Dynamic Modeling and Motion Simulation for A Winged Hybrid ... Glider.pdf · PETREL, a winged hybrid-driven underwater glider is a novel and practical marine survey platform which

China Ocean Eng., Vol. 25, No. 1, pp. 97 – 112 © 2011 Chinese Ocean Engineering Society and Springer-Verlag Berlin Heidelberg DOI 10.1007/s13344-011-0008-7

Dynamic Modeling and Motion Simulation for A Winged Hybrid-Driven Underwater Glider*

WANG Shu-xin ( ), SUN Xiu-jun ( ), WANG Yan-hui ( )1,

WU Jian-guo ( ) and WANG Xiao-ming ( )

School of Mechanical Engineering, Tianjin University, Tianjin 300072, China

(Received 29 January 2010; received revised form 8 October 2010; accepted 16 November 2010)

ABSTRACT

PETREL, a winged hybrid-driven underwater glider is a novel and practical marine survey platform which combines the features of legacy underwater glider and conventional AUV (autonomous underwater vehicle). It can be treated as a multi-rigid-body system with a floating base and a particular hydrodynamic profile. In this paper, theorems on linear and angular momentum are used to establish the dynamic equations of motion of each rigid body and the effect of translational and rotational motion of internal masses on the attitude control are taken into consideration. In addition, due to the unique external shape with fixed wings and deflectable rudders and the dual-drive operation in thrust and glide modes, the approaches of building dynamic model of conventional AUV and hydrodynamic model of submarine are introduced, and the tailored dynamic equations of the hybrid glider are formulated. Moreover, the behaviors of motion in glide and thrust operation are analyzed based on the simulation and the feasibility of the dynamic model is validated by data from lake field trials.

Key words: hybrid-driven; underwater glider; autonomous underwater vehicle; dynamic modeling; momentum theorem

1. Introduction

Conventional AUVs and legacy underwater gliders have been widely utilized as important platforms in ocean exploration due to their low cost and flexibility (Bachmayer et al., 2004). Their improvements in motion performance rely on precision of dynamic model and behavior analysis. For that, a variety of theories and approaches have been employed.

The motion characteristics of MSV “Jiaolong” is investigated based on the dynamics of underwater robot, where numerical simulation of descent/ascent and helix motion is conducted, and motion analysis of motion with large drift angle is carried out (Xie et al., 2009). The dynamic equations of a Flat-fish Shaped Under-actuated AUV are derived in light of momentum theorem and the corresponding under-actuated controller is designed (Filoktimon and Evangelos, 2007). Seafloor Landing AUV is modeled with multi-body dynamics theory, where the maneuverability of lateral and longitudinal movement is investigated and the mechanical configuration is optimized according to the analysis results (Zhang et al., 2007).

* The research was supported by the National Natural Science Foundation of China (Grant Nos. 50835006 and 51005161), the Science

& Technology Support Planning Foundation of Tianjin (Grant No. 09ZCKFGX03000) and the Natural Science Foundation of Tianjin (Grant No. 09JCZDJC23400).

1 Corresponding author. E-mail: [email protected]

WANG Shu-xin et al. / China Ocean Eng., 25(1), 2011, 97 – 112 98

A nonlinear dynamic model of a thermal underwater glider has been developed by Gibbs and Appell equations, and the three dimensional motion is investigated (Wang and Wang, 2009). Leonard and Graver developed a dynamic model of a general underwater glider based on the first principles, where the glider was considered to be composed of several point masses, and the simplified hydrodynamic model in longitudinal plane was proposed (Graver, 2005; Leonard and Graver, 2001).

Although the theories and approaches in modeling underwater glider and AUV are different, the fundamental idea and procedure are coherent. First, transform the position and attitude, the forces and moments of the vehicle into the same coordinates frame. Then establish the relationship between the generalized forces, hydrodynamics and generalized position with respect to body frame. Note that, the essential difference between the dynamics equations of underwater vehicles is from the varied vehicle configuration and its hydrodynamic profile.

