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Decision Making and FiniteTime 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
Abstract
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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.
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.
• 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).
•
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.
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.
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
• Fig. 2. Concentration fluctuates at a concrete position (80 m, 0 m) from t = 0
• to t = 250.
DynamicsModels and Definition of
Finite-Time Convergence
• 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
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
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)
Algorithm 1
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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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26: for j = i; j < robot.windcount; j ++ do
27: sum = sum + robot.wind[j];
28: end for
29: if sum<robotposition+α3∧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.
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
• where hi(i ∈ lN) is a constant.
Parallel movement for three robots. “o” and “∗” denote the initial
position and the end position, respectively.
SIMULATION RESULTS
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.
REFERENCES
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