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Manipulation of 3D Enveloped Object

K e n s u k e H a r a d a , a n d M a k o t o K a n e k o * *

, N a t i o n a l I n s t i t u t e of A d v a n c e d I n d u s t r i a l Science a n d T e c h n o l o g y ( A I S T )

1-1-1 U m e z o n o , T s u k u b a , I b a r a k i 305-8568, J A P A N

* * G r a d u a t e School of E n g i n e e r i n g , H i r o s h i m a U n i v e r s i t y

K a g a m i y a m a , H i g a s h i - H i r o s h i m a 739-8527, J A P A N

A b s t r a c t This paper discusses the manipulation of 3D en-

veloped object. By assigning all fingers as either the position controlled finger (P-finger) or the torque controlled finger (T-finger), we propose a method for let- ting the object to move in the desired direction along the surface of the P-fingers. For this purpose, we obtain the joint torque command for the T-fingers. Dif- ferent from our previous approach[17], the formulation of the total force/moment set can be applied to general 3D enveloping grasp. Also, we newly provide a sufficient condition for the total force /moment set of the object moving toward the desired direction, where it can be applied to general 3D contact configurations between the object and the P-fingers.

1 I n t r o d u c t i o n The grasp style of an object by a multifingered

robot hand can be roughly divided into two groups. One is the fingertip grasp where a robot hand grasps an object only by the fingertip. And, the other is the enveloping grasp where a robot hand grasps an object not only by the finger tip but also by the inner link and the palm(Fig. 1). For the fingertip grasp, although we can expect the manipulation to be dexterous, the hand may easily fail in grasping an object by an external disturbance. On the other hand, for the enveloping grasp, we can expect the grasp to be robust against an external disturbance. Previously, there has been much research on the manipulation of an object by the fingertip grasp[11, 12, 13, 14, 16]. In this paper, we will focus on the manipulation of an enveloped object.

To realize the manipulation of an enveloped object, there are some difficulties: First, the total force/moment produced by all contact forces cannot be uniquely determined, due to the one-to-multiple mapping from the joint torque to the contact force. As a result, the direction of the object's motion cannot be uniquely determined either. Consequently, we cease manipulating the enveloped object solely by torque control. Instead, we assign all fingers into a position controlled finger (P-finger) and torque controlled finger (T-finger), respectively. The enveloping grasp can be considered a combination of the P-fingers and the T-fingers as shown in Fig.1. If we impart an appro- priate set of torque commands for the T-fingers, the

T - f i n g e r s

P- f ingers

Fig. 1" Enveloping grasp of an object

object will move along the surface of the P-fingers. We are interested in obtaining a set of torque for the T-fingers always move the object toward the designated direction along the surface of the P-fingers (or the environment).

In our previous research[17], we proposed the manipulation of an enveloped object. To realize the manipulation, we showed a sufficient condition for the joint torque of the T-fingers to make the total force set be included into the desired region. However, we assumed that the total force set produced by the T- fingers is bounded where it is always satisfied for the 2D grasp. Moreover, we did not show how to con- struct the desired region of the total force set to move the object to the desired direction along the surface of the P-fingers.

In this paper which is an extended version of our previous research, we propose the manipulation of general 3D enveloped object. The goal of this paper is to obtain the joint torque set for the T-fingers moving the object toward the desired direction along the surface of the P-fingers. We first formulate the total force/moment set produced by the T-fingers. The assumption of the bounded total force set is not need- ed since we formulate the total force set by using the breadth-first search method. We then show a condition for the joint torque of the T-fingers to make the total force/moment set be included into the desired

region. Further, we newly show a sufficient condition for the total force/moment set to move the object to the desired direction along the surface of the P-fingers. We show that this condition can be applied for general 3D contact configurations between the object and the P-fingers. Finally, to show the effectiveness of our proposed method, we newly show simulation results.

