Các thuật toán điều khiển robot kiểu ngón tay có số bậc tự do tối thiểu trong thao tác các vật th

79B - 2010 CONTROL ALGORITHMS FOR MANIPULATING AN OBJFXT BY A PAIR OF MINIMUM-DOF ROBOTIC FINGERS CAC THUAT TOAN DIEU KHIEN ROBOT KIEU NGON TAY CO SO BAC TU' DO TOI THIEU TRONG TI lAO

JOURNAL OF J>CitNCE & TECHNOLOGY * No 79B - 2010

CONTROL ALGORITHMS FOR MANIPULATING AN OBJFXT BY A PAIR

OF MINIMUM-DOF ROBOTIC FINGERS

CAC THUAT TOAN DIEU KHIEN ROBOT KIEU NGON TAY

CO SO BAC TU' DO TOI THIEU TRONG TI lAO TAG CAC VAT Till:

Nguyen Pham Thuc Anh

Hanoi University of Science and Technology

ABSTRACT

Recently control of multifingered robotic hands in dexterous manipulating an object becomes a new direction that attracts interest of researchers in robotics Observations in everyday life show that human can grasp a small object securely by using a dual pair of his thumb and index fingers, then rotate it to a desired angle and move it to the given vicinity The purpose of this paper is to present algorithms for concurrent grasping and manipulating a flat sudace object by a pair of minimum-DOF robotic fingers with soft ends in a horizontal plane Firstly, the dynamics of the fingers-object system have been formulated by applying variational Hamilton's principle Secondly, a control framework for concurrent grasping and orientation controlling of the object have been proposed Finally, the effectiveness of the proposed control inputs has been proved by theoretical analysis and reconfirmed

by computer simulation results Our research aims to contribute for developing of intelligent robots that work in assembly lines of electronic industry and welfare robots in supporting service disable and elderly people

TOM TAT

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I INTRODUCTION

It is often observed in everyday life that

human hands can implement dexterous

manipulation with excellent skill Researches in

biology science have pointed out the dexterity

of human hands is due to partly two their

special characteristics: (1) finger-tips are soft to

ensure stable grasping of objects and (2) a

thumb finger arranges a dual structure with

other fingers, especially with an index finger

This fact becomes a challenge for many

research works in designing robotic fingers and

discovering intelligent control algorithms to

manipulate surrounding objects dexterously in

analogous manner to human fingers This will

be helpful for developing of industrial robots working in assembl}' operations and welfare robots in supporting elderly persons

II SYSTEM DESCRIPTION

A pair of dual single DOF fingers with soft fips grasping an object has been illustrated

in Fig.l The index 1 is for the left finger and 2 for the right finger Finger tips are

spherical-shaped with radius r, (for /=1,2) and covered by

a visco-elastic material Symbols q=(qi,q2)^,

[=(Ij, hf m=(mi,myf l=(li,l2J^ denote the

vector of joint angles, moments on inertia, masses, and link lengths of fingers respectively The object has parallel surfaces

49

JOURNAL OF SCIENCE & TECHNOLC

with mass A/, inertia moment /, height d, and

width u' ( ir/ + tr.) fhe svmbiil L staiuls for the

distance between two base positicMis o and o' of

the two fingers Positions of end-point Ooj of

link / are denoted bv VCCIIM' /„, (.\'i,i.y,ii)' Due

to the visco-elastic characteristics ol' the soft

finger-tips, the cimlacl bclwecn the lingers and

surfaces oC the objecl is o\' area-conlacl that

supports lo grasp the object securely and

manipulate il llcxiblv When the soft lingers

grasp the object, there arise deformations with

lengths of U/ and h - respectively

Reproducing forces /, (caused by the deformed

areas arc functions of U, in the following form

/, = A',Av; ; where K is a silliness coenicicnt

The reproducing forces /, direct from the point

0(H to the point Oj, where O, is the center

point of the contact area / fhe viscous force

acting from Oo, to the point O, can be expressed

in the form of c,(.Vv ),\v , where s ( A v J is an

increasing function of Av, The fingers-object

setup is confined in a horizontal plane and

therefore is not affected by the gravitational

force We define the reference frame {oxy}

locating at the base position of the left finger

and the object coordinate {OXY} frame at its

mass center O, position of the mass center of

the object in the reference frame by vector

7.=(-Y, v) and the rotational angle of the object

by ^

V, = Vn,-(';"^,K (1)

