The analysis of asymptotical stability of robotic hand grasping

M¢ck. Mack. TIteoryVol. 29. No. 5. pp. 635--651. 1994 Col~,riilht ~: 1994 Ekevier ScienceLtd Printed in C~remBritain. &ll rights tx.served 0094-I 14X/94 $7.00 + 0.00

Pergamon

THE ANALYSIS OF ASYMPTOTICAL STABILITY OF ROBOTIC HAND GRASPING Z H E N LU Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R. China

(Receit,ed 29 March 1993: received for publication 9 July 1993)

Abstract--This paper discusses the effect of grasp impedance of robotic hands in the asymptotical stability of the grasp in robotic hand. The factors associated with the asymptotical stability include the position feedback gains and velocity feedback gains, grasp geometry and structural impedance. The relationships between the grasp impedance and the grasp geometry are analyzed in this paper. This paper develops the expressions for the potential energy, kinetic energy and dissipation function in a grasp. Based on these results, the disturbance equation is introduced. Using the disturbance equation, this paper addresses the modal analysis of the hand-object system. The result of the analysis can he used for the optimization of some characteristic parameters to set ideal asymptotical stability.

1. INTRODUCTION There are a variety of criterions that might be considered in analysis and/or planning a grasp. The common criterion is the impedance of the grasping, which is the function of the structural impedance (including the impedance in the fingers, and transmission elements), and the grasp geometry. Meanwhile, the impedance of the grasping also depends on the position feedback gains and velocity feedback gains in finger joints. When moving or manipulating the object at high speeds, a stiff grasp might be desirable, i.e. for position servo, the infinite stiffness is desired. In this case, the grasping system can reject any force disturbances acting on it. It ensures the displacements due to inertial forces (torques) will be small. For force servo, zero stiffness is suitable. The system can maintain desired forces applied. In the actual systems we usually control the stiffness to be nonzero finite values. More generally, from the disturbance rejection point of view, it is important to control the impedance of the system in order to get desired grasp stability. Generally speaking, grasp stability has at least two meanings [I, 2]. On one hand, it refers to the ability of a grasp to return the object to its static equilibrium position or desired motion state when the object is perturbed by a disturbance. This can be named "object stability", which is the target of numerous research efforts [!-5]. On the other hand, it is also the capability to maintain contact between fingertips and the surface of the grasped object, when the object is perturbed by some disturbance wrench. This is often termed "contact stability", which raised the interest in the internal grasp forces [2, 3, 5]. Most of the researches on the robotic hands only concerned the stiffness in the grasping, and neglected the affect of viscous damping and/or Coulomb friction in the joints and transmission loops. The damping and friction have no affect to grasp stability according to the theory of stability [7], but this can not be ignored in the problems associated with asymptotical stability. Meanwhile, the statical study on the stability usually is not satisfactory. We have to research the dynamic stability further. According to the stability theory, if the generalized damping matrix and the generalized stiffness matrix are both positive definite, the system is asymptotical stable. This paper first analyses two sets of linear relationships between the result force and result moment (or external forces and moment), the contact forces, and the finger's joint torques (or forces), and their dual in velocity domain. Secondly, it develops the general expressions for grasp impedance matrices, potential energy, kinetic energy, and dissipation function of the grasp. Based on these results, the disturbance equation is introduced. Using the disturbance equation, this paper addresses the model analysis of the hand-object system. 635

636

ZHEN LU

As shown in Fig. !, an object grasped by several multi-joint fingers is in its equilibrium position. When a disturbance wrench is exerted on, it will have a small displacement. If the grasp is stable, the fingers will maintain contact with the object when the object displaces, and the object will return to its equilibrium position when the disturbance wrench is removed. However, in order to perform the motion of the object as quickly as possible with acceptably small residual vibration, there are still some close relative problems to be solved, i.e. how to reduce the residual vibration amplitudes in the hand-object system, in short, how to obtain the ideal asymptotical stability, and how to control the vibration of the object within an allowable frequency bandwidth. These requests are more strict than general stability. The generalized stiffness matrix and generalized damping matrix not only depend on the control gains but also are the functions of the grasp geometry. One can control the position servo gains and velocity servo gains to get desired asymptotical stability of the object. We will analyze this problem in sequence papers. Nevertheless, there is another factor, i.e. the grasp geometry, which effects the asymptotical stability, and therefore we have an alternative approach to solve the problems by means of rearrangement the contact points about the object. To keep the analysis tractable, we make the following assumptions: (I) The contacts between the object and the fingertips are all point contacts with idealized friction (i.e. Coulomb friction). (2) There are neither sliding nor rolling of the fingertips. (3) The stiffness and damping matrices are constant matrices during the small displacement of the grasped object.

2. R E L A T I O N S H I P S IN THE G R A S P I N G As depicted in Fig. 2, a hard frictional fingertip contact grasped an object at point Pi. The centroid of the object is b. The radius vector from b to Pi is Pi. If the contact force of fingertip Pi is fi, the result force and moment of all contact forces are F-f,+fz+...+f

s,

(i)

M = P, x t"1 + " " + Ps x fs,

(2)

where k is the number of the contact points. F is the result force exerted at point b, and M is the result moment about point b• The component form of equations (I) and (2) in a Cartesian frame is

"M M2, I

""

F=

M3

_ {IPI|

-

F,

.

" .....

"f'flf13 2

[Pk]~

.

= [W]_f,

(3)

[E,],/

t=2

f,2

F3

f*3

where M 1, M 2, M 3 are the components of M, and/'1, F2,/=3 are the components of F, and fl,, f2,, f3, are the components of fo in the frame respectively. [P,] is the cross multiple matrix associated with the component column matrix of Pi i.e. [P,]=

( O p,:

-p,: 0

-p,,]

p,.,. ~

X --,o,.,.

p~,

o/

and

p,=(p,.,,p,,.,p,:)T.

