Fifth International Symposium on Marine Propulsorssmp’17, Espoo, Finland, June 2017

Comparison of Different Methods for Estimating Energetics Related toEfficiency on a UUV with Cephalopod Inspired Propulsion

Michael Krieg1,3, Kamran Mohseni1,2,3

1Department of Mechanical and Aerospace Engineering,2Department of Computer and Electrical Engineering,

3Institute for Networked Autonomous systems, University of Florida, Gainesville, Florida

ABSTRACT

In this paper we investigate an unmanned underwater ve-hicle which utilizes a set of novel cephalopod inspiredpulsed jet thrusters. Recent studies on live squid suggestthat the efficiency of this type of propulsor may be signifi-cantly higher than previously assumed Bartol et al. (2009).However, there is little experimental data of efficiency onunderwater vehicles with this style of propulsion, due totheir recent development. Furthermore, as the technologyis fundamentally different from traditional propeller stylethrusters, standard efficiency approximations are insuffi-cient. This study is a first step towards experimental testingof the propulsive efficiency of this technology. We exam-ine three different methods for estimating control forces,as will be necessary to calculate useful propulsive work.These include: modeling propulsive jet impulse, modelingpressure distribution within the thruster mechanism, andbacking out control forces from hydrodynamic force esti-mates and vehicle inertial forces measured with a motioncapture system. All three control force measurement tech-niques showed good agreement for a variety of maneuvers,validating the different assumptions of the individual mod-els.

We also look into two methods for estimating the workdone on the surrounding fluid environment required to cre-ate the propulsion. It was observed that estimating totalrequired work from the excess kinetic energy in the wakewill always under-predict the actual total work due to vis-cous dissipation of kinetic energy prior to measurement ofthe wake energy, and that a more accurate estimate can beachieved by integrating the product of pressure and bound-ary velocity on the surface of the thruster mechanism.

Keywords

Propulsive Output, Unrestrained UUVs, Control Forces,Energetics.

1 INTRODUCTION

Figure 1: Successive frames of a UUV with squid inspiredthrusters performing a mock parallel parking maneuver.

As the use of unmanned underwater vehicles (UUVs) in-creases in both autonomous missions and missions requir-ing human-robot interactions, there is an increased needfor enhanced vehicle capabilities. One such desirable ca-pability is improved position tracking accuracy in turbulentenvironments. Accurate low speed maneuvering in any en-vironment, chaotic or not, is difficult for underwater ve-hicles to achieve without sacrificing efficient forward lo-comotion. Typically autonomous underwater vehicles thatare designed for long range missions have a slender torpedo

Corresponding Author: Mohseni@ufl.edu

shaped body to maximize efficiency of forward propulsion,and control surfaces generate maneuvering forces. Whensuch a vehicle is traveling with little or no forward velocity,the forces from control surfaces become negligible, makingturning (yaw) or sideways translation (sway) impossible.Vehicles designed for missions which require higher accu-racy maneuvering; alternatively, have thrusters attached toseveral points around the surface of the vehicle providingfull control over the six degrees of freedom, but inducingsignificant drag losses during long range transit.

Our group has proposed that long range torpedo shapedvehicles can achieve high accuracy low speed maneuver-ing utilizing cephalopod inspired jet thrusters that can beplaced internal to the UUV hull surface Mohseni (2006),minimizing impact on forward drag while providing con-trol forces in the absence of forward velocity. We havedemonstrated the feasibility of such systems with multipleUUV prototypes Krieg et al. (2011). These thrusters, likesquid and jellyfish, cyclically suck in some of the surround-ing seawater then eject it as a high momentum propulsivejet. Even though there is zero net mass flux during an en-tire cycle, there is a positive flux of momentum and en-ergy associated with the jetting. The thrusters offer sev-eral advantages. Aside from the negligible impact on for-ward drag already mentioned, the thrusters provide con-trol forces instantaneously, unlike propeller based thrustersYoerger et al. (1990) or tunnel thrusters Mclean (1991) forexample.

However, jet propulsion has long been assumed to be inher-ently inefficient Lighthill (1969); Vogel (1994), as thrustscales with jet velocity squared and power scales withjet velocity cubed for steady jets. Recent studies, how-ever, have reported propulsive efficiency as high as 78% inadult L. brevis swimming at high velocities, and averaged87% (6.5%) for paralarvae Bartol et al. (2009), challeng-ing the notion that a low volume high velocity jet inher-ently negates a high propulsive efficiency. The unexpect-edly high efficiency of this mode of swimming is likelydue, in large part, to the formation of a leading vortex ringassociated with unsteady jetting, which is known to affectpressure dynamics at the nozzle exit Krueger (2005)Krieg& Mohseni (2013).

