Proceedings VCM 2012 34 Nghiên Cứu Phân Tích Động Lực Học và Ứng Dụng Giải Thuật Di TruyềnLeo Đồi Trong Việc Tối Ưu Hóa Vận Tốc Di Chuyển Thẳng Của Robot Cá 3 Khớp Dạng Carangiform

Nghiên Cứu Phân Tích Động Lực Học và Ứng Dụng Giải Thuật Di TruyềnLeo Đồi Trong Việc Tối Ưu Hóa Vận Tốc Di Chuyển Thẳng Của Robot Cá 3 Khớp Dạng CarangiformNghiên Cứu Phân Tích Động Lực Học và Ứng Dụng Giải Thuật Di TruyềnLeo Đồi Trong Việc Tối Ưu Hóa Vận Tốc Di Chuyển Thẳng Của Robot Cá 3 Khớp Dạng CarangiformNghiên Cứu Phân Tích Động Lực Học và Ứng Dụng Giải Thuật Di TruyềnLeo Đồi Trong Việc Tối Ưu Hóa Vận Tốc Di Chuyển Thẳng Của Robot Cá 3 Khớp Dạng CarangiformNghiên Cứu Phân Tích Động Lực Học và Ứng Dụng Giải Thuật Di TruyềnLeo Đồi Trong Việc Tối Ưu Hóa Vận Tốc Di Chuyển Thẳng Của Robot Cá 3 Khớp Dạng CarangiformNghiên Cứu Phân Tích Động Lực Học và Ứng Dụng Giải Thuật Di TruyềnLeo Đồi Trong Việc Tối Ưu Hóa Vận Tốc Di Chuyển Thẳng Của Robot Cá 3 Khớp Dạng CarangiformNghiên Cứu Phân Tích Động Lực Học và Ứng Dụng Giải Thuật Di TruyềnLeo Đồi Trong Việc Tối Ưu Hóa Vận Tốc Di Chuyển Thẳng Của Robot Cá 3 Khớp Dạng Carangiform

Nghiên Cứu Phân Tích Động Lực Học và Ứng Dụng Giải

Thuật Di Truyền-Leo Đồi Trong Việc Tối Ưu Hóa Vận Tốc Di Chuyển Thẳng Của Robot Cá 3 Khớp Dạng Carangiform

A Study on Dynamic Analysis and Straight Velocity

Optimization of 3-Joint Carangiform Fish Robot Using

Genetic-Hill Climbing Algorithm

Tuong Quan Vo

Ho Chi Minh City, University of Technology – Viet Nam e-Mail: vtquan@hcmut.edu.vn or quanvotuong@gmail.com

Tóm tắt

Robot phỏng sinh học là một dạng robot mới đã và đang được phát triển trong những năm gần đây Một số robot phỏng sinh học đầu tiên được nghiên cứu là robot nhện, robot rắn, robot bò cạp, robot gián,…Trong thời gian gần đây, hai dạng robot phỏng sinh học hoạt động dưới nước đang được quan tâm nghiên cứu là robot rắn

và robot cá

Đầu tiên bài báo này giới thiệu về robot cá có 3 khâu, 4 khớp dạng Carangiform Sau đó, bài báo sẽ giới thiệu việc áp dụng giải thuật di truyền và giải thuật leo đồi để tối ưu hóa vận tốc di chuyển thẳng cho robot cá Đầu tiên, giải thuật di truyền được sử dụng để tạo ra bộ thông số đầu vào tối ưu cho robot cá Sau đó, bộ thông

số đầu vào này lại được tối ưu một lần nữa bằng giải thuật leo đồi nhằm đảm bảo bộ thông số điều khiển này gần với kết quả tối ưu toàn cục của hệ thống Cuối cùng, chúng tôi sử dụng chương trình mô phỏng để kiểm tra tính đúng đắn của giải thuật đã nêu

