Modeling of USM Underwater Glider (USMUG)

Maziyah Mat Noh1, Mohd Rizal Arshad2 and Rosmiwati Mohd Mokhtar3

USM Robotics Research Group(URRG), School of Electrical and Electronic Engineering Universiti Sains Malaysia, Engineering Campus

14300 Nibong Tebal, Seberang Perai Selatan, Pulau Pinang, Malaysia Tel: +604-5996074, Fax: +604-5941023

E-mail: 1maziyah@ump.edu.my, 2rizal@eng.usm.my, 3rosmiwati@ieee.org

Abstract— This paper presents the mathematical model of

USMUG. The longitudinal plane motion is developed here. A USMUG is considered and the general kinematic and dynamic model for longitudinal plane is derived. The USMUG is designed to be operated in shallow water applications (no more than 30 meters)[1]. It has main wings, movable rudder to produce control over heading of the USMUG, ballast and sliding (movable) mass to control the buoyancy system and the pitching or diving angle of the USMUG respectively. It was designed to have a simple cylindrical shape with nose cone shape. With the model proposed, we then will apply various control algorithms to obtain the most suitable technique for the USMUG that realizes high performance maneuverability.

Keywords—USM Underwater Glider, dynamical model, kinematics, dynamics, longitudinal plane

I. INTRODUCTION Generally, underwater vehicle can be categories into three,

that are unmanned submersibles, remotely operated vehicles (ROVs) and autonomous underwater vehicles (AUVs). An underwater glider is considered a special class of AUVs. The underwater glider concept was initially proposed by Stommel (1989) [2], where later has motivated researchers to produce several operational gliders. An underwater glider is a buoyancy-propelled and fixed-winged glider that shifts internal actuators to control attitude. Underwater glider has greater range than other underwater vehicles, and does not need expensive support vessels. The first generation gliders have been developed and tested in ocean including SLOCUM glider [3], Seaglider glider [4] and Spray glider [5]. These three underwater gliders also called a ‘legacy’ underwater glider since the three platforms have been used for testing in many oceans and proved their excellent maneuverability with help of excellent control algorithms and other sensory and communication systems. Currently many laboratory-scale gliders progressively have been developed and tested including ALBAC (University of Tokyo), ROGUE (University of Princeton) and ALEX (Osaka Perfecture University). Another underwater glider named Sterne was developed by Ecole Nationale Superieure D’Ingenieurs in Brest, France with weight of 900kg and 4.5m length able to glide at 1.3m/s which much faster than smaller gliders. Recently developed is Liberdade/XRay with weight more than 800 kg which operating at nominal speed of 1.8m/s which consider the fastest with comparing to other developed gliders.

Most AUVs have six degree-of-freedom (DOFs) in motions. In order to obtain an optimum performance of the glider, normally the mathematical model of the designed glider is computed and simulated and improvement is made based on the simulation result before it can be tested on the actual platform of glider. The mathematical model can be obtained through modeling process. There are work done in the scope of modeling related to underwater glider including Graver [6], Bhatta [7], and Mahmoudian [8]. The models are derived for steady gliding and steady turning.

II. OVERVIEW OF USMUG DESIGN

A. Design Overview The USMUG is designed meant for shallow water

applications no more than 30metres depth [1]. The shape of the USMUG is circular cylindrical with nose cone shape. The detail specifications are depicted in Table 1. The hull shape design and internal frame arrangement are shown in figure 1a and 1b respectively.

TABLE I. Design Specifications [1]

Dimension 0.17m (Diameter), 1.3m (Length), 1.0m (Wing Span)

Operation Depth No more than 30metres Operation Time More than 2 hours Main Power Lithium-Ion Sensors Eco-Sounder

Transducer IMU 5 Degree-of-Freedom Gyrocompass Depth sensor Distance sensor

International Conference on Electrical, Control and Computer Engineering Pahang, Malaysia, June 21-22, 2011

978-1-61284-230-1/11/$26.00 ©2011 IEEE 429

Figure 1a. Hull shape design [1]

Figure 1b. Internal frame arrangement [1]

III. GLIDER MODEL

The glider model include both kinematic and dynamic of rigid body, internal actuation system (ballast and movable sliding mass) and related hydrodynamics. The glider motion equation are derived by computing the kinetic energy and using kinetic energy we determine the momenta. The Newton-Euler formulation is then used to determine the forces and moments. The resulted glider equations are for glider motion in 3-D.

A. Mass and InertiaMatrix The phenomena the affects underwater glider is added

mass. When it glides through the fluid, the immediate surrounding fluid is accelerated along with body. Added mass has fairly significant effect and is related to mass and inertial values of the vehicle. We consider the glider shape is simple with wings mounted symmetrically, Mf and Jf are diagonal. The mass matrix, M, and inertia matrix, J of the glider are given by equation (1) and (2).

fs MImmmmdiagM +== ),,( 321 (1)

fs JJJJJdiagJ +== ),,( 321 (2)

Where ms = stationary mass = mh+ mb. I = identity matrix (3 x 3), Mf = added mass matrix =diag (mf1, mf2, mf3), and Jf = added inertia matrix = diag (Jf1, Jf2, Jf3).

