Abstract
A cyber-physical system (CPS) is a system of collaborating computational elements controlling physical entities. Nowadays cyber-physical systems can be found in areas as diverse as robotics, automotive, chemical processes, civil infrastructure, energy, healthcare, manufacturing, transportation, entertainment, and consumer appliances. In this paper, we address the problem of spatial-temporal group scheduling using Generalized nets (GN). We use GN in order to model the spatial, temporal, ordered and concurrent character of our mobile, distributed system. Our model is based on a discrete topology in which devices can change their location by moving from cell to cell. Using the GN, we model movement in a heterogeneous terrain as well as task execution or access to other resources of the devices. Intuitionistic Fuzzy Logic (IFL) are defined as extensions of ordinary fuzzy sets. All results which are valid for fuzzy sets can be transformed here too. Also, all research, for which the apparatus of fuzzy sets can be used, can be used to describe the details of IFL. In this paper we use it to obtain the Intuitionistic Fuzzy Estimation (IFE) for obtaining the degree of effectiveness, the degree of ineffectiveness of the robot and uncertainty during the robot movement.
1 Introduction
Cyber-Physical Systems (CPS) [1] are integrations of computation, networking, and physical processes. Embedded computers and networks monitor and control the physical processes, usually with feedback loops where physical processes affect computations and vice versa. The potential of such systems is greater than what has been realized. There are considerable challenges, particularly because the physical components of such systems introduce safety and reliability requirements qualitatively different from those in general-purpose computing. Moreover, the standard abstractions used in computing do not fit the physical parts of the system well.
Some researchers [2] are focused on the challenges of modeling cyber-physical systems that arise from the intrinsic heterogeneity, concurrency, and sensitivity to timing of such systems. It uses a portion of an aircraft vehicle management systems (VMS), specifically the fuel management subsystem, to illustrate the challenges, and then discusses technologies that at least partially address the challenges. Specific technologies described include hybrid system modeling and simulation, concurrent and hetero-geneous models of computation, the use of domain-specific ontologies to enhance modularity, and the joint modeling of functionality and implementation architectures.
Applications of CPS arguably have high potential [1]. They include high confidence medical devices and systems, assisted living, traffic control and safety, advanced automotive systems, process control, energy conservation, environmental control, avionics, instrumentation, critical infrastructure control (electric power, water resources, and communications systems for example), distributed robotics (telepresence, telemedicine), defense systems, manufacturing, and smart structures. Networked autonomous vehicles could dramatically enhance the effectiveness and could offer substantially more effective disaster recovery techniques. In communications, cognitive radio could benefit enormously from distributed consensus about available band-width and from distributed control technologies. Distributed real-time games that integrate sensors and actuators could change the (relatively passive) nature of on-line social interactions.
By focusing on the physical world it becomes obvious that non-computational processes (physical actions) are strongly distributed and concurrent. Thus, designing and programming those systems have to cope with those issues. Since thinking in distributed and concurrent terms is complexity-introducing and often error-prone [3], we have studied this problem and proposed a suitable programming model [4–6] that both abstracts from distribution and concurrency by allowing the programmer to develop sequential object-oriented program code. Besides the imperative code fragments, declarative annotations can be integrated into the source code for defining spatial-temporal constraints that are glued to imperative code fragments and restrict its execution. All this requires a coordination of resources in space and time.
The aim in this paper is to address the problem of spatial-temporal group scheduling by using Generalized nets [7, 8] and Intuitionistic Fuzzy Logic and based on the [9]. Our concept is to map space to time and describe physical locations based on durations needed to change locations. The computation of a schedule is based on those timed transitions.
Intuitionistic Fuzzy Logic (IFL) [10] are defined as extensions of ordinary fuzzy sets. All results which are valid for fuzzy sets can be transformed here too. Also, all research, for which the apparatus of fuzzy sets can be used, can be used to describe the details of IFL. In this paper we use it to obtain the Intuitionistic Fuzzy Estimation (IFE) for obtaining the degree of effectiveness, the degree of ineffectiveness of the robot and uncertainty during the robot movement.
2 Assumptions
In our understanding, a task is associated with duration and, e.g., a deadline at which the task has to be completed depending on hard or soft deadlines. Thus, tasks may have temporal constraints. In this paper, we extend the classical view by a new dimension: space.