Abstract

Biomimetic robot is a new trend of researched field which is developing quickly in recent years Some of the first researches on this field are spider robot, snake robot, scorpion robot, cockroach robot, etc Lately, two new types of underwater biomimetic robot called fish robot and snake robot are mostly concerned

In this paper, firstly a dynamic model of 3-joint (4 links) Carangiform fish robot type is presented Secondly fish robot’s maximum straight velocity is optimized by using the combination of Genetic Algorithm (GA) and Hill Climbing Algorithm (HCA) with respect to its dynamic system GA is used to create the initial optimal parameters set for the input functions of the system Then, this set will be optimized again by using HCA to be sure that the last optimal parameters set are the global optimization result Finally, some simulation results are presented to prove the proposed algorithm

Keywords - fish robot, dynamic, optimization, GA, HCA, maximum straight velocity, input torque functions

1 Introduction

Generally, many researches about underwater

propulsion mainly depend on the use of propellers

or thruster to generate the motion for object in

underwater environment Besides, most of the

natural solutions use the change of object’s body

shape for movement This changing shape

generates propulsion force to make the object

moves forward or backward effectively

Carangiform fish robot type is also one kind of the

changing body shape to create the motion for itself

in the underwater environment

George V Lauder and Eliot G Drucker made the

thorough surveys and analyses about motion

mechanisms of fish fin in advance in order to develop such a successful underwater robot system [1] M J Lighthill also surveyed about the hydromechanics of aquatic animal propulsion because of many kinds of underwater animal whose motion mechanisms were evolved throughout many generations to adapt to the harsh

of underwater environment [2] Based on natural movement, there are some other researches about this type of motion Junzhi Yu and Long Wang calculated the optimal link ratio of 4-link fish robot by using computer simulation and he showed the simulation results by comparing the moving speed by two cases of models One is modeled by

using the optimal link ratio and the other one is

considered without the optimal link ratio [3]

However, most of the researches about fish robot

are based on quite simplified model of fish as well

as experimental approaches In our research, we

considered a 3-joint (4 links) Carangiform fish

robot type The dynamic model of the robot is

derived by using Lagrange method The influences

of fluid force to the motion of fish robot are also

considered which is based on M J Lighthill’s

Carangiform propulsion [4] Besides, the SVD

(Singular Value Decomposition) algorithm is also

used in our simulation program to minimize the

divergence of fish robot’s links when simulating

fish robot’s operation in underwater environment

The main goal of this paper is the optimization

method to maximize straight velocity of a 3-joint

Carangiform fish robot in x direction The straight

velocity of fish robot is considered by applying

optimal input torque functions to its dynamic

model The optimal input torque functions are

gotten by optimized the parameters which are used

to build these functions Our proposed method to

solve this optimization problem is the combination

of GA and HCA with respect to fish robot’s

dynamic model and some other related constraints

This combination of GA-HCA gives better results

than our previous work [5] Another solution for

optimization was proposed by Keehong Seo et al

[6] They used the numerical optimization

software (NLPP – Non Linear Path Planning –

Tool Box for Matlab) to optimize the control

parameters for a simplified planar model of a

Carangiform fish robot

2 Dynamics Analysis and Motion Equations

In our fish robot, we focus mainly on the

Carangiform fish’s type because of fast swimming

characteristics which resemble to tuna or

mackerel The movement of this Carngiform fish

type requires powerful muscles that generate side

to side motion of the posterior part (vertebral

column and flexible tail) while the anterior part of

the body remains relatively in motionless state as

seen in Fig 1

Increasing size of movement

Pectoral fin

Posterior part Anterior part

Caudal fin

Tail fin Main axis Transverse axis

Fig 1 Carangiform fish locomotion type

We design 3-joint (4 links) fish robot in order to get smoother and more natural motion As expressed in Fig 2, the total length of fish robot is about 600mm which includes 4 links The head and body of fish robot are supposed to be one rigid part (link0) which is connected to link1 by active