Due to the novelty of this type of propulsion there has beenvery little investigation into the efficiency dynamics asso-ciated with different types of maneuvers. The goal of thecurrent study is to resolve several issues related to exper-imentally measuring the propulsive efficiency of this typeof thruster on a freely moving UUV, specifically the mostaccurate ways of measuring control forces and total powertransfer to the surrounding fluid without restricting the mo-tion of the UUV.

2 VEHICLE DESCRIPTION

The UUV used for this experiment is the most recent ofseveral vehicle platforms designed to showcase a new styleof underwater propulsion. The vehicle, shown in figure 2,has a long slim torpedo shaped body with a propeller inthe rear, and four internal pulsed jet thrusters to providemaneuvering forces. The vehicle is described in great detailin Krieg et al. (2011).

Figure 2: Picture of the newest generation UUV with squidinspired thrusters and other systems labeled. More detailof the vehicle can be found in Krieg et al. (2011).

2.1 Dimensions and Overall Performance

The UUV is 15 cm in diameter and 1.12 m in length witha total mass (including ballast) of 64.5kg. With thrustersoperating at frequencies approaching cavitation frequency,which is a limiting frequency for proper thruster operationKrieg & Mohseni (2008), the vehicle has a maximum swayvelocity of 0.12m/s (roughly equal to one body diameterper second). The vehicle has a maximum forward velocityof 0.56m/s (roughly 0.5 body lengths per second).

2.2 Cephalopod Inspired Thrusters

Much like squid, octopus, cuttlefish and other cephalopods,from which it draws inspiration, our thruster consists of aninternal fluid cavity that can deform to change the inter-nal volume. At the cavity interface with the vehicle hullthere is a small circular orifice exposed to the outer wa-ter. The cavity of the thruster provides the same function-ality as the squid mantle or the jellyfish bell, expanding andcontracting to cycle water in and out of the circular orifice(functionally similar to the squid funnel/siphon). The cav-ity has a cylindrical shape with a flat circular plate at theback called the ‘plunger’. The side wall is made of flexiblerubber, but reinforced with helical supports so that it candeform in the axial direction allowing the plunger to moveback and forth, but maintains a consistent diameter.

3 EXPERIMENTAL SETUP3.1 Testing Tank

The prototype UUV described in section 2 is tested withina large cylindrical water tank. The tank, which is shown infigure 3, is 7.6 m in diameter and 4.9 m deep with a totalvolume of 2.46 kl. An aluminum platform runs across oneedge of the tank, providing 5.5 m of access to the watersurface.

(UUV Testing Tank)

Motion Capture Cameras

Squid-Inspired Thrusters

AUV

(View Inside Tank)

Figure 3: Large water tank used to test UUV prototypes, showing the underwater motion capture system.

3.2 Motion Capture System

Global Coordinate System

Identified AUV MarkerConfiguration

Local CoordinateSystem

Figure 4: Screenshot of the virtual testing environmentrecorded by the motion capture system. It can be seen thatthe markers on the UUV are identified as a set configura-tion with an associated local coordinate system.

The exact trajectory and orientation of the UUV through-out different maneuvers are determined and recorded by anunderwater motion capture system installed within the ve-hicle testing tank. The system consists of 6 Oqus 510+underwater motion capture cameras, which can operate inboth marker identification mode and video capture mode,with a maximum frame rate of 250 fps. Multiple camerasare shown installed within the tank in figure 3. Prior toeach UUV testing session the motion capture system wascalibrated with an average residual of 1.78 mm±0.25 mm.The residual during calibration can be considered the accu-racy of the position measurement of the UUV. This averageresidual corresponds to an error of ±0.025 rad in the mea-sured orientation of the vehicle, which is determined fromthe position of markers on the vehicle surface relative toeach other. The markers on the vehicle surface are madeof retro-reflective tape, so that they do not project out from

the vehicle surface or affect the vehicle drag properties.

3.3 Maneuver DescriptionsThere are three basic types of maneuvers performed by theUUV in this study: sideways translation along the y-axis(sway), rotation about the z axis (yaw), and forward trans-lation along the x-axis (surge).

4 UUV ENERGETICS ASSOCIATED WITHPROPULSIVE EFFICIENCY

In the most general sense the propulsive efficiency, ηprop,of an underwater vehicle is the ratio of the useful propul-sive work, WP , (sometimes referred to as the thrust work)to the total work required to generate the propulsion, WT .Here, these characteristic quantities will be defined usingnomenclature consistent with Fossen (1994).

