Dynamic Modelling and Cascaded Controller Design of a Low-Speed Maneuvering ROV

7/30/2019 Dynamic Modelling and Cascaded Controller Design of a Low-Speed Maneuvering ROV

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Dynamic Modelling and Cascaded ControllerDesign Of A Low-Speed Maneuvering ROV

Cheng Siong China

, Micheal Wai Shing Laua, Eicher Low

a, Gerald Gim Lee Seet

a

aRobotic Research Centre, Department of Mechanical and Aerospace Engineering, Nanyang

Technological University, 50 Nanyang Ave, Singapore 639798

Abstract: This paper considers an analysis and cascaded controller design for a low-speed

maneuvering Remotely Operated Vehicle (RRC ROV II) designed by Robotic Research Centre

(RRC), in Nanyang Technological University (NTU). First, the vehicles, thrusters and tether

dynamic used for control system design are defined. A linear ROV model can be used since the

controller is tasked to keep the vehicle about the equilibrium position during station-keeping

condition, and therefore the vehicles dynamic and thrusters model obtained from experiment

test rig, can be linearized about this equilibrium position.

To enable a best choice of input and output units for use and to achieve a block diagonal

dominance system, pre and post compensators from Edmunds scaling and reordering algorithm

that performs both scaling and re-ordering of the input and output pairs are used. With roll and

pitch motions of the ROV being self-regulating, there are four degree of freedoms (DOF) to be

controlled using four thrusters instead of six DOF. Subsequently, a cascade structure is proposed

for the control system design. A H controller is designed for the inner velocitys control loop

where suitable weighting functions are chosen to account for tether disturbance and parametric

uncertainty while PD controller is designed for outer positions control loop.

With the proposed proportional derivative (PD) with H cascaded control for the nonlinear

vehicle model, the simulation test with disturbances show a smaller standard deviation in theoutput position compared to the PD-linear quadratic gaussian with loop transfer recovery (LQG-

LTR) and single-loop PD controller.

Keywords: Edmunds scaling, thruster, diagonal dominances, tether disturbance, parametric

uncertainty, H and cascaded control.


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1. INTRODUCTION

The RRC ROV II is designed to be a sub-compact intervention class ROV combining the

functions of an eye-ball ROV and a work class ROV. It has a man-in-the-loop as the supervisor

with complementary on-board controller for local control and decision making capabilities. The

vehicle is equipped with a suite of navigational sensors; INS, Doppler and side scan sonars- a pair

of cameras with stereo viewing capabilities. Its designed tasks include inspections and repairs

of pipe lines and structures. Station-keeping would require the vehicle to hover over a certain set

of coordinates while maintaining a fixed orientation in the presence of various types of

disturbances. The control of this task is exacerbated by the vehicle having non-linear dynamics,

tether disturbance, under-actuated and motion amongst its DOF is coupled.

Decoupling of the various DOF is useful not only for controller design but also in determiningallocation of control authority. The ability to decouple depends on the vehicle physical properties

and thrusters arrangement. For example, the slender form of the NPS AUV [1] allows for

decoupling into three different control regimes using two propellers and control surfaces. With

the ROV design, it is not possible to decouple in this sense, but in RRC ROV I [2] it is possible to

allocate control efforts to the outputs and to achieve a decentralized control through Edmunds

scaling and reordering routine. Even though the ROV is under-actuated, task associated with

inspection requiring the station keeping can be accomplished using a roll and pitch stabilized

camera platform.

In controller design, it is commonly accepted that linear PID (or PD) controller based on

linearized model does not perform as well as sliding mode control (SMC) [6],adaptive control

(AC) [4] and feedback linearization (FBLN) [7], amongst others, for instance in trajectory

tracking and when uncertainties are present [3]. It has been shown that AC can be designed to be

robust. Uncertainty can also be designed using robust controller such as the H methodology [5].

However, PID/PD controllers are commonly used in commercial ROV, as it is easily understood

amongst pilots. To overcome certain limitations, they can be used in combination, for example,

with AC dynamic compensation [8]. The use of the H controller in cascaded with PD controller

and combination of the pilot provides some robust response against disturbance and also non-

linear behaviour.

The outline of this paper is as follows: The modeling of vehicle dynamic that includes modeling

of added mass coefficient component, thrusters and three-dimensional tether dynamics are

shown. Then, the linear ROV dynamic for station-keeping condition is obtained, followed, an

Edmunds scaling and reordering routine is used to identify the degree of coupling amongst the

DOF and for decoupling the linear system into two block diagonal dominance system of nearly


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decoupled subsystems. In section 6, a cascaded control structure is defined with inner loop

consists of the H controller for disturbance attenuation and the outer loop the PD controller.

Section 7 shows a detailed approach using H controller design is introduced and lastly acomputer simulation is performed to compare the performance of the PD-H with PD-LQG/LTR

and the PD only control system design.

2. ROV RRC II OVERVIEW

The development of a twin barrel ROV named ROV RRC II (see Fig. 1) is a joint research

partnership with the British Gas. The first phase of work focuses on the implementing virtual

reality (VR) techniques in the ROV RRC II simulator/training platform, is developed with British

Gas to has better intelligent control and more data collection capabilities as compared to its

predecessor. The ROV would be a combination of the work-class and inspection-class ROV. This

immediate class ROV is small enough and able to maneuver within constrained workspace and

yet the capacity of carrying additional payloads such as additional sensors and equipment. Thus,

the ROV RRC II is slightly larger in size and heavier in weight. These sensors and equipment are

mounted on two pods as shown in Fig. 2. A brief description of the component layout of the ROV

RRC II is given.

1. 4x thrusters, each providing up to 70N of thrust;

2. 2x cylindrical floats

3. 4x balancing steel weight

4. Main pod (Pod 1)

5. Sensors and navigational pod (Pod 2)

6. 2x halogen lamps

7. External sensors including altimeter, scanning sonar and depth sensor

This ROV has a twin-pod design. The vehicles circuitry and components are housed in two

modular pods are secured using custom-made frame. The main pod (named Pod 1) is responsible

for vehicles communication and control. The first stage of power transformation is placed here.

The stepped-down power is consumed by all other devices and peripherals of the RRC ROV II.

This pod houses the host computer that communicates with the topside ROV control platform. It

interprets the topside commands and executes them and sends back vital positional and visual

data to surface which is collected from the second pod and camera platform respectively. Besides

that, it is also responsible for controlling the vehicles motion since the motion controller resides

in the main processing board. The Pod 1 consists of the following components:


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1. Thruster drivers

2. Main power supply [+48V]

3. Main QNX CPU

4. Power supply[+/-12V, +5V]

5. Main camera platform

Pod 2 that house the navigation computer together with the compass, tilt sensor and inertial

navigation unit. The navigation computer collects and analyzes data from internal and external

sensors such as sonar and pressure sensor through serial ports. Using Ethernet connection, the

navigation computer writes data to a shared memory location on the host computer. The host

computer to perform closed-loop control can then use the data. The Pod 2 consists of the

following components:1. Power supply[+/-12V, +5V]

2. Crossbow inertia navigation unit

3. Navigation QNX CPU

4. Dynamic Measurement Unit(DMU)-VG600 CA

5. Magnetic compass KHV-C100

6. Secondary camera platform

In summary the sensors used in the RRC ROV II are placed either in ROVs Pod 2 or external to

the ROV. Additionally, RRC ROV II has immersive 3D graphical display and control assistance

subsystems. The 3D display system gives a simulated environment for the pilot to rehearse his

approach and evaluate the suitability of his chosen tools, while the control assistance modules,

comprising the manual cruise, station-keeping, steering, cruising modes, assist to de-skill control

operation. This combined man-in-the-loop and automatic control of prescribed modes allow for a

much robust system due to its capability of switching to manual mode, compared to a fully

autonomous computer controlled system. In this paper, one of the controller designs for station

keeping is described.


