Extended Stigmergy In Collective Construction

    Building science n3 question papers. Structures of square building blocks in two dimensions according to a1, dept. of computer science, manchester univ., 1999. 3. y. terada and 127139. 12. p. gren, e. fiorelli, and n.e. leonard, formations with a mission: nd.edu/~mtns/papers/4615_3.pdf. related work this question for a site with given coordi-. sites.tufts.edu.

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www.computer.org/intelligent
Extended Stigmergy
in Collective Construction
Justin Werfel, Massachusetts Institute of Technology
Radhika Nagpal, Harvard University
Vol. 21, No. 2
March/April 2006
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20 1541-1672/06/$20.00 © 2006 IEEE IEEE INTELLIGENT SYSTEMS
Published by the IEEE Computer Society
Self-Managing Systems
Extended Stigmergy in
Collective Construction
Justin Werfel, Massachusetts Institute of Technology
Radhika Nagpal, Harvard University
T
ermite mounds can be towering, complex structures on the scale of several meters,
with complicated functional architectures that include features such as ventila-
tion systems and temperature regulation. The architects are vast numbers of simple insects
on the scale of millimeters, working with no centralized control or preplanning. How
swarms of social insects build the structures they do
is a fascinating topic not fully understood. Engi-
neering offers a complementary problem: How could
you program an artificial swarm to build a particular
structure?
Construction is the major human production activ-
ity from which automation has largely been absent.
Increasing automation could improve efficiency and
reduce accident rates.
1
Swarm systems—the most
impressive examples of construction in nature—are
especially suited as an approach for artificial con-
struction systems (see the “Related Work in Collec-
tive Construction” sidebar). An automated approach
could be particularly useful in extraterrestrial or
underwater environments, where human presence is
difficult or dangerous and traditional construction
methods are problematic.
One way social insects coordinate their behavior
is by using environmental modifications as cues:
actions make changes to the world, and those
changes influence further actions. Termites, for
instance, might be more likely to deposit building
material where deposits already exist, leading to
larger aggregations. This phenomenon is termed stig-
mergy.
2
A limitation of stigmergy is that a given set
of behavioral rules will typically generate qualita-
tively similar but not identical structures. Rules that
always produce one particular structure do exist;
however, finding a rule that produces the desired
structure can be difficult. We’re interested in this
inverse problem of construction: start with a desired
high-level design, and find a set of low-level rules
that will produce it.
Elsewhere we’ve described a scheme by which a
swarm of robots can automatically assemble solid
structures of square building blocks in two dimensions
according to a high-level user-specified design.
3
There
we discuss the use of extended stigmergy: augmenting
the basic notion of stigmergy by increasing the capa-
bilities of environmental elements. In this construc-
tion setting, the elements are the building blocks, the
basic information they carry is the simple fact of their
presence at a location, and extensions include cases
where they can store additional information, perform
computations, and/or communicate with physically
attached neighbors. Benefits can include increased
robustness and faster completion of a desired struc-
ture. Here we review three variants of that construc-
tion scheme that use extended stigmergy to different
degrees. We focus on analysis of the algorithms and
their comparative performance, mainly theoretical but
also through simulation experiments.
Framework
Figure 1 shows the framework in which we con-
sider the construction problem. Building blocks are
square and can be attached to each other on all four
sides. A block-sized marker indicates where con-
struction should start, acting as a seed around which
the structure grows. Robots in unspecified numbers
can fetch blocks from elsewhere in the workspace
and bring them to the growing structure, traveling
along its perimeter for some distance before attach-
ing their blocks. We assume that unattached blocks
are far enough away from the structure so as not to
interfere with the construction process, and that
A swarm of
autonomous robots
can be programmed
to build arbitrary
structures. Extended
stigmergy can improve
construction time
and increase the
opportunity to exploit
parallelism.
robots can find blocks and return with them
to the structure in progress. Other work con-
siders issues of clearing the workspace to
make room for construction
4,5
and of robots
finding the blocks and structure.
3,4
The heart of the problem—and the aspect
of construction we focus on here—is how
and when a robot decides to attach a block.
It’s necessary that the structure ultimately
takes on the desired shape, and that dead-end
states are avoided—that is, when sites meant
to have blocks can’t be physically reached.
Social insects build with materials such as
mud and wax—soft materials that come in
variable-sized quantities. Human construc-
tion projects typically use rigid materials and
involve regular shapes with right angles.
That’s both a benefit and a limitation.
An advantage of building with square
blocks is that, assembled, they form a grid
with an implicit coordinate system that the
robots can use for position reference, as we
describe later. When we specify desired
shapes, we’ll do so with respect to this struc-
ture coordinate system.
Conversely, rigid materials constrain where
and how they can be attached. For instance, it
might be difficult to fit a block into a space
MARCH/APRIL 2006 www.computer.org/intelligent 21
Classic work in insect-inspired construction
1
considers the for-
ward problem of what kinds of structures a given rule set will
produce. We focus on the inverse problem, starting with a given
structure and finding rule sets to produce it. Several studies have
considered other issues related to swarm construction, such as
interrobot communication, debris cleanup, minimalism,
2
and
hardware design.
