Grasp input optimization taking contact position and object
information uncertainties into consideration
Papat Fungtammasan, and Tetsuyou Watanabe, Member, IEEE

Abstract—This paper presents a novel approach for grasp
optimization considering contact position and object
information uncertainties. In practice, it is hard to grasp an
object at the designated or planned contact positions, as errors
in measurement, estimation, and control usually exist.
Therefore, we first formulate the influences of contact
uncertainties on joint torques, contact wrenches, and frictional
condition. We then include external wrench uncertainties in therequired external wrenches set (REWS). Based on thisformulation, we define the linear grasp optimization problem
for two kinds of frictional contact models—frictional point
contact (FPC) and soft finger contact—so that we cansuccessfully in grasp an object even if deviations in contact point,
object weight, and center of mass (CM) occur. The validity of
our approach is shown by means of numerical examples and theresult of experiments.

I. INTRODUCTIONGRASPING plays an important part in the area of robotics,
especially in industrial and household robotics. Many
researchers have tried to develop methods to overcome thechallenges currently associated with this functionality.
However, due to the complexity of the challenges, most of thegrasping methods developed so far are for use only inwell-defined situations [1–5]. For example, in situations
where the robot can grasp the target object precisely at thedesignated positions, the CM can be accurately estimated,
and the object is a rigid body.
In practical situations, these assumptions are virtually
impossible. For various reasons, differences between theplanned (or estimated) and actual values always occur (e.g.,measurement, control, and modeling errors). A controller canbe used to compensate for these errors, but there is no
guarantee that the grasping action will be successful. Several
researchers are presently tackling this problem and haveproposed various methods for dealing with uncertainty ingrasping. Chaeah et al. [5] represented all the uncertainties interms of Jacobian uncertainty and proposed an adaptive PD
controller for dealing with the problem. Schlegl and Buss [6]proposed hybrid closed-loop control for dealing with errors
due to control and measurement. Bone and Du [7] presented a
new metric that can measure the sensitivity of grasp to contact
uncertainty. Zheng and Qian [8] derived the conditions forforce closure grasps under contact uncertainty based on theanalysis of object motions for force closure. Christopoulos

This work was supported by KAKENHI(20760162)
P. Fungtammasan and T. Watanabe are with College of Science andEngineering, Kanazawa University, Kanazawa, 920-1192, Japan andte-watanabe@ieee.org

and Schrater [9] created an algorithm to find contact positions
for grasping 2D objects with two contact points under contact
uncertainty. Glover et al. [10] proposed an algorithm forgenerating probabilistic models of object geometry. Bereson
et al. [11] described how to use task space regions to deal withuncertainty and create grasp planners. However, theapproaches cited deal with uncertainties that occur before
grasping takes place, in order to plan a grasp enumeratingwhere grasping points should be and other parameters. Veryfew of them mention how to deal with the effects of theuncertainties that appear and have to be dealt with after thegrasping action occurs.
With this in mind, we propose a new grasp optimization
algorithm relevant to contact position and external wrench
uncertainties. This algorithm optimizes the internal contact
wrenches for grasping the object so that a robot can grasp an
object even if grasping is disturbed after/during the grasping
action due to these uncertainties. In the optimization, theuncertainties on frictional condition, torque limitation, andcontroller are considered. The effect of contact uncertainty isderived so that we need consider only the size of theuncertainty, regardless of direction. The main contributionsof this paper are as follows.
Formulation of the effect of contact position uncertainty:
The key issue is how to formulate the effect of contact
position uncertainty in grasp optimization. We formulate theeffect of contact position uncertainty on frictional conditionand joint torque limitations as linear conditions. Based on
the formulation, we present linear programming for
optimization, which provides the internal contact wrenches
and grasp inputs (such as joint torque input) for generating
the required contact wrenches.
Dealing with the uncertainties of the external wrench: Inpractice, it is hard to estimate the exact external wrench(before grasping). However, we can estimate the maximum
possible load and the range where the CM is likely to exist.
We consider this range in the grasp optimization. Normally,
external wrench uncertainty is partly affected by object
information uncertainty, such as that resulting from a lack of
information about the object.
Taking controller into consideration: In this paper, we
propose the use of a compliance controller. This controller
can compensate for external wrench, contact position
uncertainty, and so on, if frictional conditions can be satisfied
and the required joint torques can stay in their operational
ranges. Therefore, we plan the contact wrenches so thatcompensation is available. We subsequently derive thedesired state for the controller from the planned contact

