Optimization of grasping forces with finger deformability and joint parameter variation constraints

Preprints of the Fourth IFAC Symposium on Robot Control September 19-21 , 1994, Capri, Italy

OPTIMIZATION OF GRASPING FORCES WITH FINGER DEFORMABILfIY AND JOINT PARAMETER VARIATION CONSTRAINTS. B. BARKAT and

J. P.

LALLEMAND

lAboratoirtl de Mecanique dtls solidtls, Unite de recherche associee Q1I C.N.R.S U.R.A. 861 40 ..... pinea" 86022 POlTIERS _FRANCE. . , avenue UJ< recteur

A~

This paper deals with ~ op~on of ~ing with a multi-fingered robot band, under general

~ such. as ~ger defonnability and ob~~ P?'iti~ tolerances. Firstly a general formalism describing ~c. graspmg IS presented.

SecoodIy an optimization a1terion based on the minimization of securing forces and

=~ introduced. Lastly resulta cooceming numerical limulation of grasping with a thrco-fingcrcd gripp« are

Key wonU. Grasping. securing. compliance, optimization

1. INTRODUCTION With the improuved performance of industrial robots, the development of the grasping function has been the subject of numerous studies. One of the basic aspects of analysing the conditions that ensure the effectiveness of prehension systems concerns the study of stable grasp. A concept of stable grasp based on the minimization of a potential function of the grasping system was first introdu~ by Hanafusa and Asada (1977); this concept IS extended in study of Liangyi (1984) to acco~t for ~ction between fingers and the object, and IS applied by Baker (1985) to a generalized model of the gripper presented by Asada. Many further studies such as (Ngyen 1989, Mishra 1989; Markenscoff 1990). have been devoted to such a problem. Determination of contact positions is another significant issue in robotic grasping. Wolter (1985) dealt this problem is with using new criteria of stability, and Bologni (1988) searches for optimal application points on the basis of simple geometrical considerations. Following these works many studies are focused upon manipulation and optimal grasping force determination. In a common approach, the ev~~~~n criterion of the best grasp lies in lDlDl1DlZlng a cost function, as in (Demmel 1989; Markenscoff 1989) where the grasping forces are minimized and in (Kerr 1986, Tsuneo 1988 Zbiming 1988; Barkat 1993) algorithms ~ developed for finding optimal finger forces applied on the object for balancing generalized external fo~ces. The study presented in this paper deals WIth the problem of determining grasping forces.

225

But, unlike previous works, optimal grasping is defined by minimizing a cost function based on the projection of grasping forces on friction cones at contact points. So this function involves implicitly the optimal securing forces or torques developed by the powered joints of the gripper.

A general approach of hyperstatic threedimensional gasping is presented, accounting for finger deformability, joint parameter variations and shifting of the object constraints. Introduction of the above mentioned selecting criterion yields a quadratic programming problem whose unknowns ~~ securing forces or torques, grasping forces, Jomt parameter variations and displacement of the object while securing. This general formalism is applied to a three-fingered gripper. Numerical results concerning optimal grasping of a nonsymmetrical and symmetrical workpiece one presented. 2. GRASPING MODEL ~e gri~per ~ be schematized by a set of simple kinematic chains which are articulated on a common base, and closed on the object at contact zones considered as punctual (see fig. 1). Contact points, geometry and mass distribution of the object are supposed to be known. The object is submitted to given generalized external forces.

2.1. Notations

The notations used in this study according to figure 1 are: Cl simple kinematic chain representing the ith finger with ml mobilities or degrees of freedom,i=I, ... ,n

m, total number of mobilities of the given by m=

M(O) = (MI'M2 .M3 )&, is a given external momentatO. So, equilibrium equations of the object 0 are:

gripper

• Lint

f-

1=1

G, centre of mass of the object 0, R, gripper reference frame with respect to point 0, RO, object coordinate frame with respect to point G, RI,coordinate frame attached to the ith fingertip.

