modeling, simulation and control of a cyclic and collective pitch propeller for underwater vehicles

7/31/2019 Modeling, Simulation and Control of a Cyclic and Collective Pitch Propeller for Underwater Vehicles

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Hi ngh ton quc viu khin v Tng ha - VCCA-2011

VCCA-2011

Modeling, Simulation and Control of a Cyclic and Collective PitchPropeller for Underwater Vehicles

M hnh ha, m phng v iu khin chn vt bin bc tun hon v tphp dng cho phng tin ngm

Poowadol Niyomka, Jonathan Binns, Neil Bose and Hung Duc NguyenUniversity of Tasmania / Australian Maritime College

e-Mail:[email protected]

AbstractThis paper presents theoretical modelling, simulation

and control of a cyclic and collective pitch propeller

(CCPP) that is applied for underwater vehicles. The

collective and cyclic pitch propeller has been

designed and fabricated. It needs testing under variousconditions to construct the relationships between the

rpm, pitch angles of propeller blades and axial and

side thrusts. The propeller is driven by a brushless dc

motor. The propeller has two main controllers, one

controls the propeller motor speed and the othercontrols the blade angles. This paper is about the

development of mathematical equations by relevant

theory, computer-based simulation and design of the

controller for the CCPP. The paper focuses on the

mathematical models of two main components of the

propeller, one is the brushless dc motor driving it andthe other is the linkage mechanism controlling the

blade angles.

Tm tt: Bi bo ny trnh by m hnh ha lthuyt, m phng v iu khin mt chn vt bin

bc tun hon v tp hp p dng cho phng tinngm. Chn vt bin bc tun hon v tp hp c thit k v ch to. Chn vt ny cn th nghimtheo cc iu kin khc nhau nhm xy dng miquan h gia tc vng quay, gc cnh chn vt vcc lc y dc trc ln lc ngang. Chn vt c tibng mt ng c mt chiu khng chi than. Chnvt c hai biu khin chnh, mt iu khin tc ng c mt chiu khng chi than v mt iu khin

gc cnh chn vt. Bi bo ny vit v vic pht trincc phng trnh ton hc theo l thuyt, m phngv thit k b iu khin cho chn vt s dng mytnh. Bi bo tp trung vo m hnh ton ca hai bphn chnh ca chn vt, mt l ng c mt chiukhng chi than lai chn vt v b phn khc l c cukt ni iu khin gc cnh chn vt.

NomenclatureSymbol Unit Meaning

Li inch Displacements of linearactuators (i = 1, 2, 3)

(i,) degree the assigned collective pitchangle of a particular blade

r rad/sec BLDC motor speed

J kgm2

Moment of inertia of rotor

AbbreviationCCPP Cyclic and Collective Pitch Propeller

BLDC Brushless dc motor

AUV Autonomous underwater vehicleROV Remotely operated vehicle

FPP Fixed-pitch propeller

CPP Controllable pitch propeller

1. IntroductionA marine vehicle usually needs an appropriatepropulsion system generating thrust to propel it

through the water. For the conventional fixed pitch

propeller, the thrust/torque depends on the speed of

the propeller shaft. The controllable pitch propeller

running at a constant rpm has thrust/torque depending

on the pitch angle. Both fixed and controllable pitch

propellers generate axial thrust and side thrust whileside thrust is very small in comparison with the axial

thrust. A surface marine vessel equipped with an FPP

or CPP requires a rudder steering system to control its

heading angle. A submarine and/or underwater

vehicle often also need rudders/hydroplanes at bow,

stern and side. The operation of a helicopter rotor,

which is a kind of a cyclic and collective pitch

propeller in the air generates lift forces, propulsion

thrust, and turning moments and makes the helicopter

move in various directions. Thus, this concept is

being investigated as a means of marine propulsion.A cyclic and collective pitch propeller applied in a

helicopter generates both axial thrust and side thrustsby manipulating blade angles. If these axial and side

thrusts can be controlled as desired, it is possible to

apply the CCPP for propulsion and manoeuvring of

an AUV.

The main purpose of this paper is to:

describe a newly-built CCPP; model the CCPP using relevant theory; develop control programs for the CCPP; develop computer simulation programs in

order to learn the dynamics of the CCPP

before experiments are conducted; and

investigate the omni-directional thrusts bytheory and experiment.

