Geometric Characterization of Two-Finger Basket Grasps
of 2-D Objects: Contact Space Formulation
E. D. Rimon† and F. T. Pokorny‡ and W. Wan∀

Abstract This paper considers basket grasps, where a twofinger robot hand forms a basket that can safely lift andcarry rigid objects in a 2-D gravitational environment.
The two-finger basket grasps form special points ina high-dimensional configuration space of the object andtwo-finger robot hand. This paper establishes that all twofinger basket grasps can be found in a low-dimensional
contact space that parametrizes the two-finger contacts
along the supported object boundary. Using contact space,each basket grasp is associated with its depth that provides
a security measure while carrying the object, as well as
its safety margin away from a critical finger opening
where the object drops-off into its intended destination.
Geometric techniques that compute the depth and drop-off
finger opening are described and illustrated with detailed
graphical and numerical examples.
I. Introduction
Basket grasps offer a robust approach for picking objects
from a clatter, carrying them in a secure manner to theirdestination, then releasing the objects in a controlled manner by opening the basket formed by the robot hand.
Important applications include packing orders at logistic
centers and picking up parts that arrive in a clutter during
manufacturing operations [5, 6, 8, 9, 11, 12, 21]. This
paper establishes that all two-finger basket grasps as wellas their depth and drop-off finger opening can be found ina low-dimensional contact space, which parametrizes thetwo-finger contacts along the object boundary.
We study basket grasps of polygonal objects in a 2-D
gravitational environment. The robot hand is modeled as
two point or disc-fingers that freely move in IR2. The object,
B, is initially supported at an equilibrium stance (Fig. 1(a)).
The object is to be lifted and carried in a safe manner bya basket formed by a two-finger robot hand, where safety
is measured by the depth of the basket formed by the
supporting fingers. The fingers retain their relative distanceand orientation while carrying the object (Fig. 1(b)), andthe hand should be able to drop the object in a predictablemanner upon reaching the intended destination.
Basket grasps form a special class of caging grasps [17,20]. In basket grasps, the bounded mobility of the carried
object is based on rigid-body constrains imposed by the
supporting fingers and energy-bound imposed on the object
motions within the basket. Hence, efficient caging grasp
techniques can be leveraged to the synthesis of basket
grasps [10, 13, 23]. Mahler et al. [13] were the first to offer
an algorithm that computes basket grasps for two-finger†Dept. of ME, Technion, Israel. ‡Dept. of EECS, KTH Royal Inst. ofTechnology, Sweden. ∀ School of Eng. Science, Osaka University, Japan.

g

φ

g

fingers maintain fixed
distance and orientation
B

B
1
O10

x1

1
0
σ 0
1
FB

x
O2

112
00

1
0
1
O10

1
0
0
1

FB

O2

B’s center of mass

y obstructed edge
111111111111111111111
000000000000000000000
y0000000000000000
1111111111111111
x
000000000000000
111111111111111
FW
0000000000000000
1111111111111111
xFW

Fig. 1: A polygonal object B supported by point fingers
O1 and O2 in a basket grasp (the full robot hand is notshown). The parameter σ describes the inter-finger distance,
ϕ measures the two-fingers’ orientation relative to gravity.
robot hands. Their sampling based technique computes
a bound on the depth (and hence security) of candidate
basket grasps. Among the caging techniques surveyed by
Makita and Wan [14] and the Robot Grasping text [18],
let us shed light on two approaches. The first approach
was pioneered by Vahedi [22], and arrived to full maturity
in the work of Pipattanasomporn [15, 16]. Under thisapproach, the physical space surrounding the grasped object
is partitioned into convex cells. A search graph whose nodes
represent the placement of the k robot fingers in these cells
is then used to compute all caging grasps surrounding agiven object, often as a convex optimization problem per
grasp node. These techniques compute all two-finger caging
grasps of polygonal objects with n edges in O(n2 log n)
steps, and all three-finger caging grasps in O(n3 log n)
steps. However, this approach requires a high-dimensional
configuration space for the freely moving finger bodies.
In his PhD thesis [1], Allen pioneered an alternative approach that uses a low-dimensional contact space to compute
caging grasps. Under the contact space approach, the fingers maintain contact with the grasped object during thesearch for caging grasps, thus reducing the search space
into a low-dimensional contact space that can be intuitively
verified as motion of the fingers along the object boundary.
Allen showed that all two-finger caging grasps can becomputed in contact space using O(n2 log n) steps [2].
Bunis [4] used contact space to compute all three-finger
caging grasps that maintain similar triangular formations
using O(n3 log n) steps. Bunis [3] subsequently used contact space to compute all two-finger locking grasps of 2-D
objects against a wall using the same number of steps.
Motivated by the contact space approach, this paper describes how the layout and properties of two-finger basket
grasps can be computed in the low-dimensional contact
space. The basket grasps form special points in a fivedimensional configuration space (c-space) of the object
and two-finger hand. This space consists of the grasped