LF =0, •

I

M(O)-

L0A.xF =0

1=1

•

I

(1)

1=1

Note that if the given external force is the gravity force, then:

f = mg , M(O)=mOGxg where g is the gravity acceleration vector, and m is the mass of the object O.

All components of 3-dimensional vectors are expressed on the orthonormal base that defines the Raxis:

Vector equations (1) can be rewritten as the single matrix equation:

T

OG= (Xa.Ya.Za) ,postion vector ofG, T

O~ = ( X AJ' YAJ •ZAJ)'

(2)

position vector of the

initial contact point ~ , I

/

/

/

T

n = (nI.n:z.nJ ) , external normal vector at before securing , /

/

/ T

/

/

/ T

n; =(n; .n~ .n~) , external normal vector at after securing , I

F =(~.F;.~), object on Cl' I

/

/

T

I

/

/

T

force

exerted

by

with

~

F=(F\ ... ,F·), N=[ll

;.].

~

'AI =

the

[

0 ZAJ

-ZAJ 0

YAJ -XAJ

-YAJ

XAJ

0

1

where I is the 3x3 unity matrix, and AI is the cross-product matrix with respect to O~. N is the S
q = (qI.· .. . q,.) , mi-vector of joint coordinates of the kinematic chain Cl'

S = (SI .... . S"") , mi-vector of ~g force or torque at each joint of Cl' ( if the kth joint of Cl is

2. 3. Expression of small displacements

/

not powered then Sic =0.) ~I =tan(CPI)' Coulombian friction coefficient at ~. CP. is the angle of friction cone at contact point

2.3.1. Displacement of the ith furgerlip. The small displacement of the point ~ belonging to the ith fingertip consists of two kinds of elementary displacements.

• Displacement due to slight joint coordinate variations c

The small change M/ of the fingertip

~

with

respect to the variation Bql of the vector of the I

joint coordinate q , is expressed as: ~Ac _ ~ U

1-

L

80A. uqj

j=1

1:

axAJ

I

2.2. Static equilibrium equations

T

=U;. h, J;)

T

aZAJ

--I

--I

CXfI

CXfI

CXfI

. aYAJ

/

/

aq..

T

(4)

aZAJ /

aq.. aq..

the vector equation (3) is rewritten as:

T]T

C

I

MI =J with

where: f

aYAJ

--I

J =' aiAJ

In addition to the contact forces FI. i = 1, ... , n, we assume that the object 0 is submitted to the external generalized force:

[

(3)

8qj

Introducing the 3xma jacobian matrix of the kinematic chain Cl:

y

Fig. 1. Grasping schematization

Fat = f ,M(O)

i

i

,is a given external force,

226

I

&i.

(5)

•

i

i

Oq = (Oq ••...• Oq.. )

•

T

•

After securing, let A. (resp. G ) be the position of the point of the object at which the initial contact ~ took place (resp. of the centre of mass o G, see fig. 1). The displacement dll of ~ due to a solid movement of the object 0 can be expressed as:

• Displacement due to link bending of the kinematic chain Determination of this kind of displacement can be done using the superposition principle of the theory of elasticity which yields the total bending of the simple structure which results from locking

o.

•

••

d u ~ A.A. = A.G+GG +G A.

the joints of Cl for any given value of q •.

(11)

where we have:

••

This displacement can be formally expressed as:

c •• d. =CF,

G A. =9l GA. 9l being the rotation matrix of O. Then (11) becomes:

(6)

where et is the 3x3 compliance matrix of the tightened kinematic chain Cl .

(12)

Explicit expressions of et, i=I,2,3 are given in appendix for the three-fingered gripper shown in figure. 2.

where I stands for the 3 x3 unity matrix. In this case, a very convenient representation of 9l is obtained by introducing the Cardan angles 01 , O2 , 03 of the coordinate frame R., with respect to the gripper reference frame R ( see section 2.1).

