Modeling Compliant Grasps Exploiting Environmental Constraints

Gionata Salvietti1, Monica Malvezzi2, Guido Gioioso1,2 and Domenico Prattichizzo1,2.

Abstract— In this paper we present a mathematical frame-work to describe the interaction between compliant handsand environmental constraints during grasping tasks. In theproposed model, we considered compliance at wrist, joint andcontact level. We modeled the general case in which the hand isin contact with the object and the surrounding environment. Allthe other contact cases can be derived from the proposed systemof equations. We performed several numerical simulation usingthe SynGrasp Matlab Toolbox to prove the consistency of theproposed model. We tested different combinations of compli-ance as well as different reference inputs for the hand/armsystem considered. This work has to be intended as a tool forcompliant hand designer since it allows to tune compliance atdifferent levels before the real hand realization. Furthermore,the same framework can be used for compliant hand simulationin order to study the interaction with the environmentalconstrains and to plan complex manipulation tasks.

I. INTRODUCTION

Observing a person picking up a flat object off a table, itis easy to understand how our “grasp planner” is not veryprecise. A strategy often employed by humans is to place thefingertips on the table near to the object, and start to slidethem interacting with the table surface (Fig. 1). In this wayit becomes unnecessary to precisely regulate the distance ofthe hand from the surface. Instead, since the object rests onthe table, the interaction with the table surface is exploitedto place the fingers in an optimal grasping position. Thisapproach is in contrast with most of the robotic grasp plan-ners that try to determine possible contact points/regions andprecise wrist position, provided that an accurate descriptionof the object is available. In [1] the authors investigatedthe human grasp performance considering the interactionsbetween the hand and the environment. They also presentedseveral grasp strategies on two different robotic platforms.Each of the strategies was tailored to exploit constraints com-monly present in real-world grasping problems. Similarly in[2], Kazemi et al. reported the results of human subjectsgrasping studies which show the extent and characteristicsof the contact with the environment under different taskconditions. A closed-loop hybrid grasping controller thatmimics this interactive, contact-rich strategy was also testedusing a a compliant 7-DoFs Barrett Whole-Arm Manipulator(WAM).

These preliminary results on replicating the human strate-gies to exploit the environmental constraints are one of themotivations that are steering the design of robotic hands fromfully actuated stiff hands to more underactuated and compli-ant solutions [3]. Recently, several hands with compliance

1Department of Advanced Robotics, Istituto Italiano di Tecnologia, ViaMorego 30, 16163 Genoa, Italy. gionata.salvietti@iit.it

2Universita degli Studi di Siena, Dipartimento di Ingeg-neria dell’Informazione, Via Roma 56, 53100 Siena, Italy.{malvezzi,gioioso,prattichizzo}@dii.unisi.it

Fig. 1. The human hand sliding on the table to grasp the charger.

and underactuation have been proposed in the literature, forinstance the Pisa/IIT hand [4], the SDM Hand [5] and theRBO Hand [6]. The Pisa/IIT Hand embodied the conceptof adaptive synergies [7] using only one motor to jointlyactivate the 19 joints. Innovative articulations and ligamentsreplace conventional joint design. The SDM Hand presentedby Dollar er al. uses polymeric materials to simultaneouslycreate the rigid links and compliant joints of the gripper.Joints are coupled using tendons making the hand passivelyadaptable to the object physical properties. Finally, the RBOHand uses a novel pneumatic actuator design in its fingers.These actuators are built entirely out of flexible materialsand are thus inherently compliant. The air actuation systemalso make the hand highly adaptable to the grasped object.

This new family of underactuated and passively complianthands guarantee robust grasping performance under sensingand actuation uncertainty. At this early stage, however, it isnot clear how to design the hand/wrist stiffness to enhancethe robot grasping and manipulation capability. We believethat to gain dexterity in advanced manipulation tasks usingsuch compliant hands, it is necessary to understand how tofully exploit hand/object/environment interaction in order toplan the hand/arm motion. In other words, there is a need tomodel compliance at wrist, joint and contact level in order todesign the new generation of underactuated hand and to plancomplex manipulation tasks. Furthermore, there is a needingof modeling the environmental constraints to let the handsexploit them.

In this paper we introduce a mathematical frameworkto model compliant hands and their interaction with theenvironment. In particular, we extended the grasp analysismodel presented in [8] to include compliance at wrist,

phc,i,he

phc,i,ho!