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This paper is organized as follows: Section 1

introduces background, scope and purposes of the

paper, Section 2 gives a brief description of the

CCPP, Section 3 is about theoretical modelling of the

CCPP, Section 4 describes computer simulation of the

CCPP, Section 5 is about the control program of the

real CCPP and Section 6 highlights some concludingremarks and recommendations for future work.

2. Description of Newly-Built CCPP2.1 Prototype of a CCPPHumphrey [1] developed a prototype of the cyclic and

collective pitch propeller. This type of propeller was

aimed to provide a highly manoeuvrable capability

compared to a conventional propeller. The design of

the cyclic and collective pitch propeller used a

helicopter like linkage system. The three-dimensional

model of the completed propeller is shown in Fig. 1.

This propeller has the ability to produce 360 degrees

of thrust direction from a propeller rotating about asingle axis.

Fig. 1 3-D model of the CCPP [1]

The swash plate is the most important component in

the mechanism of the CCPP. It provides control of the

angle of the propeller blades. The three electrical

linear actuators change the angle of the swash plate.

The linear actuators are commanded by the operator.

The connecting linkages connect between the swash

plate and the propeller blades. The connecting

linkages allow the swash plate to change the angle

(pitch) of the propeller blades.To understand the relationship between the collective

and cyclic pitch controls and the swash plate, the

assembly can be explained as follows:

The collective pitch control manipulates the swash

plate assembly bodily forward or aft, towards or away

from the motor. This causes the angle (pitch) ofblades to change simultaneously.

The cyclic pitch control tilts the swash plate assembly

to one side. This causes the angle (pitch) of the blades

to change unevenly. The angle of each blade depends

on the location of each blade in the rotation cycle. The

result of the uneven pitch of each blade is that more

thrust is generated on the greater pitch angled blades

on one side and less thrust is generated on the lesserpitch-angled blades on the opposite side. The cyclic

pitch propeller can generate lateral thrust as a result of

this unbalanced thrust and the rake angle of the

propeller blades. See [1][6] for further information.

2.2 Control System of the CCPPThe CCPP consists of the following components as

shown in Fig. 2: a BLDC motor, BLDC motor drive,

3 stepper motor drives, three stepper motor-drivenlinear actuators, data acquisition card, and a linkage

mechanism to control the four blade angles.

Fig. 2 Electronics of the control system of the CCPP

Fig. 3 shows a functional block diagram of the CCPPwith various components. For development of the

control programs, the output variables are the motor

shaft rpm and 4 blade angles.

Fig. 3 Functional block diagram of the CCPPAconceptual diagram of the simulator and real control

system program

3. Theoretical Modelling of the CCPP3.1 Propeller Blades (the Relationship between

Actuator Displacements and Blade Angles)The propeller blades are shown in Fig. 4.

3.1.1 Equation for describing the angles of apropeller blade at any locationThe equation was developed to estimate the

instantaneous angles of each propeller blade at any

pitch setting. It is assumed that the change of theangle of the propeller blades is sinusoidal for a cyclic

pitch setting since it is controlled by the swash plate.The equation is presented as follows.

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Coll Cyc1 Cyc2i, sin 90 sin (1)

where

subscript i (1, 2, 3 and 4) is the blade number;

subscript (0 to 360 deg) is the location of thepropeller blade.

(i,) is the assigned collective pitch angle of aparticular blade;

Fig. 4 Rotation of the CCPP

Coll(i) is the assigned collective pitch angle of aparticular blade (Coll(i) =29 deg to +29 deg);Cyc1(i) is the first assigned maximum cyclic pitchangle of a particular blade (Cyc1(i) = 20 deg to +20deg)

Cyc2(i) is the second assigned maximum cyclic pitchangle of a particular blade (Cyc2(i) = 20 deg to +20deg).

3.1.2 Converting a Blade Angle in Degrees into aPercentage of Maximum RangeThe angle of propeller blade was converted into

percentage number for ease of control. It is assumed

that the relationship between the blade angle and thethrust is linear in the control and simulation programs.