WP =

∫ tf

t0

~FC · ~̇η dt =

∫ tf

t0

~τC · ~ν dt , (1)

~η = [X,Y, Z, φ, θ, ψ]T, (2)

~ν = [u, v, w, p, q, r]T, (3)

where η is the position/orientation of the vehicle in an in-ertial frame, [X,Y, Z] are positions in the inertial frame,[φ, θ, ψ] are Euler angles; ν is a vector for the linear andangular velocities in the body fixed coordinate system,[u, v, w] are the body fixed velocities in the surge, sway,and heave directions, and [p, q, r] are the angular velocitiesabout each respective body fixed axis; τC is a vector of thecontrol forces/torques in the body fixed frame; FC are thecontrol forces and torques acting on the body in the direc-tion of the inertial frame axes; and t0 and tf are the initialand final times, respectively, of the maneuver. Clearly thedefinition of the propulsive work in the body-fixed framehas the advantage that the propulsors on the UUV have afixed layout in this coordinate frame.The total work required to create the necessary propulsiveforces (τC and FC) is specific to the particular method ofactuation, and hence not easily determined from vehicle

kinematics alone. In the following sections we will de-scribe several different approaches for backing out the use-ful propulsive work and the total work needed to calculatepropulsive efficiency.

4.1 Estimating Control Forces on a FreelySwimming Vehicle

Calculating the useful propulsive work, equation (1), re-quires that the 6 component UUV velocity and controlforce/torque vectors are known in either the inertial orbody-fixed coordinates. The inertial velocity vector is mea-sured directly by the motion capture system, and is easilyconverted between the two coordinate systems. However,measuring control forces acting on a freely moving vehi-cle is more difficult, since directly attaching the vehicle tosome form of force balance would restrict its movement.Here we describe how the control forces can be recoveredby various methods.

4.1.1 Calculating Jet Forces From Motor Fre-quency

The total hydrodynamic impulse, I , of each expelled jet isequivalent to the net impulse transferred to the UUV overthe full pulsation cycle. Therefore, the average thrust pro-duced is equal to the product of the hydrodynamic impulseof a single jet and the jet pulsation frequency. It should benoted; however, that the instantaneous thrust is not neces-sarily equal to the rate at which impulse is created in the jetsince there are additional unsteady forces, as will be dis-cussed in the next section, that have no net contribution tothe average thrust over an entire cycle.The rate at which hydrodynamic impulse is created instarting jets, including the contributions from vortex ringformation and nozzle geometry, was derived in Krieg &Mohseni (2013) as,

dI

dt= ρπ

(g +

k?2 − k?14

)ub(t)

2R4p

R2. (4)

Here, ub(t) is the velocity of the plunger inside the cavity,Rp is the radius of the plunger, R is the radius of the noz-zle, k?1 is the non-dimensional slope of the radial velocityprofile at the nozzle exit, k?2 is the non-dimensional slopeof the radial velocity gradient, and g is a function relatedto the axial velocity profile at the nozzle. The terms k?1 ,k?2 , and g are characterized for different nozzle geometriesin Krieg & Mohseni (2013), but for the thrusters installedon the UUV the following values are a good approximationk?1 = −0.5, k?2 = 1.1, g = 1.25. Furthermore, the plungerdriving mechanism on the UUV creates a sinusoidal jet ve-locity program, so that the average thrust, T̄ , can be calcu-lated by integrating equation (4),

T̄ (f) = 2ρπ3R4

(g +

k?2 − k?14

)(L

D

)2

f2 , (5)

where f is the pulsation frequency and L/D is the strokeratio of the jet. For the thrusters studied here L/D = 3.

4.1.2 Unsteady Internal Pressure ModelingEven though the net impulse transfer during a single pulsa-tion is equal to the hydrodynamic impulse of the propulsivejet, the instantaneous forces created during jetting includeforces due to acceleration and deceleration of fluid insidethe jetting cavity. These forces are cyclical, and thus makeno net contribution to the impulse transfer over an entire cy-cle. However, work is required to create the internal fluidoscillations. Here we will describe a technique for ana-lytically modeling the unsteady pressure distribution insidethe thruster cavity, which can be used to calculate instanta-neous thrust, and the total work is calculated in the follow-ing section.It was recognized in Krieg & Mohseni (2015) that the cen-tral axis of axisymmetric jet flows is inherently irrotational,despite complicated vorticity patterns throughout the restof the domain. This allows internal pressure to be related tostagnation pressure by integrating the momentum equationalong the axis. In addition, the closed loop velocity inte-grals defining circulation in the cavity and jet regions canbe segmented into velocity line integrals that show up in theintegral of the momentum equation. We will not present thecomplete derivation of internal pressure; however, as de-fined in Krieg & Mohseni (2015), the relationship betweeninternal pressure and system circulation is,

