IEEE TRANSACTIONS ON ROBOTICS 1

Stable Precision Grasps by Underactuated FingersGert A. Kragten, Mathieu Baril, Clement Gosselin, and Just L. Herder

Abstract—The ability of underactuated hands to grasp smallobjects is very limited, because the precision grasp is normallyunstable. The goal of this paper is to achieve stable precisiongrasps by means of simple design modifications of the distalphalanges of the fingers. These modifications comprise thecurving of the contact area of the distal phalanx, the applicationof a mechanical limit to prevent hyper-extension of the distalphalanx, and the application of a compliant joint between theproximal and distal phalanges. A model is developed to calculatethe boundary conditions of the finger dimensions in order toachieve stable precisions grasps for different object sizes. Anexperimental setup is used to test the grasp stability and to verifythe calculated results. It is concluded that stable precision graspsexist for the combination of concavely curved distal phalangeswith a mechanical limit or with a compliant joint, if the boundaryconditions of the finger dimensions are satisfied.

I. INTRODUCTION

Underactuated robotic hands can grasp various objects,

because the fingers adapt themselves to the shape of the

objects by their mechanical behavior. Underactuation means

that the number of actuators is less than the degrees of

freedom. Numerous examples in the literature show that this

principle can lead to adaptive hands that are relatively cheap,

lightweight and easy to control compared to fully actuated

hands, e.g. [1], [2], [3]. The adaptation of the fingers to

the objects is especially beneficial to envelope objects in a

so-called power grasp [4]. However, sometimes enveloping

objects is not possible, for instance, when small objects are

grasped, or when objects are approached from above. In such

cases the objects need to be grasped by the distal phalanges

only. In literature this is normally called a precision grasp [4],

[5], a pinch grasp [6], or a palmar or tip grasp [7]. The term

precision grasp will be used throughout this paper to refer to

a grasp equilibrium that is achieved by the distal phalanges

only. It is reported that a precision grasp is difficult to achieve

by underactuated fingers, e.g. [8], [9], [10]. It was shown

in [11] that this is caused by the instability of a precision

grasp equilibrium, where unstable means that the potential

energy of the hand and object at this equilibrium point is

not a local minimum. This means that grasping small objects

by underactuated hands is impossible, unless specific design

modifications are applied.

Generally, a compliant, frictional contact material − like

the skin of a human finger − can improve the capability to

obtain a precision grasp. Some other solutions have been

Gert A. Kragten and Just L. Herder are with the Dept. of BioMechanicalEngineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft,The Netherlands (email: g.a.kragten@tudelft.nl; j.l.herder@tudelft.nl, fax:0031-15-2784713).

Mathieu Baril and Clement Gosselin are with the Departement degenie mecanique, Pavillon Adrien-Pouliot, 1065 avenue de la MedecineUniversite Laval, Quebec, (Quebec) Canada, G1V 0A6 (email: math-ieu.baril.4@ulaval.ca; gosselin@gmc.ulaval.ca, fax: 418-656-7415).

proposed to achieve a precision grasp by underactuated

fingers. The SARAH [8] and the TWIX-hand [12] make

use of an additional mechanism to keep the distal phalanges

parallel to each other as long as no contact with an object is

encountered. The adaptive hand of RobotIQ [13] can actively

switch between a mode for a precision grasp between parallel

distal phalanges, and a mode for a power grasp where the

objects are enveloped. The addition of an extra mechanism to

the fingers or the active switching between modes complicates

the design, which is not desired. Also triggered mechanisms

are proposed as a less complicated solution to keep the distal

phalanges parallel [14], [15]. However, sufficient contact

friction and accurate grasp planning remain necessary to

stably grasp the objects between parallel phalanges. If these

conditions are not satisfied, a stable precision grasp does not

exist.

The goal of this paper is to investigate the achievement of

stable precision grasps by underactuated fingers by means

of simple design modifications. The effect of these design

modifications to the grasp stability is studied by a model that

considers the fingers and object as a conservative system. This

means that the potential energy must be a local minimum

at a precision grasp equilibrium. The results of this model

and the effectiveness of the proposed design modifications

are verified by experiments. The relevance of this work is

that the existence of the precision grasp does not depend on

contact friction anymore. In addition, accurate grasp planning

is not necessary, because there will be a stability area where

the objects are attracted towards the stable precision grasp

equilibrium. Finally, it leads to understanding of the stability

of underactuated fingers and the grasped objects.

The structure of the paper is as follows. Section II introduces

a model to calculate the stability of a precision grip between

straight phalanges. Three design modifications for the underac-

tuated fingers are then proposed, and the stability of precision

grasps between these fingers is calculated as a function of

the design parameters. An experimental setup is described for

measuring the stability. Section III provides the results of the

calculated and measured stable precision configurations. In

addition, it provides a visualization of the boundary conditions

on the design parameters to achieve stable precision grasps.

The effectiveness of the three design modifications is discussed

in Section IV, followed by conclusions in Section V.

II. MATERIALS AND METHODS

A gripper consisting of two underactuated fingers with two

straight phalanges each is taken as a reference to develop a

model to calculate the grasp stability. Then, three design mod-

ifications are proposed, and the model is adapted accordingly

IEEE TRANSACTIONS ON ROBOTICS 2

Fig. 1. Reference gripper consisting of two fingers with two straightphalanges each, driven by a pulley-cable system with a constant tension forceFa. Only the distal phalanges have a contact point with the object.

to calculate the existence of a stable precision grasp and the

expected equilibrium position of the object. In addition, the

model will be used to obtain the boundary conditions for the

design parameters of the design modifications to achieve a

stable precision grasp. An experimental setup that was used

to test the stability and measure the equilibrium position is

described at the end of this section.

A. Reference Gripper and Model

The fingers of the reference gripper are dimensioned as

follows. They consist of a straight proximal phalanx of length

L1, and a straight distal phalanx of length L2. L0 is the

distance between the proximal joint and the center line (i.e.

half of the palm width). These dimensions are shown in Fig. 1.

The phalanges are driven by a pulley-cable system, where the

pulley at the distal joint has a radius r2 and is attached to the

distal phalanx. The pulley at the proximal joint has a radius

r1 and can rotate freely. The tension force in the cable Fa is

constant. The dimensions L0, L1, L2, r2/r1 are regarded as

design parameters. A cylindrical object with a radius Robj is

the reference object, which can freely move in the (x, y)-plane.

Friction and inertia are neglected.

