Grasp Moduli Spaces
Florian T. Pokorny, Kaiyu Hang and Danica Kragic
Centre for Autonomous Systems/CVAP, KTH Royal Institute of Technology{fpokorny, kaiyuh, danik}@csc.kth.se

Abstract—We present a new approach for modelling grasping
using an integrated space of grasps and shapes. In particular,
we introduce an infinite dimensional space, the Grasp Moduli
Space, which represents shapes and grasps in a continuous
manner. We define a metric on this space allowing us to formalize‘nearby’ grasp/shape configurations and we discuss continuous
deformations of such configurations. We work in particular withsurfaces with cylindrical coordinates and analyse the stabilityof a popular L1 grasp quality measure Ql under continuous
deformations of shapes and grasps. We experimentally determine
bounds on the maximal change of Ql in a small neighbourhood
around stable grasps with grasp quality above a threshold. In
the case of surfaces of revolution, we determine stable grasps
which correspond to grasps used by humans and develop an
efficient algorithm for generating those grasps in the case of threecontact points. We show that sufficiently stable grasps stay stable
under small deformations. For larger deformations, we develop
a gradient-based method that can transfer stable grasps between
different surfaces. Additionally, we show in experiments that our
gradient method can be used to find stable grasps on arbitrary
surfaces with cylindrical coordinates by deforming such surfaces
towards a corresponding ‘canonical’ surface of revolution.

I. I NTRODUCTION
A lot of attention in the robotics community has been
focussed on finding good representations that enable a robot
to grasp novel objects. Since the problem of determining
physically stable grasps is such a challenging one, it has been
popular to think of the object and grasp representation separately in order to gain further insights. Object representations
based on box-models [10], shape primitives [13], the medial
axis [17] and string based structure representations [12] havebeen successfully explored. A popular approach is to firstdevelop a grasp heuristic resulting in a set of grasp hypotheses
on a particular object and to then rank the synthesized grasps
by a grasp quality measure [4, 5, 7] depending on the contacts
between the object and the robot hand. Most existing methods
then proceed by evaluating their proposed grasp hypotheses
framework on a discrete set of objects [10, 13].
In this work, we do not think of a collection shapes in a
discrete manner, but instead take a step towards understanding
grasps on families of shapes varying continuously with someparameter. Many real world objects can be thought of asvarying continuously: besides simple properties such as height,
width etc., we can imagine that every-day object classes suchas bottles, tools, cars, etc. depend continuously on some ‘style’parameters. In order to state a mathematically concise theory,
we study a specific yet general class of such objects: we focus
on surfaces with cylindrical coordinates and their associatedgrasps. This yields a rich infinite dimensional space, the Grasp

Fig. 1: Samples from a finite-dimensional subset of the Grasp Moduli
Space spanned by the three surfaces displayed at the vertices of thetriangle.

Moduli Space for such surfaces, which we define here andwhich parametrizes continuous deformations of both objects
and grasps. In this work, we study some fundamental properties of this space, including a construction of finite dimensional
subsets induced by any finite set of ‘shape examples’. Ourdefinition of the Grasp Moduli Space is motivated by the belief
that, given the definitions and results presented in this paper,
it will be possible to define deterministic and probabilistic
methods for grasping which are able to utilize knowledge of
observed stable grasps on different surfaces. The contributionsof this paper can be summarized as follows:a) We introduce the Grasp Moduli Space G cyl incorporating
information about both grasps and associated surfaces.
We define a metric on G cyl , describe continuous deformations in G cyl and study finite dimensional subsets ofG cyl induced by any finite collection of such surfaces.
b) We provide experiments showing the evolution of stablegrasps under small deformations of the underlying surface, the contact positions of a grasp, or both, inside theGrasp Moduli Space G cyl .
c) We investigate the change in grasp quality under large
deformations of surfaces and grasp configurations in G cyl .
d) We formulate a grasp quality gradient approach in G cyl
for transferring stable grasps from one surface to anothersurface and evaluate our approach in simulation.
e) We develop an efficient heuristic for finding stable grasps
on surfaces of revolution corresponding to natural grasps
used by humans. We then show how to continuously
deform any surface with cylindrical coordinates to an
associated canonical surface of revolution and use theabove gradient approach to generate stable grasps basedon our heuristic.

(a) S1(b) S2(c) S3(d) S4

(a)

(b)

(c)

(d)

(e) S5(f) S6(g) S7(h) S8

(e)

(f)

(g)

(h)

(i) S9(j) S10(k) S11(l) S12

(i)

(j)

(k)

(l)

Fig. 2: Example surfaces with cylindrical coordinates. Figures a-d)Fig. 3: Samples from the Grasp Moduli Space. Figure a-d) and e-h)display surfaces of revolution, while the surfaces in e-l) do not exhibit
the rotational symmetry.

show grasp/surface combinations based on the surfaces in Fig. 2a-d)
and 2e-l) respectively, while the surfaces in i-l) are based on convex
combinations of all the surfaces in Fig. 2.

