An Object-Based Approach to Map Human Hand Synergies ontoRobotic Hands with Dissimilar Kinematics

G. Gioioso, G. Salvietti, M. Malvezzi and D. Prattichizzo

Abstract— Robotic hands differ in kinematics, dynamics,programming, control and sensing frameworks. Borrowing theterminology from software engineering, there is a need formiddleware solutions to control the robotic hands independentlyfrom their specific structure, and focusing only on the task.Results in neuroscience concerning the synergistic organizationof the human hand, are the theoretical foundation of thiswork, which focuses on the problem of mapping human handsynergies on robotic hands with dissimilar kinematic structures.The proposed mapping is based on the use of a virtual ellipsoidand it is mediated by a model of an anthropomorphic robotichand able to capture the idea of synergies in human hands.This approach has been tested in two different robotic handswith an anthropomorphic and non-anthropomorphic kinematicstructure.

I. INTRODUCTION

Among the many approaches presented in the literature,bio-inspired control of robotic hands seems to be one of themore promising approach [1], [2]. A deeper understandingon how the brain exploits the high redundancy of the humanhands could represent a key issue in the next generation ofcontrol algorithms. In the last two decades, neuroscientistshave studied the organization of the human hand in graspingand manipulation tasks using accurate hand tracking systems[3]. In particular, some studies demonstrated that, notwith-standing the complexity of the human hand, a few variablesare able to account for most of the variance in the patternsof human hands configuration and movement [4]. Theseconclusions were based on the results of experimental tests inwhich subjects were asked to perform grasping actions on awide variety of objects. Data were recorded by means of datagloves and were analysed with principal component analysis(PCA) techniques. The results showed that the first twoprincipal components account for most of the variability inthe data, more than 80% of the variance in the hand postures.In this context the principal components were referred tosynergies, to capture the concept that, in the sensorimotorsystem of the human hand, combined actions are favouredover individual component actions, with advantages in termsof simplification and efficiency of the overall system.

This work was partially supported by the European Commission withthe Collaborative Project no. 248587, “THE Hand Embodied”, withinthe FP7-ICT- 2009-4-2-1 program “Cognitive Systems and Robotics” andthe Collaborative EU-Project “Hands.dvi” in the context of ECHORD(European Clearing House for Open Robotics Development).

G. Gioioso, G. Salvietti and M. Malvezzi are with Department of Infor-mation Engineering, University of Siena, 53100 Siena, Italy [gioioso,salviettigio, malvezzi]@dii.unisi.it

D. Prattichizzo is with Department of Information Engi-neering, University of Siena and with the Advanced RoboticsDepartment of the Istituto Italiano di Tecnologia, Genova, Italy[prattichizzo]@dii.unisi.it

Mapping Algorithm

5-Finger Hand

3-Finger Hand

4-Finger Hand

Industrial Gripper

Fig. 1. Mapping between human synergies and robotic hands.

This reduction of Degrees of Freedom (DoFs) can beused to decrease the complexity of control algorithms forrobotic hands. Intuitively, these findings in neurosciencecan be used in robotics to control robotics hands with ananthropomorphic structure closely copying the structure ofhuman hands. This is easy to understand for anthropomor-phic robotic hands but what happens if the robotic hand has acompletely different kinematics? Can we still take advantageof the sensorimotor synergies as studied in neuroscience?This chapter provides an answer to these questions focusingon the development of a unified framework for programmingand controlling robotic hands with dissimilar kinematic.

This generalization can be achieved through a mappingalgorithm able to reproduce the movements due to a syn-ergistic control of a paradigmatic model of the human handonto the robotic hands. This mapping function focuses on thetask space avoiding to consider the peculiar kinematics of therobotic hands. In Fig. 1 the idea of the proposed approachis pictorially represented.

In this paper we present a method to map human synergiesonto robotic hands by using a virtual object. The targetis to reproduce deformations and movements exerted by aparadigmatic human-like hand on a virtual ellipsoid com-puted as the minimum volume ellipsoid containing suitablereference points that lie on the hand structure. This allows towork directly on the task space avoiding a specific projection

between different kinematics. The paper generalizes the mainidea described in [5], introducing the possibility to modeldifferent types of virtual object shapes and deformations.

The paper is organized as follows. In Section II a review ofthe literature is presented. Section III summarizes the mainresults concerning grasp properties in synergy actuated handsand describes the mapping method in detail. In Section IVsome simulations are shown to confirm the proposed ap-proach effectiveness, while the advantages and drawbacksof the proposed method together with conclusion and futurework are outlined at the end.

II. RELATED WORK

Mapping algorithms are frequently used in several appli-cations and in particular in tele-manipulation and learningby demonstration. The typical approach adopted to tele-manipulate robotic hands is to use datagloves to collecthuman hand motion data. In [6], for example, a DLR Hand iscontrolled through a CyberGlove, while in [7] a master-slaveteleoperation system is developed to evaluate the effective-ness of tele-presence in tele-robotics applications.In learning by demonstration applications, data collected byhuman actions are not directly used to control robotic hands,but to improve the grasping performance by teaching to therobot the correct posture to obtain stable grasps. In [8],authors evaluated human grasps during an arm transportationsequence in order to learn and represent grasp strategiesfor different robotic hands, while in [9] a programming bydemonstration system for grasp recognition in manipulationtasks and robot pre-grasp planning is developed.