{W}!

{B}!

{O}!

u!

uw!

λc,i,ho

wo

poc,i,oe

λc,i,he λ,i,oe

ww

τi qi

Fig. 2. A robotic hand interacting with an object and with the environment,main definitions.

joint and contact levels and to introduce the interactionwith the environment. The model presented in this workdescribes the contact between hand and object, hand andenvironment and the different interactions between hand,object and environment. We propose numerical simulationsto prove the consistency of the model. We analysed differentcombinations of compliance at wrist, joint and contact levelvarying the reference input for joint and wrist positions.

The final aim of this work is to give a simulation toolto test the compliance at different level in the new gener-ation of underactuated hand, as well as to plan hand/armcompliant system trajectories that take advantages from theenvironmental constraints.

The rest of the paper is organized as it follows. InSection II the main equations regulating the model areintroduced and described in details. Section III deals withthe linearized solution to the equation system introducedin Section II. In Section IV the numerical simulation andthe relative results are reported and discussed. Finally, inSection V conclusion and future work are outlined.

II. MODELING HAND/OBJECT/ENVIRONMENTINTERACTION

In this section, we summarize the main assumption wemade to obtain a mathematical model describing a complianthand interacting with a surface and with an object. Themain parameters that we adopted to define the system aresummarized in Fig. 2.

One of the most important parts of the model is thedescription of the contact between the surfaces. In this pre-liminary work, we only consider a single point with frictioncontact model. The contacts between bodies are representedas points in which pure forces (without torques) are applied.With this simplification we can consider only the translationpart of the relative motion between bodies at the contactpoints, without taking into account the rotation and thansimplifying the notation. This procedure can be generalized

TABLE IPOSSIBLE OBJECT/ENVIRONMENT/HAND CONFIGURATIONS.

n. HE OE HO Scheme Notes

1 NO NO YES Hand Grasping

2 NO YES NO Approaching phase

3 NO YES YES Obj./Env. interaction

4 YES NO YES Hand/Env.+Hand/Obj

5 YES YES NO Hand/Env. interaction

6 YES YES YES Most general case

for more complex contact models, for example includingthe spin moment, whose direction is around the normaldirection to the contact surfaces at the contact point andwhose magnitude depends on the relative rotation betweencontact bodies in this direction (this contact model is alsoreferred as Soft Finger contact [9]).

Furthermore, to solve the force distribution problem, weintroduce a compliant contact model: the contact surfacesare represented as locally deformable and we assume thatthe contact force is proportional to the local deformation[10], the proportionality constant is described by the contactstiffness matrix Kc,i ∈ <3×3, that is assumed constant. Morecomplex and reliable contact models are available in theliterature [11], and will be considered in future improvementsof this study.

We modeled robotic hand grasping exploiting the envi-ronmental constraints to reach and grasp the object. Theenvironment is described as a surface Ψ whose shape isknown in the base reference frame. Let Ψ (p) = 0, Ψ :<3 → < be its mathematical representation. Let {B} be areference frame fixed with respect to the environment.

In the more general frame involving grasping with flexiblehands exploiting environment constraints, three types ofinteraction (contact) can be identified:

• HE: hand/environment;• OE: object/enviroment;• HO: hand/object.

These three conditions can be combined to obtain up toeight possible configurations, the six more significant aresummarized in Table I.

Let us consider the interaction between the hand and theenvironment. We suppose that each contact can be repre-

phc,i,he

poc,i,he

λc,i,he=Kc,i,he(poc,i,he-ph

c,i,he)

undeformed contact surface

deformed contact surface

Fig. 3. Deformable contact surface: local deformation and contact force.The black line represents the environment surface, the blue one the handfinger surface.

sented as a point, let us indicate with nhe ≥ 0 the numberof contact points between the hand and the environment.According to the deformable contact model considered, wedistinguish between the contact points on the hand and onthe environment: let phc,i,he,p

ec,i,he ∈ <3 be the coordinates

of the i-th contact point on the hand and on the environmentsurfaces, respectively, expressed in {B} reference frame.In general, due to the local surface deformation, phc,i,he 6=pec,i,he, and the relative displacement between these twopoints is proportional to the contact force (see Fig. 3), i.e.λc,i,he = Kc,i,he

(pec,i,he − phc,i,he

), where λc,i,he ∈ <3 is

the contact force applied by the environment on the hand atthe contact i, and Kc,i,he ∈ <3×3 is the contact stiffness ma-trix. We furthermore collect all the contact point coordinatesand the contact forces in three vectors phc,he,p

ec,he,λc,he ∈

<3nhe , where phc,he =[phc,1,he

T, · · · ,phc,nhe,he

T]T

and soon.