• Total displacement of the fingertips

c

The total displacement D. of the ith fingertip is simply the sum:

As the displacements are assumed to be very small, all the terms in 9l can be linearized with respect to small values of 01 angles. So, an equivalent expression of (9l- I) yields:

(7)

Now. let us define, (i) the 3n-vector of all fingertip displacements:

Dc = (c D1 , ... ,D.C)T (8) (ii) the vector of all joint variations of the n

kinematic chains Cl: Oq = ( Oql •...• Oq.

(13)

r

which is the matrix of the cross-product with respect to vector 0=(0.,02,0))T. Therefore (12) can be written as:

,(iii) the

block matrices:

J=

[1'

J.}

c=[

!.

J.]

o • dll = GG +A.Gx 0

(9)

Then, introducing the following notations where all components are relative to the reference frame R:

c

Then, the generalized displacement D is globally defined by the 3n-vector relation: c

D =CF+JOq

(14)

d GG

=(OXa ,8Ya,8Za )T, d=( GGd,Or,

(10)

2.3.2. Small displacement of the initial point of contact A; As in the previous 'section, the small change of ~ belonging to the object 0 may consist of two elementary displacements.

expression (15) can be rewritten into matricial form as follows:

• Displacement due to a small shift of the object

o • dll =Hd

Because of the deformability of the fingers, the object may have rotated and moed after securing and. So, the object is submitted to a solid displacement.

(15)

• Displacement due to the deformation of the contact zone This term exists when the object is locally deformable. It can be formulated as:

227

o d l1

I

= -CoF

I

I_

(16)

where Oq

stands fot any virtual variation of the

I

*

where Co is the compliance matrix of 0 at the point of contact ~

vector q of joint parameters, and BAI denotes

• Total displacement

any virtual displacement of ~ related to Oq . The relation (5) holds for the previous virtual parameters, i.e.:

I

1*

I

The complete displacement Do of the point ~ belonging to the object 0 is the sum of (16) and (17) which yields:

(21) Thus (16) becomes:

I [( F I)T J+S I (I)T] Bq.1*

(17)

BW=

When Cl is at implying:

the n vector relations (18) obtained for can be recast into the single one:

(lr

i=I, ... ,n,

Do = H d - CoF

which defines 3n scalar expressions of small displacements of all initial contact points ~ belonging to the object.

I

I

I

(S"S2 , S3)

T

(23)

q5

j=i, ... ,m, m securing forces

S5, 3n components

of the grasping forces FI and 6 parameters describing the solid movement of the object (components of the vectors 0 and Gc;' ). Thus, the problem includes 2m+3n+6 unknowns which are submitted to the conditions stated in section 2, namely: the 6-vector equilibrium (2), the 3n-vector continuity equation displacements relation (19) and the m-vector securing equation (23). Which represents a total number of m+3n+6 scalar equaiities. So, we have to resolve a linear algebraic system with m arbitrary unknowns. In order to select a unique solution we introduce a discriminating criterion based on the objective function defined in the next sub-section.

T

s = C F+JOq-H d I

T

The unknown quantities introduced in the previous i=i, ... n et section consist of: m joint parameters

Note : In reality sliding may be occur at contact points in any handling operation. So the formalisme developed above can be used to treat problems of sliding during grasping by

where S =

•

3. SELECTION CRITERION OF A SOLUTION 3. 1. Basic problem

(19)

•

1

(22)

T

Accounting for the expressions (10) and (18), the continuity of displacements yields:

1

=0

J F+S=O

For the sake of simplicity, the compliance term in (18) is not accounted for in the following.

introducing a sliding vector S = (S , ... , S ) given by :

1*

I

BW = 0 for any Oq ,

Setting S = (S , ••• ,S) and using the block matrices J and F defined in the previous section, the n vector relations (23) can be brought together into the single m-vector relation:

(18)

C F+JOq-H d=O

FI +SI

rest.

is the sliding vector at~.

