To appear in the IEEE/RSJ International Conference on Intelligent Robots and Systems, 2013.
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Friction Coefficients and Grasp Synthesis
Florian T. Pokorny

Abstract— We propose a new concept called friction sensitivity
which measures how susceptible a specific grasp is to changes
in the underlying friction coefficients. We develop algorithms
for the synthesis of stable grasps with low friction sensitivity
and for the synthesis of stable grasps in the case of small
friction coefficients. We describe how grasps with low friction
sensitivity can be used when a robot has an uncertain belief
about friction coefficients and study the statistics of grasp
quality under changes in those coefficients. We also provide
a parametric estimate for the distribution of grasp qualities
and friction sensitivities for a uniformly sampled set of grasps.

I. I NTRODUCTION
Friction coefficients are important for determining thequality of a specific grasp and for understanding whether agrasp is force-closed or not. Most state of the art grasp synthesis approaches typically assume fixed friction coefficientsand evaluate an associated grasp quality measure such as theL1 grasp quality Qµ , [1]. In reality, friction coefficients mayvary depending on temperature, humidity and the presenceof dirt on an object. Also, a robot will rarely have knowledgeof precise friction coefficients to start with. Instead, we may
only be able to estimate a confidence interval of friction
coefficients. In this work, we address the following related
issues:a) We systematically study the impact of changes in friction coefficients on the stability of grasps in the context
of a popular L1 grasp quality measure Qµ .
nb) We propose the concept of friction sensitivity Sa,b(g)of a grasp g with respect to Qµ and fit a Dirichletn(g)) for
distribution to the distribution of (Qµ (g), Sa,b
uniformly sampled grasp configurations with three contact points.
c) We propose and evaluate algorithms for synthesizing
stable grasps with low friction sensitivity and for small
friction coefficients.
The paper is structured as follows: In Section II, we discuss related work and introduce preliminaries. In Section III,
we define friction sensitivity and describe our algorithms forgrasp synthesis. We discuss our experiments in Section IV.
Finally, we conclude our work and discuss future directionsin Section V.
II. BACKGROUND AND R ELATED W ORK
In the following, we review the grasp quality function Qµ
and the basics of friction coefficients and grasp synthesis.
The authors are with the Computer Vision and Active Perception Lab,
Centre for Autonomous Systems, CSC, KTH Royal Institute of Technology, Stockholm, Sweden, {kaiyuh,fpokorny,dani}@kth.se. This
work was supported by the EU project FLEXBOT (FP7-ERC-279933),
the Swedish Research Council and the Swedish Foundation for Strategic
Research.

Danica Kragic

0.15

Qµ (g)

Kaiyu Hang

0.1
0.05
0

0.2

0.4

0.6
µ

0.8

1

Fig. 1: For the red and blue contact point configurations depictedon the object, we consider how the grasp’s stability, measured interms of a popular grasp quality function, varies with changing
friction coefficients. The vertical axis depicts grasp quality, while
the assumed friction coefficient µ is varied from 0.2 to 1.0. The
blue grasp remains more stable under changes in friction, while thered grasp yields more stable grasps for higher friction values.

