JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 An Automatic Design Framework of Swarm based on Multi-objective Genetic Programming Zhun Fan, Senior Member, IEEE, Zhaojun Wang, Xiaomin Zhu, Bingliang Hu, Anmin Zou, and Dongwei Bao

Abstract—Most existing swarm pattern formation methods predetermined trajectory and maintain a specific swarm pattern depend on a predefined regulatory network (GRN) structure in the execution of tasks. For example, Jin [11] proposed a that requires designers’ priori knowledge, which is difficult to hierarchical gene regulatory network for adaptive multi-robot adapt to complex and changeable environments. To dynamically adapt to the complex and changeable environments, we propose pattern formation. In this work, the swarm pattern is designed an automatic design framework of swarm pattern formation as a band of circle, which encircles targets in a dynamic based on multi-objective genetic programming. The proposed environments. In the latter case of using adaptive formation, framework does not need to define the structure of the GRN- swarm robots follow an adaptive pattern to encircle targets in based model in advance, and it applies some basic network motifs the environment. For example, Oh et al. [12] have introduced to automatically structure the GRN-based model. In addition, a multi-objective genetic programming (MOGP) combines with an evolving hierarchical gene regulatory network for morpho- NSGA-II, namely MOGP-NSGA-II, to balance the complexity genetic pattern formation in order to generate adaptive patterns and accuracy of the GRN-based model. In evolutionary process, which are adaptable to dynamic environments. an MOGP-NSGA-II and differential (DE) are applied to In addition, swarm pattern formation has been widely ex- optimize the structures and parameters of the GRN-based model plored in recent years, which can be divided into four cate- in parallel. Simulation results demonstrate that the proposed framework can effectively evolve some novel GRN-based models, gories [1], namely , reaction-diffusion model, and these GRN-based models not only have a simpler structure chemotaxis and gene regulatory network (GRN). and a better performance, but also are robust to the complex The basic idea of morphogenesis is that the - and changeable environments. like signals can provide the information of relative locations Index Terms—Gene Regulatory Networks (GRN), Swarm for each robot in swarm robots. For example, Mamei [13] Pattern Formations, Self-organization, Multi-objective Genetic has studied swarm robots utilize morphogen diffusion to form Programming (MOGP), Differential Evolution (DE). a circular, ring or polygonal pattern formation. In addition, morphogenesis can be combined with other methods to form I.INTRODUCTION arbitrary swarm patterns. For example, Kondacs [14] combines N general, multi-robot systems (MRSs) are composed of morphology and geometry to generate two-dimensional arbi- I a large number of minimal, simple and low-cost robots, trary swarm patterns. each of which has limited functions and performance [1]. The second category is reaction-diffusion model, which In most cases, these simple robots can work together in a utilizes several morphogen in a cell to react with morphogen collaborative way to accomplish complex and changeable tasks in other neighbouring cells to generate complex patterns [15]. that single robot can not accomplish. In addition, MRSs with For example, the reaction-diffusion turing pattern [16] is a arXiv:1910.14627v2 [cs.NE] 1 Nov 2019 parallelism, scalability, stability, low-cost, strong typical example of reaction-diffusion model, which utilizes and high adaptability, are widely applied into the collaborative two hormones, an activator and an inhibitor, to form complex target search and rescue [2], [3], small satellites communica- and arbitrary swarm patterns. Moreover, Kondo [17] has used tion [4], [5], cooperation and navigation planning [6], [7] and mathematical models to explain the development patterns of source localization [8], [9], et al. Turing model in biological systems. The swarm pattern formation is a typical task for MRSs Without central control or coordination, chemotaxis can and embodies pattern generation and pattern maintenance [10]. control the movement, aggregation and sorting of swarm In different tasks and dynamic environments, swarm pattern robots in pattern formation. For example, Fate and Vlassopou- formation represents the coordination and local interaction of los [18] applied chemotaxis in swarm robots, which forms multi-robots to generate and maintain a swarm pattern forma- aggregation formation in a dynamic environment. Bai and tion with a certain shape, in which the shape of pattern can Breen [19] demonstrated that chemotaxis can form complex be either predefined or adaptively formed in a self-organised, swarm patterns. coordinate and cooperate way through local interaction with The GRN model [20] is inspired by the reaction-diffusion neighbouring robots and the environments [1]. In the former model, which can control the behavior of each robot in case of using a predefined pattern, swarm robots follow a swarm robots. For example, Jin [11] applied an evolutionary algorithm to evolve parameters of a GRN subnetwork, which gets a multi-robot pattern formation. Meng [21] combined JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 2

