Kinetostatic Analysis of Underactuated Fingers

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 2, APRIL 2004 211

Kinetostatic Analysis of Underactuated FingersLionel Birglen, Student Member, IEEE, and Clément M. Gosselin, Member, IEEE

Abstract—The aim of this paper is to establish a fundamentalbasis for the analysis of underactuated fingers with a general ap-proach. A new method to obtain the force capabilities of any un-deractuated fingers will be presented. Force capability is defined asthe ability to generate an external wrench onto a fixed object with agiven set of phalanges. This method is based on the introduction oftwo new matrices which completely describe the relationship be-tween the input torque of the finger actuator(s) and the contactforces on the phalanges. Using this tool, one can study the condi-tions under which certain phalanx forces vanish, and compare dif-ferent underactuation mechanisms with a rigorous approach.

Index Terms—Robot hand, shape adaptation, stable grasp, un-deractuation.

I. INTRODUCTION

ROBOTIC hands have been developed with the aim ofmatching the human hand in terms of dexterity and

adaptation capabilities to equip either a dextrous manipulatoror a human being as a prosthetic device. Pioneer designsinclude: the Okada hand [1], the Stanford/JPL (Salisbury’s)hand [2], the Utah/MIT hand [3], the Belgrade/USC hand [4],the BarrettHand [5], the hands from the DLR [6], [7], the LMShand [8], or the NASA Robonaut hand [9]. However, significantefforts have been made to find designs simple enough to beeasily built and controlled, in order to obtain practical systems[10], [11], particularly in human prosthetics. To overcome themitiged success of the early designs, mainly due to the costof the control architecture needed for complex mechanicalsystems with often more than ten actuators plus many sensors,a special emphasis has been placed on the reduction of thenumber of degrees of freedom (DOFs), thereby decreasingthe number of actuators. In particular, the SSL hand [12], theGraspar hand [13], the DIES-DIEM hand [14], the Cassinofinger [15], and the TBM hand [16] have followed this path.On the other hand, very few prototypes involve a smallernumber of actuators without decreasing the number of DOFs.This approach, namely, underactuation, can be implementedthrough the use of passive elements like springs or mechanicallimits leading to a mechanical adaptation of the finger to theshape of the object to be grasped [17]–[22]. A similar approachconsists of using elastic phalanges, which increase the adap-tation capability but decrease considerably the strength of the

Manuscript received November 18, 2002; revised February 26, 2003. Thispaper was recommended for publication by Associate Editor C. Melchiorri andEditor I. Walker upon evaluation of the reviewers’ comments. This work wassupported in part by Natural Sciences and Engineering Research of Canada(NSERC) and in part by the Canada Research Chair Program.

The authors are with the Department of Mechanical Engineering, LavalUniversity, Québec, QC G1K 7P4, Canada (e-mail: [email protected];[email protected]).

Digital Object Identifier 10.1109/TRA.2004.824641

grasp [23], [24]. Another way to consider shape adaptation isto use tentacle-like fingers [25].

Two main approaches dominate the literature on roboticgrasping. On one hand, purely theoretical work on graspingand manipulation and, on the other hand, the rather intuitivedesign of functional prototypes. This paper is an attempt tobridge this gap, for the special case of underactuated fingers.Indeed, if the development of underactuated fingers aims atovercoming the theoretical difficulties of general manipulationissues, and at obtaining prototypes of practical utility, thecapabilities of these fingers remain not well known. Prototypeshave often been built with intuitive design, without a genericknowledge of the resulting behavior, and based mainly onspecific computer-aided simulation. This paper presents aneffort to establish a common framework using simple theo-retical bases. The fundamental goal of underactuation beingsimplicity, the objective of this paper is to provide practicaltools for the analysis and comparison of underactuated fingers.Indeed, some issues have been overlooked in previous workand should be systematically addressed; for instance, the graspforce distribution, the capability of the finger to actually exertforces on a grasped object, the stability of the grasp, etc. Un-deractuation in robotic hands generates intriguing properties.For example, such hands cannot always ensure full whole-handgrasping. Indeed, the distribution of the forces onto the differentphalanges is partially predetermined by the mechanical designof the hand, and some phalanges may not be able to actuallyexert any effort in certain configurations. This uncontrollableforce distribution can also lead to unstable grasps, a continuousclosing motion of the actuator tending to eject the object. In[26], the authors have presented a brief illustration of the aboveissues for a 2-DOF finger. The latter reference can be seen asa more detailed introduction of this paper, which generalizesthe work to -DOF fingers with a general architecture. Hence,the paper [26] will be often referred to. A new method to studythe capabilities of underactuated fingers will be presentedthat allows rigorous comparison of different transmissionmechanisms through the definition of different indexes, similarto the dexterity in kinematic analysis. In this paper:

