Graph Thereotic Approach For Path Planning

Published Date: 02 Nov 2017

Abstract: Path planning plays vital role in various field of application such as computer graphics, CAD, molecular biology, computer games and Robotics. In its most general form, the path planning problem is termed as finding a collision-free path for a moving entity between a start position to a goal position in an environment containing obstacles.. Most of the effort has been spent on path planning in static environments where the obstacles are stationary and do not move, and in which the positions and geometry of the obstacles are assumed to be known. Less attention has been given to path planning in dynamic environments, which contain moving obstacles in addition to stationary ones. Path planning in dynamic environment can be done efficiently with the help of graph theoretic approaches. So, In this paper, a new approach is proposed for path planning in dynamic environment using voronoi diagram.Voronoi diagram is the powerful geometric structure to represent a roadmap.

Keywords: Voronoi Diagram, Dynamic Environment,path planning, graph thereotic approach

I INTRODUCTION

Path Planning is the key task in the field of Robotics. Planning path in static environment is easier than Path planning in Dynamic Envirnment.There are various methods are available for static path planning. But For dynamic Path planning very few methods are there as per the Literature survey done on the Dynamic path planning .

In a dynamic environment, path planning and motions along the designated roadmap is especially difficult due to the following issues:

Differential constraints that may arise during the planning phase (e.g. local minima, etc.),

Intractability of following a predetermined and precomputed plan due to the non-deterministic nature of dynamic environment as well as uncertainty of the state of the robot its field of operation

Limited environmental knowledge due to the incomplete and noisy sensory and perceptional data available to the robot.

The major task for path-planning for single mobile robot is to search a collision–free path, where topological map used to represent environment. In robot motion planning, robot has to identify its trajectories within the environment depending on nature of planning problems. There are two types of planning problems Offline & online planning. Offline planning refers to the determination of trajectories when the obstacles are stationary. While online planning means the dynamic path planning where the obstacles are moving (dynamic).

The Graph based representation of the robot working environment is one of the earliest and powerful attempts for creating maps of agents world for the purpose of safe path planning. The graph representation is basically used to connect all the available free spaces of the given field (places that are obstacle free) via a connected set/network of lines. so as to provide a path for robot for performing safe ,target oriented, collision free motion. Such a network is used for motion planning in robotics. The available free spaces are generally considered as vertices of graph whose edges are in fact a network of connected lines. Graph based representation is then used to find shortest, obstacle free path from robot’s current location up to target point. Some of the limitations that are due to graph based representation are

Time complexity in creation of graph as there is increment in robot’s field of operation.

Vulnerability against uncertainty introduced by the application of moving/movable objects.

From research point of view in the dynamic environment where the obstacles are moving leads to new aspect of the path planning problems. Basically path planning impies to following task:

to virtually design /planning a number of paths using Voronoi Diagram

Apply case study/Example to verify whether the obtained path are more suitable path or not,

Choose Best path as a, optimal path from the different paths planned above,

A] PATH PLANNING ALGORITHMS

Various approaches, algorithm have been proposed for path planning which are according to environment, type of sensor, robot capabilities and etc, these approaches are gradually toward better performance in term of time, distance, cost and complexity.

It is prerequisite that a successful algorithm needs to be convergent. That is, it needs to find a path to the goal if such a path exists. If no such path exists, it must stop and inform the user that the target is unreachable. If an algorithm is convergent, it is then assessed on the following attributes:

Path Length: The distance of the path from start to finish. This should be as short as possible.

Computation time: The algorithm’s total execution time excluding time spent on driving. This should be as short as possible and is driven by the following sub attributes.

Number of calls to the math-library: A factor which affects computation time is the number of calls to the math library.

Computation time per metre travelled: Algorithms which have a short path length carry this advantage into computation time calculations. Calculating computation time per metre travelled removes this advantage.

Rotation: The amount of turning which is performed along the path from start to finish. This should be as low as possible.

Inherent rotation: Some rotation is hardware dependant and this is filtered out in this measurement.

Robustness: The algorithm’s ability to tolerate PSD error, linear driving error and rotational driving error.This should be as high as possible.

Memory requirements: The amount of global memory reserved by the algorithm. This should be as low as possible.

Simplicity: This is measured by the lines of code required for implementation. This should be as low as possible.

In this paper, we propose a new path planning method. Our method plans an efficient path to avoid the collision with stationary/moving obstacles, and this planning is fast enough to apply real time control robots. By applying our new method, we can solve the above problems efficiently.

This Paper is organized as follows. Section II presents a literature review on path planning in Dynamic environment. Section III describes the Brief Introduction to Voronoi Diagram & algorithm for construction of voronoi diagram, the proposed approach for path planning in Dynamic Environment is described in section IV. Finally, Section V concludes with the discussion on the Expected results on the proposed method.