II. BACKGROUND AND R ELATED W ORK
The idea of continuously varying shapes has implicitly
surfaced in the grasping community in works using simple
primitives such as boxes and quadrics that depend continuously on their parameters. Similarly, one can consider the
space of continuously varying hand configurations. A successful technique is to represent robot hand configurations as
elements of a vector space and to employ principal component
analysis to determine postural synergies [8, 18]. These thendefine a low-dimensional representation which can be used
to control a robotic hand. Various extensions of this concept,
such as soft synergies [3] and adaptive synergies [6] have been
proposed. In another related work, [14] proposed an alternative
continuous representation of grasp configurations based on
simplicial complexes. These representations do however focus
on continuous deformations of hand postures rather thancapturing the space of both objects and contact point positions
jointly. In this work, we shall concentrate on grasping at thecontact level [19]. Hand kinematics do not feature in ourcurrent formulation, but can be integrated with our approachin a later planning stage which attempts to position the robot
hand so that a given contact configuration is realised.
Another field where the idea of continuous deformations ofobjects has recently shown promise in applications is computer
graphics. There, continuous representations of 3D objects canbe used for example for character animation, database-search[16] and mesh-morphing [1].
Most closely related to our approach in philosophy is however the use of continuous deformations in algebraic topology[9] and algebraic geometry. In algebraic topology in particular,

the notion of ‘shapes up to continuous deformations’ hasbeen used with great success for the classification of smoothmanifolds.
Several grasp quality measures have been developed inorder to evaluate and rank grasp hypotheses according to their
physical stability [4, 7, 19, 15]. In this work, we will consider
a well-established L1 grasp quality measure Ql introducedby [7] which we shall review now. For a given configuration
3{(ci , ni )}m
i=1 of m contact positions ci ∈ R and corresponding inward pointing unit normal vectors ni ∈ S2 ⊂ R3
and a centre of mass z ∈ R3 , let us denote this data by
g = (c1 , . . . , cm , n1 , . . . , nm , z) ∈ R3m × (S2 )m × R3 , sothat g lives in a 5m + 3 dimensional space. For contact ci , theCoulomb friction model states that, for any force fi applied at
ci , we have kfit k 6 µfi⊥ , constraining fi to lie in the friction
cone at ci , where fi⊥ ∈ R and fi⊥ ni denotes the component offi along the normal vector ni on the surface S containing thecontact point ci , and fit denotes the component of fi tangent
3t⊥
to S at ci . The friction cone
PlCi = {fi ∈ R : kfi k 6 µfi }can be approximated by { i=1 αij fij : αij > 0} for l > 3
uniformly spaced vectors fij ∈ Ci , j ∈ {1, . . . , l} per cone
Ci , satisfying hfij , ni i = 1. We work with
Pm l =⊥8 in thispaper. The space of wrenches satisfying
i=1 |fi | 6 1 isthen approximated by Conv(0, S(g)) using the convex hullS(g) = Conv(wij : i = 1, . . . , m and j = 1, . . . , l) ⊂ R6 ,where wij = (fij , (ci −z)×fij ). Ql (g) is defined to be the distance from the origin to the boundary of S(g) if 0 ∈ Int(S(g))
and is zero otherwise. In practice, Ql (g) is determined bycomputing a description of S(g) as an intersection of affine

