3.1. Colored Petri Nets
Definition 1. Let S be a set. A multiset [45] over S is defined by a mapping and is represented aswhere is an element of set S and denotes the number of occurrences of the element s in the set S. A Colored Petri Net (CPN) is a quintuple , where P is a set of n places; T is the a of m transitions; is a set of colors; is a function that associates a set of colors from to each element in . For each , is the set of possible token colors in , and denotes the total number of possible token colors in . Analogously, for each , is the set of possible occurrence colors of , and denotes the total number of possible occurrence colors of .
Matrices , are the pre- and post-incidence matrix. Each element is mapped from the set of occurrence colors of to a non-negative multiset over the set of colors of , i.e., , for and . We use to represent a matrix of non-negative integers, where denotes the weight of the arc from place with respect to color to transition with respect to color . The mapping for can be analogously defined. The incidence matrix is an matrix, where , , , and .
The
marking of place
is a
non-negative multiset over
. The mapping
is associated with each possible token color in
, which is a non-negative integer representing the number of tokens of that color contained in place
, and
where marking
is denoted as a column vector of
non-negative integers that has an
h-th component
equal to the number of color tokens
contained in place
.
The marking
is defined as an
n-dimensional column vector of multisets, where
A colored Petri net system is a colored Petri net N with a marking .
A transition
is
enabled at a marking
with respect to color
if for each
it holds
(
), and an enabled transition
may fire yielding a new marking
, denoted by
, where
,
. A
firing sequence from
is a sequence of transitions, each one firing with respect to a given color such that
Marking
is said to be
reachable in
if there exists a firing sequence
such that
, and it holds that
where
is called the
firing count vector of
, and
(
) is a multiset that indicates the firing times of transition
with respect to each of its colors.
3.2. Timed Colored Petri Net for the Multi-Type Robot Systems
A timed Colored Petri net (TCPN) is a pair
[
44], where:
is a CPN,
is a non-negative multiset over . The mapping associates to each occurrence color of transition a non-negative integer fixed firing duration. We denote by the firing delay vector, where .
We denote by a TCPN system, where is an initial marking.
In the following, it is assumed that the
moving environment of an MTRS is divided into a set of
n triangular regions that connect to each other via several predefined line segments [
47]. Each type of robot can move freely inside any triangular region except the forbidden one. We model the MTRS and its working environment by a TCPN according to Algorithm 1. To illustrate the construction of a TCPN better, a simple MTRS is discussed in the following example.
Example 1. Let us consider the MTRS in Figure 1. Its moving environment is divided into ten triangular regions, i.e., . Two robots with different functions are initially deployed in regions (represented by blue circle ) and (represented by red circle ), respectively, i.e., . The TCPN system of the MTRS is shown in
Figure 2. It consists of 10 places,
, each of which represents a region, and 26 transitions,
, each of which represents the movement of a robot from one region to an adjacent one. The description of transitions and places is presented in
Table 1.
Algorithm 1: Construct the TCPN model for the multi-type robot system |
- Require:
The working environment with K robots and n regions - Ensure:
A TCPN system and a distance vector - 1:
Let - 2:
Associate each region with a place in the place set P, i.e., each place represents a region ; - 3:
for each do - 4:
for each do - 5:
if and are adjacent, i.e., region and region are adjacent then - 6:
; - 7:
; %transition represents the movement from to - 8:
equals the average time for a type of robot moving from region to region , ; - 9:
equals the average distance for a type of robot moving from region to region , ; - 10:
, where denotes a identity matrix; - 11:
end if - 12:
end for - 13:
end for - 14:
for do - 15:
The number of robots with color initially located in region ; - 16:
end for
|
The possible color sets of each place
(
) and each transition
(
) are
and
, respectively. The initial marking
of the TCPN system is a column vector that contains ten subvectors, where
,
,
,
,
,
,
,
,
, and
. Each element in subvector
denotes the number of two different robots in region
. The incidence matrix
of the TCPN system is a
matrix, where
is a
matrix, for
and
. For instance, the sixth row of
indicates that the input (resp., output) transitions of place
are
and
(resp.,
and
).
Since every robot can have access to every region, we have
Assume that the distance and the moving time between every two adjacent regions are equal to one and two, respectively; we have the distance vector , where , and the firing delay vector , where , .