A. Grasp synthesis and L1 grasp quality
Similarly to the work reported in [2], we focus on determining contact point configurations on a surface S which
result in a force-closed grasp g. We consider grasps
g = (c1 , . . . , cm , n1 , . . . , nm , z) ∈ R3m × (S2 )m × R3
consisting of contact points ci ∈ S on some surface S ⊂ R3
and with corresponding inward-pointing unit normal vectors
ni ∈ S2 = {x ∈ R3 : kxk = 1} such that S has centre ofmass z ∈ R3 . To determine if such a grasp g can withstandexternal forces, we need to estimate if g is a force-closed
grasp [3]. Ferrari and Canny [1] introduced an L1 grasp
quality measure Qµ which can be used for this purpose. For a
fixed friction coefficient µ > 0, the Coulomb friction model
states that – under the assumption that no slippage occurs– forces applied at a contact ci ∈ R3 on some surface S
and with corresponding inward pointing unit normal vectorni ∈ R3 satisfy kfit k 6 µfi⊥ , where fit ∈ R3 denotes
the component of fi tangent to S at ci , fi⊥ ∈ R, andfi⊥ ni denotes the component along the normal directionni - i.e. these forces have to lie within the friction cone
Ci = {fi ∈ R3 : kfit k 6 µfi⊥ }. For a particular grasp g as
above, P
these friction cones Ci can then be approximated bylCi ≈ { i=1 αi fij : αi > 0} for l uniformly spaced vectors
fi1 , . . . , fil ∈ Ci satisfying hfij , ni i = 1.
In this paper, we use l = 8 such uniformly spaced vectors.
To define this L1 quality measure,
one then approximates thePm
set of wrenches satisfying i=1 |fi⊥ | 6 1 by the convex hullConv({0} ∪ S(g)), where
S(g) = Conv ({wij : i ∈ {1, . . . , m}, j ∈ {1, . . . , l}}) ,and where wij = (fij , (ci − z) × fij ). Qµ (g) is then defined
to be the radius of the largest ball inside Conv({0} ∪ S(g))
and centred at the origin. To compute Qµ (g), S(g) isrepresented as an intersection of affine half-spaces S(g) =∩sj=1 {x ∈ R6 : hx, vj i 6 λj } for some λj ∈ R, vj ∈ R6 ,kvj k = 1, which can be obtained using the Quickhull
algorithm [4]. Then Qµ (g) = max(0, minj λj ). If a grasp
g satisfies Qµ (g) > 0, it is force-closed and can withstandexternal wrenches in arbitrary direction. Furthermore, grasps
are considered more stable the larger Qµ (g) is.
B. Friction coefficients and grasp synthesis
For the purpose of robotic grasping, friction is commonlymodelled using Coulomb’s friction laws [3] for some friction coefficient µ as above. Friction coefficients depend on
various parameters: [5] discusses in particular the influenceof humidity on friction, while [6] study the dependence offriction on temperature. Further factors influencing friction
include surface properties such as dusty or oily vs. dry andclean surfaces [7].
Since robots are to work in extreme conditions suchas in search and rescue operations and in manufacturing
applications, the impact of environmental factors on friction
coefficients should be considered an important component
in the analysis of grasp hypotheses. Even in less-extreme
scenarios, such as that of a service robot in a home environment, friction can be influenced by dusty or dirty surfaces
and can vary even during a manipulation task, e.g. when arobot is washing dishes. Clearly, none of these environmental
factors can be determined exactly, and the robot hence needs
to operate with an expected friction value. In current grasp
synthesis work, such friction coefficients are often set to a
fixed value according to friction tables for various material
combinations [3], [2].
To the best of our knowledge, the problem of assessing thegoodness of a force-closed grasp with respect to robustness
under changes in friction has so far not been studied in depth.
One work which mentions the problem of uncertainty infriction coefficients is [7] where the impact of uncertainties infriction and contact positions on grasp synthesis is discussed.
In order to deal with uncertainty in friction coefficients, theauthors suggest to work instead with ‘effective friction coefficients’ which are obtained by multiplying the coefficientsof friction by some fixed reduction rate κ1 6 1 which isassumed to be known. In the work of [8], independent contact
regions are computed on discretized objects taking into
account uncertainties in friction coefficients. There, these
uncertainties are also modeled using a reduction rate. Basedon the same concept, [9] developed an algorithm to compute
minimal required friction coefficients and contact forces.
III. M ETHODOLOGY
Fig. 1 displays two examples showing how the grasp
quality measure Qµ changes with respect to changes in
the assumed friction coefficient µ for the depicted contact
configurations and for µ ∈ [0.2, 1.0]. This figure highlights
several important features of the function µ 7→ Qµ (g).