GRN and B-spline to form complex patterns in swarm robotic automatic design framework of GRN-based model for adaptive systems. Oh [12] utilized some network motifs to evolve GRN, swarm pattern formations is proposed in Section IV. Section V which generates swarm pattern formation. Oh [22] utilized a presents numerical simulation results from moving targets and two-layer hierarchical GRN to cover a desired region for target various obstacles, which verifies the adaptability, scalability entrapment. and robustness of the proposed framework. In Section VI, A substantial limitation of most existing GRN-based model conclusions and future works are discussed. for swarm pattern formation is that the structure of the network must be predefined, resulting in the inability to adapt II.PROBLEM STATEMENT AND ASSUMPTIONS to different tasks and dynamic environments. Furthermore, The studied problem mainly focus on how to generate another limitation of swarm pattern formations is that existing adaptive swarm patterns in dynamic, complex and changeable evolutionary GRN methods are mainly to optimize the param- environments. More specifically, a swarm pattern needs to eters of the network structure, and these methods are unable to form different shapes in various restricted environments. optimize topology structures of a GRN-based model, resulting In biological morphogenesis, morphogen gradients that in inability to adapt to the complex dynamic environments. guide cells migration are either directly obtained from the In order to produce a GRN-based model adapted to the mother cells or generated by a few cells known as histi- complex dynamic environments, we adopt an idea of automatic ocytic cells [15]. Inspired by these biological studies, we design [23]. The automatic design allows modules from ele- assume that there are some mother robots (a mother robot mentary building blocks and operators to automatically gener- is equivalent to a mother cell in biological morphogenesis) ate models to satisfy predefined specifications. The automatic in swarm robots. The mother robots can detect obstacles and design process is that a modular modeling language is applied targets in the environment and they are responsible for swarm to describe elementary building blocks, and then a genetic pattern formation. When the mother robots detecting targets programming (GP) is employed to automatic optimization to (or obstacles and targets) in an environment, the mother robots get an optimal model. When automatic design is applied to can form a morphogen gradient space according to the location electromechanical systems, a modular modeling language is information of the targets (or obstacles and targets). In the bond graph. For example, Erik Goodman [24] proposed an morphogen gradient space, the concentration of morphogen approach combining GP and bond graph to use in electrome- decreases as it moves away form the targets. Points whose chanical system. Jean-Francois Dupuis [25] proposed a hybrid gradient value are higher than a certain percentage of the system evolutionary design method by combining hybrid bond maximum concentration value are selected as candidate points, graph with GP. When automatic design is applied to swarm and swarm robots move to candidate points to form a swarm robots, a modular modeling language is finite-state machine. pattern. For example, Lorenzo Garattoni [26] used finite state machine In order to apply the proposed automatic design framework to control swarm robot, which could collectively sequence of GRN-based model to the swarm pattern formation. Some tasks whose order of execution was a priori unknown. assumptions are listed as follows: To tackle limitations of most existing GRN-based model for swarm pattern formation, we propose an automatic de- 1) The swarm robots can use on-board sensors (such as sign framework of swarm pattern formation based on multi- encoders and sonar sensors) to locate themselves at any objective genetic programming (MOGP). The proposed frame- time, as reported in [27]. work does not need to define the structure of GRN-based 2) The base station contains a sufficient number of robots model in advance. Some basic network motifs are created to complete the pattern formation task. By the circumfer- by the GRN-based of the model, and ence of the swarm pattern, a sufficient number of robots they are known as a modular modeling language to de- can be called from the base station. scribe elementary building blocks. The GRN-based models 3) Each of swarm robot has a limited range of perception. are automatically structured by employing these basic network Therefore, swarm robots can detect targets, obstacles and motifs. In evolutionary process, to balance the complexity other robots within the range of perception. and accuracy of GRN-based models, MOGP and NSGA-II 4) Each of swarm robot also has a limited range of commu- are employed, namely MOGP-NSGA-II. An MOGP-NSGA- nication, and the communication between swarm robots II and differential evolution (DE) are applied to optimize the and base stations is not limited. In the communication structures and parameters of the GRN-based model in parallel, range, the robot can communicate position and speed which can get some GRN-based models with simple structure information with other neighboring robots. and excellent performance. In addition, these models also have 5) The maximum movement speed of each robot is faster a good performance when they migrate directly to complex than the maximum movement speed of the targets. environments. Thus, the proposed framework can enhance 6) At least one mother robot can detect all targets and humans understanding of the structure of GRN-base model obstacles in the environment. and realize an innovative design of GRN. The remainder of this paper is structured as follows. In Sec- III.RELATED WORK tion II, the problem statement of GRN-based model is intro- In this section, a generic GRN framework using differential duced, and some assumptions are listed. Section III introduces equation model is discussed and the basic idea of MOGP and a generic GRN basic framework and the proposed MOGP. An DE is introduced. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 3