(a)

(b)

Fig. 2: (a) The s-parameterization of the object boundary.
(b) The object’s contact space U for two point fingers,
overlayed with the contours of the function σ(s1 , s2 ).
object configurations, q ∈ IR3 , the finger-opening parameterσ ≥ 0, and the angle ϕ of the two-finger system relative
to gravity (Fig. 1(a)). When the object waiting to belifted is located at a configuration q0 , the high-dimensional
configuration space contains a 2-D submanifold of all twofinger contacts along the object boundary positioned at q0 .
This submanifold is parametrized in terms of contact space,which is then used to compute the layout of all possible
basket grasps of an object waiting to be lifted.
The paper describes a geometric technique that determines the depth (and hence security) of a candidate
basket grasp. The main result of the paper, the basket
grasp theorem, states that the depth of a basket grasp isdetermined by the relative height of the object’s center ofmass at specific single- and two-finger equilibrium stances
associated with the basket grasp. These stances can bereadily found and sorted in contact space as demonstrated
with detailed examples. The paper finally addresses thedrop-off problem, where the fingers open along a fixed linein IR2 until a critical drop-off event is reached. This criticalfinger opening is associated with a frictionless two-finger
equilibrium grasp of the object (without gravity), that canbe efficiently computed by a simple contact space search.
Section II introduces the notion of contact space andcharacterizes the layout of all possible two-finger basket
grasps at a given object position using contact space. Section III describes a geometric technique that uses contact
space to determine the depth of a candidate basket grasp.
Section IV demonstrates the technique on several object
types. Section V studies the drop-off problem, where contact space is used to search for the critical drop-off finger
opening of a given basket grasp. The conclusion discusses
open problems as well as extension to 3-D basket grasps.
II. C ONTACT S PACE OF T WO -F INGER BASKET G RASPSThis section characterizes the two-finger basket grasps ofa 2-D object at a given initial position using contact space.
The object’s outer boundary is parametrized by arclength
in counterclockwise direction using the parameter s ∈ [0, L],
where L is the object’s perimeter (Fig. 2(a)). Let p(s1 ) andp(s2 ) denote the position of the finger contacts along theobject boundary, such that p1 (0)=p1 (L) and p2 (0)=p2 (L).
Definition 1. A polygonal object B is contacted by two
fingers at p(s1 ) and p(s2 ). Contact space is the parameter-