Observe, for example, that the graphs are monotonicallyincreasing with increasing friction and that they have a
distinct almost piecewise-linear looking shape which seems
to be a generic property we encountered also for other object
shapes. Furthermore, we observe that, for µ = 1, a ranking ofthese two grasps based on grasp quality alone would returnthe red grasp as a preferable grasp hypothesis, while thisgrasp is unstable for µ = 0.2 where the blue grasp remains
stable. A natural question arises: which grasp should wechoose if we only have knowledge of a confidence intervalµ ∈ [0.2, 1]?
A friction coefficient of 0.2 corresponds to the friction ofa polythene (plastic) surface in contact with a steel surface,
while a friction of 1.0 corresponds to e.g. copper against
copper. In this section, we introduce a simple sensitivitynmeasure Sa,b(g) which we will use to assess a grasp’s
stability under variations in friction. Furthermore, we devise
a parametric approach for studying the sensitivity of genericgrasps as well as grasps on specific objects. Finally, we
describe a gradient based approach for synthesizing grasps
robust under changes in friction coefficients and develop a
new algorithm that can be used to determine force-closed
grasps even for small friction coefficients.
A. Quantifying a grasp’s sensitivity to friction
To provide a computationally tractable first-order approximation of the average slope of the graph µ 7→ Qµ (g), forµ ∈ [a, b] and for a fixed grasp g, we make the following
definition:
Definition 3.1. Consider a grasp configuration g and afriction interval [a, b] ⊂ R>0 . Fix n ∈ N and consider
δ = n1 (b − a), {x0 , . . . , xn } ∈ [a, b], xi = a + iδ forn(g) of g withi ∈ {0, . . . , n}. We define the sensitivity Sa,b
respect to the parameters a, b, n to be:
n
Sa,b
(g) =

n−1
1 X
ki ,
n − m i=mwhere ki = 1δ (Qxi+1 (g) − Qxi (g)) and m is the smallest
integer i ∈ {0, . . . , n − 1} such that Q(xi ) is not zero. If nonsuch m exists, we define Sa,b(g) = 0.
n
We then consider grasps with large Sa,b(g) to be sensitiven
to changes in friction, while grasps with small Sa,b(g) areconsidered to be insensitive to such variations.
Suppose a robot has computed a set of grasp hypotheses
Hµ = {g1 , . . . , gm } of grasps gi with underlying friction
coefficient µ > 0 and such that Qµ (gi ) > 0. While traditional
ranking based approaches would select a grasp with largest
grasp quality Qµ , our definition of grasp sensitivity allows us
to react to uncertainty in the friction coefficients. Returning
20
to Fig. 1, we can compute that S0.2,1(gblue ) ≈ 0.1640 for the20
blue grasp, while S0.2,1(gred ) ≈ 0.2287 for the red grasp.
If we are working under the assumption that µ ≈ 0.6, a20
ranking by Q0.6 now favours gred , while a ranking by S0.2,1
for grasps with Q0.6 (g) > 0 returns gblue , which indeed stays
stable over the whole friction interval [0.2, 1]. To provide asimple measure balancing the benefits of large grasp quality
with low sensitivity, we define
Φna,b,µ (g) =Qµ (g)n (g) ,
Sa,b

which provides a simple scoring function for grasps. We
propose that grasps with large Φna,b,µ (g) are desirable since
they arise from a comparatively large grasp quality and low
friction sensitivity.

Algorithm 1 Search for a stable grasp for a small friction
coefficient µend > 0 on a parametric surface S.
Require: S, 0 < µend < µstart , δ > 0, N, M ∈ N
g ← SampleGrasp(S)for i ∈ {0, . . . , N − 1} doi(µstart − µend )µ ← µstart − N −1
g ← GradientAscent(Fµ , g, δ, M )end forreturn g

B. Statistical properties of grasps and friction

As we shall show, stable grasps are difficult to synthesize
with sampling based approaches such as the ones usedby GraspIT [11] if the friction coefficients are small (e.g.
µ 6 0.5). We hence propose a new procedure using ‘virtual’friction coefficients. Suppose we have a parametric form forour graspable surface S, so that points and unit normals on S
are given by coordinates (x, y) ∈ R2 as c(x, y) and n(x, y)
respectively. Since we would like to execute a gradient based
method using Qµ , we shall use a modified definition, whereQ̂µ (g) = minj λj , rather than Qµ (g) = max(0, minj λj ),where λj are the offsets of the hyperplanes defining thewrench space S(g) which we mentioned in Section II. The
advantage of Q̂ here is that we can obtain numerical gradientseven when Q̂ < 0, while Q just takes on the value zero inthose regions. Note that Q̂µ (g) = Qµ (g) when Qµ (g) > 0.
For grasps with m contacts, we then obtain a function Fµ :R2m → R mapping m contact point coordinates to the grasp