T
6
half-spaces, S(g) =: hx, vi i 6 λi }
i=1,...,s {x ∈ R
and then Ql (g) = max(0, mini=1,...,s λi ). Positive valuesof Ql correspond to physically stable grasps since arbitrary
wrenches of magnitude less than or equal to Ql (g) can thenbe resisted [7]. Note that alternative grasp quality measures[7] can be defined. One could for example consider forces
such that maxi kfi⊥ k 6 1, but the resulting L∞ quality
measure Q0l is computationally harder to determine and, sinceQl (g) < Q0l (g), stable grasps with respect to Ql are alsostable with respect to Q0l , making Ql a more desirable measurefor our current investigation. For our experiments, we willconsider the case of m = 3 contact points. While Ql provides
a physically well-motivated measure for grasp stability, it isa rather difficult function to analyse analytically due to its
complicated definition involving convex hulls. In 2D, progresson the existence of stable grasps [15] and in particular on
the synthesis of optimal two and three finger grasps has been
made, but in 3D, state of the art approaches to grasp synthesis
currently still employ various heuristics [17] combined withrandom sampling [5] in order to search for optimal grasp
configurations.
III. O UR F RAMEWORK
Representing surface and grasp configurations
Let us observe that, for the purpose of grasping, mostreal world objects can be locally approximated by smoothsurfaces in R3 . A large subset of these surfaces can furthermore be specified by cylindrical coordinates: Sf,a,b ={(f (u, θ) cos θ, f (u, θ) sin θ, (1 − u)a + ub) : u ∈ [0, 1], θ ∈S1 } for some smooth bounded function f : [0, 1] × S1 → R>0
with bounded derivatives and where the height of the surface isbounded by [a, b], with a < b. We call the set of these surfaces
with cylindrical coordinates Mcyl . A familiar subset of Mcylis the set of surfaces of revolution Mrev ⊂ Mcyl given by
surfaces of the form Sf,a,b = {(f (u) cos θ, f (u) sin θ, (1 −u)a + ub) : u ∈ [0, 1], θ ∈ S1 }, where a < b, f : [0, 1] → R>0
is again smooth, bounded and with bounded derivatives andindependent of the angular coordinate. Fig. 2 displays someexamples. Note that we can even model non-smooth shapes
such as a box as elements of Mcyl after smoothing the edges
and after removing a small neighbourhood around the z-axis at
the top and bottom as in Fig. 2e). Most everyday objects suchas bottles, glasses, handle-parts, etc. can hence be consideredeither as elements of Mcyl or can be decomposed into parts
which lie in Mcyl . While we are not investigating pointcloud data here, one can obtain coordinates for a suitable
point-cloud P , e.g. by performing regression in cylindricalcoordinates on R3 with respect to the most dominant axisof P . For Sf,a,b , Sg,c,d ∈ Mcyl and α, β > 0, we define
αSf,a,b + βSg,c,d = Sαf +βg,αa+βc,αb+βd .
Lemma III.1. Mcyl and Mrev are convex cones.
Proof: Let Sf,a,b , Sg,c,d ∈ Mcyl and α, β > 0. Clearly
αf + βg is smooth, positive and bounded and αa + βc <αb+βd. Hence, the result about Mcyl follows. The derivation
for Mrev is similar.

Note that Mcyl and Mrev are infinite dimensional. Givena set X = {Sf1 ,a1 ,b1 , . . . , Sfn ,an ,bn } ⊂ Mcyl (or Mrev ),itPis natural to consider their
Pn convex hull Conv(X) =n{ i=1 αi Sfi ,ai ,bi : αi > 0, i=1 αi = 1} which is finitedimensional and a subset of Mcyl (Mrev respectively).
Conv(X) yields a natural bounded and finite-dimensional‘shape-space’ spanned by the discrete ‘example set’ X ofobserved shapes. Since there exists natural probability distributions on any simplex (e.g. the Dirichlet distribution), it is also
possible to sample random shapes from Conv(X) as in Fig. 3and to build probabilistic methods on this space. In Fig. 1, we
display a projection of a 2-simplex spanned by the shapes
displayed at its corners together with a few uniform random
samples. Note that one could also consider
Pnthe unbounded
but finite-dimensional space Cone(X) = { i=1 αi Sfi ,ai ,bi :αi > 0}, which we shall however not study here. In this work,
Mcyl and Mrev will be used as natural ‘shapes spaces’ for the
purpose of grasping. Next, we study grasps on these surfaces
and endow the resulting space with a metric.
Given a surface S = Sf,a,b ∈ Mcyl and a pointp ∈ Sf,a,b , we can parametrize p by p = cS (u, θ) =(f (u, θ) cos(θ), f (u, θ) sin(θ), (1−u)a+ub) for some (u, θ) ∈[0, 1] × S1 . We can furthermore compute two tangent vectors∂cS
S
to S at p: t1 (u, θ) = ∂c∂u , t2 (u, θ) = ∂θ and the cor(u,θ)×t2 (u,θ)
responding unit normal vector nS (u, θ) = ktt11 (u,θ)×t.
2 (u,θ)kNote that, since f has bounded derivatives, nS takes values
in the twice-punctured sphere S2± = S2 − {±(0, 0, 1)}.
A grasp with m contact points can now be described by(u1 , . . . , um , θ1 , . . . , θm ) ∈ [0, 1]m × (S1 )m = C m using themaps cS and nS . Given a uniform mass distribution on thesurface, its centre of mass zS can be computed by the formula
R 1 R 2πcS (u, θ)kt1 (u, θ) × t2 (u, θ)kdθduzS = 0 R0 1 R 2π.
kt1 (u, θ) × t2 (u, θ)kdθdu
0 0
The grasp quality Ql and Mcyl
In order to determine the grasp quality Ql for grasps on
surfaces in Mcyl , we first need to fix our approximation of theforce-cones. For the standard normal vector n = (0, 0, 1), andfor friction coefficient µ > 0, we define the edges of the force2πjcone along (0, 0, 1) by fj = (µ cos( 2πjl ), µ sin( l ), 1), forj = 1, . . . , l. For any other unit normal vector n 6= ±(0, 0, 1),
define fj by rotating the fj for the normal (0, 0, 1) by anangle of arccos(h(0, 0, 1), ni) about the axis (0, 0, 1) × n. If
n = (0, 0, −1), we rotate by the diagonal matrix with entries(1, −1, −1) along the diagonal. The above assignment offriction cone edges to normal directions is clearly continuous
for n ∈ S2± .
Lemma III.2. Ql is a continuous function on R3m ×(S2± )m ×
R3 .
Proof: The force cones depend continuously on the input
data on the given domain and so does the convex hull. Finally,
the distance function from the origin to the nearest facet of theconvex hull is clearly continuous under continuous variationsof the convex hull.

