and G. D´ıaz-Rodr´ıguez

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Volume 2012, Article ID 949834,22pages doi:10.1155/2012/949834

Research Article

Parallel-Distributed Model

Deformation in the Fingertips for Stable Grasping and Object Manipulation

R. Garc´ıa-Rodr´ıguez

¹

and G. D´ıaz-Rodr´ıguez

²

1Facultad de Ingenier´ıa y Ciencias Aplicadas, Universidad de los Andes, Av. San Carlos de Apoquindo 2200, Las Condes, Santiago, Chile

2Departamento de Ingenier´ıa El´ectrica, Universidad de Chile, Av. Tupper 2007, Santiago, Chile

Correspondence should be addressed to R. Garc´ıa-Rodr´ıguez,rgarcia1@miuandes.cl Received 27 April 2012; Revised 13 July 2012; Accepted 25 July 2012

Academic Editor: J. Rodellar

Copyrightq2012 R. Garc´ıa-Rodr´ıguez and G. D´ıaz-Rodr´ıguez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The study on the human grip has inspired to the robotics over the past decades, which has resulted in performance improvements of robotic hands. However, current robotic hands do not have the enough dexterity to execute complex tasks. Recognizing this fact, the soft fingertips with hemispherical shape and deformation models have renewed attention of roboticists. A high-friction contact to prevent slipping and the rolling contribution between the object and fingers are some characteristics of the soft fingertips which are useful to improve the grasping stability. In this paper, the parallel distributed deformation model is used to present the dynamical model of the soft tip fingers with n-degrees of freedom. Based on the joint angular positions of the fingers, a control scheme that fuses a stable grasping and the object manipulation into a unique control signal is proposed. The force-closure conditions are defined to guarantee a stable grasping and the boundedness of the closed-loop signals is proved. Furthermore, the convergence of the contact force to its desired value is guaranteed, without any information about the radius of the fingertip. Simulation results are provided to visualize the stable grasping and the object manipulation, avoiding the gravity eﬀect.

1. Introduction

From physiological point of view, the human hands are considered as a powerful tool whereby the human brain interacts with the world, that is, how it perceives and acts with the environment1. In order to increase the dexterity in robotic hands, some intelligent human- like functions have been imitated.

In general, the dexterous manipulation in robotics, to emulate pinching motions, have been formulated in terms of the object, that is, the forces/torques exerted on it to produce the desired movements and how they behave2. The grasping and the object manipulation are

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based on the assumption that the contact between the object and fingers is frictionless, so the finger can only exert a force along the common normal axis at the contact point3. Then, to grasp the object without slipping the standard friction cone is used, generating a complex motion control since their evolution is governed by the laws of Coulomb friction, which is nonlinear and imposes constraints on the system. On the other hand, some authors have considered fingers with a very sharp curvature assuming that the contact point between the fingers and object does not change significantly. Although, some manipulation tasks are executed by robotic fingers successfully, this assumption is not valid for several manipulation tasks because the rolling between object and fingers is essential in the human manipulation tasks. Moreover, some attempts to determine the best grasp configuration and manipulation tasks are presented in 2D and 3D4–11. Unfortunately, a lot of them require an exact knowl- edge of the system parameters and the object localization12–15. Some authors, to reproduce more characteristics of the human fingertips, have considered the use of deformation models with hemispherical soft tips. Many hemispherical soft tip fingers have been designed and constructed to execute several manipulation tasks 16–19where a high-contact friction to prevent slipping and the rolling of the finger tip on the object surface are some characteristics 3,20,21. Nevertheless, in these approaches the contribution of the fingertip deformation on the manipulation tasks in a dynamic sense is not evident, considering that the rolling constraint is defined in kinematic or semidynamics sense21–24.

On the basis that human hands have fingers with soft tips, in this paper the grasping and the object manipulation is presented, using a pair of robotic fingers with n-degree of free- doms and deformable tips. The parallel deformation model is based on a virtual spring with infinitesimal section, where the normal and tangential deformations are taking into account 25. So, a tangential movement of the object without slipping and a dependency of the relative orientation between the object and the finger are considered. Inclusion of the normal and tangential deformations in the deformation model, contribute to reproduce some intrin- sic characteristics of the deformable material and a better grip. The key of our approach is to introduce the parallel deformation model to grasp an object using a pair of robotic fingers with n-degrees of freedom. Moreover, the grasping controller guaranteed that the contact force converges to the desired value, avoiding a direct dependence with the radius of the tip26,27. An approximation of the object angle based on the joint angular position of the fingers is used to control orientation of the object. Finally, a control signal for translation of the object is defined. To carry out the grasping and the object manipulation, the superposition principle is used 28,29, which allows us to separate a complex task into a set of basic tasks, where each task has a unique stationary point which represents the desired action27.

