Scientific Journal of Control Engineering October 2013, Volume 3, Issue 5, PP.301-309
The Shortest Motion Path of Multi-robot Fish Formation Based on Ant Colony Algorithm and Fuzzy Control Mechanism Susu Shan, Zhijian Ji†, Junwei Gao School of Automation engineering, Qingdao University, Qingdao 266071, China †Email:
jizhijian@pku.org.cn
Abstract For the formation control of multi-robot fish system, the paper focuses on how to reach the specified location along the shortest path during the moving process. The problem is handled by a proposed strategy which combines the leader-follower framework with ant colony algorithm. The strategy will first search for the shortest path by using ant colony algorithm. Then the formation controller will be designed according to the principle of leader-follower algorithm and fuzzy control mechanism to make robotic fish keep formation and arrive at the designated location safely along the shortest path. The feasibility of the proposed method is verified by simulation, which indicates that the multiple robot fish system can move with formation and fluency on the one hand, and as well find the shortest motion path from a given starting point to the destination point on the other hand. Keywords: Leader-follower Algorithm; Formation Control; Ant Colony Algorithm; The Shortest Path; Fuzzy Mechanism
1 INTRODUCTION 1 With the development of high-tech, MIT has successfully developed in 1994 a true sense of the world’s first bionic tuna, which opened the way for robotic fish study [1]. In practical applications, the capability of a single robot fish is limited due to the complexity, uncertainty and the concurrency of the task itself. In this case, the collaboration of multiple robot fish not only compensates for the deficiencies of the individual capabilities, but also makes the robotic fish system have the features of parallelism, robustness and flexibility. etc. that an individual one cannot match [2]. The formation control of multiple robot fish plays a fundamental role in the study of corporation, and is also a hot research topic in the literature [3, 4]. In engineering exploration and development environment with high quality performance, it often requires that the multiple robot fish system keeps a certain formation [5-8] and reaches the specified location quickly and effectively along the shortest path. The shortest path of the multiple robot fish system from the starting point to the target point can be searched out, which will not only reduce the burden of the system’s power consumption and improve the efficiency of tasks, but also have important significance for the coordinated control of multiple robot fish with low power consumption. The shortest motion path of multiple robot fish formation is to find a path from a given starting point to the destination point with obstacles in the work environment such that the robotic fish can safely bypass all the obstacles without collision, and find out the shortest obstacle avoidance path under a premise of safety. The generalized ant colony algorithm was proposed in [9]. In the case that there does not have a conflict between/among different ant groups with independent behavioural characteristics, the algorithm seeks how to find their own shortest paths which are free from sharing with other groups of ants, and avoids waiting and conflict in the combination algorithm optimization. However, the searched paths are limited by Tmax ; that is, if the value of Tmax is not large enough, the shortest path which has been searched may not be the real one. In [10], the hierarchical formation control method was proposed which combined the leader law with the behaviour-based method. More specifically, the formation of autonomous robotic fish system is divided into four levels in the paper which are character planning layer, formation This work is supported by National Nature Science Foundation of China (Nos.61075114, 41076062). - 301 http://www.sj-ce.org
design layer, behaviour control layer, and the formation evaluation layer. It should be noted that the question of how to complete the formation along the shortest path was not taken into account therein. In view of the fact that robotic fish usually cannot reach the designated position quickly and effectively along the shortest path in the running process, this paper introduces a control strategy to find the shortest path based on ant colony algorithm. The strategy takes the maximum number of iterations as the end condition in the algorithm of searching for the shortest path, which avoids the problem that the shortest path is often limited by the maximum time. Then the multiple robot fish maintains a certain formation running along the shortest path based on leader-follower algorithm and fuzzy control mechanism, and completes the collaborative task efficiently.
