# Parallel-Distributed Model Deformation in the Fingertips for Stable Grasping and Object Manipulation

Authors: R. García-Rodríguez; G. Díaz-Rodríguez
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

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 withn-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.


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## Body

## 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 environment [1]. 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 behave [2]. The grasping and the object manipulation are 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 point [3]. 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 3D [4–11]. Unfortunately, a lot of them require an exact knowledge of the system parameters and the object localization [12–15]. 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 [16–19] where 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 sense [21–24].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 withn-degree of freedoms 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 intrinsic 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 tip [26, 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 action [27]. Boundedness of all closed-loop signals is proved, while the asymptotic stability is guaranteed using the stability on the manifold [27, 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. Section2 presents dynamical equations of the fingers-object system. The blind control law is proposed in Section 3. Simulation results to confirm the validity of our approach are presented in Section 4. Finally, the conclusions are presented in Section 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 in Figure1. In the fingers-object system O is the origin for the left finger and it is considered as the reference frame, O′ is the origin for the right finger, and L is the distance between the origins of each finger. In addition, qi=[qi1,qi2,qi3]T is the joint angular positions of the finger i, ri is the radius of the soft tip finger i, l is the length of the object, lij is the length of the link j for the finger i, Oi=(xi,yi) is the center position of the deformable fingertips i with i=1,2, Oc.m.=(x,y) is the center of mass of the object, and θ is the orientation of the object. Unlike deformation model proposed by [22, 28], where the force applied to the object produce a distribute pressure and it is parametrized as a normal force fi with respect to the object surface. In this paper we use the deformation model proposed in [25], 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 fingertip i is given as
(2.1)Pi(dni,dti,ϕi)=πE[dni33cos2(ϕi)+dni2dtitan(ϕi)+dnidti2],
where dni 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 in Figure 2. Thus, the total potential energy of the deformable fingertips is expressed as
(2.2)P=P1(dn1,dt1,ϕ1)+P2(dn2,dt2,ϕ2).
The constraint between the radial deformation of the fingertip i and the object is given as
(2.3)Cni=-[(ri-dni)+li+(-1)i((x-xi)cos(θ)-(y-yi)sin(θ))]=0,
which guarantees that exists a distance that limits the grasping on the object in normal direction. A particular case of (2.3) is considered when dni=0 which represents the normal constraint for a rigid fingertip as reported in [30]. Accordingly, the normal constrained for the fingers-object system is defined as
(2.4)Cn=∑i‍fiCni,
where fi is the Lagrange multiplier and represents the contact force i.Figure 1
Rigid object grasped by a pair soft tip fingers.Figure 2
Deformation Model proposed by [25].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 deformation dni is smaller than the radius of the tip i [25], the angular displacement of the contact area and the projection displacement of the center of mass of the object is defined as
(2.5)Y˙i=-rϕ˙i,
where
(2.6)Yi=(xi-x)sin(θ)+(yi-y)cos(θ),ϕi=π-(-1)iθ-eiTqi,
and ei=[1,…,1]T of the same size as the vector qi, for i=1,2. Moreover, according to the deformation model, a tangential displacement dti on the deformable fingertip i arises when the object is rolling on the fingertip. Then, the rolling constraint between finger tip i and the object surface is given as
(2.7)Cti=Yi-cSi+riϕi+dti=0,
where cSi is the integration constant with respect to the initial conditions of contact. To avoid initial conditions in (2.7) a velocity constraint is defined as [25]
(2.8)C˙ti=Y˙i+riϕ˙i+d˙ti=0.