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Figure 1: RRC ROV II

Figure 2: Pod 1(left): Main communication and Control and Pod 2(right): Sensors and Navigation

3. Nonlinear ROV Model

As seen in the ROV design, the ROV model is composed of three main components: the ROVs

rigid body, the propeller and the DC motor. The rigid body model can be derived from the

Newton-Euler formulation. The Newton-Euler formulation is based on Newtons Second Law in

terms of conservation of both linear and angular momentum. Another important issue when

modeling 6 DOF (Degrees Of Freedom) systems is the specification of reference frames. It is

important to define two main coordinate frames: the body-fixed and the earth-fixed as shown in

Fig. 3. The body-fixed is attached to the vehicle. Its origin is normally fixed on the centre of

gravity. The motion of the body-fixed reference frame is described relative to the earth-fixed


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reference frame. The earth-fixed reference frame can be considered inertial for low velocity

vehicles such as the ROV.

Practical issues explain the need of two different reference frames. For example, it is easier to

measure position and orientation on the earth-fixed reference frame instead in the body-fixed. The

velocity is usually measured in the body-fixed reference frame. The notation defined by SNAME

(Society of Naval Architects and Marine Engineers) is as follows:

Position and orientation (earth-fixed):

[ ] 3321][ SRzyx

TT == (1)

Linear and angular velocity (body-fixed):

[ ] 621 ]v[vv RrqpwvuTT == (2)

There is also a kinetic transformation, which maps the transformation between both frames. This

transformation is based on Euler angles

v)(2

J=& (3)

where

= )(0

0)(

)(22

21

2

J

J

J (4)

with

++

++

=

)()()()()(

)()()()()()()()()()()()(

)()()()()()()()()()()()(

1

ccscs

cssscssscccs

sccssssccscc

J (5)

=

)()(

)()(0

)()(0

)()()()(1

2

cc

cs

sc

tcts

J

(6)

and s = sin(.), c = cos(.), t = tan(.). This transformation is undefined foro90= . To overcome

this singularity, a quaternion approach must be considered. However, in the project this problem

does not exist because the vehicle is not required to operate ono90= . Moreover, the vehicle

is completely stable in roll and pitch, and the thruster actuation is not enough to move the vehicle

to operate at this angle.


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Body-Fixed Coordinates

u

w

vq

r

()

()()

Figure 3: Experimental RRC ROV II

3.1 General ROV Model

A general dynamic model formulation for general underwater robotic vehicle [10] is used. The

general motion of a ROV can be described by using a body-fixed frame relative to an earth fixed

frame (see Fig. 3). These dynamic equations can be expressed in a more compact form as:

=+++ )((v)vv)v(v DC& (7)

where 6 is the actual thrust input vector consisting of control forces and moments.The thruster configuration matrix (see Fig. 4) is obtained by summation of the forces (in X,Y,Z)

and moment (in X,Y,Z) about ROVs center of gravity (CG);

(6)

(8)

with m293.0,m31.0,45,m017.0 o ==== , m016.0= and T1 to T4are the magnitude

of thrust exerted by thruster one to four respectively;

=

+ ++

+

+

=

4

3

2

1

4321

4321

43

43

43

21

sinsincoscos

00

coscos00

sinsin00

0011

)(sin)()(cos)(

)(

cos)(

sin)(

T

T

T

T

TTTTTTTT

TT

TT

TT

TT


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Figure 4: Front and side view of RRC ROV II

M=MR+MA66 is the mass inertia matrix including added mass termMA

=

rqpwvu

rqpwvu

rqpwvu

rqpwvu

rqpwvu

rqpwvu

A

NNNNNN

MMMMMM

KKKKKK

ZZZZZZ

YYYYYY

XXXXXX

M

&&&&&&

&&&&&&

&&&&&&

&&&&&&

&&&&&&

&&&&&&

(9)

and rigid body termR

M to become:

=

rzzqzypzxwvGuG

ryzqyypyxwGvuG

vxzvxypxxwGvGu

rqGpGwvu

rGqpGwvu

rGqGpwvu

NINININNmxNmy

MIMIMIMmxMMmz

KIKIKIKmyKmzK

ZZmxZmyZmZZ

YmxYYmzYYmY

XmyXmzXXXXm

M

&&&&&&

&&&&&&

&&&&&&

&&&&&&

&&&&&&

&&&&&&

(10)

withxG, yG, zG refers to the coordinate of the center of gravity. The mass, m=113.2kgand inertiaI terms are obtained from the computer-aided design software, PRO-E. The moment of inertia

value are:Ixx=6.100 kg.m2,Iyy=5.980 kg.m

2Izz=9.590 kg.m2,Ixy=-0.00016 kg.m

2,Ixz=-0.185 kg.m2

andIyz=0.0006 kg.m2.

As observed that the ROV have some planes of symmetry. The highestIterm (correspond to the

most symmetry plane) are the XZ plane, YZ plane and followed by XYplane. In matrix form, the

Ibecome:

m293.0=

o45=

MainBody

m31.0=

m017.0=

m016.0=C.B

C.G

Direction of positivethrust

Front view of RRC ROV II Side view of RRC ROV II

Y

Z

X

Z


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=

590.90006.0185.0

0006.0980.500016.0

185.000016.0100.6

I(11)

)v()v()v(AR

CCC += 66 is the Centripetal and coriolis matrix including added mass

term )v(A

C and rigid body term )v(R

C :

= (v)(v)

(v))v(

2212

1233

CC

COC

T(12)

with

++++++

+++

=)()()(

)()()(

)()()(

)v(12

qypxmuXuqzmvYvpzmuXurympxrzmwZwpym

vYvrxmwZwqxmrzqym

C

GGuGvG

uGGGwG

vGwGGG

&&

&&

&&

; (13)

+++

+++

+++

=

0

0

0

)v(22

pKpIqIrIqMqIpIrI

pKpIqIrIrNrIpIqI

qMqIpIrIrNrIpIqI

C

pxxyxzqyxyyz

pxxyxzrzxzyz

qyxyyzrzxzyz

&&

&&

&&

(14)

D(v)66 is the hydrodynamic damping matrix of linear and quadratic terms:

=

rXNqNNpNNwNNvNNuNN

rXMqMMpMMwMMvMMuMM

rXKqKKpKKwKKvKKuKK

rXZqZZpZZwZZvZZuZZ

rXYqYYpYYwYYvYYuYY

rXXqXXpXXwXXvXXuXX

D

rrrqqqpppwwwvvvuuu

rrrqqqpppwwwvvvuuu

rrvqqvpppwwwvvvuuu

rrrqqqpppwwwvvvuuu

rrrqqqpppwwwvvvuuu

rrrqqqpppwwwvvvuuu

)v(

(15)

with .|.| defines the directional dependent of the term and )(g 6 is the gravitational and

buoyancy vector:

+

+

=

sin)ByWy(sincos)BxWx(

coscos)BxWx(sin)BzWz(

sincos)BzWz(coscos)ByWy(

coscosB)(W

sincos)BW(

sin)BW(

)(

BGBG

BGBG

BGBG

g(16)


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with WandB is the gravitational and buoyancy force respectively. The xB, yB andzB refer to the

center of buoyancy.