[Es] is a 3 x 3 unit matrix. [W] is well known as the grasp matrix. The second relationship is between the contact forces and a]! finger joint torques, i.e. ~_= [ j r]f

(4)

where _~ is composed of all joint torques, and [jr] is the transpose of Jacobian matrix between contact points and joints, i.e. [J].

Stability of robot hand grasping

637

Similarly, there are also two sets of linear relationships among the generalized velocities of the object, the contact point velocities, and the joint velocities. They are

v_.,,= [ w ' l _ ~

(5)

_vp = [ j ] ~

(6)

and

in equations (5) and (6), V~ is the column matrix of contact point velocities, V_~is the generalized velocity column matrix of the object, ¢o is composed of all joint velocities. The two sets of linear relationship expressed in equations (5) and (6) are also valid for small displacements. Let q b¢ the small displacement column matrix of the object, _d the column matrix of the small displacements of all contact points, and q be the column matrix of small joint displacements. We have d = [WT]q,

(7)

LJ= [J]_0.

(8)

Generally speaking, the over constraints are desired for stable grasping. The matrix [W] has more columns than rows. The contact force _f can be divided into two parts, i.e.

(9)

_f = f_,+ f_,.

where, f, lies in the column space of [W] r, f_. lies in the nullspace of [W]. If [W] is full rank, i.e. the rank of [W] is equal to six, we have fr = [W]'_F,

(10)

where [W] + is the right inverse of, i.e. [W] [ W ] ' = [E6] ([E6] is a 6 x 6 unit matrix), or [ w ] TM r w y ( r w ] [w:]')-'. From equations (I) and (8), we have _F = [W]_f = [W]f,+

[W]f..

(I l)

According to the definition of null space, [W]f_.=O, so _F= [W]f,.

(12)

On the other hand, F is composed of two parts F =Fo+ Fq,

(13)

where ,Co is used to balance the known external wrench, that includes the gravity of the object and other known forces, and Fq is caused by the disturbance wrench in our situation. Correspondingly,

Fig. I.

638

ZXENLu

Fig. 2. _/, can also be divided into two parts, i.e. ~ = ~ + ~ , where ~ and ~ are the homogeneous forces of F__oand ~ respectively. So, we have

_F¢= [W]f¢.

(14)

Now, let us discuss the relationship between the contact force combination and the joint torques, and the relationship between the result forces and the joint torques. From equations (3) and (9), we can get

~_= [J]rCW ] "F_ + [j]rf,,.

(15)

Generally, [J] will not be square, and there will be more columns than rows. It means there will be some redundant freedom in finger joints. From equation (3), we can see that ~ must lie in the column space o f [j]r. If so, we have

_f= [ j ] t + : ~ , where [j]r. is the left inverse of [ j ] r , i.e. [J]r'[J]r=[E] ([El is a unit matrix), or [j]r. = [ [ j ] [ j ] r ] - t [ j ] . From linear algebra, we know that the left inverse of [ j ] r is equal to the transpose of the right inverse of [J], i.e. [j]r. = [ [ j ] [ j ] r ] - l j = [ j ] . t , therefore _f= [J] "rL

(16)

Combining equations (I) and (14). we have F = [ W ] [J] "'~

(17)

If we define the matrix [G] = [W] [ J ] ' T we have following formulas:

F=[G]~,

(]8)

~ = [G]rV_o,

(19)

0_= [G]rq.

(20)

For example a cylindric object with radius r is grasped by three hard frictional fingers, i.e. A, B, C (Fig. 3). The position column matrices of the three contact points in body-fixed frame bxyz are

P_, = [0, O, - r ] L P-s = [ - r cos =s, O, r sin % ] T P-c = [ - r cos =c, O, r sin ~,c] T.

The grasp matrix is

[w] =

0

r

0

0

-r

0

0

0

0

I

0

0 0

0 0

I 0 0 I

- r sin ~s

0

0

- r sin ~'c

0

rsin~s

0

•cOS=s

0 I

- r cos ~s 0

0 0

rsin%

0

rcos=c

0 I

r cos ~s 0

0 0

0

I

0

0

I

0

0

0

I

0

0

I

Stability of robot hand grasping

639

The internal grasp forces are

f~ = [0, 0, fA., f h COS~a, 0, --f=, sin =a, fc. cos ~'c, O, - f c . sin ~c] r. From static equilibrium condition, we have f b cos == + fc. cos =c = 0 f A. - f=, sin =s - fc. sin ~c = 0. It is easy to find that

[w]f_. = 0_Q. 3. THE S T I F F N E S S M A T R I C E S The stiffness matrices in object space, contact point space, and the finger joint space are defined as

[ r ] = [aF/~q].

(2 l)

[g,] = [a_f/ad].

(22)

[r,] = [a~ lae],

(23)

respectively. If _F, f_, and :~ are potential, in the neighboring region of the equilibrium position, we have = [K]~z,

(24)

f_,= [K,]_d,

(25)

~8 = [g,]~,

(26)

where ~ is the joint torque increments due to the disturbance. Substituting equation (25) into equation (14), we have ~ = [W]K_.,d. From equation (8), we get

F_,= [W] [K,] [W]'q. Comparing the equality with equation (24), we find [ r ] = [W] [K,] [W']

(27)

For stable grasping, [K] must bc positive definite as mentioned by Mason and Salisbury [5]. In order to keep [K] positive definite, one must guarantee [K,] be positive definite and [W] be full rank. if there are at least three contact points not located in the same line, [W] must be full rank. In

Y~

r---~

Fig. 3.

t z

640

Zme,~ Lu

grasping convex object, this is always true. Similarly, if we substitute equation (26) into equation (18) and consider equations (20) and (24), we get [K] = [G] [K,] [Gt].

