Tcα, u1cθu−Tsα, u2]ᵀ.
(16)
The damping matrix element of interest, Du,1, is approximatedas:
Du,1 = Cu,1Aucs,1
1
2ρa|˜ηu,1|, (17)
where Cu,1 ∈ R>0 is a constant, Aucs,1 ∈ R≥0 is the UAV’s
cross-sectional area across the zy-plane, and ρa is the air den-sity. For more details on the quadrotor UAV model, see [31].
2.6. System Constraints
In order to fully define the marine locomotive problem asa coupled UAV−buoy system, specific constraints are requiredand are presented hereafter, with their violations depicted inFig. 2.
2.6.1. Taut Cable ConstraintThis section introduces the resulting coupled dynamics of
the UAV−buoy system, which is achieved when the tether linksthe two bodies and holds positive tension, i.e. with a taut-cableconstraint that is opposite to what is shown in Fig. 2a. For thispurpose, we let W ′ = r′, α′ be a rectilinear moving polarframe fixed to Ob, shown in Fig. 1; this frame does not rotate,and it is parallel to the inertial frame W . The position of theUAV inW with respect toW ′ is defined as: r = ru− rb ∈ R2,and we let its coordinates inW ′ be r′ = r, α, such that:
r = ‖r‖, α = atan2(zu − zb, xu − xb). (18)
We also let the rates vector, r′, be:
r′ :=
[rrα
]= Rᵀ
α
[xu − xb
zu − zb
], (19)
where Rᵀα is the transformation matrix that rotates vectors in
W toW ′, and we finally let the acceleration vector, r′ be:
r′ :=
[r − rα2
rα+ 2rα
]= Rᵀ
α
[xu − xb
zu − zb
]. (20)
Definition 1. Based on Assumption 1, the cable remains taut,i.e. it maintains tension, at time t if r(t) = l. The taut-cablecondition is expressed as:
T > 0, (21)
under which the UAV−buoy system is labeled as ‘coupled’,otherwise it is labeled as ‘decoupled’.
With Assumption 1 and the taut-cable condition in (21), wehave r = l, and the polar coordinates of the UAV can be definedwith respect to the buoy’s center of mass, Ob, as:
xu = xb + lcα, zu = zb + lsα, (22)
and its velocity can be obtained as:
xu = xb − (lsα)α, zu = zb + (lcα)α. (23)
Lemma 1. When the UAV−buoy system is coupled, the UAV’sequations of motion in W ′ in polar coordinates notation areexpressed as:
−murα2 = mu(−xbcα − zbsα)−mugsα + u1sα+θu − T,
mur2α = mur(xbsα − zbcα)−mugrcα + ru1cα+θu .
(24)
Also, let Vr := ˜ηb,1 = V − Uc − vwx represent the buoy−waterrelative surge velocity. If the condition in (21) holds, the cabletension is expressed as:
T =
(Mb,11xb +Mb,12zb +Db,11Vr
+Db,12˜ηb,2 + Cb,11xb + Cb,12zb
)/cα, |α− π
2 | > εα,
(u1cθu −mug−muzu)/sα, |α− π2 | ≤ εα,
(25)
where εα ∈ R≥0 is a constant that prevents singularity in asmall region near α = π
2 .
PROOF. By differentiating r twice then multiplying it by mu,we get:
mur = muru −murb. (26)
Referring to Definition 1, we must have r(t) = l for the systemto be coupled, that is r = r = 0. Thus, by referring to (20), rreduces to the form:
r = Rα
[−rα2
rα
]. (27)
5
Referring to (15), and combining it with (26), then projectingalong the radial and tangential directions of (27) by means ofRᵀα, we can write the equations of motion of the UAV inW ′ in
the polar coordinates notation as in (24).The cable tension can be determined from the first row of
the buoy dynamics in (10), so that its expression is more rel-evant to the coupled UAV−buoy system since it shows a di-rect link with Vr, which yields the first case of (25). However,this form is not applicable near the vertical cable configuration(α = π/2) due to singularity, thus the actual cable tension, T ,is computed via (15), which yields the second case of (25).
2.6.2. No Buoy-Hanging ConstraintThe buoy is required to remain at the water surface level at
all times, that is, the UAV must not lift the buoy into the airby means of the cable tension alone, as shown in Fig. 2b. Thisconstraint can be forced by limiting the allowed cable tensionby the following inequality, deduced from (10) and (11) as:
T < mbg/sα. (28)
As noted in Remark 1, the buoy floats by itself, which meansthat no minimum cable tension is required to maintain the buoyat the water surface.
2.6.3. No ‘Fly-Over’ ConstraintThe buoy is required to remain in contact with the water
surface at all times, that is, the UAV must not force it to ‘fly-over’ the waves, as in Fig. 2c, when it encounters them withina specific frequency range. This constraint is described as:
gim > 0, (29)
which guarantees keeping the buoy partially immersed at alltimes. ‘Fly-over’ is a phenomenon that marks the flight of aplaning hull over the waves level, thus losing contact with thewater surface [32]. This phenomenon appears when the waveencounter frequency is near the resonant frequency of the hull,and is related to its Froude number [33].
To detect the occurrence of this phenomenon, the followinganalysis is presented. If the discontinuity in the buoyant forceis neglected, the buoy’s heave dynamics can be simplified andexpressed as a second-order transfer function with natural fre-quency, ωb, and damping ratio, µb, deduced from (10) as:
ωb =ρwgAb
cs,3
mb + a33, µb =
Db,33
2√
(mb + a33)ρwgAbcs,3
, (30)
where Abcs,3 is the mean horizontal cross-sectional area of the
buoy at the water surface level. We also define ωe,n, the waveencounter frequency for the nth wave component, as [29]:
ωe,n = ωn − dnω2nV
g, n ∈ Sn. (31)
The ‘fly-over’ phenomenon occurs at an exciting frequencywhere the increase in oscillations amplitude due to dynamic
magnification, ∆hamp, exceeds the mean immersed height ofthe buoy, him, that is:
∆hamp :=
N∑n
An
( 1√(1− ω2
n
)2+ (2µbωn)2
−1)> him,
(32)
where ωn = ωe,n/ωb. Further elaboration on the implicationsof this condition on the system modeling and performance re-quires knowledge of the buoy characteristics in terms of shapeand weight, as well as wave characteristics in terms of heightand wave length, which are presented in Section 4.