3.2 ROV RRC II Model

In practice, a general ROV model is used for analysis and simulation. Often it is found to be quite

good for many underwater applications. Some details and simplification on the matrices, which

compose equation (7), are presented. The inertia matrix Mhas two distinct contributions [11].

One is from the rigid body inertia. The other appears when there is water motion and is called

added mass. The added mass should be understood as pressure-induced forces and moments due

to a forced harmonic motion of the body, which are proportional to the acceleration of the body.

Some simplifications were made on theM, C, D and gmatrices in (7).

a). The origin of the body-fixed reference frame was chosen to be the gravity centre, i.e.

xG=yG=zG=0. This leave the MR to be block diagonal where the first 3 by 3 block consists of

ROVs mass and second block consisting of inertia term of the same size as shown in (17). In this

second block, the off-diagonal elements of inertia term Ixy ,Ixz in MR are much smaller than the

diagonal counterparts and it is quite difficult to get all the 36 added mass parameters, and since

the ROV is moving at low speed, the diagonal approximation is found to be quite good for many

applications. Therefore the total mass inertia in (10) becomes.

=

rzzzyzx

yzqyyyx

xzxypxx

w

v

u

NIII

IMII

IIKI

Zm

Ym

Xm

M

&

&

&

&

&

&

000

000

000

00000

00000

00000

(17)

The added mass coefficient used in MA(and CA)are computed using principle of Strip theory[11]

as shown below. It involves dividing the submerged part of the vehicle into a number of strips.

With this, two-dimensional hydrodynamic coefficients for the added mass can be computed for

each strip and summarized over the length of the body to yield three-dimensional coefficients. By

examining each parts of the ROV such as left and right float, main chassis and thrusters as a

slender body as shown in Fig. 3, the added mass coefficients for each direction is computed.

=

u

X

&

==2/

2/

11u1.0),(

L

L

dxzyAX&

(18a)


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=

v

Y

&

==2/L

2/L

2

a22v )D5.0(dx)z,y(AY& (18b)

=w

Z

&

==2/L

2/L

2

a33w )D5.0(dx)z,y(AZ & (18c)

=p

K

&])5.0([

481),(),(

2/

2/

2/

2/

5

22

2

33

2

==a

a

a

a

D

D

D

D

apDdzyxAzdyzxAyK

&

(18d)

=

q

M

&]1.0)5.0([

121),(),(

2322/

2/

2/

2/

11

2

33

2

aa

L

L

D

D

qDLDdzyxAzdxzyAxM

a

a

+==

&

(18e)

=

r

N

&

]1.0)5.0([12

1),(),(232

2/

2/

2/

2/

22

2

11

2

aa

D

D

L

L

rDLDdxzyAxdyzxAyN

a

a

+==

&

(18f)

where L, Da are the length and diameter of the ROVs components as tabulated in Table 1 and

=1024kg/m3is the density of the seawater (3.5% salinity) at 20

oC. By summing the added mass

contributed by each components (see Table 1), the total added mass coefficients for each

direction becomes: 6.0u

=&

X , 107=v

Y&

, 107=w

Z&

, ,0023.0=p

K& 23.6=qM& and

23.6=r

N&

.

Components Dimensions (Da ,, L)

Thruster 0.09m, 0.20m

Left and right barrel 0.22m, 0.90m

Left and right float 0.12m, 0.92m

Table 1: Dimension of ROV RRC II components

b) As mentioned in (a), the origin of the body-fixed reference frame was chosen to be the

gravity centre, i.e.xG=yG=zG=0. Like the previous matrix, the Cmatrix also has two distinct

contributions, one from the rigid body and other from the added mass. The Cmatrix as in

(12) become:

+

+

+

=

0

0

0

)v(12

uXmuvYmv

uXmuwZmw

vYmvwZmw

C

uv

uw

vw

&&

&&

&&

; (19)

+++

+++

+++

=

0

0

0

)v(22

pKpIqIrIqMqIpIrI

pKpIqIrIrNrIpIqI

qMqIpIrIrNrIpIqI

C

pxxyxzqyxyyz

pxxyxzrzxzyz

qyxyyzrzxzyz

&&

&&

&&

; (20)


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c) Potential damping, wave drift damping, skin friction and vortex shedding damping cause the

main damping on the ROV. However, the contribution from the potential and wave drift [11]

are negligible as compared to vortex shedding damping and skin friction. The viscosity of the

fluid mainly contributes these damping effect. Usually, these viscous damping on the

underwater vehicles are nonlinear (or quadratic) and coupled in nature. Nevertheless, if the

system has planes of symmetry, the terms higher than second-order are negligible and the

vehicle is performing a non-coupled motion at slow speed [11], the diagonal approximation

obtaining linear terms is found to be quite good for many applications.

},diag{rqpwvu

,N,M,KZ,YXD = (21)

The definition of high or low Reynolds number (Re) in literature is mainly application

dependent. Generally for flow inside a roughed-walled pipe, the Re > 2000 is considered as

high Re (and turbulent flow) but inside a small-walled pipe, the Re > 40,000 is considered as

high Re (and turbulent flow). For ROV application that is clearly not a streamlined bodies of

inviscid flow, Re of greater than 103

are treated as high Re flow (that can be a laminar,

turbulent or intermittent flow). This can be seen by the drag coefficient that is Re independent

[12] in 103

onwards (as seen in Fig.5).

Figure 5: Drag force coefficient versus Reynolds number

The contribution to the linear parts of the damping coefficients is the skin friction at low

frequency or at low Re of 103. This corresponds to low-speed maneurving that are obtained

through the drag test experiment as shown below. On the other hand, the quadratic term is

contributed by both skin frictions at high frequency (or high Re of 105) and vortex shedding.

The quadratic terms are obtained through the similitude using Oxford ROV as the prototype.


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But in this paper, the quadratic terms is negligible since the ROV is designed to be for low-

speed application. In the future, the ROV will be capable of higher speed.

For the linear hydrodynamic damping coefficients are approximated by a drag test [22]

conducted, in a confined pool, using a draw wire displacement sensor which is mounted on a

special rig designed for the drag test (see below Fig. 6). The draw-wire displacement sensor is

connected to the rear (for X-direction), the side (for Y-direction) or the top (for Z-direction) of

the ROV. The displacement, with respect to time, of the vehicle in the water is recorded and the

terminal velocities (max 0.001 m/s) of the vehicle are determined. Through the experimentation,

the net force that drives the vehicle and the terminal velocities are known. With this information,

the X, Y and Z direction drag coefficients are determined. The derivation of (22)-(24) assumedthat the vehicle is moving linearly in each direction and the maximum speeds (that correspond to

zero acceleration) were measured.