ization of all two-finger contacts along the object boundary,given by the set U = [0, L] × [0, L] in the (s1 , s2 ) plane.
Contact space consists of rectangles, each associated with aparticular pair of object edges (including same-edge pairs).
The lines bounding each rectangle represent vertex-edge contacts. The diagonal, ∆ = {(s1 , s2 ) ∈ U : s1 = s2 }, represents
all two-finger pinchings along the object boundary. The diagonal forms an obstacle in U, since the fingers may notcross each other while moving along the object boundary.
Example: Consider the inter-finger distance function
defined in U as σ(s1 , s2 ) = ∥p(s1 )−p(s2 )∥. The contoursof σ(s1 , s2 ) in U are depicted Figure 2(b). Note thatσ(s1 , s2 ) is non-negative and continuous on U, and attains
a global minimum of zero along the diagonal ∆. Also notethat σ(s1 , s2 ) forms a positive definite quadratic functionin the individual contact space rectangles.
◦
When a rigid object B is supported by finger bodies O1
and O2 at an equilibrium stance, the net wrench on B due togravity and the supporting contacts must be zero. The next
lemma characterizes the two-finger equilibrium stances incontact space U (see [19]).
Lemma II.1. Let a polygonal object B be located at
a configuration q0 in a 2-D gravitational environment. All
two-finger contacts that form feasible equilibrium stances
generically form piecewise linear curves in U.
Example: Fig. 2(a) shows a polygonal object B at a fixed
position-and-orientation in IR2 . Fig. 2(b) shows in solid black all two-finger placements along the object boundarythat form feasible equilibrium stances in contact space U. ◦To describe the object’s gravitational potential energy,
let bcm denote the position of B’s center of mass in B’s
reference frame F B (bcm = ⃗0 in Fig. 1). When B is locatedat a configuration q = (d, θ), the position of its center ofmass in the fixed world frame, F W , is given by Xcm (q)=R(θ)bcm +d, where q = (d, θ) ∈ IR3 . The gravitational potential energy of B is given by U (q) = mg (e · Xcm (q)),
where m is B’s mass, g is the gravitational constant, ande is the vertical upward direction, e = (0, 1), in IR2 . The
function U (q) simply measures the height of B’s center ofmass in F W . The gravitational wrench affecting B is givenby −∇U (q) = −mg(e, R(θ)bcm ×e) (see [19]).
Consider the free c-space, F, whose c-obstacles are
induced by fixed supporting fingers O1 and O2 . Based onconservation of energy, the non-degenerate local minimaof U (q) in F form locally stable equilibrium stances ofthe object B. This property is the basis for the following
definition of basket grasps.
Definition 2. Let a rigid object B be supported undergravity by fingers O1 and O2 at an equilibrium stance configuration q0 . The supporting fingers form a basket grasp
if q0 forms a non-degenerate local minimum of U (q) in F.
The local-minimum test of U (q) in contact space U
depends on the height of B’s center of mass as follows [19].
Lemma II.2. A two-finger equilibrium stance that involves
two edges of B forms a local minimum of U (q) when
−λ1 (ρ1 (s1 , s2 )+rO1 )−λ2 (ρ2 (s1 , s2 )+rO2 ) + ycm (s1 , s2 )< 0

where λ1 , λ2 ≥ 0 are the equilibrium stance coefficients,

(a)

(a)

(b)

Fig. 3: The two-finger equilibrium stances of the object
shown in Figure 2 are either local-minima (green) or localmaxima (red) of U (q) in contact space (see text).
ρ1 , ρ2 are the signed distances of the contacts fromthe intersection point, p, of the finger contact normals,
rO1 , rO2 ≥ 0 the finger radii, and ycm the height of B’s center of mass above p. Stances that involve a concave vertex
and an edge of B generically form local minima of U (q).1
Example: The equilibrium stances of Figure 2 are colorcoded according to their stability type in Figure 3. Greenfor the local minima and red for the local maxima of U (q)in U. The stability type of each stance describes how theheight of B’s center of mass changes when the object moves
in contact with two fixed point or disc fingers.
◦
Remark: A two-finger basket grasp of B is a nondegenerate local minimum of U (q) in the free c-space, F,
whose finger c-obstacles are CO1 and CO2 . A basket graspof B is an equilibrium stance that forms a local minimumof U (q) along the 1-D stratum S = bdy(CO1 ) ∩ bdy(CO2 )in F. The stratum S corresponds to the contour of σ(s1 , s2 )that passes through the basket-grasp point in U. Hence,
when the puncture point that determines the depth of thebasket grasp occurs at a two-finger stance, it forms a local
maximum of U (q) along the basket-grasp contour in U. ◦III. C ONTACT S PACE C OMPUTATION OF
BASKET-G RASP D EPTH
This section describes a technique that determines the depth(and hence security) of a candidate basket grasp. The technique is described for point fingers then extended to discfingers. In a basket grasp, the object’s configuration point,q0 , lies at the bottom a 3-D cavity in the free c-space F [19].
As the level-set of U (q) increases its height in F, theappearance of a puncture point, q1 , marks the depth of thebasket grasp. The basket grasp is specified as two finger
contacts on the object boundary, (s01 , s02 ) ∈ U, at the object
configuration q0 . The fingers now become fixed supports
with fixed inter-finger distance, σ 0 = ∥p(s01 )−p(s02 )∥.
The technique computes the basket grasp depth by tracing the σ 0 -contour of σ(s1 , s2 ) in U, which represents
object motions that maintain contact with the supporting
fingers. The following lemma characterizes the σ 0 -contour.
Lemma III.1 ([2]). Each contour of σ(s1 , s2 ) in U
consists of elliptical and linear segments associ1 Stances that involve disc-fingers located at convex vertices of B
typically do not form local-minima of U (q) (see [19]).