2.7. The Tethered UAV-Buoy System ModelThe formulation of the tethered UAV−buoy system, in
its coupled form, is obtained via the Euler-Lagrange formu-lation, while incorporating the results of Sections 2.4, 2.5,and 2.6. The Lagrangian function is obtained from the ki-netic (K(q, q) ∈ R≥0) and potential (U(q) ∈ R) energies asL(q, q) = K(q, q) + U(q), where q = [xb, zb, α, θu, θb]ᵀ ∈R5 is the generalized coordinates vector. The motion equationsof the UAV−buoy system can then be derived as:
d
dt
(∂L∂q
)− ∂L∂q
+∂P∂q
= τ , (33)
where τ ∈ R5 is the external forces vector; P ∈ R is a powerfunction that captures dissipative forces, such that ∂P∂q := D˜q,where D is the global damping matrix that can be formulatedbased on (12) without including a wind-induced component perAssumption 4; and ˜q is defined as:
˜q = [Vr, zb − vwz , α, θu, θb]ᵀ. (34)
To facilitate the derivation of the Euler-Lagrange equations,the kinetic energy of the system is expressed as the sum of thatof the buoy and that of the UAV:
K =1
2qᵀMq :=
1
2ηᵀ
bMbηb +1
2ηᵀ
uMuηu, (35)
where M is the global inertia matrix of the UAV−buoy sys-tem, which can be formulated by referring to (23) and using theelements of Mb and Mu. The system’s potential energy andexternal forces and moments vector can be formulated basedon (11) and (16) as:
U = mug(zb + lsα) +mbg zb,
τ = [u1sθu , u1cθu + ρwggim, u1lcα+θu , u2, fpsθu ]ᵀ.(36)
Finally, the following equations of motion that result fromEuler-Lagrange formulation (33) are obtained:
Mq + Cq + D˜q + G = τ , (37)
where Cq := 12Mq is the Coriolis matrix with M = qᵀ ∂M∂q ,
and the global vector of gravity forces G is:
G :=∂U∂q
= [0, (mb +mu)g,mu g lcα, 0, 0]ᵀ. (38)
6
Assumption 5. The buoy’s pitch dynamics are damped andstable, that is: Db,33 6= 0 and fp < 0. As a result, the buoy isassumed to remain tangent to the water surface at all times.
With Assumption 5 and the dominance of waves with rel-atively long wave period and moderate wave height, the timederivative of the buoy pitch angle, θb, is small and thus its ef-fect can be neglected in M, which yields a Coriolis matrix thatis a function of α only.
With constraints (21) and (29) satisfied, the dynamic modelequations in the coupled form are given by:
(Mb,11 +mu)xb +Mb,12zb +Db,11Vr +Db,12˜zb
−mul(cαα2 + sαα) = u1sθu ,
(39a)
(Mb,22 +mu)zb +Mb,21xb −mul(sαα2 − cαα)
+Db,22˜zb +Db,21Vr + (mb +mu)g = u1cθu
+ (ρwgim)g,(39b)
mul2α+mul(−sαxb + cαzb) +mug(lcα) (39c)
= u1lcα+θu , (39d)
Juθu = u2, (39e)
Mb,33θb +Db,33θb = fpsθb , (39f)
where ˜zb := ˜ηb,2. The UAV’s position and velocity vectors canthen be obtained from (22) and (23), respectively.
Remark 2. If the taut-cable constraint (21) is not satisfied, thesystem in (39) decouples into (10) and (15) with T = 0, andthe polar states r′, r′, and r′ are calculated from (18), (19), and(20), respectively. On the other hand, if the fly-over constraint(29) is not satisfied, the buoy’s inertia matrix in (13) reducesto Db = diag(mb,mb, Jb), the buoy’s damping matrix Db in(14) reduces to a null matrix, and fp becomes zero.
3. Control System Design
The control system design problem is defined as manipu-lating the surge velocity of the buoy, V , to track a desired ref-erence and to maintain the UAV’s elevation, zu, at a constantlevel, while ensuring that the dynamics of the UAV−buoy sys-tem remain stable and contact between the buoy and water ismaintained.
3.1. Attainable SetpointsThe control objective is to attain a steady-state mean ve-
locity of the buoy, (V ), and mean UAV’s elevation, (zu), suchthat limt→∞( 1
t
∫zu(t)dt, 1
t
∫V (t)dt) = (zu, V ). Next, we
seek to find the set of other system states, namely, θu and gim,and control inputs, u1 and u2, that will achieve the control ob-jective. Other nonzero mean system variables in a steady-statesurge motion are: T , Db,11, and Db,21. Note that the bar sign(•) refers to the mean values of the variables at equilibrium,gim implicitly represents zb, and the buoy’s pitch angle, θb, isnot considered in the setpoint analysis per Assumption 5.
Definition 2. Under specific sea conditions, namely Uc andζ(An, ωn) with n ∈ Sn, and certain safety marginsεT (Uc, ζ) ≥ 0 for the cable’s tension, which guaranteesthe coupled state of the system; and εg ∈ (0, 1) for thebuoy’s immersed volume, to ensure a minimum buoy immer-sion that is suitable to the desired system application; the setof admissible configurations consists of the equilibrium points(V , gim, α, θu), such that:
T := T (Vr, α) > εT , gim := gim(Vr, α) > εggb, (40)
where Vr = V − Uc.
Assumption 6. The equilibrium state is analyzed under the no-wave condition: An = 0 with n ∈ Sn, that is, vwx = vwz = 0.
Theorem 1. Consider the system described in (39), subject toconstraints (21) and (29), and the margins specified in (40); byAssumptions 5 and 6, the set of attainable equilibrium states isthe union of all (V , gim, α, θu) that satisfy:
θu(Vr, α) = atan( Db,11Vrcαmu g cα + Db,11Vrsα
), (41)
and the steady-state thrust value and immersed volume are cal-culated as:
u1 =
any R>0, if α = π
2
Db,11Vr/sθu , otherwise,(42a)
gim =
mb+mu
ρw− u1
ρwg , if α = π2
mb
ρw− Vr
ρwg
(Db,11tα − Db,21
), otherwise.