Boundedness of all closed-loop signals is proved, while the asymptotic stability is guaranteed using the stability on the manifold27,28,30. It is important to notice that forces-closure conditions, to grasp firmly an object, are satisfied dynamically during manipulation task execution, rather than a static equilibrium. The proposed approach is validated by numerical simulations through a pair of robotic fingers with soft tips in the horizontal plane.

This paper is organized as follows. Section 2 presents dynamical equations of the fingers-object system. The blind control law is proposed inSection 3. Simulation results to confirm the validity of our approach are presented inSection 4. Finally, the conclusions are presented inSection 5.

2. Dynamical Equations

Consider a pair of soft tip fingers, with three degree of freedom each one, grasping an object in the horizontal plane, as shown inFigure 1. In the fingers-object systemOis the origin for

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l L

O O^′

q11 q21

q22

q13

q12

r r

Oc.m.

O1 O2

q23

Figure 1: Rigid object grasped by a pair soft tip fingers.

the left finger and it is considered as the reference frame,Ois the origin for the right finger, andLis the distance between the origins of each finger. In addition,qi qi1, qi2, qi3^T is the joint angular positions of the fingeri,riis the radius of the soft tip fingeri,lis the length of the object,lijis the length of the linkjfor the fingeri,Oi xi, yiis the center position of the deformable fingertipsiwithi1,2,Oc.m. x, yis the center of mass of the object, andθis the orientation of the object. Unlike deformation model proposed by22,28, where the force applied to the object produce a distribute pressure and it is parametrized as a normal force fiwith respect to the object surface. In this paper we use the deformation model proposed in25, where a virtual spring inside the soft tip finger allows us to known the normal and tangential movements in the deformable material. In such a way, the elastic energy induced by the soft fingertipiis given as

Pi

dni, dti, φi

πE

d_ni³ 3 cos²

φi

d²_nidtitan φi

dnid²_ti

, 2.1

wheredni is the maximum radial deformation,φi is the object relative orientation angle,dti

is the contact tangential displacement of the object, and E is the Young’s modulus of the finger tip material, as shown inFigure 2. Thus, the total potential energy of the deformable fingertips is expressed as

P P1

dn1, dt1, φ1

P2

dn2, dt2, φ2

. 2.2

The constraint between the radial deformation of the fingertipiand the object is given as Cni−

ri−dni li −1ⁱ

x−xicosθ− y−yi

sinθ

0, 2.3

which guarantees that exists a distance that limits the grasping on the object in normal direction. A particular case of2.3is considered whendni 0 which represents the normal

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Oc.m.

O1

O2

θ

rY

λ1

λ2

f1

f2

Y1

Y2

dn1

dn2

dt1

dt2

rX

φ1

φ2

Figure 2: Deformation Model proposed by25.

constraint for a rigid fingertip as reported in30. Accordingly, the normal constrained for the fingers-object system is defined as

Cn

i

fiCni, 2.4

wherefiis the Lagrange multiplier and represents the contact forcei.

On the other hand, taking into account the curvature eﬀects of the fingertip, the rolling of the object on the soft tip finger will be defined as movement of the contact area, where the relative velocity of the contact area between the finger tips and the object is zero. Assuming that the normal deformationdniis smaller than the radius of the tipi25, the angular displacement of the contact area and the projection displacement of the center of mass of the object is defined as

Y˙i−rφ˙i, 2.5

where

Yi xi−xsinθ yi−y

cosθ, φiπ−−1ⁱθ−e^T_iqi,

2.6

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andei 1, . . . ,1^T of the same size as the vectorqi, fori1,2. Moreover, according to the deformation model, a tangential displacementdtion the deformable fingertipiarises when the object is rolling on the fingertip. Then, the rolling constraint between finger tipiand the object surface is given as

CtiYi−cSiriφidti0, 2.7 wherecSiis the integration constant with respect to the initial conditions of contact. To avoid initial conditions in2.7a velocity constraint is defined as25