Fig. 2. A tree in genetic programming. Fig. 1. A general framework of GRNs for swarm pattern formation. is the encode method. In GP, a hierarchical structured tree is A. The Generic GRN Framework used to encode the individuals in a population, that is, each In order to develop an automatic design framework of GRN- individual in the population is a hierarchical structured tree based model, we need to understand the basic process of GRN. consisting of functions and terminals. The structure of the tree The GRN is a network consisted by the interaction of gens in is dynamically and adaptively adjusted. The structure shown in cell-cell. That is, the expression of a gen is affected by other Fig. 2, the individual is represented by functions and terminals. , which in turn affect the expression of other genes. In this case, the model is represented as follows: The complex interaction between these genes constitutes a x1 y = + x2 − x1 (3) suitable GRN. One of the key issues is how to construct x2 GRN. The conventional mathematical models of GRN include When applying GP to solve some real-world optimization model [28], bayesian network model [29] problems [32], [24], [33], [34], [35], [36], [37], [25], ter- and differential equation model [30]. In this paper, we use minators and operators should be defined according to the the differential equation model to study the swarm pattern characteristics of a problem. GP can optimize the topologies formation. and parameters of the GRN at the same time. For swarm In the differential equation model, a general form of differ- pattern formation. We need to build some predefined network ential equation model [31] for describing GRNs is as follows: motifs. Second, the positions of targets and obstacles are dxi employed as terminals. In each iteration, the optimal individual = f (x) (1) dt i is selected by a fitness function evaluation. Where xi is the concentration of the mRNA that reflect In real-world optimization problems, there are usually more dxi the expression level of the ith gene, and dt is the rate of than one conflicting objectives. For swarm pattern formation, change of the ith gene at time t, so the model is called the the complexity and accuracy of the GRN-based model are two dxi dynamic equation. Furthermore, dt illustrates the regulatory conflicting objectives. In this paper, the proposed automatic mechanism among genes. A simple form of a linear additive design framework applies a nesting algorithm, which combines function is as follows: GP and NSGA-II, namely MOGP-NSGA-II [38], to optimize dx the swarm pattern. In MOGP-NSGA-II, NSGA-II [39] is one i = Σn w x + b (2) dt j=1 ij j i of the state-of-the-art multi-objective evolutionary algorithm, which is applied to balance the complexity and accuracy of Where wij is a real number and a weight of jth gen. Further- more, it describes the type and strength of the influence of the GRN-based models. jth gene on the ith gene. bi is the ith gen’s external stimulus. For added biological realism (all concentrations get saturated C. Differential Evolution at some point in time t), a sigmoid function may be included Differential Evolution (DE) [40] grew out of Ken Price’s into the equation. attempts to solve the Chebychev Polynomial fitting Problem We proposed a General Framework of GRN for Swarm that had been posed to him by Rainer Storn. DE is also a Pattern Formation, as illustrated in Fig. 1. First, morphogen direct search method. As the number of iterations in- concentrations are produced based on the location of targets creases, individuals adapted to the environment are preserved. and obstacles in the environment, which serve to activate genes Like other kinds of evolutionary algorithms, DE contains three g1, g2 and g3. Then, the activated genes g1, g2 and g3 are operations: mutation, crossover and selection. Among these combined through a variety of ordinary differential equations three operations, mutation operation is very important for DE. to generate swarm patterns. Mutation operation generates new individuals by combining randomly selected or given individuals, and its main purpose B. Genetic Programming is to improve the diversity of the population, so as to prevent Genetic programming (GP) is an extension of genetic al- the algorithm from falling into local optimum and premature gorithm (GA). The biggest difference between GP and GA convergence. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 4

In this paper, DE is applied to optimize the parameters of where g1 and g2 are the expression levels of gene X1 and X2, GRN-based models, which aims to find optimal parameters in respectively. a GRN-based model. 4) Logical NAND regulation A logical NAND regulation is defined as if either gene IV. THE AUTOMATIC DESIGN FRAMEWORKOF SWARM X1 or gene X2 does not express, gene Y expresses. The PATTERN FORMATION mathematical description of the logical NAND regulation is defined as follows: In this section, the automatic design framework of swarm dy pattern formation based on MOGP is introduced. In addition, = −y + 1 − sig(g1 ∗ g2, θ, k)) (8) ten predefined basic network motifs and the fitness function dt are discussed. where g1 and g2 are the expression levels of gene X1 and X2, respectively. 5) Logical OR regulation A. Basic Network Motifs A logical OR regulation is defined as if either gene X1 or gene When applying MOGP-NSGA-II to evolve GRNs, a typical X2 expresses, gene Y expresses. The mathematical description task is to define some basic network motifs. Recent researches, of the logical OR regulation is defined as follows: such as biochemistry, neurobiology, ecology and engineering, dy find that patterns of inter-connections occurring in complex = −y + sig(g1 + g2, θ, k)) (9) networks are significantly higher than those in randomized dt networks [41]. Peter M. Bowers [42] proposed that the logic where g1 and g2 are the expression levels of gene X1 and X2, analysis of phylogenetic profiles was applied to identified respectively. triplets of whose presence or absence obey certain 6) Logical NOR regulation logic relationships. In addition, the logic relationships in A logical OR regulation is defined as if both gene X1 and triplets of proteins are also frequently found in GRNs of gene X2 do not express, gene Y expresses. The mathematical a multi-cellular organism. Inspired by these researches, ten description of the logical NOR regulation is defined as follows: predefined network motifs, such as positive, negative, AND, dy OR, XOR and so on, are utilized as the basic network motifs, = −y + 1 − sig(g1 + g2, θ, k)) (10) dt which are used to construct GRNs. 1) Positive correlation regulation where g1 and g2 are the expression levels of gene X1 and X2, A positive regulation is defined as gene X activates gene Y . respectively. That is, Gene X has a positive effect on Y . The 7) Logical ANDN regulation mathematical description of the positive correlation regulation A logical ANDN regulation is defined as if gene X1 expresses from X to Y is defined as follows: and gene X2 does not express, gene Y expresses,as defined by Eq. (11). Or if gene X does not express and gene X dy 1 2 = −y + sig(x, θ, k) (4) expresses, gene Y expresses, as defined by Eq. (12). dt dy = −y + sig(g ∗ (1 − g ), θ, k)) (11) 1 dt 1 2 sig(x, θ) = (5) 1 + e−k(x−θ) dy where x represents the expression level of gene X, and y = −y + sig((1 − g1) ∗ g2, θ, k)) (12) represents the expression level of gene Y . θ represents a dt regulatory parameter for the , and k is a scale where g1 and g2 are the expression levels of gene X1 and X2 factor for the gene expression. in Eq. (11) and Eq. (12), respectively. 2) Negative correlation regulation 8) Logical ORN regulation A negative regulation is defined as gene X inhibits gene Y . A logical ORN regulation is defined as if gene X1 expresses That is, Gene X has a negative feedback effect on Y . The or gene X2 does not express, gene Y expresses, as defined by mathematical description of the negative correlation regulation Eq. (13). Or if gene X1 does not express or gene X2 expresses, from X to Y is defined as follows: gene Y expresses, as defined by Eq. (14). dy dy = −y + (1 − sig(x, θ, k)) (6) = −y + sig(g + (1 − g ), θ, k)) (13) dt dt 1 2 where x represents the expression level of gene X, and y represents the expression level of gene Y . dy = −y + sig((1 − g1) + g2, θ, k)) (14) 3) Logical AND regulation dt A logical AND regulation is defined as if and only if both gene where g1 and g2 are the expression levels of gene X1 and X2 X1 and gene X2 express, gene Y expresses. The mathematical in Eq. (13) and Eq. (14), respectively. description of the logical AND regulation is defined as follows: 9) Logical XOR regulation dy A logical XOR regulation is defined as iff gene X1 and gene = −y + sig(g1 ∗ g2, θ, k)) (7) X both express or gene X and gene X both do not express, dt 2 1 2 JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 5