(42b)
Given gim, we can solve for zb per specific buoy geometry. Inaddition, the cable tension at equilibrium is a function of Vr
and α, and expressed as:
T =
any R>0, if α = π
2
Db,11Vr/cα, otherwise.(43)
PROOF. The dynamic equilibrium of system (39) is attainedwhen xb = zb = θu = θu = θb = θb = α = α = 0, andsince we are considering surface motion of the buoy along withAssumption 6, we additionally have zb = 0. Thus, we concludethat u1 = u1, and u2 = u2 := 0, and by substituting in (39),we get:
Db,11Vr − u1sθu = 0, (44a)Db,21Vr + (mb +mu)g− u1cθu − (ρwgim)g = 0,
(44b)
mugcα − u1cα+θu = 0. (44c)
θu(Vr, α) in (41) is obtained by rearranging and dividing (44a)by (44c); in the case when α 6= π
2 , u1 can be subsequently ob-tained from (44a). As for gim, it can be obtained after substi-tuting for u1 from (42a) and for θu from (41) in (44b). Finally,the cable tension at equilibrium can be obtained from (25) by
7
canceling the zero-valued states. Note that when α = π2 , the
system of equations (44) has a solution only if Vr = θu = 0,while u1 can be any R>0 that respects the system constraints,and can be chosen to manipulate gim based on the first case of(42b).
Now we seek to define the set of possible attainable steady-state velocities, SV , under Assumption 6, that satisfy the safetymargins specified in (40). The cable tension at equilibrium, T ,can be obtained from (43); then we can determine the mini-mum absolute surface velocity, V , under a specific sea state,i.e. current and waves, that guarantees the taut-cable condition.In addition, the maximum absolute limit of SV is attained from(40) and (42b). Finally, we get SV = SV n ∪ SV p, such that:
SV p =( εT cαDb,11
+ Uc,(mb +mu − εgρw)g tθu
Db,11+ Uc
),
if α ≤ π
2
SV n =( (mb +mu − εgρw)g tθu
Db,11+ Uc,
εT cαDb,11
+ Uc
),
if α >π
2.
(45)
Remark 3. In practice, the maximum attainable absolute ve-locity can be limited by the UAV’s maximum thrust, which canbe derived from (41) and (42a), and the tether’s yield strength.It is also noted that the motion across waves of various charac-teristics may alter the velocity bounds as to be discussed in Sec-tion 4.2. In addition, the violation of the buoy’s velocity upperbound can be alternatively prevented by referring to constraint(28) and the cable tension calculation in (25), and limiting theUAV’s maximum thrust such that:
u1 <mb(1− εm)g
tαsθu, (46)
where εm ∈ [0, 1) is a safety margin that represents a fractionof the buoy’s mass, and accounts for the unmodeled dynamicforces affecting the buoy’s heave motion that might violate thesystem constraints in (28) and (29).
3.2. Operational Modes and State MachineTo achieve acceleration and deceleration motions, the
UAV−buoy system is required to manipulate the cable tension,switch between coupled and decoupled states, and achieve bidi-rectional velocity control; hence, the UAV must change its po-sitioning with respect to the buoy back and forth. Thus, thelocomotive UAV control system is to be designed to operatein both position control and velocity control modes, which ne-cessitates the use of a state machine to achieve an autonomousperformance of the UAV−buoy system. Note that a cable canonly transmit tensile forces, thus allowing only pulling actions.
Next, we provide the required definitions to describe thesystem states, present a complete cycle of the system’s opera-tional states to achieve the control objectives, and we introducea state machine that allows the execution of appropriate com-mands.
r
zu
l
ob
rrsb
zu
α0π-α0
rrsb
zuπ-2α0
α0
α
rπ-α0T
r
T
(b)
(d)
(a)
(c)
''
Figure 3: UAV−buoy system operational states in the locomotion task: (a) freeUAV motion around the buoy within the cable limit, used in initializing thesystem, (b) ready to pull forward (or backward), the UAV is in the right posi-tion to generate tension in the cable when asked to do so, (c) switching UAV’spositioning between front and rear, while following the trajectory marked indashed blue to avoid cable entanglement, and (d) coupled and pulling forward(or backward) to manipulate the buoy surge velocity.
3.2.1. Operational ModesDefinition 3. The UAV’s location with respect to the buoy isassigned one of the following two configurations:
• We call ‘front’ the configuration at which the UAV is po-sitioned to the front of the buoy, i.e. α ∈ (0, π2 ).
• We call ‘rear’ the configuration at which the UAV is po-sitioned to the rear of the buoy, i.e. α ∈ (π2 , π).
We let r be the reference radial position of the UAV with respectto the buoy. The UAV−buoy system can be in one out of fouroperational modes shown in Fig. 3:
(a) We call ‘free’ the mode during which the UAV is allowedto move freely around the buoy, while r < l.
(b) We call ‘ready to pull’ the mode during which the UAVis commanded to maintain a specific elevation (zu), and areference standby radius, rsb, which is slightly less thanthe cable length l to consume any cable slack. The eleva-tion angle is α0 if the configuration is ‘front’ and (π−α0)if the configuration is ‘rear’.
(c) We call ‘repositioning’ the mode during which the UAVmoves from one side of the buoy to the other (fore/aft),travels a total arc of (π − 2α0), while maintaining a con-stant reference radius with respect to the buoy, r = rsb,until it returns to the initial elevation, zu.
(d) We call ‘pulling’ the mode during which the UAV is per-forming a pulling action on the buoy with a reference el-evation, zu, and radius, r = l. The resulting elevationangle is α′0 if the configuration is ‘front’ and (π − α′0) ifthe configuration is ‘rear’.
8
time
Velocity
p2
n2
pulling forward
repositioning
pulling backward
ready to pullrear
pullingbackw
ard
εp1
εp2
εn2
εn1
ready to pullfront
ready to pullfront
Vp
1
n1
V
Figure 4: Demonstrative diagram showing the modes’ transition behavior andthe buoy’s velocity tracking performance during a theoretical scenario. Thedoted boundary lines govern the actions of the state machine.