C˙tiY˙iriφ˙id˙ti0. 2.8 The Lagrangian of the system with holonomic constraint is described by

LK−PCn, 2.9

wherePandCnare defined in2.2and2.4, respectively, andKis the kinetic energy of the system defined as

K Σi

1 2

q˙^T_iHi

qi

q˙imnid˙_ni² mtid˙²_ti 1

2p˙^TH0p,˙ 2.10 withHiqiis the inertia matrix of the fingeri,H0 diagm, m, I, p x, y, θ^T, andmni, mtiare the normal and tangential mass deformations, respectively. Applying the Lagrangian variational principle, incorporating the rolling velocity constraint, the equations of motion are expressed for each component of the vector as follows:

d dt

∂L

∂z˙

− ∂L

∂z ∂

∂z˙

λ1C˙t1λ2C˙t2

2.11

where z q^T₁, q^T₂, x, y, θ, dn1, dn2, dt1, dt2^T is the vector of generalized coordinates and λi

is the Lagrange multiplier which represents the tangential force exerted for the fingerion the object surface. Note that treatment of the velocity constraint should not be done in the Lagrangian2.9, but rather in the equations of motion31.

Thus, the equations of motion for the fingers are given as

Hi

qi

q¨i1 2H˙i

qi

q˙iSi

qi,q˙i

q˙i−−1ⁱfiJ_i^TrX−λi

J_i^TrY−rei

− πEd²_ni cos²

φi

dti2

3dnitan φi

eiui,

2.12

whererX cosθ,sinθ^T,rY sinθ,cosθ^T, Jiis the Jacobian of the pointxi, yiwith respect to the joint variablesqij,ei 1,1,1^T,1/2H˙iqi Siqi,q˙irepresent the matrix of Coriolis and centripetal forces, anduistands for the torque input. It is important to notice that

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the moments induced by the tangential and normal forces contribute to grasp and manipulate an object more securely. In addition, the terms of deformation model give us information about the behavior of the deformable material. Furthermore, the movement equations of the object are given as

mx¨− f1−f2

cosθ λ1λ2sinθ 0, 2.13

my¨ f1−f2

sinθ λ1λ2cosθ 0, 2.14

Iθ¨−Y1f1Y2f2−λ1dn1−l1 λ2dn2−l2

πEd²_n1 cos²

φ1

dt1 2

3dn1tan φ1

− πEd²_n2 cos²

φ2

dt22

3dn2tan φ2

0.

2.15

The last two terms of2.15represent the contribution of the deformable fingertips to assure a stable grasping through induced forces/moments. Finally, dynamical equations related to the normaldniand tangentialdtimovements on the fingertips are defined as

mnid¨niπE

d²_ni cos² φi

2dnidtitan φi

d²_ti

−fi 0, 2.16

mtid¨tiπE d²_nitan

φi

2dnidti

−λi0. 2.17

Summing the products between ˙q^T_i with 2.12, ˙x with2.13, ˙ywith 2.14, ˙θ with 2.15, ˙dniwith2.16, and ˙dtiwith2.17yields

i1,2

_t

0

q˙_i^Tui

dτEt−E0≥ −E0, 2.18

whereEKPcorresponds to the total energy of the system.

3. Controller Design

3.1. Immobilization on the Object

As first step before to execute manipulation tasks, a stable grasping must be established.

To grasp stably an object, the force-closure is used to guarantee that the object should be held securely by the fingers. This mean that maintaining the contact between the fingers and

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the object, that is,f1 >0 andf2>0 for anyt >0, the forces and torques applied on the object should immobilize it. Let the forces and torques applied on the object be defined as

− f1−f2

cosθ λ1λ2sinθ 0, f1−f2

sinθ λ1λ2cosθ 0,

−Y1f1Y2f2λ1dn1−l1−λ2dn2−l2

πEd²_n1 cos²

φ1

dt1 2

3dn1tan φ1

− πEd_n2² cos²

φ2

dt22

3dn2tan φ2

0.

3.1

Then, if we choose that

f1f2fd, λ1λ20, 3.2

the first two equations are equal to zero, while the third equation is given as

−fdY1−Y2 λ1dn1−l1dn2−l2

πEd²_n1 cos²

φ1

dt12

3dn1tan φ1

− πEd²_n2 cos²

φ2

dt22

3dn2tan φ2

0.