The stability can be determined by the sign of the eigen-

values of the Hessian matrix of the potential energy of the

mechanism [16]. The Hessian is obtained by the second

derivative of the potential energy V of the system. V can

be calculated as follows, assuming a constant input force Fa:

V = −Far1

(

θR,1 + θL,1 +r2r1

(θR,2 + θL,2)

)

(1)

where θi,j is the rotation of the phalanges with the first index

denoting the right (R) or left (L) finger, and the second index

denoting the proximal (1) or distal phalanx (2), see Fig 1. A

minus sign is put in front of the equation, because an increase

of θ1 and θ2 implies a smaller energy level. The constraint

equations that describe contact between the distal phalanx of

the right finger and the object are as follows:

L0 + L1 cos θR,1 + pR,2 cos(θR,1 + θR,2)−Xobj −Robj sin(θR,1 + θR,2) = 0

L1 sin θR,1 + pR,2 sin(θR,1 + θR,2)−Yobj +Robj cos(θR,1 + θR,2) = 0

−L0 − L1 cos θL,1 − pL,2 cos(θL,1 + θL,2)−Xobj +Robj sin(θL,1 + θL,2) = 0

L1 sin θL,1 + pL,2 sin(θL,1 + θL,2)−Yobj +Robj cos(θL,1 + θL,2) = 0

(2)

where pR,2 and pL,2 are the positions of the contact point

with the object on the distal phalanx of right and left finger,

respectively. This mechanism has four degrees of freedom

when only the distal phalanges have a contact point with the

object. Xobj , Yobj , θR,1 or θL,1 are chosen as the generalized

coordinates to obtain the most convenient geometric relations.

To express θR,2 and θL,2 as a function of the generalized

coordinates, it is needed to firstly express pR,2 and pL,2 as a

function of the other variables and coordinates by rearranging

the first and third constraint equation. These functions are

then substituted into the second or fourth constraint equation

and solved to obtain θR,2 and θL,2. Two solutions are found.

The correct solution can be selected after evaluation with real

values for all parameters. These selected formulas of θR,2 and

θL,2 are substituted in Eq. 1 to express the potential energy

V as a function of the generalized coordinates. Subsequently,

the Hessian matrix or second derivative of V to the four

degrees of freedom can be calculated. However the analytical

expression of each matrix element is very large. Therefore,

this (4×4)-matrix is numerically evaluated for a range of

design parameters to be able to calculate the eigenvalues and

check their sign.

To calculate the numerical values of the generalized coordi-

nates at a precision grasp equilibrium, the symmetry properties

of the reference gripper and object are used. This means

that Xobj = 0, pR,2 = pL,2 = p2, θR,1 = θL,1 = θ1,

θR,2 = θL,2 = θ2. The generalized coordinates are reduced to

Yobj and θ1. From static equilibrium equations it follows that

θ1+θ2 = π/2 and L0+L1 cos θ1 = Robj . The contact position

p2 is the only unknown to determine the equilibrium position

Yobj , see Eq. 2. To calculate p2, the moment equilibrium

equation of the finger about the proximal joint can be used:

Far1 = F2 (p2 + L1 cos θ2) (3)

where F2 is the contact force between the distal phalanx and

the object. F2 can be obtained with the free body diagram of

the distal phalanx:

F2 =Far2p2

(4)

Combining Eq. 3 and Eq. 4 and substituting θ2 = π/2 − θ1yields:

p2 =r2r1L1 sin θ1

1− r2r1

(5)

where 0 < p2 ≤ L2 to have a feasible contact point. It can be

checked that substitution of this equilibrium configuration into

IEEE TRANSACTIONS ON ROBOTICS 3

the first derivative of V yields indeed a zero vector. Substi-

tution of the numerical solution into the second derivative of

V yields the Hessian matrix at the equilibrium configuration.

The calculated equilibrium configuration is only stable if the

eigenvalues of this Hessian matrix are positive.

The equilibrium configuration and the four eigenvalues of

the Hessian matrix were calculated for L0 between 10 −

80 mm, L1 between 40−120 mm, and r2/r1 between 0.2−0.8while L2 = 160−L1 (constant finger length), Robj = 20 mm

and Far1 = 1000 Nmm. For all combinations of these values,

one or even two eigenvalues of the Hessian matrix were

negative. This means that no stable precision grasp was found

with two fingers with two straight phalanges. Therefore, design

modifications are needed to achieve stable precision grasps.

B. Design Modifications and Model

Three design modifications to the reference gripper are

proposed. The first modification is to change the shape of the

distal phalanges into concave curved ones. This was inspired

by the shape of, for instance, a spoon that can easily hold

objects. The second modification is to apply a mechanical

limit between the proximal and distal phalanx, which prevents

hyper-extension of the distal joint. This reduces the degrees

of freedom of the fingers at a precision grasp, while it still

allows the possibility to envelope bigger objects. The third

design modification is to apply torsional springs in the joints,

which provides the opportunity to adjust the shape of the

potential energy field of the mechanism. To simplify the

calculations, only the left finger of the gripper is considered,

while the reference object can freely move along the line

of symmetry. Consequently, stability of the object in the

x-direction is not yet considered and Xobj = 0.

1) Curved Distal Phalanges: The constraint equations for

the contact points between the object and the concave curved

phalanges can be obtained with the aid of Fig. 2. It is assumed

that the curve of the phalanges is circular with a radius Rph.

When θ2 = 0, the distal phalanx at the joint is tangential to the

proximal phalanx, but with an offset d to account for material

thickness of the phalanx. The position of the contact point on

the distal phalanx is now determined by angle Ψ (see Fig. 2).

The geometrical constraint equations, projected along x and

y, are formulated as follows:

−L0 + L1 cos θ1 + (Rph + d) cosα−(Rph −Robj) cos(α−Ψ)−Xobj = 0

L1 sin θ1 + (Rph + d) sinα−(Rph −Robj) sin(α−Ψ)− Yobj = 0

(6)

where α = θ1+θ2−π/2, and Xobj = 0. These two constraint

equations can be used to express θ2 as a function of the

generalized coordinates Yobj and θ1. It was needed to firstly

eliminate Ψ before θ2 could be calculated. This elimination

was done by firstly moving the terms containing Ψ to the other

side of the equality sign. Then both equations were squared,

added, and solved to obtain θ2. Two solutions were found,

and the proper solution was selected after evaluation with real

values for all parameters. This θ2 can be substituted into the

Fig. 2. Schematic drawing of the left finger in contact with an object. Thedrawing shows the concave curve of the distal phalanx with radius Rph andthe mechanical limit between the phalanges that applies when θ2 = θ2ML

.The object can freely move in the y-direction, while the finger is actuated bya pulley-cable system that is loaded by a constant force Fa.

potential energy function of the gripper, grasping a cylindrical

object with curved distal phalanges:

V = Far1

(

θ1 +r2r1

θ2

)

(7)

The stability of a precision grasp equilibrium is determined

by evaluation of the two eigenvalues λi of the Hessian of V at

the equilibrium configuration, where the Hessian is as follows:

H(θ1, Yobj) =

∂2V∂θ2

1

∂2V∂θ1∂Yobj

∂2V∂θ1∂Yobj

∂2V∂Y 2

obj

(8)

To determine the value of Yobj and θ1 for which a precision

grasp is achieved, two equilibrium conditions of the object

and the finger are applied. The object is in equilibrium if the

component of the contact force in y-direction is zero:

F2 sin(α−Ψ) = 0 (9)

The finger is in equilibrium if the moment about the proximal

phalanx is zero:

F2 ((Rph + d) sinΨ + L1 cos(Ψ− θ2))− Far1 = 0 (10)

where F2 is equal to:

F2 =r2Fa

(Rph + d) sinΨ(11)

Solving the constraint equations (Eq. 6) and the equilibrium

equations (Eqs. 9, 10) yields the Yobj , θ1, θ2, and Ψ for which

the finger and object are in a precision grasp equilibrium.