In the most general case, also the object interacts with theenvironment. Let us indicate with noe ≥ 0 the number ofcontact points between the object and the environment. Alsoin this case, due to the local deformation of the hand andobject surfaces due to the contact, we distinguish betweenthe contact points on the object and on the environmenttheoretically undeformed surfaces: let poc,i,oe,p

ec,i,oe ∈ <3 be

the coordinates of the i-th contact point on the object and andon the environment, respectively, expressed in {B} referenceframe. The object-environment contact force is given byλc,i,oe = Kc,i,oe

(pec,i,oe − poc,i,oe

), where λc,i,oe ∈ <3 is

the contact force applied by the environment on the object atthe contact i, and Kc,i,oe ∈ <3×3 is the object/environmentcontact stiffness matrix. Also in this case we collect all thecontact point coordinates and the contact forces in threevectors poc,oe,p

ec,oe,λc,oe ∈ <3noe .

Let us finally consider the interaction between the handand the object. Let us indicate with nho ≥ 0 the numberof contact points between the hand and the object. Dueto the local deformation of the contact surfaces, also inthis case we distinguish between the contact points on thehand and on the object: let phc,i,ho,p

oc,i,ho ∈ <3 be the

coordinates of the i-th contact point on the hand and on theenvironment theoretical undeformed surfaces, respectively,expressed in {B} reference frame. The contact force is givenby λc,i,ho = Kc,i,ho

(phc,i,ho − poc,i,ho

), where λc,i,ho ∈ <3

is the contact force applied by the hand on the object at thecontact i, and Kc,i,ho ∈ <3×3 is the contact stiffness matrix.We collect all the contact point coordinates and the contactforces in three vectors phc,ho.p

oc,ho,λc,ho ∈ <3nho , where

phc,ho =[phc,1,ho

T, · · · ,phc,nho,ho

T]T

and so on.According to the previous definition, we can express the

contact forces in the three contact types, as

λc,he = Kc,he

(pec,he − phc,he

)(1)

λc,oe = Kc,oe

(pec,oe − poc,oe

)(2)

λc,ho = Kc,ho

(phc,ho − poc,ho

), (3)

where Kc,he ∈ <3nhe×3nhe , Kc,oe ∈ <3noe×3noe , Kc,ho ∈<3nho×3nho are the stiffness matrices including all the con-tact points. In the following we will analyse a small per-turbation of system configuration from an initial equilibriumcondition, and we will indicate with ∆· the correspondingvariation of a generic system variable “·”. If a small varia-tion w.r.t. an initial equilibrium configuration is considered,equations (1-3) can be rewritten as

∆λc,he = −Kc,he∆phc,he (4)∆λc,oe = −Kc,oe∆poc,oe (5)

∆λc,ho = Kc,ho

(∆phc,ho −∆poc,ho

). (6)

The hand consists of a series of nf serial kinematicchains, the fingers. Each finger has nq,i, with i = 1, · · · , nf ,degrees of freedom. We suppose that each finger can berepresented as a series of nq,i links connected by nq,ione-degree-of freedom joints (revolute, R, or prismatic, P,joints). Let nq =

∑nf

i=1 nq,i be the total number of thehand degrees of freedom and let q ∈ <nq be the wholehand joint configuration vector, containing the values ofall the hand joint displacements (a rotation for a revolutejoint, a translation for the prismatic one). Typically in grasptheory and modeling literature, the fingers are connected toa common base (the hand palm) on which a fixed coordinatereference frame is defined. In this work, we suppose thatthe hand is connected to an arm through its palm that isnot fixed. Furthermore we suppose that the whole arm/wristsystem has a certain flexibility that is represented w.r.t. thewrist by the stiffness matrix Kw ∈ <6×6. Let us indicatewith uw ∈ <6 the generic six-dimensional displacement ofthe palm reference frame {W} with respect to the fixedbase frame {B}. The configuration vector uw is a functionof the arm kinematic structure and of the whole systemflexibility. The whole hand configuration vector will betherefore described by the joint configuration q and thepalm configuration uw. We can collect all these values ina unique vector q =