The winged, hybrid-driven underwater glider as a novel marine observation tool has gradually come upon a stage and begins to show its conspicuous role (Wu et al., 2010; Wang, 2009). The dynamic models of glider and AUV have been built by use of different principles, while those of hybrid gliders have not maturated yet. This paper tries to introduce the approaches in modeling the dynamics of legacy underwater glider, conventional AUV and hydrodynamics of submarine, and makes pertinent modification to suit the hybrid glider.

2. Fabrication and Configuration of the Hybrid Glider

The vehicle body is composed of nose dome, bow section, main section, wet section, stern section and tail cone. The various sections are modularly designed to contain each capability necessary for full functioning. The nose dome houses an external bladder for buoyancy change; the bow section possesses the buoyancy engine; the battery packs, the roll and pitch regulating apparatus and control system are mounted in the main section; two wings are symmetrically fixed to the main section and their position can be modified forward or backward; the wet section holds a droppable weight, umbilical cord, etc., outside protrudes an embedded antenna for GPS and wireless communication; the stern section includes servos for rudder deflection and the tail cone functions as an independent motorized propeller module.

It measures 3.2 m in length, 0.25 m in maximum diameter and1.8 m in wing span. It has a water

displacement of 130 kg and a volume change of 1400 ml . The maximum operating pressure is rated

to 500 dbar . Conventional torpedo shaped body with fixed wings fitted bilaterally is introduced to reduce

the water drag. The maximum forward velocity is 2 m/s in thrust operation and 0.5 m/s in glide operation.

This hybrid glider penetrates the ocean in dolphin glide path through ballast trimming and internal mass redistribution during glide operation. It behaves like traditional AUV in thrust operation, propelled by propeller and steered with horizontal and vertical rudders. This glider characterizes stealthy, low power consumption, high endurance and depth excursion capability in glide mode, and features burst speed, shallow water cruising ability and high manipulability in thrust mode.


Fig. 1. PETREL-the winged, hybrid-driven underwater

glider.

3. Dynamic Modeling of the Hybrid Glider

3.1 Frames of Reference and Kinematics Three coordinate frames are in common use for description of motions of underwater vehicles. One is

inertial coordinate system which is attached to the earth, the second is body-fixed frame and the third is wind coordinates used for expression of the hydrodynamics forces and moments (Jiang et al., 2000). As shown in Fig. 2, inertial coordinate system -E and body-fixed frame -B xyz are assigned with

standard definition. Note that, the origin of inertial frame E is on the surface of sea and coordinate axis -E points vertically downwards to the center of earth; the origin of body frame B is located on the

buoyancy center of vehicle, coordinate axis -B x , -B y and -B z are arranged by right-hand rule and

respectively point at nose, starboard side and bottom of the vehicle.

Fig. 2. Illustration of the hybrid-driven underwater glider.

The kinematics describes the position and attitude of underwater glider in inertial coordinate frame and correlates the generalized velocity in inertial frame to the one expressed in body frame. The position and orientation of the glider with respect to inertial coordinates can be expressed by using a vector from the origin of inertial frame to buoyancy center of glider b and a rotational transformed matrix from inertial frame to body frame BER . The linear and angular velocities of glider expressed in body frame are denoted

respectively as and . According to classical robotics, the kinematics for a movable rigid body can be written as:

EBb R , EB EB ˆR R , (1)


where the operator ^ maps a vector to the skew-symmetric matrix and changed the cross product representation of vector to be dot product. Generally, the orientation of glider relative to the inertial frame can be depicted with Euler angles , three crotational transforms sequentially by yaw angle , pitch

angle and roll angle . So the rotationally transformed matrix can be given by

3 2 1EB ( , , ) ( ) ( ) ( )e e eR R R R . (2)

Note that {e1, e2, e3} is the standard Euclidean basis for R3, and

ˆexp( )e iei

R , where n

n 0

1exp( )n!

