The applications of robots evolved ever since, and have taken a big leap. Making progress toward autonomous robots is of major practical interest in a wide variety of application domains including manufacturing, construction, waste management, space exploration, undersea work, medical surgery, serving and assistance for the disabled. Robots have several types including industrial robots, mobile robots, robots used in agriculture, telerobots and service robots.
The ability to navigate from one location to another location is the fundamental capability of any mobile robot. Avoiding the dangerous situation such as collision with obstacles and staying in the safe operational environment come first, but to complete any task that relates to a specific location in the robot environment, the navigation is required. In mobile robot navigation system, the robot navigation is the ability to identify its own position and then navigate towards some goal location in an environment. In order to navigate in its environment, the robot requires representation for example, a map of the environment and the ability to interpret that representation. To achieve quality of mobile robot navigation system, different approaches have been documented. In this research we develop the specification approach, which is the integration of graph theory, finite automata and Z notation. This integrated approach is applied on mobile robot navigation system in which we examine the problem of assigning a heterogeneous population of rescue robots to reach a set of target rooms, where for example a set of people may be trapped. The robots in the population are assumed to have different properties so that the allowed transitions of the different robots are different and therefore the automata and directed graph models for each robot are also different. The logical flow of the proposed solution to the mobile robot navigation system is as follow, develop the topological environment of known environment, the robots in the population are assumed to have different properties therefore topological environment for each robot are also different and modeled from the topological map of the environment, navigation automata is the generalization of the finite automata where the set of alphabets is replace by an event set and an extra event set introduced and this automat is used to navigate the mobile robots on topological map and all the work is modeled using Z notation and verified using Z/EVES.
1.2. Purpose of the research
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A new approach by integrating finite automata, graph theory and Z notation is proposed which increases modeling power of mobile robot navigation system. The purposes of the research are as follows:
a) To develop an approach of integration of graph theory, finite automata and Z notation for modeling of mobile robot navigation system in which a heterogeneous population of mobile robots can travel from an initial configuration to a target configuration.
b) To develop the topological environment model of a known environment by integrating the graph theory and Z notation.
c) To navigate the mobile robots on the topological environment, integrate the graph, navigation automata and Z notation.
d) To observe the proposed integration approach, demonstrate the two types of rescue robot problems.
e) To demonstrate the modeling power of Z notation, all models is verified using Z/EVES computer tool.
1.3. Related Work
The ability of a mobile robot to navigate in known or unknown environment is regarded as the key issue in an enormous number of research publications during the last decades. The fundamental requirement of mobile robots is to navigate from one location to another location, so that we have to think of mobile robot navigation system to accomplish such requirement. The mobile robot navigation system has attracted attention of many researchers and different navigation methodologies have been proposed. A Partially Observable Markov model to guide and keep track of a robot, so that the robot is able to complete several tasks is described in . In other works, behavior based navigation is developed using Dynamic Systems theory to generate the behaviors ,. The Petri Nets is used as a modeling tool for the navigation of a robot in unstructured environments .
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With the development of multi-robot systems, the multiple robots navigating simultaneously in a common environment raised new and challenging problems. The existing navigation of single robot was extended to multiple robots. In , potential fields are used to achieve multi-robot navigation. The navigation of multiple robots also become with interesting problems, such as cooperation and formation control, as addressed in , where a reactive behavior based approach to formation control is described.
The work carried out by the Melo, Isabel, and Lima in , ,  have examined the problem of multi-robot navigation. Further they analyze the problem of driving a robot population moving in a discrete environment from some initial configuration to a target configuration. Finite state automaton is used to model the robot population and properties of this automaton such as blocking and controllability are also discussed. Since it models the movement of the complete robot population in the environment, from a start configuration to given goal configuration, properties such as blocking and controllability have direct correspondence with the successful completion of the objective. A blocking state corresponds to a distribution of the robots from which the desired goal configuration is not achievable because one of the robots has reached a site from which it cannot leave. Controllability means that such blocking states are avoidable: it is possible to disable some actions to prevent the robots from reaching blocking configurations. In , the graph theory based on adjacency matrix is applied to the navigation problem.
The robot navigation problem has attracted considerable interest in recent years for movement through both a discrete environment and a continuous environment ,,,,,. Finite automata theory and graph theory have proved to be useful mathematical models for robot navigation through a discrete environment ,,,. In , they use the finite automata to control the navigation of mobile robot along the possible paths. Further more, they analyze the movement and control the population of a mobile robot by applying the graph theory methods is very convenient.
A number of modeling techniques for Mobile robot navigation has been developed by researchers such as Partially Observable Markov and behavior based navigation, but the existing approaches which are used to model the mobile robot navigation system have lack of the formalization. There does not exist any single approach for the navigation of mobile robot which based on integration of graph theory, navigation automata and Z notation. Thus we propose an integrated formal approach to model the mobile robot navigation system. This approach is the integration of graph theory, navigation automata and Z notation. Some efforts been done on integration of graphical based notations and formal languages such as Z notation and finite automata . Moreover a relationship between Algebraic Automata and Z is documented in . Object Z and Timed automata are integrated in , . Another work in ,  has given a constructive formalization of some important concepts of automata using Nuprl. Further work of interest is also reported in . In , a combination of Z with statecharts is established. A relationship is investigated in between Z and Petri Nets in , . Our integrated approach increased the modeling power of mobile robot navigation system.