2 THE PRINCIPLE OF ANT COLONY ALGORITHM Ant colony algorithm is a simulated evolutionary algorithm inspired by the behaviour of ants finding the path in the process of looking for food and first proposed in 1992 by the Italian scholar Marco Dorigo. When the ants travel normally, the environment may change suddenly and the obstacles are often encountered. Under these circumstances, various paths will be chosen with equivalent probability. Since pheromone concentration will gradually increase for a shorter path, more and more ants will choose this path. In this way, the ants will eventually bypass obstacles and find an optimal path. The ant colony algorithm which mainly focuses on how to find the shortest path comes from the ant foraging behaviour. When an entire ant colony is searching food, they will send some ants wandering around. The ant will return to the nest to send information to others if it finds food. Meanwhile, the pheromone will be left along the way, which will be tagged as the position of the food for the entire ant colony. Of course, the pheromone is gradually volatile. If two ants find the same food place at the same time and each of them takes a different route back to the nest, then the smell of pheromone which is left on a longer way will be relatively light. The entire ant colony will tend to edge forward along a closer way to the food location. The ant colony algorithm is to design virtual ants, allowing them to explore different routes and leaving virtual pheromones which will gradually be volatile as time goes. According to the principle that "the path with richer pheromone is closer", an optimal path can be finally selected. Ant colony algorithm is a random search optimization algorithm which is a simulation of ant colony behaviour. It can be divided into four parts [11]: strategy selection, partial update of the amount of information, local search algorithm for seeking local optimal solution, and the update of global information. The composition and model diagram of the ant colony algorithm is shown in the following Fig. 1. Node selection Search Ant Colony System
Transfer, update taboo table
Update pheromone
Partial update Global Update
FIG. 1 THE COMPOSITION AND MODEL DIAGRAM OF ANT COLONY ALGORITHM
3 THE SHORTEST MOTION PATH OF SEARCHING FORMATION BASED ON THE ANT COLONY ALGORITHM
3.1 The Representation of Pheromone In the ant colony algorithm of discrete domain, the pheromone is often left in the connection of the constructed graph. This method of storing pheromone by connection corresponds exactly to the true representation of solutions. However, the method is only applicable to solving the problem with scale not big enough. Here the node notation is in utilization. The pheromone is stored at discrete points of the environment model. Any point corresponds to a pheromone value. The size of the amount of pheromone represents the attraction of discrete points to ants. This - 302 http://www.sj-ce.org
method greatly reduces the space complexity of the algorithm. In order to simulate the behaviour of real ants, at first the following notations are introduced: Let m be the number of ants in the ant colony; d ij (i, j 1, 2, , n) represents the distance between the target point i and the target n
point j ; bi (t ) represents the number of ants at the target point i and time t . Then m
bi (t ) ; Let
ij
(t ) represent
i 1
at time t
the residual pheromone on the connection line between target point i and j . At initial time, the
pheromone on each path is identical. Assume that
ij
(0)
c (c is a constant) .
3.2 Principle of the Path Point Selection During the movement of ant k (k 1, 2, , m) , the transfer direction is determined by the pheromone on each path. The probability of ant k transferred from the target point i to the target point j at the time t is [12]: ij
[
Pijk
(t )
ik
(t )
ij
(t )] [
ik
,j
(t )]
Start
tabuk
(1)
k tabuk
0
,j
Enter the path search mode. Record the coordinates of target point and obstacles. Initialize the ant colony algorithm
tabuk
Here ij is the priori knowledge which is also called visibility. It is the heuristic information of the transfection from target point i to target 1 point j . Generally let ij is the importance of the residue dij . information on the path; and is the importance of the heuristic information on the path. Different from actual ant colony systems, artificial ants have memory capacity. tabuk (k 1,2, , m) is used to record the target point traversed by ant k . Here it is called taboo table, which will make dynamic adjustment along with the evolution process. After n moments, the taboo table should be cleared when all the ants have completed a tour. At the same time, the current ant target point should be put into tabuk , readying for the next tour. At this moment, the distance Lk of each ant passed should be calculated and the shortest path Lmin ( Lmin min Lk , k 1, , m) should be saved.
3.3 Pheromone Update Rules
(t
1)
(1
) ij (t )
Update pheromone according to formula(2)
N
Suffice condition to end iteration
Y Find an optimum solution, end path planning
Leader travels to target point along the planned shortest path
With the passage of time, the previous pheromone will gradually disappear. Let parameter 1 indicate the degree of the disappearance of information. After each cycle, the pheromone on each path should adjust according to the following formula (2) [13]. ij
Put M electronic ants on the current point. Each ant is guided to the target point by pheromone. Record accessed target point in the taboo list, then form a path
Followers track the position and direction of the leader with the formed formation
(2)
ij
Y
Encounter obstacle
m
where
k ij
ij
,
N
k 1
k
ij (t , t
1)
Q / Lk , ant K across ij 0, ant K does not across ij
;
represents the increment of pheromone in each path ij of this cycle; represents the pheromone that ant k left on path ij of this cycle [13]; Q is a constant and Lk represents the circular route of ant k . Accordingly, if the ant walks across ij , the pheromone increment is the number obtained ij k ij
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Reach the destination, task completed
End
FIG. 2 ALGORITHM FLOW CHART OF THE SHORTEST PATH OF ROBOTIC FISH FORMATION
by dividing the tour route of ant k using a constant. Here the pheromone increment is only related with the tour route, rather than the specific d ij . Finally the counter NC is set to indicate the number of travel rounds for ants, that is to say, to determine the maximum number of iterations. When the set value is reached, the program operation is ended, and accordingly, the shortest path is searched out.