Q Q for n nQ . (3)

For the following simulation, the coordinates of position and orientation of glider in inertial coordinates system and the linear and angular velocity of glider with respect to body frame are denoted as

B

B

B

b , ,uvw

,pqr

. (4)

According to the principle of coordinate transformation, we can map the Euler angles rates to the angular velocity of glider expressed with respect to body frame by using the following equation

1 1 21 2 3 B( ) ( ) ( )e e ee R e R R e R . (5)

For the typical expressions of underwater glider, they often take the form as

EBb R , BR . (6)

By rearranging Eqs. (2), (3) and (5) and substituting the vectors with components, the transform matrix

EBR and BR are obtained

EB

c c s c c s s s s c c ss c c c s s s c s s s c

s c s c cR , B

1 s t c t0 c s0 s / c c / c

R . (7)

Matrix RAB transforms vectors expressed in B frame to the ones in A frame and has the property like 1 T

AB AB BAR R R , so BER and BR can be calculated as

BE

c c s c ss c c s s c c s s s c s

s s c c s c s s s c c cR , B

1 0 s0 c c s0 s c c

R . (8)

Note that the first letters of sin, cos and tan function are used for abbreviation in this paper.

3.2 Acting Forces and Moments on the Glider The winged hybrid-driven underwater glider controls its attitude through internal mass redistribution

like legacy underwater gliders and through rudders and thruster like conventional AUV. A pitch battery pack is housed in pressure hull and moves fore and aft along the B-x axis for pitch control; an eccentric weight rolls about B-x axis like a pendulum for roll control in vehicle shell; a variable volume of water displaced by the external bladder is mounted in nose dome for buoyancy force control; the vehicle is made bottom heavy by fixing a weight inside of vehicle below the buoyancy center; the ellipsoid hull performs


reciprocal action with external fluid. In light of the fundamental configuration, the hybrid glider can be considered as a multi-rigid-body system which comprises of movable mass m and static mass sm .

Movable mass includes pitch mass pm and roll mass rm , static mass includes variable ballast mass bm ,

bottom heavy mass wm and hull mass hm . The mass of the displaced water of vehicle which produces the

buoyancy force of vehicle according to Archimedes principle is denoted as m. By defining the total vehicle mass as mv, we have the following formula:

v sm m m , s h w bm m m m , r pm m m , 0 vm m m , (9)

where 0m is the net mass of glider for ascent and descent movement control.

Through defining three base vectors of inertial coordinate frame as 1 2 3, ,i i i , the gravitational and

buoyant forces of glider can be expressed in inertial frame as:

v 3m gG i , 3mgB i . (10)

Note that the gravitational forces, buoyancy forces and the torques induced by gravitational forces are originally expressed in inertial frame. So the gravity of pitch mass, roll mass, and bottom mass can be written like ( ) 3m gi , where the subscript * denotes respectively p , r and w .

There exist reciprocal acting forces between internal masses and hull due to their direct contact and the forces is convenient to be stated with respect to body coordinates system. The forces and moments applied on the internal mass by hull of glider are written in body frame as (*) hF and (*) hT .

The propeller equipped at the end of the glider is coaxial with the longitudinal center line, i.e., rotational center line of prop is aligned with B-x axis. So the propulsion force and the rotation-induced torque are obviously suitable to be expressed in body frame. According to Wang, et al., (2009), the thrust force and rotation-induced moment can be written as

2 4prop F 1C pn DF k , 2 5

prop T 1C pn DT k , (11)

where FC and TC are respectively coefficients of force and torque, n is the rotational speed, and D

represents the outer diameter. Note that 1 2 3, ,k k k denote the base vector of body frame.