DC motor1 (joint1) Then, link1 and link2 are connected by active DC motor2 (joint2) Lastly, link3 (lunate shape tail fin) is jointed into link2 (joint3) by two extension flexible springs in order

to imitate the smooth motion of real fish The stiffness value of each spring is about 100Nm Total weight of the fish robot (in air) is about 4 kg

l0 (link0)

(link3) (link2)

(link1)

1

T1

m1 (x1,y1)

Y

X a1

l1

l2

a2

l3

a3

2

3

m2 (x2,y2)

m3 (x3,y3)

T2

Fig 2 Fish robot analytical model

In Fig 2, T1 and T2 are the input torques at joint1 and joint2 which are generated by two active DC motors We assume that inertial fluid force FV and lift force FJ act on tail fin only (link 3) which is similar to the concept of Motomu Nakashima et al [7] The expression of forces distribution on fish robot is presented in Fig 3 below FF is the thrust force component at tail fin, FC is lateral force component and FD is the drag force effecting to the motion of fish robot The calculations of these forces are similar to Motomu Nakashima et al method for their 2-joint fish robot [7]

FC V F

J F

F F

FD Direction of movement

X Y

Fig 3 Forces distribution on fish robot

We suppose that the tail fin of fish robot is in a constant flow Um so we can derive the inertial fluid force and the lift force act on the tail fin of fish robot Then we can calculate their thrust component FF and lateral component FC from the inertial fluid force and lift force We also suppose that the experiment condition of testing our fish robot is in tank so that the value of Um is chosen as 0.08m/s Fv is a force proportional to an acceleration acting in the opposite direction of the

acceleration [7] The calculation of FV is expressed

in Eq (1) The lift force FJ acts in the

perpendicular direction to the flow and its

calculation as in Eq (2) In these two equations,

chord length is 2C, the span of the tail fin is L and

 is water’s density

V

2

J

F  pr LCU a a

(2)

These fluid force and lift force are divided into

thrust component FF in x direction and lateral force

component FC in y direction as presented in Fig 4



U U

Y

X

FFV

V

F FCV



J

F Y

X

FCJ

FJ

F

Fig 4 Model of inertial fluid force and lift force

In Fig 4, U is the relative velocity at the center of

the tail fin,  is the attack angle Based on Fig 4,

the value of FF and FC can be calculated by these

two Eqs (3)-(4):

0

0

sin 360 sin 360

J

F

0

0

cos 360 cos 360

J

F

The above two equation can be simplified as

follows:

F F q  q q F q  q q (6)

If we just consider the movement of fish robot in x

direction, so the relative velocity in y direction at

the center of tail fin is calculated by Eq (7)

cos

a

   (7)

Since Um and u are perpendicular as in Fig 5(a),

so the value of U can be calculated by Eq (8):

m

U U u (8)









u

Um

U

(a)

Fig 5 (a) Relationship between U and U m (b) Diagram of attack angle  calculation

By using Lagrange’s method, the dynamic model

of fish robot is described briefly as in Eq (9)

q q q

 







By solving Eq (9) above, we can get the value of

i

q , q i (i = 1  3) However, based on the dynamic model in Eq (9), SVD (Singular Value Decomposition) algorithm is also used in our simulation program to minimize the divergence of the oscillation of fish robot’s links when simulating the operation of fish robot in underwater environment This divergence also cause the velocity of fish robot be diverged too The motion equation of fish robot is expressed in

Eq (10) x G is the acceleration of fish robot’s centroid position mis the total weight of fish robot in water (m 11.45kg) FF is the propulsion force to push fish robot forward and FD is drag force caused by the friction between fish robot and the surround environment when fish robot swims

mx F F (10)

The calculation of FD is presented in Eq (11)

2

1 2

Where r is the mass density of water V is the velocity of fish robot relative to the water flow

D

C is the drag coefficient which is assumed to be 0.5 in the simulation program Sis the area of the main body of fish robot which is projected on the perpendicular plane of the flow  2