3
Programmed self-assembly is the problem of designing tiles
with static or dynamic binding properties such that they come
together to form a given layout when mixed,
4–7
and is similar
to our communicating-blocks case. These studies often don’t
consider constraints on tile movement, so they can be subject
to crystalline defects in the patterns they form.
Two other related problems are self-reconfigurable robots
8–10
and formation control.
11,12
In both, a collection of mobile ele-
ments must automatically rearrange itself into some desired
formation, subject to some set of movement constraints.
For reconfigurable robots, these constraints typically involve
all elements being attached in one cluster at all times and can
allow maneuvers such as blocks moving down narrow passages.
Formation-control constraints tend to take the opposite
form—that of minimum separation distances. Some approaches
assume all-to-all communication or a global coordinate system,
which can be unrealistic for large numbers of mobile robots.
References
1. G. Théraulaz and E. Bonabeau, “Coordination in Distributed Build-
ing,” Science, vol. 269, no. 5224, Aug. 1995, pp. 686–688.
2. C. Melhuish, J. Welsby, and C. Edwards, “Using Templates for Defen-
sive Wall Building with Autonomous Mobile Ant-like Robots,” Proc.
Towards Intelligent Mobile Robots, tech. report, Report UMCS-99-3-
1, Dept. of Computer Science, Manchester Univ., 1999.
3. Y. Terada and S. Murata, “Automatic Assembly System for a Large-
Scale Modular Structure: Hardware Design of Module and Assem-
bler Robot,” Proc. 2004 IEEE/RSJ Int’l Conf. Intelligent Robots and
Systems, IEEE Press, 2004, pp. 2349–2355.
4. C. Jones and M. Mataric
‘
, “From Local to Global Behavior in Intelligent
Self-Assembly,” Proc. 2003 IEEE Int’l Conf. Robotics and Automation,
IEEE Press, 2003, pp. 721–726.
5. E. Klavins, R. Ghrist, and D. Lipsky, “A Grammatical Approach to
Self-Organizing Robotic Systems,” to be published in IEEE Trans.
Automatic Control, 2006.
6. P.W. Rothemund, N. Papadakis, and E. Winfree, “Algorithmic Self-
Assembly of DNA Sierpinski Triangles,” PLoS Biology, vol. 2, no. 12,
2004, p. 424.
7. Y. Guo et al., “Designing Self-Assembly for 2-Dimensional Building
Blocks,” ESOA Workshop, 2003, www.smartspaces.csiro.au/gremlab/
grempub.htm.
8. K. Kotay and D. Rus, “Generic Distributed Assembly and Repair
Algorithms for Self-Reconfiguring Robots,” Proc. 2004 IEEE/RSJ Int’l
Conf. Intelligent Robots and Systems, IEEE Press, 2004, pp. 2362–2369.
9. K. Støy and R. Nagpal, “Self-Reconfiguration Using Directed Growth,”
Proc. Distributed Autonomous Robotic Systems, Springer, 2004.
10. M. Vona and D. Rus, “Crystalline Robots: Self-Reconfiguration with
Compressible Unit Modules,” Autonomous Robots, vol. 10, no. 1,
2001, pp. 107–124.
11. K. Sugihara and I. Suzuki, “Distributed Algorithms for Formation of
Geometric Patterns with Many Mobile Robots,” J. Robotic Systems,
vol. 13, no. 3, 1996, pp. 127–139.
12. P. Ögren, E. Fiorelli, and N.E. Leonard, “Formations with a Mission:
Stable Coordination of Vehicle Group Maneuvers,” Proc. 15th Int'l
Symp. Mathematical Theory of Networks and Systems, 2002, www.
nd.edu/~mtns/papers/4615_3.pdf.
Related Work in Collective Construction
Figure 1. Construction problem framework. (a) Elements in the environment are
mobile robots, movable building blocks (red), and a marker indicating where
construction is to begin (orange). (b) Robots collect blocks and attach them to form a
specified structure.
(a) (b)
exactly one block wide between two other
blocks, such as site A in figure 2. We disallow
that situation entirely. This makes it easier for
robots to maneuver their blocks into position
and avoids more-complicated gaps, such as
gaps requiring a block to be moved down a
long tunnel or sites that are entirely closed off
and inaccessible (such as B and C). This
restriction’s corollary is the separation rule—
separated blocks must never be attached in the
same row (that is, with the same x- or y-coor-
dinate) if all of the sites between them are ulti-
mately meant to be occupied. Otherwise, as
robots add blocks, an unfillable one-block gap
will result (D in figure 2).
Here we only consider structures without
deliberate holes. Also, because our approach
depends on perimeter following, any desired
concavities in a structure must be wide
enough for two block-carrying robots to pass
in opposite directions while following the
perimeter (see figure 3). The extent to which
the latter consideration limits admissible
structures will depend on the implementation.
