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FENE − type virtual potential. FL i. : control output for module i directed to the light source. Here, the variables of the control output functions listed above are ...
SICE Annual Conference in Fukui, August 4-6, 2003 Fukui University, Japan

Adaptive Swarming by Exploiting Hydrodynamic Interaction Based on Stokesian Dynamics Method Masahiro Shimizu1 , Akio Ishiguro1 , Toshihiro Kawakatsu2 , Yuichi Masubuchi1 , and Masao Doi1 1

Dept. of Computational Science and Engineering, Nagoya University Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan [email protected], {ishiguro/mas/doi}@cse.nagoya-u.ac.jp 2 Dept. of Physics, Tohoku University Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan [email protected] Abstract: This paper discusses a fully decentralized algorithm able to create an adaptive swarm of autonomous mobile robots from the viewpoint of “computational physics”. To this end, we particularly focus on “Stokesian Dynamics method”. Simulation results indicate that the proposed algorithm can control the shape of the swarm appropriately according to the current situation without losing the coherence of the swarm. Keywords: Molecular Dynamics, Stokesian Dynamics, Swarming, Local communication

1. Introduction

cal example. Simulation results indicate that the proposed algorithm can successfully control the shape of the swarm according to the current situation without losing the coherence of the swarm nor exchanging any global information among the modules. In addition, we will discuss the necessity of the local interaction exploiting Stokesian Dynamics method in maintaining the coherence of the swarm particularly under the unstructured environment.

Recently, reconfigurable robots consisting of many identical modules have been attracting a lot of attention due to their significant abilities such as fault tolerance, scalability, and adaptability against environmental perturbation. So far various methods have been proposed to control the morphology of reconfigurable robots (see for example 1) ). Most of these approaches, however, employ fully or partially centralized control algorithms and/or assume that global information such as the position of the mass center of the whole system is available, which is normally hard to obtain only from the local interaction among modules. This leads to the following conclusions: in order to fully exploit the advantages mentioned above, (1)each module should be controlled in a fully decentralized manner, and (2)the resultant morphology of the entire system should be emerged through the module-to-module and module-to-environment interactions. In light of these facts, this study is intended to deal with a fully decentralized algorithm from the viewpoint of computational physics, which allows us to control the behavior of a reconfigurable robot system without the use of any global information. To this end, we particularly focus on Molecular Dynamics method5) and Stokesian Dynamics method6) , the former of which is widely used in computational physics to capture the behavior of multi-body systems (e.g. polymers), and the latter of which is employed to analyze the behavior of particles in a viscous fluid. Since there are still lots of technical issues in existence that have to be resolved in order to realize a truly reconfigurable robot system, we implement the proposed algorithm inspired from Molecular Dynamics and Stokesian Dynamics methods to the control of a swarm of 2-D radio-connected autonomous mobile robots as a practi-

2. Related Works In this section, some notable works concerning the control of swarm robots are introduced. Shimoyama et al. proposed an algorithm which employs unisotropic interaction among modules, showing that this algorithm can create various shapes of the swarm2) . However, in their simulations, they assumed that a global information, i.e. the position of the mass center of the swarm, was obtainable. On the other hand, Nembrini et al. proposed a fully decentralized algorithm and implemented it to the control of the behavior of a swarm made up of wirelessnetworked autonomous mobile robots3) . Although they showed that this algorithm enabled the robots to form a swarm, it still remains unclear whether such a simple attractive/repulsive interaction dynamics can create the coherent swarm.

3. Proposed Method 3.1 The Task As a first step of the investigation, in this study we mainly explore a fully decentralized control algorithm that exploits an emergence phenomenon by tak-

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PR0001/03/0000-0709 ¥400 © 2003 SICE

sum of the following contributionsF    = FiC {rij } , riL , vi , F˜j    FiV (vi ) + FiF {rij } , F˜j + FiLJ ({rij })   (2) +FiFENE min{rij } + FiL (riL ) .

ing a swarm of 2-D radio-connected autonomous mobile robots as a practical example. It is hoped that this algorithm will be seamlessly implemented to the morphology control of hard-connected reconfigurable robots1) as well as radio-connected swarm robots discussed in this paper. Here the task of the swarm is to move toward a light source (i.e. goal) while avoiding obstacles encountered. Despite its seeming-simplicity, in this task the autonomous mobile robots (hereafter we call them modules) have to cope with the obstacles without losing the coherence of the swarm (e.g. splitting), and this has to be done without exchanging any global information among the modules. In order to achieve this task, each module is equipped with an omni-directional camera and a pair of receivers and transmitters for radio-connection: the former of which is used to detect the direction to the light source and the direction/distance to its neighbouring modules, and the latter of which is for the local information exchange between the modules1 . A detailed explanation of this local communication, which is of prime importance to maintain the coherence of the swarm, will be given later. In addition, each module is equipped with omni-directional wheels driven independently, allowing rapid change of the shape of the swarm.