0.021

3 Velocity Optimization Method 3.1 Genetic Algorithm (GA) and Hill Climbing Algorithm (HCA)

Genetic Algorithm [8] [9] is based on the process

of Darwin’s theory of evolution by starting with a set of potential population with some or many individuals and then changing them during several iterations The first potential population is generated or selected randomly or arbitrary The individual in the population is called chromosome The entire set of these chromosomes is called population The chromosomes evolve during several iterations called generations GA uses the concept of survival of the fitness by randomly initializing a population of individual in which each individual contains the parameters to reach to

a possible solution of an optimization problem

Each individual in the population is assigned a

fitness value that is used to indicate the quality of

the individual as an optimal solution for the

problem or not Then, the selected individuals

become parents based on their fitness value and

then continue to create the next generation of the

potential solution to the optimal problem The new

potential generations are generated using the

methods of crossover and mutation

Sometimes, the result of GA is just the local

optimum It is not the global optimal solution for

the whole problem In this case, we use HCA to

optimize the result of GA again to make this result

better Besides, the optimization by HCA will also

find the global optimal solution for the problem

Y

X

G lobal m axim um Local m axim um

F lat local m axim um

B eg inn ing Po sition

Fig 6 Hill climbing demonstration

However, there are two popular combinations

between these two algorithms such as: HCA-GA,

GA-HCA The first type of combination

(HCA-GA) was used in our previous work [5] which

gave worse result than the second combination

(GA-HCA) as introduced in this paper HCA [10]

[11] is one of the effective methods to find the

global optimum of a problem The conceptual

diagram of HCA can be seen in Fig 6 The

process of HCA can be expressed by the following

steps:

Step 1: Pick a random point in the search space

Step 2: Consider all the neighbors of the current

state

Step 3: Choose the neighbor with the best quality

and move to that state

Step 4: Repeat Step 2 to Step 4 until all the

neighboring states are of lower quality

Step 5: Return the current state as the solution of

the problem

3.2 Using GA-HCA to maximize the straight

velocity of fish robot

The general algorithm diagram of the optimal

problem is introduced in Fig 7

Fish robot has two active joints that are joint1 and joint2 to generate the movement for whole robot Two input torque functions equations which support for joint1 and joint2 as T1 and T2 in Fig 2 are calculated as in Eqs (12)-(13):

1 1sin 2 1

T A p f t (12)

2 2sin 2 2

T A p f tb (13)

1, 2

A A : Amplitude of input torques for motor 1, 2

1, 2

f f : Frequency of input torques for motor 1, 2

b: Phase angle between input torques of two motors

From two Eqs (12)-(13), there are five parameters

1, 2, 1, 2,

input functions T1 and T2 for the system

Fig 7 General algorithm of the optimization

problem

The propulsive speed of fish robot in steady state

is presented by Eq (14):

F

P F

h

h: Propulsive efficiencyh 0.4

P: Average consumed power 1 1 2 2

2

In GA and HCA, fitness function is used to evaluate the suitable parameters’ value for the system The main ideas of fitness function for GA and HCA is: maximum propulsive speed gives maximum velocity of fish robot Besides, the

algorithm of this fitness function is also based on

the dynamic model of fish robot in order to check

the suitability of these parameters to the fish robot

or not The algorithm of fitness function is

expressed in Fig 8

2

1,

 

F

F

P

 

F

ig 8 Fitness function of GA-HCA

In order to use the optimization method by

GA-HCA, it requires 6 constraints criteria of the

optimized parameters as expressed in Eq (15)

below:

1

2

1

2

6 Dynamic model of fish robot

A

A

f

f

b

 

(15)