Basic approach
Our approach to the construction problem
involves several issues, which each of the
variants we’ll describe will need to address:
• Shared coordinate system. One basic prob-
lem is to determine whether a given site in
the grid should ultimately have a block
attached, or should be left empty. Our
approach is to specify the desired shape as
an occupancy matrix, of which every agent
has a copy. If all agents can agree on a
shared coordinate system, they can resolve
this question for a site with given coordi-
nates simply by consulting the occupancy
matrix. The trick, then, is agreeing on a
shared coordinate system—generally not
an easy task for mobile robots.
• Dead-end configurations. A partial order-
ing on block attachments is necessary to
prevent situations in which robots physi-
cally can’t reach sites meant to be occu-
pied. As outlined earlier, the separation
rule lets us avoid these situations. So, we
need a scheme to acquire the nonlocal
information regarding whether distant
blocks are attached in a given row.
• Multiple robots. Any approach intended to
work with a swarm must be able to handle
the vagaries of an unspecified number of
robots acting with no specified order or
centralized coordination. Robots will
break, timing of individual actions will be
unpredictable, and the system as a whole
must still self-organize to produce the
desired result. Our solutions to the first two
issues take this consideration into account.
Furthermore, our solutions use no explicit
communication between robots. Although
such communication can be helpful, we’d
rather not have to rely on it. This eliminates
dependence on teams: robots can build inde-
pendently, and a single robot could com-
plete the construction task if the rest of the
swarm becomes disabled. Not requiring
robots to communicate also avoids difficul-
ties associated with ad hoc mobile net-
works. Instead, robots coordinate their
actions only implicitly through the structure
in progress.
Physical realizability
Our approaches treat the world as a cellu-
lar grid—an approximation not always
appropriate for a noisy world in which posi-
tions are not discrete but continuous valued.
However, we’re careful to make this simpli-
fication in such a way that a physical real-
ization remains feasible (as we’ve demon-
strated in hardware
3
).
Robots’ precise positions are not impor-
tant; robots only need to know their position
when along the structure perimeter, and then
only as precisely as the nearest block coor-
dinate. The physical landmarks represented
by blocks and their boundaries let robots
establish and maintain position estimates to
that extent.
Block alignment on the grid must be fairly
precise. One way to achieve precise align-
ment is to equip blocks with self-aligning
connectors (active or passive), so a robot need
only get a block close to an attachment site,
and the connectors ensure fine adjustment.
These considerations let us safely make
the abstraction to the cellular-world model
we describe.
Algorithms
Our approach involves a partial ordering
on block attachment to prevent dead ends
while still allowing parallelism. Full order-
ing would require placing blocks in a partic-
ular sequence, where none could be attached
until the previous attachment was complete.
Putting restrictions instead on where blocks
may not be attached, rather than specifying
where they must be attached, potentially
allows many more blocks to be attached at
once, and also lends itself more readily to
decentralization.
Disallowing attachment at sites that are
supposed to be left empty according to the
occupancy matrix or that break the separa-
tion rule is sufficient for the reliable com-
pletion of any solid structure. Intuitively, the
separation rule prevents unfillable gaps and,
as a result, any other concavities that would
prevent robots from following the perimeter.
Together with the requirement that planned
concavities be wide enough to allow perime-
ter following, this ensures that no partial
structure will restrict access to sites meant to
be occupied. Furthermore, until the structure
is completed, there will always be some site
along the perimeter where a block can be
attached. It’s clear that this is also the least
restrictive possible partial ordering: any sites
restricted according to this scheme would
lead to a dead-end state if they were occu-
pied following some other scheme.
A centralized approach is then to monitor
the entire structure’s progress and to allow
or disallow attachment at sites according to
the occupancy matrix and the separation rule.
(This approach only interacts with robot
actions to the extent of allowing or disal-
lowing attachment at sites along the perime-
ter. Most robot actions can be autonomous
even in this “centralized” method.)
Self-Managing Systems
22 www.computer.org/intelligent IEEE INTELLIGENT SYSTEMS
A
BC
D
Figure 2. Physical constraints on block
placement. The desired structure here is
a solid 5  5 square of blocks. We assume
that robots can’t maneuver a block into a
constrained site like A. To avoid such
gaps, separated blocks must never be
attached in the same row (as in the
bottom row) if all sites between them are
meant to be occupied; otherwise, later
addition of blocks (light shading) will
result in an unfillable gap (D). Eliminating
such situations also eliminates more-
complicated problematic cases like those
at B and C.
Communicating blocks
A single centralized controller isn’t ideal
for swarm systems because it can require a
high communication load, it represents a
potential failure point that should be avoided,
and its performance might not scale with sys-
tem size.
Fortunately, we can implement the con-
troller in a decentralized way using processors
embedded in each block. Once attached to the
structure, blocks have a physical connection
to their immediate neighbors, forming the basis
for a data line. When blocks can store state and
communicate with neighbors in this way, the
structure can ensure the constraints on block
attachment. Blocks indicate to robots passing
along their perimeter whether they allow
attachment. Robots behave autonomously as
before: they fetch a block, take it to the struc-
ture, and follow the perimeter until they find a
valid attachment site.
We can outline the distributed algorithm
as follows.
3
Each block maintains the fol-
lowing state:
• a copy of the occupancy matrix,
• its location in the shared coordinate sys-
tem, and
• two bits of state per side associated with
whether blocks are attached in the row
adjoining that side.
