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An Experimental Study on a Self-repairing
Modular Machine
Eiichi Yoshida aSatoshi Murata aKohji Tomita a
Haruhisa Kurokawa aShigeru Kokaji a
aMechanical Engineering Laboratory, 1-2 Namiki, Tsukuba-shi, 305-8564 Japan
Abstract
A self-assembling and self-repairing mechanical system is experimentally studied
to demonstrate its effectiveness. We developed a 2-D model of autonomous me-
chanical unit capable of dynamic reconfiguration and inter-unit communication.
Self-assembly and self-repair experiments have been carried out using a distributed
algorithm developed under the constraints of the system’s homogeneity and locality
of information exchange. In experiments, more than ten units successfully config-
ured themselves and recovered from a fault. Besides the research of the 2-D model,
a model of 3-D system and its self-assembly algorithm are also developed.
Key words: Distributed Autonomous System, Homogeneous Modular Machine,
Self-assembly, Self-repair
1 Introduction
We have been developing a distributed mechanical system composed of many
identical autonomous units. The key issues to build such a hyper-distributed
mechanical system are homogeneity of both hardware and software, locality
of information exchange and dynamic reconfigurability. Owing to its homoge-
neous structure, our system can realize self-assembly and self-repair functions.
It can be used as a versatile self-maintainable machine, which can work as a
mobile robot, as well as a static structure in hazardous environments.
Many studies have been made on reconfigurable mechanical systems. Polypod
[1] and Tetrobot [2] are mobile robots with variable structures. CEBOT [3]
and modular robots [4] [5] are dynamically reconfigurable.
Preprint submitted to Elsevier Preprint 1 February 2006
In contrast to the above studies, our approach is characterized by its strict
homogeneity in both hardware and software, which makes the whole system
flexible and robust against faults. Another important point is the simplicity
of the hardware and the usage of local communication.
This paper focuses on hardware experiments to confirm self-assembling and
self-repairing capability. A hardware system composed of 20 mechanical units
called “fracta” was used for the validation.
We are extending the concept of modular machine to 3-D space. The last part
of the paper addresses its hardware realization and self-assembly software.
2 Unit Hardware
We built a system with twenty units whose structure is shown in Fig. 1.
The unit is mainly composed of actuation part using electro- and permanent
magnets and information processing part using a microcomputer.
A unit has three-layered structure, with three pairs of permanent magnets
(arranged concentrically with 120 degrees) in the top and the bottom layer and
three electro-magnets (rotated by 60 degrees from outer layers) in the middle.
By changing the polarity of an electro-magnet, it is attracted into, or repulsed
from the gap between outer layers of another unit. Two units change their
connecting position by an appropriate sequence of electro-magnet operations.
A unit can connect with maximum six units. The unit is also equipped with
serial optical channels for bilateral local communication.
optical communication
channel
information
processing part
(CPU, control unit)
electro-magnet
(male)
permanent magnet
(female)
Fig. 1. Autonomous mechanical unit “fractum”
2
3 Self-assembly and Self-repair Algorithm
In self-assembly and self-repair, each unit has the same software and decides
its movement only according to local information. Under these constraints,
we have been developing a difference-based algorithm [6] and a nucleation
algorithm [7]. The latter, however, needs rather large amount of information
processing, and does not suit the current stage of hardware development. Thus
we extend the difference-based algorithm to enable self-repair function.
In the algorithm, all the units are assumed to be connected and synchronized
in communication. The synchronization is realized by a simple distributed
method [8].
3.1 Connection Types
The “connection type” of a unit is introduced to describe the global configu-
ration. A unit can be formally described as a hexagon with six possible bonds.
The connection of a unit is classified into 12 connection types shown in Fig. 2.
The link between two types denotes that they are transferable by a single step
of unit motion. The notion of “distance” between two types is defined as the
number of links between them.
A “movable unit” is a unit which can rotate without carrying other units and
keeps the whole system connected after the movement. A unit with connection
type “e”, “o”, or “ε”ismovable.
3.2 Self-assembly Algorithm
The algorithm consists of the following three procedures. Each unit
(i) calculates such measures as
•“difference” between the current state and the target.
•“irritation” which increases during deadlock state.
(ii) estimates the average difference around the unit using a diffusion pro-
cess through the inter-unit communication.
(iii) if the unit has relatively large difference, it moves towards the direction
to make it smaller.
A “control step” of the algorithm is composed of above (i)–(iii) procedures.
In the algorithm, this control step is iterated by each unit in parallel.