3.3

Thus, a force-closure can be established, if the forces acting on the object are defined as, fi−→fd, Y1−Y2 −→0, λi−→0,

πEd²_n1 cos²

φ1

dt12

3dn1tan φ1

− πEd²_n2 cos²

φ2

dt22

3dn2tan φ2

−→0 fori1,2, 3.4

wherefdis the desired normal force. Once fingers grasp an object and hold it securely, we are in conditions to execute manipulation tasks on the object as orientation and move it atx−y coordinates.

Inspired that humans can execute some manipulation tasks without any object information, in this paper a stable grasping and the object manipulation based only on the center position of the soft tip fingerxi, yiare presented, so that the orientation θand the object parameters are avoided. Using the superposition principle, the joint torqueuiapplied to each finger can be defined as

uiufciuθci

−ciq˙i −1ⁱfd

l J_i^T x1−x2

y1−y2

−1ⁱ βΔtanθ x2−x1 J_i^T

tanθ 1

, i1,2, 3.5

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whereciis a diagonal symmetric positive definite matrix,β >0,fd>0 is the desired contact force,Δtanθ tanθ−tanθd,θdis the desired angle of rotation, and

tanθ y1−y2

x2−x1, l² x1−x2²

y1−y2

₂

l_w² Y1−Y2² where lwr1r2l−dn1−dn2.

3.6

Notice thatufcirefers to the stable grasp exerted on the object, whileuθciindicates the orientation control of the object. In the former case, a stable grasp is achieved minimizing the distance between the centers of the fingertips through the normal and tangential forces, which guarantee the control objectivesΔfi → 0,ΔY → 0. In the latter case, to avoid the measure- ment ofθ, an approximation ofθis proposed by the trigonometric tangent function tanθ.

This approximation has the feature that the convergence of tanθto tanθis guaranteed onceΔY → 0 is satisfied. This implies that the conditions of the stable grasping Δfi → 0,ΔY → 0, and tanθshould be satisfied.

To start the analysis and to ensure a stable grasping, it is considered that the joint torque is defined asui ufci. Substitutinguiin2.12, the closed-loop system equations are defined as

Hi

qi

q¨i1 2H˙i

qi

q˙iSi

qi,q˙i

q˙i−−1ⁱΔfiJ_i^Trx−Δλi

J_i^Try−riei

− πEd²_ni cos²

φi

dti2

3dnitan φi

ei−−1ⁱfd

l ΔY riei−ciq˙i, mx¨−

Δf1−Δf2

cosθ Δλ1 Δλ2sinθ 0,

my¨

Δf1−Δf2

sinθ Δλ1 Δλ2cosθ 0,

Iθ¨−Δf1Y1 Δf2Y2−Δλ1dn1−l1 Δλ2dn2−l2 ²

i1

−1ⁱ πEd²_ni cos²

φi

dti2

3dnitan φi

−fd

l ΔYr1r2 0, mnid¨niπE

d²_ni cos² φi

2dnidtitan φi

d²_ti

−Δfi−lw

l fd0, mtid¨tiπE

d²_nitan φi

2dnidti

−Δλi −1ⁱfd

l ΔY 0,

3.7

whereΔf fi−fdlw

l ,ΔY Y1−Y2, andΔλiλi −1ⁱfd

l ΔY.

Expressing the closed-loop system equations3.7in a vector-matrix equation we have that