Algorithm 1 State machine for the locomotive UAV’s controlsystemInput: V , V , configurationOutput: MODE
Initialization :MODE⇐ ‘free’LOOP Process
1: if (V < V − εth1) and configuration == ‘front’ then2: MODE⇐ ‘pulling’3: else if (V > V + εth1) and configuration == ‘front’ then4: MODE⇐ ‘ready to pull’5: else if (V > V + εth2) and configuration == ‘front’ then6: MODE⇐ ‘repositioning’7: else if (V > V + εth1) and configuration == ‘rear’ then8: MODE⇐ ‘pulling’9: else if (V < V − εth1) and configuration == ‘rear’ then
10: MODE⇐ ‘ready to pull’11: else if (V < V − εth2) and configuration == ‘rear’ then12: MODE⇐ ‘repositioning’13: end if14: return MODE
3.2.2. State MachineThe UAV−buoy system is supervised by a state machine
that governs switching between different control modes andcommanded actions. Fig. 4 illustrates a typical velocity pro-file, with a hypothetical tracking performance, and thresholdlines that govern the state machine actions to showcase the stateswitching mechanism. Let the threshold levels be denoted by p1
and p2 for the top two lines, and n1 and n2 for the bottom twolines. The first and second velocity error thresholds are denotedby εth1 and εth2 respectively.
The proposed state machine benefits from the threshold ve-locity lines to choose the suitable mode of action as describedin Definition 3, in a way that respects the system dynamics andinsures the system safety [16], with a pseudo-code provided inAlgorithm 1.
3.3. Controller DesignThe control system of the tethered UAV−buoy system con-
sists of an outer-loop and an inner-loop controller in a cascadedstructure. The outer-loop controller has two functions: 1) itcontrols the UAV’s relative position when the system mode is‘free’, ‘ready to pull’, or ‘repositioning’ by controlling r andα, with setpoint (zu0, r0); and 2) it controls the buoy’s veloc-ity when the system mode is ‘pulling’, by regulating the ele-vation angle, α, and the cable tension, T , which are two flatoutputs of the coupled system [17], with setpoint (zu0, V0). Onthe other hand, the inner-loop controller controls and stabilizesthe UAV’s pitch angle, θu. The proposed controller (SVCS) in-corporates the state machine in Section 3.2.2, and it is designedbased on polar coordinates. The SVCS architecture is presentedin Fig. 5, which can be summarized as follows:
• A setpoint is defined and the state machine returns thesystem mode.
• A preprocessing unit generates 1) an elevation angle αthat accounts for the actual buoy elevation variation, 2) areference radial distance r, 3) a smoothed reference ve-locity V , and 4) an estimate for the required cable ten-sion, Tc, to compensate for water drag, if applicable.
• The outer-loop controller generates radial and tangentialcomponents of the desired force that is needed for cabletension control, uv
r , in case of velocity control or simplyradial force, up
r , in case of position control (radial), andthe elevation angle control, uα (tangential). Note that theswitching between up
r and uvr is governed by the opera-
tional mode of the system such that:
ur =(1− fplH(s)
)upr + fplu
vr , (47)
where H(s) is the transfer function of a low-pass filter,and fpl ∈ 0, 1 is the ‘pulling’ mode flag.
• The outer-loop controller outputs are decoupled into acommand total thrust, u1, and a desired pitch angle, θu,c.
• Finally, the inner-loop attitude controller stabilizes thepitch angle of the UAV and produces the moment com-mand input, u2.
3.3.1. Reference Signals and Velocity SetpointIt is desired for the UAV to maintain the same altitude dur-
ing operation in order to respect aviation safety margins andsave energy by reducing unnecessary vertical motion. The ca-ble length and nominal elevation angle are chosen accordingly.However, due to the vertical oscillatory motion of the buoy inaccordance with the encountered waves, we must actively pro-vide the controller with suitable elevation angle, α, to hold thedesired UAV elevation, which is computed as:
α = asin((zu − zb)/r
). (48)
Note that the preprocessing unit outputs the supplementary an-gle of α, i.e. α ⇐ π − α, if the system configuration is ‘rear’.
9
SVCS
Preprocessingzu0 r0 V0 α V
θu,c
Tethered UAV-Buoy
Buoy-WaterInteraction
u1
u2
Outer-Loop
uα
(radial: buoy manipulation)
(tangential: UAV elevation)
(radial: relative position)
Setpoint
zb
Inner-Loop(UAV attitude)
θ ,θu uα,α, V, V, zbr, r,
Decoupling(UAV thrust)
State Machine
urv
urp
r
V
ur
Tc
Figure 5: Architecture of the Surge Velocity Control System (SVCS) for thetethered UAV−buoy system.
Furthermore, the velocity setpoint, V0, and the radial position,r0, are smoothed by second-order and fourth-order low-pass fil-ters, respectively, in order to respect the system dynamics interms of buoy−water friction and the UAV’s maximum thrust,thus preventing excessive coupling and decoupling of the sys-tem [17]. Finally, in order to improve the performance of thestate machine, V is sent to the controller only when the UAV isready to enter the ‘pulling’ mode.
3.3.2. UAV-Buoy Relative Position Control LawConsider the UAV−buoy’s relative position dynamics in
(24) for the generic case, i.e. nonzero tension, and theUAV’s attitude dynamics in (15), with states vector X1 =[r, α, θu]ᵀ and X2 = [r, α, θu]ᵀ, and control input vectorU = [up
r , uα, u2]ᵀ, such that:
ur = u1sα+θu , uα = u1cα+θu , (49)
and subject to unknown external disturbances like wind gusts,gravity waves, and water currents. Note that the relation be-tween up
r and ur was given in (47). When represented as a non-linear second-order time-varying system, the state space formis described as:
X1 = X2,
X2 = H + ΦΘ + bU + δ,(50)
where b = diag(mu, mur, Ju)−1 is the input-multiplied vec-tor, Φ = [1/(mucα), 0, 0]ᵀ is the regressor vector, Θ = T isthe parameters vector; δ = [δr, δα, δθ]
ᵀ is the vector of lumpedsystem disturbances and modeling errors across each channel,where δ = [δr, δα, δθ]
ᵀ is its estimate; and H ∈ R3 denotesthe nonlinear and gravitational terms vector defined as:
H =
rα2 − xbcα − zbsα − gsα(−2rα+ V sα − zbcα − gcα)/r
0
.
