IEEE Robotics & Automation Magazine - September 2018 - 90

naturally assumes a consistency that is thicker at the ends and
thinner in the middle, and 3) as the spinning dough freely falls, the outside of the dough dries, making it crunchy on
the outside but soft in the middle. The pizza chef is trained to
perform a streamlined hand motion to toss and catch the
dough, and a similar feat is desired for the RoDyMan robot.
The combined model of the dough grasped with robotic
fingers through unilateral constraints and the kinematics and
dynamics of the robot manipulator is derived in [4], on which
a control law achieving the desired tossing motion can be
designed. Furthermore, with a perfect knowledge of the
motion of the dough, optimal trajectories can be generated in
the special Euclidean group SE(3) for the catching phase. The
optimal trajectory generation is repeated as new sensor information is available. The trajectories are generated in such a
way that the initial position, velocity, and acceleration and
final velocity and acceleration are matched, and therefore it is
continuously differentiable at least three times. An optimal
trajectory, whose initial and final accelerations are prescribed,
has to satisfy a sixth-order boundary value problem (BVP).
Such BVP is generated by using the necessary conditions for a
path to minimize a convex combination of the jerk functional
and the acceleration functional. While minimizing the jerk
functional reduces the vibrations in the structure of the robotic manipulator, minimizing the acceleration functional reduces the total amount of energy expended during the catching
motion. The only case that we consider is the one where the
final position is left free and is part of the minimization problem. More details can be found in [4].
Experimental validations are in progress. Nevertheless,
preliminary results show that such tasks require high-peak
currents in the motors to toss the dough for more than 10 cm.
As anticipated, the motors of RoDyMan do not have such
skills. Analogies between tossing and walking gaits can be
found within mathematical models. Similar to robotic legs,
hydraulic actuators seem to have better perfomance, and the
same might hold true for tossing tasks. The stretching-thedough task can also be performed in alternative manners,
which will be explored in the future.
Batting/Juggling Skills
A very challenging primitive from the control viewpoint is the
one involving impacts. Inside, batting an object (a ball) is
intercepted by the end effector (a paddle) without grasping it,
and the object is thrown toward a precise goal. This motion
primitive is typically used by athletes, i.e., baseball or tabletennis players. Jugglers use this primitive when their hands
control the continuous motion of one or more objects through
intermittent contacts. These dynamic motions require high
velocity and precision. The design of planning and control
methods to deal with this would strongly enhance capabilities
of robot manipulators, extending the workspace size and
enhancing dexterity.
The batting task dynamic is typically defined as hybrid
because it consists of the continuous aerodynamics of the
manipulated ball (a differential equation) and the discontinu90

IEEE ROBOTICS & AUTOMATION MAGAZINE

september 2018

ous reset of the velocity at impact time (two difference equations), given by
pp b = - g - k d || po b || po b + k l S (~ b) po b,

(3a)

po +b = v p + Ca (po -b - v p) + Cb ~ -b ,

(3b)

and
+

= C c (po -b - v p) + Cd ~ -b ,

(3c)

where p b ! R 3 and ~ b ! R 3 are the position and the spin
of the ball, respectively; v p ! R 3 is the paddle velocity;
C j (R p) ! R 3 # 3 and j = " a, b, c, d , are transformation matrices dependent on the rebound parameters and on the orientation of the paddle R p ! SO (3) at the impact time;
k d (po b, ~ b) ! R and k l (po b, ~ b) ! R are, respectively, drag
and lift parameters; g ! R 3 is the gravity acceleration vector;
$ denotes the Euclidean norm; S ($) ! R 3 # 3 is the skewsymmetric matrix; and superscripts - and + represent the
state before and after the impact, respectively. The matrices
Cj can be detailed on the basis of the addressed rebound (ball
impacting the table and/or a rubber paddle). Their expression
may become complicated in nontrivial situations (like nonspherical objects), leading to the use of some (strong) assumptions and model reductions.
Five different phases have been considered to solve the batting problem by using the RoDyMan platform simulator. First,
a vision system is assumed to measure the trajectory of the ball.
By assigning the impact time, the prediction of the impact position and preimpact velocity of the ball are obtained numerically
by solving the aerodynamic model (3a). Then, the postimpact
velocity of the ball, such that it goes toward a desired goal in a
predefined time, is computed solving (3a) backward in time.
The analytic solution of the discontinuous part of the ball paddle model, given by (3b) and (3c), determines the configuration
of the paddle to generate such velocity of the ball. Thereafter,
the motion of the paddle to reach the desired configuration is a
result of the minimization of its linear and angular acceleration
with a coordinate-free approach, assuming that the path is generated on an arbitrary Riemannian manifold, similar to the
tossing primitive. Finally, the motion of the RoDyMan joints is
derived from a classical second-order closed-loop kinematic
inversion. More details can be found in [6].
A similar algorithm can be used to accomplish different
juggling patterns. The lesson learned is that these techniques
may also be applied to other dynamic tasks that share the
same hybrid nature with impacting manipulations, such as
walking or running tasks.
Final Discussion
Despite the progress made with the RoDyMan project thus
far, including real-time tracking of deformable objects
employed in tossing and sliding tasks, several problems
remain. The mechatronic platform should be revised to cope
with issues caused by the high velocity of some nonprehensile
manipulation primitives. In general, experiments involving
nonprehensile actions are not easy to solve because of the

IEEE Robotics & Automation Magazine - September 2018

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