• two matrices that completely define the contact forces asa function of the actuator(s) torque(s) are defined;

• using these matrices, configurations of the finger leadingto stable grasps are presented;

• indexes to quantify the ability of the finger to create thesestable grasps are introduced;

• different mechanisms used in underactuated hands arecompared, using the latter indexes;

• finally, considerations on other aspects not covered in thispaper are discussed.

1042-296X/04$20.00 © 2004 IEEE

212 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 2, APRIL 2004

Fig. 1. Closing sequence of a 2-DOF underactuated finger, courtesy of ThierryLaliberté, modified with permission.

The first part of the paper (Section III) establishes the fun-damental background of our analysis and requires knowledgeof screw theory and mechanical transmission design. This is thetheoretical aspect of our work. One then illustrates in the secondpart (Sections IV and V), how these results can be used to de-sign and characterize underactuated fingers. Sections IV and Vrepresent the practical aspect of the paper.

II. UNDERACTUATION IN ROBOTIC HANDS

Underactuation in robotic fingers is different from the con-cept of underactuation usually presented in robotic systems, andthese notions should not be confused. An underactuated robot isgenerally defined as a manipulator with one or more unactuatedjoints. On the other hand, underactuated fingers generally useelastic elements in the design of their “unactuated” joints. Thus,one should rather think of these joints as uncontrollable or pas-sively driven instead of unactuated. In an underactuated finger,the actuation wrench is applied to the input of the fingerand is transmitted to the phalanges through suitable mechan-ical design, e.g., four-bar linkages, pulleys and tendons, gears,etc. Since underactuated fingers have many DOFs, say , andfewer than actuators, passive elements are used to kinemat-ically constrain the finger and ensure the shape adaptation ofthe finger to the object grasped. To this end, springs and me-chanical limits are often used, though inertial properties of themechanism can also be used [13]. An example of an underac-tuated 2-DOF finger using linkages and its closing process isillustrated in Fig. 1. Notice the mechanical limit that allows apreloading of the spring to prevent any undesirable motion of thesecond phalanx due to its own weight and/or inertial effects, andalso to prevent hyperflexion of the finger. Springs are useful forkeeping the finger from incoherent motion, but when the graspsequence is complete, they still oppose the actuator force. Thus,springs shall be designed with the smallest stiffness possible,however, sufficient to keep the finger from collapsing.

III. -DOF 1-DOA FINGER

A. Introduction

A general -DOF, 1-degree-of-actuation (DOA) finger withfour-bar linkages will be considered for initial analysis. If, usu-ally, only three phalanges are used in robotic fingers, the resultspresented here can be extended to other mechanical devicesusing underactuation, such as elephant trunks or snake-likerobots. Indeed, one of the first underactuated grippers consisted

Fig. 2. Model of underactuated n-DOF finger: geometric and forceparameters.

Fig. 3. Detailed modeling of a single stage of the finger.

of two ten-phalanx fingers [17]. Results will be subsequentlyextended to more general types of transmission. This design isa generalized version of the three-phalanx finger used in theMars and Sarah prototypes [27], hence, more detailed resultsfor and will also be given. The model of the fingeris presented in Fig. 2, more details about a single stage arefound in Fig. 3. Note that the finger is planar, i.e., abduction/ad-duction is not considered. Indeed, such motions are generallynot used during the final grasp, but rather for reconfigurationof the finger or manipulation. Moreover, contacts happen notonly with the fingertip but with all phalanges, the finger beingdesigned for grasping and not for dextrous manipulation.