B. Kinematics The glider is modeled as a rigid body with fixed wings

and tail, controlled movable sliding mass, ballast control an external control and immersed in a fluid. We establish two reference frames called inertial frame and body-fixed frame. The inertial frame is composed of orthogonal axes xyz. The x and y axes lie in horizontal plane orthogonal to gravity and z lies has the direction of gravity with a unit vector of i, j, and k. The body fixed frame is a moving coordinate that is fixed to the glider. The origin of the glider body is chosen to be at the centre of the glider centre of buoyancy (CB) and it’s aligned with the axes of the glider. The body axis e1 is the longitudinal axis (directed from aft to fore), e2 is the transverse axis (directed to starboard), and e3 is the normal axis (directed from top to bottom). The 2 shows configuration of the reference frames.

Let the position of the glider b = [x y z]T is a vector from the origin of inertial frame to the origin of a body-fixed frame as shown in figure 1. The glider glide with translational velocity of v = (v1,v2,v3)T and angular velocity of ω=(ω1,ω2,ω3)T. The rotation matrix R which maps vectors the body-fixed frame into initial frame coordinates is parameterized using Euler angles (Ѳ) roll (φ), pitch (θ) and yaw (ψ) and transformation matrix, TѲ (Ѳ) which shows the relation between angular velocities, ω with respect to body frame and transformation matrix from body axes current frame to, RBC are given by equation (3), (4) and (5) respectively. The kinematics of the glider are given by equation (6)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−+

++−=

=Θ=

φθφθθφψθφψψθφφψθψ

θφψφψφθψθψθψ

ψθφ φθψ

ccscscssscssscccs

cccssssccscc

RRRRR xyz ,,,)(),,(

(3)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−Θ= Θ

θφ

θφ

φφθφθφ

ψθφ

cc

cs

sctcts

TT001

)(),,( (4)

TBC

yzBC

RRCB

csssccs

scsccRRR

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−== −

αααββαβ

αββαβ

αβ0

,, (5)

ω

ω

.

.

.

Θ

∧

=Θ

=

=

T

vRb

RR

(6)

430

Where s(.)= sin(.) , c(.) = cos(.) and t(.) = tan(.) . We can use quaternions parameterize orientation using four parameters and one constraint [9], to avoid singularities occur in (4).

Figure 2. Glider reference frames

C. Dynamics The forces and moments equations in body coordinates

are determined by Newton-Euler formulation. The glider model is derived based on simplified internal mass with internal movable, mp, ballast mass, mb fixed at centre of buoyancy (CB), and no fixed point mass offset from CB (mw = 0). The glider dynamic equations are expressed in body coordinates are given by the following equations:

viscT

p FkgRmPxP ++= ω (7a)

viscT

pp TkxRgrmPxvx +++Π=Π )ˆ(ω (7b)

∑+=k

kT

pp FRxPP ω (7c)

Where ∑∑ +=k

Tp

kk

Tk

fkRgmFR int)( Let

∑+=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

kk

Tp FRxP

uuu

u ω

3

2

1

Hence uPp =

The total linear momentum (P), angular momentum (П) and linear momentum of internal moving mass (Pp) are computed by determine the gradient of the kinetic energy equation with respect to linear velocity (v), angular velocity (ω) and internal moving mass velocity ( pr ) as given equation 6(a – c).

)()( pppssTffs rxrvmxrmDvMImP ++++++= ωωω (8a)

)(

)(

pppp

ssfsf

rxrvrm

xvrmJJvD

+++

+++=Π∧

ω

ω

(8b)

)( pppp rxrvmP ++= ω (8c) Where I = 3x3 identity matrix. We assume mw = 0 so that rs = 0 and Df = 0. Therefore equations 6(a-c) are reduced to:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

−

−+

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Π∧

∧∧∧∧

∧

ppppp

ppppppp

pppp

p r

v

ImrmIm

rmrrmJrm

ImrmImM

P

Pω (9)

The body velocities in terms of body momenta are obtained by inverting (9) as given by (10)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Π=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

pp P

PI

r

v1ω (10)

By differentiating (10), yields

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Π+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Π=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

ppp P

PI

dtd

P

PI

r

v11ω (11)

Let the ballast pumping rate, bm be one of the control inputs is given by (12)

4umb = (12) Substitute 8(a-c), I-1 and d(I-1)/dt into (11), the complete equations of motion for USMUG moving 3-D space are obtained are given in (13).

4

1

1

..

.

um

uPTJFMv

TvRb

RR

b

p

=

===

=Θ=

=

−

−

Θ

∧

ω

ω

ω

(13) Where

uFkgRmxPMvF viscT

emp −+++= ω)( (14)

I

R

ω

i j

k

e1

e2

e3

431

urxxrT

kRrgmxvPMvxPrJT

ppvisc

Tppppp

ˆ)(

ˆ)()ˆ(

−−+

++++=

ωωωω

(15)

Fvisc and Tvisc refer to external forces and moment due to lift (L), drag (D) and viscous moment (MDL2).