The Lagrangian of the system with holonomic constraint is described by
(2.9)L=K-P+Cn,
where P and Cn are defined in (2.2) and (2.4), respectively, and K is the kinetic energy of the system defined as
(2.10)K=Σi12[q˙iTHi(qi)q˙i+mnid˙ni2+mtid˙ti2]+12p˙TH0p˙,
with Hi(qi) is the inertia matrix of the finger i, H0=diag(m,m,I),p=[x,y,θ]T, and mni, mti are 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:
(2.11)ddt(∂L∂z˙)-∂L∂z=∂∂z˙(λ1C˙t1+λ2C˙t2)
where z=[q1T,q2T,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 finger i on the object surface. Note that treatment of the velocity constraint should not be done in the Lagrangian (2.9), but rather in the equations of motion [31].Thus, the equations of motion for the fingers are given as(2.12)Hi(qi)q¨i+12H˙i(qi)q˙i+Si(qi,q˙i)q˙i-(-1)ifiJiTrX-λi(JiTrY-rei)-πEdni2cos2(ϕi)(dti+23dnitan(ϕi))ei=ui,
where rX=[cos(θ),sin(θ)]T,  rY=[sin(θ),cos(θ)]T,Ji is the Jacobian of the point (xi,yi) with respect to the joint variables qij, ei=[1,1,1]T, (1/2)H˙i(qi)+Si(qi,q˙i) represent the matrix of Coriolis and centripetal forces, and ui stands for the torque input. It is important to notice that 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
(2.13)mx¨-(f1-f2)cos(θ)+(λ1+λ2)sin(θ)=0,(2.14)my¨+(f1-f2)sin(θ)+(λ1+λ2)cos(θ)=0,(2.15)Iθ¨-Y1f1+Y2f2-λ1(dn1-l1)+λ2(dn2-l2)+πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))=0.
The last two terms of (2.15) represent the contribution of the deformable fingertips to assure a stable grasping through induced forces/moments. Finally, dynamical equations related to the normal dni and tangential dti movements on the fingertips are defined as
(2.16)mnid¨ni+πE(dni2cos2(ϕi)+2dnidtitan(ϕi)+dti2)-fi=0,(2.17)mtid¨ti+πE(dni2tan(ϕi)+2dnidti)-λi=0.Summing the products betweenq˙iT with (2.12), x˙ with (2.13), y˙ with (2.14), θ˙ with (2.15), d˙ni with (2.16), and d˙ti with  (2.17) yields
(2.18)∑i=1,2‍∫0t‍(q˙iTui)dτ=E(t)-E(0)≥-E(0),
where E=K+P corresponds 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 the object, that is,f1>0 and f2>0 for any t>0, the forces and torques applied on the object should immobilize it. Let the forces and torques applied on the object be defined as
(3.1)-(f1-f2)cos(θ)+(λ1+λ2)sin(θ)=0,(f1-f2)sin(θ)+(λ1+λ2)cos(θ)=0,-Y1f1+Y2f2+λ1(dn1-l1)-λ2(dn2-l2)+πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))=0.
Then, if we choose that
(3.2)f1=f2=fd,λ1+λ2=0,
the first two equations are equal to zero, while the third equation is given as
(3.3)-fd(Y1-Y2)+λ1(dn1-l1+dn2-l2)+πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))=0.
Thus, a force-closure can be established, if the forces acting on the object are defined as,
(3.4)fi→fd,Y1-Y2→0,λi→0,πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))→0fori=1,2,
where fd is 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 at x-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 finger(xi,yi) are presented, so that the orientation θ and the object parameters are avoided. Using the superposition principle, the joint torque ui applied to each finger can be defined as
(3.5)ui=ufci+uθci=-ciq˙i+(-1)ifdl-JiT(x1-x2y1-y2)+(-1)iβΔtan(θ)¯x2-x1JiT(tan(θ)¯1),i=1,2,
where ci is a diagonal symmetric positive definite matrix, β>0,  fd>0 is the desired contact force, Δtan(θ)¯=tan(θ)¯-tan(θd),  θd is the desired angle of rotation, and
(3.6)tan(θ)¯=y1-y2x2-x1,l-2=(x1-x2)2+(y1-y2)2=lw2+(Y1-Y2)2wherelw=r1+r2+l-dn1-dn2.