In order to determine the configurations where the finger canapply forces to the object grasped, one shall proceed with aquasi-static modeling of the finger. Equating the input and theoutput virtual powers, one obtains

(1)

where is the input torque vector exerted by the actuator and thesprings, is the corresponding velocity vector, is the vectorof contact wrenches, and is the vector containing the twist ofeach of the contact points, i.e.,

(2)

BIRGLEN AND GOSSELIN: KINETOSTATIC ANALYSIS OF UNDERACTUATED FINGERS 213

where is the stiffness of the torsional spring located at point. One also has

(3)

where is a shorthand notation of the wrenchwhere the moment has been written before the force coordi-nates [28]. is the twist of the th contact point.The contact twist can be simply expressed as the product of aJacobian matrix and the derivatives of the phalanx joint co-ordinates, which is a natural choice, i.e., with

. Similarly, one can also establish the matrixwhich relates the vector to the derivatives of the phalanx jointcoordinates, and contains the elements function of the trans-mission mechanisms used, i.e., . Equation (1) thenbecomes

(4)

However, one should remember that matrix is generally notsquare. Therefore, is not unique for a given , even if Coulombfriction is considered, an infinite number of wrenches satisfy (4)if is of rank . Indeed, the general solution of (4) is ofthe form

(5)

where and is the pseudoinverse of matrix .On the other hand, for a given , is uniquely defined by (4).To obtain a bijective relationship between and , each contactwrench will be assumed to be a pure force normal to the phalanx,i.e., , where is the unit wrenchcorresponding to a pure force along , where is a unit vectororthogonal to the th phalanx. This assumption can seem veryrestrictive. However, our primary aim is to study the capabilityof the finger (and only the finger) to exert a wrench onto a fixedobject. We are, therefore, not interested in the local propertiesof the contact, such as friction or geometry of the contact. Inorder to obtain simple and general results that can characterizethe finger itself and not the system formed by the finger andthe object, pure normal forces should be considered. Obviously,the omitted local properties can have a tremendous impact interms of stability of the grasp but exceed the scope of this paper.Therefore, one shall modify vector and as

(6)

where the operator stands for the reciprocal product of screwsin the plane. Thus, is the vector of contact force magnitudes,and is the vector of the projections of the contact twists ontothe respective normals of the phalanges. One shall now proceedwith the definition of matrices and corresponding to theupdated expressions of and defined in (6). One shall keep inmind that the expression of matrix is modified so that

is preserved with (6). The twist at the th contact point is givenby

(7)

where and is the joint twist associated towith respect to the contact point considered [29]. The latter

can be written as

(8)

since every joint between two phalanges is assumed to be of therevolute type, and where is the vector from to the contactpoint on the th phalanx. is the matrix representation of thecross product in the plane, i.e.,

(9)

Thus, the virtual power can be expressed as

(10)

therefore, finally

(11)

where is a unit vector tangent to the th phalanx. With, one can finally obtain the matrix in a lower triangular form

(12)

noticing that . The elements of the matrix can begiven explicitly as

(13)

Note that the expression of matrix is not specific to underactu-ated fingers and still holds in more general cases, provided thatabduction/adduction is not considered. One can also establishthe matrix , which relates the vector to the derivatives ofthe phalanx joint coordinates, i.e., or

(14)

Note that the form of this matrix is characteristic of underactua-tion. If the finger is fully actuated, should be the identity ma-trix of dimension . Also, is a function of the linkage used totransmit the actuator torque to the th phalanx. The linkage usedin our model is most common, since it allows both compact-ness and simplicity. For the model discussed here, the principleof transmission gives the angular velocity ratio of a four-barlinkage, also known as Kennedy’s Theorem [30], [31] (consider


with constant) and by superposition,one obtains

(15)

since the velocity output of each stage is the input of the nextstage, where

(16)

is the signed distance between point and the geometric in-tersection of lines and . This distancecan be negative if the intersection point is on the same sideas with respect to . Thus, can also vanish.Geometrically, it corresponds to the case where links and

are aligned. In such a configuration, it is easy to see thatpower cannot be properly transmitted, and infinitesimal motion

is possible with . One can also obtainas a function of the parameters of the system, i.e.,

