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Decision Making and Finite- Time Motion Control for a Group of Robots Qiang Lu, Member, IEEE, Shirong Liu, Xiaogao Xie, Member, IEEE, and Jian Wang IEEE TRANSACTIONS ON CYBERNETICS, VOL. 43, NO. 2, APRIL 2013 student: 4992c034 TSAI WEN CHENG

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Abstract This paper deals with the problem of odor source localization by designing and analyzing a decision–control system (DCS) for a group of robots. In the decision level, concentration magnitude information and wind information detected by robots are used to predict a probable position of the odor source. Specifically, he idea of particle swarm optimization is introduced to give a probable position of the odor source in terms of concentration magnitude information. Moreover, an observation model of the position of the odor source is built according to wind information, and a Kalman filter is used to estimate the position of the odor source, which is combined with the position obtained by using concentration magnitude information in order to make a decision on the position of the odor source. In the control level, two types of the finite-time motion control algorithms are designed; one is a finite-time parallel motion control algorithm, while the other is a finite-time circular motion control algorithm. Precisely, a nonlinear finite-time consensus algorithm is first proposed, and a Lyapunov approach is used to analyze the finite-time convergence of the proposed consensus algorithm. Then, on the basis of the proposed finite-time consensus algorithm, a finite-time parallel motion control algorithm, which can control the group of robots to trace the plume and move toward the probable position of odor source, is derived. Next, a finite-time circular motion control algorithm, which can enable the robot group to circle the probable position of the odor source in order to search for odor clues, is also developed. Finally, the performance capabilities of the proposed DCS are illustrated through the problem of odor source localization.

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INTRODUCTION ODOR source localization, which is a type of ill-posed and dynamical optimization problem, has received much attention from researchers and engineers due to its practical significance for human security, e.g., searching for the sources of wastes and locating victims. In the last two decades, how to locate an odor source based on a single robot has been widely studied. Three typical approaches, namely, chemotaxis [23], [30], anemotaxis [8], [10], [13], [27], and infotaxis [32], have been proposed. For chemotaxis where the local concentration information is used, the robot is guided to move along the gradient direction of concentration [23], [30]. For anemotaxis where the local wind information is used, the main idea is to use the local wind direction and detection events about odor to orient the robot to locate the source of odor [8], [10], [13], [27]. For infotaxis where information gain instead of concentration gradient is used, the reader is referred to [32] and the references therein.

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Recently, particle swarm optimization (PSO) [16], which can make effective use of swarm information and individual information to guide a particle swarm to search for the optimum [28], has been used to coordinate a group of robots to deal with the problem of odor source localization [15], [17], [21], [24]. To avoid trapping into local maximal concentrations, for instance, Jatmiko et al. [15] improved the commonly used PSO algorithm based on an electrical charge theory charged particle swarm optimization (CPSO). In the improved algorithm, two types of robots (neutral and charged robots) are used. Among neutral robots, there is no repulsive force, while among charged robots, the mutual repulsive force is generated in order to maintain the positional diversity of robots. To conveniently use the PSO algorithm for odor source localization, Lu and Han [17] proposed a distributed coordination control architecture where the PSO algorithm is divided into three parts (prediction, plan, and control).

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Accordingly, the cooperative control system consists of three levels: a group level, a trajectory level, and a robot level. In the group level, swarm information and individual information are used to predict the probable position of the odor source. In the trajectory level, a movement trajectory of the robot is planned from the current position to the probable position of the odor source. In the robot level, a control law is designed to enable the robot to move along the planned trajectory. This control architecture makes the control system robust and evolvable [14]. In terms of this control architecture, the search performance of the robot group coordinated by the CPSO algorithm [15] is improved. To quickly locate the odor source, Lu and Han [21] proposed a probability PSO with information-sharing mechanism. Due to introducing the ideas of distribution estimation algorithm and niche, each robot can be provided an opportunity to choose an appropriate position in the search space such that the search performance of the robot group can be improved. To sum up, one can conclude from aforementioned research results in [15], [17], [21], and [24] that the PSO algorithm provides a mechanism to predict a probable position of the odor source through swarm and individual concentration information and then to adjust the movement direction of robots to move toward the probable position of the odor source.

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Fig. 1. Instantaneous plume where black dots denote the filaments that form a plume and arrows denote the wind speed and direction. Notation: lN denotes the index set {1, 2,...,N}. Let sig(r)α = sign(r)|r|α, where 0 < α < 1, r ∈ R, and sign(·) is a sign function.

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PROBLEM STATEMENT AND PRELIMINARIES In this section, we will illustrate the problem of odor source localization and introduce several preliminaries. A. Odor Source Localization Odor source localization is a type of ill-posed and dynamical optimization problem, which can be stated as follows. Problem 1: An odor source localization problem consists of the following: 1) a set N of N mobile robots or vehicles; 2) a set X of positions in a 2-D search space R2; 3) a setM ⊆ N ×X of possible pairs; 4) a map f : X ×[0,∞] → R giving

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Fig. 2. Concentration fluctuates at a concrete position (80 m, 0 m) from t = 0 to t = 250.