Pbρ

=P∞ρ

+dΓJetdt

+dΓ?Cavdt

+1

2u2b , (6)

where P∞ is stagnation pressure, Pb is the pressure on theinternal surface of the cavity plunger, ub is the velocity ofthe plunger, ΓJet is the circulation of the propulsive jet,Γ?Cav is the circulation inside the cavity neglecting the cir-culation created by boundary stretching.With the exception of a few locations where vorticity isgenerated (i.e. at the nozzle edge and cavity surface) vortic-ity is just diffused without increasing or decreasing overallcirculation. Therefore, the evolution of system circulation,dΓJet

dt and dΓCav

dt , can be calculated by modeling the finitenumber of vorticity sources.Unsteady cavity-jet systems have four distinct sourcesof vorticity that are identified and modeled in Krieg &Mohseni (2015). Though all the details of these models aretoo lengthy to discuss in this paper, here we will summarizeeach vorticity source and how they depend on thruster op-erating conditions. The first source is the flux of vorticitycarried in the shear layer created at the nozzle edge by fluideither exiting or entering the cavity. The rate of vorticityflux is proportional to the square of the jet velocity squared.The second source of vorticity is associated with fluid con-verging before it passes through the nozzle. This vorticitycan be modeled by the potential flow of a disc shaped ve-locity sink and increases system circulation proportional tothe jet acceleration. The third vorticity generation mech-anism only exists during the refill phase of the pulsation

cycle. It occurs when the leading vortex ring from incom-ing fluid impinges upon the inner cavity surface, which re-sults in a boundary layer being formed on the surface ofopposite sign vorticity. The fourth source is due to stretch-ing of the cavity surface, but cancels with velocity integralterms in the momentum equation integration and does notaffect pressure dynamics. Therefore, both the vorticity fluxand half-sing vorticity terms can be calculated from the jetvelocity program, but the refill impingement vorticity de-pends on both cavity geometry and jet velocity program.It should be noted that both the pressure modeling of thissection and the impulse modeling of section 4.1.1 were de-rived and validated for jetting systems in a fixed location.Neither has been validated for thrusters on a moving vehi-cle.

4.1.3 Modeling Hydrodynamic Inertial andDamping Forces

An alternative approach is to recognize that the total forceacting on the UUV, τRB , is the sum of the control forces,τC and the hydrodynamic forces, τH , resulting from vehi-cle motion, ~τRB = ~τC + ~τH . As such, if the total forces,~τRB , and the hydrodynamic forces, ~τH , are known then thecontrol forces can be calculated as the difference betweenthe two. The sum of forces acting on the UUV can be de-termined from the vehicle kinematics and finite rigid bodydynamics. The motion capture system described in section3.2 captures the 6-DOF velocity, ~ν, and acceleration, ~̇ν, ofthe UUV. The rigid body dynamics described in a frame ofreference attached to a rigid body are given using similarnotation to Fossen (1994),

~τRB = MRB~̇ν + CRB(~ν)~ν , (7)

where MRB and CRB are the mass matrix and cen-tripetal/Coriolis force matrix for the rigid body. When thegeometric center of the body lies on the center of massMRB is the diagonal matrix, diag{m,m,m, Ix, Iy, Iz},where Ii is the moment of inertial about the i’th axis andCRB is defined by,

CRB =

[∅3×3 C1

C1 C2

]C1 = m

0 w −v−w 0 u

v −u 0

C2 =

0 Izr −Iyq−Izr 0 Ixp

Iyq −Ixp 0

Thus the total forces acting on the vehicle can be calculatedfrom the motion capture kinematic data and the UUV massand moments of inertia.The most accurate method for determining hydrodynamicforces would be to measure them directly from a hypotheti-cal pressure/shear sensor networks distributed on the UUVsurface Xu & Mohseni (2013), however, that system is stillin the process of being developed. In this study we model

the hydrodynamic forces with respect to vehicle kinemat-ics, which is fairly standard in the underwater vehicle com-munity. The general equation for hydrodynamic forces onan UUV, as summarized in Fossen (1994) is,

~τH = −MA~̇ν − CA(~ν)~ν −D(~ν)~ν − g(~η) , (8)

where, MA and CA are the added mass matrix and addedcentripetal/Coriolis matrix, respectively, D is the drag ma-trix, and g(~η) is a term representing the restorative forces(i.e. buoyancy). As a first order approximation of the vari-ous hydrodynamic force coefficients, we treat the UUV asa perfect cylinder with uniform density equal to the sur-rounding water, and assume symmetry of CA(~ν) and D(~ν)

with respect to the radial directions. For such a system,

MA = diag {A11, A22, A33, A44, A55, A66}

CA(~ν) =

[∅3×3 C3

C3 C4

]