obs gene Y expresses. The mathematical description of the logical between the pattern and the obstacles. In this paper, dmin is XOR regulation is defined as follows: set to 2. Hence, the fitness function is set up as follows: sig(dij , d , k ) + sig(d , dij , k ) dy Np Nt pt max 1 min pt 2 = − y + sig(g1 ∗ (1 − g2), θ, k)) f2 =Σi=1Σj=1 dt (15) NpNt (18) obs ik + sig((1 − g1) ∗ g2, θ, k)) sig(d , d , k ) Np No min po 3 + Σi=1Σk=1 NpNo where g1 and g2 are the expression levels of gene X1 and X2, respectively. where Np, Nt and No are the number of swarm robots, the 10) Logical XNOR regulation number of targets and the number of obstacles, respectively. ij A logical XNOR regulation is defined as iff one of either gene In addition, dpt is the distance from the ith swarm robot to the ik X1 or gene X2 express, gene Y expresses. The mathematical jth target. Similarly, dpo is the distance from the ith swarm description of the logical XNOR regulation is defined as robot to the kth obstacle. follows: C. Automatic Design Framework Embedded in GRN-based dy = − y + 1 − sig(g ∗ (1 − g ), θ, k)) Model dt 1 2 (16) The proposed automatic design framework of GRN-based − sig((1 − g1) ∗ g2, θ, k)) model creates ten basic network motifs according to the where g1 and g2 are the expression levels of gene X1 and X2, interaction of gens in cell-cell. Then this framework employs respectively. the automatic design method to design the GRN-based model When applying MOGP-NSGA-II to evolve GRNs, these ten automatically according to the requirements of the scene. predefined basic network motifs are employed to constitute Finally, an MOGP-NSGA-II and DE are applied to optimize the GRNs. Furthermore, a regulatory parameter θ will be the structures and parameters of the GRN-based model in optimised by DE. Here, the scale factor k in each basic parallel, as so to get an optimal GRN-based model. The main is set to 1. difference between the proposed framework and the manual design framework is that the proposed framework does not need to predefine different GRN-based model structures under B. Fitness Function various environments, which solves the limitation of GRNs for swarm pattern formations. In other words, the proposed When applying MOGP-NSGA-II to evolve GRN-based framework applies MOGP-NSGA-II and DE to optimize the models, another typical task is to define fitness functions. automatic design GRN-based model in order that the opti- Since MOGP-NSGA-II may create some complex models in mized model can cross the restricted environment without evolutionary process. The fitness functions are set to balance colliding obstacles. the complexity and accuracy of GRN-based models. In addi- The General structure of the proposed automatic design tion, literatures [11], [12] suggest environmental restrictions framework is illustrated in Fig. 3, which is divided into into a fitness function in order to make the swarm pattern three stages. The three stages are the input stage, automatic can get cross the restricted environment without colliding any generation of initialization model stage and the optimization obstacles. stage. In the input stage, the location information of targets and In order to explain the GRN-based model, i.e, the fewer obstacles is translated into an integrated morphogen gradient nodes these models have, the better the models are. To define space. It is noticeable that this transformation behavior is the complexity of the model, an important indicator is the activated when mother robots detect targets or obstacles. This number of nodes in the GRN-based model, and this indicator integrated morphogen gradient space is served as the input to can be regarded as a fitness function, as follow: the proposed framework. In the automatic generation of initial- ization model stage, GRN-based models based on swarm pat- f1 = node(mi) (17) tern information are automatically generated by an integrated morphogen gradient space and ten basic network motifs. The where mi is the ith GRN-based model and node(mi) is the automatically generated GRN-based models are constituted by number of nodes of the ith model. activate genes g1, g2, ..., gn. Once the GRN-based models are For swarm pattern formation, an important task is to en- generated, they will function as the input of the optimization circle targets without colliding them. To satisfy this task, the stage to trigger their optimization. MOGP-NSGA-II and DE nearest and furthest distances of swarm robots from targets are employed in the optimization stage, MOGP-NSGA-II is are set. That is, dmin and dmax are the allowed minimum responsible for structural optimization of GRN-based models and maximum distances between the pattern and the targets, and DE is responsible for parameter optimization of each respectively. In this paper, dmin and dmax are set to 1 and 2, GRN-based model. Through the optimization process, the respectively. In addition, another important task is that swarm automatically generated GRN-based models are optimized to robots can not collide obstacles in the restricted environment. the optimal models satisfying the restricted environment. To satisfy this task, the nearest distance of swarm robots from The framework of the proposed algorithm is introduced in obs obstacles is set. That is, dmin is the allowed minimum distance Algorithm 1.In Algorithm 1, the algorithm is initialized at lines JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 6

Fig. 3. An automatic design framework of swarm pattern formation on MOGP.