Notice that ufci refers to the stable grasp exerted on the object, while uθci indicates 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 objectives Δfi→0,  ΔY→0. In the latter case, to avoid the measurement of θ, an approximation of θ is proposed by the trigonometric tangent function tan(θ)¯. This approximation has the feature that the convergence of tan(θ)¯ to tan(θ) is guaranteed once ΔY→0 is satisfied. This implies that the conditions of the stable grasping Δfi→0,  ΔY→0, and tan(θ)¯ should be satisfied.To start the analysis and to ensure a stable grasping, it is considered that the joint torque is defined asui=ufci. Substituting ui in (2.12), the closed-loop system equations are defined as
(3.7)Hi(qi)q¨i+12H˙i(qi)q˙i+Si(qi,q˙i)q˙i-(-1)iΔfiJiTrx-Δλi(JiTry-riei)-πEdni2cos2(ϕi)(dti+23dnitan(ϕi))ei-(-1)ifdl-ΔYriei=-ciq˙i,mx¨-(Δf1-Δf2)cos(θ)+(Δλ1+Δλ2)sin(θ)=0,my¨+(Δf1-Δf2)sin(θ)+(Δλ1+Δλ2)cos(θ)=0,Iθ¨-Δf1Y1+Δf2Y2-Δλ1(dn1-l1)+Δλ2(dn2-l2)+∑i=12‍(-1)iπEdni2cos2(ϕi)(dti+23dnitan(ϕi))-fdl-ΔY(r1+r2)=0,mnid¨ni+πE(dni2cos2(ϕi)+2dnidtitan(ϕi)+dti2)-Δfi-lwl-fd=0,mtid¨ti+πE(dni2tan(ϕi)+2dnidti)-Δλi+(-1)ifdl-ΔY=0,
where Δf=fi-fdlwl-,  ΔY=Y1-Y2, and Δλi=λi+(-1)ifdl-ΔY.Expressing the closed-loop system equations (3.7) in a vector-matrix equation we have that
(3.8)Hz¨+Cz˙+P(z)-AΔλ-DΔY+Fz˙=0,
where
(3.9)H=diag(H1(q1),H2(q2),m,m,I,mn1,mn2,mt1,mt2),C=diag(12H˙1(q1)+S1(q1,q˙1),12H˙2(q2)+S2(q2,q˙2),0,0,0,0,0,0,0),F=diag(c1,c2,0,0,0,0,0,0,0),D=[-fdl-r1e1,fdl-r2e2,0,0,fdl-(r1+r2),lwl-ΔYfd,lwl-ΔYfd,fdl-,-fdl-]T,Δλ=[Δf1,Δf2,Δλ1,Δλ2]T,P(z)=[-πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))e1-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))e200∑i=12(-1)iπEdni2cos2(ϕi)(dti+23dnitan(ϕi))πE(dn12cos2(ϕ1)+2dn1dt1tan(ϕ1)+dt12)πE(dn22cos2(ϕ2)+2dn2dt2tan(ϕ2)+dt22)πE(dn12tan(ϕ1)+2dn1dt1)πE(dn22tan(ϕ2)+2dn2dt2)],A=[-J1TrX03×1JSq103×103×1J2TrX03×1JSq2cos(θ)-cos(θ)-sin(θ)-sin(θ)-sin(θ)sin(θ)-cos(θ)-cos(θ)Y1-Y2(dn1-l1)-(dn2-l2)1000010000100001],
with JSqi=JiTrY-riei, for i=1,2.Taking the inner product betweenz˙ and (3.8) yields
(3.10)ddt(K(z,z˙)+P(z)+Part(z))=-z˙TFz˙,ddtET(z,z˙)=-∑i=1,2‍(q˙iTciq˙i),
where Part(z)=(fd/2l-)ΔY2-∑i‍∫0dni‍fdd(ξ) is called artificial potential energy. Then, the system satisfys the passivity condition in closed loop.AlthoughE˙T(z,z˙) is negative definite along the solution trajectories of fingers-object system, E(z,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 space (z,z˙) nor in 18-dimensional constrained manifold M18 defined by
(3.11)M18={(z,z˙):Cti=0,C˙ti=0,Cni=0,C˙ni=0},fori=1,2.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 and Cti=0 we have that (d/dt)Cni=z˙T(∂/∂z)Cni=0 and (d/dt)Cti=z˙T(∂/∂z)Cti=0, respectively. In fact, if the matrix A=[(∂/∂z)Cn1,(∂/∂z)Cn2,(∂/∂z)Ct1,(∂/∂z)Ct2] from previous expressions we have that 0=ATz˙. Then,
(3.12)0=ddt(ATz˙)=ATz¨+A˙Tz˙.Now, if we multiply (3.8) by ATH-1 we obtain that
(3.13)Δλ=(ATH-1A)-1[-A˙Tz˙+ATH-1(Cz˙+P(z)-DΔY+Fz˙)],
where ATz¨=-A˙Tz˙ from (3.12). Taking into account that z˙, ΔY, dni, dti, and ϕi are bounded due to ET(z,z˙)≤0; we have that Δλ is bounded under assumption that the matrix A in nondegenerate. Then, from (3.8) we have that z¨ is uniformly bounded which implies that z˙ is uniformly continuous. Now, given that q˙∈L2 from (3.10) by Barbalat lemma [32] we have that q˙→0 as t→0 which implies that z˙→0 as t→0. Due to z˙ and ΔY are uniformly continuous, we have that z¨ is uniformly continuous too. This in turn implies that z¨→0 at t→∞ and (3.8) will be defined as
(3.14)P(z)=[A,D][ΔλΔY].