Preprint submitted to IEEE Transactions on Robotics. Received: April 23, 2012 08:45:17 PST

wrenches.
In contrast to our previous research [12], the techniquepresented in this paper can deal with cases in which the object
grasp by the robot is oval in shape. Additionally, this paper
compares the results for FPC and soft finger contact.
II. PROBLEM DEFINITION
Contact Model: We consider two contact models: FPC, thecontact wrench at contact point Cij, which consists of three
dimensional force (wcij = [fx fy fz]Tij , fx, fy, fz denote force
component in 3 directions): and soft finger contact, thecontact wrench consisting of three dimensional force andmoment in the normal contact direction (wcij = [fx fy fz ]Tij, denotes moment component)[13]. There are two models forsoft finger contact constraint: the linear model and the ellipse
model. However, we consider only the linear model of soft
finger contact for two main reasons. First, contact wrenches
that satisfy the linear model automatically satisfy the ellipse
model also. Therefore the conditions formulated for the linear
model (as will be seen later) can also be used for the ellipse
model. Second, the complexity of the equation for the ellipse
model is much higher when we consider the effect of contact
uncertainty.
We assume that frictional coefficient and nominal contact
positions are all given. The number of contact points does notchange due to contact position perturbation.
Controller: We use a compliance controller for grasping.
This can be defined as
τ=KΔq

(1)

where τ denotes joint torque of all joint of robot hand, K
denotes diagonal controlling gain matrix, and Δq denotes thedifference between desired and actual joint angle of all joint.
Note that we assume that K is relatively large. Thus the Δq,
resulting from the compensation of external wrenches, isrelatively small and so we can ignore its effect on the
Jacobian matrix defined below.
Hand Posture: the nominal joint angle of the hand of therobot for grasping (grasping posture), q, is assumed to begiven. The corresponding nominal contact positions, pCij, arealso given. Here, pCij denotes the position for the jth contact
point at the ith finger.
Object coordinate frame: The object coordinate frame isgiven at the estimated CM position. Normally, the CM is
unknown before grasping. The distance between theestimated and actual positions is considered the CM deviation.
The CM deviation causes the external moment with respect tothe object frame.
Contact point uncertainty: In practice, it is very difficult for
the robot to grasp an object at the planned or designatedcontact positions due to measurement and control errors. We
regard this error as contact point uncertainty, and take it into
consideration in grasp optimization. Here, we consider therange of the uncertainty regardless of direction. We denotethe contact point uncertainty for pCij on the object surface as

ΔppoCij and uncertainty on the finger as ΔppFCij. If we set the
maximum values of ΔppoCij as Pomax and ΔppFCij as PFmax,
regardless of contact points, we get
|

|

, |

|

(2)

Required External Wrench Set (REWS) (expanded from[4]) is a set of wrenches that contains all the possible external
wrenches (i.e., the estimated external wrenches and all of
their uncertainties). We assume that REWS is given as aconvex polyhedron and that the estimated external wrench
can be presented as a convex representation
∑

∑

(3)

where wex denotes external wrench or given task, wvl denotes
the lth vertex of REWS, l denotes the proportion of eachvertex, and nrew denotes the number of the vertices.
Equation (3) indicates that if the robot can support all wvl,
then it can support wex and all of its uncertainties. Therefore,
we will balance the set of wvl instead of wex in the rest of thispaper.
Scenario: We consider the following situation. We makecontact with an object to grasp it with a nominal grasping
posture (q). We apply only internal contact wrenches, which
do not affect the resultant wrench to the object, since thecompliance controller can compensate for the external
wrench if the frictional condition is satisfied and the required
joint torques stay in their operational ranges. However, there
can be differences between the actual and the desired contact
positions. Hence, we must apply internal contact wrenches sothat we can keep grasping regardless of the contact
uncertainty.
Based on the above setting, we consider the following
problem.
Problem: Nominal grasping posture and the (designated)
contact position are given. REWS and the coefficient offriction are also given. For this case, derive the desiredinternal contact wrenches for grasping the target object so that
we can grasp the object even if there are contact positions andexternal wrench uncertainties. The optimization is under thecondition that the contact wrenches must satisfy frictional
conditions and the available grasp input can generate thesecontact wrenches.
III. BASIC FORMULA
In the ideal situation, the relation between external wrench,contact wrenches and joint torques can be represented by
[