For drag force in X-direction,

2

maxmax21/20)( uTXuuXTT

uuuu==+ (22)

For drag force in Y-direction,

2

maxmax21/20)( vTYvvYTT

vvvv== (23)

For drag force in Z-direction,

2

maxmax43/20)( wTZwwZTT

wwww==+ (24)

However, since no measurements were possible for the angular directions due to the limitation of

the test rig as shown in Fig. 6, the following equations are used. Since the roll and pitch motions

are self-stabilizable by the restoring components in (28), the pitch and roll velocity obtained from

(22) to (23) are used to solve for the unknown drag coefficient in (25) to (26).

For drag force about X axis or roll motion,

pWzppKI Bppxx &=+ sin)(1 (25)

For drag force about Y axis or pitch motion,

qWzqqMIBqqyy

&=+ sin)(1 (26)

For drag force about Z-axis, the designated four thrusters are actuated to provide the yaw motion.

Assuming maximum displacement (or zero velocity) in X and Y direction.


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0)()(1

4321=+ rrNITTTT

rrzz(27)

The estimated linear drag coefficients of the RRC ROV II are:

22.98.252 smNXuu

= , 22.51.1029 smNY vv= , 22.51.1029 smNZ ww

= ,

,.78.97 22smNKpp

=22.22.142 smNM

qq

= and 22.11.71 smNNrr

= .

Figure 6: Drag test using draw-wire displacement sensor in pool

d) For the gravitational and the buoyancy matrix, g the simplifications are as follows:

i) The vehicle is designed to be neutrally buoyant in the water, that is, the gravity

force equal to the buoyancy force (W=B).

ii) The center of buoyancy zB will be located directly above the center of gravity zG

( GB xx = , GB yy = ). HencezB-zG=0.048m with W=1110.5N.

iii) The origin of the body-fixed reference frame was chosen to be the gravity centre, i.e.

xG=yG=zG=0.

=

0

sin)z-(z-

sincos)Wz-(z-

00

0

)(

GB

GB

W

g(28)

.


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3.3. Propeller and Motor Shaft Model

This section gives the derivation of a dynamic model for the thruster used in control system

design. Through experiment [13] shown below, a steady state thrusters model approximates the

actual characteristic of the thruster in the water. Basically, the dynamics model of the thruster

consists of propellers hydrodynamics and motors electromechanical behavior. The former can

be derived from basic Newtonian fluid mechanics theory, in particularly Bernoullis Equation[12]

(assuming the flow is incompressible, inviscid and irrotational).

Instead of determining the final force and moment of the thruster as a function of propellers

rotational speed and axial velocity [3], the proposed method described below treats the axial

velocity as a function of the propellers rotational speed. Hence we can obtained a precise andsimpler equation that is a function of the propeller rotational speed.

Unsteady state thruster model is derived using the Bernoullis Equation applied to the streamlines

upstream and downstream of the propeller and assuming that the flow is incompressible, inviscid

and irrotational. The thrust output from the propeller can be defined as:

aapaptuuAulAT 2+= & (29)

wherea

u is the axial fluid velocity at the propeller (fluid advance speed) in m/s, Ap is the axial

projected area of the propeller in m2

, l is the length of the area projected in m and is thedensity of the seawater (3.5% salinity) at 20

oC in kg/m

3. It is not difficult to see that Apl is

actually the control volume in m3.

The relationship between axial velocity and rotational speed can be obtained from the energy

balance equation. Since there is a difference in the thrust due to disturbances in the water inflow

to the thruster blades, thruster-to-ROV surfaces interactions, underwater current and ROV

velocities, a factorKRis included in the energy from input to output:

QuTKatR

=

tR

aTK

Qu

= (30)

where Q is the shafts torque in N/m, is the rotational speed in rad/s and Tt is the thrust

generated from propeller rotation in Newton. The relationship between the torque and thrust is

obtained from the water tank experiment [13] that is: Tt = 40Q (forward) and Tt=-54Q (reverse)

as shown in Fig.7. The KR =2.14 is an average value obtained from the ratio of atuTQ at


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ROVs surge velocity ofu = 0.01 to 2 m/s under open loop condition. While the ua, is defined as

function ofthevehicles surge velocity u:

ua = (1-W)u (31)

where Wrefers to the wake fraction number between 0.1 to 0.4.

Figure 7: Experimental result of Thrust vs torque

By examining (30) it is not difficult to see that a

u . By applying Tt=40Q(forward), Tt= -

54Q(reverse) into (30) and substituting it into (29) gives:

TdTdot

KKT += & (32)

where the constant for both forward and reverse are:

Forward:

R

p

TdoK

lAK

40

= ,

2800

R

p

TdK

AK

=

Reverse:

R

p

TdoK

lAK

54

= ,

21458

R

p

TdK

AK

=

Similarly the torque become:

QdQdo

KKQ += & (33)

where the constant for both forward and reverse are:

Forward:

R

pp

QdoK

DlAK

40

= ,

2800

R

pp

QdK

DAK

=

Reverse:

R

pp

QdoK

DlAK

54= ,

21458

R

pp

QdK

DAK

=

T t = 4 0 Q

T t = - 5 4 Q

- 4 0 . 0

- 3 0 . 0

- 2 0 . 0

- 1 0 . 0

0 .0

1 0 . 0

2 0 . 0

3 0 . 0

4 0 . 0

5 0 . 0

6 0 . 0

-1 .0 -0 .5 0 .0 0 .5 1 .0 1 .5

F o r w a r d T h r u s t( u p s c a l e r e a d i n g )

F o r w a r d T h r u s t( d o w n s c a l er e a d i n g )

R e v e r s e T h r u s t( u p s c a l e r e a d i n g )

R e v e r s e T h r u s t( d o w n s c a l er e a d i n g )


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17

From the experimental results, the thruster model without considering the time dependent term,

0=& portion of the thrust and torque in (32) and (33) can be obtained as

Tdt KT = (34)

Qd

KQ = (35)

where the constant (see Fig.8) for both forward and reverse are: KTd= 0.0022(forward), KTd= -

0.0012(reverse), KQd = 0.00022(forward), KQd =-0.00012 (reverse). By substituting KR=2.14,

l=0.01m,Dp=0.1 and =1024kg/m3

into (32), the forward and reverse thrust in steady and non-

steady condition coincides (due to theKTdo in the equation is small).

Forward: 0022.000094.0 += &t

T (36)

Reverse: 0012.000069.0 = &t

T (37)

Similarly, this applies to the forward and reverse torque, Q in (33).

Figure 8. Experimental result ofTtvs. 2

Since the propeller is attached onto the shaft motor, the dc motor shaft dynamics has to be

considered. The electromechanical model assuming small electrical time constant as compared to

the mechanical time constant. In general, the unified hydro-electromechanical dynamic model for

both the dc motor shaft speed and propeller dynamic can be written as:

2Qdm

m

m

vmKV

R

KKJ =+& (38)

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

40.0

50.0

60.0

-40000 -20000 0 20000 40000

Speed2

(rad2/s

2)

Thrust (N)

Forward Thrust(upscale)

Forward Thrust(downscale)Forward Thrust(average)

Reverse Thrust

(upscale)Reverse Thrust(downscale)

Reverse Thrust(average)Linear (ForwardThrust (average))Linear (Reverse

Thrust (average))

20022.0 =Tt

20012.0 =Tt


18/36

18

where KQd is obtained from (33) at steady state, is the rotational speed in rad/s, Vm is the

armature voltage,Kv is the viscous friction coefficient in Nms/rad,Rmis the armature resistance in

Ohms,Km is the motor torque constant Nm/A, andJmis the rotor moment of inertia in Nm.s

2

/rad.Equation (36) and (37) imply that the unsteady state model in (32) to (33) can be approximated

by the steady state model in (34) to (35) respectively. As shown in Fig. 9, the model steady state

response match those observed in the experiments [13]. However, the approximated transient

response does not match very well with the observed oscillatory response. It is observed that most

oscillations diminish within 0.5s. This is relatively shorter than ROV dynamics response time.