IEEE TRANSACTIONS ON ROBOTICS 4

Note that the following inequality must be satisfied to have

the contact point on the physical part of the curved phalanx:

0 < Rph sinΨ ≤ L2 (12)

When Rph is very large, the distal phalanx is almost

straight. It can be verified that the calculated grasp equilibrium

configurations and their stability then approximate the values

as calculated for the reference gripper.

2) Mechanical Limits: It was also proposed to apply a

mechanical limit to the extension of the curved or straight

distal phalanx. The degrees of freedom of the system reduce

to one when the distal phalanx is against the mechanical

limit. Then θ2 is a constant value (i.e. θ2 = θ2ML, see

Fig. 2), where θ2MLis a new design variable. The constraint

equations of Eq. 6 still apply, and are used to express θ1 as a

function of the single generalized coordinate Yobj in a similar

way as described in the previous section. Substitution of this

expression of θ1 into Eq. 7 yields the potential energy V as a

function of Yobj . The stability of a grasp equilibrium is now

verified by the sign of ∂2V/∂Y 2

obj .

The distal phalanx is only pushed against the mechanical

limit if no moment equilibrium about the distal joint can be

achieved, i.e. the actuation torque applied to the distal phalanx

is smaller than the torque applied to the distal phalanx by the

contact force F2:

r2Fa ≤ F2(Rph + d) sinΨ (13)

where F2 is calculated by moment equilibrium of the finger

about the proximal joint:

F2 =r1Fa

(Rph + d) sinΨ + L1 cos(Ψ− θ2)(14)

If the constraint given by Eq. 13 is not satisfied, it means that

the distal phalanx is not against the mechanical limit. Then

Eqs. 10 and 11 apply as if there were no mechanical limit.

For a precision grasp, the equilibrium condition of the

object must also be satisfied (Eq. 9). To calculate the

equilibrium configuration, Eqs. 6 and 9 are solved to obtain

the value of Yobj , θ1 and Ψ. Note that for the calculated

equilibrium configuration, the inequalities Eq. 12 and Eq. 13

must be satisfied. Otherwise, the contact point does not exist

on the physical part of the phalanx, or the distal phalanx is

not against the mechanical limit.

3) Compliant Joints: When springs are mounted at the

joints, the formula of the energy function V is extended. This

study only investigates the effect of a linear torsion spring at

the distal joint, while the distal phalanx is curved or straight.

Assuming that this spring has a torsion stiffness K and an

initial angle θ20 , V is as follows:

V = Far1

(

θ1 +r2r1

θ2

)

+1

2K(θ2 − θ20)

2 (15)

Note that this design modification adds a term to V that is not

dependent on Fa. This means that the effect of the addition of

TABLE ICONSTANT AND THE INTERVAL OF THE VARIABLE PARAMETERS THAT

WERE APPLIED TO DETERMINE THE RELATION BETWEEN THE DESIGN

PARAMETERS AND THE BOUNDARY CONDITIONS TO ACHIEVE STABLE

PRECISION GRASPS FOR OBJECTS WITH A RADIUS OF Robj = 20, 40, AND

55 MM.

Constant Parameters:

L0 40 mm Half of the palm widthL1 80 mm Length of the proximal phalanxL2 85 mm Length of the distal phalanxFa 12.8 N Tension in the cabler2 4.7 mm Radius of the distal pulley

Variable Parameters:

r1 [6 20] mm Radius of the proximal pulleyRph [80, 250] mm Radius of the distal phalanxθ2ML

[-1, 1.5] rad Max extension angle distal phalanxK [0, 200] Nmm/rad Stiffness of the spring at distal jointθ20 [-1, 1.5] rad Initial angle of the spring

a torsion spring with a specific stiffness also depends on the

magnitude of the actuation force.

To determine the value of Yobj and θ1 at a precision

grasp equilibrium, the constraint equations (Eq. 6) and the

equilibrium condition of the object and the finger are applied

(Eqs. 9, 10). Note that for this design modification, the torque

of the spring is now taken into account to calculate the distal

contact force F2:

F2 =r2Fa +K(θ2 − θ20)

(Rph + d) sinΨ(16)

An equilibrium configuration is valid if Eq. 12 is satisfied,

and if F2 is positive (i.e. compressive). An equilibrium

configuration is stable if the Hessian matrix of Eq. 15 is

positive definite.

C. Boundary Conditions for Stable Precision Grasps

It was initially desired to achieve analytic formulas that

express the relation between the design parameters and the size

of the object to achieve stable precision grasps. Although sym-

bolic formulas of the potential energy of the mechanism and

the equilibrium positions were obtained, the desired analytic

formulas are too large to be informative. Therefore, numerical

evaluations of these formulas were used to determine the

precision grasp equilibrium configuration and its stability for

a range of values of the design parameters. The chosen values

of the design parameters are shown in Table I, where the

design parameters that were kept constant are distinguished

from the parameters that were varied. Objects with a radius of

Robj = 20, 40, and 55 mm were taken as reference objects.

These dimensions were chosen to have an object smaller,

equal, and larger than the palm width L0. After an exploration

of the grasp stability for these dimensions of the parameters,

the boundary conditions to obtain stable precision grasps were

obtained. These results are visualized in Section III.

D. Experiments

The goal of the experiments is to verify the equilibrium

configurations and their stability that were calculated with

IEEE TRANSACTIONS ON ROBOTICS 5

Fig. 3. Picture of the experimental setup, where (1) is the actuation cable;(2) is the proximal phalanx; (3) is the curved distal phalanx (Rph = 90 mm);(4) is the object (Robj = 20 mm); and (5) is the force sensor mounted ona linear stage. The mechanical limit and the torsion spring are not shown onthis picture. The proximal pulley is just visible but the distal one is shieldedby the phalanx bodies. The reference frame and the line of symmetry aresuperimposed to this picture.

the grasp model. Furthermore, the setup was used to observe

the causes of unstable grasps and the interpretation of the

potential energy functions. The experimental setup consisted

of one underactuated finger and a cylindrical object that was

able to move along the line of symmetry, see Fig. 3. The

underactuated finger consisted of two phalanges, where the

first phalanx of length L1 = 80 mm was straight. The distal

phalanx could easily be exchanged to get different curvatures

Rph, but with a constant thickness d. A pulley with a radius

r2 = 4.7 mm was fixed to the shaft of the distal phalanx. A

cable was attached to this pulley, and was wrapped around a

free-rotating pulley at the proximal phalanx of radius r1. A

constant tension was applied to the cable by a dead weight

of 12.8 N. The distance from the proximal joint to the line

of symmetry was L0 = 40 mm. A mechanical limit could be

screwed on top of the proximal phalanx to limit the extension

of the second phalanx to θ2ML= 0.78 rad (or 1.22 rad when

the distal phalanx was straight).

