Abstract— Morphogenesis is the biological process in which a fertilized cell divides, cells migrate and interact with each other, and finally resulting in the mature body plan under the control of gene regulatory networks (GRNs). Recently, biological discovered that 85% of the gene-gene regulation networks are composed of frequently recurring network patterns, which are called network motifs. Inspired by these biological studies, in this paper, we propose a developmental approach, i.e., network motifs based gene regulatory network model (NM-GRN), for self-organization of swarm robots to autonomously generate dynamic patterns to adapt to uncertain environments. First, a general GRN model is proposed with several predefined network motifs as basic building blocks, then an evolutionary algorithm is applied to evolve parameters and the structures of the NM-GRN model based on these basic building blocks. Experimental results demonstrate that the proposed bio-inspired model is effective for complex shape generation and robust to environmental changes in complex environments.

I. INTRODUCTION In this paper, we will mainly focus on the self-organization

and distributed control of swarm robots for complex shape generation. Multi-robot pattern formation has been studied extensively, which can be roughly categorized into two groups. The first group is heuristic rule-based method, such as geometrical methods, potential field-based methods, or leader-follower based methods [5, 11,15]. The second group consists of some biological inspired methods [16,19]. For example, in [19], a digital hormone model (DHM) was proposed as a decentralized control method for robot swarms, which is inspired from biological hormone diffusion systems based on Turing’s reaction-diffusion model [20] to describe the interactions between the hormones. However, most existing multi-robot pattern formation algorithms rely on predefined patterns [1], which is not suitable for dynamic uncertain environments. On the other hand, biological morphogenesis is a self-organizing process through interactions between genes, proteins, metabolites, and cells and is governed by a gene regulatory network, which is a network of the interactions among various cellular substances [3]. Inspired by biological morphogenesis, some distributed GRN-based controllers have been developed for swarm robots pattern generation and boundary coverage in our previous work [6-9, 12-13]. However, these approaches still have some issues remained.

*Y. Meng is with the Department of Electrical and Computer

Engineering, Stevens Institute of Technology, NJ 07670, USA. (Phone: 201-216-5496; e-mail: yan.meng@stevens.edu).

H. Guo is with Almende Organizing Networks, Westerstraat 50, 3016 DJ Rotterdam, Netherlands. (e-mail: scottr@almende.org).

First, the target shapes have to be predefined, which cannot adapt to unknown environments. Second, partial differential equations have been applied to describe the motion dynamics of each robot, which is only efficient for specific tasks. If the environment and tasks change, new differential equations have to be developed from scratch, which is not robust enough for dynamic environments.

Although various mathematical models of GRNs have been proposed by biological researchers [4] to describe the relationship between GRNs and the organism’s developmental process, these models always have some specific assumptions or only focus on some aspects of the GRNs. More universal or generic model development for GRNs remains to be solved. Recently, it is found that 85% percent of the gene-gene regulation networks are composed of frequently recurring network patterns, called network motifs [2]. And the property of large GRNs can be understood by analyzing these network motifs [17]. It is also found that such motifs were found in various complex networks such as biochemistry, neurobiology, ecology, and engineering [14]. Inspired by these biological studies, a new developmental approach is proposed in this paper, which is called network motifs based gene regulatory network (NM-GRN) model. Compared to other alternative swarm robot pattern formation methods, the major advantage of this new NM-GRN model is that it can automatically generate new patterns for swarm robots to adapt to uncertain environments by evolving some predefined network motifs.

The rest of the paper is organized as follows. Sec. 2 describes the general framework of the NM-GRN model. Some basic network motifs are introduced in Sec.3. In Sec. 4, an evolutionary algorithm is applied to evolve the parameters and structures of the NM-GRN model. Experimental results using e-puck robots are provided in Sec. 5. Conclusions are discussed in Sec.6.

II. THE MODEL

A. Problem Statement Here we aim to develop a new distributed model which

allows swarm robots to automatically generate different yet suitable shapes to traverse an unknown environment. If the path in the environment is wide enough, the robots should form a circular shape. Otherwise, the robots need to automatically generate appropriate shape to fit in the environment and traverse though the environment. It is assumed that the environment can be represented by a grid-based map, where that each robot occupies one grid at a time, and the distance between robots is one grid. Since the

A Bio-Inspired Developmental Approach to Swarm Robots Self-Organization

Yan Meng and Hongliang Guo

2012 IEEE/RSJ International Conference onIntelligent Robots and SystemsOctober 7-12, 2012. Vilamoura, Algarve, Portugal

978-1-4673-1736-8/12/S31.00 ©2012 IEEE 3512

number of robots does not change during the traversing process, the overall size of the covered area should keep the same even though different shapes may be generated at different times. Furthermore, the generated shapes should not be too close to the obstacles (otherwise may easily collide with obstacles), or too far away from the obstacles (otherwise a thin line may be generated all the time to traverse all the paths, which is not ideal for exploration tasks either).