IV. E VALUATIONIn the following, we describe an evaluation of our proposed methodology.
A. The impact of friction on grasp stability
Let us now study the impact of changes in friction
coefficients on Qµ . For this purpose, we sampled 10 setsof 10000 uniform samples, U1 , . . . , U10 , from the uniform
distribution on D(2). Fig. 2 displays the mean percentage
of stable grasps (i.e. Qµ (g) > 0) among the grasps in thesesets Ui against friction coefficients ranging from 0 to 5 inincrements of 0.1.
80%
stable grasps

C. Synthesizing stable grasps with small friction coefficientquality Q̂µ (g) of the corresponding grasp at those contact
points. We then proceed by iterating M steps of a standard
gradient ascent of Fµ with a small decrease in µ until adesired target friction value µend is reached. Alg. 1 provides
details, where GradientAscent(Fµ , g, δ, M ) executes M
gradient ascent steps with step size δ with respect to Fµ andstarting configuration g. Here, gradients are approximated using finite differences. SampleGrasp(S) returns a uniformly
sampled grasp configuration on the surface S.

60%
40%
20%
0%
−1

stable grasps

Let us now describe how we shall study some of the basic
statistical properties of grasp quality and friction sensitivity.
Generic random sampling: To study grasps with mcontact points generically, that is without a notion of an
object, we consider the set D(r) = B(r)m × (S2 )m × {0},where B(r) = {x ∈ R3 : kxk 6 r} and S2 = {x ∈ R3 :kxk = 1} and 0 denotes the origin in R3 . An element g =(c1 , . . . , cm , n1 , . . . , nm , z) ∈ D(r) then represents a grasp
with contacts ci , inward pointing unit contact-normals ni andwith centre of mass z at the origin, and where the contacts areconstrained to lie in the ball B(r) around the origin. Using
the uniform probability distribution on D(r), we can now
produce an arbitrary number of random grasps in this set.
We employed a similar approach of grasp sampling in ourwork [10]. The grasp quality Qµ and our sensitivity measurenSa,bcan in this context be considered as random variableson this space whose properties we can study statistically. In
our experiments, we will in particular show that a Dirichletn(g)).
distribution provides a good fit to (Qµ (g), Sa,b
Random sampling and surfaces: To study grasps onan arbitrary surface S, we shall employ uniform random
sampling on S (as in [10]) to obtain a set C = {c1 ,. . . , cl }of contact points. We can then study the set of ml tuples
of distinct configurations of m such contact points as grasp
candidates. Using this procedure, we shall then obtain inforn
mation about the distribution of Qµ and Sa,b
for a specificsurface.

0

1

2

3

4

5

6

0.4

0.5

µ

1%
0.5%
0%
−0.1

0

0.1

0.2
µ

0.3

Fig. 2: Percentage of stable grasps for uniformly sampled grasps
from D(2).

In this and all the following experiments, we used l = 8
edges to approximate the fiction cones used in the calculation

of Qµ . Fig. 2 additionally indicates the standard deviationfor our 10 sets of grasp samples U1 , . . . , U10 . Observe thatthe percentage of stable grasps increases substantially as
the friction coefficient is increased and that this percentage
decays rapidly as µ tends to zero as can be seen in the second
plot. A friction coefficient of µ = 0.2 corresponds to thefriction of polythene plastic against steel, while a friction of
1.0 corresponds to copper against copper. For µ = 0.2, only
about 0.084% of the grasps were stable, while for µ = 0.5,
2.4% were stable and, for µ = 1.0, about 16.2% of the grasps
were stable.
Friction coefficients hence significantly influence the success of sampling based grasp synthesis algorithms such as[2]. While previous work has certainly been aware of thisphenomenon, the above simple ‘generic’ sampling based
approach provides us with a first quantitative analysis of
this phenomenon which, to the best of our knowledge, has
not previously been formalized in this way. Fig. 2 provides
clear evidence that ‘straight-forward’ sampling approaches
for grasp synthesis used e.g. by the popular simulation
environment GraspIT [11], are inappropriate for low friction
coefficients.
B. Friction sensitivity for generic grasps
Recall that, when Qµ (g) of a grasp g is relatively smalln(g), forfor the expected friction coefficient µ, a big Sµ−,µ+ > 0, indicates that it may be inappropriate to use the grasp
g when we are uncertain about the exact value of µ. Let us
now investigate the relationship between friction sensitivity
and grasp quality Qµ for a generic set of grasps.
For this purpose, we sampled a set W of 1 million grasps
with three contact points uniformly from the set D(2). We
assume that the true underlying friction coefficient µ lies in
the interval [0.2, 1.0] with an expected value of 0.6, and wehence compute Q0.6 (g) for all grasps g ∈ W . Let us consider
the set of grasps W 0 = {g ∈ W : Q0.6 (g) > 0.001}. W 0
contained 29236 stable grasps. For each g ∈ W 0 , we now
20
compute an associated fiction sensitivity S0.2,1.0(g), using
a partition of the interval [0.2, 1.0] into 20 equally spaced
sub-intervals.