Fig. 4: Continuous combined surface and grasp deformation γ :[0, 1] → G cyl (3) between two points in G cyl (3).

where dR3 (x, y) = kx − yk is the standard Euclidean metricand we think of ni , n0i as embedded in R3 , in order to measure
distances between normal vectors.
So far, we have defined a metric on the space of grasps
without any notion of a surface. We now consider a distanceon our shape space Mcyl which can easily be related to our
distance on grasp configurations. For S, S 0 ∈ Mcyl ,d(S, S 0 ) =Shape and grasp deformations

suph∈[0,1],θ∈S1

In order to be able to talk about continuous grasp/surface
deformations, we make the following definition:
Definition III.3. The Grasp Moduli Space for surfaces withcylindrical coordinates and grasps with m contact points
G cyl (m) is given by G cyl (m) = Mcyl × C m , where C m =[0, 1]m × (S1 )m . Similarly, we define the Grasp Moduli Space
for surfaces of revolution with m contact points by G rev (m) =Mrev × C m and note that G rev (m) ⊂ G cyl (m).
Note that the spaces G rev (m), G cyl (m) are infinite dimensional since the spaces Mrev , Mcyl are. Since a point(Sf,a,b , g) ∈ Mcyl × C m = G cyl (m) (or G rev (m)) describes
both a particular surface Sf,a,b and a grasp g on that surface,
we are now able to describe various interesting continuous
deformations in that space: suppose that (S, u, θ), (S 0 , u0 , θ0 ) ∈G cyl (m), with u, u0 ∈ [0, 1]m , and θ, θ0 ∈ (S1 )m . Consider the
curve γ : [0, 1] → G cyl (m) given by
γ(t) = ((1 − t)S + tS 0 , (1 − t)u + tu0 ,(III.1)0
ϕ(θ1 , θ10 , t), . . . , ϕ(θm , θm, t)),

where we let t 7→ ϕ(θi , θi0 , t) be an interpolation between θiand θi0 along a shortest path on the circle, i.e. we pick coordinates θi , θi0 ∈ [0, 2π] on the circle and define ϕ(θi , θi0 , t) =(1 − t)θi + tθi0 if |θi − θi0 | 6 π and otherwise(1 − t)(θi + 2π) + tθi0 mod 2π, θi < θi0
ϕ(θi , θj , t) =(1 − t)θi + t(θi0 + 2π) mod 2π, θi > θi0 .
We can now distinguish three interesting types of deformationsin G cyl (m). Surface-only deformations, where u = u0 , θ = θ0 ,so that the cylindrical coordinates of the contact points stay
constant, but S 6= S 0 , grasp-only deformations, where S = S 0 ,but we vary the contact points, and combined deformations,where both the surface and the grasp are being deformed. Notethat the simple curve γ described above is only one example(as in Fig. 4) of how to continuously interpolate between two
grasp/surface configurations. Optimal curves, e.g. under taskconstraints could be another interesting future line of research.
We now define a simple metric on the space of grasps. Let
g, g 0 ∈ R3m × (S2 )m × R3 , withg = (ci , . . . , cm , n1 , . . . , nm , z),g0= (c0i , . . . , c0m , n01 , . . . , n0m , z 0 ).