Owing to both operation modes in thrust and glide, the winged hybrid-driven underwater glider needs a tailored hydrodynamics model which either suits conventional AUV or legacy glider. Before establishing hydrodynamic model, the wind frame with base vector 1 2 3, ,w w w needs to be introduced, which is

generally utilized to express hydrodynamics forces and moments. According to the standard in aerodynamics literatures of aircraft, the angle of attack and sideslip angle are used to describe the

orientation of wind reference frame relative to body frame. By rotating the body frame about 2k axis for

and then about 3w for , the wind frame can be obtained. So the rotational transform matrix from wind

frame to body reference frame is given by

2 3BW ( , ) ( ) ( )e eR R R , (12)

where


2 2ˆ( ) exp( )e eR ,3 3̂( ) exp( )e eR . (13)

Rotational matrix RBW also has the property like 1 TBW BW W BR R R , so the components for the

rotation matrixes can be written as

BW

c c s s cc s c s s

s 0 cR , W B

c c c s ss c 0

s c s s cR . (14)

According to the definition of wind frame, the velocity of glider is aligned with w1 of wind frame when the water around the glider is still.

BW 1R w . (15)

Simultaneously, the angle of attack and sideslip angle can be calculated by the following formula

1tan ( )wu

, 1tan ( )v . (16)

In light of the classical hydrodynamic model of underwater vehicle and typical underwater glider (Mahmoudian et al., 2009; Li, 1999), the hydrodynamic forces and moments of the hybrid glider with respect to wind reference frame are constituted of viscous hydrodynamic components and inertial hydrodynamic components.

hydro viscous f ( , )f F , hydro viscous ( , )M . (17)

The viscous hydrodynamic terms including lift, side force and drag, and moments about three axes of wind frame are the functions of aerodynamics angles, angles of control surface, linear and angular rates of glider. The expressions of viscous hydrodynamic forces and moments in wind frame are

2

viscous f h v f

1 ( , , , ) ( , , )2

A p q rF C C , 2

viscous h v

1 ( , , , ) ( , , )2

A p q rM C C , (18)

in which, 212

is the dynamic pressure, A represents the characteristic area of the glider, h and v

are respectively the revolved angles of control surfaces of horizontal and vertical rudders. In addition,

fC and fC denote the coefficients vectors of viscous hydrodynamic force, andC andC denote the

coefficients vectors of viscous hydrodynamic moments. Note that, the four coefficients vectors are all associated with linear and angular rates.

Actually, the hydrodynamic forces and moments should be transformed into the body reference frame for the dynamic modeling of glider. In light of hydrodynamics literature of submarine (Jones et al., 2002), the viscous hydrodynamic forces and moments expressed in body frame can be written as

v

h

0

BW viscous v p r

h q

XY Y Y p Y r

Z Z Z qR F ,

v

h

v

v p r

BW viscous h q

v p r

K K K p K rM M M q

N N N p N rR M , (19)

where 2/p pl , 2/q ql , 2/r rl and l is the characteristic length of glider. ( )X , ( )Y , and

( )Z are the coefficients of viscous hydrodynamic forces along three axes of body frame, ( )K , ( )M ,


and ( )N are the coefficients of viscous hydrodynamic moments about three axes of body frame. And the

inertial hydrodynamic forces and moments with respect to body reference frame are produced by the generalized acceleration of motion of glider and can be denoted as:

uu

BW f vv vr

ww wq

uv rw q

R ,pp

BW wq qq

vr rr

pw qv r

R , (20)

where ( ) represent the added mass, added static moment and added moment of inertial, which are also

called damping caused by the water around the glider.

3.3 Dynamic Modeling of Hybrid Glider Let p represent the translational momentum of the glider and represent the angular momentum of

glider about the original of inertial frame. Likewise, ( )p and ( ) respectively represent the linear and

angular momenta of internal mass ( )m , where the subscript notation denotes p, r, b, and w. Here the

momenta are all expressed with respect to inertial frame. According to momentum theorem, we have

EB prop EB BW hydrop G B R F R R f , (21)

G EB prop EB BW hydro EB BW hydrob G b B R T b R R f R R , (22)

( ) 3 EB hm gp i R F , (23)

3 h EB h EB hm gb i b R F R T , (24)

where Gb and b are position vectors of center of gravity and buoyancy of glider in inertial coordinate

system, and ( ) hb is the vector (with respect to inertial frame) of the acting point of force on internal

mass ( )m by pressure hull. ( )b represents the inertial-frame vector of center of mass ( )m .

