c 2005 Institute for Scientific
°
Computing and Information
INTERNATIONAL JOURNAL OF
INFORMATION AND SYSTEMS SCIENCES
Volume 1, Number 3-4, Pages 364–371
A METHOD OF MULTI-ROBOT FORMATION
WITH THE LEAST TOTAL COST
MEIPING SONG, GUOCHANG GU, RUBO ZHANG, AND XINGCE WANG
Abstract. A method of formation used in the “pursuit-evasion” game is presented in this paper. It reduces the computational complexity efficiently with
the least total cost. This method takes the advantages of adaptation and cooperation in multi-robot, and introduces the roles of real leader and virtual
leader. The method can work well with any type of formation, especially with
complex ones. It can overcome the problem of unnecessary total cost in the
fixed role formation when rotating wholly, and decrease the complexity of role
assignment in formation caused by the number of robots. The efficiency of
the method is illuminated from the viewpoints of geometry, complexity and
simulation.
Key Words. formation, multi-robot, role assignment, the least total cost.
1. Introduction
Today the research of robotics has been focused on multi-robotics, and the most
commonly used platforms are formation, pursuit-evasion and Robot world cup.
These all need the robots to cooperate to form a particular shape to achieve some
special task. For some special cooperative task, the efficiency of the formation
directly influences the performance of systems, such as maritime search and rescue,
and searching and driving of an invader in a certain environment. Therefore a
good method of formation is critically important for the efficiency of multi-robot
cooperation.
There have been several formation methods, such as time-optimal [1], formation
vector [2] and adaptive team formation [3]. These algorithms were concerned much
with the problems of time and the adaptability of formation, but little with the
total travel cost of multi-robot. When robots perform a task, the problem of energy
supply is still underlying. So, a method requiring the least cost is indeed necessary.
On the other hand, when multi-robot cooperation in the dynamic environment on
line, the real-time quality is important. A good method of formation should solve
the problem emerged in formation without ignoring the real-time quality. Therefore,
a distributed optimal formation method is proposed in this paper, which improves
the adaptability of formation without reducing the real-time quality of system.
It decreases the total travel cost of multi-robot greatly at the same time. The
efficiency of the method is illustrated from the viewpoints of geometry, complexity
and simulation.
The paper is organized as follow. The shortcomings of fixed role formation in
“pursuit-evasion” game are detailed firstly. Secondly, a new method with the least
total cost is prompted, and the efficiency is illustrated. Lastly, the simulation
results and conclusions are shown.
Received by the editors June 1, 2004 and, in revised form, January 22, 2005.
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2. Motivation
In “pursuit-evasion” games, the common methods to construct robots’ behavior
are based on the form of graph [4][5] or grid [6][7].
In the graphical case, the environment is known in advance and the games are
mostly based on visibility, that is, the task is achieved once the invader is detected[8][9]. In this case, there is no need for robots to form a special shape to
catch the evader. But in the grid case, the environment is often unknown, and the
pursuing robot should occupy most even all of the cells around the evader so as to
prevent from it moving. The state of “success” in grid game is shown in figure (1),
where the black squares represent pursuers and the black circle corresponds to the
evader. Every robot has five available motions: stop, left, right, up and down. So,
in the state shown in figure (1), the evader cannot move any more, and the task is
achieved.
Figure 1. The state of success in grid form
In grid games, pursuers should cooperate to achieve a shape of formation eventually. According to the model of task and the state of evaders, pursuers are partitioned into several different groups, which corresponds to particular roles. This
is what being investigated in the field of task decomposition and role assignment.
The fixed role assignment costs less calculation and can guarantee the success and
stability of formation, but it brings about the problems of unnecessary total cost
at the same time.
The problem is more obvious in tracing a target. There are four roles in the
task of figure (2): head, left, right and tail respectively. In the rhombic formation
of figure (2(a)), robot R1 is the fixed head, R2 the left, R3 the right, and R4 the
tail. When the path changes the corner as figure (2(b)) shows, the formation will
have to change into figure (2(c)) accordingly. And then this brings about more
unnecessary total cost than the situation in figure 2(d)).
