Modular robots are systems that can change its geometry or configuration when connecting more modules or when rearranging them in a different manner to perform a variety of tasks. Graph theory can be used to describe modular robots configurations, hence the possibility to determine the flexibility of the robot to move from one point to another. When the robot’s configurations are represented in a mathematical way, forward kinematics can be obtained

Aracil Santonja, Rafael

Baca Garcia, Jose Antonio

Escalera Piña, Juan Antonio

Ferre Pérez, Manuel

Yerpes, Ariadna

English

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

E. Corchado, A. Abraham, and W. Pedrycz (Eds.): HAIS 2008, LNAI 5271, pp. 649–656, 2008. 
© Springer-Verlag Berlin Heidelberg 2008 
Modelling of Modular Robot 
Configurations Using Graph Theory 
José Baca, Ariadna Yerpes, Manuel Ferre, Juan A. Escalera, and Rafael Aracil 
Universidad Politécnica de Madrid,  
Jose Gutierrez Abascal, 2, 28006 Madrid, Spain 
{jbaca,ayerpes,jescalera,aracil}@etsii.upm.es, m.ferre@upm.es 
Abstract. Modular robots are systems that can change its geometry or configu-
ration when connecting more modules or when rearranging them in a different 
manner to perform a variety of tasks. Graph theory can be used to describe 
modular robots configurations, hence the possibility to determine the flexibility 
of the robot to move from one point to another. When the robot’s configurations 
are represented in a mathematical way, forward kinematics can be obtained. 
Keywords: Modular Robots, Graph Theory, Configurations for Displacement. 
1   Introduction 
Now in days, different modular robots designs have been proposed to give a solution 
to varied fields like versatility, adaptability, robustness, costs, etc. It is possible to find 
module robot designs from 1 degree of freedom (DOF) [1], 2 DOF [2], [3] and 3 DOF 
[4], [5]. Each system shows diverse relevant advantages between them, a few of them 
are reconfigurable and others are autoconfigurable systems, but something in common 
is that they can form new robot structures when combining modules. When attempt-
ing to obtain the model of a defined robot is just a matter of following a systematic 
procedure. With a defined robot, the number of degrees of freedom, length´s links, 
masses and geometry are normally well defined and constant, facilitating its model-
ing. On the contrary, modular robots are more complex to model, due to its capability 
of increasing the DOF, and links when adding modules to the structure. The kinemat-
ics of the robot will be defined by the number of modules and the way they are con-
nected. Therefore it is necessary to use a method to easily represent the structure´s 
configuration. 
2   RobMAT Architecture 
The desire of any modular system is that the whole is greater than the sum of its parts. 
The main advantage of using modular robots lies in the possibility of constructing 
different kinematics chains by combining modules. 
RobMAT’s system architecture is divided into modules, molecules and colonies. A 
module (figure 1 (a)) is the base component of the system and it has movement and 
650 J. Baca et al. 
communication capacity. A molecule (figure 1 (b)) is an autonomous entity made out 
of n-modules connected together (n>1). The molecule holds higher capabilities of 
movement, communication and manipulation than the module. Various molecules 
cooperating in the area to fulfill a task is defined as colony. 
   
       a)                                                                      b) 
Fig. 1. a) Base component of the RobMAT system. b) Six module configuration. 
The system has three communication channels in order to properly transmit com-
mands to the whole system and to be able to work in a cooperative mode. These 
communication channels are the Intra-Module Communication, the Inter-Modules 
Communication and the Video Communication. 
The first type is in charge of the communication between modules belonging to 
the same molecule and it uses a CAN bus [6]. The Inter-Modules Communication is 
in charge of communication between molecules or between a molecule and the con-
trol station and it is based on Bluetooth. Finally, the Video Communication is used to 
send radio frequency video signals from the two mini cameras that can be attached to 
one of the molecules and that would be used to control de system in a remote working 
environment using teleoperation. 
 