The role of HCA is used to find the global optimal

values of parameters set which is based on the

population of parameters set generated by GA and

fitness function HCA algorithm is expressed by

Fig 9 below

Fig 9 HCA

By using GA, if the result is the real optimal one,

it will keep in similar values in many next generations We can depend on this characteristic

of GA to make the stop condition for the optimized process

3.3 GA-HCA implementation

The simulation results of the algorithms in this paper are carried out by using Matlab program with the toolbox of GA [12] The GA simulation program runs with the population of 500 and the generation of 500 During the heredity process, we use the selection method as normalized geometry select, the multi-non-uniform mutations as the mutation method and the arithmetic crossover as the crossover method Then, the optimal population generated by GA will be optimized by HCA to be sure that the last optimal parameters set

is the global optimal set for the system

Besides, in order to prove the effective of this proposed optimization method we also carried out the survey about the behavior of fish robot velocity which input torque functions’ parameters set are chosen arbitrary (arbitrary case) Normally, there is no method to choose the suitable values of

a five parameters set of A A f1, 2, 1,f b2, to build two

input torque functions for the system Besides, the

range value of each parameter in the set also does

not know exactly By surveying the relationship

between velocity and each parameter in the set, we

know the divergent range of velocity value So,

base on this result we can choose suitable the

range value of each parameter for the optimal case

as expressed in Eq (15) Actually in arbitrary case,

we can also base on this survey to choose the

parameters’ value for the system which relies on

their range However, this is not an optimal

method because if one of the values in the set is

chosen unsuitable, these input functions will make

the system halt or divergence Or we can choose

the most maximum parameters for the input

functions to create the maximum velocity for fish

robot This method is also not good because the

maximum value of parameters set can also be

harmful to the mechanism structure of fish robot

Therefore, by using the optimal algorithm by

GA-HCA, the combination values of these five

parameters A A f1, 2, 1,f b2, are chosen and

evaluated suitably to be sure that those values will

make fish robot swim at the maximum velocity

Moreover, by comparison our optimal results done

by simulation method with other results carried out

by experiment method of other researchers, the

advantage of our proposed methods can be proved

4 Simulation Results

4.1 Survey the influence of input torque

functions’ parameters to the velocity of fish

robot

The influences of those parameters to fish robot

velocity are considered by observing the relation

graphs between velocity and those parameters In

these cases, we consider the behavior and velocity

value of fish robot with respect to amplitudes (A1,

A2), frequencies (f1, f2), or phase angle 

4.1.1 The relationship between fish robot

velocity and amplitudes

In order to survey this relation, the values of

amplitude A1, A2 are changed while the value of

frequencies and phase angle are kept as constants

Then we input some different input functions pairs

which are built by this rule to the dynamic model

of fish robot to consider the behavior of its

velocity as presented in Fig 10 In Fig 10 the

value of f1 = f2 = 0.2Hz and  = 300 are kept as

constants

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 The relationship between amplitude and velocity - Beta = 30 Degree, f1 = f2 = 0.2 Hz

Time (s)

A1 = A2 = 1 Nm A1 = A2 = 2.5 Nm A1 = A2 = 4.5 Nm A1 = A2 = 5 Nm

(a)

0 2 4 6 8 10 12 14 16 18 The relationship between amplitude and velocity - Beta = 30 Degree, f1 = f2 = 0.2 Hz

Time (s)

A1 = A2 = 5.5 Nm

(b)

Fig 10 (a) The relationship between amplitudes

and velocity with f 1 = f 2 = 0.2Hz,  = 30 0 (b) Divergence case

In Fig 10(a), when A1 = A2 = 5Nm (the topmost continuous line), velocity of fish robot has the trend to be diverged So, if the amplitudes A1 and

A2 are increased to over 5Nm (for example 5.5 Nm), the velocity will be diverged as expressed in Fig 10(b) Besides, in Fig 10(a), the performance

of velocity has big oscillation when amplitude values reach to 5Nm Therefore, we can choose the value of amplitudes be smaller or equal than 5Nm And the value of amplitudes equal to 5Nm can be called the limited amplitude of divergence

in this case

0 1 2 3 4 5

6 The relationship between amplitude and velocity - Beta = 30 Degree, f1 = f2 = 0.6 Hz

Time (s)