We defined(g, g 0 ) = max(dR3 (ci , c0i ), dR3 (ni , n0i ), dR3 (z, z 0 )),i

max {dR3 (cS (u, θ), cS 0 (u, θ)),dR3 (nS (u, θ), nS 0 (u, θ))} .
Note that, thinking of S1 as the unit circle in R2 , we canendow S1 with the simple metric dS1 (x, y) = kx − yk, where
k.k denotes the Euclidean distance in R2 (alternatively, we
could measure differences in angles). Similarly, on I = [0, 1]
dI (x, y) = |x − y| provides us with a distance measure.
Summing these distances and the above metric d componentwise, we can hence obtain a metric on G cyl (m) = Mcyl ×C m ,which can be used to quantify distances between grasp/surface
combinations.
Definition III.4. Let S1 = Sf1 ,a1 ,b1 , S2 = Sf2 ,a2 ,b2 be
surfaces with cylindrical coordinates and let
g = (cS1 (u1 , θ1 ), . . . , cS1 (um , θm ),nS1 (u1 , θ1 ), . . . , nS1 (um , θm ), zS1 )be a grasp configuration on Sf1 ,a1 ,b1 . The naı̈ve transfer of g
to Sf2 ,a2 ,b2 is the grasp on Sf2 ,a2 ,b2 given by
g 0 = (cS2 (u1 , θ1 ), . . . , cS2 (um , θm ),nS2 (u1 , θ1 ), . . . , nS2 (um , θm ), zS2 ).
Lemma III.5 (Transfer lemma). Suppose two surfaces withcylindrical coordinates S1 , S2 satisfy d(S1 , S2 ) 6 ε for some
ε > 0 and let
g = (cS1 (u1 , θ1 ), . . . , cS1 (um , θm ),nS1 (u1 , θ1 ), . . . , nS1 (um , θm ), zS1 )be a grasp configuration on S1 . Then the naı̈ve transfer g 0 ofg to S2 satisfies d(g, g 0 ) 6 ε.
Proof: This follows directly from the definition of the
distances.
Having related distances of shapes to distances of grasps, we
now come to the connection with grasp quality. Let us consider
grasps g such that the contacts ci and the centre of mass z are
constrained to lie within the ball B(r) = {x ∈ R3 : kxk 6 r}.
The resulting set D(r) = B(r)m × (S2 )m × B(r) is bounded
and we can consider, for 0 6 δ− < δ+ , the quantity
εl (δ− , δ+ , r) = sup{ε : Ql (g) > δ− for all g such thatd(g, g 0 ) 6 ε for some g 0 ∈ D(r)with Q(g 0 ) > δ+ }.
This quantity, which could be studied analytically or - as
we shall do - numerically, can be used to give guarantees
on the existence of stable grasps near a known stable graspconfiguration. This applies both to the case where we would
like to find a nearby grasp on a single given surface, or
when we want to transfer a grasp to a nearby position ona nearby surface. We can hence already see the usefulness of
specifying a single space such as the Grasp Moduli Space,since we are now able to precisely formulate questions suchas ‘when can a stable grasp on surface X be transformed toa stable grasp on surface Y ’ as problems in G cyl (m). We areespecially excited about the potential future prospect of using
non-parametric Bayesian methods such as Gaussian Processes
on G cyl (m), which could provide a step towards a continuous
non-parametric Bayesian approach to grasp synthesis.
IV. E XPERIMENTS
For our experiments, we approximate friction cones by
a convex hull of l = 8 uniformly spaced edges. We areinterested only in grasps with three contact points lying in thebounded domain D(1). All grasps on convex combinations of
the surfaces depicted in Fig. 2 lie in this set. We work with afriction coefficient of µ = 1 throughout.

In this section, we experimentally determine a lower bound
ε for the quantities ε8 (0, 0.05, 1) and ε8 (0, 0.20, 1). This
provides us with a first numerical measure for how much thequality Q8 of a grasp varies with reasonably small changes ingrasp configurations. This stage of our analysis is completely
generic and does not require any notion of an object shapeyet.
To determine a lower bound, we sample 100 million points
S from the uniform distribution on the bounded set D(1), form = 3 contact points. This results in 83.9% unstable grasps
D0 = {g ∈ S : Q8 (g) = 0}. 4.1% of all grasps lied in theset D0.05 = {g ∈ S : Q8 (g) > 0.05}, while approximately
0.064% lied in the set D0.2 = {g ∈ S : Q8 (g) > 0.2}.
The stable grasps were distributed as displayed in Fig. 5 andare increasingly concentrated towards the origin. The maximal
grasp quality encountered was approximately 0.4013. Since
·106
4
2
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1
0.5

0.5
0

0
−0.5

−0.5
−1
1
1
0

−1

1
0

−1

0

0
1

−1 −1

−1

Fig. 6: On the left, we display resampled grasp configurations ina neighbourhood of size ε = 0.1 around a particular grasp g.
Inward pointing resampled normal vectors based at the resampledcontact point positions are displayed together with the resampledcentre of mass. On the right, normals of a grasp g resampled ina neighbourhood of size ε = 0.1 are displayed separately and are
based at the origin. For comparison, the original normals are drawn
slightly longer than the samples.
0.04
0.03
0.02
0.01

A. Local safe grasp deformation ε-neighbourhoods

0

1

0.4

Fig. 5: Grasp quality histogram for Q8 > 0, as determined from
100 million random grasps in D(1).
our grasp set S is rather large, computing a straight-forward estimator e.g. for ε8 (0, 0.05, 1) by ε = ming∈D0 ,g0 ∈D0.05 d(g, g 0 )is computationally too expensive. Since, for any g ∈ D(1), we
can however efficiently sample new points from the uniform
distribution on the neighbourhood Nε (g) = {g 0 : d(g 0 , g) 6ε}, we can sample new points from the uniform distribution