[qT,uT

w

]T ∈ <nq+6. Let us indicatewith τ ∈ <nq the set of generalized forces (a force foreach prismatic joint, a torque for each revolute joint) actingon hand joints, and with ww the total wrench that the armapplies to the hand, expressed w.r.t. {B} frame origin point.We can thus collect all the actions on the hand in the vectorτ =

[τT,wT

w

]T ∈ <nq+6.From the kinematic analysis of the hand/arm system, we

can express the position of the contact points as a function of

hand/arm configuration. Considering the hand/environmentinteraction we get

phc,he = κhe(q), (7)

while, considering the hand/object interaction

phc,ho = κho(q), (8)

where κhe : <nq+6 → <3nhe , κho : <nq+6 → <3nho

are the direct kinematic relationships. The displacementsof the contact points due to a small variation of the handconfiguration is given by

∆phc,he = Jhe∆q, (9)

∆phc,ho = Jho∆q, (10)

where Jhe ∈ <3nhe×nq+6 and Jho ∈ <3nho×nq+6 arethe complete Jacobian matrices, mapping hand configura-tion vector on contact point displacements. Complete handJacobian matrices can be evaluated as

Jhe = [Jh,he,Jw,he] ,

Jho = [Jh,ho,Jw,ho] ,

where Jh,he, Jh,ho are the hand Jacobian matrices, that canbe evaluated with the classical methods proposed for graspanalysis [8], [12], while Jw,he, Jw,ho relating contact pointdisplacement to hand palm displacement, are two matricescomposed of nhe and nho (3×6) blocks Jw,i given by Jw,i =[I3×3,−S(pi − uw), ]. 1

Applying the Principle of Virtual Work to the system com-posed of the hand connected to the arm and the environment,we get

τ = −JTheλc,he + JT

hoλc,ho, (11)

mapping the contact forces to the joint torques and palmwrench. If we consider a small perturbation from an initialequilibrium condition, we can linearize the above staticrelationship, obtaining

∆τ = −JThe∆λc,he + JT

ho∆λc,ho + H∆q, (12)

where

H =∂JT

heλhe∂q

+∂JT

hoλho∂q

.

H ∈ <(nq+6)×(nq+6) takes into account the variation ofJacobian matrices with respect to hand configuration. Thisterm is present when in the initial equilibrium condition thesystem is loaded, i.e. λc,he 6= 0 and/or λc,ho 6= 0. In thispaper we suppose that the hand and the arm are controlledin position with a proportional controller. The hand andarm system are not perfectly stiff and can be describedby means of a compliant model. The flexibility is in themore general case due to: (1) the gain of the proportionalposition control of hand joints and wrist position, (2) theflexibility of the joints, and (3) the flexibility of the links,that can be represented in the joint space [13]. The torqueapplied to hand joints and the wrench applied to the wrist

1S(v) represents the skew matrix corresponding to the vector v.

are proportional to the difference between the reference andactual hand/arm configuration, i.e.

τ = Kq (qr − q) + τ 0, (13)

where qr =[qTr ,u

Tr,w

]T ∈ <nq+6 is a vector containingthe reference hand joint displacements qr and the referencepalm configuration ur,w, and

Kq =

[Kq 00 Kw

].

Kq ∈ <nq×nq is the joint stiffness matrix, while Kw ∈<6×6, previously introduced, represents the stiffness of theconnection between the hand and the arm and thus takes intoaccount the stiffness of the whole arm. Both the matrices aresymmetric and positive. If we consider a small variation withrespect to a reference equilibrium configuration we get

∆τ = Kq (∆qr −∆q) . (14)

Let us consider the interaction between the environment andthe object. The position of the i-th contact point on the objectcan be expressed as

poc,i,eo = po + RBo p

oc,i,eo, (15)

where po represents the coordinates of {O} reference frameorigin w.r.t. {B}, RB

o is the rotation matrix between {O}and {B}, poc,i,eo are the coordinates of the i-th contact pointsin the {O} frame. If we consider a small displacement, thefollowing relationship can be written

∆poc,i,eo = ∆po − S(poc,i,eo − po)∆φ, (16)

where ∆φ = [∆φx,∆φy,∆φz]T represents the relative

elementary rotation between {O} and {B}. This equationcan be synthesized as

∆poc,i,eo = GTi,eo∆u, (17)

with u =[∆pT

o ,∆φT]T

, and

GTeo =

[I,−S(poc,i,eo − po)

].