If the considered robotic hand is strongly anthropomor-phic, the data collect from the humans can be used straight-forward in both tele-manipulation and learning by demon-stration applications. When robotic hands have differentkinematics, mapping strategies have to be taken into account.There are three main approaches in literature to deal with thisproblem: joint to joint mapping, Cartesian Space or fingertipmapping and pose mapping.The aim of the first approach is to make the poses ofhuman and robotic hands look similar. Ciocarlie and Allen,for instance, used this method to take advantage of humanhand synergies in robotic grasp pre-shaping [10]. They testedfour different robotic hands with this procedure: the humanhand joint values were directly mapped into the joints ofthe anthropomorphic hands, while some empirical solutionswere adopted with the non-anthropomorphic hands. This rep-resents the simplest way to map movements between hands.Anyway, joint to joint method has to be redefined accordingto the kinematic characteristics of the hands making difficulta generalization of the control strategy considering also thatthe performance notably decrease with non-anthropomorphicstructures.Cartesian Space mappings focus on the geometric relationsbetween the two workspaces. This solution is suitable forrepresenting the fingertip positions and it is a natural ap-proach when, for example, precision grasps are considered.In [11] a point-to-point mapping algorithm is presented

for a multi-fingered telemanipulation system where fingertipmotion of the human hand is reproduced with a three-fingerrobotic gripper. In [12] authors used a virtual finger solutionto map movements of the human hand onto a four-fingeredrobotic hand. However, these solutions do not guarantee acorrect mapping in terms of forces and movements exertedby the robotic hand on a grasped object.The pose mapping can be considered as a particular way ofindirect joint angle mapping. The basic idea of the pose map-ping is trying to establish a correlation between human handposes and robotic hand poses. For example, Pao and Speeter[13] developed an algorithm that tries to translate humanhand poses to corresponding robotic hand positions, withoutloss of functional information and without the overhead ofkinematic calculations. In [14] neural networks were usedto learn the hand grasping postures. Anyway, the proposedsolutions can produce unpredictable motions of the robothand, and thus in our opinion are only exploitable in thosecases where basic grasp postures are required.

Besides the above mentioned methods, combinations ofthem and some original solutions, as the method proposedin [15], are also present in literature. In [16] a virtualobject-based mapping is proposed. The object based schemeassumes that a virtual circle is held between the user’s thumband index finger. Important parameters of the virtual object(the size, position and orientation) are scaled independentlyand non-linearly to create a transformed virtual object in therobotic hand workspace. This modified virtual object is thenused to compute the robotic fingertip locations that in thiscase is a simple two-fingers, four-DoFs gripper.A 3D extension of the last method is presented in [17]. Evenif this extension allows to analyse more cases, this method isstill not enough general for our purposes. In particular, it isconstrained by the kinematics of the master and slave hand,the number of contact points (three) and their locations (thefingertips) which have to be the same for both the hands.Then, it can be used only for a given pair of human androbotic hands and for precision grasp operations.

The approach proposed in this paper is inspired by thelast two mentioned methods. The main contributions of ourwork with respect to the methods proposed in [16], [17] arehere summarized: our approach generalizes the mapping toa generic number of contact points that can be different inthe human and robotic hands and there are no constraints onpositions of contact points on the master and on the slavehand. Moreover, to the best of our knowledge, none of thepresented papers deal with mapping synergies of the humanhand onto robotic hands, defined as a reduced dimension ofthe configuration space.

III. OBJECT-BASED MAPPING

The main equations necessary to study hands controlled bysynergies from a kinematic point of view are here introduced.A more detailed presentation of the problem is described in[18], further details on grasping can be found in [19].

Let us consider a hand grasping an object with no distinc-tion between human or robotic. The hand can be represented

as a mechanical interface composed of a series of kinematicchains, the fingers, that share a common base, the palm.The fingers and/or the palm is connected to the manipulatedobject by some contact patches, usually represented as points(even though, due to the finger and object compliance, theyare extended to finite areas).Once the contact points on the hand are defined, the con-ventional robotic analysis tools allow to find a relationshipbetween their locations with respect to hand palm and thefinger joint variable (direct kinematics). This relationship canbe differentiated with respect to time, giving the differentialkinematic equation that relates the contact point velocitiesp ∈ℜnc to the joint velocities q ∈ℜnq

p = Jq (1)

where J is the hand Jacobian J ∈ℜnc×nq , nq is the numberof actuated joints and nc is the number of contact variables(components of the contact point twists) constrained by thecontact model.The transpose of hand Jacobian relates the grasping contactforces λ ∈ℜnc to the hand joint torques τ ∈ℜnq

τ = JTλ (2)

Contact point twist components on the grasped object arerelated to the object reference frame twist νo = [oT ωT]T bythe following relationship

p = GTνo. (3)

where GT denotes the grasp matrix transpose, G ∈ ℜ6×nc .The Grasp matrix relates the grasping contact forces λ tothe object external wrench w ∈ℜ6

w = Gλ

For further details on hand Jacobian and Grasp matricesdefinition and computation, the reader is referred to [19].

According to a model inspired by human hand synergies,we suppose that the hand is actuated using a number ofinputs whose dimension is lower than the number of handjoints. These inputs are then collected in a vector z ∈ ℜnz

that parametrize the hand motions along the synergies. Inthis paper we define the postural synergies as a joint dis-placement aggregation corresponding to a reduced dimensionrepresentation of hand movements, according to a compliantmodel of joint torques.In other terms, the reference vector qre f ∈ ℜnq for jointvariables is a linear combination of postural synergies z∈ℜnz

with nz ≤ nqqre f = fz(z) (4)

This approach differs from other works where the synergiesare considered perfectly stiff and the actual joint variables area linear combination of synergies [20], [21]. In a compliantactuation model, the torques applied by the joint motors isproportional to the difference between the reference and theactual joint displacement

τ = Kq(qre f −q

)(5)

where Kq is the joint stiffness matrix. It represents jointactuator stiffness, given by the joint position control staticgain, and the structural compliance, due to the mechanicaldeformation of hand elements (joints, drive cables, links,etc.). The problem of computing contact force distributionand object movements that can be controlled acting onsynergies has been studied in [18].The relationship in eq. (4) can be differentiated with respectto time, obtaining

qre f = Sz (6)

where the synergy matrix S ∈ℜnq×nz , is defined as

S =∂ fz

∂ z.

Matrix S columns describe the shapes, or directions, of eachsynergy in the joint velocity space. It is worth noting that,if the relationship defined in eq. (4) is not linear, synergymatrix S is not constant, but depends on synergy value z.

In the following the main contribution of the paper ispresented.