Hz¨Cz˙Pz−AΔλ−DΔYFz˙0, 3.8

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where

Hdiag H1

q1

, H2

q2

, m, m, I, mn1, mn2, mt1, mt2

,

Cdiag 1 2H˙1

q1

S1

q1,q˙1

,1 2H˙2

q2

S2

q2,q˙2

,0,0,0,0,0,0,0

, Fdiagc1, c2,0,0,0,0,0,0,0,

D

−fd

l r1e1,fd

l r2e2,0,0,fd

l r1r2, lw

lΔYfd, lw

lΔYfd,fd

l ,−fd

l T

, Δλ

Δf1,Δf2,Δλ1,Δλ2

_T ,

Pz

⎡

⎢⎢

⎢⎣

− πEd²_n1 cos²

φ1

dt12

3dn1tan φ1

e1

− πEd²_n2 cos²

φ2

dt22

3dn2tan φ2

e2

0 0 ₂

i1−1ⁱ πEd²_ni cos²

φi

dti2

3dnitan φi

πE

d²_n1 cos²

φ1

2dn1dt1tan φ1

d_t1²

πE

d²_n2 cos²

φ2

2dn2dt2tan φ2

d_t2²

πE

d²_n1tan φ1

2dn1dt1

πE

d²_n2tan φ2

2dn2dt2

⎤

⎥⎥

⎥⎦ ,

A

⎡

⎢⎢

⎢⎣

−J₁^TrX 0_3×1 JSq1 0_3×1 0_3×1 J₂^TrX 0_3×1 JSq2

cosθ −cosθ −sinθ −sinθ

−sinθ sinθ −cosθ −cosθ Y1 −Y2 dn1−l1 −dn2−l2

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥⎥

⎥⎦ ,

3.9

withJSqiJ_i^TrY −riei, fori1,2.

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Taking the inner product between ˙z and3.8yields d

dtKz,z ˙ Pz Partz −z˙^TFz,˙ d

dtETz,z ˙ −

i1,2

q˙_i^Tciq˙i

,

3.10

where Partz fd/2lΔY² −

i

_d_ni

0 fddξ is called artificial potential energy. Then, the system satisfys the passivity condition in closed loop.

Although ˙ETz,z˙ is negative definite along the solution trajectories of fingers-object system,Ez,z˙ cannot play a role of a Lyapunov function for the closed-loop system. Because it is neither define positive in the 26-dimensional state spacez,z˙ nor in 18-dimensional constrained manifoldM18defined by

M18

z,z˙ :Cti0,C˙ti0, Cni 0,C˙ni0

, fori1,2. 3.11 In order to define a constrained manifold where the trajectories of the system guarantees a stable grasping; a first step is to show the boundedness of solutions in the closed loop. Considering thatCni 0 andCti 0 we have thatd/dtCni z˙^T∂/∂zCni 0 and d/dtCti z˙^T∂/∂zCti 0, respectively. In fact, if the matrixA ∂/∂zCn1,∂/∂zCn2,

∂/∂zCt1, ∂/∂zCt2from previous expressions we have that 0A^Tz. Then,˙

0 d dt

A^Tz˙

A^Tz¨A˙^Tz.˙ 3.12

Now, if we multiply3.8byA^TH⁻¹we obtain that Δλ

A^TH⁻¹A₋₁

−A˙^Tz˙A^TH⁻¹Cz˙Pz−DΔYFz˙

, 3.13

whereA^Tz¨ −A˙^Tz˙ from3.12. Taking into account that ˙z,ΔY,dni,dti, andφi are bounded due to ETz,z˙ ≤ 0; we have that Δλ is bounded under assumption that the matrix A in nondegenerate. Then, from3.8we have that ¨zis uniformly bounded which implies that ˙z is uniformly continuous. Now, given that ˙q∈L2from3.10by Barbalat lemma32we have that ˙q → 0 ast → 0 which implies that ˙z → 0 ast → 0. Due to ˙z andΔY are uniformly continuous, we have that ¨zis uniformly continuous too. This in turn implies that ¨z → 0 at t → ∞and3.8will be defined as

Pz A, D Δλ

ΔY

. 3.14

Assuming that the matrixA, Dis nondegenerate, a point that minimizes the right side of the3.14onM18exists. The unique critical pointz^∗that minimizes3.14can be defined as

Δλ0_4×1, ΔY 0. 3.15

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This means that the closed-loop trajectories converge to critical point in the equilibrium point manifoldEPdefined as

M4

z:Cti0, Cni 0, ΔY → 0, Δfi → 0, Δλi → 0

, fori1,2. 3.16

Hence, the total potential energy is minimizing and the object is held securely. It is important to notice that initial conditions of the system and the length of the object are very important parameters to guarantee the convergence on M4. Furthermore, the matrix A, D is nondegenerate ifJ1andJ2are full rank matrices, that is, the matrix is degenerate ifqi2 qi3 0 orqi2 qi3 πfori1,2. These joint values represent a special configuration of the fingers which can be excluded as a possible configuration in the manipulation tasks. At the same time, it is possible to show that the matrixAis nondegenerate if theJ1 andJ2are full rank matrices.