A1 = A2 = 0.5 Nm A1 = A2 = 1 Nm A1 = A2 = 2 Nm A1 = A2 = 4 Nm

Fig 11 The relationship between amplitudes and

velocity, f 1 = f 2 = 0.6Hz,  = 30 0

In next example, we increase the value of frequencies to 0.6Hz and keep phase angle’s value does not change to consider the behavior of this relationship In Fig 11 above, with the value of f1

= f2 = 0.6Hz and  = 300, if amplitude values are

greater than 4Nm (the topmost dash line) the

velocity will have the trend to be diverged So, the

amplitudes value equal to 4Nm can also be called

the limited amplitude of divergence in this case

Generally, from two Figs 10-11, if frequencies are

increased, the amplitudes need to be decreased to

keep the velocity of fish robot not diverges

Moreover, bigger amplitude also makes bigger

velocity of fish robot

4.1.2 The relationship between fish robot

velocity and frequency

Similar to previous case, to survey about the

relationship between frequency and velocity, we

keep the values of amplitudes and phase angle are

constants while changing the value of frequencies

to consider the performance of fish robot velocity

as expressed in Fig 12

In Fig 12(a), the velocity has trend to be diverged

when f1 = f2 = 0.7Hz (the top dash dot line) or f1 =

f2 = 0.9Hz (the topmost dash line) So, if

frequencies’ value are greater than 0.9Hz (for

example 0.98Hz), the velocity will be diverged as

expressed in Fig 12(b) Therefore, 0.98Hz can be

said to be the limited frequency of divergence in

this case

0

0.2

0.4

0.6

0.8

1

1.2

1.4

The relationship between frequency and velocity - A1 = A2 = 3 Nm, Beta = 30 Degree

Time (s)

f1 = f2 = 0.1 Hz

f1 = f2 = 0.5 Hz

f1 = f2 = 0.9 Hz

(a)

0

5

10

15

20

25

The relationship between frequency and velocity - A1 = A2 = 3 Nm, Beta = 30 Degree

Time (s)

f1 = f2 = 0.98 Hz

(b)

Fig 12 (a) The relationship between frequencies

and velocity with A 1 = A 2 = 3Nm,  = 30 0 (b)

Divergence case

By reducing the amplitudes value, the divergent

frequency value will be increased as expressed in

Fig 13 So, with smaller amplitudes value A1 = A2

= 1Nm, the velocity of fish robot will be diverged when frequencies’ value are greater than or equal 1.6Hz (the topmost dash line) Similarly, the value

of 1.6Hz is also called limited frequency of divergence in this case

Therefore, in the relationship between frequency and velocity, the bigger frequency will make the bigger velocity of fish robot Besides, frequency range will be extended when amplitude range is shrunk

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 The relationship between frequency and velocity - A1 = A2 = 1 Nm, Beta = 30 Degree

Time (s)

f1 = f2 = 0.3 Hz f1 = f2 = 1 Hz f1 = f2 = 1.4 Hz

Fig 13 The relationship between frequencies and

velocity with A 1 = A 2 = 1Nm,  = 30 0

4.1.3 The relationship between fish robot velocity and phase angle

The phase angle  between two input torque functions also has big influence to the velocity of fish robot The relationship between velocity of fish robot and phase angle is carried out by keeping in constant the values of amplitudes (A1,

A2) and frequencies (f1, f2) The performance of velocity based on this relation is expressed by Figs 14 below

0 1 2 3 4 5 6 7 8 9 10

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

The relationship between phase angle and velocity - A1 = A2 = 1.5 Nm, f1 = f2 = 0.3 Hz

Time (s)

Beta = 1 Degree

Beta = 10 Degree

Beta = 30 Degree

Beta = 60 Degree

0

0.2

0.4

0.6

0.8

1

1.2

1.4

The relationship between phase angle and velocity - A1 = A2 = 3 Nm, f1 = f2 = 0.7 Hz

Time (s)