2 R e l a t e d W o r k Salisbury [1] has first proposed the basic concept of

whole-arm manipulation. Trinkle, Abel, and Paul [2] discussed the grasp planning issue for enveloping a planar frictionless object. Song, Yashima, and Kumar[3] constructed a dynamic simulation model of the enveloping grasp. Mirza and Orin [4] pointed out that power grasps maximize the load capability and are highly stable in nature due to a large number of dis- tributed contact points on the grasped object. Bicchi [5] addressed the problem of force decomposition in power grasps. Zhang, Nakamura, Goda, and Yoshi- moto [6] provided the measure of the robustness of the power grasp. Zhang, Nakamura, and Yoshimoto [7] discussed the region of the external force by using the polyhedral convex theory. Zhang and Gruver [8] defined the power grasp mathematically and analyzed the force distribution at the contact points. Omata and Nagata [9] highlighted the possible area of contact forces by utilizing the constraint condition obtained by the kinematic relationship. Yu, Takeuchi, and Yoshikawa [10] proposed a procedure for achiev- ing the power grasp optimization. However, in the above papers of the enveloping grasp, they did not considered the manipulation of the grasped object.

As for the manipulation of an object grasped by the fingertip grasp, Cole, Hauser, and Sastry[11] s- tudied the simultaneous control of the object motion and the internal force under the rolling contact. Cole, Hsu, and Sastry[12] also researched the control under the sliding contact. Paljug, Yun, and Kumar[13] researched the simultaneous control of object motion and the contact position under the rolling contact in 2D space. Sarkar, Yun, and Kumar[14] extended the research to the 3D rolling model. However, they did not consider the manipulation under the enveloping grasp.

On the other hand, in this paper, we address the manipulation issue of an object under the enveloping grasp. Trinkle and Paul[15] extended their former result[2] to the 2D manipulation of an object with sliding contacts. In our previous work [17], we proposed the manipulation of an enveloped object taking the friction at each contact point into consideration.

3 R e p r e s e n t a t i o n of Tota l Force / Mo- m e n t

The goal of this section is to express the total force/moment set produced by T-fingers (abbreviated as T-total F / M set) and to provide the constraint condition where the T-total F / M set should satisfy.

3.1 M a p p i n g f r o m t o r q u e t o c o n t a c t f o r c e Let us consider the i-th T-finger of a robot

hand as shown in Fig.l(a), where f i j and Ti =

[Til, ' '" ,Tim] T C R rex1 denote the contact force vector at the j - th contact point of the i-th T-finger, and the joint torque vector of the i-th T-finger, respectively. The relationship between the contact force and the joint torque for the i-th T-finger is expressed by,

~-~ - JTf~ , (1)

where f i [ f~ , . • T / ~ 3 m x 1 _ ", f i~]T ¢ , and

I gTi11 gTilm 1 ... c R (2)

o J ~

J ~ denotes the transpose of the Jacobian matrix mapping the contact force into the joint torque. 3 .2 C o n t a c t f o r c e s e t

In order to change from nonlinear to linear friction constraint, we approximate the j - th friction cone of the i-th T-finger by the L faced polyhedral convex cone[16].

L

Z ' ' > ° ) (3) ~ijVij , __ , / = 1

= v~j ,x~j , (4)

where V i i - [ v i l , . . . , v L] ¢ R a×L and Aij = [~ i l , . . . , ~L]T C t~ L × 1. V~j and ,~j denote the / t h

unit span vector and the corresponding magnitude of contact force, respectively, of the convex polyhedral cone at the j - th contact point of i-th T-finger. For the i-th T-finger, we obtain the following form.

f i - V i A i , Ai_>o, (5)

where Ai - [ A ~ , . . . , A f ~ ] T C t~ Lmxl and V i = diag [Vii . . . Vim] ¢ R 3~×L~. From eqs.(1) and (5),

r~ - J ~ V ~ , X ~ , ,X~ >_ o. (6)

For n fingers, we obtain

T - J T V A , A_>o, (7)

where T [TT1 . . . TT] T t~ mn × l - - C , J =

diag [ J l " ' " J~] ¢ R "~×3"~, V - diag IV1"" V~] ¢ T

;1 given ' k 1 7~ 3

joint torque, eq.(7) can express the set of A under the friction cone constraint. The vertices and the edges of this set can be calculated by using the Breadth-first search[18]. By applying the Breadth-first search to eq.(7), the following equation can be obtained"

£" -- E Yk~k[~l -+- " " " -+- ~r -- 1, k = l

~k >_ O,r <_ h , k - 1 , . . . , h ~ , (s) J

f __

(9)