(b)

Fig. 4: (a) A polygonal object supported at a two-finger
basket grasp. (b) The σ 0 -contour of σ(s1 , s2 ) in U.
ated with the contact-space rectangles, with vertices on
the rectangles’ bounding lines (Figure 4(b)).
The elliptical contour segments can be parametrized
in terms of B’s orientation, θ, in each rectangle Rjk asfollows. Let tj and tk denote the unit tangents to the edges
ej and ek of B. Then each contour segment is given by()()
s1 (θ)+= A−1 RT (θ)(x1 −x2 ) + p0k −p0jθ ∈ [θ−jk , θ jk ]s2 (θ)(1)
where A = [tj −tk ], p0j , p0k are the initial vertices of ej andek , and x1 , x2 are the point-finger positions. The endpoint+
angles θ−
jk and θ jk occur when one of the two fingers
touches an endpoint of the respective object edge. Whenthe two fingers support the same edge (or parallel edges)of B, the object maintains fixed orientation θ along a linear
contour segment. The segment satisfies the constraint s2 +s1 = c, where c = tj ·(RT (θ)(x1 −x2 )+p0k −p0j ) is constant.
Example: Figure 4(a) shows a polygonal object B supported by two fingers at a local minimum (i.e. a basket
grasp) of U (q). The fingers remain fixed in IR2 while B
moves along the σ 0 -contour depicted in Figure 4(b).
◦Starting at the basket-grasp point, v0 = (s01 , s02 ), the technique traces the σ 0 -contour in stages that correspond tothe contact-space rectangles and their bounding lines. The
contour is searched sequentially along its two directions.
The θ-parametrization of Eq. (1) can be used to computethe endpoints of the initial contour segments that start at v0
and reach the contact-space rectangle bounding lines, at v1−
and v1+ . In all subsequent stages, the technique constructs
the contour segments that start at the points vi− and vi+−+
and end at the points vi+1
and vi+1
located on the next
contact-space rectangles bounding lines. This process ends
when the first two-finger equilibrium stance is encounteredalong each direction, as next described.
Edge-edge equilibrium stance: Consider a stance thatinvolves two edges ej and ek of B, with inward unit normals nj and nk . The force-part of the equilibrium stancecondition requires that λ1 R(θ)nj +λ2 R(θ)nk = e for some
λ1 , λ2 ≥ 0. As described in [19], λ1 , λ2 ≥ 0 form inequality
constraints in θ that determine a θ-interval denoted I jk . The
moment-part of the equilibrium stance forms the constraint:
[

R(θ)n

R(θ)n

e

]

k
det p(s (θ)) ×j n
= 0 θ ∈ I jk .
p(s2 (θ)) × nk 0
1
j
The solutions of this equation in θ give the edge-edge equilibrium stances along the σ 0 -contour in the rectangle Rjk .