Whether a new block can be attached at a
given site is determined by the blocks adja-
cent to that site, according to the occupancy
matrix and based on the states of the adjacent
blocks’ nearest sides. To prevent communi-
cation delays from causing multiple robots
to receive near-simultaneous permission to
attach separated blocks in a single row, the
blocks must lock a row from additional
attachment before giving permission to any
robot to attach. Once attached, a new block
gets its coordinates and the occupancy matrix
from its neighbors and sets its side states
according to theirs. The old blocks update
their side states based on the new attachment
(see figure 4).
Writeable blocks
Equipping each block with a processor
increases the expense of the building mate-
rials. In addition, blocks now have more that
can go wrong with them, so each block is
more likely to be a failure point. Using pas-
sive, noncommunicating blocks can amelio-
rate both problems. We then need a distrib-
uted algorithm that lets the mobile robots
accomplish what the communicating blocks
or centralized controller achieve in the other
approaches.
The ability for robots to write information
to, and read it from, the blocks would make
this task easier. At any time, robots have
access only to local information, but need
nonlocal information to accomplish a global
task. That nonlocal information can be stored
in robot state or in the environment—differ-
ent types of information lend themselves to
one or the other. Here, the location in the
shared coordinate system is global informa-
tion that’s more natural to associate with the
blocks than the robots. One way to associate
information with blocks is to equip robots
with radio-frequency-identification trans-
ceivers and put a passive, writeable RFID tag
on each block. Such tags are small (around 1
inch), are inexpensive (around US$1), and
require no internal power source.
The robots are now the agents that main-
tain a copy of the occupancy matrix. When a
robot reaches the structure, it reads an
attached block’s coordinates to establish its
own position. The remaining task is to estab-
lish that a site has no separated blocks in the
same row. If the robot finds such a site, and
the occupancy matrix shows that the site
should be occupied, it attaches the block and
marks it with its new coordinates for robots
to refer to in the future.
A way to ensure that robots don’t attach
separated blocks in the same row is to have
all robots follow the perimeter in the same
direction (say counterclockwise). To start a
new row of blocks, a robot first checks the
entire row to make sure no blocks are already
present. After traversing a full row along the
perimeter and finding it empty, the robot can
start the new row by attaching a block at the
end. If the robot finds a partial row of blocks,
it can add a block to extend it. If all robots
follow this scheme consistently, the actions
of multiple robots won’t conflict: because
robots will only start new rows at the coun-
terclockwise end, separated blocks can’t
wind up attached.
As we’ve shown elsewhere,
3
algorithm 1
(see figure 5) results in robots only attaching
blocks at the ends of new rows or continuing
existing rows. We now show that this behav-
ior is sufficient to generate any desired solid
structure whose concavities are wide enough
to allow perimeter following.
Proof. Define an inside corner as an empty
site with blocks at two adjacent sites, and an
MARCH/APRIL 2006 www.computer.org/intelligent 23
Figure 4. (a) Blocks designate neighbor-
ing sites as available (green) or unavail-
able (red) for attachment by perimeter-
following robots. (b) When a new block
is attached, it sets its side states based on
those of its neighbors, and they update
theirs based on the new attachment.
(b)(a)
Figure 3. Examples of structures not allowed using the approach described here:
(a) structures with deliberate holes and (b) structures with alleys too narrow for
perimeter following.
(a) (b)
Self-Managing Systems
24 www.computer.org/intelligent IEEE INTELLIGENT SYSTEMS
end-of-row site as one at which either a robot
is about to turn a corner to the left, or the occu-
pancy matrix specifies that the site directly
ahead is to be left empty—that is, the site is at
the counterclockwise end of a row bordering
the structure perimeter, as figure 6 shows.
Attaching a block at an end-of-row site corre-
sponds to starting a new row; attaching a block
at an inside corner extends existing rows.
Suppose the algorithm can get stuck—that
is, it can reach a stage of construction where
no further blocks can be attached, but at least
one empty site is still supposed to be occu-
pied. This can’t be due to an unfillable gap
(such as site A in figure 2) because algorithm
1 avoids this situation. Along the incomplete
structure’s perimeter, one of the following
must be true:
• blocks are supposed to occupy all sites, or
• at least one site is supposed to be left
empty.
In the first case, consider a robot moving
counterclockwise along the perimeter of any
closed shape drawn only with right angles.
Geometrically, the robot must come to a
point where it has to take two left turns (end-
of-row sites) sequentially without taking a
right turn (inside corner) in between. But the
second left turn is an end-of-row site that can
be occupied according to algorithm 1, so we
have a contradiction.
In the second case, there can be
no inside corners where blocks
are supposed to be attached, or
algorithm 1 could attach them
there. So somewhere (after pass-
ing the previous inside corner, if
any), the robot must have passed
through a site not intended to be
occupied. Next, the robot must
encounter an end-of-row site—in
which case it can attach a block
there, a contradiction—or an
inside corner not intended to be
occupied—in which case we can
apply this argument recursively:
a finite shape can’t have inside
corners indefinitely. In all cases
we end up with a contradiction,
which completes the proof.