The object consisted of a disc of radius Robj that could

freely rotate about a shaft to emulate frictionless contact

between the object and the finger. The shaft of the object was

mounted on a force sensor that was constructed by parallel

leaf springs and four strain gauges (see [17]). It measured the

resultant of the contact forces in the direction of the line of

symmetry with a precision of 0.2 N. The object and sensor

were mounted on a spindle, which was driven with a constant

speed of v = 15.7 mm/s by a DC-motor (Maxon, A-max26,

gear GP26B 3.8:1, and encoder Enc22) and a servo amplifier

(Maxon, LSC30/2).

The following parameters were varied between the experi-

TABLE IIVALUES OF THE VARIABLE PARAMETERS AT THE EXPERIMENTS. FOR AN

EXPLANATION OF THE SYMBOLS AND THE VALUES OF THE CONSTANT

PARAMETERS, SEE TABLE I.

r1 10.0, 15.5 mmRph 80, 90, 140, 200, ∞ mmθ2ML

0.78 radRobj 20, 40, 55 mmK 55 Nmm/radθ20 0.52 rad

ments: the radius of the proximal pulley r1, the curvature of

the distal phalanx Rph, the availability of the mechanical limit

θ2MLand the radius of the objects Robj . The values are shown

in Table II, leading to 60 experiments. For each experiment,

the object was placed at Yobj = 44 mm (or 60 mm for the

object Robj = 55 mm), the finger was actuated by the dead

weight, and then the object was pulled out of the finger. The

force and position were measured at a sample frequency of 20

Hz. All experiments were repeated three times.

The precision grasp stability was also measured for a

finger with a torsional spring at the distal joint of the five

differently curved phalanges. The stiffness of the spring

was approximately K = 55 Nmm/rad, and the initial angle

was approximately θ20 = −0.52 rad, leading to another 30

experiments.

The measured data was analyzed to find the object positions

where the resultant of the contact force Fobj,y was zero, while

only the distal phalanx had a contact point with the object.

At these positions, there was a precision grasp. However, the

force perpendicular to the line of symmetry (i.e. Fobj,x) caused

a small systematic error of 35 mV/N in the reading of the pull

force Fobj,y . Considering the finger and object as one system,

it can be shown that the magnitude of this force Fobj,x is as

follows:

Fobj,x =Far1 − Fobj,yL0

Yobj

(17)

The measured output voltage was first corrected for this

systematic error for all experiments. Then, the mean of the

measured equilibrium positions was calculated for each ex-

periment.

The stability of the measured equilibrium points was

determined by the sign of the contact force difference at the

equilibrium position. This difference is an approximation

of the stiffness. A negative difference means an unstable

precision grasp, while a positive difference means a stable

one. Note that by measuring the resultant of the contact force

while the object was being moved, even unstable equilibrium

points could be observed, as long as the finger itself was stable.

The experimental results were used to verify the grasp

model. The location of stable precision grasp positions were

compared with the positions that were calculated with the

grasp model. In addition, the force needed to pull the object

out of the hand as a function of the object position was also

calculated by the grasp model. This force is equal to the

first derivative of the potential energy to the object position

(Fobj,y = ∂V/∂Yobj).

IEEE TRANSACTIONS ON ROBOTICS 6

100

105

110

115

120

125

1

1.2

1.4

1.6

1.8280

290

300

310

320

Object position Yobj

[mm]Angle of proximal phalanx θ1 [rad]

Pote

ntial E

nerg

y V

[N

mm

]

Fig. 4. Surface plot of the potential energy of an object with a radiusRobj = 20 grasped by a distal phalanx that is curved with a radius of Rph =

140 mm, while the ratio of the pulley radii of the underactuated finger isr2/r1 = 0.30. An equilibrium point is shown at Yobj = 112.6 mm. Thispoint is unstable because the energy surface is a saddle point. The solid lineshows the magnitude of θ1 when the object is displaced in the direction ofYobj . When the position Yobj = 113 mm is reached (marked by the dashedline), θ1 rapidly decreases and the finger cannot be in equilibrium against theobject anymore.

III. RESULTS

A. Potential Energy Curves

Figures 4, 5 and 6 visualize typical potential energy func-

tions of the finger grasping a moving object for the three

cases, namely where the distal phalanx is curved (Fig. 4);

where a mechanical limit applies at the curved distal phalanx

(Fig. 5); and where a torsional spring applies to the joint of

the curved distal phalanx (Fig. 6). The chosen dimensions

of the variable parameters in these figures are as follows:

the object size Robj = 20 mm; the distal phalanx curvature

Rph = 140 mm; and the ratio of the pulley radii r2/r1 = 0.30.

A mechanical limit at θ2ML= π/4 rad applies in Fig. 5. In

Fig. 6 a torsional spring with a stiffness K = 70 Nmm/rad and

an initial angle θ20 = 0.4 rad applies to the distal joint. The

dimensions of the other parameters of the finger correspond

to Table I. The rotation angle of the phalanges changes as

the object is displaced. The calculated angle of the proximal

phalanx as a function of the object position is shown by a

dashed line in these figures. It can be observed that it follows

a trajectory of minimum energy. The asterisks indicate the

calculated equilibrium positions. The equilibrium positions at

a local minimum of the potential energy are stable, while the

positions at a saddle point are unstable. Note that the graph

of the potential energy of Fig. 4 and Fig. 5 are equal, except

for the limitation of θ1 in Fig. 5 due to the mechanical limit

between the distal and proximal phalanx.

B. Experimental and Calculated Results

The measured force to pull the three objects of a radius

Robj = 20, 40, and 55 mm out of the finger is shown in

Fig. 7. These results were obtained for the finger with a

curved distal phalanx Rph = 140 mm, a mechanical limit at

105 110 115 120 125 130 135 140 145 1501.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

295295

298

298

298

298

298

301

301

301301

304

304

304

307

307

307

310

310

313

313

316

319322

Object position Yobj

[mm]

Angle

of

pro

xim

al phala

nx θ

1 [

rad]

Fig. 5. Contour plot of the potential energy of an object (Robj = 20)grasped by an underactuated finger (Rph = 140 mm, r2/r1 = 0.30) with amechanical limit between the proximal and distal phalanx at θ2ML

= π/4.Two equilibrium points are shown, namely at Yobj = 112.6 mm (unstable)and at Yobj = 140.2 mm (stable). The dashed line shows the magnitude ofθ1 when the object is displaced in the direction of Yobj . A lower limit to θ1is shown, which is caused by the mechanical limit. At the dashed-dotted partof this lower limit, Eq. 13 is not satisfied.