Now, we are in conditions to define the following.

iLet the neighborhoodN26z^∗, r0singularity-free be with a radiusr0>0 around the critical pointz^∗,0defined as

N26z^∗, r0

z,z˙ : 1

2Δp^TH0Δp

i1,2

1 2Δq^T_iHi

qi

Δqi, ≤r₀²

, 3.17

whereΔp p−pd,Δqqi−qd. That is, singular configurations of the soft finger tips are avoided inside the neighborhoodN26z^∗, r0provide thatr0is chosen ade- quately.

iiLet the neighborhoodN18z^∗, r0around the critical point be where the closed-loop trajectories remain withδ >0, that is,

N18z^∗, r0 {z,z˙ :ET ≤δ, z,z˙ ∈M18}. 3.18

iiiDefinitionsee26If for any givenε >0 there existsδε>0 and another constant r1 > 0 being less thanr0and independent ofεsuch that the solution tracking trajectories starting from any initial conditionz0,z0˙ lying onN18δε∩N26r1 remains onN18ε∩N26r0and converges asymptotically to the setM4∩N18ε ast → ∞, then it is called that the statez^∗,0is stable.

Finally, we have the following result.

Theorem 3.1. Considering that the desired reference statez^∗,0and initial statez0,z0˙ lying onM4. The trajectories in the closed-loop system remains onN18ε∩N26r0and converges asymp- totically to the setM4∩N26εast → ∞under assumption that the matrixA, DΘis nondegenerate in a neighborhoodN26r0. Thus, is assured thatΔfi → 0,Y1−Y2 → 0, Δλi → 0 ast → ∞.

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Remark 3.2. Once the forces applied to the object have been compensated to hold it stably, the object can be rotated to the desired angle using the superposition principle, that is, the control law now is defined as

uiufciuθci. 3.19

Premultiplying ˙q^T_i byuθciwe have that 2

i1

q˙^T_iuθci d

dtE0, 3.20

whereE0 1/2βΔtanθ². This means that there exists a constantηsuch that _t

0

₂

i1

q˙^T_iuθci

dψE0t−E00≥η, 3.21

whereη−E00.

Now, taking the inner product between ˙z and the closed-loop system equations with uidefined in3.19we have that

d

dtET1−

i1,2

q˙^T_iciq˙i

, 3.22

where ET1 KP Part1 and Part1 Part 1/2βΔtanθ². As in stable grasping, the closed-loop trajectories converge to critical point on a constrained manifold whereΔλi → 0, Δfi → 0,ΔY → 0 andΔtanθ → 0 ast → ∞.

Remark 3.3. Now, when the stable grasping and object orientation tasks has been established, the objective will be to move the object to desired coordinatesxdby the following control law

uxi −γx

2x−xd∂xi

∂qi, 3.23

whereγx > 0,xd > 0,xi is the Cartesian coordinates andxis the estimated position of the object which is defined as an average distance between center positions of the deformable fingertips, that is,

x x1x2

2 . 3.24

Using the superposition principle, the control law for the stable grasping and object manipulation is defined as

uiufciuθciuxi fori1,2. 3.25

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As in the previous case, if we multiplyinguxiby ˙q^Twe have that 2

i1

q˙^T_iuxi d

dtE1, 3.26

whereE1 1/2γxx−xd²is the shifting energy to move the object inx-coordinates. Con- sequently, exists a constantη1such that

_t

0

₂

i1

q˙^T_iuxi

dψE1t−E10≥η1, 3.27

whereη1−E10.

Now, taking the inner product between ˙z and the closed-loop system equations with uidefined in3.25we have

d

dtET2−

i1,2

q˙^T_iciq˙i

, 3.28

whereET2 KPPart2andPart2Part1 1/2γxx−xd². Then, the passivity condition is satisfy in the closed-loop. Hence, the closed-loop system trajectories converge to critical point on a constrained manifold whereΔλi → 0,Δfi → 0,ΔY → 0,Δtanθ → 0, andx → xd

ast → ∞.

4. Simulation Results

In order to demonstrate usefulness of our scheme for stable grasping and object orientation, numerical simulations were carried out on a pair of deformable fingertips in the horizontal plane, seeFigure 1. The simulations were implemented on stiﬀnumerical solver on Matlab R2007b, under 1 ms sampling time. Additionally, to approximate the holonomic constraints was used the Constrained Stabilization MethodCSM 33.