A task t is described by a set of properties {d, p, p′, r, T′}, with d indicating the duration of the task and p and p′ the beginning and ending location of the task, respectively. A task may also bound to a fixed location—in that case p and p′ are identical. A location is a physical position on a 2D surface. In addition, we address the problem of performing tasks jointly, i.e., a given amount of robots r ∈ R, with |R| denote the total number of robots, is required to perform a task that have to be coordinated in space and time. For simplification, we assume tasks are non-interruptable. Finally, the execution of t depends on the result of the set of predecessor tasks T′ that need to be executed prior to t.
The 2D surface in which the robots operate is discretized and mapped to a specific topology. Each cell c i in the topology indicate a space in which an arbitrary amount of robots can be placed: c i ∈ {x ∈ N |0 ≤ x ≤ |R|} and c i = |R|. We support different topologies as shown in Fig. 1 with respect to the geometry of the surface, the discretization (cell shape) and the multi-plicity of movements. A robot can change its location by moving in discrete steps to a neighboring cell along the indicated arrows. On the left hand side of the figure a cell is represented by a square and exhibits four possible movements of a robot. The middle topology doubles the degree of freedom by allowing diagonal movements. Finally, the topology on the right shows a discretization that is based on hexagons which allows for six different types of movements. During each time step a robot has different options:
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Stay in the current cell (idle)
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Move along the arrows towards a neighboring cell
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Execute a task (if the task involves movement, the robot moves towards the tasks’ ending location while executing it at the same time).
The movement model is based on a binary state: The robot does not move (idle) or simply moves (speed is not incorporated in the model). If a robot decides to move to a neighboring cell, the cell transition is associated with a given amount of time required for reaching the other cell (this again represents the worst case time needed for moving the robot). The topology does not have a homogeneous terrain, thus, times between cell transitions may vary. With this, we are able to model accessible and non-accessible obstacles. Driving uphill takes more time to process the transition than driving downhill. On the other hand, a solid formation, e.g. rocks, are not accessible and, thus, the robots have to take the longer way in terms of geographic distance. Altogether, this approach allows us to model the important properties of robots moving in a terrain without the need to deal with the physics of the actual movement actions—these are represented by the time needed for transitions between the cells.
Now, the overall goal is to find a schedule with minimal makespan such that all tasks t i are executed according to their requirements of beginning and ending location and the number of robots which includes physical positioning of robots.
3 Intuitionistic Fuzzy Logic
Intuitionistic Fuzzy Logic [10] (IFL) are defined as extensions of ordinary fuzzy sets. All results which are valid for fuzzy sets can be transformed here too. Also, all research, for which the apparatus of fuzzy sets can be used, can be used to describe the details of IFL.
On the other hand, there have been defined over IFL not only operations similar to those of ordinary fuzzy sets, but also operators that cannot be defined in the case of ordinary fuzzy sets.
Let a set E be fixed. An IFS A in E is an object of the following form:
where functions μ A : E → [0, 1] and ν A : E → [0, 1] define the degree of membership and the degree of non-membership of the element x ∈ E, respectively, and for every x ∈ E:
For every x ∈ E, let
Therefore, the function π determines the degree of uncertainty.
Obviously, for every ordinary fuzzy set π A (x) = 0 for each x ∈ E, these sets have the form:
Let a universe E be given. One of the geometrical interpretations of the IFL uses figure F on Fig. 2.
4 Generalized Net Model
The GN-model (see Fig. 3) contains 3(r + 1) transitions and 9(r + 1) places, collected in two groups and related to the three types of the tokens that will enter respective types of places:
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α-tokens and a-places represent the robots and its positions;
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β-tokens and b-places represent the tasks of the robots;
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γ-tokens and S-places represent the functions of the intelligent system.
For brevity, we shall use the notation α- and β-tokens instead of α i - and β j - tokens, where i and j are numerations of the respective tokens.
In the beginning α-, β- and γ- tokens stay, respectively, in places a r+3, b 2+r+1 and s with initial characteristics:
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\( x_{0}^{\alpha } \) = “robot r i , position of the robot r i (initial position p and end positions p′)” in place a r+3,
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\( x_{0}^{\beta } \) = “task t for the robot r i ; duration d of the task”, in place b 2+r+1,
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for i ∈ [1, …, r],
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\( x_{0}^{\gamma } \) = “intelligent system”, in place s.
Via place s 0 enter γ-token with initial characteristics:
\( x_{0}^{\gamma } \) = “robot r i , position of the robot r i (initial position p and end positions p′), task t for the robot r i ; duration d of the task, predecessor tasks T′”.