3
3.2.1 Target description
We describe the target formation using the connection types. Let us consider
a 10-unit triangle shown in Fig. 3. Any unit with the target type “K” is
connected to 4 units whose types are {o, K, K, s}. This situation is expressed
in a “statement” K{o, K, K, s}. In this example, the target shape is written
as a collection of the following statements:
o{K, K},
K{o, K, K, s},
s{K, K, K, K, K, K}.
(1)
Connection types of statements in (1) are sorted in increasing order of the
number of connecting units. Although there are cases where this description
does not correspond to unique target configuration, it suffices as long as the
target shape is not too complicated.
3.2.2 Calculation of difference
In every iteration of control step, each unit decides its current connection state
through local communication. Then each unit evaluates the sum of distances
between the types of its current state and each of the target statements. The
“difference” of unit iat t-th control step diff(i, t), is given as the minimum of
these sums. Namely, each unit evaluates the difference from the closest target
statement.
3.2.3 Diffusion process
To make the group of the units converge to the desired target configuration,
it is desirable for a unit which has relatively larger difference to get a larger
priority to move. We therefore introduce the following diffusion process. A
e
m
o
pY
K
X
fs
Fig. 2. Connection types and their
relationship
o
KK
KK
s
oKK o
Fig. 3. Description of 10-unit
triangle
4
diffusion variable x(i, t) represents the average difference around the unit i.It
is calculated as follows 1.
x(i, t +1)=x(i, t)+K¯x(i, t)−L
∆x(i, t)
,x(i, 0) = diff(i, 0) [initial state],(2)
where ¯x(i, t) : “flux” of x(i, t) in the unit i
(=
j∈neighbors of i
{x(j, t)−x(i, t)}),
K: diffusion coefficient,
L: leak constant (effective for movable units only).
If x(i, t)<0 in eq. (2), it is reset to zero. The diffusion variable Kaffects
the velocity of the diffusion, and Lhelps to converge to target by decreasing
whole diffusion variables in the system.
3.2.4 Moving strategy
A unit is activated to move clockwise or counterclockwise by the following
condition that the ratio of difference to diffusion variable (estimated average)
is larger than an activation threshold G:
Gx(i, t)<diff(i, t).(3)
This makes units of relatively large difference move earlier. The motion is
stochastic in such a way that the direction which gives smaller difference after
the movement is chosen with larger probability.
If the target configuration is accomplished, the differences become zero in all
of the units and no more motion is made.
3.2.5 Introduction of irritation
Sometimes self-assembly process is trapped in a deadlock state. The right two
configurations in Fig. 4 are in deadlock because all the movable type “o” units
have the difference 0 with proper neighbors {K, K}.
1Suppose that each unit has a water reservoir connected to neighbors with pipes.
Then, the water levels will converge to an average level as water flux passes through
the pipes. This water level analogically corresponds to the local average difference.
5
o
KK
KKs
Kss
K
ooKKK
o
KK
o
KK
ppp
oKK
o
λλ
o (K, K)
K ( o, K, K, s)
K ( K, K, s, s)
s ( K, K, K, K, s, s)
o
KK
oKKKX o
KKs
KK
o
target form deadlock
Fig. 4. Examples of deadlock
To dissolve deadlock, we add another variable called “irritation” which is
augmented if the diffusion variable x(i, t) becomes stable in undesired config-
uration [9]. When non-movable units are in deadlock, their irritation values
continue to increase and are transmitted also to movable units through local
communication.
By adding a new activation condition that “the irritation exceeds than certain
threshold,” deadlock states can be dissolved in a local way. Computer simula-
tions revealed that the performance of self-assembly was greatly improved by
the algorithm using irritation [9].
3.3 Self-repair Algorithm
The above algorithm can be used for “self-repair” operation by introducing
an extended target description. This extension makes it possible to express
target shapes including spare units [9].
In adding a spare unit to the target of 10-unit triangle, we introduce descrip-
tions with priorities shown in Fig. 5. We give the first priority to the original
triangle shape and the second to the shape with a spare unit. In this example,
up to three spares can be included by slightly modifying the description.
This simple method allows units first to build the structure with spare units,
and to repair themselves if some of units are removed. Computer simulations
showed the effectiveness of the introduced algorithm.
ε
KK
K
s
oKK o
f
o
o {ε, f}
ε {o, K, f}
K{o, K, f, s}
K{ε, K, f, s}
f {o, ε, K, K, s}
s {K, K, K, K, K, f}
Prority 1
[original shape]
Prority 2
[with a spare unit]
o {K, K}
K{o, K, K, s}
s {K, K, K, K, K, K}
spare unit
Fig. 5. Description with priorities for self-repair
6
4 Self-assembly and Self-repair Experiments
Using the developed hardware setup, self-assembly and self-repair experiments
have been conducted. The algorithm presented so far is a high-level control
which decides whether a unit moves or not and the direction of the motion. In
the real mechanism, a unit’s motion requires other units’ local cooperation. We
have developed a low-level control which consists of map generation, collision
avoidance and coil drive [7] and integrated it with the upper-level algorithm.