Fig. 2. By combining modules, distinct modular robot configurations can be formed, creating 
many ways of displacement and manipulation 
 Modelling of Modular Robot Configurations Using Graph Theory 651 
The used module had an actuator with 3 rotational degrees of freedom with its 
axes intersecting at one point creating a wrist-like joint. The design of the module 
allows accessories to be attached on one end of the module and the connector used to 
enable reconfigurations is made out of an electromagnet (male connector) and a me-
tallic plate (female connector), each part embedded in one of the molecules making 
possible to form different modular robot configurations as is shown in figure 2. 
3    Describing Modular Robot Configurations 
The flexibility of modular robots provides enormous adaptation capacity and the abil-
ity to perform a number of tasks. However, it complicates the robot’s kinematics and 
dynamic modeling [7]. The changing configuration of molecules means that, unlike 
other robots, modeling in advance is not possible. An algorithm is therefore required 
to automatically generate the model for any molecule configuration during the execu-
tion of each step of a task.  
3.1   Graph Theory 
Kinematics graphs are a widely-used tool for the theoretical analysis of mechanisms [8]. 
These graphs entail vertices and edges (or lines), which represent robot links and joints 
respectively. All elements of a mechanism belong to one of these categories, so any 
mechanism can be represented by a graph. Graph Theory [9] offers many theoretical 
tools to analyze graphs, and applied to mechanisms, specifically talking about the 
graphs of mechanisms; it provides the opportunity to analyze them in a more abstract 
manner. A graph is a mathematical object that captures the notion of connection.  
3.2   Graph Theory into Modular Robots 
A modular system can be represented by its corresponding graph, showing the rela-
tionship existing among different modules. The application of Graph Theory to modu-
lar robots requires the use of specific graph elements. This degree of specialization is 
achieved by labeling graph vertices and edges in such a way that there are as many 
labels as different types of robot links and joints. If we consider RobMAT as a ho-
mogenous robot, i.e. without considering the tools it can handle, it can be seen in 
general one type of link (each prism (p) next to the spherical joint), two types of joint 
(connector (c) and spherical (s) joint), and another element to be taken into account 
when representing module chains is the linking point between modules, which is 
called the port. Basically, a connector can have more than one place or port to join 
with other connector. Generally, a number codifies each port and is located in the 
graph close to the corresponding element where the linking is produced. The figure 3 
(a) shows representative graph elements over each robot module. Each module is 
composed of two links, one joint and its ports (e.g. module one is composed of L1, L2, 
and J1 and module two is composed of L3, L4 and J3), but when combining modules a 
joint between them is created due to the port at the link section (e.g. J2 is created when 
both modules are connected at port 4 in both cases). The figure 3 (b) shows a four 
module configuration graph and it can be noticed the joints created at each module 
union (i.e. J2, J4, J6, and J7) and its ports are showed. 
652 J. Baca et al. 
 
Fig. 3. a) Representative graph elements over each RobMAT module. b) 4-module configura-
tion graph. 
All this information is set forth in the Assembly Incidence Matrix AIM [10], so 
that it can be easily included in algorithms. The AIM is a (N+1)x(M+1) matrix with N 
vertices (v) and M edges (e). This matrix is formed by giving to each entry aij the 
number of the port that joins vi and ej, or 0 when no linking appears. The extra column 
(M+1) indicates the link type, while the extra row (N+1) shows the joint type. Figure 
3 (a) shows the RobMAT module with its ports and their assigned numbers. It can be 
seen that module has six possible connection ports with other modules. To clear up 
some final concepts, Figure 3 (b) shows the graph of the 4-module RobMAT mole-
cule, where each link is named Li with i=1,2…,8 and each joint is named Ji with 
i=1,2…,8. To distinguish between the two kinds of joints, spherical joints are repre-
sented as a single-line edge and connectors as a double-line edge. Also, the spherical 
joint edge is directed (from –1 to 1) to show the order of its three axes. The arrow 
indicates where the end-effector link is located. Also, the number at both ends sides of 
each connector indicates the port of connection between the links. The graph therefore 
contains enough information to describe the modular robot assembly.  
The AIM for the Figure 3 (b) graph is: 
 
 
3.3   Modular Robot Configurations for Displacement Determined by the 
Complexity of the Graphs 
Many configurations can be built for displacement, but just a few of them are efficient 
for the task. By analyzing the graph (links and joints structure), it can be determined 
the complexity for the robot to move from one point to another. An example of this 
situation can be noticed when comparing the six modules configuration from figure 4 
with the six modules configuration from figure 5. Although the same number of 
 