]

[ ]

(4)

where J denotes Jacobian matrix of robot hand (J=diag[Ji],Ji=col[Jij]), G=row[Gij] denotes grasp matrix, wc=col[wcij]
denote contact wrenches, col[Ai]=[AT1 AT2 …]T, row[Ai]=[A1

Preprint submitted to IEEE Transactions on Robotics. Received: April 23, 2012 08:45:17 PST

A2 …], diag[Ai] denotes diagonal matrix of A. The grasp
matrix shows the relation between the contact wrenches andthe external wrench. Note that the components G and J for the
FPC and soft finger contact are different.
For the non-slip condition, wc must satisfy the frictional
condition. The frictional constraint has been linearized into a
set of inequality equations [3][14](5)where V=diag[Vij], Vij=row[alij] as shown in Fig.1. Equation(5) is the H-representation of the convex polyhedral friction
cone at Cij. The details of Vij, for FPC, can be found in [3]. In
the case of soft finger contact, it can be obtained byconverting the V-representation from [5].

Fig. 1. Convex polyhedral friction cone for FPC at Cij.

In conventional grasp optimization, we optimize the task
function (Ftask) under three constraints. First, the actuator can
supply enough torque. Second, all contact wrenches satisfy
the frictional condition. Third, the contact wrenches can
support the external wrench. The conventional grasp
optimization can be shown as

subject to

1) |
2)
3)
4)

|

|

external wrench, while the second term should be applied by
the robot so that the grasp can be maintained, regardless of theexternal wrench. In other words, the first term is determined
after grasping while the second term should be determined
before grasping. We will set γ so that every wc can satisfy thefrictional condition and its corresponding joint torques cannot
exceed the maximum joint torque values. Then we canbalance any external wrench contained in REWS with theconstant γ.
IV. EFFECT OF CONTACT UNCERTAINTIESIn this section, we will look at the effect of contact
uncertainties on grasp optimization equation (6). First, we
will discuss the physical effect when contact uncertainties
occur (sections A and B). Next, we consider the effects oneach constraint of grasp optimization (sections C – E). Finally,
we summarize all the effects on the grasp optimization(section F) and demonstrate how to verify the required
internal contact wrench by minimizing the applied torque.
A. Contact uncertainties on object and robot hand
Consider the contact uncertainties in Fig. 2. The robot tries
to grasp the object at nominal posture (dotted line), but it
actually grasps the object at a slightly deviated posture (boldline). As a result, the contact point changes from the nominal
contact position (gray dot), to the deviated contact position(black dot). We have deviation on both the fingertip andobject surface.

|(6)

where max denotes maximum torque from actuator, n=col[nij],
nij denotes unit vector of normal direction at Cij. From thislinear equation, we can derive the contact wrenches thatsatisfy the grasp system.
In some situations, we can design the internal contact
wrenches for grasping. As mentioned above, the robot can
support the external wrench and its uncertainty if it can
support the wrench at every vertex of REWS. Thus, from (3)and (4) we get(7)
where G+ denotes the pseudo-inverse matrix, E denotes anorthogonal matrix, constructed from all null space of G and γ
denotes an arbitrary vector.
The first term on the right hand side is for balancing theexternal wrench, while the second term represents the internal
wrench, which does not affect the object motion.
Consequently, if the use of a compliance controller isconsidered, the first term will change in accordance with theFig. 2. The uncertainty of contact point on both finger andobject at the Kth link of ith finger.

With respect to contact point Cij, the relationship between
the deviations of contact point (pcij), joint angle (qi), and theposition of CM (r) are given by
,

.

(8)

B. Effect on contact normal direction of contact uncertainty
When contact uncertainty occurs, the normal contact
direction will change if the contact surface is not flat. Thischange results from curvature at the contact point and contact
point uncertainty. The relation between the deviated contact
normal direction and contact position uncertainty is depicted
in Fig. 3.
From Fig. 3, we can express the maximum deviated angle
for the contact normal direction (i.e., the boundary), called thecontact normal uncertainty, as

Preprint submitted to IEEE Transactions on Robotics. Received: April 23, 2012 08:45:17 PST(9)where maxij denotes maximum contact normal uncertainty atCij, and radij denotes minimum curvature radius from every
tangential direction at Cij, see Fig.3.
The boundary of uncertainty of the normal contact| can be found by using the law of cosines.
direction |The maximum value of the normal direction uncertainty is
|

|

√
√

.