Let P be the expression of p in body frame, and be the expression of in body frame. Note that

those terms include the translational and angular momentum of glider internal mass. Likewise, let ( )P and

( ) be the expressions of ( )p and ( ) in body frame. Then the transformation equations are

EBp R P , (25)

EBR b p , (26)

EBp R P , (27)

EBR b p . (28)

By differentiating equations above with respect to time and utilizing the kinematics’ expressions, we can get the equations like

EB ˆ( )p R P P , (29)

EB EBˆ( )R R p b p , (30)

EB ˆ( )p R P P , (31)


EB EBˆ( )R R p b p . (32)

By substituting Eqs. (25) ~ (28) into Eqs. (29) ~ (32), the dynamic equations of glider in body coordinates can be written as:

EB prop BW hydro( )p P R G B F R f , (33)

CG BE prop BW hydroP r R G T R , (34)

BE 3 hm gP P R i F , (35)

-h -h -h( )P r r F T . (36)

where ( )r and (*) hr are respectively the position of mass ( )m and acting point of force on mass ( )m by hull

with respect to body frame. Note that there is an offset from the center of buoyancy to the center of gravity for bottom-heavy design, and the offset vector with respect to body frame can be depicted as

w w h h p p b b r rCG

w h p b r

m m m m mm m m m m

r r r r rr . (37)

Owing to the uniform distribution, elliptical shape of pressure hull, the mass center of hull coincides with buoyancy center, and the position vector of mass hm with respect to body frame h 0r . The

bottom-heavy mass wm is equipped with position wr in order to keep the balance pitch angle to be zero while

hybrid glider submerged in water still, i.e., have CGr being aligned with 3k .

The equations of motion of the hybrid glider are eventually expressed with Eqs. (33) and (34), which model a submerged vehicle with general body, fixed wings, cross shaped rudders and thruster.

3.4 Synthesis and Decomposition of Dynamic Model The ballast mass bm and the bottom heavy mass wm are fixed in the hull, their translational and

rotational velocity relative to body frame are all zero, i.e., b w and b w . The pitch

mass pm travels along B-x axis without rotation relative to body frame, so p and p pd / dtr .

The roll mass rm rolls about the B-x axis, and r r h , r rd / dtr . In light of the description

of the hybrid glider (Wang and Sun, 2009), the coordinates of the ballast mass, pitch mass and roll mass in body frame are

p p p

p p

p

00

x l lyz

r ,r rx

r r r r

r rr

sincos

x ly l

lzr ,

b b

b b

b

00

x lyz

r . (38)

So p p 1d / d d / dt l tr k , r h r 1d / d t k , r r r r 2 r r r 3d / d cos d / d sin d / d t l t l tr k k ,

p h p p 1( ) l lr k , r-h rx 1lr k , b h b 1- lr k .The moment of inertia of mass of glider about the mass center is a constant and can be easily measured

by the 3D modeling software. By employing the parallel axis theorem, the moment of inertia of mass about buoyancy center of glider can be calculated as


B* G* ˆ ˆmJ J r r . (39)

where B(*)J and G(*)J are all expressed with respect to body coordinates frame.