It is assumed that the robots know the moving direction of the traversing path, but don’t know where the constraints of the traversing path are. Since we mainly focus on robot shape generation in this paper, we will use the control dynamics proposed in our previous work [7] for shape formation (i.e., to physically move the robots to the generated shapes). The robots have limited sensing and communication capabilities, which means they can only sense local environment and communicate with their local neighbors.

B. Organizing Robots In biological morphogenesis, morphogen gradients that guide the cell migration are either directly obtained from the mother cells (maternal gradients) or generated by a few cells known as organizers. Inspired by these biological studies, we assume that there is an organizing robot in the swarm robots which can initiate the pattern generation process. Only those robots who detect obstacles in the environment and have to change their traversing path to avoid obstacles will be the candidates to compete for the organizing robot through local communication.

The communication content is the path parameters of every robot. The robot with the largest path parameter is in the most forefront of the path and will be selected as the organizing robot. For example, in a 2D environment, the robot’s current position is ),( yxpos = , since we know the path representation )(up , the path parameter for the robot can be calculated as ||))((||arg

minposupu −= . Where || . || is the norm

of a vector. Once the organizing robot is selected, it will generate a

local coordinate system where itself is located at the origin of this local coordinate system and the orientation of the local coordinate system is decided by its own heading direction. Then, the organizing robot will develop/grow the target shape from a single robot to an appropriate shape (with predefined fixed number of robots) to adapt to the current environment. The organizing robot will first evaluate the neighboring grids in the local environment in an anticlockwise order. If the neighboring grids are open, it will grow into that area. If there are some obstacles in the neighboring grids, it will skip those grids and try to grow toward other open grids. Once the stop-growing condition is reached (i.e., reaching the predefined number of robots), the organizing robot stops growing and the current shape is the target shape for the robotic swarm. C. The Generic GRN Framework In biological systems, when the DNA is expressed, information stored in the genome is transcribed into mRNAs and then translated into proteins. Some of these proteins are

transcription factors (TFs) that can regulate the expression of their own or other genes, thus resulting in a complex network of interacting genes termed as a gene regulatory network (GRN). In this sense, TFs can be used to denote the inputs of the GRN framework, which are responsible to collect the environmental information and/or represent the internal states of the robots. Genes can be used as the processors of the GRN framework, which are responsible to process the inputs of the GRN and send signals to trigger the outputs of the GRN. Proteins can be adopted as the outputs of the GRN, which represent the robot actions/behaviors. In the corresponding robotic systems, we consider only one TF as an input to collect environmental information, called TF1. TF1 will gauge the minimal distance from the current robot to the nearest obstacle. TF1 can be defined as

min( )1 dTF e−= (1) Where d is the collection of the minimal distances to all the detected obstacles in the environment by the robot.

When the organizing robot initiates the shape generation process, the number of robots will be increased from one till the predefined number of robots (and will keep the same number during traversing process). TF2 represents this internal state (i.e., the number of robots within the shape at the current moment) during the shape growing phase. To reduce the system complexity and computational cost while improving the system robustness, only two genes are applied as the processors of the GRN, called G1 and G2, respectively. A robot may have three different actions: grow into an area (denoted by P1), skip an area and try to grow into another area (denoted by P2), or stop growing (denoted by P3). In summary, in this GRN framework, the genome consists of seven genetic elements: TF1, TF2, G1, G2, P1, P2, and P3. There are several assumptions on the interactions among these elements. Firstly, the environment is unknown to all the robots. The robots have to figure out the appropriate shapes dynamically when they traverse the environment. Secondly, a robot can only perform one action at a time, so there must be negative regulations among the proteins P1, P2 and P3. Thirdly, it is assumed that the environment does not change while the robot is performing any action; therefore, TF1 is not regulated by any proteins when the robot is executing any action. But the environment can be changed after the action is conducted. Finally, TF2 (the number of robots) only changes when the robot performs action P1 (grow into this area) during the shape growing phase.