1.4. Mobile Robot Navigation System
The mobile robot has the ability to move from one location to another location, but industrial robots are fixed and can not move from one location to another location. Mobile robots are used in different fields like military, industry, rescuing, security environment, entertainment and perform certain task for human. Any mobile device always moves in its environment, on the ordinary ground this usually means the robot has wheels, legs or tracks. If the robots accomplish their task in the air or underwater, their must be other methods of navigation which are used. In mobile robot navigation system, the mobile robots environment is modeled and each mobile robot can identify the location and move to the target location in its environment.
1.5. Problem Statement
Mobile rescue robots are used for rescuing people in dangerous environments as an alternative to human rescuers. The key issue of the mobile robot is to model a navigation system of a mobile robot. The problem of assigning a heterogeneous population of rescue robots to reach a set of target rooms. The robots in the population are assumed to have different properties. As a case study two types of problem are considered given by Melo et al , , . In the first type, the population of robots is required to reach a set of target rooms, each target room is reached by one rescue robot. In the second type, the population of robots is required to first reach a set of target rooms, and then each robot is required to return its rescued person to one of a specified set of safe rooms.
1.6. Formal Method
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Formal methods are based on mathematical techniques and notations used to describe and analyze the properties of software systems .System specification is written by using mathematical notations and symbols instead of describing the system informally. These notations and symbols are based on predicate logic, set theory and graph theory . Formal methods validation and verification techniques can be applied at any or all phases of software development process. The use of formal methods ensures the correctness and consistency in the development of a system. The formal specification based on well defined mathematical notations, have the similar interpretation all over the world . Thus, the use of mathematics helps to have a broad perception of system to be developed. Formal methods enforces the analyst to ask the all sort of questions which have been postponed otherwise until coding. It provides the formal description which is abstract, precise and complete. Abstraction enables us to understand the big picture of the system. By using formal methods, preciseness removes the ambiguities and completeness deals with all the aspects of behavior . Formal methods are used to reduce the error rate at the analysis phase, which will reduce the overall cost of a system. Correctness can be achieved by determining the important properties such as satisfaction and high level requirements; there is a need to integrate the other approaches with formal method to develop the feasible system.
1.7. Graph Theory
Graph theory is the important fields of the mathematics because its applications in different fields like biochemistry, electrical engineering, operation research and computer science. Graph theory can be applied on any application that has configurations of nodes and connections, such as electrical circuits, roadways or organic molecules. They also be used in less tangible application like ecosystems, sociological relationship, databases and flow of control in a computer program. Any mathematical object involving points and connections between them may be called a graph. If all the connections are unidirectional, it is called a digraph.
The problems that can be represented as a graphs are mostly in all fields and many practical problems can be represented by graphs. The environment for the robot navigation problem can also represent by a directed graph: the vertices are the rooms available in the building and the directed edge from room A to room B exists if and only if A contains a link (doors, passageways and stairs). A similar approach can be taken to problems in travel, biology, computer chip design, and many other fields. The algorithms are developed to handle the graphs is therefore a major research interest in the computer science. There are several graph rewrite systems that can represent and formalize the transformation of the graphs.
Assigning the weight to each edge of the graph is the extension of the graphs, being called as graphs with weights or weighted graphs. These graphs are used to represent the problem in which each edge having some numeric values to represent the edge cost. If a graph represents a network of rooms in a building, the weights could represent the length of each link (doors, passageways and stairs). In graph theory a directed graph with weights is called a network.
1.8. Automata Theory
Automata theory deals with the definitions and properties of mathematical models of computation. These models play a role in several applied areas of computer science. There are several automaton models that have been widely used including deterministic finite automata, nondeterministic finite automata and pushdown automata. They have many applications including text processing and searching, compilers, hardware design, definitions of programming languages and artificial intelligence. Automata theory is an excellent place to begin the study of the theory of computation. The theories of computability and complexity require a precise definition of a computer. Automata theory allows practice with formal definitions of computation as it introduces concepts relevant to other theoretical areas of computer science. Navigation automata is the extension of the finite automata, where the set of alphabets replaced with event set and an extra even set is introduced.
Develop the topological environment of known environment. The robots in the population are assumed to have different properties so that the allowed transitions of the different robots are different and therefore the automata and topological environment models for each robot are also different. Navigation automata are used to navigate the mobile robots on topological map, all the work is modeled using Z notation and verified using Z/EVES. The logical structure of our work is presented in figure 1.
In figure 1, known envirnment is to be modeled by using graph, this graph will be decomposed into subgraph's according to robot types. Graph and subgraphs are transformed into automata. To navigate the mobile robot the navigation automata is used. Navigation automata and graph automata are combined to achieve our desired mobile robot navigation.
Defining the relationship between graph, automata and Z notation, enhance the modeling power of any system. The benefits of the graph, automat and Z notation can also be obtain from this integration, such as early discovery of the ambiguities, inconsistencies and incompleteness in informal requirements. Tools assisted analysis of the correctness of specifications with respect to requirements. A useful generalization of the rescue robot problem will be modeled using formal method where the robots are required to reach in the target states. This modeling will provide us, how to build the discrete environment, environment for each robot, navigation automata and navigation of mobile robot from initial to final state.