(28)

For stabZe grasping, the conditions required to finger joints are more complicated than that required to finger contact points. To maintain [K] be positive definite, we must ensure the positive definiteness of [K,] and the full rank of[G]. Because [G] is the product of [W] by [j]+T, the rank of [G] will not be greater than the minimum value of the ranks of [W] and [d]. The hand-object system is a multi-loop chain, so there will be the possibility, that some fingers have more joint freedom than that necessary for contact point motion, and another finger joints have less joints than that necessary for contact point motion. Even the total number of the finger joints is equal to that necessary for all contact point motion, the rank of [d] will be uncertainly full rank. Furthermore, in some singular configurations of the fingers, [d] also probably lose some ranks. 4. THE DAMPING MATRICES

The damping matrices in the object space, the contact point space, and the finger joint space arc defined as [C] = [@_F#/@_V0],

(29)

[C,] = [@f_c/@Vj],

(30)

[c,] = [~Olaw_],

(3 l)

respectively, if _Fc, _fc and ~c are dissipation forces. In the neighboring region of the equilibrium position of the object, there will be three linear expressions for them, i.e. F c= [C]V0,

(32)

f.c= [C,]V,,

(33)

r ' = [C,]~.

(34)

Substituting equation (33) into equation (12), and considering equation (6), we obtain

F ' = [W] [C,] [WyVo. Comparing the equality with equation (32), we find [C] = [ W ] [C,] [ W ] ~,

(35)

We can find analogous result about [C,], [C] = [ G ] [C,] [G] T.

(36)

When [C] is positive definite, the hand-object system is termed complete dissipation system, and [C] is semipositive definite, the system is an incomplete dissipation. The complete dissipation system is asymptotically stable. For asymptotically stable grasping, the conditions are the positive definiteness of the damping matrices, and both [W] and [G] have full ranks. 5. THE INERTIA MATRIX

For a multi bodyt system, the inertia matrix is [9] [/l = (Ii/)

(i, / = I, 2 . . . . .

6),

where li] = lfi = ~. [m,(@r,/@q,) • (c=r,/@q~) + (co,/d',) * Jk * (c~,/~i,)], tHere, the term "body" refers to the grasped object or the links of the fingers.

(37)

Stability of robot hand grasping

641

where r, is the radius vector of the centroid of the body K in the inertia frame OXYZ (i.e. the global frame fixed in the palm of the robotic hand), co, the angle velocity vector of the body k, in, the mass of body k, Jk the center inertia tensor of body k, _qand q = V_0are the generalized coordinate column matrix and generalized velocity column matrices. In our case, they are the object displacement and velocity respectively. When grasping a relatively big object, the inertia of the fingers can be ignored. However, when grasping a small object, the effect of the finger inertia must be considered. [/], [K], [C] are the impedance matrices of the grasping in the object space. They directly determine the asymptoticai stability of the grasping. 6. KINETIC ENERGY OF THE DISTURBANCE The kinetic energy of the disturbed hand-object system is T = 0.5 *

_qr [l]_q

(38)

In small displacement, we have

OT/~q= 0

(39)

d(aT/~q)/dt = [l]_q = [I]oq

(40)

where [/]o is the inertia matrix of the hand-object system at equilibrium position O,

7. THE POTENTIAL ENERGY OF THE GRASP Because the generalized bias forces E0 are used to balance the gravity on the object, the work done by ~ will offset by the work done by the gravity. Since only small displacement are envisaged, the potential energy of the object U will be expanded in a Taylor series about the equilibrium position as U = Uo+ (VU)o_q + 0.5qT[H]oq + ' ' ' ,

(41)

where [H] = [V2U(q)] is the Hessian matrix of the potential energy. Because ~ = (VU) and [K] = -[OE.#/0q], we have [K] = [VeU(q)]. The stiffness matrix [K] is the Hessian matrix of the potential energy. In equation (41) the derivatives have been given the values at equilibrium position 0. The constant U0 plays no useful role in future equations because only changes in U have significance, and will therefore be omitted. As position 0 is the equilibrium position, i.e. (VU)o = 0, so that there are no terms in U which are linear in the displacements. For sufficiently small displacements, only the quadratic terms in the Taylor series need be retained, terms of higher degree being negligible. The potential energy of the object is U = 0.5qT[K]q.

(42)

8. RAYLEIGH'S DISSIPATION FUNCTION OF THE GRASPING Considering the viscous friction forces in joints and transmission mechanisms and the velocity gains, we apply the Rayleigh's dissipation function to the grasp, R = 0.5~'[C]q_',

(43)

where [C] = [V2R (q)] is the damping matrix in the object space, which is real symmetric matrix. When [C] is positivedefinite,the grasp system is termed complete dissipationsystem, and when [C] is semipositivedefinite,the system is an incomplete dissipationsystem. 9. DISTURBANCE EQUATION To formulate the disturbance equation, we substitute the expressions about T. U, R into second Lagrangian equation d(t?T /~q_) / d t + ~U / ~c/ + ~R /aq_ - aT /~q_ = 0

(44)

642

Z~N Lu

and rearrange it. We have [/]~ + [C]_q + [K]_q = 0

(45)

where [/], [C], [K] are constant matrices under small displacement.

10. MODAL ANALYSIS AND SOLVING OF THE DISTURBANCE EQUATION Using the differential operator D = d/dt. equation (45) can be written as ( [ / ] 0 2 + [C]D + K)q -- 0,

(46)

where ([/]DZ + [C]D + K) is a linear differential operator. Assume the solution of equations (45) or (46) is of the form

q = p e ~t.