The collective pitch angle is converted to percentage

of maximum range as,

Call i (29 deg to +29 deg) Call i (100% to

+100%)

P

Coll i Coll i

100% 100%

29deg 29deg

(2)

The converting constant (sensitivity) of a collective

pitch angle is,

Coll

200 % %K 3.44838

58 deg deg

(3)

Superscript P is to indicate the angle in the form of a

percentage. The converting constant of cyclic pitch

angle was determined as,

P

Cyc1 i

Cyc1

Cyc1 i

100% 100% %K 5

20 deg 20 deg deg

(4)

Similarly, the following converting constant is

obtained

P

Cyc2 i

Cyc2

Cyc2 i

100% 100% %K 5

20 deg 20 deg deg

(5)

3.1.3 Converting a Blade Angle in Percentage to

Displacement of Each ActuatorHumphrey [1] conducted an experiment to find the

relationship of the moving distance of the actuators

with the angle of the propeller. The developedequations are presented as follows.

P

Coll

ja

100

(6)

P P

Cyc2 Cyc21

j P

Cyc1

b sin tan100

if PCyc1

and

P

Cyc2 are not equal to 0 (7)

P

Cyc2jb

100

if PCyc1

or PCyc2 is equal to 0 (8)

P P

Cyc2 Cyc11

j P

Cyc1

c cos tan100

if

P

Cyc1 i and

P

Cyc2 i are not equal to 0 (9)

P

Cyc1

jc

100

if

P

Cyc1 or P

Cyc2 is equal to 0 (10)

Subscript (j) is the actuator number (j = 1,.2, 3).

Substituting the value of aj, bj and cj into the

following equations the displacement of each actuatoris estimated as

The displacement of actuator 12 2 2

1 10 11 1 12 1 13 1 14 1 15 1 16 1

3 2 2 2

17 1 1 18 1 1 19 1 110 1 1 111 1 1 112 1 1

L p p a p b p c p a p b p c

p a c p b c p c p a c p a c p b c

(11)


2 20 21 2 22 2 23 2 24 2 25 2 26 2

3 3 2

27 2 2 28 2 2 29 2 2 210 2 211 2 212 2 2

2 2 2

213 2 2 214 2 2 215 2 2


p a b p a c p b c p b p c p a b

p a c p b c p b c

(12)


3 30 31 3 32 3 33 3 34 3 35 3 36 3

3 3 2

37 3 3 38 3 3 39 3 3 310 3 311 3 312 3 3

2 2 2313 3 3 314 3 3 315 3 3


p a b p a c p b c p b p c p a b

p a c p b c p b c

(13)

The actuators are the linear stepping motors. The

stepping motor amplifiers are set to move the stepping

motors for 1 inch for every 3200 pulses.

The displacement of each actuator is converted to the

number of pulses as,

Number of pulses =i

1 inL

3200=

4

i3.125 10 L (14)

Feedback Signals: Inside each linear actuator is a

linear potentiometer. The feedback signal is in the

form of an amount of voltage Vai (see Fig. 5):The displacement of actuator 1, L1 = 0.2359Va1 (15)

The displacement of actuator 2, L2 = 0.2359Va2 (16)The displacement of actuator 1, L3 = 0.2359Va3 (17)

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The Resolution: The resolution of the data acquisition

system is 16 bit and the measuring range is between 0

volts and 10 volts.

The resolution in volts =16

10V

2=

10V

65536=

0.000152588 V (18)

and the resolution in displacement

=10 in

V 0.235965536 V

= 0.000035996 inches

= 0.0009142984 (19)

Fig. 5 Block diagram of the actuators and linkagemechanism with potentiometers

3.2 Description and Modelling of BLDC Motor

and Drive Used in the CCPPThe frameless BLDC motor (of Bayside/ParkerMotion Group) used in the CCPP is shown in Fig. 6.

Fig. 6Frameless BLDC motor used for the CCPP

The frameless BLDC motor is controlled by a

brushless servo amplifier (Advanced MotionControls) as illustrated by a block diagram in Fig. 7.

Fig. 7 Block diagram of the brushless dc pm motor system

An equivalent diagram for the BLDC motor system

with its servo amplifier and Hall sensor is shown in

Fig. 8.

Fig. 8 BLDC motor system with the servo amplifier andHall sensor (Advanced Motion Control, 2011)

Referring to Fig. 8 the equations for the BLDC motorsystem are summarised below:

Estimation of the voltage values at the star point is

given by,

a b c 1 2 3s

v v v e e ev

3 3

(20)

where

1 2 3e ,e ,e = back emf of each phase

sv = star point voltage

a b cv , v , v = the voltages at terminals a, b and c,

respectively. It equals to Vdc/2 (Vdc is the BLDC

motor input voltage).