Assumption 7. The modeling errors and external disturbancesand their derivatives are bounded.
Assumption 8. The lumped error vector δ is constant orslowly varying during a finite time interval, that is:limt1<t<t2 δα, δr, δθ ≈ 0.
Let θ′u,c be the desired UAV pitch angle to be generated bythe outer-loop controller along with the total thrust command,u1, which are calculated as:
u1 =√u2α + u2
r, θ′u,c =π
2−α−arctan(uα, ur). (51)
Let θu,c = θu,m tanh(θ′u,c/θu,m
)be a smooth and bounded ver-
sion of θ′u,c, with θu,m ∈ (0, π2 ) being the absolute upper limitof the UAV’s attitude angle. The reference state vector to befollowed is defined as X1 = [r, α, θu,c]ᵀ. Let the state errorvector be defined as:
e1 = X1 − X1. (52)
The proposed control law including the radial and tangentialthrust components for the outer-loop UAV’s relative positioncontroller, and the UAV’s pitching torque, is defined as [34]:
U = b−1[− kPe1 − kDe1 − kIeI1 + ¨X1 −H −ΦΘ
],
eI1 = e1 + k−11 e1,
(53)
where kP , kD, kI , and k1 ∈ R3×3>0 are controller gains that are
defined next.
Theorem 2. Consider the UAV−buoy’s relative position dy-namics in (24), and the state space representation of the sys-tem in (50). Suppose that Assumptions 7 and 8 hold true; thecontrol law in (51) and (53) generates the total thrust, u1, andthe UAV’s desired pitch angle, θu,c, that can stabilize the sys-tem, and reduce the tracking error to zero in finite time for a setof gains k1, k2, and γ ∈ R3×3
>0 , such that kP = I3 + k1k2,kD = k1 + k2, kI = γk1, with I3 being the identity matrix.If Assumption 8 does not hold, the tracking error reduces to asmall region neighboring the origin in finite time.
PROOF. The backstepping control design, involving two steps,is employed, and the Lyapunov function V1 = 1
2eᵀ1e1 is pro-
posed. Its derivative is expressed as: V1 = eᵀ1 e1. Since e1 doesnot explicitly include a control input, we continue the controldesign process for a second step. The virtual control input tostabilize e1 is defined as: Υ = ˙X − k1e1. Next, we define thevirtual rates error as: e2 = X1 −Υ.
By defining a second Lyapunov function:
V2 =1
2eᵀ1e1 +
1
2eᵀ2e2 +
1
2δᵀγ−1δ,
where δ = δ− δ, then by differentiating and combining it withV1, we get:
V2 = eᵀ1 e1 + eᵀ2 e2 + δᵀγ−1 ˙δ
= eᵀ1(e2 − k1e1) + eᵀ2(H + bU + ΦΘ + δ − Υ)
+ δᵀγ−1 ˙δ.
10
Next, we choose the control inputs and the lumped modelingand disturbances errors’ update rates such that V2 becomes neg-ative semi-definite:
U = b−1(−H −ΦΘ− δ + Υ− e1 − k2e2
),
˙δ = γe2,
(54)
and we get V2 = −eᵀ1k1e1 − eᵀ2k2e2. Thus, the asymptoticconvergence of V2 to zero can be obtained via Barbalat’s lemmaunder Assumption 8. If strong wind and wave disturbances ex-ist, meaning the violation of Assumption 8, the control law willstill achieve stability and finite tracking error, which can be re-duced by increasing the controller gains up to a level that over-comes the disturbances mismatch effect on V2. Finally, by sub-stituting Υ and e2 in (54), and setting eI1 := δ(γk1)−1, thePID-like control law in (53) is obtained.
3.3.3. Buoy Surge Velocity Control LawConsider the UAV dynamics inW ′, while following the po-
lar coordinates notation as presented in (24), and let the buoy’svelocity error be defined as eV = V − V . The buoy velocitymodel can be expressed in the generic case, i.e. variable radialposition, as:
V = HV − T/(mucα) + uvr/(mucα), (55)
whereHV = (rα2− r− zbsα−g sα)/cα. A control law can bedesigned for the surge velocity in a similar fashion as describedin Section 3.3.2, with a difference that only one step is requiredin the backstepping process. The resulting control law is givenby:
uvr = Tc+mucα
(−HV + ˙V −kPV eV −kIV eIV
), eIV = eV ,
(56)
where kPV and kIV ∈ R>0 are controller gains. The results ofTheorem 2 relative to stability and tracking apply.
Remark 4. Cable tension can be either directly measured (e.g.load cell) to improve the tracking performance and the system’soverall safety, or it can be estimated via an observer designbased on cable disturbance estimation methods. However, thisinternal force, T , and its estimate, T , should not be confusedwith the term Tc used in the control law (56), and representingthe required tensile force to manipulate the buoy. One simplerealization is obtained based on (43), such that:
Tc = Db,11,0V /cα, (57)
whereDb,11,0 = CS,1Awt,012ρw|V |, with Awt,0 being the zero-
tension whetted area, and CS,1 being the surge skin friction co-efficient at V , which yields a fair, yet not very accurate, esti-mate. However, the proposed controller can compensate for theestimation error as will be proven next. For a sample cable ten-sion estimation based on disturbance observation, readers arereferred to [35].
Remark 5. Practically, robust performance of the proposedcontrol laws is guaranteed by choosing large-enough k2 gainsfor a wide operating range, even if Assumption 8 is violated[36].
Table 1: Tethered UAV−buoy model parameters
Par. Value Unit Par. Value Unitlb 0.8 m mu 1.8 kghb 0.25 m Ju 0.03 kg m2
mb 12.5 kg θu,m π/4 rada11 0.625 kg l 7 ma33 12.5 kg εT 5 Nb11 0 Ns/m εg 0.05 -b33 27.5 Ns/m εm 0.1 -νw 1.78e-6 m2/s ρw 1000 kg/m3
g 9.81 m/s2 ρa 1.22 kg/m3
4. Simulations
In this section, we provide simulation results that demon-strate the fidelity of the tethered UAV−buoy system model andthe performance of the designed controller. We first define thesettings and parameters used for the devised simulation scenar-ios, which include various operating conditions to validate theproposed system. To challenge the control law’s performancetowards real-life implementation, the tethered UAV−buoy sys-tem model is incorporated in the simulator developed in thiswork, while including deviation from the described model usedby the control law, including the UAV’s propellers motor dy-namics, wind gusts, and non-exact state feedback.