(17)where and stand, respectively, for and , andwith

where can be expressed as a function of and , thusone can obtain the expressions of each recursively. Equation(15) can be used recursively to obtain as a function of vector

, i.e.,

(18)

also, for the last phalanx

(19)

one obtains

(20)

and, by identification between (20) and (14), one finally has

(21)

where and are, respectively, the identity matrix andthe zero vector of dimension . This method can also be usedto obtain elements of matrix with different types of transmis-

sion mechanisms. Examples will be given in Section IV. Finally,one obtains

(22)

which is the equation that provides a practical relationship be-tween the actuator torques and contact forces. One can note that

, thus this inverse is not computation-ally intensive, and specific inversion procedures exist for trian-gular matrices. This formulation is, therefore, very well suitedfor both analytical derivations and numerical calculations. Fur-thermore, if the finger is fully actuated, i.e., , one has

, a relationship similar to the grasp matrix linking fin-gertips to object wrenches or the Jacobian matrix that maps jointtorques to fingertip forces, both usually presented in graspingpapers [32], [33]. As for the validity of this equation, conditionsfor existence of the inverses are

(23)

(24)

which means that contacts should exist with all phalanges, andthat should exist, i.e.,

(25)

meaning geometrically that points , , and are noton the same line. If this is not true, no effort can be transmittedto upper stages of the mechanism in the current configuration.Another interesting case arises if , thenbecause of the particular form of the coefficient

and (26)

This has a simple physical interpretation: means thatthe distance between point and the geometric intersection oflines and is zero. Thus, both lines in-tersect in . If one assumes that , it means that points

, , and are on the same line. Since the actuatortorque is usually transmitted to upper stages of the mechanismthrough link , and that in this particular configuration, suchtransmission is impossible (link cannot generate any ac-tuation torque about ), no motion is transmitted, hence, notorque except for the springs, which are usually negligible. Con-sequently, an infinitesimal rotation is possible around , andif the springs are neglected, one also has that . Inconsequence, the finger cannot grasp any object with the upperphalanges since they cannot apply forces. This situation is sim-ilar to the condition of existence of matrix . Such situationsshould be avoided through suitable design, so thatwill always be positive, and .

B. Discussion on the Positive Definiteness of the Forces

Most usual cases shall now be explicitly given and studied.The following results have been verified using classical staticanalysis (free body diagrams). For , one has

(27)


(28)

(29)

And for

(30)

(31)

(32)

with

(33)

The coefficients in (32) are defined as

(34)

(35)

These expressions allow the determination, for a set of geo-metric parameters, of the contact situations defined by the pair

with

and (36)

that allow full positiveness of the vector . The set of thesecontact situations corresponds to the stable part of the spacespanned by the contact situations pair , namely, thespace of contact configurations. Stable grasps as we referredto should not be confused with form- or force-closure asusually defined in the literature. In this paper, a stable graspis a contact situation pair which corresponds to a vectorwhere no component is negative. Again, this paper tries tocharacterize the finger itself, independently from the objectbeing grasped. If springs are neglected, expressions of thelatter vectors become most simple. is obviously absent fromthe expressions because rotation about this axis leaves themechanism in the same kinematic configuration (the finger isrotated as one single rigid body). It can also be shown that signsof elements of are independent of ; the proof is, however,more cumbersome and relies on the general inverse calculus bymeans of cofactors.

The limit between the zones where all elements of are pos-itive and the rest of the space is, therefore, a hyper-surface in the contact situation space. Therefore, it is impos-

TABLE IGEOMETRIC PARAMETERS

sible to completely visualize this frontier (between all positivecomponent space and its complement), even in the three-pha-lanx case. Note, for example, that for the set of parameters pre-sented in Table I (which corresponds approximately to the pa-rameters used in our prototypes of underactuated hands [27]),the volume of the stable three-phalanx grasps is approximately14% of the whole space of contact configurations (a section ofthe configuration space is shown in Fig. 4), assuming this spaceto be bounded by a hyperparallelepiped defined by

and . This can seem abnor-mally small but, one should remember that full-phalanx graspscorrespond only to a part of the whole possible grasps. That is,fewer-than- -phalanx grasps can also be stable, as will be de-tailed in Section V. Furthermore, the stable contact situations arealmost always located in the intuitive useful workspace. For ex-ample, if one changes the angular limits to

, the percentage of stable three-phalanx grasps jumps to ap-proximately 32%. One should also note that even if the con-tact situation allows to be fully definite, the ratios betweenthe components of this vector are independent of the actuatortorque. Thus, in each contact situation, span is of dimensionone, and can be parameterized by . One cannot modify the ra-tios between the components of without changing the contactsituation. For a fully actuated finger, span is of dimension .