{ L VYk~kI~I + "" + ~r - 1, k=l

~ >_ O,r <_ h,k - 1 , . . . , h } ,

L=6

where Vy~ ¢ R a '~× l (k = 1 , . . . , r ) can be a can- didate of vertices of the contact force set. When Vy~ (k = r + 1 , . . . ,h ) exists, the contact force set does not become a bounded set since we can set ~ (k = r + 1 , . . . , h) arbitrarily as long as it is positive. 3 .3 R e l a t i o n s h i p b e t w e e n t o t a l f o r c e /

m o m e n t a n d c o n t a c t f o r c e s e t s The T-total F /M are given by:

f T -- G F f , (10) m r -- a M f , (11)

where G F = [Ia "'" Ia] ¢ R a × a ' ~ , G v - [ r l l x . . . r~,~x] C R a × a ' ~ , r ~ j x -

[ 0 1 rij~ 0 - r i jx , and ri j - [rijx rijy rij~] T - - r i j y r i j x 0

denotes the vector directing the contact point from the center of gravity of the object.

By using eqs.(9), (10), and (11), the T-total F /M sets can be obtained as a convex polyhedral set as

~r h

G F V y ~ l ~ l + . . . + ~ -- 1, k=l

~ >_ O,r <_ h,k - 1 , . . . , h } , (12)

A4T ~ GMVytc~tc[~I + " " q- ~r -- 1,

~ >_ O,r <_ h , k - 1 , . . . , h } , (13)

Since eqs.(12) and (13) are the function of the joint torque, we can discuss how to determine the com- manded torque, so that the object may move in a bounded direction along the P-fingers (or the environment). Now, recall that each friction cone is approximated by a polyhedral convex cone. By this ap- proximation, we can no more keep the exactness for both the contact force and the T-total F /M sets.

Let f(o) and f(~) (f(o) G f(~)) be the original and approximated friction cones, respectively. This relationship is illustrated in Fig.2(b). Also,

let f(o) (AA(o)) and f(~) (AA(~)), be the T-total

where y~ E t~ L m n x l (k = 1 , . . . , r ) a n d y~ (k = r + 1 , . . . , h ) denote the basic feasible solution and the basic feasible direction[18], respectively, of eq.(7). From eq.(8), we can obtain the contact force set as:

(a) Outer tangential cone

L=6 ~ ~ % ~ f (e)

,T

~ / / ~ ~ / (b) Inner tangential cone

Fig. 2" Several polyhedral convex cones

force(moment) sets obtained under f(o) and f(~), re-

spectively. Since f(o) C_ f(~), it is ensured that f(o)

(~) when each link of T-fingers ~_ 5~ °~ ~ a M~ °~ ~_ M~

h~ o ~ ~o~t~t poi~t[lrl Now, ~t / j / ~ d M~) be desired T-total F /M sets where a desired object's motion is expected. If a set of torque commands are

~ho~ ~o ~h~ 5~ °) ~_ 5~ ~) ~ d M~ °) ~_ M~), ~h~ 5~ °) ~_ 5~ ~) ~ d M~ °) ~_ M~) ~ y ~ ~ ~ d This means that the T-total F /M sets, under the o- riginal friction cone, also produce a desired motion for the object.

Now, the remaining questions are as follows:

(1) How to design f(d) and A//~)? (2) How to determine a set of torque commands so

that f(~) C_ f(d) and A//(~) C_ A//7 ), may be satisfied ?

3.4 P r o c e d u r e fo r o b t a i n i n g c o m m a n d t o r q u e u n d e r g i v e n c o n s t r a i n t s

The main purpose of this section is to consider a sufficient condition for the joint torque to put the T- total F /M set within the designated region. G F V y k and G M V y k (k = 1,. . . ,r) expressed in eqs.(12) and (13) become candidates for vertices of the T-total F /M

~ 5~ °) ~d M~ °) L~ 5~ °) ~d M~ °) b~ ~h~ T total F /M set produced by the contact force sets of T-fingers where each contact force set is expressed as a convex polyhedron including the actual friction cone.