The cause of the above problem is the fixed roles in the process, which don’t
fully take advantage of the cooperation in multi-robot. If we transfer the head role
to R3 , and the other roles clockwise to the corresponding robots in the corner of
the path, the situation will be more appealing. And the ideal situation shown in
figure (2(d)) will be achieved. Then, what attracts our attention is how to realize
role transition among robots in the distributed system. The direct method is to
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Figure 2. Analysis of fixed role formation and dynamic role assignment
match all the roles with all the robots respectively, and then selecte the best one
with some predefined criterion. But the scale of this method increases disastrously
with the number of robots. The method prompted here deals with this problem
perfectly. That is what to be discussed in the next section.
3. Optimal policy of formation with the least cost
3.1. Formulation. In order to construct continuous behavior for robots and allow
drive of evader, an extended version of grid “pursuit-evasion” game is used. The
original description of environment and task model can be found in [10]. Here,
the state of “success” is defined in figure (3) and there could be some polynomial
obstacles in the unknown environment just as in figure (5). From figure (3), it can
be seen that the pursuers should form a regular polygon, which is the inscribed one
of the predefined catching circle using the evader as center. Ri represents the ith
pursuer, i=1,. . . ,n, where n is the number of required pursuers. RT is the evader.
r is the radius of catching circle. And it is determined by the safe distance df
between robots. That is to say,
(1)
r = df /(2 × sin(π/n)).
This makes the formulation flexible for various scales of problems. When the number of required pursuers changes, the radius of catching circle changes correspondingly, which will not influence the role assignment method in this paper. By now,
the problem of task decomposition in formation is solved, and next we will go to
the problem of role assignment.
3.2. Algorithm description. The primary idea of the role assignment technique
in this paper is that, the robot whose policy causes the least total cost acts as the
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367
Figure 3. The state of success in extended version
real leader in each cycle. At the same time, the others act as assistants who fetch
their sub-tasks assigned by the real leader from the blackboard system and reset
their targets to shape a new pattern.
The process can be described as follows. Firstly, each robot acts as a virtual
leader and makes its own local decision regarding the current situation independently. Each decision is considered to cause the least cost of both itself and total
distance from the viewpoint of its possessor. Secondly,all the locally optimal policies are sent to the blackboard, where their values of total cost are compared and
the globally optimal one is selected eventually. At last, the decision maker of the
winning policy acts as the real leader, and the others act as its assistants in the
performance of this cycle. All the robots receive their new roles and continue the
task.
The instance in figure (4) is taken as an example to illustrate the algorithm
prompted here. There are 6 pursuers in the “pursuit-evasion” task, and the expected shape is the regular hexagon centered on the evader. Figure 4 shows the
local optimal decision of virtual leader R2 . RT is the evader, Ri is the ith pursuer,
i=1,. . . ,6. T0 labelled by black triangle is the nearest point to R2 on the boundary
of catching circle. Because of selfishness, R2 would prefer to take T0 as its subtarget, and then decompose the task based on T0 locally. Tj is the jth sub-target
determined by R2 according to the point of T0 , j=1,. . . ,5. That is formula (2),
(2)
angle(Tj−1 RT Tj ) = 2 × π/n.
The decision process of R2 in the distributed system can be described as the
following.
Step 1: R2 determines its sub-target T0 labelled by black triangle on the crossover
of its joining line to RT and the catching circle centered on RT as well. The other
five sub-targets are located based on T0 and RT (or any other reference point of the
formation) to form a regular hexagon centered on RT . The resulting order set of
the other sub-targets is T ={T1 , T2 , T3 , T4 , T5 }.
Step 2: Build up a polar coordination system that has the pole RT and the polar
−−−→
axis RT R2 . Compute the polar angles θi of the other pursuers in this coordination
system, i=1,3,4,5,6.
Step 3: Arrange the other pursuers so that their corresponding polar angles θi
are in an ascending order using any type of sorting method such as Bubble sort.
The final order set in figure (4) can be represented as R={R6 , R4 , R1 , R3 , R5 }.
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M. SONG, G. GU, R. ZHANG, AND X. WANG
Figure 4. The local optimal decision of virtual leader R2
Step 4: Match the corresponding elements in the sub-targets order set T and
pursuers order set R in turn, and then the locally optimal decision of R2 is accomplished. Each pursuer corresponds to a sub-target in this local policy.