4.1. Simulation SettingsTo validate the proposed UAV−buoy system with the de-
signed SVCS, a series of simulations is performed in the MAT-LAB Simulink ® environment. We consider a quadrotor UAVand a simplified homogeneous cuboid buoy with the dimen-sions and parameters listed in Table 1. The quadrotor UAV mo-tor dynamics are modeled as a first-order low-pass filter with atime constant τm = 0.05 s, and its total thrust and pitch torqueare bounded such that ‖u1‖ ≤ 160 N, and ‖u2‖ ≤ 11.2 N m.The mass of the buoy is chosen such that the buoy is one quarterimmersed under no external loads based on the balance betweenthe gravitational and buoyancy forces, that is mb := ρw gb /4.The buoy’s immersed volume is then defined as:
gim(∆h) =
gb if ∆h > hb
2 ,
0 if ∆h < −hb
2 ,
gb/2 + lbhb∆h otherwise,(58)
where ∆h = ζ(xb, t)− zb(t). The wetted area is calculated as:
Awt(∆h) =
4lbhb if ∆h > hb
2 ,
0 if ∆h < −hb
2 ,
lbhb + 2lb(hb
2 + ∆h) otherwise.(59)
The resulting added mass and damping are calculated as de-scribed in Section 2.4, and their values are presented in Ta-ble 1. The buoy’s skin friction coefficients in its body x- and
11
z-directions can be estimated as CS,i = 0.075/(log10 Re−2)2,where Re = |Vr|lb
νw∈ R≥0 is the Reynolds number, limited to
turbulent flows (Re > 105), with νw being the water’s kine-matic viscosity [37]. To detect the coupling state of the system(coupled / decoupled), we rely on the tension estimation in thesecond case of (25).
To properly evaluate the performance of the proposedSVCS design, a Cartesian-based nominal controller (CBNC)that uses a PID control law in its outer-loop, and without su-pervision of a state-machine, is implemented for benchmarkingpurposes. It consists of a velocity (x) controller and an elevation(z) controller, with gains kP,CBNC = diag(7, 3), kI,CBNC =diag(1.2, 1), and kD,CBNC = diag(5, 2), respectively. TheSVCS gains are selected as k1 = diag(16.9, 4.6, 7.5), k2 =diag(2.6, 2.4, 2.5), γ = diag(0.5, 0.3, 0.3), kPV = 25, andkIV = 12.
The feedback signals are assumed to be available from sen-sor measurements and estimations, and are modeled as follows:the UAV’s pose is virtually obtained from an on-board GlobalPositioning System / Inertial Navigation System (GPS/INS)module, and the elevation angle and the radial distance arevirtually obtained from a stereo camera system. In simula-tion, this is mimicked by augmenting the feedback states bya filtered Gaussian noise with the corresponding state-of-the-art accuracy of each sensor before being used by the con-troller. With mav(•) denoting the mean absolute value of theestimation error of entity (•), we set mav(xu) = 0.02 m =mav(zu) = 0.02 m, mav(θu) = 0.5°, mav(α) = 0.16°, andmav(r) = 0.02 m. Subsequently, the buoy’s states are deter-mined from (18), (19), and (20). More details on state estima-tion is given in Section 5.1.
4.2. Velocity Bounds
The constraints’ bounds (εT , εg, and εm) presented in Ta-ble 1 are mainly based on the expected buoy−water relative ve-locity, in addition to the buoy’s shape, weight, and skin friction.A possible command velocity range of SV = (−19.0,−3.1) ∪(2.1, 18.0)m s−1 is calculated from (45) under no-wave condi-tion (Assumption 6).
In the presence of waves, the feasible working velocity withno violation of constraint (29) reduces from above, and can bequantified by referring to (32) and (42b) as follows. We solvefor gim to get him, then find ∆hamp for some (V , α), underdifferent wave conditions. Fig. 6 provides the buoy’s heave dy-namic amplification results under excitation of a single fully-developed wave component [38], with α = 45° and Uc = 0. Tohave a unified representation of him, the Stokes drift effect isneglected in calculating Vr. The natural frequency of the buoy,calculated from (30), is ωb = 8.9 rad s−1.
Fig. 6 can be interpreted as follows: for a given sea condi-tion with wave amplitude and period An,Tn, the buoy hopsover the waves (‘fly-over’ condition) when its horizontal (for-ward or backward) velocity, V , falls outside the shaded area(dome) formed by the him curve, for a given ∆hamp,n (coloredplots corresponding to various wave amplitudes and periods).
-20 -15 -10 -5 0 5 10 15 200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
-10 18-6 131
5
Figure 6: Buoy’s heave dynamic amplification, ∆hamp,n, under excitationof different fully-developed single wave components An,Tn of the setsA = 0.27, 0.61, , 1.2, 1.5, 3.3/2 m, and T = 3, 4, 5, 5.7, 8 s. The meanbuoy’s immersed height, him, draws the boundary dome for the ‘fly-over’-freeregion.
Sample zones, where the buoy ‘fly-over’ condition does not oc-cur, are marked on top of the figure as Sfo
V ,n. The comprehen-
sive results captured by Fig. 6 show that the system operationis direction-dependent, and they also serve as a reference forpredicting the performance of the buoy in terms of heave os-cillation and ‘fly-over’ phenomenon under different wave con-ditions, ranging from high-frequency low-amplitude waves tolow-frequency high-amplitude ones, and even for superpositionof various waves, as will be demonstrated in the subsequentsections. We note that the above analysis is provided for a buoyof known characteristics (Table 1), and serves as a guideline forthe system performance.
4.3. Simulation Scenarios
We validate the fidelity of the derived system model andevaluate the performance of the designed controller in fourcases: C1, C2, C3, and C4. All cases include a wind gustof uwd = −3 m s−1 and a water current component Ul =−0.5 m s−1. The scenarios are described as follows:
• C1: wind gust and water current only.