 Modelling of Modular Robot Configurations Using Graph Theory 653 
 
Fig. 4. Each configuration can be represented by graphs and with this a mathematical way to 
describe the structure generated  
modules are used, and thanks to an additional platform (accessory) as common  
connection between each pair of modules in figure 5, the flexibility of the robot to 
execute displacement is higher when less links and joints appears at the graph. 
Therefore, during real applications, it is important to seek out for equilibrium be-
tween the complexity of the task and the configuration of the modular robot, For ex-
ample. the use of two molecules which have simple graphs due to its configuration 
(e.g. four modules configuration at figure 4) to carry on with the task, instead of using 
one molecule compound of many modules with high pull-push torque as a result, but 
forming a complex graph full of several links and joints (e.g. six modules configura-
tion at figure 4).  
When analyzing modular robot configurations in a simple way, it can be seen re-
peatable assembly patterns. This repeatability will depend undoubtedly when using 
the same configuration of a pair of modules to form chains, as is shown in figure 4. 
By this observation, it is achievable to describe the actual configuration in a simpli-
fied manner. The figure 4 shows different configurations formed by two, four, and six 
modules chains. It can be detected that the matrix (pattern) obtained by the two mod-
ules graph is repeated at four and six module matrix due to the use of a second and 
third same pair of modules configuration, the only difference is the connection be-
tween each set. 
654 J. Baca et al. 
 
Fig. 5. While smaller the complexity of the graph, the highest flexibility of the robot to move  
4   Forward Kinematics 
Once a configuration can be described in a mathematical formulation the correspond-
ing kinematical model can be obtained. All joints in the RobMAT are rotational; nev-
ertheless modules can have any kind of joint. For this reason, it is helpful to apply 
Screw Theory [11]. This allows treating prismatic and rotational joints in the same 
expressions without specific changes. Also, almost any joint can be decomposed as a 
combination of those basic ones. A forward kinematics equation of an open chain 
robot can be uniformly expressed as a product of exponentials (POE) [12]. Using 
POE and Graph Theory, forward kinematics of tree-type modular robots can be ob-
tained. The following equation is an example that solves RobMAT module direct 
kinematics (Figure 3). Let q1, q2 and q3 be the generalized coordinates of the 3-DoF 
RobMAT module. The corresponding POE formula is given as follows: 
  
)0(),,( 0ˆˆˆ3210 332211 eqSqSqSe HeeeqqqH =  
where: 
⎟⎟
⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎜⎜
⎝
⎛
−
=
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎠
⎞⎜⎝
⎛
−=
=
=
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎠
⎞⎜⎝
⎛
−=
−=
=
⎟⎟⎠
⎞
⎜⎜⎝
⎛
=
−=
=
1000
2
001
0010
2
100
0,
2
,0
)0,0,1(
ˆ
0,0,
2
)0,1,0(
ˆ
)0,0,0(
)1,0,0(
ˆ
0
3
3
3
2
2
2
1
1
1
L
L
H
L
v
w
S
L
v
w
S
v
w
S
e
 
( )iSˆ  is a screw and is expressed in the base frame (0). Furthermore, all coordinates 
are expressed in the base frame, and also in the zero position which is a fixed posi-
tion. )0(0eH is the well-known homogenous transformation between the end-effector 
frame (e) and the base frame at zero position. Once the module kinematics model has 
been determined, the kinematics of molecules can be obtained by compounding sev-
eral module kinematics. Hence, an arbitrary module is designated as root, and posi-
tion/orientation is propagated from this root to every end-side module through the 
modules in the molecule. In this way, the forward kinematical model is obtained. 
 Modelling of Modular Robot Configurations Using Graph Theory 655 
To automate this process, it is important to know how modules are connected to 
each other. Depending on where the module port is attached, orientation and/or posi-
tion changes will or will not be required. According to the port number in both links, 
a fixed homogeneous transformation can be obtained that provides the reference 
frames of neighbour modules.  
For instance, the following described forward kinematics model of link L8 (figure 3 
(b)). The root link is L1 and one possible path from L1 to L8, which is obtained from 
the graph or AIM, would be {L1, J1, L2, J2, L3, J7, L7, J8, L8}. Then, for each vertex on 
the path, a homogeneous transformation is added as shown in the next expression. 
( ) ( ) 0·102··5·14··4·0 eHHDeHeHDeHeHH −−=  
where: 
matrix Docking
1000
0100
0010
0001
connectors and
frames reference
between 
tionsTransforma
1000
0010
100
0001
1000
0010
100
0001
2
5
0
3
2
6
0
2
44
0
1
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
−
=
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
==
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
−
==
==
−
D
HH
HH
IHH
W
e
W
e
e
 