With these assumptions, the general GRN framework can be depicted in Fig. 1. This general GRN framework is embedded into each robot in a distributed manner in the system. However, only the organizing robots will initiate the framework and generate the suitable shapes virtually in its own mind. Then, this generated shape will be sent to all the other robots through local communication so that they can merge to this shape automatically using the controller we proposed in our previous work [7].

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Fig. 1. The general GRN framework.

III. BASIC NETWORK MOTIFS Recent biological studies suggest that some “network

motifs” occur significantly more frequently than other structures in the GRNs in a multi-cellular organism [14,17], and in most cases GRNs are largely composed of these motifs. Inspired by these studies, we propose four basic regulations, namely, positive, negative, AND, and OR regulations as the basic network motifs to constructing the generic GRN framework. These regulations can be applied to any regulation among genes, proteins, and TFs, for example, gene-gene, gene-protein, gene-TFs, protein-protein, TFs-TFs, protein-TFs, etc. In the following description, to make it simple, we only use gene-gene regulations as examples here.

A. Positive Regulation A positive regulation is from X to Y is depicted in Fig. 2

and defined as: when gene X activates gene Y or gene X poses a positive feedback to gene Y.

Fig. 2. Positive regulation from X to Y

The mathematical description of the positive regulation from X to Y is defined as follows:

( , )dy y sig xdt

θ= − + (2)

( )1( , )

1 k xsig xe θθ − −=

+ (3)

Where y is the expression level of gene Y; x is the expression level of gene X. θ is a regulatory parameter which will be optimized later by an evolutionary algorithm in our model.

B. Negative Regulation A negative regulation is depicted in Fig. 3 and defined as:

when gene X inhibits gene Y or gene X poses a negative feedback to gene Y. The mathematical description of negative regulation is defined in Eqn. (4).

Fig. 3. Negative regulation from X to Y

(1 ( , ))dy y sig xdt

θ= − + − (4)

Where the definitions of x, y and θ are the same as we described in the positive regulation.

C. OR-Regulation Gene Y expresses only if either gene X1 or gene X2 expresses, this is called OR-regulation from X1 and X2 to Y, which is depicted in Fig. 4. The mathematical description of the OR-regulation is defined as follows:

11 1 1( , )dg g sig x

dtθ= − + (5)

22 2 2(1 ( , ))dg g sig x

dtθ= − + − (6)

1 2( ( , ) ( , ))dy y sig g sig gdt

θ θ= − + + (7)

Where 1g and 2g are the expression levels of gene X1 and X2, respectively. 1x and 2x are the inputs to 1g and 2g , respectively. y is the expression level of gene Y.

θθθ and ,, 21 are regulatory parameters.

Fig. 4. OR-Regulation Relationship between X1, X2 and Y

D. AND-Regulation Gene Y will express only if both genes X1 and X2 express,

this is defined as the AND-regulation from X1 and X2 to Y, which is depicted in Fig. 5. The mathematical description of the AND-regulation relationship is as follows:

11 1 1( , )dg g sig x

dtθ= − + (8)

22 2 2(1 ( , ))dg g sig x

dtθ= − + − (9)

1 2( ( , ) ( , ))dy y sig g sig gdt

θ θ= − + ⊗ (10)

Where the definitions of all the variables are the same with the ones described in the AND-regulation.

Fig. 5: AND-Regulation Relationship between X1, X2 and Y

IV. EVOLVING NETWORK MOTIFS MODEL

A. Simplifications and Parameterization of the Generic GRN Framework

Based on the predefined network motifs described in the previous section, we need to construct the structure of the GRN using an evolutionary algorithm. In this manner, each link in the GRN can be mathematically modeled by one basic network motif. To reduce the fitness searching landscapes of the evolutionary algorithm, the following realistic simplifications have been made. Since there are three types of robot actions (P1, P2, and P3), the robot must respond to any outputs sent from the combinations of G1 and G2 processors (as shown in Fig. 1). There are four possible combinatory regulations of G1 and G2, namely,

TF1 TF2

P3P2P1

G2G1

x y+

x y-

x1

x2

+

-

+y

y

x1

x2

+

-

x

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positive-positive, positive-negative, negative-positive, and negative-negative regulations. For example, positive-positive regulation means that both G1 and G2 have a positive regulation on the target protein, and meanwhile, the combinatory regulation is an AND-regulation. Other types of regulations can be deducted accordingly.