Fig I Fingers-object system

Next we formulate geometric relations

for keeping contacts between two fingers and

the object The position of the center points Xi^

of the contact areas in the coordinate frames

can be calculated in the following form

In the object frame {ObJ, the points /,

have a form

'">,=[ u ):]','">,.[w,,-};]' (2)

where the term } stands for distances between

the center points Xi of contact areas and ihc

other smlacc point at which the A'-axis of the object frame crosses the object surfaces 1 he term )', must be subject to the constraints that the fingertips purely roll without slipping on the objecl surfaces, or cquivalentlv the

velocities dV/dt on the object surfaces equal to that on its corresponding finger-ends (r

lv,)df7Vd/, that is

dt dt where (/?, + q^ =7r + ( -\)'0

(3)

(4)

From the above two relations, two important formulas can be obtained

/ ; ' • ( / , - / „ : ) = - ( > ; - ) : ) (5)

where / / = [cos^,sin6?],r^'^ = [ - s i n ^ c o s ^ ] III DYNAMICS O F PHYSICAL

I N T E R A C T I O N B E T W E E N FINGERS AND AN O B J E C T

Dynamics ol" a pair of dual single DOF fingers grasping a 21) object with flat surfaces are considered, fhe kinetic energv A' of the overall svstem can be expressed as follows:

fhe potential energy P

deformation is described as:

t - ) ( 6 )

of finger-end

It should be remarked that eq (5) is non-holomic constraint due to an existence of

functions Ax, and eq (3) can not be integrated

in time However, eq (3) can be regarded as the

SO

JOURNAL OF SCIENCE & TECHNOLOGY * No 79B - 2010

form written in terms of infinitesimally small

variation of 5x as follows:

^ ' ' ^ dxj dx,'

Thus, the variational form for the

Lagrangian L=K-P with external forces of

control input // and the non-holomic constraints

can be expressed as:

\j;^{5L + ^{^x,)A.^,^cSX ^u!Sc},

-A,Kr,-Ax,)^^^]5X)dt^0

d.\ c\

(8)

where A, expresses Lagrange multiplies

The vector A' is defined as

X=(qi.q\x.y.O)^ By applying this principle to

the objective system, we can obtain the

dynamic equations of two fingers as [1]:

h'4\ + Aj'Ji -[>^oVv -''i + ^^"i]^ = "i

I-,q.,_ - Jl^J^f - Wlz'', - '2 + •^^•: ]^_ = ":

(9)

and of the grasping object as:

Mjc-{f-f.)cose-{?^+?^)^t)

My-{f-fi_)sme + iA,+A,) = 0 (10)

10 -YJ,+ YJ, - M'/^ + Ms/l, = 0

The tangential constraint forces with the

magnitude A, emerge at the center points of the

contact areas / in the direction tangential to the

object surfaces The normal contact f o r c e s / are

functions of deformation Ax, and their

derivatives as follows [2]:

/ = / + ^ , ( A x , ) A i : , (11)

Based on the dynamics of the

fingers-object systems, sensory-feedback algorithms

for grasping and manipulafing an object will be

designed in the next section

IV ALGORITHM FOR STABLE

GRASPKVG OF AN 2D OBJECT AND

CONTROLLING ITS ROTATIONAL

ANGLE

Refer to a regular manipulation of human

fingers, we can see that there are two phases in

his motion: firstly the human being has to grasp

the object securely and secondly rotates it to a

desired orientation while keeping the contact stably The time for switching from grasping to manipulating depend on fiexibility and skill of human In the first phase, he can grasp an object without knowing its physical parameters and state parameters such as position and orientation In the second phase, in order to rotate the object to the desired angle, the information about actual rotational angle is necessary With an assumption that the

rotational angle 0 of the object can be

measured, wc proposed a control framework for

two phases: the Hrsl phase 11, ii/i + ii,,,, for stable

grasping the object and the second phase

u,-u,i\uo, for controlling its orientation The

first phase is from starting time to a switching

time /, the control input u, Uii+u,,,, aims to

formulate stable grasping is:

Jcos0^

-c.q,-{-\)'

r-Ax

I('',-^v,)

1=1,2

s u i t / )

iY,-Y,)f,

(12)

where u,,,, is for torque balance, Cj are damping

coefficients, and J„, are Jacobian matrices defined as:

Jo,=

dq, ' dq, , From the switching time, the control input u,

=Uf,+Ue, aims to rotate the object can be

designed angle as

/ c o s ^ ^

U,=Uj.,+Ug, = - ( - l ) ' J o

'ys'm0 J fd-(^A,

-(-1)'

.Js:\n0^

^^ l^COSt'y

(13)

• r, + Ax K^^0

)

The closed dynamics have a form:

lA^JlrM^-^-^l^-'-'^^iY.-Yf)

dq,

= -/:v,^|-./>,.^,Ai-,

rM'

(14)

fq, -JlrM - ^ ^ 2 ^'-^^^{Y,-Y)

oq., rw

= - A : V , ^ + J o V v ^ 2 ^ 2 (15)

J O l k N A L OF S( lENCI & TECHNOLOGY • No 7915 - 2UM)

Mx -/;A/,+/vV, Mi \

dll ,

'•^-' A = ; ; ( ; , A \ , - C A \ , )

r.v

10 -\\\u + y,.\i,- -I ' A,

16)

(17)

It is possible to have

; 1 2 •^ " '

As combined with etj (1 8), we have

^-^th=-Y |r//;+^,(Av,)Ax-,^

(23)

where rw - /• -( /•, Vv, \v

It is possible to reali/c that the closed

dvnamics ensure that:

1 d

2dt /•, = - X {c,^/ + ^,(Vv),\v; I I :

where /•" A + \T t / / ( > ; > ' )

-(18)

(19) 2(/-,^':)

and \/' = X r (/('/)-./>/'/

Since £1 is positive definite with respect

to X^(qi.q:.x.y.0)' and X under nonlolomic

constraints of eq (3) and relation (5), £; plays a

role of Lyapunov function It is possible to

conclude that

Vv ^ 0

.\/| —>0 as / ->oo

AA, ^ 0

Y,-Y._^0

That implies the balance of torque and

force on the grasping object and clearly that the

stable grasping the object bv means of dual

single-DOF soft fingers can be realized in

theoretical analysis

The second phase is for controlling of

rotational angle 0 of the object Taking

derivative in time eq (5) leads to

- ( ) ; - K ) : ( Z , „ - i r ) r + ( j „ , - j ) ^ ^

cu (2(

= q'jLr,-q[-Jl,r^+w{^x,)d

and

-^r(-^oV;-''i+^^i) + ^/2(-7,L'v

-r^ + lyx^) = {M\ +w,)0

(20)

(21)

(22)

w here /••, =F, +A'^(»', ^ w,)^0^ (24)

Since /•,'• is positive definite with respect

to V (cfi.q^x.y.Ol' then E^ plays a role of

Lyapunov function It is possible to conclude that

\q, -> 0

Av, > 0

¥, >o

,\A, > 0

}; - r , >o

AO >o

Next in order to verify the effectiveness

of the proposed control scheme for stable grasping of the objecL a computer simulation has been carried out Physical parameters of fingers-object system have been reported in Table 1 and control parameters in Table 2

Table I Physical parameters of the system

Link mass Inertia moment Link lensith Object mass Inertia moment Object length Object width Object height Stiffness cocL Viscous coef

w;/=-»/;

// /:

/ / - / :

\ /

I

h

U/ + 1f;

d

A

<''AI=^'J:

0.025 [kgl 6.66E-06[kg.m-]

0.040[m]

0.03 [kg]

7.5E-06[kg.m'] 0.05[ml

0.03[m]

0.0 l[m]

150000[N/m-] 100000[Ns/m']

Table J Parameters of control input

Desired contact force Desired rotational angle

Damping coefficient Gain coefficient Switching time

.fd

0j

Ci=C2

Ko

/v

UN]

0 [rad]

0.001[msN] 0.3

0.0285 [sec]

JOURNAL OF SCIENCE & TECHNOLOG\ * No 79B - 2010 From starting time to a switching time /,