C3 =

0 A33w −A22v

−A33w 0 A11u

A22v −A11u 0

C4 =

0 A66r −A55q

−A66r 0 A44p

A55q −A44p 0

D(~ν) = diag {D11, D22, D33, D44, D55, D66}

g(~η) = 0 ,

(9)

where Aij are the coefficients of the added mass matrixand Dij are the coefficients of the drag matrix. The prin-ciple components of the added mass matrix, Aii, are cal-culated according to strip theory derived from 2D potentialflows Wendel (1956) and UUV geometry. The principlecomponents of the drag matrix are defined as standard highRe drag equations, for i = [1, 2, 3] Dii = 1

2ρSiCdi |~ν|,D44 = 0, and for i = [5, 6] Dii =

ρL3sds

24 CDy,z |νi|. Inthese expressions Si are the wetted areas of the UUV indifferent directions, Ls and ds are the length and diame-ter of the UUV, the value for CDx is taken from Hoerner(1965) for cylinders in axial flow, and the value for CDyis taken from Hoerner (1965) for cylinders in cross flow.The coefficients approximated for the UUV of this study

are listed below,

A11 −A66 = [2.0 kg, 20.7 kg, 20.7 kg,

0 kgm, 0.0030 kgm, 0.0030 kgm]

D11 −D66 =[1.87kgs , 14.03kgs , 14.03kgs ,

0kgm2

s , 3.63kgm2

s , 3.63kgm2

s

]In this experiment we operated the UUV near the surfacein order to maintain communication using a wireless an-tenna. Since the UUV is near the surface, the added massestimates will likely be too high, as added mass coeffi-cients reported for catamarans and bulb sections near thefree surface have even become negative for certain frequen-cies Fossen (1994). In order to improve our estimates ofhydrodynamic forces, we calibrate these coefficients usingspecific portions of the trajectory data when the thrustersare not active. When the UUV has performed a maneu-ver, and the propulsors are shut off, the vehicle continuesto move as the hydrodynamic forces bring it to rest. Asan example, figure 5 shows the rotational velocity of theUUV during one of the yaw maneuvers, with the region ofinactive drifting marked by the vertical red lines.

5 10 15 20Time (s)

0

0.1

0.2

0.3

0.4

0.5

r-V

eloc

ity (

rad/

s)

Figure 5: Radial velocity r of the UUV performing an ex-ample yaw maneuver. The vertical dashed lines indicate aspecific period of this maneuver which is used to calibratethe hydrodynamic force coefficients, since the vehicle ismoving through the fluid but there are no control forcesacting on it.

During these periods the hydrodynamic force vector, ~τH ,is identical to the total rigid body force/torque vector, τRB ,calculated from acceleration/velocity data. Thus, unknownhydrodynamic force coefficients can be determined by fit-ting the motion capture trajectory data for all the maneu-ver sections without active propulsion. Using an itera-tive least-squares algorithm, we fit the added mass anddrag matrix coefficients, Aii and Dii, to the form (8) withτH = τRB

(~ν, ~̇ν)

, assuming constant coefficients, for allof the trajectory sections with inactive propulsors. The ini-tial guess was set to the analytically predicted coefficients,

and coefficients were allowed to vary between zero and200% of the predicted coefficients. The resulting fitted co-efficients near the surface are listed below,

A11 −A66 = [0.52 kg, 1.97 kg, 1.97 kg,

0 kgm, 0.0029 kgm, 0.0021 kgm]

D11 −D66 =[0.87kgs , 21.05kgs , 21.05kgs ,

−0.01kgm2

s , 3.54kgm2

s , 3.54kgm2

s

]The accuracy of the hydrodynamic force modeling isshown in figure 6. In this figure the total rigid bodyforces/torques, as calculated from vehicle trajectory data, isshown along with the modeled hydrodynamic forces calcu-lated using the coefficients listed above. This figure shows5 example sections of maneuvering data out of 15 sectionsused to fit the coefficients.

4.2 Calculating Total Energy Spent on Propul-sion

The total work required to create propulsion, just like theuseful propulsive work, can be calculated using a numberof different techniques. Here we discuss the benefits andshortcomings of 3 different methodologies.