For each robot, actions P1 (grow into the evaluation area) and P2 (skip the evaluation area and grow into another area) must respond to one of the four combinatory regulations. Action P3 (stop growing) must respond to the other two unselected combinatory regulations. For example, if P1 responds to positive-positive regulation, and P2 responds to negative-positive regulation, P3 must respond to other two combinatory regulations. The regulatory parameters 1θ and

2θ introduced in the basic network motifs are the same for P1, P2, and P3. In this manner, we have 17 parameters that need to be optimized, namely 11θ , 12θ , 21θ , 22θ , 11t , 12t , 21t , 22t , 1c ,

2c , 1θ , 2θ , 1t , 2t , 12p , 13p , and 23p .

ijθ and ijt refer to regulation parameters and regulation

types from TFi to Gj (i, j = 1, 2), respectively. ijt can be 1 (positive regulation) or 2 (negative regulation). For example, if 11 2.5θ = and 11 1t = , which means that it is a positive regulation (the meaning of ijt will be specified in the following subsection). The partial dynamics (not considering the effect of TF2) of G1 can be described as:

11 1 11( , )dg g sig tf

dtθ= − + (11)

Where ic (i = 1, 2) denotes the combinatory regulation type from TF1 and TF2 to Gi. ic can be 1 (AND-regulation) or 2 (OR-regulation).

1θ and 2θ are the shared common regulatory parameters for the regulations between genes and proteins. it denotes the responding combinatory regulation of Pi, where it can be 1 (positive-positive), 2 (negative-positive), 3 (negative-negative), or 4 (positive-negative). ijp denotes the regulatory parameter for the negative regulation between proteins Pi and Pj (which corresponds to θ in Eqn. 4 ).

B. Evolving the Parameterized GRN Framework There are two requirements for the shape generation: (1) to keep the same size of the shape covered by the robots all the time; (2) to ensure robots not too close to or too far away from obstacles. Suppose the expected size of the generated shape covered by robots is denoted by 0s . The shortest distance from the generated shape to the nearest obstacle should not be smaller than a predefined distance 0d but as close to 0d as possible. To this end, the fitness function of the evolutionary algorithm can be set up as:

02 20( 2 ) (max( 1) )df TF s TF e−= − + − (12)

Where f is the fitness value. TF1 gauges the distance from the generated shape to the obstacle. TF2 denotes the number of robots within the shape during the shape growing phase.

Here, we adopt the covariance matrix adaptation evolution strategy (CMA-ES)[10] to optimize those 17 parameters. Since CMA-ES needs to perform parameter optimization in a continuous domain, we need to transform the discrete numbers (such as ijt ) into continuous values.

For example, ijt can be assigned as real values from 0 to 1.

If ijt <0.5, it is setup as 1. If ijt >0.5, it is 2. Similar procedure

will be applied for ic and it . it can be assigned to be real values from 0 to 1. If it <0.25, we set it as 1; if 0.25 0. 5it< < , it is 2, if 0.5 0. 75it< < , it is 3, if 0.75 1it< < , it is 4.

The parameters are setup as follows: 21θ and 22θ are assigned to be real numbers from 0 to 400. All the other parameters are assigned to real numbers from 0 to 1. Due to the large parameter space, the population size was set to 400. The evolutionary algorithm was run for 50 generations. During the evolutionary process, the expected size of the generate shape is set as 200, and the shortest distance from the shape to the obstacles is set as 1.

After the evolutionary process, we select the best individuals in the last generation as the optimal parameter combination for the GRN framework. The optimized parameter combination is listed as follows: 11θ = 0.3668,

12 0.0032θ = , 21 0.0891θ = , 22 199.8601θ = , 11 1t = ,

12 2t = , 21 2t = , 22 1t = , 1 1c = , 2 1c = , 1 0.2786θ = ,

2 0.0021θ = , 1 3t = , 2 4t = , 12 0.3755p = , 13 0.4025p = , and 23 0.2486p = . These parameters are applied to the GRN framework that is used in the following simulations and experiments.

From the above evolved parameters, the final GRN framework can be constructed as follow. 11 1t = ; 12 2t = ;

21 2t = ; 22 1t = . 1 1c = and 2 1c = suggest that the combinatory regulation from TF1 and TF2 to G1 and G2 are both OR-regulations. 1 3t = suggests a negative-negative combinatory regulation from G1 and G2 to P1; 2 4t = suggests a positive-negative combinatory regulation from G1 and G2 to P2. Based on the aforementioned deduction, the final GRN framework is depicted in Fig 6. Since P1 and P2 will respond to the negative-negative regulation and positive-negative regulation from G1 and G2, P3 will respond to the remaining regulation relationships (positive-positive and negative-positive), we can deduct that G1 will not have any effect on P3, so only G2 will have a positive regulation on P3. These parameters are applied to the following simulation and experimental results.