2. 4. Securing equations The relation between the securing force or torque Si at the active joints of the kinematic chain Cl ,

3. 2. Objective function and complementary constraints

I

and the grasping force -F exerted by the fingertip on the object 0 can be easily established by using the principle of virtual works. The virtual work

Among previous unknowns it seems natural to I

consider securing forces SJ as control parameters of the kinematic chains Cl. These parameters can be optimized on the basis of a criterion to be minimized subject to all the stated constraints. Thus we introduce the quadratic objective function :

I

8W of all internal and external forces applied to Cl admits the simple expression :

I (I)T BAI*+ (I)T 1* S Oq,

8W = F

(20)

228

L n

1 • 1 FobJ = {arctan!.(S)-cp.}

without sliding, according to the value of the friction coefficient at the points of contact, and in accordance with the angle of incidence of the contact forces while securing.

(24)

1= 1

where /, (S) is a sub-function depends implicitly on SI and represents the proportion of the tangentiel force with respect to the normal force at contact point~ so:

f I

(S) = T(S)I N(S)

4. 2. Finger modelization Kinematic and elastic characteristics of the fingers are described by means of the block matrices J and C defined by (9) in section 2.3

(25)

4. 2. 1 Determining the matrix J:

I

The objective function defined above expresses the projection of grasping forces on friction cones at contact points.

The structure of the matrix J is defined by (9) with n=3. The sub matrix .P, defined by (4), simply becomes:

In order to avoid solutions which minimize F obJ

l

1

with values too great for SJ ' it may be necessary to bind these unknown quantities by introducing maximum values compatible with the capacities of the securing actuators. So, we add the m scalar constraints which define the bounds of actuator's capacities:

IS~I ~ S~-a

l-a is a given positive numerical

~~" a;;.. ~~

=[

where Cl·

J.

i=J.2.3.

rI is the single joint coordinate of the finger

(26)

where each SJ value. Summarizing all previous conditions, the problem we have to solve is a quadratic program which consists in finding, for i=J •...• n. j=J •...• mi. k=J.2.3. the 2m+3n+6 unknown parameters (q~ . S; . F; . O. , GG~) minimizing the objective function (24) and satisfying the equality constraints (2, 19, 23) and inequality constraint (26) . d

Note : the parameters Ok ' GG k describing the solid movement of the object defines assembling and positioning tolerances.in robotics operations.

4. APPLICATION TO A THREE-FINGERED GRIPPER

Fig. 2. Three-fmgered gripper with a translational fmger Cl and two rotational ones ~, ~ (not represented )

4. 1. Description of tbe gripper This gripper (see fig.2) was designed at L.M.S. Eacb finger has a single degree of freedom, so m 1 = m2 = m J = 1 . The fingers C2 and C3 can rotate about two parallel axis. The third finger Cl has a translatory movement along an axis which is orthogonal to the pervious ones. Since the fingers are independently driven by individual motors, it is possible to obtain various grasping configurations corresponding to the same points of contact.

finger trajectory

y

~.<

x

The geometry of the fingertips makes it possible to consider the contacts as punctual with friction. Their frictional surface permits friction, with or Fig. 3. Example of grasping

1 Laboratoire de Mecanique des Solides, Poitiers, France.

229

With the notations offig.3, one obtains:

the grasping configuration is impossible needing modification of the contact points on the poundary of the workpiece)

Y", =0, X-"l =rcosq2' Y-"l =b+rsinQ2' ZA2 =0

Data and results for nonsyrnmetrical grasping of a workpiece (see fig.5), are summarized in tables I-a and I-b. One can notice that the greatest object movement component is about 0.13 mm, and the greatest component of the rotation vector e is about 0.084 degrees. These rather small values are mainly due to the stiffness of the fingers whose joint parameters variations are very small during securing.

X~ = rcosQ3' Y~ = -b+rsinQ3' ZA3 =0

The values ZAi=O signify that the fingertips initially belong to the coordinate plane Z=O of the reference frame R After derivation the whole matrix J becomes:

[

il o

0

o

0 0

, l = ~ o

0

0

2

2

0

0

0

i4 is

0

0

0

0

0

0

3

3

i7 i8

where

Jf = J, J; = b - YA, ' Jf Ji = -(b + YA,J. Jt = X A,

~l

We can notice also that, the variation of grasping forces with respect to number of iterations (see fig. 6), throughout the first 100 iterations, the fluctuation of grasping forces corresponds in practice to the adjustement of actuators force and torques during the handling operation. After 100 iterations, we remark the stability of the grasping forces, which correspend to the optimal grasping forces necessary to handle the workpiece. Figure 7 shows the convergence of the objective function which corresponds to the projection of grasping forces on friction cones at contact points.