The physical parameters of the fingers and object are shown inTable 1wherelij,mij

andIij are the length, mass and moment of inertia of the linkj 1,2,3 for the fingeri1,2, andM,I,lare the mass, moment of inertia and the length of the object, respectivelyTable 2.

Moreover,L 0.64mis the distance between fingers,E 50000N/m²is the Young’s modulus of the fingertips andri 0.01mis the radius of the hemispherical finger tip for i1,2.

The simulation study is divided in three steps. As first step before any manipulation task is necessary to guarantee that the stable grasp is achieved through the control law defined asui ufci. The initial conditions used in the simulations are q10 30,91.61, 71.24^T,q20 30,91.61,71.24^T,x0, y0 0.032,0.047 m,θ0 0, anddn1, dn2, dt1, dt2 0.0025,0.0025,0,0 mwhich establish the following conditions:f10 1.1 N, f20 1.1 NΔY0 0 m. For reference, these initial conditions are called normal initial conditionsCIN.

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Table 1: Physical parameters.

Parameter Value

l11l21 0.05m

l12l22 0.04m

l13l23 0.03m

m11m21 0.05kg

m12m22 0.03kg

m13m23 0.02kg

I11I21 1.4167×10⁻⁵kg m²

I12I22 6.25×10⁻⁶kg m²

I13I23 3×10⁻⁶kg m²

Table 2: Parameters of the object.

Parameter Value

M 0.02kg

I 5.67×10⁻⁶kg m²

l 0.03m

InFigure 3we observe that the forces on the objectΔfi andΔλi converge rapidly to the desired values using the normal initial conditions defined previously. At the same time, the fast convergence ofΔY to zero is shown inFigure 3. On the other hand, the Figures4,5, and6 show the performance ofΔfi,Δλi, andΔY for diﬀerent values of damping gainsci. Notice that the damping gains are related with the convergence velocity towards equilibrium point that minimizes the potential energy of the system. However, the system takes longer to converge and several oscillations arise when the damping gains are small. The control parameters used in this step arefd1Nandc1c20.01Nms/rad.

Finally, the Figures7, 8, and 9 show the convergence ofΔfi andΔY to zero under more extreme initial conditions. The initial conditions used in these simulations are CI1and CI2 which are defined as q10 30,95.65,64.93^T, q20 30,100.45,76.09^T, ΔY0 0.01m, and q10 30,93.73,68.32^T, q20 30,102.16,72.78^T, dn1, dn2, dt1, dt2

0.0013,0.0037,0,0 m,ΔY0 0.01m, respectively.

Notice that the convergence to zero of Δfi and ΔY present a transient responses for diﬀerent initial conditions, but still converge to the desired values in few seconds.

Additionally, the control objectives converge in diﬀerent times. Specially, the convergence ofΔY takes more time for establishing a stable grasping when the initial conditions are more extreme, CI₂. Thus, it is possible to describe the stable grasping in two phases. In the first phase, the stabilization of the normal and tangential forces on object is performed while the stabilization of the rotational moments to stop the angular motion of the object is carried out in a second phase.

Once the all forces applied to the object have been compensated, it is possible to execute any manipulation task. In this case the rotation of the object, to a desired angleθd, will be realized. Using the superposition principle the control law is given asui ufciuθci. The convergence to zero ofΔfi,Δλi,ΔY, andΔθusing the CI_Ninitial conditions are shown in Figures10and11. It is important to notice the special role ofY1−Y2as a parameter to increase the dexterity. In this case, the convergence ofΔY is closely associated to object orientation

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0.1 0

−0.1

0.5 0

−0.5

1 0

−1

0 0.2 0.4 0.6 0.8 1

t(s)

0 0.2 0.4 0.6 0.8 1

t(s)

0 0.2 0.4 0.6 0.8 1

t(s)

∆f1

∆f2

∆λ1

∆λ2

∆Y

∆Y(m)∆λi(N)∆fi(N)

×10⁻³

Figure 3: Convergence ofΔfi,Δλi, andΔYfori1,2.

0.15 0.1 0.05 0

−0.05

−0.1

−0.15

−0.2

ci=0.1 ci=0.01 ci=0.001

Time(s)

0 0.2 0.4 0.6 0.8 1

∆f1(N)

Figure 4: Convergence ofΔficonsidering several values ofci.