Inert blocks
It’s possible to build arbitrary
solid structures even if all blocks
are passive and identical. (This
case uses the most basic sense of
stigmergy for construction, where blocks
only carry information indicating their pres-
ence.) When robots reach the structure, they
must first establish their position, as before,
so they all agree on a coordinate system. If
position information isn’t available every-
where, as it was in previous cases, robots will
need to do more work and use more memory
to obtain it. Once they know their location,
they can avoid attaching separated blocks in
the same row in the same way as in the write-
able-blocks case.
The simplest way to establish position is
for the marker to have one distinct edge that’s
along an edge of the desired structure in the
occupancy matrix. A robot reaching the
structure follows the perimeter until it finds
this landmark. Because the edge is to lie
along the finished structure’s perimeter, it
will always be available to robots following
the perimeter at any stage of completion.
Finding the landmark establishes the robot’s
position; thereafter, it pays attention to block
boundaries and corners to keep its position
knowledge current.
Comparison
The three approaches are similar in many
ways:
• Agents use a fixed set of simple control
rules regardless of the desired structure.
• Robots act independently, asynchronously,
and in parallel, without maintaining state
based on other robots or establishing lead-
ers with special roles.
• No explicit communication between robots
is required.
Thus all three share properties such as the
ability to adapt to unexpected delays and
varying numbers of robots. However, the
algorithms differ with respect to robustness,
cost, and speed.
The inert-block variant isn’t very robust to
position errors. If a robot is momentarily dis-
placed away from the perimeter, it must travel
all the way back around to the marker to
reestablish its position with certainty. If it mis-
counts its movements, it can get out of regis-
ter with the structure and consequently attach
a block somewhere inappropriate. In contrast,
in the approaches with writeable and com-
municating blocks, position information is
available everywhere around the perimeter,
so robots can easily correct position errors.
Communicating blocks have a different
robustness problem: Because of the increased
complexity, each block becomes a more
likely failure point. For large structures, fail-
ures will occur even with very-high-reliabil-
ity components. It’s possible to recover from
a block failure, but if the components are sim-
pler, failure will occur less often. Writeable
and inert blocks are more robust in that
respect.
Computation is inexpensive enough to
think about embedding it pervasively in an
environment. However, for applications that
involve building with very cheap materials,
the added cost of embedding computation
could be significant. RFID tags, by compar-
ison, are currently on the order of US$1 each,
and manufacturers argue that the price could
soon go down as much as two orders of mag-
nitude. Inert blocks don’t require even that
additional cost, or the cost of equipping the
robots with RFID transceivers.
We can most thoroughly and quantita-
tively compare the three variants with respect
to speed. Construction speed is a function of
how far robots must travel to collect enough
nonlocal information to decide whether to
attach a block, as well as the number of
places there are to attach blocks at any given
time—in a sense, how much work exists that
multiple robots could be doing in parallel.
These two aspects aren’t completely inde-
pendent: with fewer sites simultaneously
available for attachment, robots typically
have to travel further to reach one.
Figure 5. Algorithm 1 is the robot pseudocode
procedure for assembling a structure of inert,
writeable blocks. An end-of-row site is one where
the robot is about to turn a corner to the left or
where the site directly ahead is not supposed to
have a block according to the occupancy matrix.
while structure not complete do
get block from cache
go to structure
read position from neighboring label
5:
seen-row-start
 false
while still holding block do
if (site should have a block) and
((at inside corner) or
(
seen-row-start
and (at end-of-row))) then
10: attach block here
write coordinates to block
else
if at end-of-row then
seen-row-start
 true
15: end if
follow perimeter counterclockwise
end if
end while
end while
Total distance traveled
In general, the time required to build a given
structure is a function of several factors:
• the structure’s size and shape;
• the number of robots, N;
• the time required for a robot to fetch a
block and bring it to the structure, L;
• the time required for a robot to attach a
block, A; and
• the distance a robot must travel along the
perimeter after reaching the structure
before it finds a valid attachment site, D.
We measure time in units such that it takes
one time step to travel the length of one cell
of the structure grid.
L is typically task specific, and A is imple-
mentation specific. Because D, by contrast,
depends primarily on the algorithm used, we
focus on that quantity in comparing the three
approaches’ performance. We consider a
fixed number of robots building an n
 n
square and investigate the total number of
perimeter-following steps D = D taken by
all robots during construction.
To the extent that interference between
robots can be avoided, N robots will com-
plete the structure N times faster than one
robot. However, as robot density in the work-
space increases, interference also increases
as robots maneuver to avoid each other, reach
the limit as to the amount of work they can do
at one time, and so on.
For this analysis, our model places block-
carrying robots randomly on a large circle sur-
rounding the marker and moves them inward
until they reach the perimeter. After a robot
attaches a block, it moves instantaneously
back to the surrounding circle and continues
with another block. We make the cellular
assumption that robots and blocks occupy dis-
crete positions on a grid corresponding to the
occupancy matrix. (Later, we discuss the con-
sequences of relaxing this assumption to let
robots have continuous-valued positions.)