For linear system analysis, (38) is generalized by a first-order thruster model that relates (not the

) the thrust output for certain voltage input:

TTgI

s..G

41020

970+

= (39)

with

volt10inputif0.33,

volt20inputif0.6,

volt30inputif0.82,

volt40inputif,1

=

=

=

=

=T

g

where the value of the Tg between each voltage interval is assumed to vary linearly,

i.e 910.gT = for V=35 volt, andI4 is an identity matrix of dimension four.

Figure 9: Time responses of the experimental thrust and the thruster model

0 0.5 1 1.5 2

10

20

30

40

50Thrust(N)

0 0.5 1 1.5 210

20

30

Thrust(N)

0 0.5 1 1.5 25

10

15

20

Thrust(N)

Time(s)

ApproximationExperimental

input: 40v

input: 30v

input:20v


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19

3.4. Three-dimensional Tether Dynamic

Tether interaction [14] in the ROV dynamic is a common disturbance present in the tethered

ROV. It can be more significant as compared to the water current. The internal forces present in

the tether and the wave that causes launch boat to drift from its initial can restrict the movement

of the ROV even with a good tethered management system.

As shown in Fig.10, the tether is divided into nequal length elements and the result from one

element is propagated into next till it reaches the final end point at the ROVs CG. The inertial

reference frame (X, Y, Z) is defined at surface of waterline and the first cables element is

attached to the launch boat. The tether forces in three-dimensional (3D) are computed by the

method in Sagatun [16] that uses Catenary equations with end forces estimates. The differential

equation derived in [16] is used to solve the forces at each element, i .

+=

)()(

11)(

hWfhWfEAhWf

dh

dF

ci

T

ci

ci

T(40)

where h is the distance in m,EA is the axial stiffness of the cable in N, Wc is the weight per length

of the cable in N/m,fi is the force exerted on the cables element i in N,FT is the local forces in

X, Y, Z within the cables element i and thefi-Wch represents the tension in vectorial form.

Equation (40) is solved numerically using the MATLAB routine. The initial guess for the forces

in N is [4 5 180]T

, cables length is 300m, cables weight is Wc= 1N/m, diameterdc= 0.014m,

modulus of elasticityEc=200109

N/m2, axial stiffnessEA=3 104 N and density c =662.2kg/m

3.

The launch boat positions (in X and Y directions) are assumed to move by 0 to 20 m due to the

wave that causes the boat to drift from its initial position. Note that even if the initial guess for the

estimated force deviated from the correct value, the routine managed to find a good estimate.


20/36

20

Figure 10: Schematic diagram of the tethered ROV

The forces at X, Y and Z direction (i.ezyx

fff and, ) due to the change in boats position dx

and dyare so determined. The 3rd

order polynomial equations that relate the boats position due to

the disturbance dx =[x x2

x3]

T, dy=[y y

2y

3]

Tto output cables force [fx fy fz]

Tare :

+

=

+=

300

0370

740

04702

0150006800120

00270

3

2 .

.

x

x

x

.

...

.

f

f

f

CdDF

z

y

x

xxxTx

(41)

+

=

+=

300

12

0290

04702

0061

0100051000450

3

2 .

.

y

y

y

.

.

...

f

f

f

CdDF

z

y

x

xyyTy

(42)

whereyx

dd , are the boats position in X and Y direction respectively. The vectors Cx and Cy

are the biased force in the X, Y and Z direction while the matrix DxandDy represent the cables

force distribution in the Cartesian coordinate for the dx and dy.

It is observed that the value of the disturbance force fz increases to over 300N as the input

disturbance velocity tends to dx= dy= 20 m whilefxandfy are small respectively. This drift in the

position is just a worst case consideration which in actual situation, the launch boat will be

anchored to maintain in its position, that isyx

dd , is small. The disturbance force is consistent

with [15] as well.

i-1

i

n-1

n

1n

Z

X

element 1

element n-1

element n

Boat

ROV

C.G.

i+1


21/36

21

4. LINEAR ROV MODEL

During the station keeping, the controller has to keep the vehicle about the equilibrium position or

orientation. Under this situation, the vehicle dynamics can be linearized about an equilibrium

position. As results of the linearization of (7) and (3) about the equilibrium point, oo , , a linear

time invariant model of twelve-orders in state-space model can be written as:

B +=& (43)

=& (44)

where the A1212 is the system matrix, B 412 is the input matrix, E 126 is the output

matrix:

[ ] v10

0

00

111

cT

TTTG; ;

Is

I; E

M; B

J

GMD)(CMA

+

==

=

=

+= ,

ovvv = 6 is the perturbation in velocities vector,

o = 6 is the

perturbation in position vector, T46 is the thruster configuration matrix, 6

c is the

commanded optimal thrust input vector,6 is the actual thrust input vector, GT 44 is a

linear thruster model matrix and T+=T

T(TT

T)

-1 64

is a Moore-Penrose pusedo-inverse. The

linear matrices C, D, G andJused in (46)-(52) are computed using the follow derivative.

)J(,Jg(

,GD(

, DC(

Co

ooo

=

=

=

=

)

v

v)v

v

v)v

(45)

where the matrices C(v)66 ,D(v) 66 ,g() 16 and J() 66 are stated in (3) and

(7). Note that the linear damping matrix is used in the derivative. Details of the matrices in (43)

and (44) are tabulated below.

=

zzyzxz

yzyyxy

xzxyxx

R

III

III

III

m

m

m

M

000

000

000

00000

00000

00000

(46)


22/36

22

=

r

q

p

w

v

u

N

M

K

Z

Y

X

&

&

&

&

&

&

00000

00000

00000

00000

00000

00000

M A

(47)

00

00

00

0000

0000

0000

++

++

++

=

oxxoxyoxzoyyoxyoyzoo

oxxoxyoxzozzoxzoyzoo

oyyoxyoyzozzoxzoyzoo

oo

oo

oo

R

pIqIrIqIpIrImumv

pIqIrIrIpIqImumw

qIpIrIrIpIqImvmw

mumv

mumw

mvmw

C

(48)

=

00

00

00

0000

00000000

opoqouov

oporouow

oqorovow

uov

ouow

ovow

A

pKqMuXvY

pKrNuXwZ

qMrNvYwZ

uXvY

uXwZvYwZ

C

&&&&

&&&&

&&&&

&&

&&

&&

(49)

=

r

q

p

w

v

u

N

MK

Z

Y

X

D

00000

0000000000

00000

00000

00000

(50)

000000

0cos)(0000

0sinsin)(coscos)(000

000000

000000

000000

=

oGB

ooGBooGB

zz

zzzzG

(51)

=

2

1

0

0

J

JJ (52)

where


23/36

23

++

++

=

ooooo

oooooooooooo

oooooooooooo

J

coscossincossin

cossinsinsincossinsinsincoscoscossin

sincoscossinsinsinsincoscossincoscos

1

=

oooo

oo

oooo

J

coscoscossin0

sincos0

tancostansin1

2

For simulation purpose (see Fig. 11), the voltage input to the thrusters, T1 and T2 are both set at

40V. Since T1 and T2 are used for surge velocity, the response does show high magnitude at surge

velocity as compared to the rest. However, the error in the responses between the linear and

nonlinear model is represented in maximum norm,G

. It is mainly due to the model

uncertainty such as the nonlinearity presents in the system. The velocity range for the linear

model is: 1v0 < . The model uncertainty will be included in the controller design.