297

297

298

298

298

298

299

299

299

299

299

299

302

302

302

302

302

302

305

305

305

305

310

310

310

315

315

320

3203253303353

40

Object position Yobj

[mm]

Angle

of pro

xim

al phala

nx θ

1 [ra

d]

110 120 130 140 150 1600.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

Fig. 6. Contour plot of the potential energy of an object (Robj = 20)grasped by an underactuated finger (Rph = 140 mm, r2/r1 = 0.30) witha torsional spring between the proximal and distal phalanx with a stiffnessK = 70 Nmm/rad and an initial angle θ20 = 0.40 rad. Two equilibriumpoints are shown, namely at Yobj = 116.2 mm (unstable) and at Yobj =

153.6 mm (stable). The dashed line shows the magnitude of θ1 when theobject is displaced in the direction of Yobj .

θ2ML= 0.78 rad, and a pulley ratio r2/r1 = 0.30. Note that

the measured force of the three repetitions per experiment are

plotted, showing the repeatability of the experiments. In ad-

dition, the calculated force-displacement lines are superposed

by dashed lines to show the correlation between the model

and the experiments. Where Fobj,y = 0, it means that the

contact force along the line of symmetry is zero and the system

consisting of the finger and the object is thus in equilibrium.

The lines belonging to Robj = 20 and 40 mm intersect the line

Fobj,y = 0 twice: first with a negative slope and then with a

positive slope. It can be observed that a small disturbance of

the object equilibrium at the point with the negative slope will

result in a large displacement away from the equilibrium point.

IEEE TRANSACTIONS ON ROBOTICS 7

40 60 80 100 120 140 160 180 200−2

−1

0

1

2

3

4

Object position Yobj

[mm]

Conta

ct

forc

e −

Fobj,y [

N]

Robj

= 20

Robj

= 40

Robj

= 55

Fig. 7. Measured force to pull objects with a radius Robj = 20, 40 and55 mm out of a finger with a curved phalanx Rph = 140 mm, and amechanical limit at θ2ML

= 0.78 rad. Dashed lines show the calculated

force as a function of the object position. A stable grasp is achieved at theobject positions where the contact force is zero and where the slope is positive.

This first intersection is thus an unstable equilibrium point

where the distal phalanx was not against the mechanical limit.

When the equilibrium configuration with the positive slope is

disturbed, the object will move back to the equilibrium point.

The second intersection is thus a stable equilibrium point.

In between, the proximal phalanx angle θ1 rapidly decreases

while θ2 increases until it hits the mechanical limit. The left-

most part of the lines − where Fobj,y increases − belongs

to the object positions where also the proximal phalanx was

in contact with the object. This was not considered in the

grasp model. The right-most part of the lines belongs to the

object positions where the end point of the distal phalanx is

in contact with the object. This was also not considered in the

grasp model.

The effect of the curvature Rph of the distal phalanx on the

measured force Fobj,y and equilibrium positions is shown in

Fig. 8. These results belong to an object size Robj = 20 mm.

The superimposed dashed lines show the calculated force-

displacement curves.

Figure 9 shows the experimental results for the case where

a torsional spring was applied to the joint of the curved distal

phalanges, while an object of Robj = 20 mm was pulled out

of the hand. The stiffness of the spring was approximately

K = 55 Nmm/rad, and the initial angle was approximately

θ20 = 0.52 rad. The dashed lines show the calculated force as

a function of the object position. For this stiffness and initial

angle, no stable precision grasp was measured for any of the

objects.

Table III summarizes all stable equilibrium positions that

were measured or calculated. A graphical representation of

these results is shown in Fig. 10 as a function of the curvature

of the distal phalanx. The mean of the measured stable

equilibrium points are visualized by a dot (·) or a circle (◦)

when r2/r1 = 0.30 or 0.47, respectively. Error bars show

two times the standard deviation. The calculated values are

shown by solid or dashed lines for r2/r1 = 0.30 and 0.47,

respectively.

40 60 80 100 120 140 160 180 200−2

−1

0

1

2

3

4

Object position Yobj

[mm]

Conta

ct

forc

e −

Fobj,y [

N]

Rph

= 90

Rph

= 140

Rph

= 200

Rph

= ∞

Rph

= 80

Fig. 8. The measured force to pull objects with a radius Robj = 20 mm outof a finger with a curved phalanx Rph = 80, 90, 140, 200 mm or a straightphalanx (Rph = ∞). The mechanical limit was at θ2ML

= 0.78 rad (1.22rad when Rph = ∞) and a pulley ratio r2/r1 = 0.30. The dashed linesshow the calculated force as a function of the object position. The existenceof a stable precision grasp and the equilibrium position of the objects dependson the dimension of Rph.

40 60 80 100 120 140 160 180 200−2

−1

0

1

2

3

4

Object position Yobj

[mm]

Conta

ct fo

rce −

Fobj,y [N

]

Rph

= 80

Rph

= 90

Rph

= 140

Rph

= 200

Rph

= ∞

Fig. 9. Measured force to pull objects with a radius Robj = 20 mm out ofa finger with a curved phalanx Rph = 80, 90, 140, 200 mm and a straightphalanx (Rph = ∞) and a pulley ratio r2/r1 = 0.30. A linear torsion springof K = 55 Nmm/rad and an initial angle of approximately θ20 = −0.52 radwas mounted at the joint between the proximal and distal phalanx. Dashedlines show the calculated force as a function of the object position. No stableprecision grasps were observed for the chosen dimensions of the finger, springand objects.

C. Boundary Conditions

This section summarizes the results of the calculated bound-

ary conditions on the dimensions of the design parameters to

achieve stable precision grasps for objects with various radii.

No stable precision grasps were found at all for the case with

only curved distal phalanges. Stable precision grasps do exist

when also a mechanical limit is applied to the curved distal

phalanx. The boundary conditions that apply to the case with a

mechanical limit are shown in Fig. 11. It visualizes the upper

and lower limit of the angle θ2MLwhere a mechanical limit

has to apply as a function of the curvature of the phalanges

Rph in order to achieve stable precision grasps for different

IEEE TRANSACTIONS ON ROBOTICS 8

TABLE IIITHE MEAN OF THE MEASURED STABLE EQUILIBRIUM POSITIONS FOR

DIFFERENT OBJECT SIZE Robj , PULLEY RATIO r2/r1 AND CURVATURE OF

THE DISTAL PHALANX Rph . ALL THESE STABLE POSITIONS WERE

OBTAINED WHILE THE DISTAL PHALANX WAS AGAINST THE MECHANICAL

LIMIT AT θ2ML= 0.78 RAD. THE STANDARD DEVIATION WAS SMALLER

THAN 0.58 MM. THE VALUES BETWEEN BRACKETS ARE THE CALCULATED

EQUILIBRIUM POSITIONS.