The target of this work is to determine a method to map aset of synergies defined on a reference human hand ontoa generic robotic hand. A model of the human hand isthus defined. We refer to this model as paradigmatic hand.The paradigmatic hand is a kinematic and dynamic modelinspired by the human hand that does not closely copy thekinematical and dynamical properties of the human handbut rather represents a trade-off between the complexityof the human hand model accounting for the synergisticorganization of the sensorimotor system and the simplicity,and accessibility, of the models of robotic hands available onthe market. A detailed kinematic analysis of the paradigmatichand with postural synergies is reported in [22].

Let the paradigmatic hand be described by the jointvariable vector qh ∈ ℜnqh and assume that the subspace ofall configurations can be represented by a lower dimensionalinput vector z ∈ℜnz (with nz ≤ nqh ) which parametrizes themotion of the joint variables along the synergies qh = Shzbeing Sh ∈ℜnqh×nz the synergy matrix. In terms of velocitiesone gets

qh = Shz. (7)

The ultimate goal of this work is to find a way of controllingthe joint variables qr ∈ℜnqr of the robotic hand in a syner-gistic way using the vector of synergies z of the paradigmatichand. In other terms we want to design a map Sr to steer therobotic joint variables as follows

qr = Sr z (8)

where map Sr depends on synergy matrix Sh and othervariables as explained in the following.

Remark 1: Actually, according to the compliant modelpreviously summarized, and according to eq. (6), in eq. (7)and (8) we should indicate the reference hand joint valuesqhre f and qrre f respectively. It is worth noting that themapping procedure explained in this paper considers a virtualgrasp, i.e. a grasp in which no real contact forces are present,

and then, according to eq. (2) and (5), there is no differencebetween the reference and the actual hand configuration.However, the mapping procedure can be applied even whena real grasp is considered: in this case it will provide thereference value of joint variables, that will differ from theactual one, and the difference will depend both on the contactforce magnitude and on the system compliance. In thefollowing equations, for the sake of simplicity, we will notuse the subscript re f to indicate the reference joint values.�

In this work, we propose a method of projecting synergiesfrom paradigmatic to robotic hands which explicitly takesinto account the task space. One of the main advantagesof designing a mapping strategy in the task space is thatresults can be used for robotic hands with very dissimilarkinematics. The idea is to replicate the task performed withthe paradigmatic hand using the robotic hand with projectedsynergies.

The mapping is defined assuming that both the paradig-matic and the robotic hands are in given configurations q0hand q0r.

Remark 2: Note that the mapping procedure can be ap-plied for any pair of reference configurations q0h and q0r,i.e. paradigmatic and robotic hand configurations can be setindependently. However, the choice of very dissimilar initialconfiguration may lead to hand trajectory that appears verydifferent in the configuration space, although they produce,on the virtual object, the same displacement and the samedeformation. Very dissimilar initial configuration may alsolead to other types of problems, for example one of the handsmay reach its joint limits or singular configurations while theother could further move. �

Given the configuration q0h, a set of reference pointsph = [pT

1h, · · · , pTih, · · · ]T are chosen on the paradigmatic hand.

In this work we considered the fingertip points as referencepoints. However, the algorithm can be applied also choosingother reference points, for example on the intermediate pha-langes or in the hand palm, and furthermore the number ofreference points is not a-priori fixed. Adding other referencepoint can be useful when a power grasp is taken into account.The virtual ellipsoid is then computed as the minimumvolume ellipsoid containing the reference points in ph (Fig.2). Note that in general reference points do not lie on theellipsoid surface. Let us parametrize the virtual ellipsoidby its center oh and by the lengths s jh ( j = 1,2,3) ofits semi-axes. The motion of the hand due to synergiesactivation could be described using a large set of parameters.In this paper we simplify the problem assuming a rigid-bodymotion, defined by the linear and angular velocities of theellipsoid center oh and ωh respectively, and a non-rigid strainrepresented by the variations of the semi-axes. Let s j be thederivative of the j-th semi-axis length.Although the virtual ellipsoid does not represent an objectgrasped by the paradigmatic hand, it can be easily shownthat with a suitable model of joint compliance and contactcompliance, the rigid-body motion of the virtual ellipsoidcorresponds to the motion of a grasped ellipsoidal object

However, the algorithm can be applied also choosing otherreference points, for example on the intermediate phalangesor in the hand palm, and furthermore the number of referencepoints is not a-priori fixed. Adding other reference pointcan be useful when a power grasp is taken into account.[[[CHECK THIS]] The virtual ellipsoid is then computedas the minimum volume ellipsoid containing the referencepoints in ph (Fig. ??). Note that in general reference pointsdo not lie on the ellipsoid surface. Let us parametrize thevirtual ellipsoid by its center oh and by the lengths s jh( j = 1,2,3) of its semi-axes. The motion of the hand due to

Fig. 2. Mapping synergies from the human (paradigmatic) to the robotichand: the fingertip positions of the paradigmatic hand allows to define thevirtual ellipsoid. Activating the human hand synergies, the ellipsoid is movedand strained; the same motion and strain is imposed to the virtual ellipsoiddefined for the robotic hand.

synergies activation could be described using a large set ofparameters. In this paper we simplify the problem assuminga rigid-body motion, defined by the linear and angularvelocities of the ellipsoid center oh and ωh respectively, anda non-rigid strain represented by the variations of the semi-axes. Let s j be the derivative of the j-th semi-axis length.Although the virtual ellipsoid does not represent an objectgrasped by the paradigmatic hand, it can be easily shownthat with a suitable model of joint compliance and contactcompliance, the rigid-body motion of the virtual ellipsoidcorresponds to the motion of a grasped ellipsoidal objectand that the non-rigid motion accounts for the normal com-ponents of the contact forces for a ellipsoidal object grasp.By representing the motion of the hand trough the virtualobject, the motion of the generic reference point pih can beexpressed as

pih = oh +ωh × (pih −oh)+3

∑j=1

s jh�(pih −oh)

t s jh�

s jh, (7)

where the superscript ()t represent the transpose and s jhrepresent the versor corresponding to the ellipsoid semi-axes.