4 DESIGN OF THE FORMATION KEEPING CONTROLLER 4.1 The Basic Principle In this paper, based on the principle of leader-follower algorithm [14], the controller is designed so that the multirobot fish system can effectively keep formation and avoid obstacles in the complex environment. Under the framework of leader-follower, one robot fish is selected as the leader, and the rest are followers. When the speed and angle of the leader are changed, the followers’ desired states are determined based on the data which is collected by the camera. According to the distance and the azimuth angle between follower and leader, as well as their corresponding changes, the speed and direction of the follower can be controlled through fuzzy mechanism so that the multi-robot fish can quickly and accurately reach the desired pose, and finally realize the formation maintenance. The basic principle of the designed controller is as follows. Firstly, the leader accepts the task and runs to the target point along the planned shortest path while its running state is transmitted to each follower which then calculates its relative position in the formation according to the received information, and gets the control variables to determine its speed and direction according to the external environment and the control strategy. Finally, during the normal process, the followers adjust their operation states to keep the whole formation. The algorithm flow chart of the shortest path of multi-robot fish formation is shown in Fig. 2.
4.2 Controller Design In this paper, the system consisting of two robotic fish is taken as an illustrative example. The controller is designed to keep formation by adopting the parallel structure [15]. By controlling the relative distance l and the relative angle between the two robotic fish, a desired value (ld , d ) can be obtained. In this way, a formation can be formed and maintained, as shown in Fig. 3. y y
t
vt 0
r
y
x
d t
v0
O
Rt d
l R0
0
O
O
FIG. 3 FORMATION KEEPING BASED ON l
x x
Below the algorithm is described in detail for a system composed of two robotic fish. In the control process, the status of leader and follower robotic fish is expressed as R0 ( x0 , y0 , v0 , w0 ) and Rt ( xt , yt , vt , wt ) , respectively. The positive direction of speed line is the forward direction of robotic fish, and the counter clockwise direction is the positive direction of angular velocity. Based on formation requirements, the parameter (l , ) denotes the value that the follower is relative to the leader. As shown in Fig. 3, the following equation [16] can be obtained. l
vt cos r 1 (v0 sin l
dwt sin r
v0 cos
dw0 cos
dw0 sin
vt sin r
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dwt cos r
lw0 )
(3)
Here w0 , wt denote, respectively, the angular velocity of the leader and follower; 0 , t are respectively the angle between the direction of leader and follower movement and the positive X-axis; By geometric knowledge, it is known that: r 0 t.
ld and d are respectively the ideal distance and azimuth of the follower relative to the leader. In order to make the two robotic fish keep a certain formation, the final control objectives are (l ld ) 0 , ( 0 . Now the d) control strategy is defined as [16]: l
1
(ld 2(
l) d
(4)
)
Combining (3), (4), the control variables solved are: cos r wt [ 2l ( d ) v0 sin dw0 cos lw0 sin r ] d vt dwt tan r l ) v0 cos dw0 sin 1 (ld where . cos r Similarly, in view of the limitations of (vt max , wt max ) , the control variables should be changed to
wt
wt max , wt wt , wt max wt wt max , wt
wt max wt max , vt wt max
vt max , vt vt , vt max vt vt max , vt
(5)
vt max vt max . vt max
Thus it can be seen that, in the algorithm about l [17], as long as the angular velocity, speed, position and heading are given to the pilot robotic fish, the followers can run toward the position which has relative distance l and relative direction from the pilot robotic fish. Then the formation can be realized and maintained. In order to ensure the formation stability, the robot fish should have its corresponding coordinates and azimuth at each moment. The goal of the controller is to find followers’ translational speed and rotational speed. Fuzzy control method is introduced to control the followers’ translational speed and heading angle, so that the followers can reduce the adjustment time and the shock, when the leader’ speed and heading angle are mutated. The designed controller has four input variables and the two output variables. The two output variables are parallel. Denoted by L(t ) ld l the difference of the ideal distance and the actual distance between the follower and leader is. At moment t , L(t ) and the distance difference’s change L(t ) are used as the input variables, and meanwhile, the follower’s speed stalls V (t ) is used as the output variable. Among them, the domain of L(t ) is [-30, 30], and the fuzzy set is represented as {NB, NS, ZE, PS, PB}; the domain of the distance difference’s change L(t ) is [-6, 6], and the fuzzy set is represented as {NB, NS, ZE, PS, PB}; the domain of the output variable V (t ) is [1, 15], and the fuzzy set is repgarded as {VS, LS, ME, LF, VF}. The corresponding membership function of each fuzzy control variable is respectively shown in Fig. 4, Fig. 5, and Fig. 6.