0
0.005

0.01

0.015

0.02

0.025

0.03

0.04

0.06

0.08

0.1

0.12

0.2
0.15
0.1
0.05
0
0.02

Fig. 7: The top figure displays the smallest grasp quality (verticalaxis) encountered when resampling 25 grasps in a neighbourhoodof size ε (horizontal axis) around approximately 410000 grasps
from D0.05 . The bottom figure displays the smallest grasp quality
encountered when resampling 150 points in a neighbourhood of size ε(horizontal axis) for each of the approximately 64000 grasps in D0.2 .
The experiments are repeated four times resulting in the displayed
small variations.

on Nε (g) for each g ∈ D0.05 and test if we encounter unstableconfigurations for various ε settings.
For ε8 (0, 0.05, 1), we worked with approximately
410000 samples from D0.05 and resampled 25
points from Nε (g) for each such sample g andfor ε∈{0.005, 0.01, 0.015, 0.02, 0.025, 0.03, 0.03}.
Similarly, for ε8 (0, 0.2, 1), we worked with all available
samples in D0.2 (about 64000) and resampled 150
new points for each such sample (see Fig. 6) and forε ∈ {0.02, 0.04, 0.06, 0.08, 0.1, 0.12}. To test for stability,
we repeated these experiments 4 times. Fig. 7 summarizes
the result. We found that the minimal grasp quality for eachparameter setting remained rather stable between iterationsof our resampling procedure. Our results show in particularthat 0.02 and 0.1 can be used as a lower-bound estimate for
ε8 (0, 0.05, 1) and ε8 (0, 0.2, 1) respectively. Let us remark
here that the determination of optimal approximations toεl (δ− , δ+ , r) is a very interesting topic, which we believe
should be investigated in more detail in future. Importantly,
estimates of this type do only need to be computed once forvarious parameter settings for later use - e.g. using a clustercomputer. Once such constants are determined, samplingalgorithms could then use this information to explore thespace of possible contact point configurations more rapidly
than uninformed uniform sampling.
B. An example of ε-stability under deformations in G cyl

C. Behaviour under global deformations in G cyl
Next, let us get a qualitative understanding of how grasp
quality changes for larger deformations along a continuous
path in the Grasp Moduli Space G cyl (3). To obtain a starting
set of interesting grasps, we uniformly sample 200 contact
positions on each of our model surfaces displayed in Fig. 2
using the uniform sampling procedure just described. We again

Fig. 8: Surface S with a stable grasp g (blue lines) and three
deformations in the Grasp Moduli Space in a ball of radius 0.3,
where only the contact and only the surface are deformed by a small
amount in a) and b) respectively and where both are deformed in c).
In all cases, the grasp remains stable after deformation.

Q8 (γij (t))

We now apply the local stability result just obtained tothree types of deformations in the space G cyl (3). We consider
the surface S with coordinates (0.3, 0.3, 0.4) displayed inFig. 1. We determined 200 uniformly distributed random
contact positions on S using the stratified uniform sampling
method on smooth embedded surfaces as explained in [2].
This involves an inversion of a strictly monotonic function
on [0, 1] for which we use the bisection method with 30
steps and integrations in 1D and 2D, for which we use
Simpson’s method partitioning the domain of integration intoa grid with 60 uniform steps along each dimension. We thenrank all combinations of three such contact points by grasp
quality. This results in 200= 1313400 evaluations of Q83and takes about 31 minutes on a current computer, while therandom sampling takes about half a minute. The best graspg (displayed in blue in Fig.8) satisfied Q8 (g) ≈ 0.33. We
now experimented with using the bound on ε8 (0, 0.2, 1) by
deforming the grasp and surface. One might imagine that thegrasp g has been previously learned by the robot and is stored
in a database. Suppose now that the robot encounters thefollowing three cases: a) the object is encountered again, but
due to imprecision in control is grasped slightly differently(grasp-only deformation), b) another object S 0 very similar
to S is encountered and the naı̈ve transfer grasp is executed
precisely (surface-only deformation), c) an object S 0 similar
to S is encountered but the transfer grasp is executed only
up to some imprecision (combined deformation). Due to ourbound of the previous section, we know that any grasp g 0 suchthat d(g, g 0 ) 6 0.1 should be stable and that the naı̈ve transferof the grasp g to any surface S 0 such that d(S, S 0 ) 6 0.1
will also be stable by the naı̈ve transfer lemma. We explicitly
verified each of the three cases by sampling 3 random surfaces
near the surface S in the simplex depicted in Fig. 1 and bysampling nearby grasps for the case a) and c). Fig. 8 depicts
an explicit example, where we now allow larger deformations
such that d(g, g 0 ) 6 0.3 and d(S, S 0 ) 6 0.3 respectively. Evenfor those deformations, the grasp quality of g 0 remained larger
than 0.19 in this example.