L~ 5~ ~) ~ d M ~ ) b ~ ~h~ ~ w h ~ w~ w ~ 5~ °) ~ d M~ °) ~o ~ i ~ Th~ ~d~io~ b~ ~ 5~ ~) ~ d 5~ °) is shown in Fig.3 where c~ (i = 1 , . . . ,pf) denotes a unit normal vector of a supporting hyperplane of

(d) f T directing the inside of f(T d) and the position of the center of gravity of the object, respectively. We usually select c~ for the object to simultaneously keep contact with the P-fingers and to move in the desired direction. If all the vertices of the T-total F /M satisfy

T > 0 ( i - - 1 j - - p f + c ~ f T _> 0 and T e a j m T _ , " " " , p f

1 , . . . , p f + p,~), the T-total F /M set f(r ~) and A4(r ~)

always exist inside of f(d) and A//(r d). This condition is equivalent to

T c ~ G F V Y > o, i - 1 , . . . , p f , (14)

, ~ " - ~ Total force set Total ' - l - ' - f i l ~ ~ -~, , r (d) ,,4"~ "-_~,]_~-~ .T -(~0 . ~brce set

V ( a ) (b)

Fig. 3: Relationship among F (°), F (~), and F (d) P-tinge

c T G , -- , • M V Y >_ o j pf + l .. ,pf + p,~, (15)

where Y = [Yl " ' " Yh] The common set of inequalities (14) and (15) and

the torque limitation produces the torque set T~i which ensures the object's motion along the P-finger.

c ~ i G F V Y >_ o n T m i n ~ T ~ Tm~x}(,16) T T~j - { T I c ~ j G M V Y >_ o n T m i n ~ T ~ T m a x ~ 1 7 )

[ T h e o r e m 1] A sufficient condition for simultane-

ously producing the T-total F / M sets within F (d) and

Ad(T d) are given by

T~ -~ ¢ (18)

where Tc = T~I n . . . n T~(ps+p.~ ) [Proof] If the contact force set approximated by the convex polyhedron includes the original friction cone, then the T-total F / M set approximated by the con-

vex polyhedron also includes its original cone. If ~C(Td)

and Ad(T d) include ~C(T~) and Ad(T ~), respectively, they

also include ~c(o) and Ad (°). If Tc # ¢, it is obvious that the joint torque exists by restricting the original

T-total F / M within ~C(Td) and Ad(d)T • []

Here, we give some comments on the formulations provided in this section.

The number of inequalities increases according to L which expresses the number of the supporting plane of the friction cone. However, since we only require the sufficiency be preserved, it is sufficient to assume L = 3 to approximate the friction cone. This is a great advantage in reducing the compu- tation burden.

When sliding occurs at a contact point, the contact force lies on the edge of the kinetic friction cone. On the other hand, when rolling occurs, the contact force lies within the friction cone. There- fore, regardless of the contact mode, the contact force is always included within the static friction cone. Since the contact force lying anywhere within the contact force set is considered, we do

.

.

Fig. 4: Motion of object projected on a plane

not need to be concerned with the kinds of motion actually happening during the manipulation pro- cess. This is another advantage of our method.

In our formulation, we numerically obtained the T-total F / M set by using the breadth-first search. However, when the T-total F / M set becomes a bounded set and does not have a basic feasible direction, we can obtain the joint torque set T~i in a very simple manner, which is shown in [17].

If each link of fingers does not have one contact point, we cannot provide a sufficient condition since the original T-total F / M set is not always included within the approximated ones[17]. There- fore, when applying our method to cases where each link of fingers does not have one contact point, we should make the approximated friction cone as close to the original cone as possible. In such cases, L should be large.

4 Manipulation In this section, we obtain a sufficient condition for

the T-total force/moment space for producing the desired motion of the object.

4.1 A suff icient c o n d i t i o n for m a n i p u l a - t ion

As shown in Fig.4, we consider the 2D motion of an object projected onto the plane H, perpendicular to the contact surface. We also consider the case where the object makes surface contact with the P-fingers.