3.5 Assignment of Physical and Hydrodynamic Parameters There are lots of values of parameters to determine the simulation of motion of the hybrid glider. The

masses and moments of inertia of pitch mass, roll mass, ballast mass, bottom-heavy mass and hull mass obtained from the 3D solid model of the hybrid glider belong to the physical property, where the ballast mass is variable for net buoyancy driving. The coordinates of internal masses relative to body frame involve in the configuration dimensions, where the position of pitch and roll mass are variables and for attitude adjustment. The hydrodynamics coefficients of hybrid glider include force and moment coefficients, added mass, added static mass and added moment of inertia etc. In addition, the deflection angles of horizontal and vertical rudders are changeable and for vehicle direction control.

As shown in Tables 1 to 3, the parameters can be divided into three groups: geometric and physical parameters, hydrodynamic coefficients and control variables.

Table 1 Geometric and physical parameters of the hybrid glider

Notation Value Description Notation Value Description

pm 9 kg xGrJ 0.0052 kg·m2

rm 3 kg y GrJ 0.0026 kg·m2

wm 18 kg z GrJ 0.0030 kg·m2

Moment of inertia

of roll mass

bm 100 m

Masses of

rigid bodies

xGpJ 0.0368 kg·m2

bl 1.30 m y GpJ 0.0496 kg·m2

pl 0.5 m z GpJ 0.0497 kg·m2

Moment of inertia

of pitch mass

rxl 0.63 m xGwJ 0.1956 kg·m2

rrl 0.045 m y GwJ 14.554 kg·m2

wyl –0.02 m

Configuration

of

rigid bodies

z GwJ 14.442 kg·m2

Moment of inertia

of bottom-heavy mass

l 3.19 m xGhJ 0.9782 kg·m2

A 0.06 m2Body size

y GhJ 72.770 kg·m2

D 0.18 m Prop diameter z GhJ 72.212 kg·m2

Moment of inertia

of hull mass


Table 2 Hydrodynamic coefficients of the hybrid glider

Coef. Value Description Coef. Value Description

uu 12 kg Y 3

vv 170 kg vY 0.21

ww 125 kg

Added mass

pY 0

vr 31 kg·m rY –0.58

Force coefficient along B-y axis

wq –46 kg·m

Added static moment K 0

pp 12 kg vK 0

qq 170 kg pK –0.2

rr 125 kg

Added moment of inertia

rK 0

Moment coefficient about B-x axis

FC 0.21 M –0.61

TC 0.033 Prop coefficient

hM 0.21

qM –0.157

Moment coefficient about B-y axis

0X –0.34 Force coefficient

along B-x axis N 0.4

Z 10 vN 0.21

qZ 2 pN –0.082

hZ –0.68

Force coefficient along B-z axis

rN –0.145

Moment coefficient about B-z axis

Table 3 Control variables of the hybrid glider

Variable Min value Max value Unit Description

bm –0.7 0.7 kg Ballast mass in the nose dome

pl –0.2 0.2 m Shift displacement of pitch mass

r– /2 /2 rad Pendulum angle of the roll mass

n –17 17 r/s Rotational speed of propeller

h– /4 /4 rad Deflection angle of horizontal rudders

v– /4 /4 rad Deflection angle of vertical rudders

4. Motion Simulation and Characteristic Analysis

In consideration of the low and uniform velocity of translational and rotational motion of internal masses relative to the glider, the higher order terms of the state variables in the dynamic model can be omitted. In addition, due to the small angle of attack, small sideslip angle and low maneuverability the hybrid glider will keep a steady motion, and the viscous forces and moments can be expressed in linear derivatives and be neglected. In light of the two assumptions above, the simplified equations of steady motion can be deduced.


The hybrid glider is usually operated in thrust and glide mode. For thrust operation, the hybrid glider performs level flight with the propulsion of prop and turning motion with deflection of rudders. For glide operation, the hybrid glider roams along the saw teeth path via the coordinations of the internal mass redistribution and net buoyancy change. In the following, the simulation of basic motion and helix motion as a conventional autonomous underwater vehicle, and the zigzag path motion and spiral motion as a legacy underwater glider will respectively be conducted.