The phase voltages for a brushless dc motor were

expressed in matrix form as shown in followingequation [3],

1 1 1 1

2 2 2 2

3 3 3 3

v R 0 0 i L 0 0 i ed

v 0 R 0 i 0 L 0 i edt

v 0 0 R i 0 0 L i e

(21)

where: v1, v2, and v3 are phase voltages

R is the winding resistance

i1, i2, i3 are phase currents

L is the winding inductance

ei (i = 1, 2, 3) is the back emf of each phaseThe phase voltages are estimated by the following

equation,

1 a sv v v 2 b sv v v

3 c sv v v

The equation, which was used to estimate theelectromagnetic torque, is presented by [3],

e 1 1 2 2 3 3r

1T e i e i e i

(22)

Furthermore, the equation, which was used to study

the transient behaviour of the brushless DC motor, is

presented by,

re 1

dT T J

dt

(23)

where

eT = electromagnetic torque

Linkage

mechanism

Stepper

motors and

actuators

Li i, Pulses

Potentio-

meters

Vai

1. pre-installed integral

commutation board2. rare earth magnets

3. rotor assembly

4. machined groves

6. high-density copper

winding

7. minimised end turns

8. skewed laminations

9. optimised slot fill

BLDC

PM motor

Brushlessmotor

servo amp

ii r [rpm]

Control (vdc), direction

and enable signals

Hall

sensor

(feedback)Commu-

tation feedback

(load)

DC/AC converter

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lT = load torque

J

inertia of the motor and the driven load; and

r

angular speed of the rotor

Interested readers can find more detailed information

on the mathematical models of a brushless dc motor

in [3][5][7].

4. Computer Simulation Program of theCCPP Using LabVIEW

4.1 Diagram of Computer Simulation ProgramComputer simulation program was made with NI

LabVIEW. In the preliminary investigation of the

CCPP dynamics, the simulation program wasdeveloped based on the block diagram as shown in

Fig. 9, in which there are two controllers. The rpmand blade angles were controlled by two modes: Auto

(closed-loop) and Man (open-loop).

Fig. 9Block diagram of the simulation program

In the simulation program two controllers weredesigned based on the conventional PID control law:

P I D

d tu K e t K e t dt K

dt (24)

4.2 Simulation Programs of the CCPPThe simulation program is divided into two parts. The

first part is for the brushless motor. The objective of

the simulation program of the brushless motor was tostudy the dynamics and also the steady-state

performance at various loading conditions. The main

user interface of the simulation program is given in

Fig. 10. There were six inputs, which could be

assigned: the load torque, moment of inertia, the

sampling rate, reference speed of the motor, referencecurrent, and the motor and drive parameters. There

were also several outputs, including the total

electromagnetic torque, current motor speed, an

electrical position of the rotor, phase voltages, phase

current, and phase back emf. Fig. 11 shows simulatedresults.

The second part of the simulation is for the pitch

control. The objective simulation was to study

algorithm of the control program. The program can

simulate automatic and manual control of the

propeller pitch. The pitch angle cannot be measured

directly. The pitch angle was calculated according to

the relationship between pitch angles and the

displacement of the actuators. The main user interface

is given in Fig12. Fig 13 shows a result of automaticpitch control for a collective pitch angle of 29 degrees

and cyclic pitch angles of -20 degrees and 0 degrees.

The speed of the actuator was adjustable by

increasing the pulse rate, the sampling rate and the

resolution. The manual control allows the operator tomanoeuvre each actuator.

Fig.10 The main user interface of the main motorsimulation

Fig.11The result of the automatic RPM control

Fig. 12 The main user interface of the pitch controlsimulation

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Fig.13 The result of automatic pitch control for a pitch

setting of 29 degree collective pitch and -20 degree cyclic

pitch

5. Control Program of the Real CCPPThe objective of the program was to control the cyclic

and collective pitch propeller. The program was

divided into two parts. The first part was to controlthe angle of the propeller. Controlling the angle of the

propeller blades can be done manually or

automatically. Manual control was implemented by

setting the number of pulses to be sent to each

actuator, and the moving direction of each actuator.