1-3. At line 1, a initial population P0 is created by using Algorithm 1: The framework of MOGP-NSGA-II for the ramped half-and-half method. At line 2, the regulatory swarm pattern formation parameters of each individual in the initial population are Input: optimized by DE. At line 3, the fitness functions are applied G: the integrated morphogen gradient space; to evaluate each individual in the initial population. At line 6, genmax: the maximum generation. the offspring population Qg is created by using the crossover, Output: a set of non-dominated and feasible solutions. mutation and reproduction of MOGP-NSGA-II. At lines 7-10, 1 Initialize: Randomly create an initial population P0 of the regulatory parameters of each individual q in the offspring GRN-base models form ramped half-and-half method; population are optimized by using DE, and the fitness values 2 Optimal Parameters: the parameters of these models are of q is obtained by using the fitness functions. At line 12, the optimized by DE; new population Pg+1 is selected by the NSGA-II selection. At 3 Evaluate: evaluate the population P0 by fitness functions; line 13, the generation counter is updated. At line 15, a set of 4 Set gen = 0; non-dominated and feasible solutions is selected. 5 while gen ≤ genmax do 6 Generate the offspring population Q by applying V. NUMERICAL SIMULATIONS g genetic operations; In this section, we will evaluate the proposed automatic 7 foreach q ∈ Qg do design framework of swarm pattern formation based on multi- 8 the parameters of q are optimized by DE; // q is objective genetic programming by numerical simulations. an individual in Qg; First, the proposed framework forms a swarm pattern to 9 evaluate each individual q in Qg using fitness encircle one or more targets in a channel, which verifies functions; the feasibility of the framework. Second, swarm patterns are 10 end formed to verify the adaptability of the proposed framework in 11 Rg ← Pg ∪ Qg; // Pg is gth parent population; a compound channel. Finally, swarm patterns that are formed 12 Form the new population Pg+1 from Rg by the in the compound channel migrate directly to an environment NSGA-II selection; where obstacles are randomly distributed, so as to verify the 13 gen = gen + 1; transferability of the proposed framework. 14 end 15 Output the non-dominated and feasible solutions. A. Swarm Pattern Formation in a channel To demonstrate the feasibility of the proposed automatic framework, the proposed framework and an evolving hierar- chical gene regulatory network (EH-GRN) [12] are compared 1. For the proposed automatic framework, the regulatory in a simply simulated area. The simply simulated area is that a parameter θ of each basic network motif is initialized swarm pattern encircles one or more targets without colliding randomly between 0 to 2. During optimization, θ of each a channel, and the channel is in a region of 20 by 20 meters, as basic network motif is optimized by DE. shown in Fig. 4. An evaluation criterion is that swarm pattern 2. For EH-GRN, the regulatory parameter θ of each basic can not collide with the channel while encircling all targets. network motif is optimized by the covariance matrix In evolutionary process, the detailed parameters are listed evolution strategy [43], and the regulatory as follows: parameter θ ranges from 0 to 2. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 7

20

15

10 y/m

5

0 0 5 10 15 20 x/m

Fig. 4. A channel is in a region of 20 by 20 meters, and green shadows represent the channel.

Fig. 5. The non-dominated solutions are achieved by MOGP-NSGA-II. The A, B, C and D are selected four points. The point A and D have two extreme 3. For the proposed automatic framework, the population cases. The point B and C are called knee points. size for structure optimization is 40, and the population size for parameters optimization is 10. For EH-GRN, the population size is set to 40. 4. For the proposed automatic framework, the maximum depth of tree is 4, and the minimum of depth of tree is 1. 5. The crossover rate of MOGP-NSGA-II and DE are 1.0 and 0.9, respectively. And the mutation rate of MOGP- NSGA-II and DE are 0.1 and 0.5, respectively. 6. The evaluation number of the proposed automatic frame- work and EH-GRN are both 4000. In Fig. 5, the non-dominated solutions are achieved by MOGP-NSGA-II when the evaluation number reaches 4000. Fig. 6. The syntax tree of point B is achieved by using MOGP-NSGA-II. x1 is that the location information of target forms a morphogen gradient space. Fig. 5 shows that the lower the number of nodes, the higher x2 is that the location information of obstacle forms a morphogen gradient the fitness value. In other words, the number of nodes at point space. NAND and XNOR are two basic network motifs. A is 3, but it has a high fitness value. Although the fitness value of point D is the smallest among all non-dominated solutions, the structure of point D is the most complex. To balance the complexity and accuracy of GRN-based models without a priori knowledge, point B and C are selected by using [44], they are called knee points. Furthermore, the fitness values of point B and point C are very close, and the structure of point B is simpler than that of point C. Thus, point B is selected in the task. Fig. 6 shows the syntax tree of the point B, and Fig. 7 shows the GRN structure of the point B. In addition, x1 and x2 are environmental inputs, that is, x1 is that the location information of target forms a morphogen gradient space, and x2 represents a morphogen concentration that is produced from obstacles. NAND and XNOR are Fig. 7. In the considered scenario, the automatic design framework is two basic network motifs. For point B, the mathematical optimized to get GRN-based model. x1 is that the location information of description of the syntax tree is as follows: target forms a morphogen gradient space. x2 is that the location information of obstacle forms a morphogen gradient space. G1 and G2 are activate genes, dy and the concentration of G represent a morphogen gradient space to form 1 = − y + 1 − sig(x ∗ x , θ , k)) 2 dt 1 1 1 1 the desired swarm pattern. dy 2 = − y + 1 − sig(y ∗ (1 − x ), θ , k)) (19) dt 2 1 2 2 − sig((1 − y ) ∗ x , θ , k)) To measure the performance of the proposed automatic 1 2 2 design framework, a state-of-the-art method, namely EH-GRN, where θ1 and θ2 are 0.8393 and 0.9256, respectively. Eq.(19) is tested in the scenario. Fig. 8 shows an EH-GRN structure is a mathematical description of NAND and XNOR. y1 and for swarm pattern formation, and the structure is optimized y2 are morphogen concentrations, where y2 is a morphogen to achieve by CMA-ES. In addition, p1 and p2 represent gradient that defines the swarm pattern. a morphogen concentration produced by the environmental JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 8