u NB
NS ZE 1
30 20 10 0
u PS
NB NS ZE
PB
10 20 30
L(t )
6 4 2
0
1
PS
PB
2
4
6
L(t )
FIG. 4 MEMBERSHIP FUNCTION OF THE INPUT L(t ) FIG. 5 MEMBERSHIP FUNCTION OF THE INPUT L(t )
u
VS
ME
LS
LF
VF
1
V (t ) 0
2
4
6
8
10 12 14 15
FIG. 6 MEMBERSHIP FUNCTION OF THE OUTPUT V (t ) - 305 http://www.sj-ce.org
Meanwhile, denoted by (t ) d the difference of the ideal and the actual azimuth between the follower and leader is. At moment t , (t ) and the azimuth difference’s change (t ) are used as the input variables, and the follower’s direction stalls P(t ) is used as the output variable at the same time. The azimuth difference (t ) is fuzzed by which {NB, NS, ZE, PS, PB} can be obtained; the azimuth difference’s change (t ) is fuzzed, by which {NB, NS, ZE, PS, PB} can be obtain. The domain of the output variable P(t ) is [-7, 7], and the fuzzy set is {LB, LS, ME, RS, RB}. The corresponding membership function of each fuzzy control variable is respectively shown in Fig. 7, Fig. 8, and Fig. 9.
u NB
NS
ZE
1
PB
PS
NB
15
5 0 5
ZE
u
PS
PB
1
(t ) 60
NS
15
15 10 5
60
(t) 0
5
10
15
FIG. 7 MEMBERSHIP FUNCTION OF THE INPUT (t ) FIG. 8 MEMBERSHIP FUNCTION OF THE INPUT (t ) LB
7
LS
4
u
ME
2
1
RB
RS
P(t ) 0
2
4
7
FIG. 9 MEMBERSHIP FUNCTION OF THE OUTPUT P(t )
Due to the fact that the two output variables of the fuzzy controller have a parallel relationship, it is necessary to build two parallel rule bases. The fuzzy inference rules can be expressed as: If L(t ) and L(t ) , then V (t ) ; If (t ) and (t ) , then P(t ) . After many experiments, we can get the corresponding fuzzy control rule table, as shown in the following table. TABLE 1 SPEED STALLS CONTROL RULE TABLE OF MULTI-ROBOT FISH
L(t )
V (t )
L(t )
NB NS ZE PS PB
NB
NS
ZE
PS
PB
VS VS LS LS LS
VS VS LS ME ME
VS LS ME LF VF
ME ME LF VF VF
LF LF LF VF VF
TABLE 2 DIRECTION STALLS CONTROL RULE TABLE OF MULTI-ROBOT FISH
(t )
P(t )
(t )
NB NS ZE PS PB
NB LB LB LB LS LS
NS LB LB LS ME ME
ZE LS LS ME RS RS
PS ME ME RS RB RB
PB RS RS RB RB RB
After the blurring and fuzzy reasoning are completed and the fuzzy rules are established, the output variables’ defuzzification can be realized based on the maximum membership degree method. With all these efforts, the precise values can be obtained about the speed stalls and the direction stalls. Then the follower will perform the two swimming instruction to better keep the formation. - 306 http://www.sj-ce.org
5 SIMULATION AND ANALYSIS In the experimental simulation, the effectiveness of the algorithm is verified for the system consisting of two robotic fish. The goal is to make system not only maintain longitudinal formation, but also always choose the shortest path to pass through the specified target point. Eventually, the formation will return to the starting point in the known complex environment. To this end, four target points are first determined in a complex environment. Their coordinates are (0, 0), (14, 6), (18, 9) and (8, 4.8), where (0, 0) is the starting point as well as the point to which the robot fish will eventually return. In the environment of Matlab, the fuzzy controller is established according to the membership functions of the control variables and the fuzzy rules. Next, the fuzzy controller is imported to Simulink work area, and a formation control module is built. Then, m files are written based on both the basic principles of the leader-follower formation control and the program flowchart of the ant colony algorithm, and in the m files, we call the formation control module which is built in Simulink. In the program, the parameter characterizing the degree of importance of pheromone is 2. While the parameter characterizing the degree of importance of heuristic factor is 2. The pheromone strength coefficient is 0.5 and the pheromone evaporation coefficient is 0.5. In order to verify the effectiveness of the algorithm, the parameters of the algorithm are further optimized. Based on the experiment and relevant data [18], the realization degree of the shortest path is optimized when the number of ants is three times that of target point. Therefore, with m =12, NC _ max =50, running simulation file yields the shortest path trajectory of the multi robotic fish formation, as shown in Fig. 10. Shortest path trajectory of multi-robot fish formation 10
---goal ---obstacle
8
---leader ---follower
Ordinate y(dm)
6
4
2
0
-2 -5
0
5 10 Abscissa x(dm)
15
20
FIG. 10 THE SHORTEST PATH TRAJECTORY OF MULTI-ROBOT FISH FORMATION