(47)

Substituting it in equation (43) and factoring out e~', we have ([/]22 + [C]~. + [K])~ = 0_.

(48)

The compatibility condition of equation (48) is A = I[I],~ 2 + [C]~. + [K]I -- 0,

(49)

which is an algebraic equation of 12th degree in R. Let us assume that the roots of equation (49) are distinct. From equation (48), appropriate to each roots ;t~, we have a modal column matrix/~ -- [Pu, P~ . . . . . p~]r, i -- I, 2 . . . . . 12. There are 12 such modal column matrices corresponding to the 12 roots J.'s. Combining the modal column matrices ~ to obtain a 6 x 12 modal matrix [p]. The solution of equation (45) can be expressed as

(50) where, [e ~'] --diag[e ~'', e ~2'. . . . . ea2rJ, and c = [cl, c2 . . . . . c!2] T is a column constant matrix. Differentiating equation (50) with respect to time, we have q -- [p][~][e~']c,

(51)

[2] = diag[~.t, ~.2. . . . . 2,2].

(52)

where

Let [A] = [p][~.] E R 6x ,2, equation (51)can be written as

//=

[A] [ea].~.

(53)

Combining equations (50) and (53), we have

/

\[A]/

Given the initial conditions q(0) = q0 and _q(0) = "_qo,and noticing that [e~1 -- [E,2 ] at t = 0 ([E,2] is a 12 x 12 unit matrix), we can evaluate the constant column matric c from

c

\[al}

""

I I. THE STRATEGY OF THE PLANNING The goal of the optimization is to improve the asymptotical stability of the grasp by redistribution of the contact points in the surface of the object grasped. If a system is stable and lightly damped, the roots ,t, of equation (45) are complex with negative real parts. It is evident that the disturbance motions can be expressed as sine and cosine functions with amplitudes diminishing exponentially.

Stability of robot hand grasping

643

Let

~zp-, = -sp +oapi 2zp = -sp - mpi

(p = 1,2 . . . . . 6).

(56)

The time constants arc defined as

rp= l /sp

(57)

(p = 1,2 . . . . . 6).

I I.I. Objective function

The better asymptotical stability of the grasp can be achieved by increase the value of the sp. From equation (56)

sp = -0.5(k=p_ t + X~).

(58)

We define the objective function as 6

F(P,,P2 . . . . . P , ) = - ~ opsp,

(59)

where os are the right factors, which control the vibration of the object within the favorite frequency bandwidth. 11.2. Constraints

The contact points must lie on the surface of the object grasped. Surface equation 5 (x, y, z) = 0 of the object can be accepted as equality constraints, i.e.

G,(x,y,z)=S=O

( i = 1 , 2 . . . . . k)

(60)

To keep the contact stability, the normal contact forces must be exerted towards the surface of the object, i.e.

G..(x,y,z)=g..-go
( i = 1 , 2 . . . . . k),

(6t)

where

ga= (0, 0, I)[C~.]_f,,

(62)

g0~=(0, 0, I)[Cm.] ( ~ + f-.0)

(63)

and

is the bias forces in direction ni. Thus

G..,(x, y, z) = (0, 0, 1 ) [C~.] ([K,]d), < 0

(i = 1,2 . . . . . k).

(64)

In equations (62)-(64). [ C ~ ] is the direction cosine matrix from frame bxyz to the local contact frame with origin Pi. No sliding assumption requests

Gz,,.i Ig,illlg.I -/<

O,

(65)

G=..= {g,mlllg,.I -/<

O,

(66)

=

where f is static friction coefficient. 12. ILLUSTRATIVE EXAMPLE The hand mechanism is composed of three open chains (see Fig. 4). One of them is shown in Fig. 5. According to Denavit-Hartenberg notation, the structure parameters and the joint variables are listed in Table I.

644

Zh~N Lu

ZA0

Zo

u

Fig. 4. The object grasped by three fingers.

The transformation matrix are: rcos 0, 0

sin 0,

a, cos 0~ a, sin0,

!

0

0

0

0

I

,T,]= lsiiO. 0-cosO,

[

cos0~ sii03

['r3] =

-sin03 cos03

cos02 sii02

I

-sin02 cos 02 0 0

0 a3cos033] 0 a~sin

0

I

0

0

0

1

Thus. we can obtain cos 0:~ - sin 02~ 0 sin 0. cos 023 0 0 0 I 0 0 0

[' T3] =

0~ cos 02 r[ cos sin 0~ cos 02 [°T2] = L

Sio02

COS 0~ COS 02~

[°T3] =

sin 01 cos 0. sin 02~ 0

a3cos 023-k a 2cos 02

a3 sin 023 + a2 sin 02 0

1

-cos 0, sin 02 sin01 -sin0, sin02 -cos0, cos 02 0 0 0

Hcos0s 1 Hsin01 a2 sin 02 I

-cos 0, sin 023 sin 0~ U cos 0~ - s i n 0, sin 023 -cos 0~ U sin 0~ cos 02j 0 V 0 0 I

0 a2cos02 0 a2sin02 l 0 0 i

Stability of robot hand grasping

645

ZAO oo,

•--

.3./

XAI ~

ZA2

--- XA3

XA2

Fig. 5. Finger mechanism.

where U = a 3 cos 0,3 + a z cos 02 + a,,

V = a~ sin 023 + a2 sin 0:,

H = a2 cos 0z + a,, 023 = 0~ + 05. The contact point, say A, has coordinate column matrix in frame

AoXAoYAoZAo

(A_)Ao = [U cos 0,. U~.o,. V] T. The absolute velocity of point A has its component column matrix in frame

(A_),o

=

AoXAoYAoZAo

Ida] [0,. 02.03]~.

where [ - U sin 0.