Grouping all the reference point motions one gets

ph = Ah

ohωhs1hs2hs3h

, (8)

where matrix Ah ∈ ℜnch×9 is defined as follows

Ah =

I −S(p1h −oh)�(p1h −oh)

t s1h�

s1h · · ·· · · · · · · · · · · ·I −S(pih −oh)

�(pih −oh)

t s1h�

s1h · · ·· · · · · · · · · · · ·

(9)

Matrix Ah depends on the motion that has to be reproducedon the robotic hand and consequently it depends on the task.

Considering the contact model as hard finger [?], thematrix Ah can be seen as the Grasp matrix for the virtualobject.

From (??), (??) and (??) we can evaluate the virtualellipsoid motion and deformation as a function of the synergyvector velocity z of the paradigmatic hand

ohωhs1hs2hs3h

= A#h ph = A#

hJhShz, (10)

where A#h denote the pseudo-inverse of matrix Ah. These

motions and deformations have to be mapped onto therobotic hand. Let us consider the robotic hand in a givenconfiguration q0r ∈ ℜnqr with a set of selected referencepoint location vector pr ∈ ℜncr . Note that no hypothesiswere imposed on the number of reference points on theparadigmatic human and robotic hands, in general we canconsider nch �= ncr, neither on their locations, and neither onthe initial configuration of the two hands. The same use of thevirtual ellipsoid is applied here: find the minimum ellipsoidenclosing the reference points and indicate with or its centercoordinates and with s jr the lengths of its semi-axes.

In order to take into account the differences betweenthe dimensions of the human and the robotic workspace, ascaling factor is introduced. This scaling factor is obtainedconsidering to virtual spheres computed on both the humanand the robotic hand as the minimum volume sphere con-taining reference points. The virtual object scaling factor isthen defined as the ratio between the radii of the two spheres:ksc = rr

rh. Note that the scaling factor depends on the hand

dimensions, but also on their configuration.Then, the motion and deformation of the virtual ellipsoidgenerated by the paradigmatic hand are scaled and trackedby the virtual ellipsoid referred to the robotic hand:

orωrs1rs2rs3r

= Kc

ohωhs1hs2hs3h

(11)

However, the algorithm can be applied also choosing otherreference points, for example on the intermediate phalangesor in the hand palm, and furthermore the number of referencepoints is not a-priori fixed. Adding other reference pointcan be useful when a power grasp is taken into account.[[[CHECK THIS]] The virtual ellipsoid is then computedas the minimum volume ellipsoid containing the referencepoints in ph (Fig. ??). Note that in general reference pointsdo not lie on the ellipsoid surface. Let us parametrize thevirtual ellipsoid by its center oh and by the lengths s jh( j = 1,2,3) of its semi-axes. The motion of the hand due to

Fig. 2. Mapping synergies from the human (paradigmatic) to the robotichand: the fingertip positions of the paradigmatic hand allows to define thevirtual ellipsoid. Activating the human hand synergies, the ellipsoid is movedand strained; the same motion and strain is imposed to the virtual ellipsoiddefined for the robotic hand.

synergies activation could be described using a large set ofparameters. In this paper we simplify the problem assuminga rigid-body motion, defined by the linear and angularvelocities of the ellipsoid center oh and ωh respectively, anda non-rigid strain represented by the variations of the semi-axes. Let s j be the derivative of the j-th semi-axis length.Although the virtual ellipsoid does not represent an objectgrasped by the paradigmatic hand, it can be easily shownthat with a suitable model of joint compliance and contactcompliance, the rigid-body motion of the virtual ellipsoidcorresponds to the motion of a grasped ellipsoidal objectand that the non-rigid motion accounts for the normal com-ponents of the contact forces for a ellipsoidal object grasp.By representing the motion of the hand trough the virtualobject, the motion of the generic reference point pih can beexpressed as

pih = oh +ωh × (pih −oh)+3

∑j=1

s jh�(pih −oh)

t s jh�

s jh, (7)

where the superscript ()t represent the transpose and s jhrepresent the versor corresponding to the ellipsoid semi-axes.

Grouping all the reference point motions one gets

ph = Ah

ohωhs1hs2hs3h

, (8)

where matrix Ah ∈ ℜnch×9 is defined as follows

Ah =

I −S(p1h −oh)�(p1h −oh)

t s1h�

s1h · · ·· · · · · · · · · · · ·I −S(pih −oh)

�(pih −oh)

t s1h�

s1h · · ·· · · · · · · · · · · ·

(9)

Matrix Ah depends on the motion that has to be reproducedon the robotic hand and consequently it depends on the task.

Considering the contact model as hard finger [?], thematrix Ah can be seen as the Grasp matrix for the virtualobject.

From (??), (??) and (??) we can evaluate the virtualellipsoid motion and deformation as a function of the synergyvector velocity z of the paradigmatic hand

ohωhs1hs2hs3h

= A#h ph = A#

hJhShz, (10)

where A#h denote the pseudo-inverse of matrix Ah. These

motions and deformations have to be mapped onto therobotic hand. Let us consider the robotic hand in a givenconfiguration q0r ∈ ℜnqr with a set of selected referencepoint location vector pr ∈ ℜncr . Note that no hypothesiswere imposed on the number of reference points on theparadigmatic human and robotic hands, in general we canconsider nch �= ncr, neither on their locations, and neither onthe initial configuration of the two hands. The same use of thevirtual ellipsoid is applied here: find the minimum ellipsoidenclosing the reference points and indicate with or its centercoordinates and with s jr the lengths of its semi-axes.