4.2.1 Kinetic Energy in the WakeThe Froude efficiency model, commonly used to discusspropeller efficiency, says that efficiency is inversely propor-tional to the increase in velocity of fluid passing through theactuator disc plane. Inherent in this concept is a recogni-tion that perfectly efficient propulsion adds kinetic energyto the vehicle, and any kinetic energy added to the wakeis wasted. Modern measurement techniques, such as digi-tal particle image velocimetry (DPIV) allow non-intrusivemeasurement of the wake velocity field. A technique forcalculating propulsive efficiency from the wake velocityfield is laid out in Krueger (2006), and can be summarizedas,

ηprop =T̄ x

T̄x+ Ej, (10)

where T̄ is the average thrust applied to the body over themotion, x is the distance the body travels, and Ej is the ki-netic energy in the wake. Bartol et al. used this method tocalculate propulsive efficiency of live swimming squid Bar-tol et al. (2009), challenging notions of the low efficiencyof jetting propulsion.Here we would like to point out that (10) is an accuraterepresentation of efficiency, provided that all wasted en-ergy contributes to the wake kinetic energy, Ej . This isa valid assumption for the majority of underwater loco-motory systems; however, there are two energy dissipationmechanisms, that we will discuss in this paper, invalidatingsuch an assumption. One mechanism is viscous dissipationin the wake; and we should note that efficiency calculationsin Bartol et al. (2009) corrected for this dissipation, whichwas determined through personal communication with the

0 2 4 6 8 10 12 14 16

Time (s)

-4

-2

0

2

4

6

8

10

12H

ydro

dyna

mic

x -

For

ce

FittedActual

(a)

0 2 4 6 8−0.4

−0.2

0

0.2

0.4

Hyd

rody

nam

ic R

−T

orqu

e

Time (s)

FittedActual

(b)

Figure 6: Accuracy of the hydrodynamic force estimation of the form (8) using constant coefficients fit to data whenthe propulsors were inactive. Actual hydrodynamic forces acting on the UUV calculated from vehicle accelerations areshown by the circular markers in the body-fixed (a) u and (b) r directions, along with the fitted forces shown by the solidline.

authors. The other mechanism is viscous dissipation in-side the jetting cavity (for cases where one exists), and isaccelerated by the largely stationary solid boundaries. Ingeneral, the degree to which either mechanism affects theefficiency calculation is inversely proportional to the char-acteristic Reynolds number of the propulsive system.In the absence of DPIV velocity data of the vehicle wake,models for jet kinetic energy can be used to model the wakefor vehicles using jet propulsors. The rate of generation ofkinetic energy in unsteady jets under the influence of vortexring formation is derived in Krieg & Mohseni (2013) as,

dE

dt=ρπ

2

(h+

k?22

)ub(t)

3R6p

R4. (11)

Here h is a function of the axial velocity profile, and for thenozzle used on the UUV can be set to h = 1.21. Integratingover an entire cycle with a sinusoidal jet velocity programgives the jet energy for each pulsation as a function of fre-quency,

E =16ρπ3R5

3

(h+

k?22

)(L

D

)3

f2 . (12)

4.2.2 Internal Cavity Pressure modelingOne issue with characterizing efficiency by wake kineticenergy, is that it does not account for energy lost to viscousdissipation in the surrounding fluid. For propulsors likethose in this study, where the internal cavity must be refilledin between jetting cycles, the ingested fluid sees significantviscous dissipation inside the bounded cavity before beingsubsequently ejected. Therefore total work calculations, asdescribed in the previous subsection, will generally give anunderestimate of total work spent.In section 4.1.2 we described modeling for internal cavitypressure that was derived in terms of evolution of system

circulation. The pressure distribution calculated by thismodel can be used to calculate the instantaneous powertransferred to the fluid, in addition to the instantaneousthrust. The power required to collapse the cavity boundaryis the product of pressure on the cavity surface and the nor-mal component of that surface velocity. For the thrustersof this investigation, the plunger at the back of the cavity isthe only part which moves with a velocity normal to its sur-face. Therefore the instantaneous power for these thrustersis the product of pressure on the plunger Pb, area of theplunger πR2

p, and the plunger velocity ub.4.2.3 Measure Current Draw to the Motor

The simplest way to get the total energy spent on propul-sion is to directly measure the current and voltage goingfrom the batteries to the thruster motors. However, the to-tal power going to the motors also includes energy that islost to friction in the plunger driving system and lost duringtransmission of electrical to mechanical power in the motoritself. While these losses are important, they are associatedwith the specific implementation of the propulsion technol-ogy and not the intrinsic characteristics of the propulsiontechnique.

5 RESULTS

In the previous sections we laid out several different meth-ods for calculating both the useful propulsive work, and thetotal work required for propulsion of a freely moving UUV.In this section we examine the accuracy/agreement of thedifferent techniques.

5.1 Estimating Control Forces

We presented three methods for calculating control forces.These include calculating the hydrodynamic impulse ofeach jet based on jet velocity program and motor frequencydata the dividing by the duration to get average thrust (sec-

5 10 15 20 25Time (s)

0

0.1

0.2

0.3

0.4

r -

Vel

ocity

(ra

d/s)

14 15 16 17 18 19Time (s)

0.8

1

1.2

1.4

1.6

r -

Tor

que

Hydrodynamic FittingJet ImpulsePressure Modeling

Figure 7: A comparison of control torque about the vehicle fixed z axis during a steady portion of a yaw maneuver isshown on the right. The vehicle rotational velocity during the entire maneuver is shown on the left, with vertical dashedlines showing the period for which the different torque calculation methods are compared.