5. DECOUPLING OF LINEAR ROVs MODEL

The direct benefit to obtain a diagonal dominance (or decouple) system is to obtain several small

subsystems allowing the use of control structure such as decentralized control. Besides, it

balances and enables a best choice of the input and output units to use when the knowledge about

the system is insufficient. This improves the system reliability and efficiency as shown in [26].

Thus makes the individual loop more independent prior to any controller design.

To achieve diagonal dominance, Edmunds scaling [17] and input-output(I/O) reordering routine

is used. The routine has shown to work in some structure when Perron-Frobenius [18,19] and

one-norm scaling [20] are not appropriate and inadequate to achieve the diagonal dominant

system.

During the Edmunds scaling and I/O reordering routine, the linear system is scaled prior to I/O

re-ordering to make the dominant multivariable element more obvious. In first part of the routine,

it balances the effect of each of the inputs (outputs) in each column (row) of the system gainmatrix. Followed by re-ordering the I/O into pairs to maximize the diagonal dominance. As a

whole, the scaling and I/O re-ordering attempt to minimize the multivariable nature of the

problems by balancing the interaction between loops.

Prior to Edmunds routine, the diagonal dominant of the linear ROV system needs to be

determined. Rosenbrock's row (or column) diagonal dominance [21] with Gershgorin discs [18]

superimposed on the diagonal elements of the system frequency response was performed. The

plots in Fig. 12a indicate that the system is highly interactive over all frequencies (1-100 rad/s), as


24/36

24

observed from the increasing size of the discs in the diagonal elements in all rows. The increasing

size of the disk implies that the off-diagonal terms are more dominant than its diagonal terms or

in another words, the system is not diagonal dominance.

As shown in step responses of PosteG Pree in Fig. 13, the scaling and I/O pairing using the

Edmunds scaling resulted in two diagonal blocks of higher amplitude (see the dotted line) and

more evenly distributed Gershgoin discs between the loops shown in Fig. 12b. The Pree and Poste

matrices (resulted from the Edmunds routine) for both the scaling and I/O pairings at frequency

of 2 rad/s are given by:

05283.10000

0004553.000000001598.0

1043.000000

00006485.00

001176.0000

Poste

=

(53)

0000717.000

0318.80000

000003215.0

1789.000000

00001332.00

001852.0000

Pree

=

(54)

In (53) and (54), the fourth element (input) in the first column of Pre e is paired with the fourth

element (q) in the first row of Poste while the second element (input) in the second column of Pree

is paired with the second element (v) in the second row of Poste , and similarly for the rest. This

gives the scaled input and output pairs with the following output velocities, v reordered as p, v, q,

u, r, w and the commanded optimal distribution of control inputc

becomes:c3c5,c1c6c2,c4

,,, .

To verify the degree of block diagonal dominance of the system after Edmunds scaling and

reordering routine, the following method known as the block diagonal dominance measure of the

system is used. Suppose TFM of the vehicle is partitioned into several interconnected subsystems

3333

3333

jjji

ijii

GG

GG(55)

where superscript 33 and subscript ji, refer to the size of the partitioned block and elements

in the block respectively. To determine the block diagonal dominance of G(s) , the gains of each

blocks ofG(s) are determined as follows:


25/36

25

Gofaluesingular vsmallestv

vGmin)G(

ii

ii

0ii

==x

(56)

As a comparison, the block dominance measure of the system after the Edmunds scaling becomes

more block diagonal dominance, as the value decreases from41012 to 16106 (see Fig.14).

Hence, the first 33 blocks containing thep, v, q are decoupled from u, r, w.

6. CONTROL STRUCTURE DESIGN

After the dynamic modeling and decoupling are performed on the ROV, the next task is to select

a control structure to be used. Typically, the control structure used is single loop in nature.

However it is not efficient if the DOF is of twelfth order in the case of ROV RRC II.

Furthermore, due to the ROV operation requirement, both the velocity and position must be

controlled separately during the station keeping (or localized inspection). This inevitably leads to

a cascade control structure consisting of an inner loop used for velocity control and outer loop for

position control. While the ROV is performing the station keeping, the outer loop is trying to

maintain its current position.

As shown in Fig. 15, in the control structure, the pre and post compensators from the Edmunds

scaling and re-ordering routine are included. This results in a decentralized control with only [v u

r w]T

respectively. Note that the pitch and roll velocity are not included in the feedback due to the

self-stabilizing in the roll and pitch angles [22] provided by the buoyancy and gravitational force

vectors that form a restoring couple in these direction. Hence, the outputs selected are the position

coordinates and the yaw angle that are not self-stabilizable. These are variables that have no

natural equilibrium and hence cannot be left unattended for a long period of time without control.

In the next section, the H controller design used in the inner loop control (as shown in Fig. 16)

of the vehicles velocity is illustrated. Due to the simplicity in the PD controller for outer loop

control, it is not shown in this paper.

7. H CONTROLLER DESIGN

H controller must be able to perform in the presence of tether disturbance and parametric

uncertainty. In signal terms, the objective is to minimize the maximum value of exogenous output

(labelled as) z due to exogenous input signal w. For example, if the uncertainties are given byFT

=W4 dtand u= Wy,it is desired to find a controller that minimizes the maximum norm of

closed-loop transfer function,Hsubjected to a parametric uncertainty bound of 1

. The


26/36

26

is a diagonal additive uncertainty that is used to represent the error due to the linearization and

ROV modeling:

==

6

2

1

i

}diag{O

(57)

The procedure for designing a H [23,25] controller for the ROV RRC II is shown as follow.

Consider a feedback system with diagonal additive uncertainty A. The W is a normalization

weight for the uncertainty, Wd is a weight for the tether disturbance and W1 to W2 are performance

weight. The generalized plant has inputs and outputs is derived as:

=

=

v

z

z

y

;

u

d

u

H

t

c

1

2

zw (58)

By writing down the equations or simply inspecting the Fig.16:

1111

22

GuudWV

GuWuWdWWzuWz

uWy

c

c

c

c

tdH

td

=

++==

=

(59)

The generalized plantPfrom w to z has the form:

GWI

GWWWW

W

W

P

=

4

1211

200

00

(60)

Note that the transfer function from u to y(upper left element in P) is zero because u has no

direct effect ony(except throughK). With that, to derive closed loop transfer function,Hknown

as Linear Fractional Transformation (LFT), first partition thePto be compatible withK, that is:


27/36

27

00

00

1

2

12

11

11

GW

W

W

; P

WWW

P

d

=

= ; [ ]

22421G; P-W-IP ==

(61)

and then find (P,K)FHl

= using21

1

221211PK)PK(IPP + :

11

22

21

1

221211

SWWSW

KSWWKSW

KSWWKSW

PK)PK(IPPH

d

d

d

=

+=

(62)

where the sensitivity ( ) 1+= GKIS and complementary sensitivity ( ) 1+= GKIGKT .