1.2095, θ9 = 1.6798, θ10 = 1, θ11 = 0.5385, θ12 = 0.2763, θ13 = 1.3445. Fig. 9 shows that the swarm pattern that optimized to achieve by the proposed automatic design framework encircles a target across a channel without colliding the channel. In particular, when the target does not enter the channel, the swarm pattern is a circular shape way to encircle the target, as illustrated by Fig. 9 (a) and (f). This is because the mother robots do not detect obstacles. The concentration of swarm pattern is only affected by the target input. Fig. 9 (b)-(d) show how the target passes through the channel. The swarm pattern still encircle the target in a circle shape way. This is because the width of the channel is larger than or equal to the furthest distances of swarm robots from the target. That is, the swarm pattern is not affected by the channel. Fig. 8. Illustration of an EH-GRN structure for swarm pattern formation. Fig. 10 shows that the swarm pattern that optimized to The structure of model is predefined, and CMA-ES is applied to optimise the achieve by EH-GRN encircles a target across a channel regulatory parameters. g1, g2 and g3 are activate genes. The concentration of M represent a morphogen gradient space to form the desired swarm pattern. without colliding the channel. In particular, Fig. 10 (a) and (f) show if the target does not enter the channel, then the swarm pattern is a circular shape way to encircle the target. inputs in Fig. 8. In other words, p1 and p2 represent the In addition, the biggest difference between Fig. 9 and Fig. morphogen concentrations produced by the environmental 10 is that the swarm pattern passes through the channel in inputs. g1, g2 and g3 are activate genes. The concentration of different patterns. That is, Fig. 10 (b)-(d) show that the swarm M represent a morphogen gradient space to form the desired pattern encircle a target in an elliptical shape way to cross the swarm pattern. θi (i = 1, ..., 13) is a regulatory parameter. channel and does not collide the channel. This is because the For an EH-GRN structure for swarm pattern formation restricted scenario is relatively simple and has little impact on in the considered scenario, each swarm robots follows the swarm pattern formation. following dynamic equations to generate a morphogen gradient Note that, the GRN-based model of the two swarm pattern space that define the swarm pattern. are optimized to achieve by the proposed automatic design dy framework and EH-GRN when one target is encircled, which 1 = −y + 1 − sig(p , θ , k)) (20) dt 1 1 1 directly applies to encircle two targets in the channel. Fig. 11 dy and Fig. 12 show two different swarm patterns encircle two 2 = −y + 1 − sig(p , θ , k)) (21) dt 2 2 2 targets across the channel. For the proposed automatic design dg framework, Fig. 11 (a) shows that the swarm pattern encircles 1 = −g + sig(y ∗ y , θ , k)) (22) dt 1 1 2 7 two targets in an elliptical shape and does not collide the dy channel when two targets are not far away. As the two targets 3 = −y + sig(p , θ , k)) (23) dt 3 1 3 gradually move away, the swarm pattern encircles the two dy targets in a gourd shape, as shown Fig. 11 (b) and (c). Finally, 4 = −y + sig(p , θ , k)) (24) dt 4 2 4 swarm pattern forms two separate circular shape encircling two dg targets respectively, as shown Fig. 11 (d) and (f). For EH-GRN, 2 = −g + sig(y + y , θ , k)) (25) dt 2 3 4 8 the most significant difference is when the two targets enter the dy channel, the swarm pattern collides with the obstacles to form 5 = −y + sig(p , θ , k)) (26) dt 5 1 5 two separate parts, as shown Fig. 12 (b). In Fig. 12 (e), swarm dy patterns encircle two targets in two different shapes. That is, if 6 = −y + sig(p , θ , k)) (27) dt 6 2 6 the target is in the channel, swarm pattern encircles a target in dg an elliptical shape. If the target is outside the channel, swarm 3 = −g + sig(y ∗ y , θ , k)) (28) dt 3 5 6 9 pattern encircles a target as a circle shape. One reason is that dy the structure of the swarm pattern has been predefined, which 7 = −y + 1 − sig(g , θ , k)) (29) dt 7 1 10 can not find an optimize solution in the restricted scenario, as so to the width of the channel is smaller than the maximum dy8 radius of compound swarm pattern formed by surrounding two = −y8 + sig(g2, θ11, k)) (30) dt targets at the same time. For the proposed automatic design dy9 framework, it applies some efficient basic network motifs, = −y9 + sig(g3, θ12, k)) (31) dt which makes swarm pattern adapt the considered scenario. dM In summary, the proposed automatic design framework can = −M + sig(y7 + y8 + y9, θ13, k)) (32) dt find a GRN-based model, which encircles a or two targets where the following parameter values: θ1 = 1, θ2 = 0.5328, without colliding the channel and verifies the feasibility of θ3 = 1, θ4 = 0.4448, θ5 = 0, θ6 = 0.934, θ7 = 2, θ8 = the proposed framework. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 9

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(d) (e) (f) Fig. 9. The swarm pattern that optimized to achieve by the proposed automatic design framework encircles a target across a channel without colliding the channel. (a)-(f) show that the swarm pattern encircle the target in a circle shape way, when the target passes through the channel.