• C2: wind gust, water current, and moderate waves withtwo wave components (N = 2), such that: A1 =0.135 m, d1 = 1, T1 = 3 s, and σ1 = π; A2 = 0.75 m,d2 = 1, T2 = 5.7 s, and σ2 = 0.
• C3: high-frequency small-amplitude waves (head-seas),with A1 = 0.135 m, d1 = −1, T1 = 3 s, and σ1 = 0.
• C4: high-amplitude low-frequency waves (head-seas),with A1 = 1.65 m, d1 = −1, T1 = 7 s, and σ1 = 0.
Note that the wave components definitions in each scenario isindependent from the other scenarios. Sample visual illustra-tions of the environments in C1 and C2 are given in Fig. 7,which are generated via the custom-built simulator that wespecifically developed to serve as a physics engine and pro-vide live animations for tethered UAV−buoy locomotives. In
12
zu
water surface line
Case 2: gust, current and wavesCase 1: gust and current only
two wave components
motion direction
Figure 7: Sample screenshots from animations of two simulation scenarios(C1 and C2) in true scale. Animations are generated via a custom-built sim-ulator that is specifically developed to serve as a physics engine for tetheredUAV−buoy locomotives.
both cases, C1 and C2, the buoy is commanded to accelerate toreach an inertial velocity V = 5 m s−1, after which it graduallydecelerates to 0 m s−1 then to−4 m s−1. The desired referencemean sea level altitude is zu = 5.0 m, which corresponds to amean elevation angle of α0 = 45. The system is initiated inthe decoupled state, and its velocity is initiated to be equiva-lent to the zero-time water velocity via (3) and (4). Based onAssumption 5, the buoy’s pitch angle is calculated by differen-tiating (2) with respect to xb:
θb = atan( N∑
n
Ankn cos(dnωnt− knxb + σn)). (60)
While cases C1 and C2 provide a baseline evaluation of theproposed robotic system and its controller, cases C3 and C4challenge its performance in extreme cases, i.e. under fast oscil-lations (C3), and high amplitude undulations (C4). A low-sloperamp velocity input (V = 0.25 t) is applied to carefully capturethe performance of the system at different velocities, and head-seas are considered to emphasize and validate the universalityof the buoy’s dynamic heave performance captured in Fig. 6.
4.4. Simulation Results and Discussion
The simulation results for C1 and C2 are shown in Fig. 8aand Fig. 8b, respectively. In both cases, the quadrotor UAVequipped with the SVCS is able to pull the buoy at the desiredvelocities (V ) without overshoot, with minimal fluctuations invelocity (V ) and elevation (zu), while not violating constraint(29) as indicated by the immersed volume plot (gim/gb), andwithout unnecessarily decoupling the system (as seen in the rsubplot). The resulting commands to the UAV, u1c and u2c,are bounded and free of high-frequency chattering. On theother hand, the Cartesian-based controller without state ma-chine supervision (CBNC) results in significantly larger veloc-ity (V ) and elevation (zu) fluctuations, reaching up to 2 m s−1
and 2.2 m, respectively, in the wavy environment (C2).The proposed controller adjusts the elevation angle, α,
while the buoy elevation, zb, changes − driven by the contour-following behaviour of the buoy under long waves excitation− to prevent unnecessary UAV vertical motion (zu) as evidentin Fig. 8b. The adjustments are such that α varies in responseto changes in the buoy’s elevation, zb, which is proportional tothe wave encounter frequency. It is also observed that the el-evation angle (α) and pitch angle (θu) are smooth and stable,
CBNC SVCSReference
-5-2.5
02.5
5
-0.75
0
0.75
4567
3
5
7
45
90
135
(°)
-40
0
40
u(°
)
0
15
30
0.15
0.25
0.35
0 20 40 60 80Time (s)
10305070
u 1c(N
)
0 20 40 60 80Time (s)
-3
0
3
u 2c(N
.m)
-5-2.5
02.5
5
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4567
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45
90
135(°
)
-40
0
40
u(°
)
0
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0 20 40 60 80Time (s)
10
40
70100
u 1c(N
)
0 20 40 60 80Time (s)
-3
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3
u 2c(N
.m)
(a) case 1: gust and current only.
(b) case 2: gust, current and waves.
ζ (x )b zb
ζ (x )b zbCBNC:
SVCS:
ζ (x )b zb
ζ (x )b zbCBNC:
SVCS:
Figure 8: States and control inputs for the simulation scenarios C1 and C2, withboth the state machine-supervised surge velocity control system (SVCS) and astandard Cartesian-based nominal UAV controller (CBNC). The region in redmarks when the mode is not ‘pulling’, and the region in green marks when themode is not ‘repositioning’.
13
Table 2: Tracking Error and Consumed Energy
CaseV , error zu, error cons. en.(cm/s) (cm) (kJ)
CBNC SVCS CBNC SVCS CBNC SVCSC1 36.4 5.4 28.4 2.7 111.8 58.9C2 61.3 6.1 42.8 5.9 93.4 61.2
and exhibit small tracking error for the SVCS. We note that thereference UAV pitch angle, θu,c, for the CBNC is not plottedfor figure clarity purposes, since both systems possess the sameinner-loop controller, in addition to the fact that CBNC has noreference elevation angle, α, and radial position, r.
The SVCS-controlled UAV achieves the desired surge ve-locity of the buoy by adjusting the cable tension, T , in an ap-propriate and relatively smooth manner as seen in (T ). Con-trarily, the CBNC has no direct control of the cable tension andthe radial position of the UAV, which leads to repeated largeinput pulses that deteriorate the transient performance. Finally,it is observed that the change in the immersed volume of thebuoy greatly depends on the encounter frequency. It is also no-ticed that the buoy remains in contact with the water surface((gim/gb)) for both controllers.
To quantify the performance of the two controllers, the tra-jectory tracking errors of V and zu, and the energy consumedby the UAV (calculated per [39]) are reported in Table 2. TheSVCS results in an average reduction in the tracking error of88 % and in energy consumption of 42 % versus the CBNC.
Remark 6. Expressing the SVCS in polar coordinates yields acorrelated control performance, which means that each controlchannel, (ur and uα), independently affects one control param-eter (α or V ). However, this is not the case for the Cartesian-based controller (CBNC), where each of the x- and z-controlchannels has a dual effect on each control parameter, which re-sults in a degraded performance.