4.1 Basic Motion Simulation in Thrust Operation The basic motion for a conventional autonomous underwater vehicle includes surge, swerve and dive

etc. To evaluate the performance of basic motion of the hybrid glider, the following numerical simulation is

performed. First, by setting the control variables p p r h v, , , , ,m l n as 0 kg, 0 m, 0 rad, 13.3 r/s, 0 rad,

0 rad and the initial condition 0 0 0 0 0 0 0 0 0 B0 B0 B0, , , , , , , , , , ,u v w p q r as 0.01 m/s,0 m/s,

0 m/s,0 rad /s,0 rad/s,0 rad/s,0 rad,0 rad,0 rad,0 m 0 m,0 m, we have the hybrid glider propelled forward

for 30 s . Then, by changing deflection angle of vertical rudders to 0.78 rad , the hybrid glider is kept

swerving for a circle. Finally, by resetting the vertical rudder and rotating the horizontal ones for 0.139 rad ,

the hybrid glider is made to dive downward and the duration is assigned 30 s . Note that, in order to avoid

the singularity, the initial condition can not be all zero.

Fig. 3. Behavior in basic motion simulation

From Fig. 3, the maximum forward speed is 1.92 m/ s in surge stage, the pitch angle is17.90 in the

dive procedure, and the diameter and forward speed are 18.65 m and 1.57 m/ s during swerving test,

respectively.


4.2 Helix Motion Simulation in Thrust Operation Helix motion simulation is commonly utilized for comprehensive investigation of performance of

autonomous underwater vehicle. First, by setting the control variables 0 p r h v, , , , ,m l n as

0 kg 0 m 0 rad,13.3 r/s,0.09 rad,0.35 rad, , , and the initial condition 0 0 0 0 0 0 0 0 0 B0 B0 B0, , , , , , , , , , ,u v w p q r

as 0.01 m/s, 0 m/s, 0 m/s, 0 rad /s, 0 rad/s, 0 rad/s, 0 rad, 0 rad, 0 rad, 0 m, 0 m, 0 m , the hybrid glider

hovers down for 250 s . Then by resetting the horizontal and vertical rudders, it performs level flight for

60 s . Finally, by changing the horizontal and vertical deflection angles of to be 0.09 and 0.35 rad ,

the glider swirls up for 250 s .

Fig. 4. Behavior in helix motion simulation.

According to the simulation, the diameter of swerving is 40.64 m , the resultant velocity is 1.85 m/s ,

and the glider reaches the depth of 75.77 m in 250 s .

4.3 Zigzag Motion Simulation in Glide Operation Zigzag motion is the principal working pattern for legacy gliders. By setting the control variables

0 p r h v, , , , ,m l n as 0.5 kg, 0.05 m, 0 rad, 0 rad/s, 0 rad, 0 rad , and the initial condition 0 0, ,u v

0 0 0 0 0 0 0 B0 B0 B0, , , , , , , , ,w p q r as 0.01m/s,0m/s,0m/s,0rad/s, 0rad/s, 0rad/s, 0rad, 0rad, 0rad, 0m, 0m,0m ,

the hybrid glider is made to dive. By changing ballast mass and displacement of pitch mass to be 0.5 kg

and 0.05 m , the glider is made to get surface. The simulation duration for each down/up cycle is set to be

400 s and two yos are simulated.

During the flight, the pitch and attack angles are kept as 27.02 and 4.59 , so the glide path angle is about

31.61 . The glider moves at a velocity of 0.48 m/ s , and reaches the depth of 45.75 m within 200 s .


Fig. 5. Behavior in zigzag motion simulation.

4.4 Spiral Motion in Glide Operation Spiral motion simulation is performed for comprehensive validation of glide performance of the

hybrid glider. By setting the control variables 0 p r h v, , , , ,m l n as 0.5 kg, 0.01m, 0.35 rad,

0 rad/s, 0 rad, 0 rad , and the initial condition 0 0 0 0 0 0 0 0 0 B 0 B 0 B 0, , , , , , , , , , ,u v w p q r as

0.01m/s, 0 m/s, 0 m/s, 0 rad/s, 0 rad/s, 0 rad/s, 0 rad, 0 rad, 0 rad, 0 m, 0 m, 0 m , the simulation time is assumed to be 4000 s .