Let \( x_{cu}^{\alpha } ,x_{cu}^{\beta } \) and \( x_{cu}^{\gamma } \) be current α-, β-, and γ-tokens’ characteristics, respectively. Generalized net is presented by a set of transitions:
where transitions describe the following processes:
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Z 1—Task of the intelligent system for control of the robots;
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Z 2—Activity of the robots;
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Z 3—Task performed of the robots;
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Z 4—Choice of action for the robot 1;
…
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Z 4+r−1—Choice of action for the robot r;
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Z 4,1—Movement of the robot 1;
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Z 4,2—Execution of a task of the robot 1;
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Z 4,3—Calculating the intuitionistic fuzzy estimation of the execution of a task of the robot 1;
…
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Z 4+r−1,1—Movement of the robot r;
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Z 4+r−1,2—Execution of a task of the robot r.
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Z 4+r−1,3—calculating the intuitionistic fuzzy estimation of the execution of a task of the robot r.
The forms of the transitions are the following.
where:
and
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\( W_{s,a1}^{{}} \) = “The tasks are ordered in the proper way”,
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\( W_{s,b1} = W_{s,a1} \).
The α-token obtains characteristic “robot r i, initial position p and the end positions p′ of the robot r i” in place a 1. The β-token obtains characteristic “task t for the robot r i; duration d of the task, predecessor tasks T′” in place b 1.
The β-token obtains characteristic “task t for the robot r i; duration d of the task, predecessor tasks T′” in place b 1.
Initially when still no information for robot tasks has been obtained, all estimations are given initial values of <0, 0>. When k ≥ 0, the (k + 1)-st estimation for the effectiveness of the system is calculated on the basis of the previous estimations according to the recurrence relation
where:
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\( m = \frac{{\sum\limits_{i = 1}^{r} {m_{i} } }}{r}, \)
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\( n = \frac{{\sum\limits_{i = 1}^{r} {n_{i} } }}{r} \)
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\( {<}\mu_{k} ,\nu_{k}{>} \) is the previous estimation, and <m, l> is the estimation of the latest measurement, for m, l ∈ [0, 1] and m + l ≤ 1.
where:0
and
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\( W_{r + 3,a2} \) = “There is a feedback from robot r i”,
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\( W_{r + 3,a3} \) = “There is a task for robot r 1”, …
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\( W_{r + 3,r + 2} \) = “There is a task for robot r r”.
The α-token obtains characteristic “robot r i, end positions p′ of the robot r i” in place a 2 . The α-token obtains characteristic “robot r 1, end positions p′ of the robot r 1” in place a 3 . The α-token obtains characteristic “robot r n , end positions p′ of the robot r r” in place a r+2 . The α-token obtains characteristic “robot r i , end positions p′ of the robot r i” in place a r+3 .
where:
and
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\( W_{2 + r + 1,b2} \) = “There is a task for robot r 1 for time d”, …
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\( W_{2 + r + 1,2 + r - 1} \) = “There is a task for robot r r for time d”.
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\( W_{2 + r + 1,2 + r} \) = “There is a feedback from robot r i”,
The β-token obtains characteristic “task for robot r 1, time d” in place b 2. The β-token obtains characteristic “task for robot r r, time d” in place b 2+r−1. The β-token obtains characteristic “task for robot r i, time d” in place b 2+r.
where:
and
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\( W_{4,4,a4,1} \) = “There is a feedback from robot r 1”,
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\( W_{4,4,b4,1} \) = “The current task for robot r 1 is executed”.
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\( W_{4,4,a4,2} \) =”There is a command for movement for robot 1”
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\( W_{4,4,a4,3} \) =”There is a task for robot 1”
The α-token obtains characteristic “robot r 1, end positions p′” in place a 4,1. The β-token obtains characteristic “robot r 1, executed task” in place b 4,1. The α-token obtains characteristic “robot r 1, end positions p′ of the robot r 1” in place a 4,2. The α-token obtains characteristic “task for the robot r 1” in place a 4,3.
where:
and \( W_{4,7,4,5} = W_{4,7,4,6} \) “The robot 1 performed the movement to positions p′”.
The α-tokens obtain characteristic “the result of the performans of the movement to positions p′” in places a 4,5 and a 4,6.
where:
and \( W_{4,10,4,8} = W_{4,10,4,9} \) “The robot 1 performed the task”.