In the following experiments, a control step including these two control lev-
els takes 23.25[sec]. The most time-consuming part is the control of electro-
magnets which should be operated appropriately with sufficient interval, so
that undesired motion may not occur. It is also necessary that resultant con-
nection type after the motion should not be unstable “e” type. The control
step includes 18 sequential operations of electro-magnets, each of which takes
1.0[sec].
The pictures in Fig. 6 are taken from one of the experiments. Starting from
a two-line configuration, 11 units form a 10-unit triangle with one spare unit
based on the proposed self-assembly algorithm. This target shape remains
stable as long as all the units are working without faults.
The 1000 simulations of this self-assembly resulted in the success ratio 98.3%
with the average convergence time 57.7 control steps. Figure 6 (left column)
shows snapshots taken from the shortest case of 6 control steps.
To verify the self-repair capacity, a simulated fault is given to one of the 11
units by cutting its electric power source (except for “s” type unit). Each unit
sends out an “OK” signal to its neighbors at the beginning of each control step
while working properly. The fault can be detected if a unit does not receive
the expected “OK” signal. In this case, the unit reverses the polarity of the
corresponding electro-magnet to cut off the faulty unit.
Then self-repair process proceeds automatically. Even though total units in
the system has changed, the global self-repair process is launched thanks to
its distributed characteristics of the algorithm. We can see the original 10-unit
triangle is constructed in the right column of Fig. 6.
We examined the self-repair capacity by giving the simulated fault to each
unit except the center “s” type. As a result, self-repair completed successfully
in all of these 9 experiments and the average self-repair time was 2.2 control
steps.
The experiments demonstrate that a mechanical system consisting of many
identical hardware and software can maintain themselves using self-assembly
7
and self-repair functions in a distributed manner.
starting self-assembly
spare unit
defective unit
(power source cut)
Self-assembly experiment
cutting off the
defective unit
self-repair using
the spare unit
self-repair completed
Self-repair experiment
Fig. 6. Experiment of self-assembly and self-repair
8
5 Development of 3-D System
We are currently attempting to extend the concept of our self-assembling
modular machine to 3-D space. This section outlines its present development
of hardware realization and software for distributed self-assembly.
5.1 Design of 3-D Unit
The difficulty of design lies in how both geometric complementarity and system
homogeneity are satisfied in 3-D space. A solution we reached is illustrated in
Fig. 7. The unit has six connecting arms in three orthogonal axes which can
rotate independently. Only one DC motor of 7[W] is used as power source in a
unit to realize compact implementation. Its torque is delivered by controlling
solenoids and electro-magnetic clutches. We confirmed that a unit generates
enough power to lift another unit.
The motion of units is basically made by pairwise operation. Consider two
connected units (shaded) on a plane Amade by 4 units in Fig. 8 (a). Unit Y
moves to the position in Fig. 8 (b) when unit Xrotates about the axis b–b
after releasing the connection of Yto Z. By repeating this elementary step
motion, various 3-D structures can be configured without external help. We
have constructed four 3-D hardware units as shown in Fig. 9 and verified their
reconfiguration capacity by experiments [10].
5.2 Self-assembly of 3-D System
A distributed self-assembly algorithm for 3-D units has been developed in
parallel with the hardware development.
Rotating armConnecting hand
Fig. 7. Schematic view of 3-D unit
)
X
Y
Z
b
bb
X
Y
b
Z
(a) (b)
Fig. 8. Typical motion of 3-D units
9
Fig. 9. Hardware setup of the four 3-D mechanical units
One of major difficulties of developing 3-D self-assembly algorithm lies in the
multiplicity of degrees of freedom, whereas 2-D system has only to choose
one of two directions, clockwise or counterclockwise. The algorithm should be
implemented in a distributed manner in order to avoid premature convergence
to undesired shape. We introduce a stochastic relaxation process based on
simulated annealing to cope with these problems.
We classified the connection types of 3-D unit in 1-neighborhood system in
cubic lattice as shown in Fig. 10. We neglect rotational transformations in this
classification.
Assuming that a unit rotates by 90 [deg] at one step, a unit with type C1
has maximum two reachable positions. Other than C1, types C21 and C31 are
also regarded as movable here.