Rph [mm] Robj = 20 mm Robj = 40 mm Robj = 55 mm

r2/r1 = 0.3080 119 (119.4) 133 (133.6) 140 (141.5)90 124 (123.1) 137 (138.4) 146 (147.0)

140 141 (140.2) 159 (160.2) -200 161 (158.3) - -∞ - - -

r2/r1 = 0.4780 - - -90 - - -

140 - - (160.2) -200 166 (158.3) - -∞ - - -

80 100 120 140 160 180 200 220110

120

130

140

150

160

170

Curvature of the distal phalanx Rph

[mm]

Equili

brium

positio

n Y

obj [

mm

]

Robj

=55

Robj

=20

Robj

=40

Fig. 10. Visualisation of the measured and calculated stable equilibriumpositions as a function of the curvature of the distal phalanx Rph for differentobject sizes Robj . The mean of the measured equilibrium points are visualizedby a dot (·) or a circle (◦) when r2/r1 = 0.30 or 0.47, respectively. Errorbars show two times the standard deviation. The calculated values are shownby solid and dashed lines for r2/r1 = 0.30 and 0.47, respectively.

sizes of objects. At the upper limits of θ2ML, the contact point

on the phalanx with the object is at the tip of the phalanx

(Rph sinΨ = L2, compare to Eq. 12). At the lower limit of

θ2ML, the contact point on the distal phalanx is so close to the

joint, that the phalanx starts to flex (r2Fa = F2(Rph+d) sinΨ,

compare to Eq. 13). At the lower limit of θ2ML, the stable

and unstable equilibrium positions that were shown on the

energy contour lines in Fig. 5 coincide. When the ratio of

r2/r1 increases, the upper limits remain the same, while the

lower limits shift upward.

Stable precision grasps are also possible when a torsional

spring applies to the joint of a curved distal phalanx. Such sta-

ble grasps were not measured, but are obtained by calculation

with stiffer springs than the one applied in the experiments.

Figure 12 visualizes the upper and lower limit of the initial

angle θ20 of the torsion spring as a function of the spring

stiffness K in order to achieve stable precision grasps for

80 100 120 140 160 180 200 220 240−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Curvature of distal phalanx Rph

[mm]

Angle

of

mechanic

al lim

it θ

2M

L

[ra

d]

Robj

=20

Robj

=40

Robj

=55

Fig. 11. Upper and lower limits to the angle θ2MLwhere the mechanical

limit applies as a function of the curvature of the distal phalanx Rph.The upper and lower limits are shown for different objects sizes, while thedimensions of all other design parameters were kept constant according toTable I.

different sizes of objects. These limits apply to the finger

with a curvature Rph = 140 mm, r2/r1 = 0.30 and all

other dimensions according to Table I. It turned out that

the upper limits of θ20 are attained when the contact point

on the phalanx with the object is at the tip of the phalanx

(Rph sinΨ = L2, compare to Eq. 12). At the lower limits

of θ20 , it turned out that the precision grasp equilibrium was

neutrally stable (i.e. the Hessian was semi positive). In other

words, the position where the force-displacement line is in a

local minimum coincides with the position where the resultant

of the contact force on the object is zero. Alternatively, the

stable and unstable equilibrium positions, which were shown

on the energy contour lines in Fig. 6, coincide at the lower

limit of θ20 . When Rph is smaller, both limits move upwards.

Effectively the vertical distance between the two lines becomes

larger. A comparable effect can be observed when r2/r1becomes larger. This effect is stronger at low stiffness K.

IV. DISCUSSION

It was decided to use the principle of conservation of

energy to investigate the possibilities to achieve stable

precision grasps with underactuated hands. Three typical

plots of the energy V by a surface plot or contour lines

were shown by Figs. 4, 5 and 6. This approach turned

out to be very powerful to firstly understand the boundary

conditions to the dimensions of the design parameters in

order to achieve stable precision grasps, as it was explained

in Section III. Secondly, the stability of the equilibrium

points can easily be visualized by the contour lines of the

potential energy V , or by calculation of the eigenvalues of

the second derivative of V . Unfortunately, calculation of the

equilibrium points by solving the first derivative of V −

which is zero at equilibrium − did not succeed analytically.

However, it was used to check the results by substitution of

the numerical solutions. In some cases, unstable equilibrium

points were calculated that were not observed during the

experiments. Inspection of the potential energy then showed

IEEE TRANSACTIONS ON ROBOTICS 9

20 40 60 80 100 120 140 160 180 200−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Spring stiffness K [Nmm/rad]

Initia

l angle

θ2

0

[ra

d]

Robj

=20

Robj

=40

Robj

=55

Fig. 12. Upper and lower limits on the initial angle θ20 of the compliantjoint as a function of its stiffness K. The upper and lower limits are shownfor different objects sizes, while the dimensions of all other design parameterswere kept constant according to Table I. The curvature of the distal phalanxis Rph = 140 mm, and the pulley ratio is r2/r1 = 0.30.

that these equilibrium points were unstable because of the

finger mechanism (∂2V/∂θ21< 0). Indeed, such equilibrium

points can not be observed by the experimental approach of

this study. The unstable ones caused by an unstable object

(∂2V/∂Y 2

obj < 0) are observable, since the displacement of

the object was controlled by the spindle.

A good correlation was found between the experiments

and calculations, as can be observed in Figs. 7, 8, 9. The

measured and calculated force-displacement lines show the

same trends, and the difference between the measured and

calculated resultant of the contact force is generally less than

0.3 N. Especially the difference between the measured and

calculated stable equilibrium positions is small: generally less

than 1 mm, but at most 2.7 mm (see Fig. 10 or Table III).

Two remarkable differences can be observed:

(1) Figures 7 and 8 show a difference in the measured and

calculated object position where the resultant contact force

Fobj,y drops and becomes negative. At this position, θ1 rapidly

decreases, while θ2 increases until it hits the mechanical limit

(compare to the dashed line in Fig. 5 just after the unstable

equilibrium point). We observed that the finger was more

prone to hitting the mechanical limit in the experiments if

a higher cable force Fa was chosen. This means that the

observed difference is caused by friction or inertia of the

system.

(2) Table III shows a calculated stable precision grasp at

Robj = 40 mm, Rph = 140 mm, and r2/r1 = 0.47 which

was not measured. Inspection of the results showed that one

measurement actually does intersect the zero twice shortly

after each other, while the lowest measured contact force of

the other two repetitions remained just above zero (0.02 N).