Vertex-edge equilibrium stance: Let the fingers support
the j’th vertex of B at p(s1 ), and the edge ek of B at p(s2 ).
The object orientation, θ, is fixed at this particular (s1 , s2 )point. To check equilibrium stance feasibility, one models
the finger at the vertex as a disc of small radius ϵ > 0. Let
ej1 and ej2 be the two edges of B that meet at the vertex,
with inward unit normals nj1 and nj2 . The disc-finger canrealize any contact force direction within the generalized
contact normal at the vertex, λ11 nj1 + λ12 nj2 such that
λ11 , λ12 ≥ 0. The equilibrium stance force-part requiresthat λ11 R(θ)nj1 + λ12 R(θ)nj2 + λ2 R(θ)nk = e for some
λ11 , λ12 , λ2 ≥ 0. The solution determines a sector of force
directions at the object vertex, λnj1+(1−λ)nj2 , such that λvaries in an interval I j contained in [0, 1]. The moment-partof the[ equilibrium stance forms the constraint:]
λn̄ +(1−λ)n̄

R(θ)n

e

j1
j2
k
det p(s (θ)) ×=0(λn̄j1 +(1−λ)n̄j2 ) p(s2 (θ)) × nk 0
1
where n̄j1 = R(θ)nj1 and n̄j2 = R(θ)nj2 . A solution λ∈I jidentifies the feasibility of a vertex-edge equilibrium stanceat an endpoint of the σ 0 -contour segment in Rjk .
Example: The σ 0 -contour depicted in Figure 4(b) contains the basket-grasp point (green dot no. 1). The neighboring local maxima of U (q) along the σ 0 -contour form
vertex-edge equilibrium stances (red dots no. 2 and 3). ◦The technique compares the heights of B’s center of massat the two-finger equilibrium stances located on the σ 0 -contour, taking into account the relevant single-finger equilibrium stances as summarized in the following theorem.
Theorem 1 (Basket Depth). Let two finger contacts at(s01 , s02 ) form a basket grasp of a 2-D object B at q0 .
−+ +Let (s−
1 , s2 ) and (s1 , s2 ) be the two-finger equilibrium
stances encountered along both directions of the basketgrasp contour in U. Let h− and h+ be the height of B’s center of mass at these stances. Let hi be the minimum heightof B’s center of mass over the single-finger equilibrium+
stances along the boundary segment [s−i , si ] for i = 1, 2.
The basket-grasp depth is given by min{h− , h+ , h1 , h2 },measured relative to the height of B’s center of mass at q0 .
Remark: One only needs to check single-finger stances thatform saddle points of U (q) in F. These are single-finger
stances that involve a horizontal edge of B, and certain
single-finger stances that involve a vertex of B (see [19]). ◦Proof sketch: The 3-D cavity of a basket grasp at q0 issurrounded by the finger c-obstacles CO1 and CO2 and thelevel-set of U (q) that passes through the puncture point, q1.
The puncture point q1 can appear on the individual cobstacle boundaries, S 1 =bdy(CO1 ) and S 2 =bdy(CO2 ), oron their intersection curve S = bdy(CO1 ) ∩ bdy(CO2 ). The
curve S corresponds to the σ 0 -contour in U that passes
through (s01 , s02 ). Hence, h− and h+ measure the height of
B’s center of mass at the first local-maxima of U (q) encountered along S, which form saddle points of U (q) in F.
If the lowest saddle point of U (q) lies on S i = bdy(COi ),consider the rim curve defined as the intersection curve S i ∩{q : U (q) = U (q1 )}. It forms a closed loop with constantheight in F that passes through q1 . Since q1 is the lowest
saddle point of U (q) just above q0 , the rim curve intersects

h−

(a)

h+

(b)