We define the manipulation of an object as follows:

[Definit ion] Let b, tl and c~1 be the vectors express- ing the direction of motion of the object, the upward direction along the surface of the P-finger, and the direction to push the object against the P-finger, respectively, as shown in Fig.5. The following two cases can be considered as the manipulation of an object along the surface of the P-finger:

T Posit ive Manipulat ion" bTtl > O, % 1 f T > 0

T Negative Manipulat ion" bTtl < O, c~l f T > 0

' /J P~ r~/P2

T (a)

~ mTc2

\ P-finger (b)

Fig. 5" Manipulation of object on the projected plane

Regardless of the motion of the object at the contact point, we focus on whether the motion at the center of gravity projected onto 12 is in the positive or the negative direction. We assume that the object is not moving or moving slowly which means that the direction of motion coincides with the direction of the total force/moment acting on the object. We show a sufficient condition for the positive manipulation as:

[ T h e o r e m 2] Consider the object as shown in Fig.5.

F (d) is spanned by t l and t2 where t2 - - v 2 ; Vl and v2 are unit vectors of the edges of the friction cone projected onto 12. Assume tTl rp1 > O. As a sufficient condition for the positive manipulation, we can cons- der the following two cases:

[Case 1]

(i) The T-total force space is produced inside of F (d).

(ii) m r c 1 < 0 and m r c 2 < 0 where m r c i (i = 1,2) is the T-total moment at both edges of the contact segment projected onto 12. Both are positive for the C C W direction(Fig.5).

[Case 2]

(i) T-total force space is produced inside of F (d).

(ii) m f c 1 > 0 and m f c 2 < O.

[Proof] See Appendix. [] We note that, although we do not consider the effect

of gravity in the above theorem, the theorem holds if the sum of the T-total force and the gravity force in the tangential direction of the surface of the P-finger is produced inside of ) c(d).

Here, we give some comments on the sufficient condition for manipulation.

1. We note that, in the [Case 2], the object will maintain surface contact with the P-finger since

rlitee

(a) Surihce comact with a P-finger (b) Surface contact with multiple P-fiagers

-fi)~ger line (C) Line contact with a P-finger (d) Point contact with a P-finger

Fig. 6" Several contactconfigurations with P-fingers

A

i i:il ....... i

Fig. 7: Motion of object restrected by two constraints

both mTC1 > 0 and mTC2 < 0 are satisfied. Un- der t2 = - v 2 , all forces acting on the object cannot balance, and it is ensured that the object begins to slide along the P-fingers with maintaining the surface contact.

2. As shown in Fig.6 (a) and (b), we can regard the following cases as the surface contact: (1)The surface of the object contacts with the surface of a P-finger (Fig.6(a)), (2) The surface of the object makes line contact with multiple P-fingers (Fig.6(b)). Moreover, under m r c l = m r c 2 ,

[Case 1] of T h e o r e m 2 can be applied to the case where the object makes line contact (Fig.6(c)) or point contact (Fig.6(d)) with P-fingers since those cases can be regarded as a special case of the surface contact where the area of contact becomes zero.

3. To extend the manipulation to 3D space, we consider multiple projection planes as shown in Fig.7. If Theorem 2 holds for both 121 and 122, the T- total force space is restricted within the region shown in the figure. It is considered that the object will move within the region.

~ ~ Z x

171 • ~

T-Finger P-Finger 24

II1 C a l c •

(a) 3D Model (b) View from x-z plane (c) View from projection plane

Fig. 8" Overview of system used for numerical calculations

1

T l l

0.7

0.4 0.02

m

• •

• •

Hll m

5,

0

% ' w

-15t " -~" 1 0.05 0.1 0.15 -20 -15 -10 -5 0 5

T 12 [Nm] /T~ [N]

(a) Joint torque set (b) T-total force set (ZlF1.2, z12--0.1)

Fig. 9: Results of numerical calculations

4.2 S i m u l a t i o n As shown in Fig.8, we consider the 3D manipula-

tion of a sphere by three fingers, where two of them and the last one are assigned as T-fingers and a P- finger, respectively. We set same joint angle and joint torque for two T-fingers where 011 = 37r/4[rad] and 012 = 7r/2[rad]. The radius of the sphere, the distance between two T-fingers, and the friction angle at each contact point are set as r = 0.1[m] and 1 = 0.1[m], and 7r/8[rad], respectively. Each friction cone is approximated by a four faced polyhedral convex cone which includes the actual friction cone. We assume two projection planes 121 and 122 as shown in Fig.8(b), where the desired region of T-total force UT within the projection plane is shown in Fig.8(c). From the definition of )CT, the condition [Case 1] (ii) in Theorem 2 can always be satisfied unless the T-total force is produced within fT .