This equilibrium point might not be observed because of the

precision of the sensor (0.2 N).

It is concluded that the grasp model is valid, and it can

be applied to calculate boundary conditions on the design

parameters of the fingers to achieve stable precision grasps.

A. Effectiveness of the Design Modifications

Three design modifications were proposed to achieve stable

precision grasps with underactuated fingers. Only changing

the curvature of the distal phalanges does not lead to stable

precision grasps. However, it can be achieved when also

a mechanical limit or a compliant joint is added between

the proximal and distal phalanx. Then, certain boundary

conditions have to be respected. This section discusses the

effect of the actual radius of the curvature, the location of

the mechanical limit, or the stiffness and the initial angle of

the compliant joint on the range of object sizes that can be

stably grasped, the equilibrium position of the objects, and

the force-displacement curves.

Curved phalanx and mechanical limit: The curvature Rph

of the distal phalanx itself has a small effect on the resultant

contact force as function of the object position. That can be

observed by the first part of the experimental results that were

shown in Fig. 8. In a previous study [11], the maximum

magnitude of the force-displacement line was related to the

maximal allowable force disturbance on the grasped object.

The curvature does thus not affect this aspect of the grasp

performance. The upper and lower limits on Rph and on the

angle θ2MLwhere the mechanical limit has to apply were

shown in Fig. 11. To achieve a stable precision grasp for a

range of objects, Rph and θ2MLmust be chosen above the

lower limit of the smallest object, and below the upper limit

of the biggest object that has to be grasped. Thus, a feasible

design space for Rph and θ2MLis obtained. Note that collision

between opposing fingers of a hand are not considered. This

can occur when θ2MLis large while L0 is small. Modification

of r2/r1 in the calculations showed that this design space

becomes smaller when r2/r1 increased.

The choice of the dimensions of Rph and θ2MLdetermines

the stable equilibrium position of the objects. To grasp small

objects from, for instance, a flat surface, it might be desired

to have the equilibrium position at the tip of the phalanx.

However, the equilibrium position of smaller objects is more

towards the palm, as it can be observed in Fig. 7 and 10.

Since this equilibrium is stable, the object is attracted to

this position, even if the initial contact with the object is at

the tip of the phalanx. This means that it is not necessary

to accurately position the hand with respect to the object in

order to achieve a precision grasp.

Curved phalanx and compliant joint: Figure 12 shows that

a minimum stiffness K is required to achieve stable precision

grasps. To stably grasp a range of objects, K and the initial

angle θ20 must be chosen above the lower limit of the smallest

object, and below the upper limit of the biggest object that

has to be grasped. It can be observed that a high K is

needed to stably grasp a range of object sizes. However, a

high K negatively affects the possibility to envelope objects,

since the flexion of the distal phalanx becomes limited (i.e.

θ2,min = θ20 − r2Fa/K, compare to Eq. 16). Indeed, the

magnitude of the actuation force can be increased to achieve

power grasps, but the distal contact force will be less than

IEEE TRANSACTIONS ON ROBOTICS 10

without a compliant joint.

The stable equilibrium position of smaller objects is more

towards the palm of the hand, like in the case with the

mechanical limit. In addition, the equilibrium of small objects

is weakly stable, because one of the eigenvalues of the

stiffness matrix for these objects is close to zero. This means

that small force disturbance lead to large position changes of

the object, and it can lead to ejection of the object.

Based on these considerations it is concluded that the

application of curved phalanges and a mechanical limit is more

effective to achieve stable precision grasps than the application

of a compliant joint in combination with curved phalanges. It is

expected that the combination of all three design modifications

lead to an even better precision grasp performance.

B. Extension and Application

This study to achieve stable precision grasps was limited

to a planar case with circular objects having one degree of

freedom. We performed also a preliminary study to investigate

the stability of circular objects of Robj = 20, 40, and 55mm having two degrees of freedom (i.e. moving in x- and y-

direction). In this study, also the right finger was modeled. The

dimensions of the fingers were according to Table I, while they

had a mechanical limit at θ2ML= π/4 rad, distal phalanges

with a curvature Rph = 80, 90, 140 and 200 mm, and a

pulley radius r2/r1 = 0.3. The results showed that whenever

a precision grasp existed with the distal phalanx against the

mechanical limit, it was also stable in the x-direction.

To extend this study to other kind of underactuated finger

mechanisms, the potential energy function (Eq. 1) has to be

generalized. This generalization can be summarized as the

addition of the energy change that is caused by the rotation

θi,j of each ith phalanx of the jth finger:

V =

N∑

j=1

M∑

i=1

∫ θi,j

θi,j,0

Ti,jdθi,j (18)

where N is the total number of fingers, M is the number of

phalanges per finger, θi,j,0 is a reference rotation angle of the

phalanx, and Ti,j is the torque that the actuation mechanism

applies to the (i, j)th phalanx. This torque is generally a

function of θi,j and numerical integration techniques are

needed to obtain V . V can be further extended with terms

that account for compliance in the joints as in Eq. 15, or other

potential energy fields.

Extension of this study to other kind of object shapes

or phalanx shapes requires modification of the geometric

constraint equations like the ones in Eq. 2. However, the

non-linearity and complexity of these equations will strongly

increase for objects that do not have a continuous shape.

The relevance of such detailed calculations are disputable,

considering the goal of this paper. In this study the shape of

the object was convex. Flat or concave objects are normally

easier to hold in a precision grasp. The shape of the object

was also rotation symmetric. For non-symmetric objects, the

stability also depends on the orientation of the object. This

means on the one hand that the results cannot be extrapolated

to any object shape. On the other hand, it is likely that for other

objects there exist an orientation where a stable precision grasp

exists, when the curvature of the phalanx and the location of

the mechanical limit satisfy the design constraints.

Friction was not considered in this study, since the fingers

and object were considered as a conservative system. Friction

hinders the attraction of objects towards the stable grasp

configuration. On the other hand, it also increases the force

that is needed to disturb a stable precision grasp. It is expected

that concavely curved distal phalanges with a moderate friction

coefficient will further improve the ability of underactuated

fingers to stable hold small objects in a precision grasp.

V. CONCLUSIONS

The goal of this paper is to investigate the achievement of

stable precision grasps with underactuated hands by means of

simple design modifications of the fingers. This contribution

allows underactuated hands to grasp and hold small objects

without the necessity of an additional mechanism to the

fingers, high contact friction between the distal phalanges and

the objects, or an accurate grasp planning.