With communicating blocks, the best pos-
sible case for D(n) is 0: robots can always
randomly end up on the perimeter at sites
where blocks may be attached, making travel
along the perimeter unnecessary. The worst
case is O(n
3
): if the robots build one full row
of the structure first and thereafter are espe-
cially unlucky as to where they reach the
structure, they might have to travel a distance
O(n) for each of the O(n
2
) blocks they attach.
A case worse than O(n
3
) isn’t possible for a
square structure because a robot will never
have to travel further than O(n) to attach any
block, and only n
2
blocks exist.
With writeable blocks, the best case for D
is O(n
2
): new rows must be started n times,
and each time a robot must travel distance
O(n) to survey the entire row to verify that
it’s empty. However, a robot can attach addi-
tional blocks in existing rows without trav-
eling if the robot always reaches the struc-
ture at an inside corner. Here, too, the worst
case is O(n
3
).
With inert blocks, the best case occurs if
the robot always hits the structure adjacent
to the marker’s labeled side. Because robots
only become eligible to attach blocks after
passing that landmark, the structure grows in
a stereotyped way, and we can write an exact
expression for the number of steps. This
expression is O(n
3
), because a robot must
typically travel the length of a side (n) to find
a place to attach a block for each of n
2
blocks.
The worst case is when the robot always hits
the structure just past the landmark. Then it
not only must travel the same distance to find
an allowed attachment site but must also first
circle the entire structure to find the land-
mark, adding O(n
3
) steps to D .
Table 1 summarizes these results. In addi-
tion to the best and worst cases for scaling
behavior, the average case is of interest and can
be measured experimentally. For structures of
side lengths {10, 20, 30, 40, 50} and 10 robots,
we performed 10 independent runs in which
we measured D (see the table and figure 7a).
With writeable and inert blocks, average scal-
ing is close to the worst case. Furthermore, the
multiplicative factor is considerably better for
writeable than for inert blocks: although the
two scale similarly with structure size, the for-
mer is consistently about three times faster.
With communicating blocks, the average scal-
ing behavior is significantly better than for the
other two approaches, though still far from the
theoretical best case.
Parallelism
At any stage of construction, there will be
some number of simultaneously eligible sites
where robots could attach blocks. This num-
ber reflects the shape in which the structure
grows. It also affects how much robots can do
at any given time—that is, the extent to which
the system can exploit the swarm’s parallelism.
Figure 7b shows the maximum number of
available sites over the course of a run for our
experiments on square structures of varying
side lengths. Because of the stereotyped way
that structures grow with the inert-blocks
algorithm, only one site is ever admissible for
attachment at a time. Writeable blocks allow
slightly more parallelism: not being limited
to a single landmark to establish position lets
robots work on all four sides of the structure
at once. Still, no more than four sites can be
simultaneously available, regardless of the
structure’s eventual size. Communicating
blocks let many more sites be simultaneously
available for attachment. Moreover, that avail-
ability scales up with structure size.
We can explain this as follows. If we
assume that N is arbitrarily large and ignore
interference between robots, some robot will
always be present to attach a block at any
allowed site. So, we can see construction as
taking place in a series of steps of length A,
where, at each step, blocks are attached at all
allowed sites. As figure 8a shows, with write-
MARCH/APRIL 2006 www.computer.org/intelligent 25
E
E
E
I
E
E
EE
Figure 6. A sample structure in progress.
Blocks have darker shading; empty sites
meant to be occupied have light shading.
E indicates an end-of-row site; I, an inside
corner.
Table 1. Best, worst, and average cases for the total number of steps robots must take
along the perimeter when building an n  n square.
Case Inert blocks Writeable blocks Communicating blocks
Best 2
n
3
–
n
– 1
n
2
+
n
– 1 0
Worst 5
n
3
+ 3
n
2
– 4
n
– 5
O
(
n
3
)
O
(
n
3
)
Average ~(3.03 ± 0.03)
n
2.981±0.003
~(1.1 ± 0.2)
n
2.93±0.07
~(2.5 ± 0.6)
n
2.36±0.06
able blocks, each step will have four allowed
sites. This is because robots must first pass
through an end-of-row site before they may
attach a block at another end-of-row site. The
time required to attach n blocks thus scales
like nA. With communicating blocks (figure
8b), the number of allowed sites can increase
linearly with the step number (along with the
perimeter) if the structure’s design doesn’t
limit growth. At step t, then, O(t) blocks will
be attached, so the time to attach n blocks
scales like .
Total construction time
The minimum possible number of steps
required to assemble any given structure,
assuming an unlimited number of noninter-
fering robots, is equal to the longest shortest
distance from the marker. That is, if we start
at any block in the desired final structure and
take one-cell hops to adjacent structure
blocks, we must make some minimum num-
ber of hops d
min
to reach the marker; the
fastest possible construction time is the max-
imum d
min
among all blocks in the structure.
This is because a robot can only attach a
block if there are neighbors in the structure to
attach it to, and an inductive approach shows
that the earliest step in which any block could
possibly be attached is the one equal to its
shortest distance from the marker.
The communicating-blocks approach
achieves this minimum-possible construc-
tion time.