(10)

In practice, it is hard to detect contact uncertainty,
especially if it is small. We take this into consideration when
we design or plan the constant internal wrenches for grasping(since we assume a compliance controller is used), so thatfrictional condition can be satisfied when we have Δpwc.
Now we will consider Δpwc, given in (12). First, weconsider the effect of contact uncertainty on the componentsof the grasp matrix. If the object coordinate frame does notchange during grasping, this effect can be expressed bywhere θ denotes angle between designed and actual directions,ΔpA denotes uncertainty of A due to contact uncertainties. We
will now use the information from equations (8)–(10) to
derive the effect of contact uncertainties in the sections
below.

Fig. 4 The relation between nominal wcd and actual contact wrench wca

Fig. 3 Relation between maximum deviated angle maxij and contactoposition uncertainty P max. Note that in this picture radij lie on 1st
tangential direction since the radius on this direction is the shortest.

C. Static relationship considering contact uncertainty
When contact uncertainties occur, the grasp matrix andcontact wrenches will be affected. In order to grasp
successfully, equation (4) has to be satisfied even under theeffect of uncertainties. Consequently, we have
(

)(

).

(11)

[[

]]

[[

]

].

(soft) (13.2)

From this equation, it can be seen that only the momentcomponent of the external wrench is influenced. The
uncertainty of this moment, Δpm=[0 I]ΔpGwc, is expressed
by
∑ |

|[

̂

]

(FPC) (14.1)

∑ |

|[

̂

]

(soft) (14.2)

where fij =[fx fy fz]Tij, ̂If we neglect the higher order term, we get(FPC) (13.1)

denotes unit vector of direction of. First, let us consider the moment uncertainty related to fij.
.

(12)

∑ |

| |[

̂

]

∑ |

|

| (15)

As can be seen, the contact uncertainties lead to theuncertainties of contact wrenches. If we consider grasp
optimization equation (6), the change in the contact wrenchaffects every constraint. We will consider its effect in section
IV(D) and IV(E) below.

For FPC we can get

D. Effect of contact uncertainty on the frictional condition
From equation (11) and (12), we can see that contact
uncertainty causes uncertainty in the contact wrenches. The
total contact wrenches may not satisfy the frictional condition,
as shown in Fig.4. Moreover, the friction cone will be
disturbed if contact normal uncertainty occurs. Thus, insummary, the effect of contact uncertainty on frictional
condition can be divided into two parts: the effect on thecontact wrenches and the effect on the friction cone. We willshow how to deal with each effect below.

where nwij denotes the normal contact force component at Cij,sijdenote original static coefficient of friction at Cij. In thecase of soft finger contact, the frictional constraint of thelinear model can be shown as

|

|

|

|

√

.

(16.1)||

where mijdenote original moment coefficient of friction at| denotes magnitude of tangential contact force at
Cij , |

Preprint submitted to IEEE Transactions on Robotics. Received: April 23, 2012 08:45:17 PST

Cij. Since |
|

|

|

|

(

|

, then)||.

(16.2)

From (12), the boundary of the contact wrench uncertainty
becomes,|||[ ]| |

|

|

|

[

|

E. Effect of contact uncertainty on joint torques
We now consider the deviation of joint torques due tocontact position perturbation. From (4), the kth joint torque ofthe ith finger becomes

(FPC)

]| |

|

(soft)

∑ ([

|

(18)

[
[ ]|

|

[

[√
)

(FPC)
(

) (

)] (soft)

where ncol[nij], and ∊{1,2}. Note that for soft finger
contact, equation (18) indicates two inequality equations.
Now consider the effect of contact uncertainty on thefriction cone. If ∆pnij (from (10)) occurs, the friction cone will
deviate from the designated position and orientation. Thechange in the friction cone can be represented as an
adjustment to the coefficient of friction. The new coefficientof friction, called the equivalent coefficient of friction, isexpressed by
(
(

[[
[

)

∑[

]

] ]

[[

]

[

] ]

(22)

]

where Δpqi denotes vector of joint angle deviation for finger
ith( Δpqi =col[Δpqij]).
The first term indicates the deviation due to the changing of
the joint angles without any change in pFCij, while the otherterm indicates the deviation due to the contact point Cij on thefingertip without any change in the joint angles. Note that thecontact position uncertainty causes deviation on both joint
angles and contact position on fingertips, as shown in Fig. 2.
We consider the two components of the Jacobian deviation
separately, as in (22).
The joint angle deviation consists of two factors: contact
deviation and low resolution. As assumed in section II, we
ignore the deviation due to the resolution. Thus, from (8), thejoint angle deviation can be expressed by
.