Assuming that the matrix [A,D] is nondegenerate, a point that minimizes the right side of the (3.14) on M18 exists. The unique critical point (z*) that minimizes (3.14) can be defined as
(3.15)Δλ=04×1,ΔY=0.
This means that the closed-loop trajectories converge to critical point in the equilibrium point manifold (EP) defined as
(3.16)M4={(z):Cti=0,Cni=0,ΔY→0,Δfi→0,Δλi→0},fori=1,2.
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 non-degenerate if J1 and J2 are full rank matrices, that is, the matrix is degenerate if qi2=qi3=0 or qi2=qi3=π for i=1,2. 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 matrix A is nondegenerate if the J1 and J2 are full rank matrices.Now, we are in conditions to define the following.(i)
Let the neighborhoodN26(z*,r0) singularity-free be with a radius r0>0 around the critical point (z*,0) defined as
(3.17)N26(z*,r0)={(z,z˙):12ΔpTH0Δp+∑i=1,2‍12ΔqiTHi(qi)Δqi,≤r02},
where Δp=p-pd, Δq=qi-qd. That is, singular configurations of the soft finger tips are avoided inside the neighborhood N26(z*,r0) provide that r0 is chosen adequately.(ii)
Let the neighborhoodN18(z*,r0) around the critical point be where the closed-loop trajectories remain with δ>0, that is,
(3.18)N18(z*,r0)={(z,z˙):ET≤δ,(z,z˙)∈M18}.(iii)
Definition(see [26]) If for any given ε>0 there exists δ(ε)>0 and another constant r1>0 being less than r0 and independent of ε such that the solution tracking trajectories starting from any initial condition (z(0),z˙(0)) lying on N18(δ(ε))∩N26(r1) remains on N18(ε)∩N26(r0) and converges asymptotically to the set M4∩N18(ε) as t→∞, then it is called that the state (z*,0) is stable.Finally, we have the following result.Theorem 3.1.
Considering that the desired reference state(z*,0) and initial state (z(0),z˙(0)) lying on M4. The trajectories in the closed-loop system remains on N18(ε)∩N26(r0) and converges asymptotically to the set M4∩N26(ε) as t→∞ under assumption that the matrix [A,DΘ] is nondegenerate in a neighborhood N26(r0). Thus, is assured that Δfi→0,  Y1-Y2→0,  Δλi→0 as t→∞.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(3.19)ui=ufci+uθci.
Premultiplying q˙iT by uθci we have that
(3.20)∑i=12‍q˙iTuθci=ddt(E0),
where E0=(1/2)β(Δtan(θ)¯)2. This means that there exists a constant η such that
(3.21)∫0t‍(∑i=12‍q˙iTuθci)dψ=E0(t)-E0(0)≥η,
where η=-E0(0).
Now, taking the inner product betweenz˙ and the closed-loop system equations with ui defined in (3.19) we have that
(3.22)ddtET1=-∑i=1,2‍(q˙iTciq˙i),
where ET1=K+P+Part1 and Part1=Part+(1/2)β(Δtan(θ)¯)2. As in stable grasping, the closed-loop trajectories converge to critical point on a constrained manifold where Δλi→0, Δfi→0, ΔY→0 and Δtan(θ)¯→0 as t→∞.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 coordinatesxd by the following control law
(3.23)uxi=-γx2(x--xd)∂xi∂qi,
where γx>0,  xd>0,  xi is the Cartesian coordinates and x- is the estimated position of the object which is defined as an average distance between center positions of the deformable fingertips, that is,
(3.24)x-=x1+x22.
Using the superposition principle, the control law for the stable grasping and object manipulation is defined as
(3.25)ui=ufci+uθci+uxifori=1,2.