Fig. 5: (a) A two finger vertex-edge saddle point of U (q)with relative height h− . (b) A two finger vertex-edge saddle
point of U (q) with relative height h+ .
the 1-D stratum S at points lower than the local-maxima
encountered along S. Hence, starting at q1 , the object B canmove along the rim curve while maintaining contact withthe single finger Oi , until the other finger is encounteredat some point of S. At this instant the object is located ata point (s∗1 , s∗2 ) on the σ 0 -contour in U. Moreover, (s∗1 , s∗2 )lies on the piece of the σ 0 -contour that contains the basketgrasp point (s01 , s02 ) and bounded by the higher endpoints
−
+ +
(s−
1 , s2 ) and (s1 , s2 ).
When B moves along the rim-curve segment that starts at
q1 , its contact point with Oi moves in a continuous manneralong B’s boundary, until the other finger is encountered at(s∗1 , s∗2 ). If the contact point started outside B’s boundary+
segment parametrized by [s−i , si ], it must have passed+
through one of the interval’s endpoints, s−i or si , before∗
reaching the point si that lies inside the interval. Since
B maintained a single contact with Oi during this motion,
the contact point at q1 must be located inside B’s boundary+
segment parametrized by [s−i , si ]. The heights h1 and h2
thus capture all single-finger equilibrium stances that canpossibly form the puncture point of the basket grasp. Extension to disc fingers: The disc fingers can beconverted into point fingers by expanding the boundary of B
outward by the disc-fingers’ radius. The boundary of theresulting c-space object, CB, contains straight line edges
as well as circular arcs. Every circular arc can be approximated by a regular k-segment polygonal arc. The
basket-grasp depth can now be computed on the polygonal
approximation of CB. The output will closely approximate
the exact basket-grasp depth, and will usually identify theexact object features that determine the basket grasp depth.
IV. R EPRESENTATIVE E XAMPLESThis section illustrates the technique of the previous section
on three examples.
Example 1: Consider the basket grasp of the convex
object B depicted earlier in Figure 4 (green dot no. 1).
The two local maxima of U (q) along the σ 0 -contour ofthe basket grasp form vertex-edge equilibrium stances (reddots no. 2 and 3). These stances are shown in Figure 5
with their relative heights h− and h+ . The object also
possesses two single-finger equilibrium stances along the
red boundary segments (see Theorem 1). These stances areshown in Figure 6 with their relative heights h1 and h2 .

h2

h1

(b)

(a)

(a)

(b)

Fig. 6: (a) A single-finger saddle point of U (q) with relative
height h1 . (b) A single-finger saddle point of U (q) withrelative height h2 .

Fig. 9: (a) A cup-like object supported at a two-finger
basket grasp. (b) The σ 0 -contour of σ(s1 , s2 ) in U, withthe local minimum (no. 1) and two local maxima (no. 2
and 3) of U (q).

In this example, the minimum height and hence the basket
grasp depth is h−, depicted in Figure 5(a).
◦
h

h+

−

(a)

(b)

Fig. 10: (a) A vertex-edge saddle point of U (q) with relative
height h− . (b) A vertex-edge saddle point of U (q) withrelative height h+ . The basket grasp depth is h− .

(a)

(b)

Fig. 7: (a) A non-convex object having a shifted centerof-mass, supported at a two-finger basket grasp. (b) The
σ 0 -contour of σ(s1 , s2 ) in U, with the local minimum (no.
1) and two local maxima (no. 2 and 3) of U (q).

h+

h−

(a)

(b)

Fig. 8: (a) An edge-edge saddle point of U (q) with relative
height h− . (b) An edge-edge saddle point of U (q) withrelative height h+ . The basket grasp depth is h+ .
Example 2: Figure 7(a) shows a non-convex object B
having a shifted center of mass. The object is supported
in a basket grasp via an edge and a concave vertex of B.
Figure 7(b) shows the basket-grasp point in contact space(green dot no. 1). The two local maxima of U (q) along theσ 0 -contour of the basket grasp are edge-edge equilibrium
stances, located inside the contact-space rectangles (reddots no. 2 and 3). These stances are shown in Figure 8
with their relative heights h− and h+ . The object has no
single-finger equilibrium stances along the red boundary
segments (see Theorem 1). The minimum height and hence
the basket grasp depth is h+, depicted in Figure 8(b). ◦Example 3: Figure 9(a) depicts a cup-like object supported in a basket grasp via an edge and a concave vertexof B. Figure 9(b) shows the basket-grasp point in contact
space (green dot no. 1). The two local maxima of U (q)along the σ 0 -contour of the basket grasp are located onbounding lines of the respective contact-space rectangles(red dots no. 2 and 3). These stances are shown in Figure 10