In order to take into account the differences betweenthe dimensions of the human and the robotic workspace, ascaling factor is introduced. This scaling factor is obtainedconsidering to virtual spheres computed on both the humanand the robotic hand as the minimum volume sphere con-taining reference points. The virtual object scaling factor isthen defined as the ratio between the radii of the two spheres:ksc = rr

rh. Note that the scaling factor depends on the hand

dimensions, but also on their configuration.Then, the motion and deformation of the virtual ellipsoidgenerated by the paradigmatic hand are scaled and trackedby the virtual ellipsoid referred to the robotic hand:

orωrs1rs2rs3r

= Kc

ohωhs1hs2hs3h

(11)

However, the algorithm can be applied also choosing otherreference points, for example on the intermediate phalangesor in the hand palm, and furthermore the number of referencepoints is not a-priori fixed. Adding other reference pointcan be useful when a power grasp is taken into account.[[[CHECK THIS]] The virtual ellipsoid is then computedas the minimum volume ellipsoid containing the referencepoints in ph (Fig. ??). Note that in general reference pointsdo not lie on the ellipsoid surface. Let us parametrize thevirtual ellipsoid by its center oh and by the lengths s jh( j = 1,2,3) of its semi-axes. The motion of the hand due to

Fig. 2. Mapping synergies from the human (paradigmatic) to the robotichand: the fingertip positions of the paradigmatic hand allows to define thevirtual ellipsoid. Activating the human hand synergies, the ellipsoid is movedand strained; the same motion and strain is imposed to the virtual ellipsoiddefined for the robotic hand.

synergies activation could be described using a large set ofparameters. In this paper we simplify the problem assuminga rigid-body motion, defined by the linear and angularvelocities of the ellipsoid center oh and ωh respectively, anda non-rigid strain represented by the variations of the semi-axes. Let s j be the derivative of the j-th semi-axis length.Although the virtual ellipsoid does not represent an objectgrasped by the paradigmatic hand, it can be easily shownthat with a suitable model of joint compliance and contactcompliance, the rigid-body motion of the virtual ellipsoidcorresponds to the motion of a grasped ellipsoidal objectand that the non-rigid motion accounts for the normal com-ponents of the contact forces for a ellipsoidal object grasp.By representing the motion of the hand trough the virtualobject, the motion of the generic reference point pih can beexpressed as

pih = oh +ωh × (pih −oh)+3

∑j=1

s jh�(pih −oh)

t s jh�

s jh, (7)

where the superscript ()t represent the transpose and s jhrepresent the versor corresponding to the ellipsoid semi-axes.

Grouping all the reference point motions one gets

ph = Ah

ohωhs1hs2hs3h

, (8)

where matrix Ah ∈ ℜnch×9 is defined as follows

Ah =

I −S(p1h −oh)�(p1h −oh)

t s1h�

s1h · · ·· · · · · · · · · · · ·I −S(pih −oh)

�(pih −oh)

t s1h�

s1h · · ·· · · · · · · · · · · ·

(9)

Matrix Ah depends on the motion that has to be reproducedon the robotic hand and consequently it depends on the task.

Considering the contact model as hard finger [?], thematrix Ah can be seen as the Grasp matrix for the virtualobject.

From (??), (??) and (??) we can evaluate the virtualellipsoid motion and deformation as a function of the synergyvector velocity z of the paradigmatic hand

ohωhs1hs2hs3h

= A#h ph = A#

hJhShz, (10)

where A#h denote the pseudo-inverse of matrix Ah. These

motions and deformations have to be mapped onto therobotic hand. Let us consider the robotic hand in a givenconfiguration q0r ∈ ℜnqr with a set of selected referencepoint location vector pr ∈ ℜncr . Note that no hypothesiswere imposed on the number of reference points on theparadigmatic human and robotic hands, in general we canconsider nch �= ncr, neither on their locations, and neither onthe initial configuration of the two hands. The same use of thevirtual ellipsoid is applied here: find the minimum ellipsoidenclosing the reference points and indicate with or its centercoordinates and with s jr the lengths of its semi-axes.

In order to take into account the differences betweenthe dimensions of the human and the robotic workspace, ascaling factor is introduced. This scaling factor is obtainedconsidering to virtual spheres computed on both the humanand the robotic hand as the minimum volume sphere con-taining reference points. The virtual object scaling factor isthen defined as the ratio between the radii of the two spheres:ksc = rr

rh. Note that the scaling factor depends on the hand

dimensions, but also on their configuration.Then, the motion and deformation of the virtual ellipsoidgenerated by the paradigmatic hand are scaled and trackedby the virtual ellipsoid referred to the robotic hand:

orωrs1rs2rs3r

= Kc

ohωhs1hs2hs3h

(11)

However, the algorithm can be applied also choosing otherreference points, for example on the intermediate phalangesor in the hand palm, and furthermore the number of referencepoints is not a-priori fixed. Adding other reference pointcan be useful when a power grasp is taken into account.[[[CHECK THIS]] The virtual ellipsoid is then computedas the minimum volume ellipsoid containing the referencepoints in ph (Fig. ??). Note that in general reference pointsdo not lie on the ellipsoid surface. Let us parametrize thevirtual ellipsoid by its center oh and by the lengths s jh( j = 1,2,3) of its semi-axes. The motion of the hand due to

Fig. 2. Mapping synergies from the human (paradigmatic) to the robotichand: the fingertip positions of the paradigmatic hand allows to define thevirtual ellipsoid. Activating the human hand synergies, the ellipsoid is movedand strained; the same motion and strain is imposed to the virtual ellipsoiddefined for the robotic hand.

synergies activation could be described using a large set ofparameters. In this paper we simplify the problem assuminga rigid-body motion, defined by the linear and angularvelocities of the ellipsoid center oh and ωh respectively, anda non-rigid strain represented by the variations of the semi-axes. Let s j be the derivative of the j-th semi-axis length.Although the virtual ellipsoid does not represent an objectgrasped by the paradigmatic hand, it can be easily shownthat with a suitable model of joint compliance and contactcompliance, the rigid-body motion of the virtual ellipsoidcorresponds to the motion of a grasped ellipsoidal objectand that the non-rigid motion accounts for the normal com-ponents of the contact forces for a ellipsoidal object grasp.By representing the motion of the hand trough the virtualobject, the motion of the generic reference point pih can beexpressed as

pih = oh +ωh × (pih −oh)+3

∑j=1

s jh�(pih −oh)

t s jh�

s jh, (7)

where the superscript ()t represent the transpose and s jhrepresent the versor corresponding to the ellipsoid semi-axes.