Fig. 3: The distribution of grasp quality Q0.6 (horizontal axis)

20
against sensitivity S0.2,1(vertical axis) is displayed on the left andthe mapping of this data onto the standard simplex in R3 is shownon the right.

20
Fig. 3 displays the distribution of (Q0.6 , S0.2,1.0) for ourset of stable grasps W 0 . We will now study this distribution
in more detail.
A parametric density estimate: Observe that the data in
the left part of Fig. 3 is located in a cone with apex at theorigin. We can see that grasps with low sensitivity and highgrasp quality are sparse in this data-set. In order to be able
to quantify statements about the likelihood of encountering
grasps with prescribed grasp quality and friction sensitivity,
we propose a parametric density fit as follows: as a firststep, we determined edges e1 , e2 of the smallest triangle in
R2 enveloping all the samples and with apex at the origin.
Both e1 , e2 have one end-point at the origin and satisfy
he1 − e2 , di = 0, where d is the vector which equallydivides ∠(e1 , e2 ). Moreover, the length of e1 , e2 is chosen
as small as possible, and such that the triangle still contains
all the samples. In our case, e1 = (0.5756, 0.9094) and e2 =(0.0017, 1.0762). The triangle containing the edges e1 , e2 ismapped to the standard 2-simplex ∆ = {(x1 , x2 , x3 ) : xi >0, x1 + x2 + x3 = 1} by an affine map mapping the origin to
the vertex (0, 0, 1) ∈ ∆. The right part of Fig. 3 displays theimage of our data-points on ∆. Given our transformed data
points in ∆, we determined a Dirichlet distribution fit to thedata. Recall that a Dirichlet distribution Dir(α1 , α2 , α3 ) on
∆ is determined by three concentration parameters αi > 0.
We performed a maximum likelihood fit of the parameters to
the data using the fastfit Matlab toolbox [12]. The estimatedparameters were (α1 , α2 , α3 ) = (1.0001, 2.2273, 9.8739).
40
1.0

20
0
0.5

1.0
Fig. 4: Comparison between the fitted Dirichlet distribution andthe observed data.

The surface plot in Fig. 4 shows the resulting density
function of Dir(1.0001, 2.2273, 9.8739) over the projectionof ∆ onto the xy plane together with a standard histogram
density estimator. As we can see in that figure, the chosen
Dirichlet density provides a visually satisfying fit to the data
following the histogram density estimate closely.
To further quantify the quality of the fit, we ran a Pearson
χ2 test [13] to test the difference between the observed andexpected frequencies. For this purpose, we used Mathematica’s Monte-Carlo-based χ2 testing function to evaluate thegoodness of our fit and used a significance level of α = 0.05.
After repeating the test 10 times, the resulting average pvalue was 0.833, indicating that our fit is of high quality. The
usefulness of our parametric fit lies in the fact that it provides
a summary of the data enabling us to compute probabilities
for the occurrence of samples in different regions of thequality/sensitivity parameter space.
Table I provides examples of computed probabilities forencountering grasps in selected parameter regions based on
our Dirichlet distribution fit. The bracketed expressions in

TABLE I: Probabilities for encountering a grasp in the selected20
parameter regions in (S0.2,1.0, Q0.6 ) space for uniform samples in{g ∈ D(2) : Q0.6 (g) > 0.001} determined using our Dirichlet
distribution fit. The corresponding relative observed frequencies
from our data-set are displayed in brackets below each such value.