[JA] = i

Uc;sO,

- (a3 sin 0,5 cos 0, + az sin 02 cos 0, ) - (a3 sin 0z5 sin 0j + a2 sin 0: sin 0, )

- a3 sin 023 cos O, \

a3 cos 02~ + a 2 cos 02

a 3 cos 023

-a3 sin 023 sin 01 J •

Similarly, we can obtain

Table I I 2 3

• ,_,.,

a,

~

O,

90' O' 0~

a,

0

Oi

a:

0

a~

0

0z O~

A

Z~EN Lu

646 and /-U

sin 0,

[ J ' ] = ~ UoSO,

-

(a3 sin 02~ cos 0, + a2 sin 02 cos 0, ) (a3 sin 023 sin 0~ + a2 sin 0z cos 0r ) a 3 COS 023 + a 2 COS 02

- U sin 0,

[Jc]=

U cosOr 0

- (a 3 sin 023 cos Oj + az sin 02 cos Or ) - ( a 3 s i n 023 sin Oi + a2 sin Oz sin 01 ) a3 cos Oz3 + a2 cos Oz

The components o f the contact point velocities in frame

-- a3 sin O~ cos Oj - a3 sin 023 sin 0, ] • a3 cos O. /. --a3 sin 023 cos 0t --a3 sin 0z3 sin 0 , / • a3 cos 023

/c

OXYZ are

(A--)To= [AoAo [JA] [0., 02, 03]~, (_B)o' = [Ao0o[Je] [0,, #2, #3g,

(C )~ = [Aoco [Jc] [Or, 02, 03]Tc, where [AoAo], [,4oo0], [Aoco] are the direction cosine matrices from frames AoX,~oYAoZAo, BoX0o Y0oZ0o, CoXco YcoZco to frame OXYZ respectively. Let 01=O,A,

03=0~,

02=02A,

O, = Ola, 05 = 02s, 06 = O~a, 07 = O,c, Os= 02c, O~= 03c. We have

Vp = (J](o, where [3] = diag[AoA0] [JA], [,'400o] [Je], [Aoc0] [Jc], o~ = [0,. 02 .....

0,1'.

The nonzeros elements of matrix [G] are:

6(I, 3) = ,fir/a G(I, 4) = - x / ~ sin an cos

aar/a

G(I, 6) = v/2 sin z anr/a G(I. 7) = - x/~ sin ac cos

acr/a

G(I. 9) = x/~ sin 2 acr/a G(2. i) = (2 -

x//2)r/a

G(2.4) = (2 - x/~) - 3 cos' G(2, 5) = -cos a8

=nr/a

r/a

G(2, 6) = 3 sin an cos =a

r/a

G(2, 7) = [(2 - v/2) - 3 cos = %]r/a G(2, 8) = -cos

=cr/a

G(2, 9) = 3 sin =c cos G (3.4) = - ~

acr/a

cos 2 a8

r/a

G(3.6) = x/~ sin a8 cos a 8 r/a G(3.7) = - ~

cos 2 acr/a

G(3, 9) = x/~ sin ac cos

,,cr/a

647

Stability of robot hand grasping

G (4, i) = - (2 - ~/2)/a G(4, 4) = (2 - , ~ ) s i n =m/a G(4, 6) = (2 - ~/2)cos =s/a

G(4. 7) = (2 - ~/2)sin =c/a 6(4,

9) =

(2

-,/2)cos =c/a

G(5, 3) = , / 2 / a G(5, 4) == ~

cos =s/a

G(5, 6) = -x/r2 sin us/a G(5, 7) --- ,/2 cos =c/a G(5, 9) = - V ' ~ sin =c/a

G (6. 2) = - !/a G(6, 3) = - ( I + ,v/2)/a

G(6, 4) = - (I + ~/2)¢os =s/a G(6, 5) = - l/a G(6, 6) = (I + x/~)sin aa/a G(6, 7) = - ( I + ~/2)cos =c/a G(6, 8) =

-

I/a

G(6. 9) -- (I + v/2)sin =c/a. Using equations (28) and (36), we can get the matrices [K] and [C] as following: K(I, I) = 2(k3 + sin 2 % cos 2 % k4 + sin 4 as k6 + sin 2 = cos' ~,ck7 + sin 4 ack9)r2/a 2 K(I. 2) = (sin =e cos =,(~/~ + 2 - 3v/2 sin 2 as)k4 + 3v/2 sin' % cos a, k6 + (~/~ + 2 - 3~r2 sin 2 %)sin a¢ cos ack 7 + 3v/2 sin 3 % cos ack9)r2/a 2 K ( I , 3) = 2(sin % cos 3 aek4 -t- sin 3 0¢ecos % k 6 + sin a¢ cos 3 =ok 7 + sin 3 0¢c cos K ( I , 4) = 2(I - .~r2)(sin2 =e cos % k 4 + sin 2 % cos

~ck9)r2/a 2

%k6

+ sin2 =c cos =ck7 + sin 2 ac cos ack9)r/a 2 K ( I , 5) = 2(K3 - sin a8 cosZ =8 k4 - sin3 =e k6 - sin =c cos2 =c k 7 - sin 3 =ck9)r/a z

K(I. 6) = ( - ( I + ~r2)k 3 + (V/2 + 2)sin =8 cos2 ask4 + (v/2 + 2)sin' % k 6 + (.v/~ + 2)sin =c cos' =ck7 + (V/2 + 2)sin 3 =ck9)r/a = K(2, I) -- (sin =a cos =,(v/2 + 2 - 3v/2 sin 2 =re)k4 + 3v/2 sin 3 =a cos =ak6