with their relative heights h− and h+ . There are also twosingle-finger equilibrium stances along the red boundary
segments (see Theorem 1), having higher relative heights
h1 and h2 . The minimum height and hence the basket grasp
depth is h− , depicted in Figure 10(a).
◦V. C RITICAL D ROP -O FF F INGER O PENING
This section characterizes the critical finger opening ofa given basket grasp that allows the release of the object B
at its intended destination. To analyze this problem, we
assume that the fingers open along a fixed line in IR2 ,determined by the inter-finger segment at the given basket
grasp (Figure 11(a)). The angle between the inter-finger
segment and the direction of gravity is thus fixed at ϕ = ϕ0 .
Recall that (q0 , σ 0 ) are the object configuration and interfinger distance at the initial basket grasp. The configuration
space of the drop-off problem consists of the coordinates(q, σ) ∈ IR4, where q ∈ IR3 and σ ≥ σ 0 .
The (q, σ)-space can be thought of as a one-parameterfamily of 3-D spaces, q ∈ IR3 , parametrized by σ. For eachσ, the fingers form fixed supports while the object moves
freely in IR2 . Let CO1 |σ and CO2 |σ denote the finger
c-obstacles in each σ-slice of IR4. The composite finger c-obstacles in IR4 are thus ∪σ ≥σ 0 CO1 |σ and∪σ ≥σ 0 CO2 |σ . When the fingers move apart along a fixedline in IR2 , the finger c-obstacles move apart as rigid shapes
in the corresponding σ-slices of IR4 . At σ = σ 0 , the finger
c-obstacles overlap in a local neighborhood of q0 (Figure 11(b)). As the fingers move apart, a drop-off puncture
point will eventually appear between the finger c-obstacles,
at a point (q1 , σ 1 ) such that σ 1 > σ 0 (Figures 11(c)-(d)).
The finger opening difference, σ 1−σ 0 , measures the basket
grasp’s safety margin away from the critical drop-off event.
The remainder of this section describes a scheme forcomputing the drop-off puncture point in contact space U.

gB
center of mass

σ1

0
1
1
0
0
1

O2

σ0

11
00
00
11
00
11

y O1

x

^
η
1
CO 2

U
q0

^
η
2

g

CO1

Β
1
0
0
1

f1

σ1

(c)

drop−off grasp(b)puncture pointat σ=σ1(a)O1basket grasp

f2

1
0
1
0

O2

CO 2

^
η
2

^
η
1
q1

(d)

(a)