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

αδβ

α

αδβ

LrSS

DR

LSFD

RFv rCBCBisc

)(

(16)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

++=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

rNNML

DMMM

T

rDL

DL

DL

viscδβ

αβ

ω

δβαβ

ω3

2

1

(17)

Where Dω is matrix contains terms which parameterize viscous angular damping. We use coefficients-based model for hydrodynamic forces and moments that similar to literature [10]

IV. DERIVATION OF THE DYNAMICS EQUATIONS OF LONGITUDINAL PLANE

In this section the motion equations are specialized for motion in longitudinal plane, i-k plane of inertial coordinates and e1-e3 plane of the body coordinates.. Following conditions (18) are required for glider to remain in the longitudinal at all time:

0,0,0,0,0.

0,0,0,0

2

22312

=====

=====

ϕφβ

ωω

andujkR

PrvT

pp (18)

With above condition we have (19). The rotation matrix, R in (3) is reduced to:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

3

1

3

1

3

1

2

3

1

00

00

00

0cos0sin

010sin0cos

u

uu

P

PP

r

rr

v

vv

z

xbR

p

p

p

p

p

pωω

θθ

θθ

(19)

Where θ is pitching angle Substitute (19) into (13) we obtain the following motion equations restricted to longitudinal plane

θθ sincos 31 vvx += (20a) θθ cossin 31 vvz +−= (20b)

2ωθ = (20c)

)

)(sin

cos)((1

31132

233113

131132

2

ururM

PrPrgrm

grmvvmmJ

PPDL

PPPPPp

Pp

+−

++−

+−−=

ωθ

θω

(20d)

)cossin

sin(1

1

232331

1

uDL

gmPvmm

v emP

−−+

−−−=

αα

θωω (20e)

)sincos

cos(1

3

212113

3

uDL

gmPvmm

v emP

−−−

++=

αα

θωω (20f)

231111 ωPP

pP rvP

mr −−= (20g)

213331 ωPP

pP rvP

mr +−= (20h)

11 uPP = (20i)

33 uPP = (20j) 4umb = (20k)

Where α is angle of attack, D is drag, L is lift and MDL is viscous moment. These forces and moment are modeled as follows

2)(21 AVCD D αρ= (21a)

2)(21 AVCL L αρ= (21b)

ωαρ ωDAVCM MDL += 22 )(

21

(21c) Where CD, CL, and CM are the standard aerodynamic drag, lift and moment coefficient, A is the maximum cross sectional area, and ρ is the fluid density. The coefficients may be estimated by using data of generic aerodynamic bodies or through CFD analysis [1]

V. CONCLUSION In this paper the motion equations of an underwater glider,

called USMUG is successfully derived. The model obtained is specific for longitudinal plane. We plan later we will further develop the model for the lateral plane. Using these two models later the gliding control algorithms can be developed to obtain an optimum performance of the glider motion. Another plan is to implement the control strategy in actual platform of underwater glider that is developed in URRG laboratory [1].

ACKNOWLEDGMENT The authors would like to thank the National Oceanographic Directorate (NOD) of the Ministry of Science and Innovation (MOSTI) for the research grant awarded.

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REFERENCES [1] Nur Afande Ali Hussain, et al., Design of an underwater glider

platform for shallow-water applications. Int. J. Intelligent Defence Support Systems, 2010. 3(3).

[2] Stommel, H., The Slocum mission. Oceanography, 1989. 2(1): p. 22-25.

[3] Webb, D.C., P.J. Simonetti, and C.P. Jones, SLOCUM: an underwater glider propelled by environmental energy. Oceanic Engineering, IEEE Journal of, 2001. 26(4): p. 447-452.

[4] Eriksen, C.C., et al., Seaglider: a long-range autonomous underwater vehicle for oceanographic research. Oceanic Engineering, IEEE Journal of, 2001. 26(4): p. 424-436.

[5] Sherman, J., et al., The autonomous underwater glider "Spray". Oceanic Engineering, IEEE Journal of, 2001. 26(4): p. 437-446.

[6] Graver, J.G., Underwater Gliders: Dynamics, Control, and Design, in Mechanical and Aerospace Engineering. 2005, University of Princetone: Princetone.

[7] Bhatta, P., Nonlinear Stability and Control of Gliding Vehicles, in Department of Mechanical and Aerospace Engineering. 2006, University of Princetone: Princeton.

[8] Mahmoudian, N., Efficient Motion Planning and Control for Underwater Gliders, in Department of Aerospace and Ocean Engineering. 2009, Virginia Polytechnic Institute and State University: Blacksburg, Virginia.

[9] Fossen, T.I., Marine Control Ststems: Guidance, Navigation and Control of Ships, Rigs, and Underwater Vehicles. 2002, Trondheim, Norway: Marine Cybernetic.

[10] Geisbert, J.S., Hydrodynamic Modeling for Autonomous Underwater Vehicles Using Computational and Semi-Empirical Methods. 2007, Virginia Polytechnic Institute and State University: Blacksburg, Virginia. p. 99.

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