Q0.6> 0.02> 0.05> 0.10> 0.1520
S0.2,1.0(g) 6 0.2

20
S0.2,1.0(g) 6 0.4

20
S0.2,1.0(g) 6 0.6

0.10030.43170.6002(0.1127)(0.4576)(0.6109)0.01960.2097

0.2913(0.0189)(0.1994)(0.3062)0.00010.0352

0.0923(0.00007)(0.0379)(0.0902)0(0)0.0042(0.0046)0.0203(0.0236)

Fig. 5 analogously to the generic case in Fig. 3. Observe
that, while the general concentration of the grasp qualities
and sensitivities towards the origin remains a dominantfeature, we can observe object-specific properties in these
distributions such as the sparse fringes for the box object anda stronger concentration towards the origin for the top left
and bottom right object. Fig. 5 additionally displays sample
grasps which corresponds to the respective red points in
Fig. 6. These initial investigations provide evidence that such
distributions in terms of grasp quality and friction sensitivity
could be used also for the classification of the graspabilityof various objects under varying friction assumptions.

the table indicate the number of samples lying in thoseregions divided by the total number of samples. Since
these are very close to the probabilities predicted by our
Dirichlet distribution fit, this provides further assurance thatthe parametric representation can be used to calculate theseprobabilities without the knowledge of the full sample set.
C. Friction sensitivity for example objects
Having studied properties of grasp quality and friction
sensitivity in a generic setting, we now concentrate on grasps
on the four surfaces displayed in Fig. 5. For the purposeof this experiment, we assume a parametric representationof these surfaces allowing us to compute normals at eachsurface point p. We used four of the surfaces studied in[10] and followed the same uniform contact point sampling
procedure as outlined in that paper.

20
Fig. 6: Distributions of Q0.6 (horizontal axis) and S0.2,1(verticalaxis) for random grasps on the 4 surfaces displayed in Fig. 5 areshown in the same order as in that figure. The red dots correspondto the example grasps in Fig. 5 respectively.

D. Gradient ascent on Φna,b,µ (g)(a) Bottle surface evaluation

Fig. 5: Example surfaces and grasps on them
In particular, to study the surface-specific distributionsof grasp quality and friction sensitivity, we sampled 100
contact points C uniformly from each of these surfaces andcomputed the resulting 100= 161700 distinct grasps with3
three contact points chosen from C. Fig. 6 displays theresulting distributions for each of the surfaces depicted in(b) Box surface evaluation

Fig. 7: Results of gradient ascent on Φna,b,µ (g) represented in (Qµ ,nSa,b) parameters. Original grasps (blue points) are improved bygradient ascent resulting in the green points. The pairs of red dots
correspond to initial and final grasps displayed in Fig. 9.

We now experimentally verify that, for any grasp g withQµ (g) > 0, a simple fixed step-size gradient ascent candramatically improve the value of Φna,b,µ (g) and hence resultin a more desirable grasp. In the following, unless specifiedelsewhere, we set a = 0.2, b = 1.0, n = 20 and µ = 0.6.
Analogously to the proposed algorithm Alg. 1, we consider aparametric surface representation ϕ : R2 → S of our objectn
S and perform gradient ascent of the function Ha,b,µ
sendinga grasp g specified by the centre of mass z of the surface anda triple of surface contact point coordinates to the resulting
value of Φna,b,µ (g).
We consider the bottle and the box surface depicted in theleft column of Fig. 5. The grey points in Fig. 7 display the
distribution of grasp quality and sensitivity values for thebottle and the box surface which we computed previously
and which are also displayed in Fig. 6. We divide theparameter region [0.01, 0.285] × [0.01, 0.385] into uniformly
spaced boxes of size 0.025 × 0.025 and picked a grasp fromthe grey sample points for each non-empty box. This results
in a set of grasps G for each of the two surfaces. G is
depicted by blue dots in Fig. 7. We then apply 200 steps of anstandard fixed step-size gradient ascent with respect to Ha,b,µ
for every grasp in G and compute gradients numericallyusing small finite differences.
Fig. 7 shows the result of this gradient ascent on boththe bottle and the box surface by green points. We can seethat almost all the blue dots in Fig. 7 have been moved
towards the right edge of the distribution cone, indicating an
improvement in Φna,b,µ (g). Fig. 8 illustrates the performanceof the gradient ascent using bar-plots, with black and blue
bars showing Φna,b,µ (g) values before and after gradientascent respectively. It is worth mentioning that, looking at
Fig. 8, the final value of Φna,b,µ seems to bounded by similar
upper bounds, both for the bottle and the box surface. Fig. 9
displays two examples of gradient ascent on both surfaces
and corresponding to the red dots in Fig. 7. The trajectory on
the object surface represents the location of contacts in each
iteration of the gradient ascent. Note that, if we imagine thebottle to be wet or slippery, the red grasp is intuitively lessstable than the blue grasp, which is confirmed by the graphof the grasp quality depicted next to the bottle.