The results of the numerical calculations are shown in Fig.9. The set of joint torque of the T-fingers satis- fies Theorem 2 and the T-total force space is projected onto 121, shown in Fig.9 (a) and (b), respectively. If we choose the joint torque of the T-fingers, shown in Fig.9(a), the object is guaranteed to move upward a- long the P-finger.

5 C o n c l u s i o n s We have discussed the manipulation of 3D en-

veloped object, by assigning all chains to either the

T-chain or the P-chain. We showed to sufficient con- ditions: 1. The condition for the joint torque to make the T-total force/moment set be included into the desired direction, 2. The condition for the T-total force/moment set to realize the desired manipulation. Real-time implementation of our algorithm is considered to be our future research topic.

Finally, the authors would like to express their sin- cere thanks to Mr. Tatsuya Shirai and Mr. Mitsushi Sawada for their simulation and experimental works.

R e f e r e n c e s [1] Salisbury, J. K., Whole-arm manipulation, Proc.

of the 4th Int. Syrup. of Robotics Research, San- ta Cruz, CA, 1987. Published by the MIT Press, Cambridge MA.

[2] Trinkle, J. C., J. M. Abel, and R. P. Paul: En- veloping, frictionless planar grasping, Proc. of the IEEE Int. Conf. on Robotics and Automation, 1987.

[3]

[4]

Song, P., M. Yashima, and V. Kumar: Dynamic Simulation for Grasping and Whole Arm Manipu- lation, Proc. of 2000 IEEE Int. Conf. on Robotics and Automation, pp. 1082-1087, 2000.

Mirza, K., and D. E. Orin" Control of force distribution for power grasp in the DIGITS system, Proc. of the IEEE 29th CDC Conf., pp1960-1965, 1990.

[5]

[6]

[7]

Is]

[9]

Bicchi, A: Force distribution in multiple whole- limb manipulation, Proc. of the IEEE Int. Conf. on Robotics and Automation, pp196-201, 1993.

Zhang, X-Y., Y. Nakamura, K. Goda, and K. Yoshimoto: Robustness of power grasp, Proc. of the IEEE Int. Conf. on Robotics and Automation, pp2828-2835, 1994.

Zhang, X-Y., Y. Nakamura, and K. Yoshimoto: Mechanical analysis of grasps with defective contacts using polydedral convex set theory, J. of Robotics Society of Japan, vol. 14, no. 1, pp105- 113, 1996.

Zhang, Y., and W. A. Gruver: Definition and force distribution of power grasps, Proc. of the IEEE Int. Conf. on Robotics and Automation, pp1373- 1378, 1995.

Omata, T., and K. Nagata: Rigid body analysis of the indeterminate grasp force in power grasps, IEEE Trans. on Robotics and Automation, vol. 16, no. 1, pp46-54, 2000.

[10] Yu, Y., K. Takeuchi, and T. Yoshikawa: Op- timization of robot hand power grasps, Proc. of the IEEE Int. Conf. on Robotics and Automation, pp3341-3347, 1998.

[11] Cole, A. B. A., J. E. Hauser, and S. S. Sastry: Kinematics and Control of Multifingered Hands with Rolling Contacts, IEEE Trans. on Automatic Control, vol. 3~, no. ~, pp. 398-~0~, 1989.

[12] Cole, A. B. A., P. Hsu, and S. S. Sastry: Dynamic Control of Sliding by Robot Hands for Regrasping, IEEE Trans. on Robotics and Automation, vol. 8, no. 1, pp. ~2-52, 1992.

[13] Paljug, E., X. Yun, and V. Kumar: Control of Rolling Contacts in Mult i-Arm Manipulation, IEEE Trans. on Robotics and Automation, vol. 10, no. ~, pp. ~1-~52, 1994.

[14] Sarkar, N., X. Yun, and V.Kumar: Dynamic Con- trol of 3-D Rolling Contacts in Two-Arm Manipu- lation, IEEE Trans. on Robotics and Automation, vol. 13, no. 3, pp. 36~-376, 1997.

[15] Trinkle and R. P. Paul: Planning for Dexterous Manipulation with Sliding Contacts, The Int. Y. of Robotics Research, vol. 9, no. 3, pp. 2~-~8, 1990.