)

(19)
Then [[[])

where seqij and meqij denotes equivalent static and moment
coefficient of friction at Cij respectively. Next, we linearize
this friction cone and create matrix Veq with the seqij andmeqij, in the same way as the derivation of (5).
In summary, we must consider the effect of contact wrench
uncertainty (18) and the effect of the changing of friction
cone (19). We compensate for (19) by increasing the contact
force in the normal direction, and using Veq instead of V tocompensate for the change in the friction cone. Consequently,(5) becomes(] )((soft)

]

[(

]

(FPC)
]|

[

(21)
where [A]k denotes the kth row vector of the matrix A and we
ignore higher order terms. Σj indicates the summation of allcontact points on ith finger. Since the Jacobian is a function of
the joint angle and the contact point on the finger(Jij=Jij(qi,pFCij)), [ΔpJTij] k can be written as

where
|

]

∑[

(17)

where
denotes separation matrix for Cij ().
The magnitude of each contact wrench uncertainty isbounded|frictional condition can be satisfied even when the grasping
force (unexpectedly) changes due to the deviation.

]

)

.

(20)

We use (20), instead of (5), to verify the frictional
condition for the grasping system with uncertainty so that the

] ]

|[[

is bounded as follows

] ]

|

|

[

]

|

.

(23)

Next, we show that the second term of (22), [JijT] k, can be
expressed by
[

]

[[

](

)]

(FPC)

[

]

[
[

](

)]

(soft)(24)

where pFiK denotes position of proximal joint for Kth link
where Cij exists, and ziK denotes rotation axis of FiK frame

Preprint submitted to IEEE Transactions on Robotics. Received: April 23, 2012 08:45:17 PST

[15], as shown in Fig 2. Since we are considering the casewhen there is no Δpqi, (24) becomes
[[

] ]

[[

]

[[

] ]

[

[

]

] (FPC)] (soft)

Then we get
|[[

] ]

|

|

|

.

(25)

By combining (23) and (25), we can define the maximum
Jacobian deviation due to contact position uncertainty as
|[

] |

|

[

]

|

.(26)

The magnitude of the contact wrench is bounded as follows
|

|

√

(27)

where MAXij=max(sij,mij). We here leave the proof of (28)
as an exercise for you, dear reader.
From (19), (22), (26), and (27) the uncertainty of the kth
joint torque of the ith finger resulting from the deviation of theith finger can be expressed by
|

|

∑(

)
√
|[

(28)

,
] |

.

Equation (28) shows that when contact uncertainties occur,
they act as an additional load on each robot joint. We have tosubtract them from the maximum torques supplied by theactuator. As a result, the torque limitation constraint in
optimization equation (6) will change such that the required
torque, which includes the effect from contact uncertainties,
will be lower than the maximum torque.
F. Grasp optimization
Consider optimization equation (6). From the above
derivation, the contact uncertainties will act as additional loadon joint torque and affect the frictional condition. Therefore,
we adjust the optimization constraint by using (20) as thefrictional constraint and substitute (28) into the joint torquelimitation. If we assign the task function as the ratio of therequired and the available torque and then solve for theinternal contact wrenches, we get the linear optimization
equationsubject to
1)
|
| |
|
2) | |
3)
(
[
]
)
4)
5)
6)
(j=1,2,...,Mi , i=1,2,…,N , l=1,2,….,nrew)(29)| is maximum value of | | . If the required
where |
joint torque is above this value, we conclude that we cannot
grasp the object. In this algorithm, we try to find the internal
contact wrench that produces the minimum under thederived grasping conditions (where denotes the ratio
between the required torque and the maximum torque).
Conditions 1, 2, and 3 indicate joint torque limitations:conditions 4 and 5 indicate frictional conditions: andcondition 6 indicates internal and external contact wrenches.
If linear programming equation (29) cannot be solved, itmeans that the probability that at least one of the constraints
cannot be satisfied is high. The grasp posture is not good forsupporting defined uncertainty. Therefore a new grasp
posture should be considered. If we can obtain a solution γ =γ*, we will derive the corresponding desired joint angle qd =[qd11 qd12 …]T using the following relationship:
qd=K-1JTEγ*+qnominal(30)