+ (v/2 + 2 - 3V/2 sin' =c)sin =cCOS =ck 7 + 3v/2 sin 3 ac cos' =ck9)rZ/a z K(2. 2) = (2(3 - 2V/2)k I + (9 sin' =m - 6(v/2 + l)sin 2 =e + 3 + 2V/2)k4 + cos aek5 + 9 sin z =e cos z =ink6 + (9 sin' =c - 6(v/2 + I) sin2 =c + 3 + 2~/~)k 7 + cos ==ck8

+ 9 sin = =c cos= =ck9)r2/a= K(2, 3) = 2((3~/2 sin' =, - 4v/2 sin' =, + v/2 + 2 cos' = , ) k 4 + 3v/2 sin =8 cos 2 =ek6

+ (3v/2 sin' =¢ - 4V/2 sin 2 =c + ~/2 + 2 cos 2 =c)k 7 + 3../~ sin 2 =c cos' =ck9)r'/a"

648

Z ~ N Lu

K(2. 4) = (2(2x/~ - 3)k I + (3(2 - ../~)sin' % - x/~)sin ~sk4 + 3(x/~ sin" :Is - x/~ + 2 cos' %)sin ~,sk6 + (3(2 - x/~)sin' :cc - x//2)sin %k 7 + 3(x/~ sin' ac - x/~ + 2 cos: ~c)sin ack9)r/a' K(2, 5) = (cos as (3v/2 sin" as - x/~ - 2)k4 - 3x/~ sin z as cos % k 6 + cos ac(3x/~ sin: ~c - V/~ - 2)k7 - 3x/~ sin: ac cos =ck9)r/a z K(2, 6) = ((3 + 2x/~)cos 3 a s k 4 + 3(v/2 + I)sin' as cos a s k 6 + (3 + 2x/~)cos' ack7 + 3(x/~ + I)sin' ac cos ack9)r/a z K(3, I) = 2(sin ~8 cos~ aak4 + sin 3 as cos :¢sk6 + sin ac cos3 ack7 + sin ~ ac cos ack9)r2/a z r ( 3 , 2) = 2((3x/~ sin' as - 4x/'2 sin 2 as + V/2 + 2 cos: as)k4 + 3x/~ sin as cos: ask6 + (3V/2 sin' ac - 4x/~ sin" ac + x/~ + 2 cos 2 ac )k 7 + 3x/~ sin: =c cos z ack9)rZ/a" K(3. 3) -- 2(cos ~ a s k 4 + sin" as cos" a s k 6 cos 4 =ok7 + sin' =ck9)r2/a 2 K(3.4) -- 2((I - v/2)sin as cos" a s k 4 + (x/~ - x / ~ sin as - cos 2 = , ) k 6 x (I - x/~)sin ac cos: ack7 + (x/~ - x/~ sin ~c - cos2 =c)k9)r/az K(3. 5) -- - 2(cos j a s k 4 + sin" as cos a s k 6 + cos 3 a c k 7 + sin z =c cos ack9)r/a z K(3.6) = (x/~ + 2)(cos ~ a s k 4 + sin z =8 cos a a k 6 + cos ~ a c k 7 + sin z ac cos =ck9)r/a z K(4. I ) -- 2( I - V/2) (sin: aa cos a8 k 4 + sin" aa cos as k 6 + sin 2 =c cos a c k 7 + sin' ac cos ack9)r/a" /('(4. 2) = (2(2x/~ - 3)k I + (3(2 - x/~)sin" as - x/~)sin a s k 4 + 3(x/~ sin" as - x/~ + 2 cos' as)sin a s k 6 + (3(2 - x/~)sin: ac - x/~)sin a c k 7 + 3(x/~ sin' ac - v/2 + 2 cos' at)sin ack9)r/a 2 K(4, 3) = 2((! - v/2)sin as cos" ask4 + (v/2 - x / ~ sin ~s - cos' =s )k6 x (I - x/~)sin acCOS 2 ack7 + (x/~ - x/~ sin a¢ - cos' =c)k9)r/a' K(4. 4) = 2(3 - 2v/2)(k I + sin: a s k 4 + cos' a s k 6 + sin' a c k 7 + cos' ack9)/a z K(4. 5) = 2(x/~ - !)(sin as cos a s k 4 - sin "Is cos ask 6 + sin ac cos =ck 7 - sin a¢ cos =ck9)/a' K(4. 6) = - x / ~ ( s i n as cos a s k 4 - sin as cos a s k 6 + sin ac cos ack7 - sin =c cos =¢k9)/a' K(5. I) - 2(K3 - sin as cos: a s k 4 - sin ~ a s k 6 - sin ac cos" =ck7 - sin ~ =ck9)r/a' K(5. 2) = (cos =s(3x/~ sin' as - v / 2 - 2)k4 - 3x/~ sin' an cos a s k 6 + cos ac(3x/~ sin" =c - x/~ - 2)k 7 - 3x/~ sin' ac cos =ck9)r/a'

K(5, 3) =

-

2(cos 3 ank4 + sin' as cos a s k 6 + cos 3 =ck 7 + sin 2 =c cos ack9)r/a'

K(5.4) = 2(x/~ - I)(sin ~s cos a s k 4 - sin ~s cos ank6 + sin a¢ cos ack7 - sin a c c o s ack9)/a z K(5, 5) = 2(k3 + cos' a s k 4 + sin: a s k 6 + cos' =ck7 + sin' ack9)/a" K(5, 6) = - ( 2 + x/~) (k 3 + cos: a s k 4 + sin' a s k 6 + cos: ack7 + sin" =ck9)/a 2