The target shape is described in a similar manner as the 2-D algorithm. For
instance, the target shape of a box with twelve units is simply written as:
C31{C31, C31, C41},
C41{C31, C31, C41, C41}.
(4)
The 2-D self-assembly algorithm is applied to the 3-D system with many
123456
0
1
C1
C21
C20
C31
C30
C41
C40 C5 C6
Valence
Type
Fig. 10. Connection types for 3-D units
10
degrees of freedom by introducing stochastic relaxation method. When a unit
has multiple reachable positions, it decides where to move according to the
probability Piof moving to the alternative ibased on Markov Random Field
(MRF) [11] as follows:
Pi=1
Zexp (−Di
T)Z=
i∈reachable positions
exp (−Di
T),(5)
where Diis the difference between the current and the target connection
state (4) and Tcorresponds to the “artificial temperature” in the simulated
annealing method.
The probability Piis uniform for each reachable position iwith large Tvalue,
while Pifor the lowest potential approaches 1 if Tis small. Tis decreased
according to the following formula:
T=C
log (1 + t),(6)
where tand Care the number of control steps and constant parameter con-
cerning system size, respectively.
By decreasing Tgradually in this way, units search uniformly in “searching
space” in earlier steps. As time elapses, the units come to make more con-
centrated motion to the closest position to complete target structure. The
premature convergence to undesired shapes with local minima can be avoided
in this way.
initial state after 20 steps
after 40 steps after 70 steps (completed)
Fig. 11. Self-assembly simulation of 12 units
11
Figure 11 shows a graphical view of a simulation of 12-unit box described in (4)
from initial ladder shape. The system succeeded in self-assembly in 70 steps.
The effectiveness of the algorithm has been verified through a quantitative
evaluation [12].
5.3 Towards Self-assembly of Large-Scale System
The self-assembly algorithm based on stochastic process has such advantages
as flexibility and simple implementation, and works well for small number of
units, say, up to twenty units. However, it becomes less effective for larger sys-
tem in which ambiguity in description and importance of assembly “sequence”
are significant. Since few studies has been made on distributed self-assembly
of large-scale 3-D structure, we are currently working on this issue based on
a developmental scheme like nucleation method in 2-D system [7].
We introduce primitive structures which determines the geometrical relation-
ship between “nodes”, which denotes a group of units here, as shown in Fig. 12.
By assigning another sub-structure to each nodes of a primitive structure, var-
ious complex shapes as shown in Fig. 13 can be described. Figure 14 illustrates
the recursive representation of the shape in Fig. 13(a).
Self-assembly proceeds by constructing first top-level structure, then down to
sub-structure, and so forth. Using this algorithm, rough shaping is performed
in the early stages of self-assembly, and then detailed shape is assembled in a
line square octahedron cube
RefVect:
direction vector
RefVect:
normal vector
Fig. 12. Primitive types of description
(a) 48 units (b) 56 units
Fig. 13. Examples of large-scale structure
12
Level 1
Level 2
Level 3
x
y
z
Fig. 14. Recursive description of structure in Fig. 13(a)
concurrent and distributed manner later as shown in Fig. 15.
Figure 16 shows a self-assembly process of 56-unit structure in Fig. 13(b),
where dark colored units have reached a position in the target shape. As
can be seen, a large structure composed of more than 50 units is successfully
constructed in a distributed way.
Possible applications of such a system include machines used in environments
inaccessible to humans, for instance, planetary exploring vehicles or satellite
antennae. Transported in a compact folded form, they can expand to the
original structure when working, and repair themselves if some part becomes
damaged.
group of units
[initial state]
nodes upper
lower
detailed forms appear
by extending lower nodes
final shaperough shaping
Fig. 15. Large-scale self-assembly process
13
initial state after 10 steps
after 20 steps after 106 steps (completed)
Fig. 16. Self-assembly simulation of many-unit structure
6 Conclusions
We presented experimental research on self-assembly and self-repair of a dis-
tributed mechanical system. A 2-D model of identical units called “fracta” is
used as hardware which realizes dynamic reconfiguration and inter-unit com-
munication. Based on a distributed algorithm that allows a group of units
to transform themselves into a desired shape, we have demonstrated the self-
assembly and self-repair functions successfully. This experimental results con-
firmed the hardware feasibility of the self-repairing mechanical system and
opened its way to applications such as long-running explorer or surveillant in
hazardous environments.
As a further challenge, we are currently on the way to 3-D system. We designed
a prototype unit and confirmed its basic function of reconfiguration capacity.
Self-assembly algorithms for small and large systems have been implemented
as well, whose effectiveness was shown by computer simulations.
14
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