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Since the dynamics of each dimension of robots is independent of others, we assume that the dimension number of robots n = 1without loss of generality in the following. A continuoustime dynamics model of N identical robots is considered and given by DynamicsModels and Definition of Finite-Time Convergence

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Communication Topology Let G = (ν, E,A) be a weighted undirected graph of order N with the set of nodes ν = {ν1, ν2,..., νN}, the set of edges E ⊆ ν × ν, and a weighted adjacency matrix A = [aij ] with nonnegative adjacency elements aij. For the undirected graph G, the adjacency matrix A is symmetric, i.e., AT = A. Let L(A) = [lij ] ∈ RN×N denote the graph Laplacian of G = (ν, E,A), which is defined by

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OPTIMIZATION AND DECISION MAKING IN THE DECISION LEVEL In order to introduce the idea of the PSO algorithm for decision making on the position of the odor source, we will first describe a commonly used form of the PSO algorithm given by vi(k + 1) =f (vi(k), ui(k)) xi(k + 1) =xi(k) + vi(k + 1) (3) with f (vi(k), ui(k)) = ωvi(k) + ui(k) (4) ui(k) = α1 (xl(k) − xi(k)) + α2 (xg(k) − xi(k)) (5)

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Algorithm 1 1: / ∗ Initialization ∗ / 2: Initialize parameters α3, c1, and c2; 3: Initialize a posteriori estimate error covariance Pi(0) and a measurement noise covariance matrix R(0); 4: Initialize the source estimate position ˆrs(0) using the current position ri(k) when the first concentration detection event occurs; 5: Set robot.l = 0 and robot.windcount = 0; / ∗ Releasing time and an accumulator that can record the number of wind velocity ∗ / 6: / ∗ Main Body ∗ / 7: repeat 8: Perform (8) to obtain pi(k); 9: / ∗ Store wind velocity within 100s ∗ / 10: robot.wind[robot.windcount] = wind; 11: / ∗ Concentration detection events occur ∗ / 12: if robot.snsd > 0 then

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13: Perform (20) and (21) to obtain a priori position of the odor source ˆr−i s (k) and a priori estimate error covariance P−i(k); 14: / ∗ Calculate measurement noise variance ∗ / 15: Set kk = 0 and sum = 0; 16: for i = robot.l; i < int(CurrentTime); i++ do 17: sum=sum+(int(CurrentTime)−i) ∗ R(k−1); kk = kk + 1; 18: end for 19: R(k) = sum/kk; 20: Perform (22) to calculate the Kalman gain; 21: Perform (24) to calculate a posteriori estimate error covariance Pi(k); 22: / ∗ Calculate the movement distance of filaments ∗ / 23: Set sumtemp = 0 and kk = 0; 24: for i = 0; i < robot.windcount; i++ do 25: Set sum = 0; 26: for j = i; j < robot.windcount; j ++ do

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27: sum = sum + robot.wind[j]; 28: end for 29: if sum robotposition− α3 then 30: sumtemp = sumtemp + sum; kk = kk + 1; 31: end if 32: end for 33: Calculate the measurement zi(k) = robotposition − sumtemp/kk; 34: Perform (23) to generate a posteriori position estimate of the odor source ˆris (k); 35: Let qi(k) = ˆris (k); 36: Perform (10) to calculate the final position hic(k); 37: end if 38: until Termination conditions are satisfied.

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FINITE-TIME PARALLEL MOTION CONTROL ALGORITHM In this section, we will propose a parallel motion control algorithm, which can coordinate the robots to form a parallel motion through the interaction with its neighbors and environment. We first give a finite-time formation algorithm that can keep a certain distance among robots. This finite-time formation algorithm is described by

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where hi(i ∈ lN) is a constant. Parallel movement for three robots. “o” and “ ∗ ” denote the initial position and the end position, respectively.

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SIMULATION RESULTS

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CONCLUSION A DCS has been proposed to coordinate a team of robots to locate the odor source. In the decision level, we have employed the idea of PSO to give a probable position of the odor source in terms of concentration magnitude information. Moreover, we have built an observation model according to wind information and used a Kalman filter to estimate the position of the odor source in order to improve the prediction performance about the position of the odor source obtained by utilizing concentration magnitude information. In the control level, we have designed a finite-time parallel motion control algorithm and a finite-time circular motion control algorithm. Accordingly, a Lyapunov approach is used to analyze the convergence property of the proposed motion control algorithms. Finally, this study has shown the performance capabilities of the proposed DCS for the problem of odor source localization.

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REFERENCES [1] E. Balkovsky and B. Shraiman, “Olfactory search at high Reynolds number,” Proc. Nat. Acad. Sci. U.S.A, vol. 99, no. 20, pp. 12 589–12 593, Oct. 2002. [2] S. P. Bhat and D. S. Bernstein, “Continuous finite-time stabilization of the translational and rotational double integrators,” IEEE Trans. Autom. Control, vol. 43, no. 5, pp. 678–682, May 1998. [3] S. P. Bhat and D. S. Bernstein, “Finite-time stability of continuous autonomous systems,” SIAM J. Control Optim., vol. 38, no. 3, pp. 751–766, 2000. [4] R. T. Bupp, D. S. Bernstein, V. S. Chellaboina, and W. M. Haddad, “Finite settling time control of the double integrator using a virtual trapdoor absorber,” IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 776–780, Apr. 2000. [5] Y. Cao and W. Ren, “Distributed coordinated tracking with reduced interaction via a variable structure approach,” IEEE Trans. Autom. Control, vol. 57, no. 1, pp. 33–48, Jan. 2012. [6] J. Cortés, “Finite-time convergent gradient flows with applications to network consensus,” Automatica, vol. 42, no. 11, pp. 1993–2000, Nov. 2006.

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