IEEE Robotics & Automation Magazine - September 2017 - 92

θx
xg

y

yg

og

Pin-1

θy

c1

zg

c2 Pin-2
x

o

Gripper

c3

Gripper

Pin-3
(a)
Rod

Rod

(a)

(b)

Figure 12. (a) The gripper as it moves to the caging configuration
and (b) the object after it has been firmly grasped.

Definition

xg yg og zg

The gripper reference frame

og

The center of the gripper's base plane

Input: The vertices of the polyhedron
Step 1: Set i x = i )x , i y = i )y , and project the object to the
xoy plane.
Step 2: Construct the subfunction g1 using

o

The projection of og on the support plane

xoy

The support plane

X

The state of the object

(x, y, z)

The position of the center of the object on
the xg yg og zg frame

(ix, iy, iz)

The rotational angles of the polyhedron
around the xg-, yg-, and zg axis, respectively

ci

The point of intersection of the edge with
the line passing through og and pin i

|c i og|

The distance between ci and og

g1

The subattractive function corresponding
to (x, y, iz)

g2

The subattractive function corresponding
to (ix, iy)

d 1i

The minimum of subattractive function g1
under i x = i ix, i y = i iy

g 1 = max ^| c 1 o g | , | c 2 o g | , | c 3 o g | h
x, y, i z ! U 1

Step 3: Find the attractive region in subspace (x, y, i z, g 1) ,
and compute the minimum of the attractive function using

^ | c 1 og | , | c 2 og |, | c 3 og |h .
g 1 = max
x, y , i
z

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*

SEPTEMBER 2017

x, y, i z

Step 5: Construct the attractive subfunction g2 using
g 2 = {d 1i }
Step 6: Find the attractive region in subspace (i x, i y, g 2)
using the algorithm presented in Table 1.
Output: A set of form-closure grasping cages

(11)

Note that the definition of g 1 is similar to that in (3). Note
also that g1 is determined by the projection of the object. In
subspace (i x, i y, g 2), we define
*

d 1i = min ^ g1 (x, y, i z) h
Step 4: Change the orientation (i x, i y) ! X 2 of the object,
then return to Step 2.

problem by converting higher-dimensional problems into a
set of lower-dimensional ones. We present a method to divide
the high-dimensional configuration space described by an
attractive function into two subfunctions.
In subspace (x, y, i z, g 1), we define

92

Figure 13. (a) The coordinate frame of the three-pin gripper,
where the y g axis passes through the gripper's center o g and
pin-3, and the z g axis, is perpendicular to the gripper's palm.
(b) The projection of the object and the pins on the x g o g y g plane,
where the x axis is parallel to the x g axis, the y axis is parallel to
the y g axis, and o is the projection of o g .

Algorithm 2

Table 2. Symbols used in the discussion of
caging 3-D objects using a three-pin gripper.
Symbols

(b)

g 2 = {d 1i }

(i x , i y ) ! X 2

,

(12)

where
^ g1 (x, y, i z)h
d 1i = min
x, y , i
z

i

i

i x = i x, i y = i y

.

(13)

Detection of a Cage in Low-Dimensional Subspaces
As described in (10)-(12), we first establish the attractive
function in the subspace (x, y, i z, g 1) by setting (i x = i x),
)
iy = i y). We then construct the attractive function in subspace (i x, i y, g 2), using the minimum of the attractive function in subspace (x, y, i z, g 1) . Algorithm 2 shows the
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