In summary, the CBNC does not cope with the introduc-tion of waves to prevent them from disturbing the system in anunpredictable manner, nor it respects the system configuration,whereas the SVCS shows its disturbance-rejection property insignificantly attenuating the waves’ effect even without know-ing their characteristics. All of the above factors, combined,justify the design of the relatively complex SVCS for the pro-posed marine locomotive UAV system.
The performance of the SVCS-equipped locomotive systemagainst high-frequency and high-amplitude waves are shownin Fig. 9a and Fig. 9b, respectively. The first separation ofthe buoy from the water surface occurs at t = 21 s and V =5 m s−1 in C3, and t = 41 s and V = 11 m s−1 in C4, whichare marked by the yellow strips in their respective subplots. Thetracking accuracy in V and zu demonstrates that the proposedSVCS performs well in the considered extreme scenarios, aslong as they are within the working zones established in Fig. 6.Beyond these zones, i.e. after the instances marked by the yel-low strips in Fig. 9, the buoy ‘fly-over’ deteriorates the system
SVCSReference
0 20 40 60 0 20 40 60
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(°)
0 20 40 60Time (s)
(a) case 3: high frequency, head seas.
00.5
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0 20 40 60Time (s)
(b) case 4: high amplitude, head seas.
-1.50
1.53
ζ (x )b zb
0 20 40 60 0 20 40 60
ζ (x )b zb
Figure 9: Simulation scenarios C3 (high-frequency small-amplitude waves) andC4 (high-amplitude low-frequency waves) to illustrate the SVCS performanceagainst extreme sea conditions. The yellow strips mark the first buoy−waterseparation (‘fly-over’) in each case.
performance, which manifests as jumps of the buoy above thewaves as exhibited in the z subplot.
5. Practical Considerations
For the proposed system to lend itself well to physical im-plementation, we target in this section critical aspects that areessential to experimentally validate the proposed system, inpreparation for its deployment in real-life.
5.1. States EstimationThe SVCS requires the following states for feedback: r, r,
r, α, α, θu, θu, V , xb, and zb. Unlike most applications of thetethered UAV problem, the tether of the UAV−buoy system isnot anchored to a fixed point as in [40, 12, 17]; furthermore,knowing that the system is allowed to decouple, resulting ina slack cable, observer-based methods that target system stateestimation in the taut cable case, as in [16], cannot be solelyemployed for state feedback. However, since the UAV−buoysystem works above the water surface, it is practical to assumethat GPS coverage is available, which allows for UAV pose es-timation in the inertial frame using a GPS sensor and an inertialmeasurement unit (IMU) that is equipped with a magnetometer[16]. We also note that if the system is designed to operate inthe vicinity of a marine structure, Real-Time Kinematic (RTK)GPS solutions can also be utilized to attain more accurate iner-tial state estimation [41]. To solve the state estimation problem
14
when the cable is slack, the UAV can be equipped with a stereocamera to detect and estimate the buoy location in the cameraframe using special-purpose algorithms [42, 43], from whichthe UAV’s relative radial coordinates to the buoy (r and α), andthe buoy’s velocity (V ) can be estimated [44]. We note that amonocular camera can provide adequate accuracy for the con-trol problem on hand only if the buoy’s dimensions are knowna priori [45]. As for laser-baser sensory equipment, they aresusceptible to sun rays exposure and water surface refraction,which deems them unsuitable for such applications. Last butnot least, using encoders placed on the UAV can help with mea-suring the cable’s length and elevation angle in the taut-cablecase, and a force sensor (e.g. load cell) allows measurement ofthe cable’s tension [16], thus providing the control system withadditional information to improve its performance.
5.2. Power considerations
To make the system more energy efficient, it can be de-signed to allow the UAV to land on the buoy, or to float di-rectly on the water surface, during long standby periods [27].Another alternative to further extend the work-time of the sys-tem is to integrate an umbilical power cable within the tether.Furthermore, using an umbilical power line with power banksstationed on the buoy can be more efficient than increasing theon-board power capacity of the UAV under specific conditions.Also, a small relative buoy−water velocity and a streamlinedbuoy shape can result in better energy efficiency. We note thatin case of large umbilical power transmission cables, their masscannot be neglected and must be compensated for in the controllaw of the coupled dynamics model as in [10] and [46]; and inthe decoupled form, the UAV controller should be modified tocompensate for the cable mass as in [47].
5.3. Platform
The locomotive UAV system can be deployed from shipsand marine structures. It can be an independent system, and ifdesigned and equipped to work autonomously, it can link itselfto the target floating object using an on-board cable and performmanipulation afterward. Such designs have higher mobility,easier deployability, and independence from potentially-bulkybuoys.
6. Conclusion
The novel problem of a marine locomotive UAV system isdefined, in which a quadrotor UAV is tethered to a floating buoyto control its surge velocity. The system dynamics are sepa-rately modeled for each subsystem including the water medium,the buoy, and the UAV, then combined via the Euler-Lagrangeformulation. The attainable setpoints and constraints of the pro-posed system are defined, then a precision motion control sys-tem is designed to manipulate the surge velocity of the buoywithin certain limits, which require maintaining the cable in ataut state and keeping the buoy in contact with the water sur-face. A simulation environment is defined, and the proposedSVCS is validated and compared to a nominal Cartesian-based
UAV controller, while showing superior tracking performanceand disturbance rejection in certain waves, surface currents, andwind conditions.
The proposed system paves the way in front of a wide va-riety of novel marine applications for multirotor UAVs, wheretheir high speed and maneuverability, as well as their ease ofdeployment and wide field of vision, give them a superior ad-vantage. It best suits applications that require remote and fastmanipulation with minimal water surface disruption.
Going forward, we aim to extend the problem to the three-dimensional (3D) space, perform energy minimization tech-niques for the system’s path-planning under different wave andcurrent conditions, and study the buoy’s stability by introducingadditional constraints to the system. The controller and buoystability will be further challenged in more complex wave sce-narios to uncover the buoy’s shape effects on the system perfor-mance. A parallel future path entails building a prototype of theproposed system and performing experimental validation of theproposed controller.
ACKNOWLEDGMENT
This work is supported by the University Research Board(URB) at the American University of Beirut (AUB).
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