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0
0

0.2

0.4

0.6
t

0.8

1

0

0.2

0.4

0.6

0.8

1

t

Fig. 9: Plots of Q8 (γij (t)) against t ∈ [0, 1], where γij (t) denotes
the naı̈ve transfer of a fixed grasp gi ∈ Si to Sj for all i 6= j on
the left and the interpolation transfer from gi ∈ Si to the grasp withcoordinates u = ( 12 , 12 , 12 ), θ = (0, 2π, 4π) on Sj , i 6= j, on the3
3right. The red, black and blue lines correspond to starting from thesurfaces displayed in Fig. 2a, i) and j) respectively.

calculate the grasp quality of all triples of such contacts up to
permutation and use the best grasp gi on Si , for i = 1, . . . , 12
as a model grasp on each surface. The left part of Fig. 9
displays the variation in grasp quality when we try to apply
the naı̈ve transfer of the grasp gi from Si to Sj , i 6= j, by
simply keeping the same cylindrical coordinates of the grasp
while deforming the underlying surfaces via γij (t) = (1 −t)Si +tSj for t ∈ [0, 1]. The right part of that figure shows theinterpolation transfer between gi ∈ Si and the grasp on surface
4π
Sj with coordinates u = ( 12 , 12 , 12 ), θ = (0, 2π
3 , 3 ) and using
γ(t) as in Eq. III.1. As can be seen from the figure, Q8 behavesin fact rather tamely in our naı̈ve transfer examples. While Q8
is not a smooth function, it seems to be Lipschitz continuous as already indicated by our experiment a). For the interpolationtransfer, we note a higher amount of variability in our example

and that a large amount of target grasps with coordinates givenby (u, θ) are in fact stable at the end of the deformation. Let us
also remark here that there are many interesting open questions
about interpolation transfers that can be asked, such how tocharacterize subsets of grasp/surface configurations in G cyl (3)for which such transfers lead to stable grasps.
D. Gradient methods on G

cyl

Given the ability to apply infinitesimal deformations tograsps and surfaces, variational and gradient based methods
immediately come to mind. In the case of a fixed surface, a
gradient ascent with respect to grasp quality can for example
be considered. To the best of our knowledge, this idea has only
been investigated in the simple case of a sphere [11]. Since
we are however also able to deform the surface, we focus on
the interesting question of how to optimally transfer a stablegrasp g ∈ Si to the surface Sj while maintaining stability andwhere S1 , . . . S12 again denote the surfaces depicted in Fig. 2.
We propose to use a sub-division {t0 , . . . , tN1 } ⊂ [0, 1] with
ts = Ns1 and N1 = 100 and consider the curves of surfaces
αij (t) = (1−t)Si +tSj . While Q8 is not differentiable everywhere, we none-the-less evaluate a gradient descent inspired
Q8 (x+hei )−Q8 (x)8approach on −Q8 by approximating ∂Q
,
∂xi ≈
h
where ei denotes the i-th standard basis vector. We choseh = 10−5 in our experiments. We now propose the following
simple algorithm in Grasp Moduli Space, transferring a grasp
g ∈ S to a stable grasp on S 0 : In step s, we iterate N2 = 20
gradient descent steps, with xn+1 = xn + δ∇Q(xn ) and step
length δ = 0.001 on the surface αij (ts ) to reach a local
maximum on αij (ts ). We then apply the naı̈ve transfer tothe nearby surface αij (ts+1 ) and proceed to step s + 1 until
αij (1) = Sj is reached. In order to measure the improvementof this approach relative to the global naı̈ve transfer consideredin the previous experiment, we apply our gradient method
to the grasps gi ∈ Si considered previously. We foundthat our method typically improves the quality of the grasp
during the deformation when compared to the naı̈ve method,
but the absolute grasp quality can undergo non-monotonic
variations during the deformation. Fig. 10 shows the final
relative improvement in grasp quality. As we can see, ourapproach typically improves the grasp quality substantially.
Furthermore, all the resulting grasps on the target surfaces
were stable in our experiment. Let us remark here also that
we did not exclude the case of a transfer ‘from Si to Si ’, whichcorresponds to performing gradient based grasp optimization
on a fixed surface.
E. The role of Mrev ⊂ Mcyl
Recall that we can think of surfaces of revolution as forminga natural infinite-dimensional subset Mrev ⊂ Mrev . When we
determined best grasps on the surfaces of revolution displayedin Fig. 2a-d) by brute-force search for experiment c), we
observed that the best grasps on surfaces of revolution seemed
to occur along a single plane containing the axis of rotation,
which corresponds to a natural grasp used by humans as well.
We hence propose exploiting the rotational symmetry of these