Extending this kinematic relationship to all the contact pointswe get

∆poc,eo = GTeo∆u, (18)

where

GTeo =

GT1,eo

· · ·GTneo,eo

.Similarly, considering the contacts between the hand and theobject, the following relationship can be written

∆poc,ho = GTho∆u, (19)

where GTho ∈ <3nho×6 is the Grasp Matrix [9]. By

applying the Principle of Virtual Works to the object, thefollowing relationship describing object static equilibriumcan be obtained

wo = −Geoλc,eo + Ghoλc,ho, (20)

where wo ∈ <6 is the external wrench applied to theobject. If a small perturbation w.r.t. an initial equilibriumconfiguration is considered, the static equilibrium in the newconfiguration is described by

∆wo = −Geo∆λc,eo + Gho∆λc,ho + L∆u, (21)

whereL = −∂Geoλc,eo

∂u+∂Ghoλc,ho

∂u.

L ∈ <6×6 takes into account the variation of Geo and Gho

with respect to object displacement. It can be neglected if inthe initial reference configuration the system is unloaded, i.e.if λc,eo = 0 and λc,ho = 0, or if the matrices do not dependon u. This condition can be obtained by expressing contactdisplacements and contact forces w.r.t. {O} reference frameand assuming that the contact points are fixed on the object.

Summarizing all the equations describing a small variationof system configuration w.r.t. an initial equilibrium config-uration, the following homogeneous linear system can beobtained

∆λc,he + Kc,he∆phc,he = 0∆λc,oe + Kc,oe∆poc,oe = 0∆λc,ho −Kc,ho∆phc,ho + Kc,ho∆poc,ho = 0

∆phc,he − Jhe∆q = 0

∆phc,ho − Jho∆q = 0

∆τ + JThe∆λc,he − JT

ho∆λc,ho −H∆q = 0

∆τ − Kq∆qr + Kq∆q = 0∆poc,ho −GT

ho∆u = 0

∆poc,eo −GTeo∆u = 0

∆wo + Geo∆λc,eo −Gho∆λc,ho − L∆u = 0(22)

III. LINEARIZED SOLUTION

The linear system in eq. (22) can be written in this form

A∆ξ = ∆υ, (23)

where

∆ξ =[∆λT,∆pT

c ,∆qT,∆τT,∆uT]T,

with∆λT =

[∆λT

c,he,∆λTc,oe,∆λT

c,ho

],

∆pTc =

[∆phTc,he,∆poTc,oe,∆phTc,ho,∆poTc,ho

].

The matrix A is square so, if it is not singular, the linearsystem can be solved to find ∆ξ.

The solution of the linear system in eq. (22), obtainedwith ∆wo = 0, ∆q 6= 0 allows to understand, for a givenconfiguration, which are the possible controllable forces andmovements that can be obtained acting on hand and armreference configuration [10]. The solution can be expressedas

∆λ = Vq∆qr,∆pc = Pq∆qr,∆q = Qq∆qr,∆τ = Tq∆qr,∆u = Uq∆qr.

(24)

Each of the above matrices can be partitioned in order tohighlight the contribution of arm reference variation and handjoint reference variation in the whole result. For instance, theactual joint configuration ∆q can be expressed as follows[

∆q∆uw

]=

[Qqq Qqu

Quq Quu

] [∆qr∆uw,r

]where Qqq describes hand joint variation ∆q when thereference values of hand joints are varied and the handpalm reference displacement is constant, Qqu describes handjoint variation ∆q when the reference values of hand palmreference displacement is varied, while the reference valuesof the hand joints are constant, and so on.

The solution of the linear system in eq. (22) when ∆wo 6=0 and ∆q = 0 allows to investigate on how the system is ableto resist to external disturbances, represented as variations ofthe wrench applied to the object. The solution in this casecan be expressed as

∆λ = Vw∆wo,∆pc = Pw∆wo,∆q = Qw∆wo,∆τ = Tw∆wo,∆u = Uw∆wo.

(25)

In particular, Kgrasp = U−1w maps the variation of the

external wrench applied to the object to object displacement.It is the generalization of the Grasp Stiffness matrix definedin [14], [15], [16] to hands grasping an object and interactingwith a surface.