= X A, '

4. 2. 2. Compliance matrix et Each finger is considered as an angled elastic beam (see fig. 11). First, the compliance coefficients of the finger Ci are expressed on the local base associated with the local frame Ri. Explicit expression of such coefficients is given in the appendix. But the compliance matrix in relation (6) is related to the fixed base Ri associated with the global reference frame R So we have to express Ci by the matrix relation: C

liT

= (B)

i

CB

i

(27)

where C is the compliance matrix of the finge~ Ci expressed in the local base (see appendix), BI is the coordinate rotation matrix betwen R and Ri. 4. 3. Numerical simulatioDS For the three-fingered gripper under consideration, the set of constraints is made up of 18. The problem to solve consists of 21 unknown parameters, three of these being the securing force and torques to be minimized. The quadratic programming problem which results, is numerically resolved by using Box (1965) routine. We assume that three contact points were chosen on the grasped object. Following the study presented in (Barkat 1991; Markenskoff 1989) it is also assumed that the normals at the contact points are convergent. Nevertheless, friction introduction doesn't really require this assumption in order to ensure the feasibility of the grasp. A computationnel algorithm is given in fig. 4.

Fig.4. Algorithm

As above, (see fig. 8), tables 2-a and 2-b, illustrate

respectivly grasping configuration, data and results of a symmetrical piecweork. Figure 9 confirms the three egual grasping forces for the symmetrical configuration. Figure 10 represents the influence of the friction coefficient on the joint

Note: Algorithm non convergence occurs in the following situations: (i) if bound on securing force and torques are reached (heavy workpiece) (ii) if

230

parameter variations. The more the friction coefficient is small, The more securing is needed, The greater the joint parameter variation.

UJlI"'pUl~

forces F F

12.

11-.-..................................... .. .. ... ~.L .......

8. F

!

,,~J,; ~----------~- . 4.

JC

.ao.J\ \............

\. ........../ · .. ··········:·:~ .oo

O.

o

.. '

100

200

300

400

iteration I

Fig. 5. Nonsymmetrica1 grasping

Fig. 6. Grasaping force variations Fi(I)

Table 1. Optimal solution for grasj>ing

Fobj

a nonsymmetrical workpiece l-a-Data

2.

Contact point ooordinatcs [mml

OAt=( -.6634E+02, -.1210E-03, .OOOOE+OO) OA1 =( -.S247E+Ol, .4967E+02, .OOOOE+OO) OAJ =( -.3167E+02,-.1246E+02, .OOOOE+OO) Centre ofmasa ooordinatcs Imml OG=( -O.3216E+02, 0.12S3E+02, O.OE+OO)

L

Object weight: IIlI - 8 [NJ

o.

Normal vector compooents -before securingDJ=( -.9766E+OO, -.21S2E+OO, .OOOOE+OO) DZ=( .S384E+OO, .8427E+OO, .OOOOE+OO) .J=( .2493E-Ol, -.9997E+OO, .OOOOE+OO) Bound on securing force and torques Stmax: 8.00 IN I S2max :0.290 [Nm I S3max:0 .29O INm I Friction coefficients "I = 0.90,

o

100

Fig.7. Variation ofF

200

300 Iteration I

400

'(I)

Table 2. Optimal solution for grasj>ing a

symmetrical workpiece

"1 = 0.S89 , "J= 0.70

2-a-Data Contact point ooordinatcs

lnunl

OA1=(-.SS88E+02, O.OOOOOOO,.OOOOE+OO) OA1 =(-.3188E+02, .1386E+02,.OOOOE+OO) OAJ = (-.3188E+02,-.1386E+02,.OOOOE+OO)