The conditions for to become zero, which can be consid-ered as the edge of positiveness, become more complicated asdecreases. For example

(37)

or (38)

or(39)

or

(40)

The first condition on , namely, , has a simple geo-metrical meaning, i.e., the contact point is located on the equi-librium position of the last phalanx [26] [Fig. 5(b)]. However,to the best of the knowledge of the authors, higher order caseshave no simple interpretation and can correspond to situationswhere intuitively, one would expect that a full-phalanx graspis stable [Fig. 5(c)]. Illustration of configurations where one ormore forces become negative are shown in Fig. 5.


Fig. 4. Contact forces and associated stability loci (k = l =2 and k = l =2). Each force component is shown with the plane f = 0 in grey, intersections withthe latter are outlined.

Fig. 5. Contact situations where one or more forces vanish (parameter set 1).If k exists, it is assumed that k = l =2. Vanishing forces are indicated bydashed lines.

A remarkable result is that the vanishing of is most rare,because it can only happen when one is zero, whichcan be prevented by means of mechanical limits or careful de-sign. One can also remark that if one and only one is zero fora given , there is an implicit relationship between the ’s for

only. If more ’s are zero, one has a set of implicit equa-tions that must be satisfied by certain components of the contactsituation pair. The latter can have one, many, or no solutions,determining if this equilibrium configuration is possible. To ob-tain a final stable grasp with the general -phalanx finger, oneshould have a contact situation where the contact forcevector has only positive components (however, not strictly).

It is trivial to demonstrate that such situations are a subset ofthe set described by , meaning that the finger cannot bein equilibrium if the last phalanx is not. This simple statementyields more consequences than it seems, as to ensure the finalstability of the grasp, one has to obtain a stable last phalanx.Thus, one can design the last phalanx to obtain an optimallystable design [26]. That way, under the assumption that the vari-ation of position of point is negligible and no internal pe-riodic mobility is possible, contact made with the last phalanxwill never be lost inside the finger workspace, and such an ini-tial contact will tend to converge to a stable contact situation.The evolution of the contact situation trajectory can be rathercomplex, depending on the geometry of the grasped object andthe dynamic properties of the phalanges. Contacts can be made,lost, remade, etc., during the grasping sequence. Nevertheless,if one uses an optimal last phalanx design, a stable configura-tion will be attained under the assumptions previously stated.It should be noted that the second type of unstable frontier de-scribed in [26] should be avoided at any cost, because it can leadto a roll-back phenomenon, where the last phalanx slides againstthe object with a continuous closing motion, resulting in a sit-uation where the finger grasps nothing but itself (Fig. 6). Thisphenomenon is the -phalanx generalization of the ejection phe-nomenon presented in [20]. The design presented in [20] insistson the mechanical joint limits of to avoid thelatter type of ejection, which is the reason why we have previ-ously named this range “useful workspace.” Mechanical limitsare key elements in the design of underactuated fingers consid-ering stability issues, because they limit the shape adaptation to


Fig. 6. Roll-back phenomenon.

reasonable configurations (thus avoiding ejection). In the worstcase where all joints except the first one are locked, the fingersimply acts as a basic rotational gripper (with a quite complexshape), but still can firmly grasp the object. From that point ofview, underactuated hands could be considered more like grip-pers with shape adaptation than traditional robotic hands.

