Biomedical Engineering Reference
In-Depth Information
barrier that is two or more blocks in height cannot
be scaled, a block deposits itself just in front of
that barrier, creating a step that other blocks can
climb. Other steps will be deposited using this
same rule, and in this way a new “layer” of the
staircase is laid down, beginning with the top step
and propagating down the emerging staircase, as
shown on the right side of Figure 10. Irrespective
of its height, the assembly of a staircase can thus
be accomplished in a very parsimonious fashion,
with a single rule.
It should be noted that the staircase will not be
built at a wall location other than what is shown
in Figure 10, due to the requirement that the three
vertical grid locations to the “upper left” of the goal
location must be empty. However, it is still pos-
sible that blocks will begin to assemble staircases
adjacent to the internal columns, rather than the
wall. This illustrates a limitation of a stigmergic
approach that relies strictly on geometric pattern
matching. We developed an extension of earlier
stigmergic models (Bonabeau, 1999, Theraulaz,
1995) to arbitrary structures, by allowing a block to
have a limited memory
M
, and augmenting it with
three integer variables
reaches a goal location that was matched by this
rule, and becomes stationary, it sets the value of
the
substructure type
variable
[
M
to
wall_ stair
.
This allows blocks to determine the part of the
structure to which another block belongs. As
blocks come to rest at different locations in the
environment, their memory variables become
markers for other blocks, somewhat similar to the
pheromone markers employed in other systems.
In this way a form of stigmergy is achieved.
The presence of physical constraints such as
gravity and block impenetrability imposes higher-
level ordering constraints on the self-assembly
process; for example, the columns of the building
must be assembled prior to the walls. Further-
more, the assembly and disassembly of staircases
has to be appropriately sequenced. To address
this, we once again integrate low-level behaviors
(namely, stigmergic pattern matching and move-
ment) with a higher level coordination scheme.
Specifically, these behaviors are influenced by
a block's
mod
e. Mode changes generally occur
in response to the completion of certain parts of
the structure. These locally detected completion
events are communicated via
message
variables:
when a block sees that a message variable of a
nearby block is set to some particular value, it
may choose to set its own message variable to
that value. Such modifications to
M
are governed
by v
ariable change rules
, which are triggered by
observations of particular variable values within
the memory of
b
itself, as well as the memories
of nearby blocks. Implicitly, these rules define a
finite state machine, somewhat akin to those dis-
cussed in previous sections. Each state is a mode
value that corresponds to the specific subset of
stigmergic rules that is applied when attempting
to match a pattern, as well as a particular form
of movement dynamics.
The methodology above has been successfully
applied in simulations towards the self-assembly of
a number of specific target structures. Full details
are presented at Grushin, 2006. While different
structures required the design of distinct sets
M
, for construction in three dimensions. A seed
block is assigned arbitrarily chosen constants
x
,
y
and
z
as values. As another block
b
deposits
itself adjacent to the emerging structure, it reads
the memory of a nearby stationary block
b
' and
sets its own variables to the correct relative posi-
tion within the overall structure; for example, if
b
places itself directly to the left of the seed block,
it can set
[
z
]
M
[
x
]
,
M
[
y
]
and
M
=
. A
coordinate system thus emerges in this fashion,
through simple, purely local interactions (Grushin,
2006, Werfel, 2006). The columns' positions
relative to the seed are such that their blocks have
values of
M
[
x
]
=
x
0
−
1
,
=
and
M
[
y
]
y
[
]
z
0
0
0
y
. We thus
prevent the placement of staircase blocks next to
the columns by imposing the
memory condition
1
that are either
y
or
M
[
y
]
1
yM
on blocks adjacent to the goal loca-
tion, within the rule's antecedent (Figure 10).
The rule's consequent states that when a block
[
]
<
y
0
−
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