Lemma. No two adjacent blocks can have
the same distance from the seed d
min
.
Proof. Because the seed is unique and
occupies one site, we can tile the plane like a
checkerboard, where white squares are an
even distance from the seed and red squares
are an odd distance, as figure 9 shows.
Theorem. With an unlimited number of
noninterfering robots, the communicating-
blocks algorithm results in each site that’s
meant to be occupied becoming occupied on
the step that’s equal to its shortest distance
from the marker, d
min
.
Proof. The proof is by induction. For sites
adjacent to the seed, the proof is trivial.
Next, consider a structure after t steps,
where every perimeter block has d
min
≤ t.
Less than t is possible only if sites neighbor-
ing those blocks are intended to be left
empty; otherwise, blocks would already have
been attached at those sites, by assumption.
Then, for any sites bordering the perimeter
that are meant to be occupied, d
min
= t + 1.
As we now show, every one of those sites will
have a block attached at the next time step
(t + 1) without conflict.
Figure 10 shows the three possible cases.
(b)(a)
8
7
6
5
4
3
2
1
Placed
at step:
Figure 8. If the number of robots N is arbitrarily large, then at the end of each step
of length A, we can assume that blocks will be attached at every site where it is
legal to do so (blocks attached at successive steps are shown successively lighter):
(a) writeable blocks, (b) communicating blocks.
Self-Managing Systems
26 www.computer.org/intelligent IEEE INTELLIGENT SYSTEMS
10 20 30 40 50
0
10
20
30
40
50
Side length
Max number of allowed sites
Inert blocks
Writeable blocks
Communicating blocks
10
1
10
3
10
4
10
5
Side length
Steps along perimeter
Inert blocks
Writeable blocks
Communicating blocks
(b)(a)
Figure 7. (a) Total number of steps taken along structure perimeter as a function of structure side length. (b) Maximum number of
sites during construction where robots could simultaneously attach blocks.
In the first case (figure 10a), a block can
be attached at an inside corner, extending two
existing rows. Such an attachment can’t result
in a violation of the separation rule. Because
the new block starts no new rows, if a viola-
tion were to occur at the next time step, it
would also occur in the block’s absence.
Thus, attachment at such a site won’t result
in conflict and is always allowed.
In the second case (figure 10b), a block starts
a new row. If only one such block is to be
attached in the new row at the next time step, the
attachment won’t violate the separation rule.
The third, and the only potentially prob-
lematic, case (figure 10c) involves adding
two blocks in the same new row at the same
time. Using the lemma presented earlier, at
least one site must separate these two blocks
(marked by question marks in the figure). If
they’re meant to be separated, they can both
be attached at once without conflict. A prob-
lem, then, can only arise if all sites between
them are meant to be occupied.
The two blocks must border the structure
on the same side, as figure 10 shows; other-
wise, either the partial structure violates the
separation rule or the structure design has a
loop in it, neither of which is allowed.
The two new blocks’ old neighbors (X in
figure 10c) have d
min
= t, by hypothesis.
Because all sites between the new blocks are
meant to be occupied, all sites between the
X blocks must also be meant to be occupied,
or the structure design would have a loop.
Furthermore, by step t, all sites between the
X blocks must be occupied, or the structure
would have violated the separation rule.
The occupied sites adjacent to the X blocks
(Y in the figure) must have d
min
= t – 1. If they
had d
min
= t + 1, they would still be unoccu-
pied by hypothesis. According to the lemma,
they can’t have d
min
= t, and, if they had d
min
< t – 1, the X blocks would have been occu-
pied earlier, by hypothesis. So, the sites adja-
cent to both the Y blocks and the ? blocks (Z
in figure 10c) have d
min
= t, and thus must be
occupied by hypothesis. In this case, how-
ever, the ? blocks aren’t starting a new row.
This contradiction completes the proof.
With writeable or inert blocks, the structure
typically has fewer sites where blocks can be
attached at any given time, as we described
earlier. Using the cellular assumption, write-
able blocks allow only one row per side to be
simultaneously under construction. As figure
11 indicates, robots can work on separated
rows on the same side at the same time, but
they can’t start on the next layer outward until
they complete the previous row. Using the
inert-blocks algorithm, robots attach blocks
no more than one at a time in a predictable
order. Thus, completing an n-block structure
will require n attachment steps, no matter the
number of robots. (Alternative algorithms for
inert blocks, not covered here, can somewhat
improve the opportunity for parallelism, but
not better than writeable blocks.)
If we relax the cellular assumption, robots
might be able to reach the structure between
grid sites such that they recognize an end-of-
row but not an inside corner (see figure 12).
Robots building with writeable blocks can
then end up with two or more layers on the
same side under construction at once. This
can change some of our results for writeable
blocks (results for communicating and inert
blocks aren’t affected). It improves the best-
possible case for D(n) to O(n). More than
four sites can become available for block
attachment at a time, allowing better use of
swarm parallelism and faster construction.
