IEEE Robotics & Automation Magazine - June 2012 - 48

GCP

Approach
Direction
α
pgrasp

Figure 5. The computation of the grasping pose pgrasp .

from all RRT nodes will result in a nonuniform distribution of approach directions.
Computing the Target Pose
For computing the target grasping pose pgrasp , a virtual
representation of the hand, including a preshape, a grasping point, and an approach direction, is used. Based on the
work of [23], the grasp center point (GCP) and the
approach direction are defined for the hand that should be
used for grasping. The preshape, the definition of GCP,
and the approach direction of the anthropomorphic hand
that is used in our experiments can be seen in Figure 5.
Based on this GCP definition, the target grasping pose
pgrasp 2 SE(3) is determined by searching for the point
pto 2 R3 on the object's surface that has the shortest distance to the GCP. pto defines the translational part of
pgrasp , and the rotational component is derived by rotating the coordinate system of the GCP by a so that the
approach direction points toward ptobj (see Figure 5).
Approaching the Object
In Algorithm 3, the generation of an approach movement
can be seen. Beginning with the selected RRT node
nApproach , intermediate RRT nodes are created by iteratively moving the end effector toward the grasping pose
pgrasp . This is done by computing the workspace difference
Dp between the current TCP pose (stored in the RRT node
n:p) and the target pgrasp . To ensure small steps in the

(a)

(b)

workspace, the method LimitCartesianStepSize is
applied to limit the resulting displacement. Later, the corresponding movement Dq in C-space is computed by using
the pseudoinverse Jacobian. The pseudoinverse J þ is
derived via singular value decomposition, although other
approaches, which may be more efficient, can be chosen as
well. If the resulting configuration n0 :q, which is computed
by applying the movement to the current configuration
n:q, is in collision or joint limits are violated, the last valid
RRT node n is returned. Otherwise, the corresponding TCP
position of n0 :q is computed, and the RRT is extended by
n0 . This is performed until the distance to the target pose
pgrasp falls below Threshold Cartesean.
Algorithm 3. Approach(RRT , nApproach , pgrasp )
1n

*

nApproach

2 repeat
3

Dp

pgrasp Á ðn:pÞÀ1

4

Dq

J þ ðn:qÞ Á LimitCartesianStepSizeðDp Þ

5

0

n :q

6

if ðCollisionðn0 :qÞjj!InJointLimitsðn0 :qÞÞ then

7

n:q þ Dq

return n

8

end

9

n0 :p

ForwardKinematicsðn0 :qÞ

10

RRT:AddNodeðn0 Þ

11

n

n0

12 until ðLengthðDp Þ < ThresholdCartesean Þ
13 return n

Determining the Grasp Quality
To evaluate the quality of the resulting pose ngrasp , the
fingers are closed and the set Cg of resulting contacts are
determined. Closing the hand is performed by iteratively
moving the finger joints with small steps until (self-)collisions are detected. Based on this contact information (consisting of positions and normals on the object's surface), a

(c)

(d)

Figure 6. The Bimanual Grasp-RRT planner is used to search a collision-free grasping trajectory for 14 DoF of both arms of
ARMAR-III. (a) The initial setup. (b) A workspace visualization of the RRT. (c) The generated grasping configuration. (d) The friction
cones are visualized at the contacts.

48

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

JUNE 2012
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2012