[16] Kerr, J., and B.Roth: Analysis of multifingered hands, Int. or. of Robotics Research, vol.~, no.~, pp3-17, 1986.

[17] Kaneko, M., K. Harada, and T. Tsuji: A Suffi- cient Condition for Manipulation of Envelope Fam- ily, Proc. of 2000 IEEE Int. Conf. on Robotics and Automation, pp. 1060-1067, 2000.

[18] Chvatal, V • Linear Programming, W.H.Freeman and Company, New York and Oxford, 1983.

A P r o o f o f T h e o r e m 2 We define some variables as shown in Fig.10 where

f p , wp, and vpi (i = 1,2) denote the sum of the contact forces applied by the P-fingers, the angular velocity of the object, and the translational velocity of the object at the edge of the contact segment, respectively. Since both the T-fingers and the P-fingers provide contact forces for the object, the total force acting on the object (abbreviated as TP-total-force) is given by

f lTP = f iT nt- f P" (19) Although we use the same variable f r as in Section 3, the vectors express the 2D variables projected onto H in this section. By using the relation of acceleration acting on the object, we can also define the TP- to ta l force as

f T p - - m ( ~ ) P ~ - - r P i x & P + w ~ r P ~ ) , ( i - - 1, 2). (20)

Since we assume that the object is not moving or that the motion of the object is slow, the direction of acceleration coincides with the direction of motion of the object. Therefore, we consider the direction of TP- to ta l force to indicate whether the positive manipulation can be realized or not. For both [Case 1] and [Case 2], depending on the motion of the objec- t at the contact point, we can consider the following three cases: [a]The object slips in the positive direction, [b]The slip does not occur, and [c]The object slips in the negative direction.

We first prove [Case 1]. In this case, since the object cannot keep equilibrium, the object begins to

~.t- ~VF1 ~.~ WF~ vI~2

Fig. 10" Definition of variables

rotate around the edge of the contact segment. We consider the instant of time when the object begins to

rotate. By the definition of F(T d), f r -- t lk t l + t2kt2, ktl > 0, kt2 > 0, and &p _< 0 are always satisfied. For [a], subst i tut ing iJp1 - eykc, k~ > 0, and f p - v2k~2,k~2 > 0. into eq.(19), we can see that the TP-total-force can occur both in the positive and in the negative directions. However, substi tut ing into eq.(20) and taking the inner product with ey - t l , the following equation can be obtained:

tTl S T P -- m k c -- meTx ~'Pld2p -Jr- mtTl ~'Pla)2p, (21)

where ex = C~l. Since the right hand side of eq.(21) is always positive, we can see that neither the negative nor zero TP- to ta l force can occur. For [b], we can prove that neither the negative nor zero TP- to ta l force can occur, similar to the outcome in [a]. For [c], we can set ~)P1 = - eykc , kc > 0, and f p = vlk~l ,k~l > O. Substi tut ing these relationships into eq.(20), we can see that the TP-total-force can occur both in the positive and in the negative directions. However, substi- tut ing into eq.(19) and taking the inner product with ey = t l, the following equation can be obtained:

tTl S T P -- tTl ( t l k t l + t2kt2)+ tTl V l k v l • (22)

Since tTlt2 > 0, we can see that the right hand side of eq.(22) is always positive which means that the positive TP- to ta l force occurs.

Then we prove [Case 2]. For [b], substi tut ing /JP1 = /JP2 = o, c~p = 0, and f p = vlkv l + v2kv2, k~l > 0, k~2 > 0 into eq.(20), we obtain f T P = o. However, subst i tut ing into eq.(19), we can see that f T P cannot be zero. Therefore, the object cannot keep its equilibrium. For [c], we can prove that the object cannot slip in the negative direction, thus similar to the outcome in [b]. For [a], substi tut ing Vp1 = Vp2 = eykc , kc ~ O, COp = O, and f p = v2kv2, k~2 _> 0 into eqs.(19) and (20), we see that the TP- total force occurs in the positive direction. Conse- quently, these discussions hold the theorem.

manipulation of 3d enveloped object - hlab-osaka-u ... · manipulation of 3d enveloped object ......

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