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Fig. 6. The evolved GRN framework.

We use the Non-Uniform Rational B-Spline

(NURBS)[18] model to represent the environment and the traversing path of the robots. NURBS is a mathematical model commonly used in computer graphics and structural design for generating and representing curves and surfaces. NURBS can offer one common mathematical form for both analytic and freeform shapes. A NURBS curve is defined by its order, a set of weighted control points, and a knot vector. The control points define the shape of the curve, and the knot vector is a set of parameters that determines where and how the control points affect the NURBS curve. A NURBS model can represent both curve and surface in a two- or three-dimensional Cartesian space. Please refer to [18] for the more detailed description of NURBS.

V. EXPERIMENTAL RESULTS

A. Adaptive Shapes in an Unknown Environment To evaluate whether the robots can generate different

suitable shapes (with the same size of the initial shape) using the proposed GRN-based framework to traverse through a complex unknown environment, a set of snapshots of this simulation is shown in Fig. 7.

Fig. 7: Different generated shapes in different environment.

We conducted the simulation with 40 robots, and each

robot will occupy an area size of 5. Since it is very congested to show individual robot in Fig. 7, we only show the constructed shape here. We can see that the robots can automatically generate various suitable shapes to adapt to different environmental constraints using the proposed GRN framework.

B. Experimental Results using E-puck Robots To evaluate the proposed pattern generation algorithm, a

proof-of-concept experiment has been performed for a swarm robotic system consisting of six e-puck education robots (http://www.e-puck.org/). Each e-puck robot (as shown in Fig. 8) is approximately six centimeters in diameter with a circumferential ring of eight infrared proximity sensors, a pair of step motors in a differential-drive configuration, three microphones, one Infrared sensors, and one ZigBee wireless communication card. Infrared proximity sensors are used for distance detection and simple local communications between robots. Microphones are used to trigger the start of the experiments, and wireless card is used for debugging and uploading the software on the e-puck robots. In the experiments, each robot is provided with a starting position in a global coordinate system.

Fig. 8: The e-puck educational robot.

To implement the proposed model on physical robots, we

have to consider a few real-world constraints. First, it is assumed that robots are holonomic in the simulations. However, e-puck robots are differential-drive robots and non-holonomic. Second, self-localization of the robots in an indoor environment may become an issue. The GRN-based model does not consider the nonholonomic constraints imposed by the differential-drive robot. Therefore, there must be a translation between the desired motion of the GRN dynamics and the robot’s actual motion. For this proof-of-concept implementation, self-localization is performed by an open-loop estimation using an odometry method with the onboard encoders. As we know, the measurement errors using the odometry method

TF1 TF2

P3P2P1

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may get accumulated over time. To mitigate errors in measurement of the e-puck’s geometry, a scaled version of the UMBMark calibration procedure was performed on each e-puck robot. Fig. 9 shows a set of snapshots of the experimental results. The robots move forward with an initial pattern. When they traverse through a narrow area, some of the robots detect that the initial pattern is too big to traverse the narrow area. These robots then compete for the organizing agent. Once the organizing agent is selected, it triggers the pattern generation process. Once the new pattern is generated by the organizing robot, it sends the target pattern information to others, so that all the robots form the target pattern to traverse through the narrow area. When they pass the narrow area, the robots will reorganize to their original pattern and continue moving.

(a) t = 1sec (b) t = 11 sec

(c) t = 16 sec (d) t= 20sec

(e) t = 29 sec (f) t = 37 sec

Fig. 9. A set of snapshots of an experiment showing that six e-puck robots are able to traverse through a narrow environment using the proposed NM-GRN framework.

VI. CONCLUSION AND FUTURE WORK Inspired by the biological gene regulatory networks and network motifs, we proposed a new developmental approach for swarm robot pattern generation to automatically adapt to uncertain complex environments. Firstly, a general GRN model has been proposed. Then, several network motifs were proposed as the basic building blocks for the general GRN model. Lastly, an evolutionary algorithm is applied to evolve parameters and the structures of the general GRN model. A proof-of-concept experiment using e-puck robots has demonstrated the efficiency and feasibility of the proposed model. Currently, the evolutionary process is still an off-line process and is independent of the developmental process (i.e., pattern generation process). We will investigate this issue in our future work by introducing an online evolving GRN

framework. Another issue we plan to investigate in the future is how to deal with robot failures during traversing.

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