Grouping all the reference point motions one gets

ph = Ah

ohωhs1hs2hs3h

, (8)

where matrix Ah ∈ ℜnch×9 is defined as follows

Ah =

I −S(p1h −oh)�(p1h −oh)

t s1h�

s1h · · ·· · · · · · · · · · · ·I −S(pih −oh)

�(pih −oh)

t s1h�

s1h · · ·· · · · · · · · · · · ·

(9)

Matrix Ah depends on the motion that has to be reproducedon the robotic hand and consequently it depends on the task.

Considering the contact model as hard finger [?], thematrix Ah can be seen as the Grasp matrix for the virtualobject.

From (??), (??) and (??) we can evaluate the virtualellipsoid motion and deformation as a function of the synergyvector velocity z of the paradigmatic hand

ohωhs1hs2hs3h

= A#h ph = A#

hJhShz, (10)

where A#h denote the pseudo-inverse of matrix Ah. These

motions and deformations have to be mapped onto therobotic hand. Let us consider the robotic hand in a givenconfiguration q0r ∈ ℜnqr with a set of selected referencepoint location vector pr ∈ ℜncr . Note that no hypothesiswere imposed on the number of reference points on theparadigmatic human and robotic hands, in general we canconsider nch �= ncr, neither on their locations, and neither onthe initial configuration of the two hands. The same use of thevirtual ellipsoid is applied here: find the minimum ellipsoidenclosing the reference points and indicate with or its centercoordinates and with s jr the lengths of its semi-axes.

In order to take into account the differences betweenthe dimensions of the human and the robotic workspace, ascaling factor is introduced. This scaling factor is obtainedconsidering to virtual spheres computed on both the humanand the robotic hand as the minimum volume sphere con-taining reference points. The virtual object scaling factor isthen defined as the ratio between the radii of the two spheres:ksc = rr

rh. Note that the scaling factor depends on the hand

dimensions, but also on their configuration.Then, the motion and deformation of the virtual ellipsoidgenerated by the paradigmatic hand are scaled and trackedby the virtual ellipsoid referred to the robotic hand:

orωrs1rs2rs3r

= Kc

ohωhs1hs2hs3h

(11)

Fig. 2. Mapping synergies from the human (paradigmatic) to the robotichand: the fingertip positions of the paradigmatic hand allows to define thevirtual ellipsoid. Activating the human hand synergies, the ellipsoid is movedand strained; the same motion and strain is imposed to the virtual ellipsoiddefined for the robotic hand.

and that the non-rigid motion accounts for the normal com-ponents of the contact forces for a ellipsoidal object grasp.By representing the motion of the hand through the virtualobject, the motion of the generic reference point pih can beexpressed as

pih = oh +ωh× (pih−oh)+3

∑j=1

s jh

[(pih−oh)

T s jh

]s jh, (9)

where s jh represent the versor corresponding to the ellipsoidsemi-axes.

Grouping all the reference point motions one gets

ph = Ah

ohωhs1hs2hs3h

, (10)

where matrix Ah ∈ℜnch×9 is defined as follows

Ah =

I −S(p1h−oh)[(p1h−oh)

T s1h

]s1h · · ·

· · · · · · · · · · · ·I −S(pih−oh)

[(pih−oh)

T s1h

]s1h · · ·

· · · · · · · · · · · ·

(11)

Matrix Ah depends on the motion that has to be reproducedon the robotic hand and consequently it depends on the task.Considering the contact model as hard finger [19], the matrixAh can be seen as the Grasp matrix for the virtual object.From (1), (7) and (10) we can evaluate the virtual ellipsoidmotion and deformation as a function of the synergy vectorvelocity z of the paradigmatic hand

ohωhs1hs2hs3h

= A#

h ph = A#hJhShz, (12)

where A#h denote the pseudo-inverse of matrix Ah. These

motions and deformations have to be mapped onto therobotic hand. Let us consider the robotic hand in a givenconfiguration q0r ∈ ℜnqr with a set of selected referencepoint location vector pr ∈ ℜncr . Note that no hypothesis

were imposed on the number of reference points on theparadigmatic human and robotic hands, in general we canconsider nch 6= ncr, neither on their locations, and neither onthe initial configuration of the two hands. The same use of thevirtual ellipsoid is applied here: find the minimum ellipsoidenclosing the reference points and indicate with or its centercoordinates and with s jr the lengths of its semi-axes.

In order to take into account the differences betweenthe dimensions of the human and the robotic workspace, ascaling factor is introduced. This scaling factor is obtainedconsidering two virtual spheres computed on both the humanand the robotic hand as the minimum volume sphere con-taining reference points. The virtual object scaling factor isthen defined as the ratio between the radii of the two spheres:ksc =

rrrh

. Note that the scaling factor depends on the handdimensions, but also on their configurations.The motion and deformation of the virtual ellipsoid generatedby the paradigmatic hand are scaled and tracked by thevirtual ellipsoid referred to the robotic hand:

orωrs1rs2rs3r

= Kc

ohωhs1hs2hs3h

(13)

where the scale matrix Kc ∈ℜ9×9 is defined as:

Kc =

kscI3,3 03,3 03,303,3 I3,3 03,303,3 03,3 I3,3

. (14)

According to eq. (10) and (11), the corresponding robotreference point velocity is given by

pr = Ar

orωrs1rs2rs3r

, (15)

where matrix Ar ∈ℜncr×9 is defined as follows:

Ar =

I −S(p1r−or)[(p1r−or)

T s1r

]s1r · · ·

· · · · · · · · · · · ·I −S(pir−or)

[(pir−or)

T s1r

]s1r · · ·

· · · · · · · · · · · ·

(16)

Recalling eq. (12) and (13) we can express the robotichand reference point velocities pr as a function of thesynergy velocities z:

pr = ArKcA#hJhShz (17)

and, considering the robot hand differential kinematics pr =Jrqr, where Jr ∈ℜncr×nqr is its Jacobian matrix, the followingrelationship between robot hand joint velocities and synergyvelocities is defined:

qr = J#r ArKcA#

hJhShz. (18)

Then the synergy mapping Sr in (8) for the robotic hand isdefined as

Sr = J#r ArKcA#

hJhSh. (19)

Note that the paradigmatic hand synergy matrix Sh is mappedto the synergy matrix for the robotic hand Sr through matrixJ#

r ArScA#hJh which is function of paradigmatic and robotic

hand configurations (q0h and qrh) and, of location of thereference points for the paradigmatic and robotic hands (phand pr).