Stability of robot hand grasping

K(6, !) = ( - ( ! + ,v/2)k 3 + (V/2 .6 2)sin =e cos2 = . k 4 + (~./~ .6 2)sin; % k 6 + (~//2 + 2)sin accos 2 ack7 + (~/2 + 2)sin 3 ~'ck9)r/a 2 K(6, 2) -- ((3 + 2,v/2)cos 3 aek4 + 3(~//2 + l)sin: ~'s cos % k 6 + (3 + 2,v/2)cos ~ :[ck7 + 3(,v/2 + l)sin 2 ac cos "*ck9)r/a 2 K(6, 3) -- (~//'2 + 2)(cos 3 otek4 + sin 2 ae cos aek6 + cos 3 % k7 + sin:~c cos ~,ck9)r/a 2 K(6, 4) -- - x/~(sin % cos % k 4 - sin ~e cos aek6 + sin "c cos % k 7 - sin ~c cos ack9)/a 2 K(6, 5) -- - ( 2 + ~V/2)(k3 + cos: % k 4 + sin 2 ask6 + cos 2 % k 7 + sin: ",ck9)/a: K(6, 6) = (k2 + k5 + k8 + (2~//2 + 3)(k3 + cos 2 % k 4 + sin: % k 6 + cos 2 ack7 + sin 2 % k 9 ) ) / a 2 C ( I, I ) = 2( c 3 + sin 2 ~e cos2 as c 4 + sin 4 as c 6 + sin 2 ~c cos" ac c 7 + sin 4 ac c 9)r 2/a2

C(I, 2) = (sin as cos ~0(~/~ + 2 - 3,v/2 sin 2 ~,)c4 + 3~/~ sin ~ as cos ~'ec6 + (x/~ + 2 - 3x/~ sin 2 ac)sin accos acc7 + 3x/~ sin ) ac cos acc9)rS/, 2

C(I. 3) = 2(sin as cos ) a e c 4 + sin ) as cos aeC6 + sin ac cos ) a c c 7 + sin ) ac cos acc9)rS/a 2 C(I. 4) = 2(I - x/~) (sin 2 as cos as c 4 + sin 2 as cos as c 6 + sin 2 ac cos acC7 + sin 2 ac cos acC9)r/a' C(I. 5) -- 2(c3 - sin as cos' aeC4 - sin ) aoC6 - sin ac cos: acC7 - sin ) acC9)r/a 2 C(I. 6) -- ( - ( I + x/~)c 3 + (x/~ + 2)sin as cos' aoC4 + (x/~ + 2)sin' a o c 6 + (x/~ + 2)sin ac cos' a c c 7 + (x/~ + 2)sin ) acC9)r/a'

C(2. I) = (sin as cos ae(.¢/2 + 2 - 3x/~ sin' a s ) c 4 + 3x/~ sin ) as cos aoC6 •6 (x/~ 4- 2 - 3x/~ sin' ac)sin ac cos a c c 7 .6 3x/~ sin ) ac cos acc9)r:/a 2

6"(2. 2) =

(2(3 - 2x/~)c I + (9 sin4 as - 6(x/~ + I)sin 2 as + 3 + 2.¢/2)c4 + cos' aeC5 .6 9 sin' as cos" a e c 6 .6 (9 sin4 ac - 6(x/~ .6 l)sin 2 ac + 3 .6 2x/~)c7 .6 cos' acC8 + 9 sin' ac cos' acc9)r'/a"

C(2. 3) = 2((3x/~ sin4 as - 4x/~ sin" as + x/~ + 2 cos' as)c4 + 3x/~ sin as cos= aac6 .6 (3x/~ sin' ac - 4x/~ sin z ac .6 x/~ .6 2 cos' ac)c 7 .6 3x/~ sin' ac cos' acc9)r'/a 2 C(2.4) -- (2(2x/~ - 3)cl + (3(2 - x/~)sin: as - x/~)sin aaC4 6 3(x/~ sin 2 as - x/~ .6 2 cos 2 as)sin as c 6 + (3(2 - x/~)sin 2 ac - x/~)sin acc7 + 3(x/~ sin' ac - x/~ + 2 cos' ac)sin acc9)r/a'

C(2, 5) = (cos as (3x//2 sin' ae - x/~ - 2)c4 - 3x/r2 sin' ae cos aoc6 .6 cos ac(3xF2 sin' ac - x/~ - 2)c 7 - 3x/~ sin' ac cos acc9)r/a' C(2. 6) = ((3 + 2x/~)cos ) a e c 4 + 3(x/~ + I)sin 2 as cos a e c 6 + (3 + 2x/~)cos ) acc7 + 3(x/~ + I)sin 2 ac cos acC9)r/a 2 C ( 3. i ) = 2(si n as cos ) as c 4 .6 sin ) as cos as c 6 .6 sin ac cos ) ac c 7 4- sin ) ac cos ac c 9)r '/a 2 C(3. 2) = 2((3x/~ sin' as - 4x/~ sin" as + x/~ .6 2 cos 2 as)c4 .6 3x/~ sin as cos" asc6 .6 (3x/~ sin' ac - 4 x / ~ sin" ac .6 x/~ 4- 2 cos 2 ac)C 7 .6 3x/~ sin' accos 2 acc9)rZ/a z C(3. 3) = 2(cos' asc4 + sin' as cos" asC6 cos' acC7 + sin' ac cos: acC9)r'/a' MMT ~/~--I