j

{rij } riL

3.2 The Algorithm

d2 ri = Fi + FiC , dt2

set of relative position vectors between modulus i and j observed by module i : relative position vector between the light sourse and module i : velocity of module i

v i  F˜j

:

FiV

:

FiF

:

FiLJ

:

FiFENE

:

FiL

:

set of control output vectors for module j except for the hydrodynamic force control output for module i due to friction control output for module i due to hydrodynamic effect control output for module i due to Lennard − Jones − type virtual potential control output for module i due to FENE − type virtual potential control output for module i directed to the light source

Here, the variables of the control output functions listed above are locally obtainable from the sensory information for each module. The first and second terms on the right-hand side of Equation (2) mimic the dynamics of a particle moving in a viscous fluid. The third term is a virtual interaction potential force between the modules, for which we assume Lennard-Jones-type interaction for uniatomic molecular systems 3 . The fourth term is also a virtual interaction potential force, which acts to constrain the distance between the modules to its equilibrium value. The fifth term is the driving force toward the light source. In the following, we will elaborate each component of the control output function FiC : FiV , FiF , FiLJ , FiFENE , and FiL .

Let us denote the control output acting on the i-th module (hereafter we call it module i) dertermined by the proposed algorithm as FiC . Then, the equation of motion of module i is given as follows 2 : mi

:

(1)

where mi is the fictious mass of module i, ri is the position vector of module i in the absolute coordinate system, and Fi is the force acting on module i except for FiC created by the algorithm discussed below (e.g. reaction forces from obstacles). One of the central issues in controlling the shape of the swarm is to design FiC by including correlations and interactions among the modules. Here, we propose a scheme to accomplish this task by employing the concept of Molecular Dynamics and Stokesian Dynamics methods, and discuss its advantages. In this study, we assume that FiC is expressed as a

Control output due to FiV This control output stabilizes the motion of the module by generating a friction force that is proportional to its velocity. This can be explained using the analogy from the viscous friction acting on a particle moving in a visous fluid. The explicit form of FiV is given by FiV (vi ) = −6πηavi ,

1 The

detectable range of the omni-directional camera and the range of the radio-connection of each module are assumed to be considerbly less than the span of the swarm. This restriction allows us to discuss the exploitation of emergence phenomena in the course of the change of the swarm shape. 2 Although Equation (1) is described in terms of the absolute coordinate of the position of the module, i.e. ri , each module employs locally obtainable information in its determination of the control output FiC .

(3)

3 The Lennard-Jones potential is normally used to describe the interatomic van der Waals interaction of microscopic molecular systems. In our model, however, the Lennard-Jones-type interaction between the modules does not correspond to such a physical interaction. Instead, the interaction is introduced to generate a control output able to maintain the swarm that is stable in a similar manner as a liquid droplet in a molecular system. In this sense, we label this Lennerd-Jones interaction as a “virtual interaction”.

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6 times as big as the module radius a4 . Note in passing that for the calculation of FiF , local communication among the modules is indispensable. Control output due to FiLJ This control output stabilizes the shape of the swarm in such a way that the equilibrium inter-module distance equals σ. When σ is twice the module radius a, the modules come to in contact with each other. The Lennard-Jones-type potential, therefore, tends to maintain the swarm as a circular shape on a 2-D plane if there is no external force. This corresponds to the surface tension effect of a liquid droplet, which minimizes the total surface area of the droplet. In our robotic case, the swarm maintains its compact shape when it moves around obstacles or passes through a narrow aisle. In such a collective behavior, the motion of each module is minimized so that the total energy comsumption of the swarm is minimized. The explicit form of FiLJ is given as follows5) F

Figure 1: Velocity field induced by FiF acting on module i.

FiLJ ({rij }) = −

φLJ (rij ) = φLJ 0 FiF

Control output due to This control output enables a moving module to induce motion of the surrounding modules in the same direction, leading to a cooperative motion of the whole swarm, similar to the hydrodynamic interaction between particles in a viscous fluid. When a particle is moving in a fluid, this particle imposes a force on the fluid. The effect of this force is propagated to a distant point through the Oseen tensor, and induces a velocity field that drives another particle at that point4)6) (see Figure 1)D With such a cooperative effect, we can realize spontaneous emergence of the systematic motion of the whole swarm. The explicit expression of FiF is given as follows:

{rjk }



σ rij

12

 −2

σ rij

(7)

6

(8)