Solution for the sub-problems

The above presented solution is relative to the more gen-eral case, in which a contact exists between the hand and theobject, the hand and the environment surface, and the objectand the environment. This case is indicated with number 6 inTable I. A linear system can be similarly obtained in all theother cases, by choosing the proper equations and unknownsfrom the system in eq. (22). For instance, the equationsthat have to be solved in the case indicated with number5 in Table I, i.e. hand/environment and object/environmentinteraction, are

∆λc,he + Kc,he∆phc,he = 0∆λc,oe + Kc,oe∆poc,oe = 0

∆phc,he − Jhe∆q = 0

∆τ + JThe∆λc,he −H∆q = 0

∆τ − Kq∆qr + Kq∆q = 0∆poc,eo −GT

eo∆u = 0∆wo + Geo∆λc,eo − L∆u = 0

(26)

All the cases in Table I can be thus derived with thesame approach. The proposed procedure can be adopted toevaluate and analyse, for a given configuration, the maingrasp properties. Even if eq. (22) and (26), represent alinear quasi–static approximation of the generally nonlinearequations describing the system dynamics, they can be thebasis of a procedure for grasping simulation.

(a) Modular hand, initial configura-tion.

−80 −60 −40 −20 0 20 40 60 800

20

40

60

80

100

120

x (mm)

y (m

m)

(b) Planar gripper, initial configura-tion.

Fig. 4. Starting configurations for the simulation of the modular 3-fingeredhand and of the planar 4DoFs gripper.

IV. NUMERICAL SIMULATION

The proposed linear quasi–static model has been imple-mented and verified by means of several numerical simula-tions using SynGrasp [17]. SynGrasp is a Matlab Toolbox forthe simulation and the analysis of grasping in which severalhand models are available. It also includes functions for thedefinition of hand kinematic structure and of the contactpoints with a grasped object, the coupling between jointsinduced by a synergistic control, compliance at the contact,joint and actuator levels. For this work, we integrated thehand models available considering also compliance at wristlevel and the contact with a fixed surface. In particular,we used the models of a 4 DoFs planar gripper and athree-fingered modular hand. The gripper is composed of 2symmetric 2 DoFs fingers, whose link lengths are 50 mm andwhose base length is 60 mm. The links are connected eachother and to the base by 4 revolute joints with parallel axes,so that the mechanism is planar. Concerning the modularhand, each module (42× 33× 16)mm has one DoF and it isconnected to the others obtaining kinematic chains that weconsider as fingers. These chains are connected to a commonbase defined as a palm. In the proposed configuration eachfinger has three DoFs, thus the hand has globally nine DoFs.The main goal of the numerical simulations was to verifythe behavior of the compliant hands during the interactionwith objects and environment. In particular, in the first set ofsimulations we considered the modular hand interaction onlywith the environment, while in the second set the gripperis interacting both with the object and the environmentalconstraints.

A. Modular hand interacting with the environment

In the first set of simulations we focused on the in-teraction between a compliant modular hand, and an en-vironmental constraint represented by a flat surface, e.g.,a table. We considered the modular hand already incontact with the table at the beginning of the simula-tions. The initial configuration of the hand joint vector isqinit =

[0 0.2 0.6 0 0.2 0.6 0 0.2 0.6

]rad.

The hand in the initial configuration is reported in Fig. 4.We tested different combination of compliance values andreference inputs. The result showed in Fig. 5 are obtainedusing a fixed reference input

∆qr =[

0.1 0 0 0.1 0 0 0.1 0 0],

for 20 simulation steps. This reference input results in themotion of the joints more close to the palm of all the fingers.The six components of the vector representing the wristreference input, was set to zero in this case. We set thefollowing contact stiffness values

Kc,he = kcI3×nhe, Kw = kwI6×6, Kq = kqInq×nq .

In Fig. 5-a we used kc = 1000 N/mm, kq = 1000 Nmm/radand kw = 0.001 N/mm, simulating a stiff hand interactingwith a stiff surface and connected to a flexible arm. Sincethe both hand joints and contact have an high stiffness thereference input produce a displacement of the wrist towardthe direction perpendicular to the table. In Fig. 5-b we testedthe behavior of the hand when the joint stiffness in muchlower that the contact stiffness. We set kc = 1000 N/mm,kq = 1 Nmm/rad, and kw = 1 N/mm. From the resultswe can observe a bigger deformation of the hand. Finallyin Fig. 5-c we simulated a soft contact with a stiff hand.The coefficients where fixed to kc = 1 N/mm, kq = 1000Nmm/rad and kw = 1 N/mm. We did not represent thedeformation of the table: in the diagram a penetration ofthe hand fingers onto the constraint surface can be observed.