5 10 15 20 25Time (s)

-0.15

-0.1

-0.05

0

v -

Vel

ocity

(m

/s)

14 16 18 20Time (s)

-6

-4

-2

0

2

y -

Forc

e

Hydrodynamic FittingJet ImpulsePressure Modeling

Figure 8: A comparison of control force in the vehicle fixed y axis during a steady portion of a sway maneuver is shownon the right. The vehicle velocity during the entire maneuver is shown on the left, with vertical dashed lines showing theperiod for which the different force calculation methods are compared.

tion 4.1.1); calculating the instantaneous force from thepressure on the inner surface of the thruster cavity from jetvelocity program and motor frequency data (section 4.1.2);and by modeling hydrodynamic forces in terms of vehiclekinematic data, then calculating control forces as the dif-ference between inertial forces and hydrodynamic forces(section 4.1.3). Here we will show that all 3 techniquesactually produce similar results for control forces duringdifferent maneuvers.

Figure 7 shows the control torque on the vehicle about thez axis during an example yaw maneuver. On the left is thevehicle angular velocity, r, for the entire maneuver. Thehydrodynamic forces are calculated with the three differentmethods after the UUV has reached a more-or-less steadyrotational velocity marked by the vertical dashed lines. Thetorques calculated from the 3 methods over this period areshown on the right. It can be seen that the different meth-ods produce nearly identical estimates of the torque act-ing on the unrestricted UUV. This not only validates the jetimpulse and pressure models for thrusters on moving plat-forms, but also validates the accuracy of the hydrodynamicforce model while the thrusters are active. However, thereis significant noise on the motor frequency signal whichneeds to be corrected in the hardware in order to improveforce estimates.

Figure 8 shows the force comparison during a section ofa sway maneuver. It can be seen that all 3 models stillhave good agreement, but the noise on the motor frequencysignal is more significant.

The total propulsive work over any given maneuver is theproduct of these instantaneous control forces and the vehi-cle velocity, ~ν , (measured by the motion capture system)and integrated over the duration of the maneuver.

5.2 Estimating total Power/Energy

Here we compare the total required work calculations dur-ing unrestricted maneuvers. These methods include calcu-lating the kinetic energy of each expelled jet from the motorfrequency data and adding that to the the propulsive workdone on the vehicle (section 4.2.1), and calculating the in-stantaneous power transfer to the fluid using the internalpressure model and plunger velocity, then integrating overthe pulsation cycle to get total work (section 4.2.2).

Figure 9 shows the total required power output calculatedby these two methods during the same yaw maneuver de-picted in figure 7. It can be seen that the total power outputcalculated using the pressure model is nearly 50% greaterthan the required power calculated from jet kinetic energy.This is mostly due to the fact that there is a non-zero workassociated with refilling the thruster cavity in between jet-ting cycles. This work required to refill the cavity addskinetic energy to the incoming fluid which is mostly lostto viscous dissipation inside the confined cavity before thefluid is ejected as a propulsive jet. So the total power cal-

culated from the circulation based pressure model shouldbe considered the more accurate method when calculatingvehicle propulsive efficiency.

Although the refill work increases the total required workby 50% for this particular velocity program and thruster it-eration, it should not be considered a standard proportion-ality for unsteady jetting propulsion. The work required forrefilling is in fact heavily dependent on internal cavity ge-ometry and jetting velocity program. In fact it was shownin Krieg & Mohseni (2015) that energy required to refill thecavity using an impulsive velocity program was nearly halfthe energy required to fill using a sinusoidal velocity pro-gram; and that refill work can be further reduced by induc-ing impingement earlier on in the refill phase. For a thrusterwith geometry designed to minimize the refill work, thisproportion could be reduced significantly. In the case ofthe live squid studied in Bartol et al. (2009), there was somenon-zero work put into the refilling phase that did not showup in the wake kinetic energy, which would decrease theoverall efficiency. However, it is possible, and even quitelikely, that squid morphology has evolved to minimize therequired refill work, so the decrease in efficiency may notbe significant.