CO1

Fig. 11: (a) A 2-D object B supported at a two-finger basket
grasp with inter-finger distance σ 0 . (b) Cross section of(q, σ) ∈ IR4 at the basket grasp point (q0 , σ 0 ). (c) The critical drop-off finger opening, σ 1 , is a frictionless two-finger
equilibrium grasp of B. (d) Cross section of (q, σ) ∈ IR4 at
the drop-off puncture point (q1 , σ 1 ).
The σ-slices of (q, σ)-space are level-sets of the scalarvalued function π : IR4 → IR, defined as π(q, σ) = σ. Consider the free c-space defined as IR4 − int(CO), where
CO = ∪σ ≥σ 0 ( CO1 |σ ∪ CO2 |σ ) and int(·) denotes set interior. According to Stratified Morse theory [7], the puncture
point (q1 , σ 1 ) is a critical point of π(q, σ) having thefollowing properties [17]. All critical points of π(q, σ) in
the free c-space occur at frictionless two-finger equilibrium
grasps of the object B (without gravity). And all puncture
points are saddle points of π(q, σ) in the free c-space.
The set of all two-finger contacts with the object B forms
a 2-D submanifold of IR4 , parametrized by contact space U.
Allen [2] established that the behavior of π(q, σ) in the freec-space, IR4−int(CO), is in one-to-one correspondence withthe behavior of the inter-finger distance function, σ(s1 , s2 ),in U. In particular, the saddle points of π(q, σ) (which
occur at two-finger frictionless equilibrium grasps) appear
as saddle points of σ(s1 , s2 ) in U. Hence, the computationof the critical drop-off finger opening reduces to a search
for the lowest saddle point of σ(s1 , s2 ) in U that retracts
under decreasing σ to the basket-grasp point (s01 , s02 ).
The saddle points of σ(s1 , s2 ) are of two possible
types [2]. Equilibrium grasps where one finger contacts
a convex vertex while the other finger contacts an opposingedge of B (Figure 11(c)); or equilibrium grasps that involvea convex vertex and an opposing concave vertex of B. All
saddle points are thus located on the bounding lines of
the contact-space rectangle (at most one on each bounding
line segment). Let Rjk be the contact-space rectangle that
contains the basket-grasp point (s01 , s02 ). First determine thetwo endpoints of the σ 0 -contour segment of σ(s1 , s2 ) on
bounding linesof Rjk . The search can now be thought ofas gradually filling contour-segments of σ(s1 , s2 ) inside
Rjk with water, while searching for two-finger equilibrium
grasps on the rectangle’s bounding lines. As the water rises
between successive contour segments according to their σ(b)Fig. 12: (a) The search for the critical drop-off event starts
with the basket-grasp contour segment. (b) The search ends
at the lowest saddle point of σ(s1 , s2 ), located on the left
bounding line of a contact space rectangle R13 .
values, they will eventually reach the lowest saddle pointof σ(s1 , s2 ) that retracts under decreasing σ to the basketgrasp point (s01 , s02 ). This search takes O(n2 log n) steps,
where n is the number of edges of the object B [2].
Example: Consider the two-finger basket grasp of theconvex object B depicted earlier in Figure 4 (green dot
no. 1). The water filling analogy starts with the σ 0 -contour
segment of the basket grasp in the rectangle R13 (Figure 12(a)). As the water rises with increasing σ, they reachthe frictionless two-finger equilibrium grasp located on theleft bounding line of R13 . It is a saddle point of σ(s1 , s2 )that marks the critical drop-off finger opening of σ 1 .
◦VI. C ONCLUSION
The paper established that the basket grasps can be analyzed and computed in a low-dimensional contact space.
First, contact space can be used to compute the layout ofall possible basket grasps of an object waiting to be lifted
at a given position. Second, the depth (and hence security)of a candidate basket grasp can be computed along thebasket-grasp contour in contact space. The main result of
the paper, the basket grasp theorem, states that the depthof a basket grasp is determined by the relative height ofthe object’s center of mass at specific single and twofinger equilibrium stances associated with the basket grasp.
These stances can be readily found and sorted in contact
space as demonstrated with detailed examples. Third, thecritical drop-off event that allows release of the object at its
intended destination is associated with a frictionless twofinger equilibrium grasp of the object (without gravity), thatcan be computed as a simple search in contact space.
In future research, it is natural to seek quality measuresthat will guide the selection of optimal basket grasps. It
seems that the basket-grasp depth (which affects quality)
increases as the fingers open apart. However, the allowed
finger opening must satisfy a user-specified margin of
security away from the critical drop-off event. Finally, ourfuture work will focus on synthesizing basket grasps using
3-D robot hands. It is a natural extension of this paper,with possible follow-up on the role of the hand’s palmin performing safe transition from basket grasps to fully
secure immobilizing grasps of 3-D objects.
Acknowledgement: This work was supported by grant no.
1253/14 of the Israel Science Foundation.

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