Finally, one can also characterize the capability of the under-actuated finger to create full-phalanx grasps with an index

(41)

where is a Kronecker-like symbol for positiveness ofvector that eliminates non-whole-phalanx grasps

ifotherwise

(42)

and is the workspace of the finger in terms of , i.e.,the hyperparallelepiped defined by and

or (to bepreferred). This index physically represents the percentage ofthe workspace that is achievable by full-phalanx grasps, namely,whole-hand grasping workspace. One shall modify the indexdefinition to obtain a measure of how far one is to lose contactwith one phalanx by including a “distance” from the vanishingof the smallest component, i.e.,

(43)

One can also use an index based on the grasp-forces distributionto provide the grasp with roughly equally distributed contactforces, such as (based on [34], slightly modified)

(44)

Numerical values of the latter indexes will be presented inSection IV-D.

IV. OTHER TRANSMISSION MECHANISMS

The benefit of the method presented in this paper is that itallows one to obtain very quickly the expression of the contactforces developed by an underactuated finger. This can be espe-cially useful if different mechanisms are considered, matrixis independent of the type of the transmission, and only the el-ements of matrix need to be modified. A method to expressthe coefficients of follows. One has to remember the generalform of (15) obtained by superposition, i.e.,

(45)

where is the transmission factor of the th stage. The latterequation means that the velocity output of each stage is the inputof the next stage (Fig. 3 illustrates this concept). To obtain thefactor , one simply has to virtually “lock” and deter-mine the ratio between and , which can be obtained forany mechanism. Most usual transmissions (e.g., gears, pulleys)are already well known. Once this factor is defined, a recursivescheme can be easily derived from (15), and therefore, the co-efficients can be written as

(46)

Finally, expressions for the contact forces are obtained by (22)where the inversion of both and matrices is needed. In thefollowing sections, examples of usual transmission mechanismswill be given. Notice that different types of mechanisms can bemixed in the same finger without changing the method.

A. Double-Stage Mechanism

The so-called double-stage mechanism, i.e., a mechanismsimilar to the one represented in Fig. 2 but with two consecutivephalanges merged into a single one, has been considered forcompactness [18]. Analyzing such mechanisms is very similarto the initial study, the matrix remains unchanged since thefinger is still planar (keeping in mind that phalanx lengthsshould be adjusted), and only revolute joints are used. Matrix

becomes

(47)

It corresponds simply to locking every even-numbered jointfrom the previous design to zero degree. The expression of thematrix is straightforward and can be used to design a particular


Fig. 7. Tendon-actuated finger.

finger with a specific desired grasp-force distribution. By usingtwo stages instead of one, one can obtain specific ratios withouthaving large and link lengths. Thus, this mechanism willindeed generally lead to more compact designs. However, suchdesigns will also have some drawbacks, such as the limitedspace available for sensors.

B. Tendon-Pulley Transmission

A legacy of the tradition in robot finger actuation, i.e., atendon-actuated finger, is now studied. Such designs havealready been used in underactuated fingers [13], [17], [22],[35]. The proposed layout is shown in Fig. 7. The expressionof matrix remains again unchanged. Matrix can be simplyexpressed as

(48)

where and for are, respectively, the radius of thepulley located at the base and the end of the th phalanx. Notethat, in this case, the latter matrix is constant for any configura-tion of the finger, as opposed to what was obtained in the anal-ysis presented above for fingers actuated with linkages. One canagain proceed with the expression of the contact force vector as

Fig. 8. Equivalence between pulley-tendon and four-bar transmission.

presented in (22). An interesting result arises if the pulley radiusratio is constant, i.e.,

(49)

Usually, for compactness purposes. The latter expressionconsiderably simplifies the expression of matrix . The expres-sion of the last phalanx forces can then be expressed as

(50)

with . Similar to the previous study, conditions forthe forces to become zero are implicit functions that cannot beeasily solved. Interestingly, the contact force on the last pha-lanx is only the function of the ratio and the location of thecorresponding contact , not the configuration of the finger it-self. Moreover, one can define an equilibrium point [26] for thismechanism as

(51)

Furthermore, note that the limit case where is againan almost always unstable design (equilibrium point pushedto infinity) for the two-phalanx finger, like its mechanical“evil twin,” the parallelogram [26]: both have the exact samecondition for positiveness! Similar to what was obtained inSection III-B, a tendon-actuated finger with pulley radii equiva-lent to link lengths, i.e., and , has a possiblestable three-phalanx grasp occurrence of approximately 56%,to be compared with the 14% of the mechanism first studied.If limits are placed on elements of , similar to what wasdone in Section III-B, that percentage drops to approximately41%, which is still superior to the mechanical linkage. Thus,for small forces and with appropriate control to compensateelasticity and friction, this mechanism can be very efficient, as,for instance, for lightweight prosthetic devices. The kinematicsimilarity between both mechanisms is pointed out with ouranalysis technique, highlighting the inherent generality of thepresented method. Instantaneous equivalence is illustrated inFig. 8.