The current position of each actuator was measured

by the linear potentiometer. The automatic control

system utilized the on-off control method with the

feedback signals from the linear potentiometers. To

change the manual system to an automatic system a

toggle switch was used. The manipulation of the

angles of the propeller blade can be done by simplychanging the setting value at the slide bars. The first

slide bar was for changing the angle of all propeller

blades simultaneously. This slide-bar was used to

command the propeller to generate the forward or

backward thrusts. The other two slide-bars were tocontrol the side thrusts. One slide was for changing

the thrust vector vertically. Another one was for

changing the thrust vector horizontally. The user

interface window on the angle control program is

given in Fig. 14.

Fig.14 The user interface window of the pitch controlprogram

The second part of the program was for controlling

the speed of the main motor. The motor could be

controlled manually or automatically. Controlling the

motor manually was undertaken by turning the knob

to adjust the value of the reference voltage at the

brushless servo amplifier. The range of the reference

voltage is from zero to ten volts. The PID controlmethod was selected for automatically controlling the

speed of the motor. The feedback signal was from the

hall-effect sensors. The operator could adjust the

desired speed. The user interface window on the

speed control program is shown in Fig. 15.

Fig.15 The user interface window of the speed controlprogram

6. ConclusionsThe control program for a cyclic and collective pitchpropeller was developed to control the pitch of the

propeller blades and the RPM of the propeller shaft.The simulation of the brushless dc motor was also

developed to estimate the electrical parameters such

as the phase voltages, the phase current and the back

emf. The shaft velocity was successfully controlled

with PID control law.

Recommendations for future work: In the future, thesimulation program will be developed to have

capability to predict the thrusts and moments, which

are generated by the cyclic and collective pitch

propeller. Currently as scheduled in the research plan,

captive experiments are planned. The propellers truepropulsion performance will be conducted in the

Circulating Water Channel, at Beauty Point at the

Australian Maritime College. In this experiment, the

cyclic and collective pitch propeller will be attached

behind a body of an underwater vehicle. The self-

propulsion test will also be conducted using a load-varying test [4]. The information of the captive test

will be analysed and used to implement the control

program, which can control the direction of the thrusts

of the propeller. After the control program is

modified, the control system of the vehicle with the

cyclic and collective pitch propeller will be tested in

an untethered trial. The untethered test will be used toimprove the control algorithm and to verify the

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simulation program. The underwater vehicle and its

fairings were designed and built as shown in Fig. 14.

Fig. 16 Photo of the underwater vehicle with the attachedCCPP for experiments

After conducting experiments mathematical models of

the CCPP dynamics will be developed and verified.

The main objective of the control program in future isto control the axial and side thrusts as desired as

illustrated in Fig. 17.

Fig 17Future control program for CCPP

AcknowledgementsThis paper is part of the 2010s Institutional ResearchScheme Grants (IRSG) funded project. The authors

would like to thank the University of Tasmania,

Office of Research Services for funding.

References[1] Humphrey, T. C.. Design and Fabrication of a

Collective and Cyclic Pitch Propeller.Newfoundland, Memorial University of

Newfoundland. Master of Engineering Thesis,

2005.

[2] Humphrey, T. C., Bose. N. and Williams, C.Improving AUV Manoeuvrability: Development

of a Collective and Cyclic Pitch Propeller. IEEE

Ocean Engineering Society Newsletter. Vol

XXXIX-1, 2005. Accessed on 20/9/2011 at:

http://www.ieee.org/organizations/society/oes/ht

ml/winter05/auv.html.

[3] Nesimi, E.. LabVIEW for Electric Circuits,Machines, Drives and Laboratories, Pearson

Education, 2002.[4] Bose, N. Marine Powering Prediction and

Propulsors. The Society of Naval Architects

and Marine Engineers (SNAME), 2008.

[5] Baldursson, S.. BLDC Motor Modelling andControl A MATLAB / SimulinkImplementation, Master Thesis. Institutionen fr

Energi och Milj, 2005.[6] Niyomka, P.. Designing the Tests for Collective

Pitch and Cyclic Pitch Propeller, BE (Naval

Architecture) Thesis. Australian Maritime

College, Launceston, 2009.

[7] Ying, L. and Ertugrul, N.. The DynamicSimulation of the Three-Phase BrushlessPermanent Magnet AC Motor Drives with

LabVIEW. Australasian Universities Power

Engineering Conference (AUPEC), Darwin,

1999. Accessed on 20/10/2011 at

http://itee.uq.edu.au/~aupec/aupec99/ying99.pdf

Biography

Poowadol Niyomka received

the B.S. degree in Production

Engineering from King

Monguts Institute of

Technology North Bangkok,Bangkok, Thailand in 2005

and the B.S. degree in Naval

Architecture from Australian

Maritime College, Tasmania,

Australia in 2009, where he is

currently working toward a

PhD degree. His research interests include pitch

control and brushless motor control of a cyclic and

collective pitch propeller for an autonomous

underwater vehicle.