I-b-Result!! 0ptima1 grasping forces expressed in the global frame

Centre ofmass ooordinatcs

[N]

it=( -.7934E+Ol, .4147E+Ol, -.2227E+Ol) fj= ( .lS82E+Ol, .S422E+Ol, -.2772E+Ol) IJ = ( .63S2E+Ol, -.9S69E+Ol, -.3002E+Ol) Cattle ofrrr- diJplacancot [-I GG =( .1292E+OO, .6S87E-Ol , -.9991E-(2)

lnunl

OG=(-.3988E+02, 0.0000000, O.OOOOE+OO) Object weigbt: l1li -8 [N]

Normal vector compooents -before securingDJ=(-O.l000E+Ol, O.OOOOE, O.OOOOE+OO) D~ O.SOOOE+OO, .8660E+OO, O.OOOOE+OO) .J=( O.SOOOE+OO,-.8660E+OO, O.OOOOE+OO)

Rotation of the object [n111 e = ( .1466E-02, .10SlE-02, .4906E-(3) Joint parameter variatiOOl while securing

Bound on securing force and torques Slmax : 10 [N I S2max : 0.lS0 [Nm I

[DIIII, ..... n111 .2027E+OO, -.2034E-02, -.8833E-(2) 0ptima1 securing force and torques SI: 7.9340 [N I S2: 0.7861[Nml

S3max: 0.lS0 [Nm I

~9:

Friction coefficients "I =0.8391, I'l =0.8391 , "J=O.8391

S3 : -0.268 INml

231

reduced friction angles
2-b-Results 0ptima1 grasping forces expressed in the global frame

[NI fl=( .3178E+{)1. -. 1676E-Q4•. 2667E+{)l) fj=( .3178E+{)1 •. 9218E-OS •. 2667E+{)l) h = ( .3178E+{)1 • .7S44E-OS • .2667E+{)l) Centre ofmass displaccmcnt [mml GGd =( .4092E-Ol. -.S701E-06. -.1077E-Ol) Rotatioo of the object [radl 9 = (-.S383E-08 • .8241E-03. -.2323E-07)

occur.

5_ CONCLUSION In the present work, the problem of hyperstatic grasping is dealt with, taking into account the finger deformability, frictional contacts and joint parameters variations constraints. The approach we have developed, is based on the introduction of a securing force minimization criterion and leads to a formulation which is perfectly capable of mastering the determination of interaction forces at finger joints and on the object. The formulated quadratic prograrruning problem accepts a large range of constraints which can be easily adapted to the specifications of the object and the gripper. In a similar way, an optimal grasp, as defmed in this paper, can be made stable simply by recomputing the optimal solution with a slightly decreased angle of cone friction.

Joint parameter variations while securing

[mm, racl, radl oq=(0.6603E-Ol. 0.1996E-02. -0. 1996E-02) Optimal securing force and torques 8 1:3.17790

[N I

82 :0.081211 [Nml 8 3 :-0.081211 [Nml

y

..... / 40.00

;t

0.30-

•

"", .. ;:40,00 ' ....

\

\

•"

\

Fig. 8. symmetrical grasping

0.20-

"~,

'-. .. '

'

0.10-

...........

..-.... . ........ -

8. 0.00 L....-.....--.----r-r-.....--.---r-r--.....--rII ... 1 6.20 0'.40 0 .60 6.80 Friction coefficient f.1

1'.00

6.

Fig. 10. Variation of joint parameter Oql with respect to friction coefficient for grasping a symmetrical worlcpiece

4.

2.

6. FUTURE WORKS In futur works we will take into account the 3D sliding variables in relation (19), friction constraints and sliding conditions related to this kind of problem. We will also formulate another cost function and use a computational algorithm for optomization with nonlinear constraints to find the optimal securing and grasping forces.