In the second set of simulations, we tested differentwrist compliance values choosing as reference joint vectorcomponents the following values

∆uw,r =[

0 0 −0.2 0 0 0],

which corresponds to a motion of the hand toward the tablewithout closing the hand joints, whose reference vector wasset to zero. In Fig. 6-a we considered kw = 0.001 N/mm,while in Fig. 6-b kw was equal to 1 N/mm. The othercoefficients where kept fixed at kc = 1000 N/mm, kq = 10Nmm/rad. As expected in the case with high wrist stiffness,the hand deforms, since the palm is pushed toward the tableand the hand joints are compliant. In the second case, whenthe wrist stiffness is low, all the motion is “captured” by thewrist producing an almost zero effect in the hand joints.

B. Planar gripper interacting with the environment and withan object.

The second numerical example is realized with the modelof a planar 4 DoFs gripper, whose SynGrasp representationin the initial configuration is shown in Fig. 4(b). Duringthe simulation, in the first phase the gripper moves towardsthe flat surface, by changing wrist reference configuration,Fig. 7(a). Once the contact with the surface is reached, thegripper is closed by acting on the reference values of theproximal joints (Fig. 7(b)), until the gripper links reachesthe object (Fig. 7(c)). In the final part of the simulation,(Fig. 7(d)) the gripper interacts both with the surface andwith the object: the reference values of the proximal jointsare increased until the object is lifted up.

The gripper is planar and has two fingers, each fingeris composed of two phalanges with the same lengths a =50 mm. Let J1, · · · , J4 be the joint axis traces on thegripper plane, and let θ1, · · · , θ4 be the joint angles. A fixedreference frame {B} is fixed on a flat horizontal surfacethat represents the environment. The y axis is supposed

(a) The modular hand interacting with a plane withhigh joint stiffness and low contact stiffness.

(b) The modular hand interacting with a planewith low joint stiffness and high contact stiffness.

(c) The modular hand interacting with a planewith high joint stiffness and low contact stiffness.

Fig. 5. Different hand configuration obtained varying compliance at joint and contact level.

(a) The modular hand interactingwith a plane with high wrist stiffnessand low joint stiffness.

(b) The modular hand interactingwith a plane with low wrist stiffnessand low joint stiffness.

Fig. 6. Different hand configuration obtained varying compliance at wristlevel.

−80 −60 −40 −20 0 20 40 60 800

20

40

60

80

100

120

x (mm)

y

(mm

)

(a) Initial configuration.

−80 −60 −40 −20 0 20 40 60 80

−20

0

20

40

60

80

100

x (mm)

y

(mm

)

(b) The gripper reaches the environ-ment surface.

−80 −60 −40 −20 0 20 40 60 80

−20

0

20

40

60

80

100

x (mm)

y

(mm

)

(c) The gripper slides on the surfaceuntil it reaches the object.

−80 −60 −40 −20 0 20 40 60 80

−20

0

20

40

60

80

100

x (mm)

y

(mm

)

(d) The gripper grasps the object.

Fig. 7. Four steps of the simulated grasping.

orthogonal to the surface. Let W a reference frame on thegripper base, let uw = [uw,x, uwy]T be the coordinates ofits origin w.r.t {B}, and let φ the angle between x axes. Inthis simulation, we suppose that the gripper interacts witha the horizontal plane y = 0 and with a spherical object,with radius r = 30 mm, whose center O has coordinatesuo = [0, r] w.r.t. {B}.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.075

0.08

0.085

0.09

0.095

0.1

time [s]

wrist

ve

rtic

al d

isp

lace

me

nt

[m

]

kw

= 10 N/m

kw

= 100 N/m

kw

= 1000 N/m

uwr

(a) Vertical displacement of the wrist, for different values of kw .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.035

0.036

0.037

0.038

0.039

0.04

0.041

0.042

time [s]

ob

ject

ve

rtic

al d

isp

lace

me

nt

[m

]

kw

= 10 N/m

kw

= 100 N/m

kw

= 1000 N/m

(b) Vertical displacement of the object, for different values of kw .