Experiments and calculations show that only curving the

contact area of the distal phalanges is not sufficient to achieve

stable precision grasps. They do exist when curved phalanges

are combined with a mechanical limit between the proximal

and distal phalanx that prevents hyper-extension of the distal

phalanx. Then the curvature of the phalanx must be within an

upper and lower limit, which mainly depends on the relative

actuation torque of the phalanges, the maximum extension

angle of the distal phalanx, the length of the distal phalanx,

and the object size. Stable precision grasps also exist for

the combination of curved phalanges and a compliant joint

between the proximal and distal phalanx. The existence is

strongly dependent on the initial angle of the spring, while

the stiffness must be higher than a lower limit. This limit also

depends on the actuation torque of the fingers.

VI. ACKNOWLEDGMENT

This work was carried out as part of the FALCON project

under responsibility of the Embedded System Institute with

Vanderlande Industries as the industrial partner. This project

is partially supported by the Dutch Ministry of Economic Af-

fairs within the framework of the Embedded System Institute

(BSIK03021) program.

REFERENCES

[1] L. Birglen, T. Laliberte, and C. M. Gosselin, Underactuated Robotic

Hands, ser. Springer Tracts in Advanced Robotics, B. Siciliano,O. Khatib, and F. Groen, Eds. Springer-Verlag Berlin Heidelberg, 2008,vol. 40, ISBN 978-540-77458-7.

[2] A. M. Dollar and R. D. Howe, “The highly adaptive SDM hand: Designand performance evaluation,” The International Journal of Robotics

Research, vol. 29, no. 5, pp. 585–597, 2010.[3] C. Meijneke, G. A. Kragten, and J. L. Herder, “Design and performance

assessment of an underactuated hand for industrial applications,” in Proc.

of the 1st Int. Workshop on Underactuated Grasping, 2010.[4] J. R. Napier, “The prehensile movements of the human hand,” The

Journal of Bone and Joint Surgery, vol. 38B, pp. 902–913, 1956.[5] M. R. Cutkosky and I. Kao, “Computing and controlling the compliance

of a robotic hand,” IEEE Trans. on Robotics and Automation, vol. 5,no. 2, pp. 151–165, 1989.

IEEE TRANSACTIONS ON ROBOTICS 11

[6] W. P. Cooney and E. Y. Chao, “Biomechanical analysis of static forcesin the thumb during hand function,” The Journal of Bone and Joint

Surgery, vol. 59, pp. 27–36, 1977.[7] C. L. Taylor and R. J. Schwarz, “The anatomy and mechanics of the

human hand,” Artificial Limbs, vol. 2, pp. 22–35, 1955.[8] T. Laliberte, L. Birglen, and C. M. Gosselin, “Underactuation in robotic

grasping hands,” Machine Intelligence and Robotic Control, vol. 4, no. 3,pp. 1–11, 2002.

[9] M. C. Carrozza, G. Cappiello, S. Micera, B. B. Edin, L. Beccai, andC. Cipriani, “Design of a cybernetic hand for perception and action,”Biological Cybernetics, vol. 95, pp. 629–644, 2006.

[10] C. M. Gosselin, F. Pelletier, and T. Laliberte, “An anthropomorphicunderactuated robotic hand with 15 dofs and a single actuator,” in Proc.

of the 2008 IEEE Int. Conf. on Robotics and Automation, May 2008,pp. 749–754.

[11] G. A. Kragten and J. L. Herder, “The ability of underactuated handsto grasp and hold objects,” Mechanism and Machine Theory, vol. 45,no. 3, pp. 408–425, 2010.

[12] V. Begoc, S. Krut, E. Dombre, C. Durand, and F. Pierrot, “Mechanicaldesign of a new pneumatically driven underactuated hand,” in Proc. of

the 2007 IEEE Int. Conf. on Robotics and Automation, 2007, pp. 927–933.

[13] RobotIQ, “Robotiq adaptive gripper: specification sheet,” retrieved Oc-tober 13, 2010 from http://www.robotiq.com.

[14] S. J. Bartholet, “Reconfigurable end effector,” 1992, US Patent No.5108140.

[15] L. Birglen, “The kinematic preshaping of triggered self-adaptive linkage-driven robotic fingers,” in Proc. of the 1st Int. Workshop on Underactu-

ated Grasping, 2010.[16] V.-D. Nguyen, “The synthesis of stable grasps in the plane,” in Proc.

of the 1986 IEEE Int. Conf. on Robotics and Automation, vol. 3, 1986,pp. 884–889.

[17] G. A. Kragten and J. L. Herder, “A platform for grasp performanceassessment in compliant or underactuated hands,” Journal of Mechanical

Design, vol. 132, no. 2, pp. 024 502–1–6, 2010.

Gert Kragten received the B.Sc. degree (2004,cum laude) and M.Sc. degree (2006, cum laude)in Mechanical Engineering from Delft Universityof Technology, The Netherlands. His thesis waslaureated with the UfD-StuD Award, Best Graduateof Delft University of Technology, 2005/2006. Heis currently a PhD-candidate at Delft University ofTechnology. His research interests are underactuatedgrasping, prosthetic and robotic hands, and compli-ant mechanisms.

Mathieu Baril received the B. Eng. degree inMechanical Engineering from Universite Laval in2009. He has been awarded an under-graduate and apost-graduate scholarship, both by the Natural Sci-ences and Engineering Research Council of Canada(NSERC). He is currently at Master’s degree in theRobotics Laboratory of the Mechanical EngineeringDepartment at Universite Laval. His research inter-ests are underactuated anthropomorphic prosthetichands and cable-driven mechanisms.

Clement Gosselin received the B. Eng. degreein Mechanical Engineering from the Universite deSherbrooke, in 1985. He then completed a Ph.D. atMcGill University. In 1989 he was appointed by theDepartment of Mechanical Engineering at UniversiteLaval, Quebec where he is now a Full Professorsince 1997. He is currently holding a Canada Re-search Chair on Robotics and Mechatronics sinceJanuary 2001.

His research interests are kinematics, dynamicsand control of robotic mechanical systems with a

particular emphasis on the mechanics of grasping and the kinematics anddynamics of parallel manipulators and complex mechanisms.

His work in the aforementioned areas has been the subject of numerouspublications in International Journals and Conferences as well as of severalpatents. He has been an associate editor of the ASME Journal of MechanicalDesign, the IEEE Transactions on Robotics and Mechanism and MachineTheory. Dr. Gosselin is a fellow of the ASME and he received, in 2008, theASME DED Mechanisms and Robotics Committee Award for his contribu-tions to the field.

Just L. Herder received his M.Sc. degree (1992,cum laude) and his Ph.D. (2001, cum laude) inMechanical Engineering at Delft University of Tech-nology. He is currently part-time full professor atthe University of Twente and part-time associateprofessor at Delft University of Technology. He haspublished over 90 full papers in international peer-reviewed journals and conferences and has receivedseveral international awards. He is or has been boardmember of five international conferences and isassociate editor in three international journals. Three

start-up companies have emerged from his research and he holds a dozeninternational patents in different areas of mechanism design.