Fig. 6. Behavior in spiral motion simulation.


Fig. 6 shows that the hybrid glider performs a spiral motion with a turning diameter of 224.51 m , the

motion velocity is 0.39 m/ s , the vertical velocity is 0.18 m/s , the attack angle is 7.33 and the pitch

angle is 20.35 .

5. Field Trial in Thrust and Glide Operation

The swerving experiment in thrust operation was performed in Qingnian Pool. Note that the control

variables b p r h v, , , , ,m l n = 0 kg, 0 m, 0 rad, 13.3 rad/s, 0 rad, 0.78 rad and the upper vertical

rudder is beyond the surface of water during the swerving experiment. The turning diameter in simulation with contribution of double rudders is 18.65 m , and according to the trajectory of the hybrid glider in Fig.

7, the turning diameter in experiment with one rudder submerged in water is about 40 m , the velocity of the

hybrid glider is about 1.6 m/ s with prop rotating at a speed of 13.3 rad /s . By comparing with the

simulation and experiment, the dynamic model coincides approximately with the real model. The glide operation experiment was completed in Fuxian Lake. Note that the control variables

0 p r h v, , , , ,m l n are set as 0.5 kg, 0.05 m, 0 rad, 0 rad/s, 0 rad, 0 rad to dive and as

0.5 kg, 0.05 m, 0 rad, 0 rad/s, 0 rad, 0 rad to surface, and the duration for one yo is 400 s . From

Fig. 8, the equilibrium pitch angle is 32 , the steady vertical velocity is about 0.22 m/ s , and the dive

depth is around 42 m within 200 s . The experiment data fit well with the glide simulation.

Fig. 7. Navigation trajectory in thrust operation experiment. Fig. 8. Flight path in glide operation experiment.

6. Conclusions

The winged hybrid-driven underwater glider characterizes high maneuverability and long endurance etc., and works well in ocean exploration with both glide and thrust operation. This paper considers it as a multi-rigid-body system with floating base and unique external profile, and then builds the dynamic model of the hybrid glider by utilizing linear momentum and angular momentum equations. In addition, by comparing with the typical motion simulations and experiments of hybrid glider, the feasibility of the


dynamic model was validated. By getting through the whole paper, the concluding remarks can be synthesized as three points:

(1) The underwater glider was modeled as a multi-particle system by Graver and Leonard, where the control effect of translational motion of internal mass on attitude of underwater glider was taken into consideration. Whereas, the winged hybrid-driven underwater glider is established as multi-rigid-body model, and the effect of both translational and rotational motion of internal mass on the controlled orientation of hybrid glider is considered.

(2) By combining together the approaches for modeling legacy underwater glider and conventional autonomous underwater vehicle, and introducing hydrodynamic coefficients of submarine, the tailored dynamic equations of the hybrid glider are established.

(3) Through suitable simplification of the dynamic model of hybrid glider, the equations of steady motion of hybrid glider are deduced and the simulations of motion in thrust and glide operation with all states observable were performed. In addition, the dynamic model is validated by comparing the simulation results with the field experiment.

Acknowledgements – This work is supported by the National Natural Science Foundation of China (Nos. 50835006 and

51005161), the Science & Technology Support Planning Foundation of Tianjin (No. 09ZCKFGX03000) and the Natural Science

Foundation of Tianjin (No. 09JCZDJC23400). The authors appreciate all staff members in R&D team for PETREL.

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Dynamic Modeling and Motion Simulation for A Winged Hybrid ... Glider.pdf · PETREL, a winged hybrid-driven underwater glider is a novel and practical marine survey platform which