Thus, by evaluating all paths from the root to the pending modules of each mole-
cule, the full forward kinematics model is solved. This model is very important in 
order to obtain molecule workspace and to determine its functionality. 
5   Conclusions 
The possibility of modelling modular robot configurations can be achieved by graph 
theory. Its study generates a mathematical object so it can be manipulated to perform 
complex analyzes, like forward kinematics. Analyzing the graph of each correspond-
ing robot configuration and comparing them with tested robot displacements a con-
clusion show up. The more complex the graph is (i.e. high number of links and 
joints), the robot flexibility to move from one point to another decreases, due to the 
reduced workspace of the DOF when having many modules so close. Although it is 
possible to represent any modular robot structure with graphs, just some configura-
tions will be useful for real applications. Therefore, the search for equilibrium be-
tween complexity of the task and modular robot configuration must be achieved.  
656 J. Baca et al. 
References 
1. Yim, M., Duff, D., Roufas, K.D.: Walk on the Wild Side. IEEE Robotics & Automation 
Magazine 9(4), 49–53 (2002) 
2. Murata, S., Yoshida, E., Kamimura, A., Kurokawa, H., Tomita, K.Y., Kokaji, S.: M-
TRAN: Self-Reconfigurable Modular Robotis System. IEEE/ASME Trans. Mechatron-
ics 7(4), 431–441 (2002) 
3. Castano, A., Behar, A., Will, P.M.: The CONRO Modules for Reconfigurable Robots. 
IEEE/ASME Trans. Mechatronics 7, 403–409 (2002) 
4. Shen, W.-M., Krivokon, M., Chiu, H., Everist, J., Rubenstein, M., Venkatesh, J.: Multi-
mode Locomotion for Reconfigurable Robots. Autonomous Robots 20(2), 165–177 (2006) 
5. Escalera, J.A., Ferre, M., Aracil, R., Baca, J.: ROBMAT: Teleoperation of a Modular Ro-
bot for Collaborative Manipulation. In: Proc. 11th International Conference, Knowledge-
Based Intelligent Information and Engineering Systems, Part 2, pp. 1204–1213 (September 
2007) 
6. Crespo, A., Baca, J., Yerpes, A., Ferre, M., Aracil, R., Escalera, J.A.: CAN Application  
in Modular Systems. In: 12th International CAN Conference, Barcelona, Spain, pp. 01-1 – 
01-7 (2008) 
7. Chen, I.-M., Yang, G.: Automatic generation of dynamics for modular robots with hybrid 
geometry. In: Proceedings of 1997 IEEE International Conference on Robotics and Auto-
mation, April 20-25, vol. 3, pp. 2288–2293 (1997) 
8. Freudenstein, F., Dobrjanskyj, L.: On a Theory for the Type Synthesis of Mechanisms. In: 
Proc. 11th Int. Congress of Applied Mechanics, Munich, pp. 420–428 (1964) 
9. Gross, J.L., Yellen, J.: Graph Theory and Its applications. Chapman & Hall/CRC (2006) 
10. Chen, I.-M., Yang, G., Kang, I.-G.: Numerical inverse kinematics for modular reconfigur-
able robots. Robotic Systems J. 14, 213–225 (1999) 
11. Ball, R.: A Treatise on the Theory of Screws. Cambridge University Press, Cambridge 
(1900) 
12. Brockett, R.: Robotic Manipulators and the Product of Exponential Formula. In: Mathe-
matical Theory of Networks and Systems, pp. 120–129 (1984) 


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