B. Pattern Formation in a compound channel fitness value of point B is lower than that of point A. That is, the GRN-based model of the point B is more suitable for the To verify the adaptability of the proposed automatic design restricted scenario than that of point A. Fig. 17 and Fig. 18 framework, the framework is tested in a compound channel. show the GRN structure of point A and point B, respectively. The four parts, a channel, a narrow channel, a circular narrow channel and a T-shape channel, constitute the compound channel, as shown in Fig. 13. In addition, the restricted In addition, the mathematical description of the point A is scenario is in a region of 25 by 25 meters. When a target as follows: pass through the restricted scenario, the swarm pattern needs to generate various shapes to adapt to the restricted scenario. An evaluation criterion is that swarm pattern can not collide with the compound channel while encircling a target. dy 1 = − y + sig(x ∗ (1 − x ), θ , k)) In evolution process, except for evaluation number, all the dt 1 2 1 1 other parameters are the same as the previous simulation dy 2 = − y + 1 − sig(y ∗ (1 − x ), θ , k)) (33) experiment. The evaluation number of the proposed automatic dt 2 1 1 2 framework and EH-GRN are both set to 8000. In Fig. 14, − sig((1 − y1) ∗ x1, θ2, k)) the non-dominated solutions are achieved by MOGP-NSGA-II when the evaluation number reaches 8000. Fig. 14 shows that point A and B are selected as knee points. That is, the number of nodes at point A and point B are 5 and 7, respectively. In where θ1 and θ2 are 1.3072 and 0.7472, respectively. Eq.(33) other words, they all have a simple the structure of GRN-based is a mathematical description of ANDN and XNOR. y1 and model. Fig. 15 and Fig. 16 show the syntax tree of point A y2 are morphogen concentrations, where y2 is a morphogen and point B, respectively. For Fig. 15 and Fig. 16, there is gradient that defines the swarm pattern. a difference: the right structure of the syntax tree at point B is more complex than that at point A. Because of this, the In addition, the mathematical description of the point B is JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 10

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(d) (e) (f) Fig. 10. When a target pass through a channel, the swarm pattern is formed by an EH-GRN structure, which encircles a target without colliding the channel. (a)-(f) show that the swarm pattern encircle the target in an elliptical shape way, when the target passes through the channel. as follows: For an EH-GRN structure for swarm pattern formation dy in the considered scenario, each swarm robots follows the 1 = − y + sig(x ∗ (1 − x ), θ , k)) dt 1 2 1 1 following dynamic equations to generate a morphogen gradient dy space that define the swarm pattern. 2 = − y + 1 − sig(x ∗ x , θ , k)) dt 2 1 1 2 (34) dy dy1 3 = − y + 1 − sig(y ∗ (1 − y ), θ , k)) = −y1 + sig(p1, θ1, k)) (35) dt 3 1 2 3 dt dy2 − sig((1 − y1) ∗ y2, θ3, k)) = −y + 1 − sig(p , θ , k)) (36) dt 2 2 2 where θ1, θ2 and θ3 are 1.5441, 0.0904 and 0.2414, re- dg 1 = −g + sig(y ∗ y , θ , k)) (37) spectively. Eq.(34) is a mathematical description of ANDN, dt 1 1 2 7 NAND, and XNOR. y1, y2 and y3 are morphogen concen- dy3 = −y3 + sig(p1, θ3, k)) (38) trations, where y3 is a morphogen gradient that defines the dt swarm pattern. dy 4 = −y + 1 − sig(p , θ , k)) (39) To measure the adaptability of the proposed automatic dt 4 2 4 design framework, EH-GRN is tested in the compound chan- dg 2 = −g + sig(y ∗ y , θ , k)) (40) nel. Fig. 19 shows an EH-GRN structure for swarm pattern dt 2 3 4 8 formation, which is optimized to achieve by CMA-ES in the dy5 = −y5 + sig(p1, θ5, k)) (41) restricted scenario. In addition, p1 and p2 represent a mor- dt phogen concentration produced by the environmental inputs dg2 = −g2 + sig(y3 ∗ y4, θ8, k)) (42) in Fig. 19. In other words, p1 and p2 represent the morphogen dt concentrations produced by the environmental inputs. g1, g2 dy5 = −y5 + sig(p1, θ5, k)) (43) and g3 are activate genes. The concentration of M represent a dt morphogen gradient space to form the desired swarm pattern. dy7 = −y7 + sig(g1, θ10, k)) (44) θi (i = 1, ..., 13) is a regulatory parameter. dt JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 11

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(d) (e) (f) Fig. 11. When two targets pass through a channel, the swarm pattern that formed by the proposed automatic design framework encircles two targets without colliding the channel. (a)-(e) show that the swarm pattern encircle the two targets in an elliptical shape way, when the targets passes through the channel.