FiLJ ({rij }) : Lennard − Jones potential force φLJ : measure of the depth of φLJ (rij ) 0 Control output due to FiFENE This control output constrains the distance between neighbouring modules to its equilibrium value. In polymer simulations, a polymer chain is modelled as a sequence of coarsegrained objects called segments that are connected by bonds. Such bonds are assumed to be non-stretchable beyoned a prespecified upper bound. In order to model this characteristic, the concept of the FENE potential is frequently employed5) . The explicit form of FiFENE is expressed as:   FiFENE min{rij } = j     −∇i φFENE min{rij } × Θ 6a − | min{rij }|

(4)

i=j

F˜j = FjV (vj ) + FjLJ ({rjk }) + FjL (rjL ) 1 (I + rˆij rˆij ) H (rij ) = 8πηrij

∇i φLJ (rij ) Θ (6a − |rij |)

i=j

where η is the viscosity coefficient, and a is the radius of the module.

   = FiF {rij } , F˜j   6πηa H (rij ) · F˜j Θ (6a − |rij |)



j

(5)



(6)

j

set of relative position vectors between modules j and k observed by module j I : unit tensor, Iαβ = δαβ : unit vector parallel to rij rˆij H (rij ) : Oseen tensor, (ˆ rij rˆij )αβ = rˆij α rˆij β

(9)



φFENE min{rij }

:

j

= 



1  − k(R0FENE )2 × ln 1 −  2

min{rij } − σ j

R0FENE

2     (10)

FENE

(rij ) : FENE potential φ R0FENE + σ = 6a : upper bound of the length of FENE bond

In Equation (4), Θ (6a − |rij |) accounts for the detectable range of the distance sensor implimented to each module, i.e. the omni-directional camera. In the present study, this range is assumed to be 6a, which is

4 In real physical robots, this value depends on the characteristics of the sensor implemented.

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φLJ and φFENE are repulsive when rij < σ, and are attractive for rij > σ, respectively. As shown in Figure 2, φLJ accounts for the short-range attractive interaction whilst φFENE denotes the long-range attractive interaction: the former of which governs the motion of the neighbouring contacting modules, and the latter of which guarantees the stability of the swarm when an external perturbation is applied. Control output due to FiL This control output drives the modules toward the light source. In the following simulations, for simplicity we assume that this driving force depends only on the direction to the light source. If there is another module and/or an obstacle on the way to the light source, the module cannot detect the light source and thus the driving force will vanish. The functional form of FiL is given by FiL (riL ) = F0L rˆiL F0L rˆiL

(11)

:

coefficient depending on the intensity of the light source : unit vector to the light source

4. Simulation Results In order to verify the feasiblity of the algorithm discussed above, we have carried out simulations. The following simulations have been conducted to investigate the validity of this algorithm particularly in terms of (1)the adaptability against environmental changes, and (2)the fault tolerance.

4.1 Verification of the adaptability In this experiment, we employed two types of environment to investigate the adaptability: one is an environment in which a circular obstacle exists on the way to the light source (test environment 1: see Figure 3), and the other contains obstacles forming a wide-to-narrow aisle (test environment 2: see Figure 4). In the following simulations, the number of the modules in the swarm was set at 16, and the situations depicted in Figure 3 and Figure 4 were used as the initial conditions. Shown in Figure 5 and Figure 6 are the simulation results, indicating how the swarm moves under test environment 1 and test environment 2, respectively (time evolves from left to right in each figure). Note that the slightly-shaded circles denote the modules that percieve the light source at the moment. As in the figures, the swarm can successfully negotiate the environmental changes and finally reach the light source without losing its coherence. Interestingly, in both cases the swarm eventually converges to a shape expanding along the moving direction. This phenomenon is mainly due to the effect of the Oseen tensor implemented in the term FiF , which represents the hydrodynamic interaction in Stokesian Dynamics. Further discussion of this effect will be given in the following section.

Figure 2: Shapes of the Lennard-Jones potential φLJ (rij ) (top), φFENE (rij ) (middle), and φLJ (rij ) + φFENE (rij ) (bottom), respectively.

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Figure 5: Simulation results under test environment 1 (see left to right).

Figure 3: Test environment 1.

Figure 6: Simulation results under test environment 2 (see left to right).

A typical simulation result is illustrated in Figure 8. As shown in the figure, the swarm was frequently observed to get stuck around the entrance of the narrow aisle. This strongly indicates that this local communication plays an essential role in negotiating such an unstructured environment. In order to investigate the above consideration quantitatively, we have focused on the following terms in Equation (2): FiLJ , FiFENE , FiV , and FiF , and have analized how these terms contribute to the global behavior of the swarm. For this purpose, we have measured the correlation between each of these terms and the velocity vector of the mass center of the swarm, which is calculated by

Figure 4: Test environment 2.

