IEEE Robotics & Automation Magazine - September 2018 - 88

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5. The results of the tracking process with the (a)-(d) input images (four different objects), and (e)-(h) the corresponding
registered reprojected mesh.

them with the internal forces computed using the physical model of L ag rang ian mechanical eq uations:
..
.
Mx + Cx + K lx + f0 = fext, where x ! R n contains the positions of the n vertices, M ! R n # n, C ! R n # n, and K l ! R n # n
are the mass, damping, and stiffness matrices, fext ! R n is the
external forces vector, and f0 ! R n is an offset on the internal forces due to rotational effects. A Euler implicit integration scheme and a conjugate gradient method are used to
solve the system with respect to x. The elastic forces fext are
based on geometrical correspondences between the point
cloud and the mesh [17].
To validate the method, some results have been obtained
with various soft objects that include deformations due to
bending and stretching or compression actions and fractures obtained under challenging conditions, like occlusions
or fast motions (see Figure 5). Some preliminary experiments integrating the method into a robotic manipulation
task with a silicon pizza dough have also been carried out
[see Figure 5(g) and the "Friction-Induced Manipulation
Primitive" section].
Nonprehensile Motion Planning and Control
Four manipulation primitives have been considered so far
within the RoDyMan project: nonprehensile rolling, sliding,
tossing, and batting/juggling. Sliding and tossing take into
account deformable objects, while rolling and batting/juggling only consider rigid objects. It is difficult to find relevant
applications involving deformable objects in pure rolling and
juggling tasks. In the following sections, the controller and/or
motion planner designed for the aforementioned primitives
are described.
Nonprehensile Rolling
An actuated manipulator of a given shape, referred to as the
hand, manipulates an object purely through rotations and
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september 2018

without grasping or caging it. Therefore, the object can only
roll on the shape of the hand. Case studies like the manipulation of a ball on a plate, a ball on a beam, and so on are thoroughly examined in the literature [27]. In this article, only
planar rolling is described.
Because highly geared harmonic drives are present within
the RoDyMan mechatronic platform, we use the acceleration
of the hand ip h ! R as input a h ! R for the system. The
dynamic model for a nonprehensile planar rolling manipulation system, in which the hand can only rotate around its center of mass, is described by
ip h

= a h,

(1a)

and
sp h = - b -221 ^b 12 a h + c 21 io h + c 22 so h + g 2h,

(1b)

where s h ! R is the contact position of the object on the
hand, whose shape is parametrized through arc length,
b 12 ! R and b 22 ! R are  entries of the inertia matrix
B ! R 2 # 2, while c 21 ! R and c 22 ! R are entries of the
(2 # 2) Coriolis matrix of the mechanical system, and g 2 ! R
is the second element of the (2 # 1) gravity-force vector.
Detailed expressions of each term are provided in [18], which
also notes that, if the Coriolis terms are zero, a nonprehensile
planar rolling manipulation system is differentially flat with
the output (b 12 b 22) i h + s h . Among the class of systems for
which the aforementioned assumption is true, we recall the
ball-on-disk (BoD) system, which is mathematically equivalent to the disk-on-disk (DoD) system in the transversal
plane [11].
The BoD consists of a ball rolling on top of a disk, as
shown in Figure 6. The disk is the actuated hand, and the ball
rolling on the hand is the object. The control problem for the
BoD is to balance the object in the upright position while
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