MQM Paper: [Gupta/Struss 95b]

Vineet Gupta, Peter Struss

Representing a Paper Path in a Photo Copier.
**(extended abstract of [Gupta/Saraswat/Struss
95])**

In: Leonie Dreschler-Fischer, Simone Pribbenow (Eds.): KI-95 Activities: Workshop, Posters, Demos. Gesellschaft für Informatik GI, pp. 119-120, 1995.

**Abstract**

Our work addresses a problem of spatial representation
arising in a real domain: modeling paper transportation in a photo copier.
Actually, it is more a case study on how to *avoid *sophisticated
spatial reasoning, or, at least, to simplify it significantly.

In principle, a sheet transported in a copier corresponds
to a 2-dimensional surface being deformed and moving around in a 3-dimensional
space in a continuous way (unless it tears). A model based on this spatial
representation would be very complex. It turns out that a simpler representation
suffices to solve many interesting problems in the domain. It is obtained
by considering a *cross-section* of the actual 2-dimensional paper
path, as for instance depicted in Fig. 1, which is a set of contiguous
curve sections. We ignore the curvature of these sections and consider
distances only along the trajectory. This gives us locally a representation
by an interval of the real number line-except at branch points, which we
have to take into account as they occur in our domain as illustrated by
our introductory example. Indexing the maximal linear sections of the paperpath
by integers *i * **N**,
and denoting the length of the section by *length***_{i}**,
we represent a section by a pair

Thus the paperpath is represented by

with an appropriate topology on PATH that treats the branching
points as least upper bounds or greatest lower bound resp. of the section
intervals. Thus each branching point *b** _{k}*
determines a set of (potentially) connected sections, i.e. all sections
that intersect with arbitrarily small open neighborhoods of

To describe the motion of points along PATH, we need a
generalized *derivative* of pathpoints that takes transitions at branching
points into account: for *p*
PATH,

where *v* is the real-valued velocity, and *trans*
= (*i*, *j*) specifies a transition from one section to another
one. Of course, this is interesting only at branching points, as for any
point on section *i*, *trans *= (*i*, *i*). *trans*
has to be continuous at branching points in the sense that it can change
from (*i*, *i*) to (*j*, *j*) only by going through
(*i*,* j*).

With this representation of the paperpath the problem of modeling the motion of a sheet is basically turned into a 1-dimensional locally linear problem which can be handled by real-valued variables, except for branching points, where a decision has to be made about the continuation of motion.

The essential characterization of the motion of a sheet
is obtained from the motions of its leading and trailing edge. Gates, which
control the branching, are located at branching points and characterized
by a (possibly dynamically changing) set of physically possible connections,
a subset of the connections determining the topology at the branch. If
the leading edge reaches the gate, *trans* in its derivative is restricted
to the (currently) possible connections of the gate. Also the occurrence
of buckling can be determined, namely if the length of the sheet segment,
i.e. the integral of the difference between the speeds at its left-hand
and right-hand bounds, becomes greater than the length of the paper path
interval covered by this segment.

Libraries of primitive, local model fragments that can be composed to form a model of a particular scenario are described in detail in [1], [2].

**References**

1. [Gupta/Struss 95] Gupta, V., Struss, P.: Modeling a Copier Paper Path: A Case Study in Modeling Transportation Processes. In: QR-95, Working Papers of the Ninth International Workshop on Qualitative Reasoning, Amsterdam, 1995. (Also: Tech. Report TR-95-019, International Computer Science Institute, Berkeley, 1995)

2 [Gupta/Saraswat/Struss 95] Gupta, V., Saraswat, V., Struss, P.: A Model of a Photo Copier Paper Path. In: Second Engineering Problems for Qualitative Reasoning Workshop, IJCAI-95, Montreal, Canada, 1995.

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