Beta = 1 Degree Beta = 10 Degree Beta = 30 Degree Beta = 60 Degree

Fig 14 The relationship between phase angle and

velocity

4.2 Optimal results created by GA-HCA

In this optimization method, the optimal value of

five parameters A A f f b1, 2, 1, 2,  will be generated

by GA-HCA simultaneously base on each

parameter’s range values as in Eq (15) The input

torque functions built by these optimal parameters

will make fish robot swim at the maximum

velocity with respect to its dynamic model In this

optimization program, we will consider the two

frequencies of two motors are similarf1 f2 f

and it is called same frequencies case Therefore,

the optimization method will optimize four

parameters A A f b1, 2, , 

Table 1: Optimal value of parameters set by

GA-HCA

A1 A2 f1 = f2 Beta Fitness

1.36 1.38 1.24 14.62 7.46

The two input torque functions for the system are

introduced as Eq (17)

1

2

1.36sin 2 *1.24

1.38sin 2 *1.24 14.62

p

p



By applying two input torque functions in Eq (17)

to the dynamic model of fish robot, the average

velocity value of fish robot is about 0.59 (m/s)

during the concerning time of 20 seconds Fig 15

simulates the operation of fish robot’s mechanism

system during concerning time in same

frequencies case

The relationship graph between velocity-time and moving distance-time of fish robot by inputting

Eq (17) to the dynamic system are expressed in Fig 16 below In this case, fish robot swims at distance at about 11.69m after 20 seconds After about 14 seconds, velocity of fish robot will be kept stable at the average value of 0.62 (m/s)

-20 -10 0 10 20

Link 1

Time (s)

-4 -2 0 2 4

Link 1

Time (s)

-10 -5 0 5 10

Link 2

Time (s)

-2 -1 0 1 2

Link 2

Time (s)

-2 -1 0 1 2

Link 3

Time (s)

-0.2 -0.1 0 0.1 0.2

Link 3

Time (s)

Fig 15 Fish robot links ‘oscillation and their

angular velocities created by using Eq (17)

0 2 4 6 8 10 12 14 16 18 20 0

5 10 15 The relation between moving distance and time

Time (s)

0 2 4 6 8 10 12 14 16 18 20 0

0.2 0.4 0.6 0.8

The relationship between real velocity and time

Time (s)

Fig 16 Fish robot velocity and moving distance

with respect to time by using Eq (17)

In Fig 16, the velocity of fish robot takes about 12 seconds to reach to the stable state Or, it can be said that there is no big change in velocity’s value after the steady state as in Fig 16

Generally, in our proposed method, the optimal result is generated by GA-HCA The results of this method are just the simulations Our next step is to carry out some experiment which is based on these simulation results Besides, some researchers as Motomu Nakashima et al [7] and Koichi Hirata et

al [13] used experiment method to find the optimal velocity for their fish robot The maximum velocity of fish robot done by Motomu Nakashima

is about 0.5 (m/s) and Koichi Hirata’s fish robot velocity is around 0.2 (m/s) By comparison our proposed method and these two experiment method of two researchers above, our simulation results are nearly similar This means; by using optimization algorithm by GA-HCA, we know that