Fig. 9. Equivalence between gears and four-bar transmission (gears arerepresented by their primitive circles).

TABLE IICHARACTERISTIC INDEXES

C. Gears

Consider that instead of tendons, one has a gear train con-necting the principal gears located on the axis of the fingers.With such a mechanism, matrix becomes

(52)

where and are, respectively, the number of teeth ofthe gear located at the base and the end of the th phalanx. Bothgears are connected through a train that can be described bytwo equivalent gears with tooth ratio , that is, thegear with teeth is in contact with the gear , whichis rigidly connected with gear , which itself finally trans-mits motion to gear . Note that this form is very similar tothe double-stage mechanism presented in Section IV-A. In fact,this mechanism is to pulleys and tendons what double-stage isto the four-bar mechanism. A comparison is again illustrated inFig. 9.

D. Comparison

To compare the different mechanisms presented in the pre-vious sections, one can use the indexes presented in (41), (43),and (44). Results are presented in Table II. In order to obtaincomparable indexes, equivalence between link lengths, pulleyradii, etc., are established with the same rule as in Section IV-Band with the parameters defined in Table I. When only two pha-langes are considered, geometrical parameters will correspond

to the last two phalanges in the latter table. The workspace usedis defined by and .

It can be seen that in general, tendon-actuated fingers havebetter overall performances than linkage-actuated ones. This isdue to the fact that they do not suffer from mechanical singularconfigurations (for example, when three points of the transmis-sion mechanism are on the same line, see Section III-B). This isnot the case with multiple-stage fingers, because such singularconfigurations become most rare if they exist at all, since bothstages contribute to attaining the same kinematic configuration.However, larger forces can be applied with linkages than withtendons, since power transmission is performed through rigidbodies, resulting in very firm grasps. With more phalanges, onetends to achieve more adaptive grasps, but it becomes less likelythat all phalanges simultaneously perform the grasp.

V. LESS-THAN- PHALANX GRASPS

In order for a less-than- phalanx grasp to be stable, everyphalanx in contact with the object should have a correspondingforce strictly positive. For phalanges not in contact with the ob-ject, the corresponding generated forces, defined in (22), shouldbe zero, since the latter forces can also be seen as the externalforces needed to counter the actuation torque; equilibrium is im-plicitly assumed by (1). Thus, the relationships discussed in Sec-tion III-B for positiveness of forces also give us the equilibriumconditions for less-than- phalanx grasps. For example, if onewants to know under which conditions a stable grasp can beachieved with no contact on phalanx, say , one has to sat-isfy the condition

(53)

Such conditions are of greater importance than it seems at firstglance, since, as it has been shown, only part of the fingercontact situation space is stable with full-phalanx stable grasps.Thus, full-phalanx grasps are only a subset of the possiblestable grasp contact configurations. Many questions remainunanswered to characterize the behavior of the finger, e.g., isthere a global index that would allow us to describe the overallstability of the finger? Generalization of the ejection phenom-enon seems a promising way to characterize the stability ofan underactuated finger, but this theory should be extendedto cases where the hypotheses presented in Section III-B arerelaxed. Moreover, nontrivial less-than- phalanx contactequilibrium is possible, nontrivial, meaning with nonsimplegeometrical relationships, like two links on the same line. Thus,it is theoretically possible to fully constrain a general -DOFfinger with a single contact in such nontrivial configurations.