After graduating with first

class honours with Bachelorof Engineering from the

University of New South

Wales, Dr Binns completed aMaster of Science from

Curtin University of

Technology. After a time in

industry Dr Binns then

completed his Doctor of

Philosophy at the AustralianMaritime College. Dr Binns has worked in yacht

design firms as a designer and researcher. He has been

employed on two Americas Cup campaigns workingon experimental and computational hydrodynamics.In 2007 Dr Binns joined the Australian Maritime

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College as a post-doctoral research fellow. Now he is

managing a large range of research projects all related

to hydrodynamics for the marine industry at the

AMC.

Neil Bose is the Director of

the AMC National Centrefor Maritime Engineering andHydrodynamics at the

Australian Maritime College

and a Professor of Maritime

Hydrodynamics. Neil obtained

his B.Sc. in Naval

Architecture and OceanEngineering from the

University of Glasgow in

1978 and his Ph.D. also from Glasgow in 1982.

During 1974-1980 he was a partner in the Cape Wrath

Boatyard, a builder of wooden and fibreglass boats

and yachts. In 1983 he was appointed as a NewBlood lecturer in Naval Architecture and OceanEngineering at the University of Glasgow. He moved

to the Memorial University of Newfoundland,

Canada, in 1987. In 2003 he was appointed to a Tier 1

Canada Research Chair in Offshore and Underwater

Vehicles Design and he led the purchase and

commissioning of an International Submarine

Engineering Explorer class autonomous underwater

vehicle (AUV) with a 3000m depth rating. He came

to AMC in Tasmania in May 2007. His research

interests are in marine propulsion, autonomousunderwater vehicles, ocean environmental

monitoring, ice/propeller interaction and aspects ofoffshore design.

Dr. Hung Nguyen is a lecturer

in Marine Control Engineeringat National Centre for

Maritime Engineering and

Hydrodynamics, Australian

Maritime College, Australia.

He obtained his BE degree in

Nautical Science at Vietnam

Maritime University in 1991,

then he worked as a lecturer

there until 1995. Hecompleted the MSc in Marine Systems Engineering in

1998 at Tokyo University of Marine Science andTechnology and then the PhD degree in Marine

Control Engineering at the same university in 2001.

During April 2001 to July 2002 he worked as a

research and development engineer at Fieldtech Co.

Ltd., a civil engineering related nuclear instrument

manufacturing company, in Japan. He moved to theAustralian Maritime College, Australia in August

2002. His research interests include guidance,

navigation and control of marine vehicles, self-tuning

and optimal control, recursive system identification,

real-time control and hardware-in-the-loop simulationof marine vehicles and dynamics of marine vehicles.

Appendix

Numerical values of parameters of the CCPP aregiven in Tables 1 and 2.

Table 1 Numerical values (p1ij, p2ij and p3ij) of the

linkage mechanism

p1ij p2ij p3ji

-0.000059371 -0.000513070 -0.003414267

-0.268200000 -0.268044444 -0.269644444

-0.001200000 -0.300481481 0.300796296

0.347129630 -0.173296296 -0.173648148

-0.005824407 -0.006598452 -0.002267316

0.001195201 -0.006716099 -0.005483003

-0.003942054 0.005323117 -0.011051630

0.032803922 -0.028352941 0.028588235

0.005392157 -0.016411765 -0.016254902

0.065587464 0.001509804 -0.005352941

-0.038000000 -0.056289459 0.058565242

-0.118492308 -0.033406268 -0.032764672

0.032882051 -0.033989744

0.019374359 0.019210256

0.058512821 0.059825641

0.101774359 -0.102635897

Table 2 The parameters and their measured values forthe brushless dc motor

Parameters Value

Torque constant, kt 0.80057 Nm/A

Back emf constant, ke 0.80195 V/rad/s

Moment of inertia, J 0.000465815875 kgm2

Number of poles 12

Winding resistance, R 0.8

Equivalent winding

inductance, L

4.65 mH

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modeling, simulation and control of a cyclic and collective pitch propeller for underwater vehicles

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