II RELATED WORK

Path planning has been subject of in-depth studies and analysis during the past decades. Much more focus has been given to the path planning in static environment but less attention has been given to dynamic path planning. But still there are several methods are available till date.Fig 1. Represents the classification of the techniques for path planning in robotics. The Robot motion planning is basically divided into two main categories i.e. Path planning in Static environment and Path planning in Dynamic environment. The Several methods have been used for motion planning in dynamic environment which are Artificial potential Field approach, methods based on fuzzy logic, biologically inspired methods, and a graph theoretic approach. The graph theoretic approaches used for both static as well as dynamic environment. In our literature review, main focuses is on path planning in Static and Dynamic Environment using Graph based modelling.

In this paper, our main focus is on the roadmap-based path planning and utilizes a powerful computational geometry data structure, the Voronoi diagram, to obtain shortest path. The advantage of using a Voronoi diagram as a roadmap over alternative methods, among which the visibility graph prevails, is its efficiency. The Voronoi diagram can be constructed in just O (nlogn) time, whereas even the fastest known algorithm for constructing visibility graph [1] can take O (n2) time in the worst case when the visibility graph has O (n2) edges.

A general method for refining a path obtained from a roadmap based on classical numerical optimization techniques can be found in [2]. The authors apply costs to each edge and use an augmented Dijkstra’s algorithm to determine an optimal path. The edges that are nearer to obstacles are assigned higher costs. However, there is no guarantee that the method will generate an optimal path, as the path is constrained to the edges in the roadmap.

Mobile Robot Navigation using Voronoi Diagram and Fast Marching [3] proposed path planning in two steps. First it creates voronoi diagram by extracting safest areas in the environment and second step is the Fast Marching method that applied on Voronoi diagram. Path planning in Robot Navigation using Tube Skeletons structure and Fast Marching [4] is a new sensor based non-holonomic Path Planner which consist of the global motion planning and local obstacle avoidance capabilities. In the first step the safest areas in the environment are modelled by means of a tube skeleton similar to a Voronoi diagram but with tubular shape. In the second step Fast Marching Method is applied to the tube skeleton extracted areas in order to obtain the best path in terms of smoothness and safety.

Path Planning based on Voronoi Diagrams and Genetic algorithms [5] method was proposed for static path planning. the path planning was based on Voronoi diagrams, where obstacles in the environment are considered as the generating points of the diagram and the environment is static, and a genetic algorithm is used to find a path without collisions from the robot source to target position & uses Fitness function which consider the length, safety and smoothness of the path for path planning.

Roadmap-Based Path Planning Using the Voronoi Diagram using parameter Clearance-Based Shortest Path[6][7] was proposed by Priyadarshi Bhattacharya and Marina L. Gavrilova which creates a roadmap from the Voronoi Diagram and path planning is based on roadmaps. Optimal path is obtained from different paths using minimum clearance criteria. a minimum clearance value is initially set by user. Here it finds the quality path based on clearance from obstacles, overall length and smoothness.

Fig 1.Classification of path planning in Dynamic Environment

III METHODOLOGY

1. Voronoi Diagram

Voronoi diagrams are one of the main structures in the computational geometry area, commonly used as visibility graphs and for finding networks collision-free paths, they are also one of the most common techniques for building trajectory maps.

They have been extensively used in the field of robot motion planning with combination of other methods for finding a optimal path between source to destination.

Fig.2. Voronoi Diagram

Voronoi diagrams have two interesting properties in the context of path planning with mobile robots:

Every Voronoi edge belongs to the bisector of the line segment determined by the two generating points of the regions that determine the edge.

Every Voronoi vertex is located exactly at the circumcenter of the polygon defined by the generating points of the regions that determine the vertex.

These properties make the Voronoi diagrams a good alternative for computing maps of paths with maximum security.

1.1 DEFINITION:

Let P = {p1, p2, …, pn} be a set of n given point sites in the plane. We define the Voronoi diagram of P as the subdivision of the plane into n cells, one for each site in P, with the property that a point q lies in the cell corresponding to a cell pi iff dist(q, pi) < dist(q, pj) for each pj in P with j ¹ i. We denote the Voronoi diagram of P by Vor(P). The cell that corresponds to a site pi is denoted V(pi).

The Voronoi diagram is a planar subdivision whose edges are straight line segments. Some edges are line segments and others are half-lines. Unless all sites are collinear there will be no edges that are full lines.

1.2 PROPERTIES OF THE VORONOI DIAGRAM:

 For the Voronoi diagram Vor(S) of a set of points S the following holds:

A point q is a vertex of Vor(S) iff its largest empty circle C (q) contains three or more sites on its boundary.

The bisector between sites pi and pj defines an edge of Vor(S) iff there is a point q in R2 such that C (q) contains both pi and pj on its boundary but no other site.