(0.0285 second), we apply the control input of

eq (12) for stable grasping and after U the

control input of eq (13) for controlling of

object's rotational angle is applied The

response of the overall fingers-object dvnamics

governed by the proposed control inputs have

been shown in the following Figures

i \

A A

-V \

; ;

i • •

1 , :

i ' :

; ;

: : • ; : ' : : i '

' • ' ' , ' , I

: ' ] ' ]

I ', ',

I \ ' - - f

0 0.1 0 2 0 3 0 4 0 5 0 6 0 7 0.8 0 9 1

/

1

1

1

time[ second]

lg-2 Contact force f J at

Vf

;

) 0.1 0

the leftflnger

• ' ' : • '

1 1 •

; ; ;

.2 0.3 0 4 0 5 0

; ; :

6 0 7 0.8

- ' I

09 1 tinne[second]

Fig 3 Contact force f2 at the right finger

0.15

• 9

• 0 0.1 0 2 0.3 0.4 0.5 0.6 0.7 0 8 0.9 1

time[second]

Fig 4 The rotational angle 0 of the object

0 0.1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 t

IJmolBecond]

Fig 5 Deformation AX/ of the left-finger

=/\A^

0 0.1 0 2 0,3 0 4 0 5 0.6 0,7 0.8 0.9 1

time[secondl

Fig 6 Deformation AX2 of the right-finger

The responses have verified the effectiveness of the proposed control input of eqs (12 ,13) The normal contact f o r c e s / (for /=1,2) converge to the desired v a l u e / / in 0.06 second (Figs 2,3)r"The rotational angle of the

object ^converges to the desired value 0d in

0.12 second (Fig 4) The fingers keep contact with the object during whole manipulating time

when Ax, is always positive (Figs 5,6) Clearly

that the simulation results have verified the effectiveness of the proposed control algorithm The noteworthy point that the fingers always maintain the contact with the object during manipulation time It is not a serious matter when the overall system is confined in a horizontal plane, but in a vertical plane, the contact loosing of only one finger will make the object to drop freely Then in the next work, we will develop the proposed control input for manipulating the object under the effect of gravity

JOURNAl O F S( IKN( F "i FIX HNOLOGY * No 79B - 2010

variables in both grasping and orientation control of the object to the desired states The simulation results have reconfirmed the validity

o\' the control methods It is interesting to

reali/c a dexterity of the simplest fingers with soft-ends The basis of study can be developed for the case of grasping and manipulating an objecl under the effect of gravity

The paper dealt with the two problems:

(1) stable grasping an 2D fiat surface objecl

without an\ i>bjeel sensing information and (2)

concurrent reali/.ation '.•'I' secure grasp and

orientation ol" the object bv means ol" dual

single-DOF fingers with soft ends in a

horizontal plane Control frameworks have been

found out in dynamic sense and designed Irom

joint sensing information so that they become

applicable in experiments Theoretical analvsis

REFERENCES

1 S Arimoto P.I".A Nguven II.-Y Han, and /, Doulgeri; "Dynamics and control of a set of dual fingers with soft-tips""; Robotics 2000, 18, Part 1, pp 71-80

2 Suguru Arimoto; "Dexterity and Control fheory of Multi-fingered Hands: A Differential-Geometric Approach"; Springer, 2007

3 P.T.A Nguven, S Arimoto, and H.-Y Han; "Computer simulation of controlled motions of dual fingers with soft-tips grasping an object"; Proc Japan-USA Symposium on Flexible Automation,

2000, pp 1039-1046

4 P.T.A Nguyen and S Arimoto; "Dexterous manipulation of an object by means of multi-DOF robotic fingers with soft-tips"; J of Robotic System, 17, No.7, 2002, pp 349-362

5 S Arimoto and P.T.A Nguyen; "Principle of superposition for realizing dexterous pinching motion of a pair of robot fingers with soft-tips"; lEICE Trans, on Fundamental Electronics Communication and Computer Sciences, E84-A, No 1, pp 39-47

Author s address: Nguyen Pham Thuc Anh fel.: (+844) 3869.2306

Email: thucanhnguyen@mail.hut.edu.vn Department of Industrial Automation Hanoi University of Science and Technology

No 1, Dai Co Vict Str., Ha Noi, Viet Nam^