As in the previous case, if we multiplying uxi by q˙T we have that
(3.26)∑i=12‍q˙iTuxi=ddt(E1),
where E1=(1/2)γx(x--xd)2 is the shifting energy to move the object in x-coordinates. Consequently, exists a constant η1 such that
(3.27)∫0t‍(∑i=12‍q˙iTuxi)dψ=E1(t)-E1(0)≥η1,
where η1=-E1(0).Now, taking the inner product betweenz˙ and the closed-loop system equations with ui defined in (3.25) we have
(3.28)ddtET2=-∑i=1,2‍(q˙iTciq˙i),
where ET2=K+P+Part2 and Part2=Part1+(1/2)γx(x--xd)2. Then, the passivity condition is satisfy in the closed-loop. Hence, the closed-loop system trajectories converge to critical point on a constrained manifold where Δλi→0, Δfi→0, ΔY→0, Δtan(θ)¯→0,  and x-→xd as t→∞.


## 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 the object, that is,f1>0 and f2>0 for any t>0, the forces and torques applied on the object should immobilize it. Let the forces and torques applied on the object be defined as
(3.1)-(f1-f2)cos(θ)+(λ1+λ2)sin(θ)=0,(f1-f2)sin(θ)+(λ1+λ2)cos(θ)=0,-Y1f1+Y2f2+λ1(dn1-l1)-λ2(dn2-l2)+πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))=0.
Then, if we choose that
(3.2)f1=f2=fd,λ1+λ2=0,
the first two equations are equal to zero, while the third equation is given as
(3.3)-fd(Y1-Y2)+λ1(dn1-l1+dn2-l2)+πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))=0.
Thus, a force-closure can be established, if the forces acting on the object are defined as,
(3.4)fi→fd,Y1-Y2→0,λi→0,πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))→0fori=1,2,
where fd is 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 at x-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 finger(xi,yi) are presented, so that the orientation θ and the object parameters are avoided. Using the superposition principle, the joint torque ui applied to each finger can be defined as
(3.5)ui=ufci+uθci=-ciq˙i+(-1)ifdl-JiT(x1-x2y1-y2)+(-1)iβΔtan(θ)¯x2-x1JiT(tan(θ)¯1),i=1,2,
where ci is a diagonal symmetric positive definite matrix, β>0,  fd>0 is the desired contact force, Δtan(θ)¯=tan(θ)¯-tan(θd),  θd is the desired angle of rotation, and
(3.6)tan(θ)¯=y1-y2x2-x1,l-2=(x1-x2)2+(y1-y2)2=lw2+(Y1-Y2)2wherelw=r1+r2+l-dn1-dn2.
Notice that ufci refers to the stable grasp exerted on the object, while uθci indicates 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 objectives Δfi→0,  ΔY→0. In the latter case, to avoid the measurement of θ, an approximation of θ is proposed by the trigonometric tangent function tan(θ)¯. This approximation has the feature that the convergence of tan(θ)¯ to tan(θ) is guaranteed once ΔY→0 is satisfied. This implies that the conditions of the stable grasping Δfi→0,  ΔY→0, and tan(θ)¯ should be satisfied.To start the analysis and to ensure a stable grasping, it is considered that the joint torque is defined asui=ufci. Substituting ui in (2.12), the closed-loop system equations are defined as