After obtaining the desired hand posture, we can find thenecessary torque based on compliance controller mentioned
in section II. This grasp input (joint torque input
corresponding to *) can generate the required internal
contact wrenches.
V. SIMULATION AND EXPERIMENTAL RESULTS
In this paper, we use examples to verify our approach. We
used the robot hand called “Kanazawa Hand” [16], developedin our laboratory. This hand has 4 fingers and 12 degree of
freedom (DOFs). The selected target object was a 3.8 cm
radius, 2.5 cm high cylinder. The designed contact points
were as depicted in Fig. 5.
This robot grasps objects from the +y direction, as
illustrated in Fig. 5. The simulation conditions are shown in
Table I. The frictional constraint was represented by anoctahedral convex cone. The gravitational direction was –y.
The object’s frame was set at origin (0,0,0). For simplicity,
we assumed that (PFmax) was equal to (POmax), and they wererepresented as Pmax, in this simulation.
TABLE ISIMULATION CONDITION
Coefficient of friction ( s &  m) 0.3 & 0.1
Nominal CM position(0,0,0)Maximum torque0.805 N.m (for proximal joints)
Maximum torque0.232 N.m (for the other joints)

Preprint submitted to IEEE Transactions on Robotics. Received: April 23, 2012 08:45:17 PSTyzxFig 5 Contact position and used grasp posture with Kanazawa Hand

A. Allowable deviation of CM
First, we looked at the effect of contact uncertainty on
graspable load and the deviation of center of mass (CM).
Normally, we cannot accurately estimate the position of CM
for general objects and this causes the grasping system tobecome more burdened. This deviation affects the momentcomponent of the external wrench, which is equal to the crossproduct of its force component and the CM deviation.
Therefore, we set wvi for REWS as follows:
nrew=5, wv1=[0,0,0,0,0,0]
wv(1+2+2)=wmax[0,-1,0, PCMmax (-1),0, PCMmax(-1)],{1,2}where wmax represents maximum supportable object’s load
and PCMmax represents the maximum CM deviation. Since theonly external force is in the y-direction, the external moment
due to CM deviation in the y-direction was set to zero.
We also considered the relation between the graspable load
and the deviation of CM at one constant Pmax. We investigatedthis relation by choosing Pmax and wmax then finding the
maximum PCMmax that could be achieved by (29). Then wevaried wmax. In this simulation we considered the cases when
Pmax = 0, 1, 2, and 3 mm. The simulation results are shown in
Fig. 6.

Fig 6 Relations between maximum supportable object’s load andmaximum CM deviation under different maximum contact
uncertainties (above; FPC, below; Soft finger contact)

From the simulation results, it can be seen that at a constantPmax the supportable load decreases when the CM deviation
increases. When Pmax ≠ 0, the grasping ability of robot

decreased. In other words, when Pmax increases, moreaccuracy in CM position estimation is required for supporting
the same load. When Pmax increases, the robot hand has tograsp a lighter load at the same accuracy of CM estimation.
The figure also indicates the maximum contact uncertainty
that the grasp system can support at the specific load andaccuracy of CM estimation. For example, suppose that the
robot hand grasps the 2 N object with perfect CM estimation.
The maximum contact uncertainty for this system would be 2
mm (if we consider FPC). However, if the contact uncertainty
is higher than 2 mm, the robot cannot grasp the object at thissetting.
When FPC and soft finger contact are compared, we see that,in contrast to FPC, soft finger contact can support more
uncertainty for both CM deviation and contact uncertainty.
B. Experimental results
In this section, we show the validity of our approach (29)and compare it to conventional approaches (6). We applied
our algorithm to a real robot. The experiment was set up as
illustrated in Fig. 5. The experimental conditions set were as
follows. The weight of the object is 1 N. A contact uncertaintyof 3 mm was assumed, and the CM position was precisely
obtained. From linear programming equation (29), we
obtained the internal contact wrench Eas in Table II. Note
that, we can obtain the joint torque from both (1) and (30), but
we show the internal contact wrench, which is easier tounderstand.
TABLE IIREQUIRE CONTACT WRENCHES AT EACH CONTACT POINT