Fig. 8. Vertical displacement of the wrist and of the object during thesimulation, for different values of wrist stiffness kw .

In the initial configuration, the gripper is over the surfaceand does not interact neither with the surface nor withthe object. We considered the following initial referenceconfiguration for the gripper joints: θ1 = π

6 rad, θ2 = π2 rad,

θ3 = π6 rad, θ4 = π

2 rad, the load applied to the object isw0 = 0 N. In the first simulation phase the gripper wristis moved to reach the environment surface and at the sametime the reference values of the gripper joints are increasedso that the gripper closes. When the gripper reaches the planethe joint angles are θ1 = 1 rad, θ2 = 1.3 rad, θ3 = 1 rad,θ4 = 1.3 rad,

When the gripper reaches the surface, due to the systemgeometry, the contact points correspond to the fingertips,let us indicate them with T1 and T2 and let us indicatewith t1, t2 ∈ <2 their coordinates. The hand/environmentJacobian matrix can be evaluated in this case as

Je =

Je,1,1 Je,1,2 0 0 1 0 −t1,y + uw,y

Je,2,1 Je,2,2 0 0 0 1 t1,x − uw,x

0 0 Je,3,3 Je,3,4 1 0 −t2,y + uw,y

0 0 Je,4,3 Je,4,4 0 1 t2,x − uw,x

in which the matrix terms can be expressed as:

Je,1,1 = sφ(ac12 + ac1)− cφ(as12 + as1)Je,1,2 = ac12sφ− as12cφ

etc., with s1 = sin θ1, c1 = cos θ1, s12 = sin(θ1 + θ2), andso on.

Once the gripper reaches the contact surface, we startedto close it by increasing the reference values of θ1 and θ2angles, until the object is reached. The contact points areindicated with C1 and C2, let c1, c2 be their coordinates, andlet l1 and l2 be the distances from J2 and J4, respectively.The object center coordinates and object orientation aredefined with respect to the base reference system by thevector u = [ux uy φ]T, where φ represents the anglebetween the local and base reference systems abscissae axes.The hand/environment Jacobian matrix can be evaluated inthis case as

Jo =

Jo,1,1 Jo,1,2 0 0 1 0 −c1,y + uw,y

Jo,2,1 Jo,2,2 0 0 0 1 c1,x − uw,x

0 0 Jo,3,3 Jo,3,4 1 0 −c2,y + uw,y

0 0 Jo,4,3 Jo,4,4 0 1 c2,x − uw,x

,in which the matrix terms can be expressed as

Jo,1,1 = sφ(l1c12 + ac1)− cφ(l1s12 + as1)Jo,1,2 = l1c12sφ− l1s12cφ

,

etc.. The hand/object grasp matrix is given by

Go =

1 0 1 00 1 0 1

−c1,y + uy c1,x − ux −c2,y + uy c2,x − ux

.The geometric terms, are considered by defining the ma-

trices H and L, that can be evaluated as a function of Je, Joand Go derivatives. Their expressions are not reported herefor the sake of brevity.

Different simulations were performed changing, also inthis case, the values of system stiffness. Fig. 8 shows, fordifferent values of wrist stiffness, the corresponding wrist andobject vertical displacement. As expected, during the slidingphase, the wrist displacement decreases as wrist stiffnessincreases.

V. CONCLUSION

The diffusion of underactuated and compliant robotichands and the need of operating in unstructured and uncertainenvironments is leading to grasp planning procedures inwhich the robustness and adaptability is more important thanprecision and accuracy. In this framework, an interestinggrasping strategy could be based on the exploitation ofenvironmental constraints, which could be seen as support forthe robotic hand, rather than an obstacle to be avoided. In thispaper we proposed a mathematical representation of roboticgrasping in which a compliant hand exploits the environmentsurface to reach the object in a reliable and robust way.Using this model the user is able to evaluate the main graspproperties, taking into account the interaction between thehand and the object, the object and the environment, thehand and the environment. In particular, the effect of handand arm properties can be evaluated: the proposed modelcan be used in the design phase to define hand properties

necessary to obtain the desired performance during graspingoperations involving the exploitation of environmental con-straints. Future development of the study will be devoted todefine a more general description including dynamics effectsand rolling between contact surfaces, and a more detaileddescription of contact model, including friction limits. TheSynGrasp functions developed for this work are availableon-line at http://syngrasp.dii.unisi.it.

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