CONCLUSIONS

In this study we provided the preliminary work towards ex-perimental testing of the propulsive efficiency of cephalo-pod inspired jetting technology. We examined threedifferent methods for estimating control forces; model-ing propulsive jet impulse, modeling pressure distributionwithin the thruster mechanism, and backing out controlforces from hydrodynamic force estimates and vehicle in-ertial forces measured with a motion capture system. Allthree control force measurement techniques showed goodagreement for multiple maneuvers, validating the differ-ent assumptions of the individual models. We also lookinto two methods for estimating the power transferred tothe surrounding fluid environment required to create thepropulsion. It was observed that estimating total requiredwork from the excess kinetic energy in the wake will al-ways under-predict the actual total work due to viscous dis-sipation of kinetic energy prior to measurement of the wakeenergy, and that a more accurate estimate can be achievedby integrating the product of pressure and boundary veloc-ity on the surface of the thruster cavity.

Future work includes examining accuracy of the hydrody-namic force modeling during cross-couple maneuvers, ex-amining the accuracy of the jet pressure models when ve-hicle velocities begin to approach the jet velocity, findingways to reduce the refill work through cavity geometry,and comparing the overall propulsive efficiency of UUVsemploying the cephalopod inspired thrusters to UUVs em-ploying traditional propeller thrusters.

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Figure 9: A comparison of total power output spent to generate the propulsion during a steady portion of a yaw maneuveris shown on the right. The vehicle rotational velocity during the entire maneuver is shown on the left, with vertical dashedlines showing the period for which the different energy calculation methods are compared.

ACKNOWLEDGMENTSThis work was sponsored by the Office of Naval Research,Grant #N00014-16-1-2083

REFERENCESBartol, I. K., Krueger, P. S., Stewart, W. J. & Thompson,

J. T., 2009. ‘Hydrodynamics of pulsed jetting in ju-venile and adult brief squid Lolliguncula brevis: Evi-dence of multiple jet ‘modes’ and their implications forpropulsive efficiency’. Journal of Experimental Biology,212(12), pp. 1889–1903.

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Krieg, M., Klein, P., Hodgkinson, R. & Mohseni, K., 2011.‘A Hybrid class underwater vehicle: Bioinspired propul-sion, embedded system, and acoustic communicationand localization system’. Marine Technology SocietyJournal: Special Edition on Biomimetics and MarineTechnology, 45(4), pp. 153–164.

Krieg, M. & Mohseni, K., 2008. ‘Thrust characterizationof pulsatile vortex ring generators for locomotion of un-derwater robots’. IEEE Journal of Oceanic Engineering,33(2), pp. 123–132.

Krieg, M. & Mohseni, K., 2013. ‘Modelling circulation,impulse and kinetic energy of starting jets with non-zeroradial velocity’. Journal of Fluid Mechanics, 719, pp.488–526.

Krieg, M. & Mohseni, K., 2015. ‘Pressure and work anal-ysis of unsteady, deformable, axisymmetric, jet produc-ing cavity bodies’. Journal of Fluid Mechanics, 769, pp.337–368.

Krueger, P. S., 2005. ‘An over-pressure correction to theslug model for vortex ring circulation’. Journal of FluidMechanics, 545, pp. 427–443.

Krueger, P. S., 2006. ‘Measurement of propulsive powerand evaluation of propulsive performance from thewake of a self-propelled vehicle’. Bioinspiration andBiomimetics, 1(4), pp. 49–56.

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DISCUSSIONQuestion from Tom Van Terwisga Could you explain tome how you separate the hydrodynamic forces on the AUVfrom the control forces in method of ‘extraction of the forcefrom motion dynamics’?Author’s Response Here we apply a standard model forthe hydrodynamic forces assuming that they are a functionof the vehicle kinematics as described in section 4.1.3. Thevarious coefficients (stability derivatives) that relate the dif-ferent components of hydrodynamic forces to various com-ponents of vehicle velocity and acceleration are calibratedfrom AUV trajectory data during periods when all thrustershave been turned off, hence zero control forces. Once thismodel has been calibrated, it is used to calculate hydro-dynamic forces over the entire testing period, and controlforces are calculated as the difference between the totalrigid body forces and the modeled hydrodynamic forces.This is then validated by comparing with the other meth-ods for estimating control forces.

Follow up Question Do I correctly understand that youassume that the thrust influence of the jet-hull interaction isequal to zero? Do you know what sort of error is involvedin that assumption?

Author’s Response Yes we are assuming that the drag co-efficients of the vehicle are unaffected by the jet flow whenthe thruster is turned on. If we examine the 2D flow overcross sections of the vehicle at the thruster opening locationthe jet flow will eliminate the rear stagnation point, whichwill certainly have an effect on the drag profile of that sec-tion. Since these thrusters are being used for maneuvering,the jet flow will only affect the flow over a small portionof the length, and thus the effect on hydrodynamic forceswill be negligible. However, if the thrusters are being usedfor forward propulsion, the jet flow will have a significanteffect on the vehicle drag and the method for extractingcontrol forces by subtracting off hydrodynamic forces willnot be as effective.