The proposed algorithm is consequently a non-linearmapping between the paradigmatic human-like hand and therobotic hand. The obtained synergy matrix is not constantand depends on hands configurations.

IV. SIMULATIONS

We validated the proposed approach on a modular three-fingered 9 DoFs robotic hand and on a DLR-HIT II Handmodel with five fingers and 15 DoFs [23]. We comparedour results with the joint to joint mapping and the fingertip-mapping methods [10], [11]. Other mapping methods [14],[16] were not taken into account since they can not beeasily extended to kinematic structures that differ from thoseproposed in the relative applications.

The grasp of two different objects was considered: asphere and a cube. The paradigmatic and robotic hand jointvariables, and the contact points in the initial configurationswere known. Starting from this initial given grasp, we mod-ified the reference joint values according to the previouslydescribed methodology. Since the hand is grasping an object,by activating the synergies, both the contact forces and thegrasped object position and orientation vary. The details ofthe relationships between input synergy and output variablevalues are detailed in [18]. Algorithm performances wereevaluated comparing the object motion directions and graspquality obtained controlling the robotic hands with the abovementioned algorithms [18], [22].

Grasp quality evaluation was performed using both quali-tative and quantitative metrics in order to evaluate the force-closure properties of the grasp as described in [24]. Thequalitative metric returns a boolean value that shows if theobtained grasp is force-closure. The quantitative aspect ofthe grasp quality is expressed using a penalty function. Theresulting index represents the inverse of the distance of thegrasp from violating contact constraints. All details of theused indexes can be found in [24].

In the first simulation, a spherical object was consideredand the reference points for the human and robotic handswere chosen on their respective fingertips.We considered the paradigmatic hand grasping a spherewith the fingertips of the thumb, index, medium and ringfingers, while for the modular three-fingered hand and theDLR-HIT II hand we considered three and four contactpoints respectively. This emphasizes the independence ofour method to the number of selected contact points. Theparadigmatic and robot hand grasps that were analysed areshown in Fig. 3. The computed scaling factors ksc were 1.7

Fig. 3. The human-like hand (top-left), the modular three- finger robotichand (top-right) and the DLR-HIT II Hand (bottom) grasping a sphere withrespectively four, three and four contact points.

and 1.9 for the modular and the DLR-HIT hand respectively.

The obtained results are summarized in Table I for theDLR hand and for the modular hand, respectively. Each rowcorresponds to the case of controlling hands with one synergyor combinations of synergies. This analysis was carried outconsidering the first three synergies and their combinations.

The second column shows the grasp quality indexes forthe human-like hand controlled with synergies, while thethird one reports those of the robotic hand controlled withthe reference joint values obtained with the proposed virtualellipsoid mapping. The fourth and the fifth columns referto the joint to joint mapping and to the fingertip mapping,respectively [10], [11]. The performance is expressed bymeans of the cost function measuring the grasp qualityas described in [24]. The selected cost function basicallyrepresents a sort of distance between the contact forces andthe direction normal to the contact surface, then lower valuesrepresent set of contact forces that are farther from thefriction cone boundaries and then are better from the graspstability point of view. This cost function can be evaluatedonly if the grasp is force-closure, empty values in the tablemean that, for that method and those selected synergies, noforce closure is achievable. More details on the evaluationof grasp quality measures in synergy actuated robotic handscan be found in [22].

Let us analyse, for example, results shown in Table I,relative to the DLR hand. In the paradigmatic hand (secondcolumn), force closure can be obtained by activating onlythe first, the first two or the first three synergies, no forceclosure can be obtained if we activate only the second or thirdsynergy once. By increasing the number of synergies fromone to three clearly grasp quality increases (cost functionvalue decreases). If we consider the virtual ellipsoid mappingmethod described in the preceding section, we obtain betterresults with respect to the joint to joint and the fingertipmethods. The same quality indexes were evaluated consid-

ering the grasp of an object with different shape, a cube.The obtained results are summarized in Tables II for theDLR hand (the modular hand presents qualitatively similarresults). By observing the obtained results, we can concludethat, concerning the grasp quality index, the virtual ellipsoidmapping, both for spherical and cubic objects gets closer tothe human-like grasp behaviour in all the analysed cases.

TABLE IGRASP QUALITY EVALUATION FOR THE SPHERICAL OBJECT.

DLR-HIT II hand

Synergies P VE JtJ FSyn 1 0.2 − − −Syn 2 − − − −Syn 3 − − − −Syn [1-2] 0.14 0.046 − 0.079Syn [1-3] 0.09 0.037 0.057 0.049

Modular hand

Synergies P VE JtJ FSyn 1 0.2 − − −Syn 2 − − − −Syn 3 − − − −Syn [1-2] 0.14 0.02 0.113 −Syn [1-3] 0.09 0.02 0.020 −

P: paradigmatic hand, VE: Virtual Ellipsoid,JtJ: Joint to Joint, F: Fingertip

TABLE IIGRASP QUALITY EVALUATION FOR THE CUBIC OBJECT.