4.2 Verification of the fault tolerance In order to verify the fault tolerance of the proposed algorithm, we have conducted simulations under test environment 2. A representative simulation result is illustrated in Figure 7. In the figure, the empty circle denotes a malfunctional module which cannot send the control signals to its wheels. Thus, this module cannot move autonomously. As in the figure, this malfunctional module is eventually ejected from the swarm after passing through the narrow aisle. It should be noted that the shape of the entire swarm is almost similar to the one shown in Figure 6. This strongly suggests that this algorithm ensures gradual degradation, which is indispensable for the realization of swarm intelligence.

N mod



 v cog (t) · Fiele (t) =



v cog (t) · Fiele (t)

i

Nmod

, (12)

where v cog (t) is the velocity vector of the mass center of the entire swarm at time step t and Nmod is the number of the modules. Fiele (t) is a target for this correlation, which takes one of the following terms: FiLJ (t), FiFENE (t), FiV (t), and FiF (t). The result obtained under environment 2 is illustrated in Figure 9. For the ease of explanation, the time evolution of |v cog (t)| is also depicted in the figure. This result provides us the following three points that have to be noted. First, the control output from FiLJ (t) and FiFENE (t) show no direct correlation with the global behavior of the swarm, i.e. v cog (t). This is due to the fact that FiLJ (t) and FiFENE (t) act as internal forces, and thus they do not contribute to v cog (t). Second, FiV (t) has an inverse correlation with v cog (t). This is simply because this control output acts as a friction, stabilizing the global behavior of the swarm.

5. Discussion In this section, we mainly discuss the effect of local interaction exploiting Stokesian Dynamics, which remarkably differs from the conventional control algorithms such as 3) . More specifically, we investigate the influence of this local communication on the coherence of the swarm. For the sake of the following discussion, we have conducted simulations without this local communication under test environment 2 as a comparative experiment.

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6. Conclusion This paper investigated a fully decentralized algorithm from the viewpoint of computational physics, mainly paying attention to Molecular Dynamics and Stokesian Dynamics methods. We implemented this algorithm to the control of a swarm of 2-D radio-connected autonomous mobile robots. The simulation results support several conclusions and have clarified some interesting phenomena for further investigation, which can be summarized as follows: first, the proposed algorithm can control the shape of the swarm without losing the coherence of the swarm; second, the local interaction exploiting Stokesian Dynamics is indispensable in maintaining the coherence of the swarm particularly under an unstructured environment. A significant feature of this algorithm is that it is designed in such a way that the resultant morphology of the entire system emerges through the module-tomodule and module-to-environment interactions without utilizing any global information. We therefore expect that this algorithm will be seamlessly implemented to the morphology control of various types of reconfigurable robots as well as the swarm robot discussed in this paper. We are currently constructing real experimental robots for the validation of the proposed algorithm in a real environment.

Figure 7: Simulation results under the existence of a malfunctional module.

Figure 8: Simulation results without the StokesianDynamics-based local interaction under test environment 2. Third, and the most crucial point here, the control output from FiF (t) shows significantly strong correlation around time step 1500. Interestingly, around this time step the swarm is moving slowly due to its approach to the entrance of the narrow aisle. It should also be noted that this correlation disappears immediately after the swarm has successfully passed through the aisle. This strongly suggests that the unisotropic effect created by the hydrodynamic interaction in Stokesian Dynamics enables the swarm to pass through the narrow aisle by actively deforming its shape. This also indicates the necessity of the local communication among the modules in the course of negotiating the unstructured environment.

References [1] A. Kamimura, S. Murata, E. Yoshida, H. Kurokawa, K. Tomita, and S. Kokaji: “Self-Reconfigurable Modular Robot – Experiments on Reconfiguration and Locomotion –”, in Proc. of 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2001), pp.590-597 (2001) [2] N. Shimoyama, K. Sugawara, T. Mizuguchi, Y. Hayakawa, and M. Sano: “Collective Motions in a System of Motile Elements”, Phys. Rev. Lett., 76, pp.38703873 (1996) [3] J. Nembrini, A. Winfield, and C. Melhuish: “Minimalist Coherent Swarming of Wireless Networked Autonomous Mobile Robots”, From animals to animals 7, MIT Press, pp.373-382 (2002) [4] M. Doi and S.F. Edwards: “The Theory of Polymer Dynamics”, OXFORD SCIENCE PUBLICATIONS (1986) [5] Nagoya University Doi Project Research and Development of the Platform for Designing High Functional Materials: “OCTA COARSE-GRAINED MOLECULAR DYNAMICS PROGRAM COGNAC USER’S MANUAL”, http://octa.jp (2002) [6] R. B. Jones and R. Kutteh: “Sedimentation of Colloidal Particles Near a Wall : Stokesian Dynamics Simulations”, Phys. Chem. Chem. Phys., 1, pp.2131-2139 (1999)

Figure 9: The obtained correlation under test environment 2.

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