![](bg6.png)
purely by noise in the observations. Second, the solu-
tion to the problem implies stacking objects of differ-
ent size with some restrictions, i.e, a big object cannot
be stacked onto a small one.
Thanks to the proposed space state and action
space, the codification of the different positions and
sizes of the glasses lets the planner generate a pol-
icy able to deal with the container-contents problem
stacking the glasses in the best way, maximizing the
free space on the tray. The solution we propose to
be able to handle the unknown number of objects is
based on focus the attention on only one of the ob-
jects present in the each one of the different regions
on the table, and it has turned out to be particularly
adequate and powerful.
5.1 Future Work
Here we have considered that the objects can be al-
ready stacked on the table, and that the the robot can
perform stacking actions on the table. However this
condition has hardly raised in our experiments. One
challenge we are facing now is to incentive object
stacking actions on the table, before putting them in
the tray. A new variable is needed here to balance
the cost between two transportation actions and one
stacking plus a transportation operation. In the com-
putation of this cost the trajectory from the table to the
tray in the transportation action becomes important,
as we want to stack closer objects, or more important,
stack objects that are in the transportation trajectory
to the tray. Our formulation is general in the number
defined zones on the table, so we face this new condi-
tion as a natural extent of the presented algorithm.
We have considered to add an additional action
when observations are not enough to decide for one
action, i.e. in order to gather information about
the number of glasses stacked on each position. A
promising option is an asking action to an opera-
tor where the answer could modify the agent’s belief
state (Armstrong-Crews and Veloso, 2007).
ACKNOWLEDGEMENTS
This work has been partially supported by the Span-
ish Ministry of Science and Innovation under project
DPI2008-06022 and the Generalitat de Catalunya un-
der the consolidated Robotics Group. G. Aleny
`
a was
supported by the CSIC under a JAE-Doc Fellowship.
REFERENCES
Armstrong-Crews, N. and Veloso, M. (2007). Oracular par-
tially observable markov decision processes: A very
special case. In Proc. IEEE Int. Conf. Robot. Au-
tomat., Rome., pages 2477–2482.
Glashan, R., Hsiao, K., Kaelbling, L. P., and Lozano-P
´
erez,
T. (2007). Grasping POMDPs: Theory and experi-
ments. In RSS Workshop: manip. for human env.
Hsiao, K., Kaelbling, L. P., and Lozano-p’erez, T. (2007).
Grasping POMDPs. In Proc. IEEE Int. Conf. Robot.
Automat., Rome., pages 4685–4692.
Hsu, D., Lee, W., and Rong, N. (2007). What makes some
POMDP problems easy to approximate? In Advances
in Neural Information Processing Systems (NIPS).
Hsu, D., Lee, W., and Rong, N. (2008). A point-based
pomdp planner for target tracking. In Proc. IEEE Int.
Conf. Robot. Automat., Pasadena.
Kahlmann, T., Remondino, F., and Ingensand, H. (2006).
Calibration for increased accuracy of the range imag-
ing camera Swissranger
T M
. In ISPRS Commission V
Symposium, pp. 136–141, Dresden.
Kemp, C., Edsinger, A., and Torres-Jara, E. (2007). Chal-
lenges for robot manipulation in human environments.
IEEE Robot. Automat. Mag., 14(1):20–2.
Kolb, A., Barth, E., and Koch, R. (2008). ToF-sensors:
New dimensions for realism and interactivity. In Proc.
IEEE CVPR Workshops, vol. 1-3, pp. 1518–1523.
Kuehnle, J. U., Xue, Z., Stotz, M., Zoellner, J. M., Verl,
A., and Dillmann, R. (2008). Grasping in depth maps
of time-of-flight cameras. In Proc. Int. Workshop
Robotic Sensors Environments, pp. 132–137.
LaValle, S.M. (2004). Planning Algorithms. Cambridge UP
Lindner, M. and Kolb, A. (2006). Lateral and depth calibra-
tion of PMD-distance Sensors. In Proc. 2nd Int. Sym.
Visual Computing, vol. 4292, pp. 524–533.
Saxena, A., Driemeyer, J., and Ng, A. Y. (2008). Robotic
grasping of novel objects using vision. Int. J. Robot.
Res., 27:157–173.
Thrun, S., Burgard, W., and Fox, D. (2005). Probabilistic
Robotics. MIT Press, Cambridge.
Trilla, L. (2009). Planificaci
´
o de moviments en entorns amb
incertesa per a manipulaci
´
o d’objectes. Master’s the-
sis, Universitat Polit
`
ecnica de Catalunya.
PLANNING STACKING OPERATIONS WITH AN UNKNOWN NUMBER OF OBJECTS
353