The simulation results Fig. 11 and Fig. 12 depict the shortest motion path of the multi robotic fish formation corresponding to the travelled route length of the generation. Fig. 11 shows the average length of the path corresponding to the current generation traversed after each iteration. Fig. 12 shows the shortest path length of the current generation after each iteration. It can be seen that the algorithm can get the optimal solution of the problem after 20 iterations. That is to say, the algorithm can find the shortest path to complete the task. Average length of multi-robot fish formation path
Shortest path length of multi-robot fish formation
49
41.25
48 41.2
Shortest length of path
Average length of path
47 46 45 44
41.15
41.1
41.05
43 41 42 41
0
5
10
15
20 25 30 Number of iterations
35
40
45
50
FIG. 11 AVERAGE LENGTH OF THE FORMATION PATH
40.95
0
5
10
15
20 25 30 Number of iterations
35
40
45
50
FIG. 12 THE SHORTEST PATH LENGTH OF THE FORMATION
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In the process of simulation, the distance and orientation angle between follower and leader deviate from the ideal value, when the robot fish encounters obstacles. The following situation may also occur that distance deviates from ideal values or azimuth deviates from ideal values, when the robot fish arrives at each target point. The curves of follower’s speed stalls and direction stalls are respectively shown in Fig. 13, Fig. 14. As seen, when the distance or range between follower and leader deviates from the ideal value, under the regulation of the formation keeping controller, the follower can timely respond to the speed and direction. In this way, the whole formation can run normally again as soon as possible. Speed stalls of Follower
Direction stalls of Follower
14
8
12
6 4 2
8
Direction
Speed(cm/s)
10
6
0 -2
4
-4
2
0 0
-6
50
100
150 200 Time(s)
250
300
350
FIG. 13 SPEED STALLS OF FOLLOWER
-8
0
50
100
150 200 Time(s)
250
300
350
FIG. 14 DIRECTION STALLS OF FOLLOWER
The simulation results show that the algorithm has a good performance in the enhancement of the stability of formation control as well as in finding the shortest path quickly and efficiently.
6 CONCLUSIONS This paper studies how the multiple robot fish system arrives at the designated location along the shortest path and keeps a certain formation under the environment with obstacles. In traditional formation control, multiple robotic fish cannot arrive at the designated location along the optimal path quickly and efficiently. To overcome the problem, we have adopted the strategy of combining the leader-follower algorithm with ant colony algorithm, and introduced the fuzzy control mechanism in multi-robot fish system. Simulations show that the proposed method is feasible, which can make the multi-robot fish system realize formation control effectively. The method cannot only n ensure the smooth movement of formations but also find the shortest path from a given starting point to the destination point. As a consequence, the robotic fish can bypass all the obstacles in the movement without collision and meanwhile arrive at the designated location safely.
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AUTHORS 1
3
Master. Her research interest is the multi-
Professor, PhD, Master's tutor.
Susu Shan was born in 1988, Female,
Junwei Gao was born in 1972, Male,
agent system.
He received his Ph.D. degree in Traffic
Email: shansusu@126.com
Information and Control Engineering from China Academy of Railway Sciences in 2003. He participated in the Gansu Province
2
Natural
Science
Fund,
the
Zhijian Ji was born in 1973, Male, Ph. D,
National Natural Science Foundation and Beijing City
Professor, Tutor of doctoral students. His
Committee of science and technology projects and other topics.
research interests are the population
He published over 30 papers, including then articles published
system dynamics and control, complex
on EI, ISTP included. His research interests are intelligent
network, switching power system analysis
system, intelligent control, hybrid system theory and engineering
and control, system biology and control
application. Email: qdgao163@163.com
systems based on network. In the Journal of international control field and in the important meeting of the control community, he has published more than 40 papers, including 16 SCI articles, more than 30 papers indexed by EI. He presided over two National Natural Science Funds, and participated in many National Natural Science Fund and 973 and 863 project. Email: jizhijian@pku.org.cn
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