(3.7)Hi(qi)q¨i+12H˙i(qi)q˙i+Si(qi,q˙i)q˙i-(-1)iΔfiJiTrx-Δλi(JiTry-riei)-πEdni2cos2(ϕi)(dti+23dnitan(ϕi))ei-(-1)ifdl-ΔYriei=-ciq˙i,mx¨-(Δf1-Δf2)cos(θ)+(Δλ1+Δλ2)sin(θ)=0,my¨+(Δf1-Δf2)sin(θ)+(Δλ1+Δλ2)cos(θ)=0,Iθ¨-Δf1Y1+Δf2Y2-Δλ1(dn1-l1)+Δλ2(dn2-l2)+∑i=12‍(-1)iπEdni2cos2(ϕi)(dti+23dnitan(ϕi))-fdl-ΔY(r1+r2)=0,mnid¨ni+πE(dni2cos2(ϕi)+2dnidtitan(ϕi)+dti2)-Δfi-lwl-fd=0,mtid¨ti+πE(dni2tan(ϕi)+2dnidti)-Δλi+(-1)ifdl-ΔY=0,
where Δf=fi-fdlwl-,  ΔY=Y1-Y2, and Δλi=λi+(-1)ifdl-ΔY.Expressing the closed-loop system equations (3.7) in a vector-matrix equation we have that
(3.8)Hz¨+Cz˙+P(z)-AΔλ-DΔY+Fz˙=0,
where
(3.9)H=diag(H1(q1),H2(q2),m,m,I,mn1,mn2,mt1,mt2),C=diag(12H˙1(q1)+S1(q1,q˙1),12H˙2(q2)+S2(q2,q˙2),0,0,0,0,0,0,0),F=diag(c1,c2,0,0,0,0,0,0,0),D=[-fdl-r1e1,fdl-r2e2,0,0,fdl-(r1+r2),lwl-ΔYfd,lwl-ΔYfd,fdl-,-fdl-]T,Δλ=[Δf1,Δf2,Δλ1,Δλ2]T,P(z)=[-πEdn12cos2(ϕ1)(dt1+23dn1tan(ϕ1))e1-πEdn22cos2(ϕ2)(dt2+23dn2tan(ϕ2))e200∑i=12(-1)iπEdni2cos2(ϕi)(dti+23dnitan(ϕi))πE(dn12cos2(ϕ1)+2dn1dt1tan(ϕ1)+dt12)πE(dn22cos2(ϕ2)+2dn2dt2tan(ϕ2)+dt22)πE(dn12tan(ϕ1)+2dn1dt1)πE(dn22tan(ϕ2)+2dn2dt2)],A=[-J1TrX03×1JSq103×103×1J2TrX03×1JSq2cos(θ)-cos(θ)-sin(θ)-sin(θ)-sin(θ)sin(θ)-cos(θ)-cos(θ)Y1-Y2(dn1-l1)-(dn2-l2)1000010000100001],
with JSqi=JiTrY-riei, for i=1,2.Taking the inner product betweenz˙ and (3.8) yields
(3.10)ddt(K(z,z˙)+P(z)+Part(z))=-z˙TFz˙,ddtET(z,z˙)=-∑i=1,2‍(q˙iTciq˙i),
where Part(z)=(fd/2l-)ΔY2-∑i‍∫0dni‍fdd(ξ) is called artificial potential energy. Then, the system satisfys the passivity condition in closed loop.AlthoughE˙T(z,z˙) is negative definite along the solution trajectories of fingers-object system, E(z,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 space (z,z˙) nor in 18-dimensional constrained manifold M18 defined by
(3.11)M18={(z,z˙):Cti=0,C˙ti=0,Cni=0,C˙ni=0},fori=1,2.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 and Cti=0 we have that (d/dt)Cni=z˙T(∂/∂z)Cni=0 and (d/dt)Cti=z˙T(∂/∂z)Cti=0, respectively. In fact, if the matrix A=[(∂/∂z)Cn1,(∂/∂z)Cn2,(∂/∂z)Ct1,(∂/∂z)Ct2] from previous expressions we have that 0=ATz˙. Then,
(3.12)0=ddt(ATz˙)=ATz¨+A˙Tz˙.Now, if we multiply (3.8) by ATH-1 we obtain that
(3.13)Δλ=(ATH-1A)-1[-A˙Tz˙+ATH-1(Cz˙+P(z)-DΔY+Fz˙)],
where ATz¨=-A˙Tz˙ from (3.12). Taking into account that z˙, ΔY, dni, dti, and ϕi are bounded due to ET(z,z˙)≤0; we have that Δλ is bounded under assumption that the matrix A in nondegenerate. Then, from (3.8) we have that z¨ is uniformly bounded which implies that z˙ is uniformly continuous. Now, given that q˙∈L2 from (3.10) by Barbalat lemma [32] we have that q˙→0 as t→0 which implies that z˙→0 as t→0. Due to z˙ and ΔY are uniformly continuous, we have that z¨ is uniformly continuous too. This in turn implies that z¨→0 at t→∞ and (3.8) will be defined as
(3.14)P(z)=[A,D][ΔλΔY].
Assuming that the matrix [A,D] is nondegenerate, a point that minimizes the right side of the (3.14) on M18 exists. The unique critical point (z*) that minimizes (3.14) can be defined as
(3.15)Δλ=04×1,ΔY=0.
This means that the closed-loop trajectories converge to critical point in the equilibrium point manifold (EP) defined as
(3.16)M4={(z):Cti=0,Cni=0,ΔY→0,Δfi→0,Δλi→0},fori=1,2.