our fish robot will swim at the maximum velocity

at about 0.62 (m/s) in optimal cases

5 Conclusion

In this paper, a model of 3-joint Carangiform fish

robot type is presented From this type of fish

robot, its dynamic model is derived by using

Lagrange’s method Besides, the influence of fluid

force which exerts on the motion of fish robot in

underwater environment is also considered in

robot’s dynamic model by using the concept of M

J Lighthill’s Carangiform propulsion Besides,

SVD algorithm is also used in our simulation

program as an effective method to reduce the

divergence of fish robot links when solving the

matrix of its dynamic model

By inputting different arbitrary values of input

torque functions T1, T2 to fish robot’s dynamic

system, we made a survey on the relationship

between fish robot velocity and other parameters

of two input functions For example, some of the

relationships are amplitude-velocity,

frequency-velocity and phase angle-frequency-velocity The results of

this survey are used to define the range value of

each parameter in the input function Besides,

these ranges of parameters’ value are also the

constraints criteria for the optimization program of

HCA Then, by using the combination of

GA-HCA simulation program, the optimal parameters

set which are built two input torque functions for

two active motors at joint1 and joint2 are gotten

These two optimal input torque functions can

make fish robot swim at the maximum velocity at

about 0.62 (m/s) with respect to its dynamic

model

6 Future works

In continuing to this problem, some experiments

are going to be carried out to check the agreement

between simulation results and the experiment

results Besides, another control problems will also

considered in the next steps

References

[1] Lauder, G.V and Drucker, E.G., Morphology

And Experimental Hydrodynamics Of Fish Fin

Control Surfaces, IEEE Journal of Oceanic

Engineering, Vol 29, No 3, pp 556-571, July

2004

[2] M J Lighthill, Hydromechanics Of Aquatic

Animal Propulsion, Annual Review of Fluid

Mechanics, January 1969, Vol 1, pp 413-446

[3] Junzhi Yu and Long Wang, Parameter

Optimization Of Simplified Propulsive Model For Biomimetic Robot Fish, Proceeding of the

2005 IEEE, International Conference on Robotics and Automation, Barcelona, Spain, pp 3306-3311, April 2005

[4] M J Lighthill, Note On The Swimming of

Slender Fish, Journal of fluid mechanics, Vol 9,

pp 305-317, 1960

[5] Tuong Quan Vo, Byung Ryong Lee, Hyoung

Seok Kim and Hyo Seung Cho, Optimizing

Maximum Velocity of Fish Robot Using Hill Climbing Algorithm and Genetic Algorithm, The

10th International Conference on Control, Automation, Robotics & Vision, 17-20 December 2008, Hanoi, Vietnam

[6] Keehong Seo, Richard Murray, Jin S Lee,

World Congress in Prague, 2005

[7] Motomu NAKASHIMA, Norifumi OHGISHI

and Kyosuke ONO, A Study On The Propulsive

Mechanism Of A Double Jointed Fish Robot

International Journal, Series C, Vol 46, No 3,

pp 982-990, 2003

[8] Colin R Reeves, Jonathan E Rowe, Genetic

Algorithms – Principles And Perspectives, A Guide to GA Theory, Kluwer Academic

Publishers, 2003

[9] Randy L.Haupt, Sue Ellen Haupt, Practical

Genetic Algorithms – Second Edition, A John

Willey & Son, Inc., Publication, May 2004

[10] Andrew W Moore, Iterative Improvement

Search Hill Climbing, Simulated Annealing, WALKSAT, and Genetic Algorithms, School of

Computer Science Carnegie Mellon University

[11] Masafumi Hagiwara, Pseudo Hill Climbing

Genetic Algorithm (PHGA) for Function Optimization, Proceeding of 1993 International

Joint Conference on Neural Networks

[12] Christopher R Houck, Jeffery A Joines,

Michael G Kay, A Genetic Algorithm for

University

[13] Koichi HIRATA*, Tadanori TAKIMOTO** and

Kenkichi TAMURA***, Study on Turning

Performance of a Fish Robot, *Power and

Energy Engineering Division, Ship Research Institute, Shinkawa 6-38-1, Mitaka, Tokyo

181-0004, Japan, **Arctic Vessel and Low Temperature Engineering Division, Ship Research Institute, *** Japan Marine Science and Technology Center

Vo Tuong Quan was born in

1979, Ho Chi Minh City, Viet

Nam In 2005, he received his

MSE in Ho Chi Minh City University of Technology, Viet Nam about Machine Building Engineering And, in 2010, he received his PhD in University

of Ulsan, Ulsan, Korea about Mechanical and Automotive Engineering

He has been a Lecturer in the Department of

Mechatronics, Faculty of Mechanical Engineering,

Ho Chi Minh City University of Technology from

2002 until present His currently researches are

about the underwater robots, biomimetic robots,

bio-mechatronics systems and automatic control

systems in industry