The α-tokens obtain characteristic “the result of the execution of the task” in places a 4,8 and a 4,9.
where:
and \( W_{4,8,4,12} \) = “The ituitionistic fuzzy estimation was calculated”,
The α-tokens obtain characteristic: “intuitionistic fuzzy estimation \( \left\langle {m_{1} ,l_{1} } \right\rangle \)”.
The estimations \( \left\langle {m_{1} ,l_{1} } \right\rangle \) ∈ [0, 1] × [0, 1] reflects the degree of effectiveness (m 1) and the degree of ineffectiveness of the first robot (l 1) for a time t.
where:
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p 1 is the number of successfully movements and tasks performans of the robot 1,
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s 1 is the number of unsuccessfully movements and tasks performans of the robot 1,
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n 1 is the total number of movements and tasks performans of the robot 1.
The degree of uncertainty π1 = 1 − m 1 − n 1 reflects the cases when the robot have not completed a movement or task, which is the number of α-tokens in both places a 4,7 and a 4,10.
where:
and
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\( W_{14,11} \) = “There is a feedback from robot r r”,
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\( W_{14,11} \) = “The current task for robot r r is executed”.
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\( W_{14,13} \) = “There is a task for movement for robot r”
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\( W_{14,b11} \) = “There is a task for robot r”
The α-token obtains characteristic “robot r r, end positions p′” in place a 4+r−1,1. The β-token obtains characteristic “robot r r, executed task” in place b 4+r−1,1. The α-token obtains characteristic “robot r r, end positions p′ of the robot r 1” in place a 4+r−1,3. The α-token obtains characteristic “task for the robot r r” in place a 4+r−1,2.
where:
and \( W_{17,15} \) = \( W_{17,16} \) = “The robot n performed the movement to positions p′”.
The α-tokens obtain characteristic “performed movement to positions p′” in places a 4+r−1,5 and a 4+r−1,6.
where:
and \( W_{110,18} \) = \( W_{110,19} \) = “The robot r performed the task”. The α-tokens obtain charac-teristic “performed task” in places a 4+r−1,8 and a 4+r−1,9.
where:
and \( W_{4,8,4,12} \) = “The ituitionistic fuzzy estimation was calculated”.
The α-tokens obtain characteristic: “intuitionistic fuzzy estimation \( \left\langle {m_{r} ,l_{r} } \right\rangle \)”.
The estimations \( \left\langle {m_{r} ,l_{r} } \right\rangle \) ∈ [0, 1] × [0, 1] reflects the degree of effectiveness (m r) and the degree of ineffectiveness of the robot with number n (l r) for a time t.
where:
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p r is the number of successfully movements and tasks performans of the robot n,
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s r is the number of unsuccessfully movements and tasks performans of the robot n,
-
n r is the total number of movements and tasks performans of the robot n.
The degree of uncertainty πr = 1 − m r − n r reflects the cases when the robot have not completed a movement or task, which is the number of α-tokens in both places a 4+r−1,7 and a 4+r−1,10.
5 Conclusion
The proposed GN introduces model the spatial, temporal, ordered and concurrent character of mobile, distributed system. The model is based on a discrete topology in which devices can change their location by moving from cell to cell. With GN, we model movement in a heterogeneous terrain as well as task execution or access to other resources of the devices.
On the other side a cyber-physical system (CPS) is a system of collaborating computational elements controlling physical entities. Nowadays cyber-physical systems can be found in areas as diverse as robotics, auto-motive, chemical processes, civil infrastructure, energy, healthcare, manufacturing, transportation, entertainment, and consumer appliances. In this paper, we address the problem of spatial-temporal group scheduling using Generalized nets (GN). The IFE is used to estimate the degree of effectiveness, the degree of ineffectiveness of the robot and uncertainty during the robot movement.
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Acknowledgments
The authors are grateful for the support provided by the project NTS Germania 11, funded by the DAAD.
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Sotirov, S., Sotirova, E., Werner, M., Simeonov, S., Hardt, W., Simeonova, N. (2016). Ituitionistic Fuzzy Estimation of the Generalized Nets Model of Spatial-Temporal Group Scheduling Problems. In: Angelov, P., Sotirov, S. (eds) Imprecision and Uncertainty in Information Representation and Processing. Studies in Fuzziness and Soft Computing, vol 332. Springer, Cham. https://doi.org/10.1007/978-3-319-26302-1_24
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