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Time(s)

0 0.2 0.4 0.6 0.8 1

ci=0.1 ci=0.01 ci=0.001 0.3

0.2 0.1 0

−0.1

−0.2

−0.3

−0.4

∆λ1(N)

Figure 5: Convergence ofΔλiconsidering several values ofci.

∆Y(mt)

Time(s)

0 0.2 0.4 0.6 0.8 1

10 8 6 4 2 0

−2

−4

−6

×10⁻⁴

ci=0.1 ci=0.01 ci=0.001

Figure 6: Convergence ofΔYconsidering several values ofci.

through tanθ. The control parameters using in this second step arefd 1N,c1 c2 0.01Nms/rad,β0.1N/rad, andθd−10π/180 rad.

As the final step of this study, a stable grasping and object manipulation which include the orientation and translation of the object to a desired reference is presented. Using a superposition principle, the control law for this step is defined asuufciuθciutras.

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CIN

CI1

CI2

0 2 4 6 8 10

Time(s) 0.8

0.6 0.4 0.2 0

−0.2

−0.4

−0.6

∆f1(N)

−0.8 1.2 1

Figure 7: Convergence ofΔficonsidering diﬀerent initial conditions.

∆λ1(N)

CIN

CI1

CI2

0 2 4 6 8 10

Time(s) 0.8

0.6 0.4 0.2 0

−0.2

−0.4

−0.6

Figure 8: Convergence ofΔλiconsidering diﬀerent initial conditions.

The Figures 12,13, and 14show the convergence of Δfi, Δλi,ΔY, and Δθ to zero using CI_N initial conditions. The control parameters used in this step arefd 1 N,c1 c2 0.01Nms/rad,β 0.1 N/rad,θd −10π/180 rad,γx 50N/m, and xd x00.01m.

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∆Y(mt)

CIN

CI1

CI2

12 10 8 6 4 2 0

−2

×10⁻³

0 2 4 6 8 10

Time(s)

Figure 9: Convergence ofΔYconsidering diﬀerent initial conditions.

t(s)

∆f1

∆f2

∆λ1

∆λ2

0 2 4 6 8 10

t(s)

0 2 4 6 8 10

0.4 0.2 0

−0.2

−0.4

1 0.5 0

−0.5

∆λi(N)∆fi(N)

Figure 10: Convergence ofΔfiandΔλifori1,2.

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0.3 0.2 0.1 0

−0.1

∆θ(rad)

∆θ

∆Y 2 1 0

−1

−2

t(s)

0 2 4 6 8 10

t(s)

0 2 4 6 8 10

∆Y(m)

×10⁻³

Figure 11: Convergence ofΔYandΔθfori1,2.

∆Y(m)∆θ(rad)

2 0

−2

0.2 0

−0.2

0.02 0

−0.02

∆Y

∆x

∆θ

∆x(m)

t(s)

0 2 4 6 8 10

t(s)

0 2 4 6 8 10

t(s)

0 2 4 6 8 10

×10⁻³

Figure 12: Convergence ofΔY,Δθ, andΔxfori1,2.

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t(s)

∆f1

∆f2

∆λ1

∆λ2

0 2 4 6 8 10

t(s)

0 2 4 6 8 10

0.4 0.2 0

−0.2

−0.4

1 0.5 0

−0.5

∆λi(N)∆fi(N)

Figure 13: Convergence ofΔfiandΔλifori1,2.

∆Y 1.5

1 0.5 0

−0.5

−1

−1.50 2 4 6 8 10

(mt)

×10⁻³

(s)

Figure 14: Convergence ofΔYfori1,2.

5. Conclusions

A scheme to grasp and manipulate an object using soft tip fingers is presented. In order to include more characteristics of the human fingertips a parallel deformation model is used. The control law proposed ensures stability on a constrained manifold. Furthermore,

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the control law avoids information of the radius of the tips and the convergence to the desired force value is guaranteed.

Numerical simulations, on a pair of deformable fingertips in horizontal plane, allow us to visualize the convergence of the closed-loop trajectories to the desired point. In addition, the special role ofΔY in the manipulation task and the eﬀects of the superposition principle that have been observed.

Acknowledgments

This work was partially supported by CONICYT, Departamento de Relaciones Interna- cionales, Programa de Cooperaci ´on Cient´ıfica Internacional, CONICYT/CONACYT 2011- 380, and the Universidad de los Andes, Chile, FAI project ICI-002-11.

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