649

650

ZH~N LU

C(3, 4) = 2((1 - V/2)sin % c o s : a , c 4 + ( v / 2 - V/2 sin % - cos: a , ) c 6 x (! - v / 2 ) s i n ac cos2 ~c c 7 + (V/2 - V/2 sin ~c - c o s2 "c )c 9 ) r / a : C(3, 5) -- - 2(cos 3 ~ 8 c 4 + sin: ae cos a 8 c 6 + c o s ~ ~cC7 + sin: ac cos ~cc9)r/a e C(3, 6) = ( v / 2 + 2 ) ( c o s 3 a o c 4 + sin e as c o s a a c 6 + cos ~ ~cc7 + sin e ac cos ~,cc9)t/a: C(4, !) = 2(! - ~/'2)(sin 2 us cos u s c 4 + sin e u s c o s u 8 c 6 + sin e Uc cos ~'cc7 + sin e ~'c cos ucc9)r/a 2 C(4. 2) -- ( 2 ( 2 v / 2 - 3)¢1 + (3(2 - v/2)sine us - ~ / ~ ) s i n u s e 4 + 3 ( v / 2 sin 2 us - V/2 + 2 cos e u s ) s i n u a c 6 + (3(2 - ,v/~)sin e u c - ~ / ~ ) s i n u c c 7 + 3 ( , j ~ sin e Uc - v / 2 + 2 cos e Uc)sin UcC9)r/a e C(4. 3) = 2((I - ~ / 2 ) s i n us cos e us c 4 + ( ~ / 2 - ~ / 2 sin us - cos e u s ) c 6 x (I - v / 2 ) s i n u c c o s 2 uc c 7 + ( v / 2 - v / 2 sin Uc - cos e Uc)C 9 ) r / a e C(4, 4) -- 2(3 - 2 V / 2 ) ( c I + sin 2 u e c 4 + c o s e u o c 6 + sin e ucc7 + cos 2 uccg)/a e C(4, 5) -- 2(V/2 - I ) ( s i n u8 cos ~'8 c 4 - sin uo cos uo c 6 + sin Uc cos u c c 7 - sin uc cos ~ccg)/a e C(4, 6) -- - ~ / 2 ( s i n uo cos u a c 4 - sin ~o cos u o ¢ 6 + sin uc cos UcC7 - sin uc cos u c c 9 ) / a ' C(5, 1) -- 2(c3 - sin u8 cos e u o c 4 - sin 3 u a c 6 - sin uc cos e ucc7 - sin ~ ucc9)r/a e C ( 5 . 2 ) = (cos u , ( 3 ~ / 2 sin 2 ua - V/2 - 2 ) c 4 - 3,~/~ sin e us cos u o c 6 + cos uc(3~//2 sin e Uc - V ~ - 2 ) c 7 - 3 v / 2 sin 2 uc cos UcC9)r/a e C(5. 3) -- - 2 ( c o s 3 us c 4 + sin e us cos us c 6 + cos 3 ucc 7 + sin e uc cos u c c 9 ) r / a e

C (5, 4) = 2 ( x / ~ - I ) (sin us cos us c 4 - sin us c o s us c 6 + sin Uc c 7 - sin uc cos uc c 9)/a e C(5, 5) = 2(c3 + cos e uec4 + sin 2 uec6 + cos 2 UcC7 + sin 20tcC9)/a 2

C(5, 6) = - ( 2 + x/2)(c3 + cos e usc4 + sin e uec6 + cos e ucc7 + sin 2 uccg)/a e C(6, !) -- ( - ( I

+ V/2)c3 + (v/2 + 2)sin ~,, cos 2 u a c 4 + (,ff~ + 2)sin' usc6

+ (v/2 + 2)sin uc cos 2 ucc7 + (V/2 + 2)sin ~;ccg)r/a e C(6, 2) = ((3 + 2v/2)cos 3 ~sc4 + 3(v/2 + l)sin e us cos usc6 + (3 + 2v/2)cos 3 UcC7 + 3(v/2 + i)sin e ~c cos UcC9)r/a 2 C(6, 3) = (v/2 + 2)(cos 3 uac4 + sin e us cos uec6 + cos j ucc7 + sin-' Uc cos UcC9)r/a e C (6, 4) = - v/2(sin us cos ua c 4 - sin ua cos 0~s c 6 + sin Uc cos Uc c 7 - sin ~c cos u c c 9)/a-' C(6, 5) = - ( 2 + v/2)(c3 + cos e ~ac4 + sin e uoc6 + cos e otcc7 + sin" UcC9)/a e C(6, 6) = (c2 + c5 + c8 + (2V/'2 + 3)(c3 + cos 2 u8c4 + sin 2 uac6 + cos 2 ucc7 + sin" UcC9))/a 2, where k l , k 2 . . . . . k 9 and c l . . . . . c9 are the diag elements of [Kz] and [CT]. The inertia matrix [/] = diag[ll i , / 2 2 , / 3 3 , m, rn, m].

,o = [0,, O: .....

O,]'.

13. C O N C L U D I N G

REMARKS

The impedance control is an essential problem in grasping planning. The analysis presented in this paper has revealed that the impedances of grasping not only depend on the position servo gain

651

Stability of robot hand grasping

and velocity servo gain, but also concern with two sets of linear relationships indicated by grasping matrix W, and Jacobian matrix J. By rearrangement of the contact points, one can have relatively ideal asymptotical stability of grasping. A method that alters the time constant is proposed. Using the method, we can select the optimal robotic hand finger configuration during the control process. REFERENCES J. W.

:

and L. T. tiler. Roe. 1986 IEEE ht. Con/. Roborics aad Aulomarion. pp. 876-883. Y. Nakamura cf a/.. Robot. Res. 8(2).4441 (1989). J. Kerr and B. Roth. Robot. Rrs. Y4). 3-17 (1986). M. R. Cutkosky and 1. Kao. /EEE Trims. Robot. AUIOIWI S(2). 151-165 (1989). M. T. Mason and J. K. Salisbury. Robot Han&v and the hfechanics 01 Mantjndarion. MIT Press. Cambridge. Mass. (1985). D. T. Greenwood. Clussical Dynamics. Prentice-Hall. Englewood Cliff.. NJ (1977). W. Gao. The Stability of Motion. Beijing Institute of Aero. and Astro (1980). In Chinese. Jameson

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