The upper left block, H11 in (62) is the transfer function from u to y. This is the transfer

function, for evaluating robust stability due to parametric uncertainty:

1

11

+= GK)K(IWH (63)

The weighting function is used to shape the sensitivity of the closed-loop system,Hto the desired

level. A typical type of performance function used is the low pass filter, high pass filter and a

constant weight. The tuning of the weighting function, irrespective of the type used, is performed

iteratively. The sequence is to tune the first entry of the weighting function so that the nominal

performance is still enforced. Retain this value for the first entry and repeat the procedure for the

second entry, and so on.

W1. From the frequency response of the disturbance, the disturbance are dominant up to 100

rad/s. To reduce the disturbance up to this frequency, W1 is selected to be a high-pass filter at

frequency range 100 rad/s onward. Different gains for each entry were selected, sequentially.

W2. A low-pass filter was selected to shape the input sensitivity of the closed-loop system. A

similar first-order low pass filter was used in each channel with a corner frequency of 0.1 rad/s, inorder to limit the input magnitudes at high frequencies and thereby limit the closed-loop

bandwidth.

W .A gain a low-pass filter was designed for shaping the additive uncertainty to the ROV and

hence shaping the input sensitivity of the closed-loop system. It limits the size of the input to the

ROV.

Wd. The weight here was chosen to be sufficiently small in order to prevent the appearance of

some badly damped modes in the closed-loop system due to disturbance.


28/36

28

One of the difficulties in advanced control techniques, based on optimisation, is the high order of

the optimal controller. It is usually necessary to reduce the order of the controller so that it can be

easily implemented. To determine the lowest possible controller order to be used, Hankel

singular value [24] is applied to measure the state controllability and observability at different

controllers order .

Hankel SVs for modal truncation are larger than those of the Schur balanced truncation which

indicates that the former has better state controllability and observability properties than latter. In

other words, the modal truncation contains more information than the Schur balanced truncation.

It therefore seems to be preferable to use the modal truncation method. As result, the order of the

controller has reduced from 54 to 20.

In the S plots, the weighting function used has indeed shaped the sensitivity of the system belowthe 0-dB margin. Since the singular value of the sensitivity function is low, it shows that the

closed-loop system with a disturbance signal has less influence on the system output. Conversely,

a large value ofS means the system has a poor stability margin.

From the input sensitivity, KS plot, it can be seen the effect of the input disturbances is quite

negligible on the ROV output. The KS after applying the shaping function is well below the

margin of 0 dB. The robust stability due to parametric uncertainty as indicated in (35) is below

the margin of 0 dB and hence is robust against parametric uncertainty.

8. COMPUTER SIMULATION RESULTS

In this section, a comparison between the cascaded control of the PD-PD, HPD controller

and single loop PD were performed on the nonlinear model with the disturbances. By doing so, it

tests the robustness of each controller against the parametric uncertainty and tether disturbance.

As the vehicle is currently fitted only with limited sensors the desired position commands values,

[ ]T0015.000 are chosen with this purpose in mind. The constant parameters used for

PD controllers in HPD cascaded control are:Kp= 50 KD=50 while constant parameters used

for PD controllers in PDPD cascaded control are:Kp= 50KD=50.

As observed in Fig. 17, the response is regulated about the desired position set points and its

velocity components are small. This represents the station-keeping condition whereby the

velocity is kept at zero while the position is allowed to vary. Note that the velocities and its

corresponding positions are rearranged as the result of the Edmunds scaling and re-ordering

routine.


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29

As shown in Table 2, the standard deviation of the ROVs position for different controllers are

illustrated. It is obvious to see that PDH has a smaller standard deviation as compared to its

counterparts LQG/LTR and the PD using the single-loop structure.

Standard DeviationController Type

x(m) y(m) z(m) (rad) (rad) (rad)

PD-H 0.0068 0.6441 0.0652 0.0001 0.0007 0.0019

PD-LQG/LTR -0.0332 0.4394 -0.0880 0.0690 0.0108 -0.050

PD only -0.0456 1.2076 1.0409 0.0113 0.0015 0.009

Table 2: Standard deviation of the ROVs position

9. CONCLUSIONS

The vehicles and thrusters dynamic used for control system design were defined. The added

mass coefficient due to the pressure-induced forces and moments caused by the submerged

vehicles acceleration was derived using the Strip theory. The linear ROV model with the added

mass components for the station-keeping condition was shown. The Edmunds scaling and

reordering algorithm that enables a best choice of input and output pairs and a block diagonal

dominance system was performed.

Due to the roll and pitch motions of the ROV being self-regulating, only four degree of freedoms

(DOF) were controlled instead of six. The cascaded form of control structure was defined.

The H controller subjected to a pre-determined tether disturbance and parametric uncertainty

was designed for the inner velocity control loop and the PD controller for the outer position

control loop. The simulation test on the proposed proportional derivative (PD) with H cascaded

control demonstrates a smaller standard deviation in output position as compared to the PD-linear

quadratic gaussian with loop transfer recovery (LQG-LTR) and single-loop PD controller.

Although the used of H and PD controller are not new in the current ROV control application,

the cascaded structure with the pre and post compensators in the inner loop to obtain a more

block diagonal dominance control have not been discussed. Thus, this paper provides an insight

to the above.


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30

ACKNOWLEDGEMENTS

The authors would like to acknowledge the contributions by all the project team members from

NTU Robotics Research Centre especially Mr. Lim Eng Cheng, Ms. Agnes S.K. Tan, Ms. Ng

Kwai Yee and Mr. You Kim San.

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8. Antonelli, G., Chiaverini, S., Sarkar, M. and West, M. Adaptive control of an autonomous

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New Jersey.


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underwater robotic vehicle. Proceedings of the 11th

International Symposium on Unmanned

Untethered Submersible Technology (UUST'99), Autonomous Undersea Systems Institute,

New Hampshire, August 1999, 455-466.

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system.Proceedings of Oceans 96, 1996, pp 517-523.

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Journal of Offshore Mechanic and Arctic Engineering, Vol 123, pp 43-45, 2001.

17.Edmunds, J.M. Input and output scaling and reordering for diagonal dominance and blockdiagonal dominance. In Proc. of IEE, 1998, Part D(145), 523-530.