Step 5: Put this local optimal decision result into the blackboard system, select
the global optimal decision and then the real leader is determined according to the
criterion of the least total cost.
Step 6: According to the real leader’s policy, each robot may fetch its new role
and sub-target through the blackboard system, modify the position of sub-target
(detailed next) and forms in the new pattern. Then the transfer of leadership is
accomplished.
Note that because of the fact that there are obstacles in the environment, some
of the original sub-targets may fall into the area occupied by the obstacles. If
the sub-target of a certain pursuer is in the obstacle, it should withdraw the subtarget along the radius vector shown in figure(5), until the sub-target is safe for
the robot to avoid collision with the obstacle. When obstacles are so close that
the safe distance can’t be satisfied, the robot will head in a line formation. In the
figure, the dark polygons are static obstacles, and Ti0 is the real sub-target after
modification, i=2,3,5,6. This technique makes the shape of formation flexible to
the environment, especially when the obstacles are dense.
3.3. Brief proof. Next we briefly prove that the combination of the dotted line
R1 → T3 or R3 → T4 in figure(4) is better than that of R3 → T3 and R1 → T4 with
the principle of least total cost from the viewpoint of geometry. These generally
represent different types of role assignment, and the former is what implied by the
method in this paper.
Firstly, the former avoids the crossover of lines, which means avoiding the path
conflict between R1 and R3 implicitly. To resolve the conflict in multi-robot system,
it needs not only the computation work of replanning but also the communication
between the involved robots. This will degrade the real-time performance of the
A METHOD OF MULTI-ROBOT FORMATION WITH THE LEAST TOTAL COST
369
Figure 5. Modification of target when in the obstacle
system, especially in the situation that communication is difficult, such as underwater.
Secondly, there are the following four relationships for the triangles 4R1 OT3
and 4R3 OT4 in terms of geometry.
(3)
kR1 Ok + kOT3 k > kR1 T3 k.
(4)
kR3 Ok + kOT4 k > kR3 T4 k.
(5)
kR1 Ok + kOT4 k = kR1 T4 k.
(6)
kR3 Ok + kOT3 k = kR3 T3 k.
Then there is
(7)
kR1 T4 k + kR3 T3 k > kR1 T3 k + kR3 T4 k.
The inequality(7)can be drawn from the above four ones. It can then be proved
that the current decision is the locally optimal one on the analogy of this. On
the other hand, step 5 of the algorithm avoids the local optimization, which takes
the advantage of cooperation between robots. So the final decision would be the
globally optimal one.
3.4. Analysis of complexity. This algorithm introduces the roles of real leader
and virtual leaders. Given n robots, the most costly work of each virtual leader is
composed by two parts. The first one is to make its local optimal decision, which is
actually sorting the (n-1) polar angles and matching them with the corresponding
pursues. The other is to detect the globally optimal policy within the n policies
of all virtual leaders’. The time complexity for this process is O(n2 ) when Bubble
sort method is used.
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M. SONG, G. GU, R. ZHANG, AND X. WANG
But without virtual leaders and a real leader, the robot should firstly repeat
combining the n robots with the n sub-targets in (n!) possible patterns, and (n!)
policies are established. Then the robot would select the global one from these
(n!) policies. The time complexity for this process is O(n!). It can be drawn
consequently that the algorithm prompted in this paper can reduce time complexity
greatly.
4. Simulation and conclusions
The feasibility of this algorithm can be easily tested by a simulation system.
The result is that the robots transfer the power of leading in the situation shown
in figure (2), and come up with the shape shown in figure (2(d)). In condition that
the obstacles are dense, the robots can modify the shape of formation to go through
them safely, just as shown in figure(6).
Figure 6. The simulating result
It can be concluded that this method can avoid unnecessary total costs caused
by wholly rotating through changing roles according to the dynamic reference point
and environment flexibly. Secondly, it reduces the computations greatly in contrast
to the case without a leader. Thirdly, this method overcomes the problem of combination explosion caused by the increasing number of robots, and is more robust
than the method of matrix selection because it can avoid the final unexpected awful
result [1].
This method can be easily extended to complex shape of formation, but its
advantage is not obvious with the linear shape.
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College of Computer Science and Technology, Harbin Engineering University, Harbin, Heilongjiang 150001, CHINA
E-mail: songmeiping@hrbeu.edu.cn