0.4
0.2
0
0

20

40

60

80

100

120

140

Fig. 10: Sorted differences in final grasp quality between our
gradient approach and the naı̈ve transfer approach when transferringfrom Si to Sj for all 122 = 144 combinations of surfaces and startingfrom our stable grasps on Si (note, we include the case i = j here).
In all but one case, the final grasp quality using the gradient descentis higher. Remarkably, all final grasps, even those grasps that were
unstable under the naı̈ve transfer (see t = 1 on the left part of Fig. 9)
are stable on the target surface when the gradient method is used.

surfaces and suggest the following heuristic for determining
stable grasps on such surfaces:1, . . . , m−1a) Let U = {0, mm , 1}, for some m ∈ N.
b) Compute grasp quality Q8 for the triples of contact points
with cylindrical coordinates (u1 , 0), (u2 , π), (u3 , π) ∈[0, 1] × S1 , where u1 , u2 , u3 ∈ U and u2 < u3 .
c) Select a grasp with optimal grasp quality Q8 from theabove triples as a grasp candidate.
To test our heuristic, we considered the finite dimensional
subset of G cyl (3) spanned by the four surfaces of revolution
displayed in Fig. 2a-d) and sampled 100 uniformly distributed
convex combinations of those surfaces and 100 uniformly
distributed contact points on each such surface using stratified
random sampling. We then computed the grasp quality forevery three-element subset of the contact points on each of
the 100 surfaces and stored the best grasp for each surface
resulting from this brute-force method. This computation
took

about 250 seconds per surface, resulted in 100= 161700
3evaluations of Q8 per surface and yielded stable grasps distributed as displayed in Fig. 11 and with a mean grasp qualityof 0.2435. We then applied our heuristic with only m = 10
height sub-divisions, which took about
 0.89 seconds per surface and which required just 10 10= 450 evaluations of2
Q8 per surface. Again, all resulting grasps had positive grasp
quality distributed as in Fig. 11. Interestingly, the mean quality
obtained using our heuristic was 0.2521 and hence higher
than the mean quality obtained using the time-consuming brute
force method.
20
10
0

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

Fig. 11: Quality histogram for grasp hypotheses over 100 random
surfaces of revolution spanned by the examples in Fig. 2a-d) using
brute-force evaluation on 100 contact points per surface (red) andusing our heuristic (blue).

Note that our simple heuristic now gives us an example for
how we can reason efficiently about a subset (e.g. G rev (3)) of
the Grasp Moduli Space G cyl (3). It seems hence natural to beable to apply this strategy for surfaces that are close to surfacesof revolution in our Grasp Moduli Space. Furthermore, there
exists a canonical projection π : Mcyl → Mrev defined asfollows: Given a surface Sf,a,b , we define π(Sf,a,b ) = Sfˆ,a,b ,R 2π
1f (u, θ)dθ. We can continuously movewhere fˆ(u) = 2π0towards this projection using the curve γ(t) = (1−t)S+tπ(S).
Let us now consider the following deformation based grasp
synthesis strategy: for any surface S ∈ Mcyl , we determine
an optimal grasp g on π(S) using our heuristic and then apply
our gradient method to transfer g back to a grasp on S. Fig. 12
displays the differences in grasp quality when compared to thebrute-force search for best grasps on the given surfaces using
200 uniform random samples per surface. Our method canidentify stable grasps in these cases, while requiring only about
4 seconds for each surface when we chose N1 = N2 = 10,
δ = 0.01, h = 10−5 and m = 10 for our gradient descentand heuristic parameters and when approximating centres ofmass during the deformation using Simpson’s integral formulaon a 20 by 20 grid. As we can see, the grasp qualities are
all positive but on average not as high as using brute force
search. However, our method took only 4 seconds per surface
as compared to 31 minutes using brute force search.
0.4
0.2
0

S5

S6

S7

S8

S9

S10

S11

S12

Fig. 12: Comparison of grasp quality obtained using brute-force
search on 200 random contact points on the surfaces in Fig. 2e-l)(blue) and using our deformation to a surface of revolution approach(red).

V. C ONCLUSION AND F UTURE W ORK
In this work, we introduce a new view on contact based
grasp analysis based on a single infinite dimensional space –the Grasp Moduli Space – in which continuous deformationsof surface and grasp configurations can be described by continuous curves. We study finite-dimensional subsets of this spaceinduced by data as well as the local stability of a grasp quality
function. A new grasp synthesis approach based object/grasp
deformations to a canonical surface of revolution is discussed
as well as a gradient based approach for transferring andinterpolating between grasps on differing surfaces. We believethat our framework could in future be used to develop novel
probabilistic as well as optimization based techniques forgrasping based on the concept of continuous deformations ofboth objects and grasps. While our method is currently basedon contact-level information, we believe that the inclusion ofrobot hand kinematics in our framework could provide anotherpromising future research direction.
VI. ACKNOWLEDGEMENTS
This work was supported by the Swedish Foundation for
Strategic Research and the EU grants FLEXBOT (FP7-ERC279933) and TOMSY (IST-FP7-270436).

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