DLR-HIT II hand

Synergies P VE JtJ FSyn 1 0.3 − − −Syn 2 − − − −Syn 3 − − − −Syn [1-2] 0.20 0.156 − −Syn [1-3] 0.14 0.146 − −

P: paradigmatic hand, VE: Virtual Ellipsoid,JtJ: Joint to Joint, F: Fingertip

We further evaluated the error arising from the mapping ofa rigid body motion of an ellipsoidal object. The rigid bodymotion was obtained by applying a constant synergy rate, i.e. z = const. We firstly evaluated the trajectory of the virtualellipsoid with the paradigmatic hand. We then compared thistrajectory with those obtained with the DLR-HIT II hand,by applying the proposed virtual ellipsoid (VE), the jointto joint (JtJ) and the fingertip (F) mapping procedures. Inthe virtual ellipse algorithm, we used a scale factor ksc = 1,in order to compare the trajectories for the paradigmaticand robotic hand respectively. According to the geometricanalysis presented in [18], that relates the dimensions ofcontrollable internal contact force and rigid body motionsubspaces to the number of actuated synergies, we observedthat, if an object rigid body motion has to be produced,

0.50

0.51

1.5

0.5

0

0.5

1

1.50

1

2

3

4

5

ParadigmaticVirtual EllipsoidFingertipJoint to Joint

Fig. 4. Trajectories in the workspace of the center of the ellipsoidal objectfor the modular (left) and DLR-HIT II (right) hand. The blue lines representthe mapped motion obtained with the proposed algorithm; the red linerepresent the motion with the paradigmatic hand, while green and violetlines represent joint to joint and fingertip mapping respectively. Axis unitsare in cm.

(a) (b) (c)

Fig. 5. The first synergy mapped onto the modular hand: (a) starting posi-tion, (b) middle position, (c) end position. In the top-left the correspondingparadigmatic hand postures.

at least four synergies have to be considered with threecontact points (with a Hard Finger contact model) and sevensynergies with four contact points. For the DLR-HIT II handanalysis we then considered the first seven synergies. Thearising trajectories of the center of the ellipsoidal object formodular and DLR-HIT II hand are shown in Fig. 4. As itcan be seen from the plots, the trajectories obtained for therobotic hand with the virtual ellipsoid procedure is very closeto those obtained with the paradigmatic hand, with respectto the other mapping methods. This result is clearly due tothe fact that the mapping itself is based on the replication,in the workspace, of the same rigid body motion of a virtualobject.

V. EXPERIMENTS

The proposed mapping procedure has been validated bysome experiments performed with a fully-actuated robotichand with a modular structure. Each module (42× 33×16mm) has one DoF and it can be easily connected to theothers obtaining kinematic chains that we can consider asfingers. These chains are connected to a common base thatcan be thought as a palm. In the proposed configuration eachfinger has three DoFs, thus the hand has globally nine DoFs.

The first two synergies of the paradigmatic hand, mappedon the modular robotic hand according to eq. (19), are shownin Fig. 5, 6.

(a) (b) (c)

Fig. 6. The second synergy mapped onto the modular hand: (a) startingposition, (b) middle position, (c) end position. In the bottom-right thecorresponding paradigmatic hand postures.

(a) (b) (c)

Fig. 7. Object rigid body motion obtained controlling the first foursynergies: (a) starting position, (b) middle position, (c) end position. The redarrows represent the motion of the object, while the black arrows representthe movement imposed by the paradigmatic hand.

Concerning the rigid body object motion obtained control-ling the robotic hand with synergies calculated according tothe method described in the preceding sections and summa-rized in eq. (19), the results are shown in Fig. 7.

Although the used device represents a trivial example ofrobotic hand, the complexity and the high number to DoFsto control are, in our opinion, a possible benchmark tovalidate our approach. Furthermore its kinematic structureis significantly different from the paradigmatic hand one, soit could be useful to test how the proposed mapping methodbehaves with very dissimilar hand structures.

VI. DISCUSSION

The proposed mapping strategy, based on mimicking be-haviour of human hand synergies, could be the basis of aninterface between a higher level control, that defines thesynergy reference values z, and the robotic hand. The highlevel can be thought as independent from the robotic handstructure. The interface, based on the proposed mappingstrategy, represents the low level control stage whereby theinput synergies are translated into reference joint valueswhich actually control the robotic hand.

This mapping has been tested in manipulation tasks. Workis in progress to validate the virtual ellipsoid mapping alsofor the approaching phase of grasps. Simulation resultsare very interesting in terms of performances as shown inthe previous section. However this approach presents somedrawbacks.

The proposed mapping is based on a heuristic approach:we choose to reproduce a part of the hand motion, whichpractically corresponds to move and squeeze an ellipsoidalobject. Although squeezing and moving an object explainsa wide range of tasks, many other possibilities exist in

manipulating objects which are not modelled with this map-ping. Work is in progress to generalize the proposed methodenriching the possible motions to be reproduced.

Differently from the joint to joint mapping, with respectto which the proposed method gets better performances, hereSr is not a constant matrix but it depends on both the humanand robotic hand configurations and by the reference positionpoints of the human and robotic hands which should be givenin this work.

VII. CONCLUSION AND FUTURE WORK

Designing synergy-based control strategies in the paradig-matic hand domain can dramatically reduce the dimension-ality of the grasping and manipulation problems for robotichands. However, an efficient mapping is needed to deal withrobotic hands with dissimilar kinematics. We proposed amethod for mapping synergies that using a virtual objectallows to work directly in the task space thus avoiding theproblem of dissimilar kinematics between human-like handand robotic hands. We compared our solution to the mostused solutions existing in literature and we evinced that theproposed method is more efficient in terms of mapped graspquality and direction of motion. Our preliminary results seemto be very promising, they were performed on a roboticthree fingers 9 DoFs hand, a modular hand with a kinematicstructure very different from that of the human hand, andalso on a 15 DoFs DLR HIT II robotic hand. The resultsshowed the effectiveness of the proposed method. Furtherinvestigation on different robotic hands is in progress. Oneof the main issues of our approach is that the mapping isnon-linear and that its implementation could need a highcomputational burden. Ongoing research is focused on theevaluation of the conditions whereby some simplification canbe applied to get constant or slowly varying mappings. Asfuture work, moreover, an integration with grasping simulatorlike Grasp-it! [25] is expected in order to use its graspplanner to determine initial configurations of the human andthe robotic hands. The choice of the initial configurationswill be based on performance indexes such as those used tomanipulability analysis [26].

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