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25.Zhou, K., Doyle, J. and Glover, K. Robust and optimal control. 1995, Prentice Hall, New

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32

FIGURES

Figure 11: Body velocity response using thruster input: T1 = T2= 40V

)1.0,,,1.0,,( rad/srqpm/swvuoooooo

==

0 20 400

0.1

0.2

0.3

0.4

0.5

Time(sec)

surge vel(m/s)

nonlin

lin

0 20 40-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

Time(sec)

sway vel(m/s)

nonlin

lin

0 20 400

0.01

0.02

Time(sec)

heave vel(m/s)

nonlin

lin

0 20 40-0.3

-0.2

-0.1

0

0.1

Time(sec)

roll vel(rad/s)

nonlin

lin

0 20 40-0.3

-0.2

-0.1

0

0.1

Time(sec)

pitch vel(rad/s)

nonlin

lin

0 20 40-0.3

-0.2

-0.1

0

0.1

Time(sec)

yaw vel(rad/s

nonlin

lin


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33

Figure 12a: Direct Nyquist Array with Gershgoin disk before Edmunds scaling

Figure 12b: Direct Nyquist Array with Gershgoin disk after Edmunds scaling

-5 0 5

x 10-3

-5

0

5x 10

-3

-2 0 2

x 10-3

-1

0

1x 10

-3

-2 0 2

x 10-3

-1

0

1x 10 -3

-2 0 2

x 10-3

-2

0

2x 10

-3

-2 0 2

x 10-4

-2

0

2x 10

-4

-0.02 0 0.02-0.01

0

0.01

-5 0 5

x 10-5

-5

0

5x 10

-5

-5 0 5

x 10-5

-2

0

2x 10

-5Nyquist Diagram with Gershgorin Disk

-2 0 2

x 10-5

-2

0

2x 10

-5

-2 0 2

x 10-4

0

2

4x 10

-5

-2 0 2

x 10-4

0

2

4x 10

-5

-5 0 5

x 10-5

-2

0

2x 10

-5

-2 0 2

x 10-5

-1

0

1x 10

-5

-5 0 5

x 10-4

0

2x 10

-4

-5 0 5

x 10-7

-2

0

2x 10

-7

-5 0 5

x 10-4

-2

0

2x 10

-4

-5 0 5

x 10-5

-5

0

5x 10 -5

-2 0 2

x 10-4

-5

0

5x 10 -5

-5 0 5

x 10-5

-2

0

2x 10 -5

-2 0 2

x 10-5

0

0.5

1x 10 -5

-2 0 2

x 10-5

-2

0

2x 10 -5

-2 0 2

x 10-5

-2

0

2x 10

-5

-5 0 5

x 10-3

0

2

4x 10

-3

-2 0 2

x 10-4

-2

0

2x 10

-4

-5 0 5

x 10-6

-2

0

2x 10

-6

-2 0 2

x 10-4

-5

0

5x 10

-5

-5 0 5

x 10-4

-5

0

5x 10

-4

-2 0 2

x 10-5

-2

0

2x 10

-5

-2 0 2

x 10-4

0

0.5

1x 10

-4

-1 0 1

x 10-5

-5

0

5x 10

-6

-2 0 2

x 10-4

-2

0

2x 10

-4

-2 0 2

x 10-3

-1

0

1x 10

-3

-2 0 2

x 10-4

-2

0

2x 10

-4

-1 0 1

x 10-5

-1

0

1x 10

-5

-2 0 2

x 10-4

-5

0

5x 10

-5

-2 0 2

x 10-5

-2

0

2x 10

-5

-2 0 2-1

0

1

-2 0 2-1

0

1

-2 0 2-1

0

1

-2 0 2-1

0

1

-1 0 1-0.5

0

0.5

-1 0 1-0.5

0

0.5

-1 0 10

0.2

0.4

-0.01 0 0.01-5

0

5x 10

-3

Nyquist Diagram with Gershgorin Disk

-5 0 5

x 10-3

-5

0

5x 10

-3

-0.05 0 0.05-0.02

0

0.02

-0.05 0 0.05-0.02

0

0.02

-1 0 10

0.2

0.4

-0.1 0 0.1-0.1

0

0.1

-0.05 0 0.05-0.05

0

0.05

-0.02 0 0.02-0.01

0

0.01

-0.02 0 0.02-0.02

0

0.02

-0.05 0 0.05-0.02

0

0.02

-0.05 0 0.05-0.02

0

0.02

-0.5 0 0.5-0.2

0

0.2

-0.2 0 0.2-0.1

0

0.1

-2 0 2

x 10-3

-2

0

2x 10

-3

-0.01 0 0.01-0.01

0

0.01

-0.01 0 0.01-0.01

0

0.01

-0.2 0 0.2-0.1

0

0.1

-1 0 10

0.5

-0.01 0 0.01-0.01

0

0.01

-0.05 0 0.05-0.05

0

0.05

-0.05 0 0.05-0.05

0

0.05

-0.01 0 0.01-5

0

5x 10

-3

-0.05 0 0.05-0.05

0

0.05

-1 0 1-0.5

0

-0.05 0 0.05-0.05

0

0.05

-0.05 0 0.05-0.05

0

0.05

-0.2 0 0.2-0.2

0

0.2

-1 0 1-1

0

1

-0.5 0 0.5-0.5

0

0.5


34/36

34

Figure 13. Step responses after Edmunds scaling

Figure 14: Block diagonal dominance measure of the system before(left) and after(right)

Edmunds scaling

Step Response

Time (sec)

-1

0

1tau_c 4

p

-1

0

1

v

0

1

2

q

-1

0

1

2

u

-1

0

1

r

0 1 2 3-1

0

1

w

tau_c 2

0 1 2 3

tau_c 6

0 1 2 3

tau_c 1

0 1 2 3

tau_c 5

0 1 2 3

tau_c 3

0 1 2 3

Command optimal thrust input ( c )

Outputvelocities

( )

10-10

100

2

3

4

5

6

7

8

9

10

11

12x 10

-4

Block Diagonal Dominance Measure of the System using min (G)

Frequency (rad/s)

Gain

10-10

100

0

1

2

3

4

5

6x 10

-16

Frequency (rad/s)

Gain

First diagonal dominance block

Second diagonal

dominance block


35/36

35

Figure 15:Decoupled cascaded control structure design

Figure 16: H block diagram with PD controller

+

- 1 x 1

3 x 3

DecoupledROV system

v

u

r

wPD

+

-

Input

H E

output

matrix

inner loop

outer loop

Disturbance

(tether)

+

+

z

x

y

z

x

y

)PreG(Post ee

G

z1

+

K

z2

ycu

Hv

td

E

Outputmatrix

PD+

-

+

-

ontrollercH

DecoupledROV

W

dW

2W

1W

)PreG(Post ee

+

+ +

TF

u y


36/36

Figure 17: Position (top) and velocity (bottom) response using PDH cascaded control

0 20 40 60-0.2

0

0.2

0.4

0.6

Time(sec)

surge vel(m/s)

PD-Hinf

PD-LQG/LTR

PD

0 20 40 60-0.2

0

0.2

0.4

0.6

0.8

Time(sec)

sway vel(m/s)

0 20 40 60-2

-1

0

1

2

3

4

Time(sec)

heave vel(m/s)

0 20 40 60-1

0

1

2

3

4

Time(sec)

roll vel(rad/s)

0 20 40 60-1

0

1

2

3

4

Time(sec)

pitch vel(rad/s)

0 20 40 60-2

-1

0

1

2

3

4

Time(sec)

yaw vel(rad/s

0 20 40 60

-1

-0.5

0

0.5

1

1.5

2

Time(sec)

surge pos(m)

PD-Hinf

PD-LQG/LTR

PD

0 20 40 60

-0.1

0

0.1

0.2

0.3

Time(sec)

sway pos(m)

0 20 40 60

-0.2

-0.1

0

0.1

0.2

0.3

Time(sec)

heave pos(m)

0 20 40 60-0.1

0

0.1

0.2

0.3

Time(sec)

roll angle(rad)

0 20 40 60-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Time(sec)

pitch angle(rad)

0 20 40 60-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Time(sec)

yaw angle(rad

Documents

Dynamic Modelling and Cascaded Controller Design of a Low-Speed Maneuvering ROV