0.2
0.15
0.1
0.05
0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.1

0

Fig. 9: An example of gradient ascent for Φna,b,µ corresponding to

the red points in Fig. 7. The red grasps converge to the blue ones
under our gradient ascent. The trajectories are depicted as faintlines.

E. An evaluation of Algorithm 1
We now come to an evaluation of Alg. 1 which we
proposed in order to synthesize force-closed contact configurations on surfaces with low friction coefficients. Again,
we consider the bottle and box surfaces displayed in Fig. 5.
100%
80%
60%
40%
20%
0%

2

4

6

8

10

2

4

6

8

10

Φn
a,b,µ (g)

40%
0.6
20%

0.4
0.2
0

0%
0

10

20

30

40

Fig. 10: Percentages of stable grasps for each of the 10 runs of

Φna,b,µ (g)

µ
0.6
0.4
0.2
0

0

10

20

30

40

50

60

70

our experiment with 120 grasps per experiment for the bottle (top)and box (bottom) surfaces and for µstart = 1 and µend = 0.2.
In red, we display the percentages of stable grasps for the original
random grasps, in green, the percentages of stable grasps after asimple gradient ascent of Fµend and, in blue, the percentage of
stable grasps synthesized using Alg. 1.

µ
nFig. 8: Results of the gradient ascent on Ha,b,µ
on the bottle (top)and the box (bottom) surface. Each bar represents a grasp sample
shown in Fig. 7. Bars are sorted in ascending order of the final
Φna,b,µ (g) values which is depicted along the vertical axis. Blackbars depict Φna,b,µ values of the original grasp samples and blue
bars are values after gradient ascent.

We sampled 10 contact
 points uniformly on these surfaces
and computed all 10= 120 distinct 3-contact grasp
3
configurations for these contacts, resulting in a grasp set
G for each surface. Next, we studied the effectiveness of
Alg. 1 for these grasps, setting µstart = 1.0, µend = 0.2 anddescending from µstart to µend in N = 50 steps and using agradient ascent with step size δ = 0.05 and M = 40 steps periteration according to Alg. 1. We repeated this experiment 10
times for each surface, resulting in the percentages of stablegrasps (Qµend (g) > 0) depicted by blue dots in Fig. 10.
To compare our result to a more straightforward approach,
we tested the alternative approach of simply performing a
gradient ascent of Fµend for each grasp, and with step-sizeδ = 0.05 and for M = 200 iterations which resulted in themuch lower percentages of stable grasps depicted in green.
If we simply use a sampling based approach and evaluate
the grasp quality for each grasp in our set with frictionµend , almost none of the sampled grasps had positive grasp
quality as indicated by red dots. Our results hence indicatethat Alg. 1 can be used to successfully synthesize stablegrasp configurations on objects with low friction coefficientsby repeating the algorithm a few times with different random
starting grasps until a stable grasp is found.
V. C ONCLUSION
Studying grasping under uncertainty is an important areain robotics [14], [15], [16], [17], [18]. While most current
state of the art approaches concentrate on aspects of imperfect object or robot models, we studied another fundamental
problem in grasp synthesis in this work: the dependence ofgrasp stability on friction coefficients. We believe that thisis an important problem when robots are to operate in openended environments with changing conditions.
We have in particular studied the statistics of stable grasps
under changes in friction coefficients and have introduced thenotion of friction sensitivity measuring the susceptibility of agrasp’s quality to changes in friction. Furthermore, we haveproposed and evaluated two gradient ascent algorithms forsynthesizing force-closed contact configurations on parametric surfaces with potentially low friction and for the synthesisof stable grasps with low friction sensitivity.
In our future work, we would like to evaluate our approachwith a real robot and study changes in friction coefficientsin a real application such as a household robot cleaning
dishes which might be dirty or wet, impacting heavily on
the resulting friction properties. Further directions might also

include the study of alternative friction models and grasp
quality scoring functions.
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