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 non-degenerate if J1 and J2 are full rank matrices, that is, the matrix is degenerate if qi2=qi3=0 or qi2=qi3=π for i=1,2. 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 matrix A is nondegenerate if the J1 and J2 are full rank matrices.Now, we are in conditions to define the following.(i)
Let the neighborhoodN26(z*,r0) singularity-free be with a radius r0>0 around the critical point (z*,0) defined as
(3.17)N26(z*,r0)={(z,z˙):12ΔpTH0Δp+∑i=1,2‍12ΔqiTHi(qi)Δqi,≤r02},
where Δp=p-pd, Δq=qi-qd. That is, singular configurations of the soft finger tips are avoided inside the neighborhood N26(z*,r0) provide that r0 is chosen adequately.(ii)
Let the neighborhoodN18(z*,r0) around the critical point be where the closed-loop trajectories remain with δ>0, that is,
(3.18)N18(z*,r0)={(z,z˙):ET≤δ,(z,z˙)∈M18}.(iii)
Definition(see [26]) If for any given ε>0 there exists δ(ε)>0 and another constant r1>0 being less than r0 and independent of ε such that the solution tracking trajectories starting from any initial condition (z(0),z˙(0)) lying on N18(δ(ε))∩N26(r1) remains on N18(ε)∩N26(r0) and converges asymptotically to the set M4∩N18(ε) as t→∞, then it is called that the state (z*,0) is stable.Finally, we have the following result.Theorem 3.1.
Considering that the desired reference state(z*,0) and initial state (z(0),z˙(0)) lying on M4. The trajectories in the closed-loop system remains on N18(ε)∩N26(r0) and converges asymptotically to the set M4∩N26(ε) as t→∞ under assumption that the matrix [A,DΘ] is nondegenerate in a neighborhood N26(r0). Thus, is assured that Δfi→0,  Y1-Y2→0,  Δλi→0 as t→∞.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(3.19)ui=ufci+uθci.
Premultiplying q˙iT by uθci we have that
(3.20)∑i=12‍q˙iTuθci=ddt(E0),
where E0=(1/2)β(Δtan(θ)¯)2. This means that there exists a constant η such that
(3.21)∫0t‍(∑i=12‍q˙iTuθci)dψ=E0(t)-E0(0)≥η,
where η=-E0(0).
Now, taking the inner product betweenz˙ and the closed-loop system equations with ui defined in (3.19) we have that
(3.22)ddtET1=-∑i=1,2‍(q˙iTciq˙i),
where ET1=K+P+Part1 and Part1=Part+(1/2)β(Δtan(θ)¯)2. As in stable grasping, the closed-loop trajectories converge to critical point on a constrained manifold where Δλi→0, Δfi→0, ΔY→0 and Δtan(θ)¯→0 as t→∞.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 coordinatesxd by the following control law
(3.23)uxi=-γx2(x--xd)∂xi∂qi,
where γx>0,  xd>0,  xi is the Cartesian coordinates and x- is the estimated position of the object which is defined as an average distance between center positions of the deformable fingertips, that is,
(3.24)x-=x1+x22.
Using the superposition principle, the control law for the stable grasping and object manipulation is defined as
(3.25)ui=ufci+uθci+uxifori=1,2.
As in the previous case, if we multiplying uxi by q˙T we have that
(3.26)∑i=12‍q˙iTuxi=ddt(E1),
where E1=(1/2)γx(x--xd)2 is the shifting energy to move the object in x-coordinates. Consequently, exists a constant η1 such that
(3.27)∫0t‍(∑i=12‍q˙iTuxi)dψ=E1(t)-E1(0)≥η1,
where η1=-E1(0).Now, taking the inner product betweenz˙ and the closed-loop system equations with ui defined in (3.25) we have
(3.28)ddtET2=-∑i=1,2‍(q˙iTciq˙i),
where ET2=K+P+Part2 and Part2=Part1+(1/2)γx(x--xd)2. Then, the passivity condition is satisfy in the closed-loop. Hence, the closed-loop system trajectories converge to critical point on a constrained manifold where Δλi→0, Δfi→0, ΔY→0, Δtan(θ)¯→0,  and x-→xd as t→∞.