Iteration I Fig. 9. Grasping forces variation F

For the approach presented, it must be emphasized that the computed optimal solution yields grasping forces which are at the sliding limit for most of the contact points, if not all of them (i.e. grasping forces on friction cones). This is true for the examples presented. In order to ensure the stability of the grasp, a new solution can be computed with the previous solution as a guess solution, and with

7. ACKNOWLEDGMENTS The authors would like to thank Dr. S. Zegloul, Dr. G. Alfonso Pamanes and Dr. G. Bessonnet for Their help and suggestions.

232

8. REFERENCES

fingrertip prehension grasp, ASME PUB 153,125-138.

Baker, S., Fortune, S. and Crosse, E. (1985). Stable prehension with three fingers. Proc. Symp. Theory of computing. Barkat B et Lallemand, 1. P. et Vuliez P. (1991). Developpement d'un prehenseur tridigital configurable, approche de la prise optimale. loem congres Francais de Mecanique , Paris 26 sept. Barkat B et Lallemand, 1. P. (1993). Detennination d'une relation de compatibilite et optimisation de la prise plane par trois points vis cl vis d'une perturbation exterieure. ll em congres Francais de Mecanique , Lille 611 sept., 365-368 Bologni, L. (1988). Robotic graspins: How to detennine contact positions IFAC Robot control Karlsruhe. FRG, 395-400. Box ,M. 1.,(1965). A new methode of constrained optimization and a comparaison with other methods. Computer Journal, 8, 42-52 . Demmel, 1. and LafIerriere, G. (1989). Optimal three finger grasps Proc. IEEE Int. Con! on Robotics and Automation. Hanafussa, H. Asada. H. (1977). Stable prehension by Robot Hand with elastic fingres . Proc. of 7th intoSymp. on Industrial Robots. 361-368. Kerr, ] and Roth, B. (1986). Analysis of multifingered hands » Int. Journal of robotics research. 4-4,3-17. Liangyi , D. Kohli (1983). Analyse of conditions of stable prehension of a robot hand with elastic fingers. Proc. of intelegence and productivity con! Markenscoff, X. and Papadimitriou, C. H. (1989). Optimum grip of a polygon Int. Journal of robotics research. 8-2, 17-29. Markenscoff, x., Luquin, N. and Papadimitriou. C. H. (1990) The geometry of grasping. Int. Journal ofrobotics research. 9-1,61-74. Mishra, B. and Naoumi, S. (1989). Some discution of static gripping and its stability. IEEE Transactions on Cystems Man. and Cyber.,194, 783-796. Nguyen, N (1989). Constructing stable grasp. Int. Journal ofrobotics research. 8-1,26-37. Timoshenko, N. (1953). Resistance des materiaux. Paris et Liege librairie polytechnique ch. Beranger. Tsuneo Y. and Kiyoshi, N. (1988). Evaluation of grasping force for multifingered hand. Proc. IEEE Int. Con! on Robotics andAutomation. WoIter, 1. D., Voltz, R. A. and Woo, A. C. (1985). Automatic generation of gripping positions. Proc. IEEE Int. Con! on Robotics and Automation, 204-213 . Zhiming, 1. and Roth, B. (1988). Direct computation of grasping force for three-

9. APPENDIX Compliance coefficients of the finfers The finger geometry is described (see fig. 11 ). Each finger is considered as an angled elastic beam. The compliance coefficients are expressed with resPect to the followi~ constants: S : finger section S =ab, E : Young's modulus, G : shear modolus, I :quadratic moment of a

rectangular section,

3

1= ab /12, 3

10: polar quadratic moment, 10 = [Jab (jJ=1 for a square section), where j3 is a numerical coefficient that depends on (a/b) .

....... <

Rigid part

Fig. 11 . Finger Geometry

The components of the compliance matrix C are easily formulated by means of the energetic method [TIM 53] . One obtains: 'I

h] (h

i Cl I

= ES + El

C;2

=

o.

=

hr/ El

i

CH

"3 + 'I

)

h + h GS •

C;I = 0 ci = 22

h 3 +'1 3 h2 'I h 'I +--+ 3El Glo GS

C;3 =0 i

h'i]

c3J = 2E! C~2 = 0 3

h '1 h - 'I ci =-+--+ 33 ES 3E1 GS

233