dy8 = −y8 + sig(g2, θ11, k)) (45) optimal GRN-based model according to different restricted dt environments. dy 9 = −y + 1 − sig(g , θ , k)) (46) dt 9 3 12 Fig. 21 shows that the swarm pattern of point B that dM optimized to achieve by MOGP-NSGA-II encircles a target = −M + sig(y + y + y , θ , k)) (47) dt 7 8 9 13 across a compound channel without colliding the channel. Fig. 21 (a)-(i) show when a target passes through the compound where the following parameter values: θ1 = 0.1438, θ2 = 1, channel, the swarm pattern is a circular shape way to encircle θ3 = 0.3457, θ4 = 0.8571, θ5 = 0.3827, θ6 = 1, θ7 = 1.5841, the target. This is because the width of the narrow channel θ8 = 1.1972, θ9 = 0.4208, θ10 = 0, θ11 = 0, θ12 = 0.5977, and the width of the circular narrow channel are the narrowest θ13 = 0.6777. Fig. 20 shows that the swarm pattern of point A that part of the composite channel In Fig. 21 (c)-(g). As long as optimized to achieve by the proposed automatic design frame- the swarm pattern can pass through the two parts in a certain work encircles a target across a channel without colliding shape, it can also pass through the other parts of the composite the channel. In particular, when the target does not enter the channel in this shape. The optimized swarm pattern can pass channel, the swarm pattern is a circular shape way to encircle through the narrowest part of the composite channel in a circle the target, as illustrated by Fig. 20 (a) and (i). Fig. 20 (b)- shape, so the circle shape can also be used to encircle the target (h) show the swarm pattern encircles the target in different in other parts. shapes way. For example, when the target moves from the Fig. 22 shows when a target pass through a compound channel to the narrow channel, the shape of swarm pattern channel, the swarm pattern that formed by EP-GRN encircle gradually changes from elliptocytosis to circular, as shown the target in a circle shape. Fig. 22 (d) and (f) are the narrowest Fig. 20 (b)-(c). As the target moves from the circular narrow part of the composite channel. That is, the width of the narrow channel to the T-channel, the shape of swarm pattern gradually channel and the width of the circular narrow channel are the becomes trapezoid shape, as shown Fig. 20 (f)-(h). The reason narrowest part of the composite channel. In these the narrowest for the dynamic change of swarm pattern with different target parts, swarm pattern collides with obstacles when encircling position is MOGP-NSGA-II can automatically optimize the the target. It shows that EH-GRN optimized swarm pattern JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 12

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(d) (e) (f) Fig. 12. When two targets pass through a channel, the swarm pattern that formed by an EH-GRN structure encircles two targets. (a)-(f) show that the swarm pattern encircle two targets across a channel.

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Fig. 14. The non-dominated solutions are achieved by MOGP-NSGA-II. The can not adapt to these two scenarios well. The limitations of point A and B are called knee points. predetermined GRN structure are revealed from the side.

In summary, the proposed automatic design framework VI.CONCLUSION can find two GRN-based models, which encircles a target without colliding the channel and verifies the adaptability of This paper proposes an automatic design framework of the proposed framework. swarm pattern formation based on multi-objective genetic pro- JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 13

Fig. 15. The syntax tree of point A is optimized to achieve by MOGP-NSGA- II. x1 is that the location information of target forms a morphogen gradient space. x2 is that the location information of obstacle forms a morphogen Fig. 18. The evolutionary design framework gets GRN. x1 is that the location gradient space. ANDN and XNOR are two basic network motifs. information of target forms a morphogen gradient space. x2 is that the location information of obstacle forms a morphogen gradient space. G1, G2 and G3 are activate genes, and the concentration of G3 represent a morphogen gradient space to form the desired swarm pattern.

Fig. 16. The syntax tree of point B is optimized to achieve by MOGP-NSGA- II. x1 is that the location information of target forms a morphogen gradient space. x2 is that the location information of obstacle forms a morphogen gradient space. ANDN, NAND and XNOR are three basic network motifs.

Fig. 19. Illustration of an EH-GRN structure for swarm pattern formation. The structure of model is predefined, and CMA-ES is applied to optimise the regulatory parameters. g1, g2 and g3 are activate genes. The concentration of M represent a morphogen gradient space to form the desired swarm pattern.

obstacles have demonstrated that the proposed approach is able to automatically generate complex patterns highly adaptable and robust to dynamic and unknown environments. As future work, a proof-of-concept experiment will be performed to evaluate the proposed pattern formation algorithm using e- Fig. 17. The evolutionary design framework gets GRN. x1 is that the location puck education robots in real-world environments. information of target forms a morphogen gradient space. x2 is that the location information of obstacle forms a morphogen gradient space. G1 and G2 are activate genes, and the concentration of G2 represent a morphogen gradient space to form the desired swarm pattern. ACKNOWLEDGMENT

This work was supported by the Key Lab of Digital gramming. The proposed framework improves the flexibility Signal and Image Processing of Guangdong Province, by of the swarm pattern generation to be applied in various the Key Laboratory of Intelligent Manufacturing Technology complex and changeable environments as it applies some (Shantou University), Ministry of Education, by the Science basic network motifs to automatically structure the GRN- and Technology Planning Project of Guangdong Province based model. In addition, by introducing obstacles as one of China under grant 180917144960530, by the Project of of environmental input sources along with that of targets, Educational Commission of Guangdong Province of China we address the weakness of the previous H-GRN pattern under grant 2017KZDXM032, by the State Key Lab of not being adaptable to obstacles in the sense that a target Digital Manufacturing Equipment & Technology under grant entrapping pattern itself changes as an obstacle approaches. DMETKF2019020, and by the National Defense Technology Numerical simulations considering static/moving targets and Innovation Special Zone Projects. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 14

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Fig. 20. When a target pass through a compound channel, the swarm pattern of point A is formed by MOGP-NSGA-II, which encircles a target without colliding the channel. (a)-(e) show that the swarm pattern encircle the target in different shapes way, when the target passes through the channel. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 16

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Fig. 21. When a target pass through a compound channel, the swarm pattern of point B is formed by MOGP-NSGA-II, which encircles a target without colliding the channel. (a)-(e) show that the swarm pattern encircle the target in a shapes way, when the target passes through the channel. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 17

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Fig. 22. When a target pass through a compound channel, the swarm pattern is formed by EP-GRN, which encircles a target without colliding the channel. (a)-(e) show that the swarm pattern encircle the target in a shapes way, when the target passes through the channel.