VI. CONCLUSIONS

This paper has presented and analyzed the force capabilitiesof underactuated fingers. The main contribution of this paper isthe introduction of matrices (Jacobian) and (Transmission).These matrices allow immediate characterization of the finger,provide the expressions of contact forces developed by thefinger, lead to considerations on equilibrium with any number


Fig. 10. Tendon-actuated mechanism, contact forces, and stability loci (k = l =2 and k = l =2). Each force component is shown with the plane f = 0 ingrey, intersections with the latter are outlined.

of phalanges in contact, and provide tools for comparisonbetween different designs. Such comparisons can be veryuseful to choose a particular design of underactuated finger.However, one should keep in mind other considerations that arenot easily mathematically quantifiable. For example, frictionand elasticity can eliminate tendon-actuated fingers, despitetheir good performance indexes. Furthermore, if theoretically,an infinite number of phalanges is ideal for grasping, sincesuch a finger would tend to approach the spatial complement ofthe shape of the object, which is the perfect grasper, practicalconsiderations of cost and complexity of the design limit thenumber of phalanges to generally three or four, at most. Hence,the authors believe that the tools presented in this paper canhelp refine underactuated finger designs in terms of geometricparameters, although the general layout of the finger remainsapplication driven. In order to properly design underactuatedfingers, one should be aware of the ejection phenomenon, inorder to achieve stable grasps and phalanx force distribution,avoiding weak last phalanges that cannot ensure sufficient forceto secure the grasp. Material not included here because of lackof space and under current investigation includes the in-depthstudy of a three-phalanx finger considering friction, geometryof the contact, and optimal phalanx force distribution.

ACKNOWLEDGMENT

The first author expresses his most grateful thanks to theoriginal designer of our prototypes of underactuated hands,

T. Laliberté, for providing him with such stimulating material.Merci!

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Lionel Birglen (S’00) received the B.Eng. degree inmechatronics from École Nationale Supérieure desArts et Industries de Strasbourg, Strasbourg, France,in 2000. He is currently working toward the Ph.D. de-gree in mechanical engineering at the Robotics Lab-oratory, Université Laval, Québec, QC, Canada afteran accelerated promotion from the M.Sc. level.

His work focuses on the kinematic analysis andcontrol of complex mechanical hands and parallelmanipulators, with a particular emphasis on forcecontrol and general theory of real-time control.

Mr. Birglen is a student member of ASME.

Clément M. Gosselin (S’88–M’89) received theB.Eng. degree in mechanical engineering fromthe Université de Sherbrooke, Sherbrooke, QC,Canada, in 1985, and the Ph.D. degree from McGillUniversity, Montréal, QC, Canada in 1988.

In 1988, he accepted a post-doctoral fellowshipfrom the French government and joined INRIA,Sophia Antipolis, France, for a year. In 1989, hejoined the Department of Mechanical Engineering,Laval University, Quebec City, QC, Canada, wherehe has been a Full Professor since 1997. He has held

a Canada Research Chair on Robotics and Mechatronics since January 2001. In1995, he received a fellowship from the Alexander von Humboldt Foundationwhich allowed him to spend six months as a Visiting Researcher in the Institutfür Getriebetechnik und Maschinendynamik, Technische Hochschule, Aachen,Germany. In 1996, he spent three months at the University of Victoria, Victoria,BC, Canada, for which he received a fellowship from the BC AdvancedSystems Institute. His research interests are kinematics, dynamics, and controlof robotic mechanical systems with a particular emphasis on the mechanicsof grasping and the kinematics and dynamics of parallel manipulators andcomplex mechanisms. His work in the aforementioned areas has been thesubject of several publications in international conferences and journals. Heis the French language editor for the international journal Mechanism andMachine Theory.

Dr. Gosselin received the Gold Medal of the Governor General of Canada in1985, the D. W. Ambridge Award from McGill University for the best thesisof the year in Physical Sciences and Engineering in 1988, and the I. . Smithaward from the Canadian Society of Mechanical Engineering in 1993. He is amember of the Institute for Robotics and Intelligent Systems (IRIS), one of thenetworks of the Canadian Centres of Excellence, and a member of the AmericanSociety of Mechanical Engineers (ASME) and the Canadian Committee for theTheory of Machines and Mechanisms (CCToMM).

Kinetostatic Analysis of Underactuated Fingers

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