FPCSoft finger
proposednormal
fx
2.53
1.564
fy
-0.136
-0.13
fz
0.535
-0.01300

C2fx-1.563-0.907-1.166-0.805
fy
0
0
0.078
0.055
fz
0.9027
0.456
0.673
0.465
-0.01
-0.009C3
fx-1.5634-0.774-1.365-0.758
fy
0
0
0.058
0.075
fz
-0.9027-0.4466-0.726-0.452
0.01
0.009

0.59
0.275
0.486
0.271

The unit for force component is Newton, and for moment component is
Newton-meter.
point
C1

proposed
3.127
00

normal
1.58
0
-0.09

It can be seen that there is an approximate twofold increasein the required contact wrench when we consider contact
uncertainty for this grasping condition. In addition, therequired contact wrenches are lower when we use soft finger
contact as the frictional constraint. We applied grasp input tothe robot so that the robot could generate the contact wrenchin Table II, and we verified the result. In this experiment, theresult was categorized into four groups (A–D), as shown in
Fig. 7. The explanation of each group is as follows.
Group A: The robot successfully grasped and lifted theobject and the object had the same orientation as it did before
it was grasped.

Preprint submitted to IEEE Transactions on Robotics. Received: April 23, 2012 08:45:17 PST

Group B: The robot successfully grasped and lifted the object
but the object tilted due to slippage.
Group C: The robot successfully grasped and lifted theobject but the object fell after a period of time had elapsed.
Group D: The robot was not able to lift the object.

Fig 7 The number of grasping results categorized into each group

From the experimental results, it can be seen that theprobability of successful grasping was higher when we used
our proposed grasp optimization algorithm. The robot was,for the most part, able to grasp the object successfully (groups
A and B), while with the conventional grasp optimization
technique, the robot could hardly grasp the object. However,in this experiment, our proposed grasp optimization
technique still could not guarantee a 100% success rate andthe results were not consistent for all 10 times the experiment
was conducted. A possible reason for this may be attributed tofactor affecting the performance of the robot hand such as
backlash and static friction of the robot. Static friction meansthat there is a threshold for robot movement. When there isstatic friction, the robot does not response if the input torqueis smaller than the threshold. Moreover, when there isbacklash in the system, we cannot guarantee that the robot canmove to the required posture, qd, even when it receives thedesired position from the encoder. The backlash can becompensated for but the exact range of the backlash varies
and is very hard to specify.
The results show that the chance of successful graspingwhen using soft finger contact is a little bit lower than that of
FPC. However, it does not mean that the robot can grasp
better with FPC. The reason is that theoretically, if we use soft
finger contact analysis, the grasp system requires less contact
wrenches to support the same external wrench. In other words,
the results show that the possibility of a successful grasp
increases if the internal contact wrenches increase. Note that
we used the same experimental setup while the contact modelchanged.
VI. CONCLUSION
This paper presented new grasp optimization techniquepertaining to contact position and object information
uncertainties. In section III, we reviewed the basic formulanecessary for a grasping system. In section IV, we first
verified the boundary of contact uncertainties and its effect on
normal contact direction (if any). We then verified the effectof contact uncertainty on each grasp constraint (i.e., grasp
equation, frictional condition, and joint torque limitation) andits boundary. Finally, we summarized all of the affected

constraints into linear grasp optimization. In this paper, we
optimized the applied joint torque for solving internal contact
wrenches. We included the external wrench uncertainties in
REWS and applied REWS to linear grasp optimization
instead of one exact external wrench. Numerical solutions
and experiments were then conducted to verify the validity of
our approach. We found that, theoretically, increasing theinternal contact wrenches can compensate for small contact
uncertainties. The size of the uncertainties that can becompensated for vary for grasp posture, size of external
wrench, and available torques. By considering torquelimitation and contact position uncertainty, we evaluatedwhether a robot hand can grasp an object successfully, withgreater accuracy. In the future, the imperfections in the robot
hand will be taken in to account so that grasp optimization canbe guaranteed even when contact uncertainty occurs.
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Preprint submitted to IEEE Transactions on Robotics. Received: April 23, 2012 08:45:17 PST