2. Algorithm for Construction of Voronoi Diagram:

The Voronoi diagram for a collection of given points (called sites, here sites means Point approximation of obstacles) is the graph formed by the boundaries of specially-constructed cells. Each of these cells surrounds one of the given sites and has the property that all points within the cell are closer to the enclosed site than to any other site. In this type of problem, the boundaries of the specially-constructed cells are equidistant between the two nearest obstacles. These cell boundaries form the Voronoi diagram and are ideal for a mobile robot's path. By having the robot travel midway between the closest obstacles, actually minimizes the chance that an error in the localization of the robot or an inaccuracy in the base map will result in a robot-obstacle collision. The method of voronoi diagram for mobile robot motion planning reduces the chances of collision with obstacle comes in the path.

Computation of voronoi Diagram in 2D is a well known problem. The simplest method to construct diagram is the method of intersecting half planes. One connects each site, with straight lines, to every other site. For a specified given set of points, perpendicular bisector between the two points leads to halfplanes & perpendicular bisector of all the lines connected to other sites yields to the set of halfplanes. The intersection of theses half planes leads to the voronoi cell for the specified site.As shown in the figure process repeats for the n no of points.The boundaries of the computed cell are straight lines and said to be a voronoi edges,Though this method is straight forward but has complexity O((n^2)log n).

The process

Fig.3 Basic method for Voronoi Diagram construction

There much more efficient methods than above described method are available for generation of voronoi diagram which is a collection of discrete points. Steven Fortune presented a novel and extremely efficient method for the computation in 1986. By using a sweep-line algorithm, complexity reduces to order O (n(log n)).In proposed method, this fortune’s sweepline algorithm is used for construction of voronoi diagram.

In general, a sweep-line algorithm makes use of a horizontal line which extends across a planar map and which moves from the top to the bottom of the plane. As the sweep line travels, it gathers information above the given structures on the map. The sweep line may pass the location of needed Voronoi edges before it encounters the point sites which actually generate these edges. Fortune solved this problem by continually updating the Voronoi diagram above the sweep line as this line encounters new structures.  the algorithm does not actually have the sweep line travel down the map in small infinitesimal steps. Instead, the line jumps from the location of a special event down to the location of another special event. Two types of events occur in this algorithm: site events and circle events. The site events help determine the Voronoi diagram's edges, while the circle events determine the diagram's vertices.

http://www.codeproject.com/KB/recipes/411242/SiteEvent.png

Fig. 4 Fortune’s Sweepline algorithm for construction voronoi diagram

The relationship between the variables that determine the number of rows, columns and the side of each cell of the voronoi diagram reproduces the relationship between the size of the robot, obstacles and the total area where the robot can move. So, the representation and the algorithms consider the robot as a single particle (point) & also the obstacles are considered as single point which is moving not steady. Now the starting and target vertices on the Voronoi diagram have been chosen, we have to search for a practical path between these two vertices. 

IV PROPOSED METHOD

Path planning in dynamic environment is tricky task. There are several methods available for the path planning in dynamic environment using graph theoretic approaches. One of the good geometric structure voronoi diagram has been used for the purpose of path planning. But still there is need to develop more strong technique for path planning in dynamic environment. So, this paper proposed a method to find path between source & destination with the help of voronoi diagram. Since the environment is dynamic we need to calculate path after specific interval of time.

Fig. 6 Flowchart for Proposed method

Proposed method uses the voronoi diagram for path planning in dynamic environment. Above Flowchart shows the steps for implementing proposed method.

Step 1: For robot motion planning , it considers point approximation of random obstacles.

Step 2: From that random sites (points) it creates voronoi diagram with the help of Fortune’s sweepline Algorithm.

Step 3: Selecting two random points as Start & destination points.

Step 4: Generate different paths using different searching algorithm like Dijkstra’, A* search algorithm.

Step 5: Choose optimal path from obtained path by considering one suitable factor(time, space covered etc.).

Our proposed method is to improve the path planning in dynamic environment. With this proposed method mobile robots will be able to achieve safe path planning in presence of moving obstacles.The proposed methods basically divided into three main task.

A] Construction of voronoi diagram :

For finding safe path , proposed method suggest to design a roadmap for the given workspace using voronoi diagram.For voronoi map generation we will be using the fortune’s sweepline algorithm which is best suitable method for construction for voronoi diagram costruction.

B] Finding paths using shortest path algorithm:

After creation of voronoi map ,the path is generated on voronoi map using shortest path algorithm i.e. Dijkstra’s algorithm. But path needs to calculate in subparts. Because, whenever obstacles move the voronoi diagram is generated after specific time interval. The subpoint needs to be decided on the generated voronoi maps. We can generate two or more paths for comparison.

C] Calculating optimal path:

Finally, one example will help us to prove whether given method is able to generate suitable path or not. The method generates a path which is collision free. A mobile robot is able to reach from source position to destination. Final path is shown in following fig.

VI] CONCLUSION

This paper, presented an approach for path planning to a target for mobile robot in a dynamic environment; it allows a safe path planning for mobile robot to navigate through dynamic obstacles in order to reach the target without collision. The path finding strategy is designed using.voronoi diagram in dynamic environment where the obstacles are moving. The given method try to generate the optimal path but it may be time consuming due to the dynamic environment.So,Our Future planning will be of generating path in more suitable time.