However, the way the robot has to reach the
structure for this case to occur makes it a low-
MARCH/APRIL 2006 www.computer.org/intelligent 27
2
4
*
2
24
2
2
4
1
1
3
3
3
3
33
31
1
2
2
2
4
3
Figure 9. Sites labeled with their distance
from the seed (marked with an *).
Figure 11. A desired structure under construction using writeable blocks (lightly
shaded cells show the desired structure; darker shading indicates blocks added so far).
Arrows indicate that one row per structure side can be simultaneously under
construction. Separations caused by areas meant to be left empty, such as those at the
top and bottom, allow rows on the same side to be treated independently.
ZZ????
(b) (c)(a)
YYXX
Figure 10. Three possible cases where robots might attach blocks (marked with a ?) at
step t + 1: (a) attaching a block at an inside corner, (b) attaching a single block in a new
row, and (c) adding two blocks in the same new row.
probability event. Thus, although relaxing
the cellular assumption can substantially
improve best-case performance, it’s unlikely
to significantly affect average performance.
I
mportant work remains to extend this
approach to other classes of structures,
including structures with deliberate holes
and 3D structures. Another important issue
is the use of explicit communication between
robots.
6
More-extensive cooperation could
let robots handle tasks that they couldn’t
accomplish alone,
7
help reduce interference,
or achieve better than a factor of N speedup.
An open question is to what other prob-
lems extended stigmergy might usefully be
applied. We expect it to be a useful frame-
work for coordinating other multiagent sys-
tems that can affect their environment, poten-
tially offering advantages like those shown
here over the basic stigmergy that’s been used
in foraging and other swarm applications.
We also hope that approaches like those we
described will be useful in large-scale con-
struction systems and will help to enable human
presence in unfriendly environments. In addi-
tion, researchers could apply these principles
to other production technologies in which
mobile robots construct objects on smaller
scales. Ultimately these could include nanofab-
rication or rapid-prototyping systems.
References
1. A. Warszawski and R. Navon, “Implementa-
tion of Robotics in Buildings: Current Status
and Future Prospects,” J. Construction Eng. and
Management, vol. 124, no. 1, 1998, pp. 31–41.
2. P.-P. Grassé, “La Reconstruction du Nid et les
Coordinations Interindividuelles Chez Belli-
cositermes natalensis et Cubitermes sp. La
Theorie de la Stigmergie: Essai d’Interpreta-
tion du Comportement des Termites Con-
structeurs” (“Nest Rebuilding and Coordina-
tion between Individuals for Bellicositermes
natalensis and Cubitermes sp. Theory of Stig-
mergy: Attempting to Interpret the Behavior
of Builder Termites”), Insectes Sociaux, vol.
6, 1959, pp. 41–81 (in French).
3. J. Werfel et al., “Distributed Construction by
Mobile Robots with Enhanced Building
Blocks,” to be published in Proc. 2006 IEEE
Int’l Conf. Robotics and Automation, IEEE
Press, 2006.
4. J. Werfel, “Building Blocks for Multiagent
Construction,” Distributed Autonomous
Robotic Systems, Springer, 2004.
5. C. Parker, H. Zhang, and R. Kube, “Blind
Bulldozing: Multiple Robot Nest Construc-
tion,” Proc. 2003 IEEE Int’l Conf. Intelligent
Robots and Systems, IEEE CS Press, 2003,
pp. 2010–2015.
6. C. Jones and M. Mataric, “Automatic Syn-
thesis of Communication-Based Coordinated
Multirobot Systems,” Proc. 2004 IEEE/RSJ
Int’l Conf. Intelligent Robots and Systems,
IEEE Press, 2004, pp. 381–387.
7. B. Gerkey and M. Mataric, “Pusher-Watcher:
An Approach to Fault-Tolerant Tightly Cou-
pled Robot Coordination,” Proc. 2002 IEEE
Int’l Conf. Robotics and Automation, IEEE
Press, 2002, pp. 464–469.
For more information on this or any other com-
puting topic, please visit our Digital Library at
www.computer.org/publications/dlib.
Self-Managing Systems
28 www.computer.org/intelligent IEEE INTELLIGENT SYSTEMS
The Authors
Justin Werfel is a doctoral student in the Computer Science & Artificial
Intelligence Laboratory at the Massachusetts Institute of Technology. His
research interests include swarm engineering, evolutionary theory, and rein-
forcement learning. He has an SM in computer science from MIT. He is a
member of the IEEE Robotics and Automation Society and Sigma Xi. Con-
tact him at MIT 32-386, 32 Vassar St., Cambridge, MA 02139; jkwerfel@
mit.edu.
Radhika Nagpal is an assistant professor of computer science at Harvard
University. Her research interest is in biologically inspired approaches to
multiagent and distributed systems. She received her PhD in computer sci-
ence from MIT. Contact her at 235 Maxwell Dworkin, 33 Oxford St., Cam-
bridge, MA 02138: [email protected]
B
C
A
Figure 12. If a robot reaches a structure
between grid sites as in the case illustrated,
and its sensing is sufficiently limited such
that it can perceive the block at B but not at
A, it could start following the perimeter
counterclockwise, register the end-of-row
as it turns the corner, and attach a block at
C. Such an attachment isn’t possible under
the cellular assumption.
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