## 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, see Figure1. The simulations were implemented on stiff numerical solver on Matlab R2007b, under 1 ms sampling time. Additionally, to approximate the holonomic constraints was used the Constrained Stabilization Method (CSM) [33].The physical parameters of the fingers and object are shown  in Table1 where  lij, mij and  Iij are the length, mass and moment of inertia of the link j=1,2,3 for the finger  i=1,2, and M, I, l are the mass, moment of inertia and the length of the object, respectively (Table 2). Moreover, L=0.64[m] is the distance between fingers, E=50000[N/m2] is the Young's modulus of the fingertips and ri=0.01[m] is the radius of the hemispherical finger tip for i=1,2.Table 1
Physical parameters.
Parameter
Value
l
11
=
l
21
0.05 (m)
l
12
=
l
22
0.04 (m)
l
13
=
l
23
0.03 (m)
m
11
=
m
21
0.05 (kg)
m
12
=
m
22
0.03 (kg)
m
13
=
m
23
0.02 (kg)
I
11
=
I
21
1.4167
×
1
0
-
5
(kgm2)
I
12
=
I
22
6.25
×
1
0
-
6 (kgm2)
I
13
=
I
23
3
×
1
0
-
6 (kgm2)Table 2
Parameters of the object.
Parameter
Value
M
0.02 (kg)
I
5.67
×
1
0
-
6 (kgm2)
l
0.03 (m)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 q1(0)=(30,91.61,71.24)T, q2(0)=(30,91.61,71.24)T, (x(0),y(0))=(0.032,0.047)[m], θ(0)=0, and (dn1,dn2,dt1,dt2)=(0.0025,0.0025,0,0)[m] which establish the following conditions: f1(0)=1.1[N], f2(0)=1.1[N]ΔY(0)=0[m]. For reference, these initial conditions are called normal initial conditions (CIN).In Figure3 we observe that the forces on the object Δfi and Δλi converge rapidly to the desired values using the normal initial conditions defined previously. At the same time, the fast convergence of ΔY to zero is shown in Figure 3. On the other hand, the Figures 4, 5, and 6 show the performance of Δfi, Δλi, and ΔY for different values of damping gains ci. 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 are  fd=1[N] and c1=c2=0.01[Nms/rad].Figure 3
Convergence ofΔfi, Δλi, and ΔY for i=1,2.Figure 4
Convergence ofΔfi considering several values of ci.Figure 5
Convergence ofΔλi considering several values of ci.Figure 6
Convergence ofΔY considering several values of ci.Finally, the Figures7, 8, and 9 show the convergence of Δfi and ΔY to zero under more extreme initial conditions. The initial conditions used in these simulations are CI1 and CI2 which are defined as q1(0)=(30,95.65,64.93)T, q2(0)=(30,100.45,76.09)T, ΔY(0)=0.01[m], and q1(0)=(30,93.73,68.32)T, q2(0)=(30,102.16,72.78)T, (dn1,dn2,dt1,dt2)=(0.0013,0.0037,0,0)[m], ΔY(0)=0.01[m], respectively.Figure 7
Convergence ofΔfi considering different initial conditions.Figure 8
Convergence ofΔλi considering different initial conditions.Figure 9
Convergence ofΔY considering different initial conditions.Notice that the convergence to zero ofΔfi 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 of ΔY 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 as ui=ufci+uθci. The convergence to zero of Δfi, Δλi, ΔY,  and Δθ using the CIN initial conditions are shown in Figures 10 and 11. It is important to notice the special role of Y1-Y2 as a parameter to increase the dexterity. In this case, the convergence of ΔY is closely associated to object orientation through tan(θ)¯. The control parameters using in this second step are  fd=1[N], c1=c2=0.01[Nms/rad],  β=0.1[N/rad], and θd=-10(π/180)[rad].Figure 10
Convergence ofΔfi and Δλi for i=1,2.Figure 11
Convergence ofΔY and Δθ for i=1,2.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 asu=ufci+uθci+utras.The Figures12, 13, and 14 show the convergence of Δfi, Δλi, ΔY, and Δθ to zero using CIN initial conditions. The control parameters used in this step are fd=1[N], c1=c2=0.01[Nms/rad], β=0.1[N/rad], θd=-10(π/180)[rad], γx=50[N/m], and x¯d=x¯0+0.01[m].Figure 12
Convergence ofΔY, Δθ, and Δx for i=1,2.Figure 13
Convergence ofΔfi and Δλi for i=1,2.Figure 14
Convergence ofΔY for i=1,2.


## 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, 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.



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*Source: 949834-2012-10-17.xml*